It is experimentally verified. hence it is true

My questions are: if this is true, can some one show me the gravitational lensing by the center of our Milky Way galaxy? Can some one upload the gravitational lensing (Einstein Ring) photos and let's analyze if they are truly caused by gravitational lensing?

Dear Preston Guynn ,

Thanks for the link. Just looked at it. It didn't mention the effect of solar atmosphere. The Sun's atmosphere extends quite far and it will be denser near the Sun's surface, so any light including radio waves will be bent by it as it will appear like a prism. So light sure will be bent by the prism indirectly affected by Sun's mass through the density variation of its atmosphere. No body seems to care about it, at least should have some study if this effect is negligible.

The gravitational lensing is not convincing to me. For example, the attached Hubble image of Abell 370 galaxy cluster can hardly be viewed as the effects of lensing because it doesn't seem to stretch the stars in the so-called lensed galaxies. There are multiple centers if the lensing is coming from centralized masses, meaning multiple black holes or galaxies are doing lensing. But not all stars or galaxies appear to be lensed in the presence of multiple centralized masses. You can say that the galaxies are in front of the black holes. But the problems is those unlensed galaxies are sometimes smaller than the lensed ones, meaning the are farther than the lensed ones. Why the farther galaxies are not lensed? The lensing is a very hard sell to me.

Dear Preston Guynn ,

Thanks for the links. I have gone through them.

There are many unknown observations indicates that gravity alone cannot explain astronomical observations, and the gravitational lensing is suspicious because it only treats gravity as attractive forces.

Here is link to a paper showing that the attempt to use gravitational lensing doesn't work very well for supposedly lensed gamma ray bursts. https://iopscience.iop.org/article/10.3847/2515-5172/abfdbd

This paper is based on the assumption that gravitational lensing won't change the spectral distribution of photons, but the data analysis shows otherwise, meaning that the supposed lensing is not conclusive.

As of the galactic velocity curve puzzle, I have a different explanation. I believe there is antimatter around the galaxies that exerts repulsive gravity to the outside of the galactic arms, therefore helping the galactic disk maintain its shape when rotating around GC at higher speed than GC's gravity can sustain. Namely, there exists antimatter ring to provide concentric force in addition to GC's gravity pull.

The ring is astronomically observed if you check the link at http://www.solstation.com/x-objects/gal-ring.htm.

Dear Preston Guynn ,

Thank you for the links. The ring in the http://www.solstation.com/x-objects/gal-ring.htm has a radius beyond the halo or the HI gas. The ring is possibly different than normal matter. They might be antimatter created later after the Milky way took shape and pushed away by the antigravity. HI gas or halo might still be the old material from the Big Bang but drifted slowly to the galaxy.

Dear Hong,

On the bending of light by the Sun - the deflection angle was measured very precisely during solar eclipses and the angle agrees perfectly with the prediction of the GR, i.e. gravitational lensing. On the other hand, nobody has presented a theory of bending light due to a change of the refractive index in the Solar atmosphere (I assume that this is what you are referring to), which agrees with the data. Hence, we accept the first explanation (gravitational lensing) as the more probable, but everybody is welcome to try to disprove it by experiments or to try to figure out another theory which agrees perfectly with the data. Before this happens there is no point saying that another explanation is better (no prove for that).

The cluster - what do you mean by "it doesn't seem to stretch the stars in the so-called lensed galaxies"? Lensed galaxies are the arcs and they are extremely stretched. You don't see individual stars in them, because they are too distant.

Lensing galaxies (orange ones belonging to the cluster) are closer than the lensed galaxies (arcs), which has been confirmed by te measurements of redshifts, so I am not sure why you say that they are more distant. Just the fact that they are smaller on the sky doesn't imply larger distance - they are just intrinsically small galaxies. They cannot be lensed by the cluster, because they belong to them. Are you referring to lensing by individual BHs in their centres? Well. this effect is not measurable for such distant galaxies. The defection angle for stars far away from the BH is tiny, so the stretch is almost zero. You won't see this effect on this image, because the resolution is too poor.

Lensing by our own BH in the centre of the Milky Way. Yes - this should happen, but how do you think this can be measured (or disprove by a failed measurement)? In order to notice this you would need either a) the measurement of the shape of stars which should appear stretched - but the angular sizes of the stars around (and behind!) the BH are too small to measure, so this cannot be done; b) the measurement of the displacement of the image of a star with respect to the unlensed position. This can be done for the Sun, because you can wait half a year, measure the position of the star without the effect of lensing, and then compare it with the lensed position. However, this cannot be done for our BH - how can you measure the "unlensed" position?

In summary - all experiments we have agree with gravitational lensing. No theory has been worked out to explain all the data in a different way. Then, the scientific method allows us to accept the gravitational lensing scenario and to keep investigating other options.

Dear Michal Michalowski ,

If none mentions the deflection by the Sun's atmosphere, the result is doubtful for different reasons. Don't you think physics should be strict and leave no stone unturned?

Lensing of galaxies theoretically implies stretched stars. This is really a good chance to prove lensing. Trying to avoid this great proof by saying the stars are indistinguishable because they are too far seems to be avoiding a good proof. I don't think astronomers are trained to avoid good proofs. There surely will be black holes in front of galaxies that has distinguishable stars. I look forward to seeing if anyone can provide such images so that I can favor the gravitational lensing theory, so that I can move forward to figure out if my theory has problems.

In the center of Milky Way, there is a supermassive black hole. It is at the center of our galaxy, therefore there should be plenty of stars behind it. The stars that orbit around it should also sometimes affected by the lensing when they go behind the black how. However, none talks about the gravitational lensing. See the attached image from UCLA. The orbits of the stars around the black hole are perfectly non-lensed. Do you think this is a good proof that gravitational lensing simply doesn't work for Milky Way?

Michal Michalowski I am of the opinion that gravitational lensing is a true phenomenon. You mentioned a concern that stars appear as point like objects but this is what I would expect with the light coming from a specific star being bent around a substantial mass. As a similar example, the light from a star which is observed "behind" the sun during a solar eclipse appears as a circular image.

Gravitational lensing is caused by a substantial body of mass in the line of sight between us and the galaxy being lensed. That body of mass could be a single galaxy or a cluster of galaxies. These galaxies will themselves contain smaller black holes and a supermassive black hole at the centre so, in this sense, black holes do contribute to gravitational lensing.

It is true that gravitational space bending can occur around black holes just as it occurs around the sun. One of the images that we actually have of a black hole was taken in 2019 in M87. The light that we observe surrounding the dark centre of the image must be coming from behind the black hole and being lensed. This supermassive black hole is 6.5 billion solar masses.

Richard

You don't have to have a black hole to get gravitational lensing. Any sufficiently large mass can do it; in fact, the vast majority of examples of gravitational lensing are due to rich clusters of galaxies, NOT black holes. However, if you move around a black hole, there are bound to be lines of sight that put the black hole in front of some source of light, so although WE might not be able to see gravitational lensing by a specific black hole, SOMEONE can (presuming there are any "someones" out there who are IN the right line of sight).

Of course, since you have created a theory that denies the possibility of gravitational lensing, I am sure that you can also devise explanations that deny all known examples of gravitational lensing, which means that any answer you receive will be, at least as far as you are concerned, completely unconvincing, and therefore a waste of the time and effort put in by the person answering you. So that's all I feel a need to say on the subject.

Dear Hong Du

Thank you for your answer. Nobody is avoiding any test. What you are proposing (measuring the shapes of stars) is just impossible with the current instrumentation (and unfortunately likely for many years to come). The Sun has a radius of 696340 km so such star close to the Galactic centre (8 kpc away) has the angular size of ~696340/(8*3.086e+16) = 2.8e-12 radians or 5.8e-7 arcsec (58 micro arcsec). This is the very best resolution ever obtained by humans with the Earth-wide array of telescope - EHT. However, this is at mm wavelengths at which stars don't emit much light, so we can't use it for our purpose. You would need such resolution in the optical or near-IR at which the currently best resolution is 0.002 arcsec obtained by VLTI. Hence, you see that we are many orders of magnitude behind what is needed to measure the sizes (and stretching) of stars. One would need to build an optical interferometer with the size of the Earth, but currently we don't know how do do this. I am 100% sure that once we do this in the future, the test which you propose will be done.

From these calculations it is obvious that the circular size of the stars close to the Galactic centre on the image you have attached is not the reflection of their sizes, but this is just the effect of the atmosphere and the diffraction limit of the telescope. From these data we have no information about the shapes of these stars.

Also we need to remember that lensing signature is minimal if a source is only slightly behind the lens (unless it is perfectly aligned). The source-lens distance would need to be comparable to the observer-lens distance for the effect to be strong, so now we are talking about stars on the other side of the Galaxy, which would be even smaller and the dust attenuation would prohibit doing what you propose.

Measuring the sizes and shapes of stars in external galaxies (it seems that you are proposing this) is even less possible because the distances are many orders of magnitudes larger, so the angular sizes are many orders of magnitude smaller.

By the way, the only size measurements of stars that can be done is for the very nearby giants.

Coming back to the lensing by the Sun - I am sure it doesn't go this way. Gravitational lensing with GR explains the data perfectly, so now it is the turn of people claiming that this is the deflection by the Sun's atmosphere to prove that such theory can agree with the data as well. Before this happens nobody will take such claims seriously.

Oh, I noticed a mistake in my calculations. 5.8e-7 arcsec is of course 0.58 micro arcsec not 58. Hence, the Sun close to the Galactic centre would appear 100 times smaller than I claimed and hence there is even less chance to measure the sizes and shapes of stars there (and beyond) in the near future.

Hi Michal Michalowski ,

You are trying to say that any stars shall be points and shall not appear stretched at the galactic center. This is true if there is no lensing. Lensing will stretch stars to arcs that are easily larger than detectable points.

With lensing theory, some stars will appear to be circular ring because lensing is circularly symmetrical. The stretching I am talking about is that the lensing will cause point source appear stretched to arcs or rings. If a galaxy has distinguishable stars, all stars will appear stretched when lensed. See the horseshoe galaxy typically used as an example of gravitational lensing. But there really isn't typical stars been stretched, but a rather fuzzy galaxy.

Another example is the HE 0435-1223 at https://cerncourier.com/a/gravitational-lens-challenges-cosmic-expansion/

The four stars are not stretched, but still dots. I consider the lensing theory invalid in this case.

As of the deflection by solar atmosphere, it is so easy to create the deflection with a density parameters and match the measured deflection. As there is no direct measurements to verify the density of the solar atmosphere, thus the theory is arbitrary and easy to match any numbers the lensing theory can give. I am not interesting in writing down such man made parameters.

Nevertheless, I believe the lensing needs more proof because it is not just about the Sun, but about many many black holes and galaxies that are in the universe everywhere. The lensing theory causes so much trouble because it means all the images taken by any telescope are just the artifacts of the results of numerous visible or invisible lensing products. How can you tell or prove which image is lensed and which is not, and how can you believe what astronomers talked about is from lensed measurements and non-lensed measurements.

Hi Courtney Seligman ,

Accepting lensing theory is so easy and won't cause trouble because most people believe it. What I am asking is that believers can provide images of the lensing pictures and lets discuss if lensing is true or not. Show the known examples, the more the better.

There's a nice picture just above your reply, and theEuropean Hubble Facebook page shows images on a regular basis (here's the URL of their most recent one: https://esahubble.org/images/potw2120a/ ). I also recall seeing an image of a four-point Einstein 'star' recently, but don't remember when or on what site.

Btw, the fact that the image of the Einstein Ring shown above is gravitational lensing was confirmed by measuring the redshift of the ring, which is much greater than that of the central galaxy, showing that the galaxy whose image was turned into a ring is billions of light-years further away than the lensing galaxy.

Hi Hong Du

>> You are trying to say that any stars shall be points and shall not appear stretched at the galactic center. This is true if there is no lensing. Lensing will stretch stars to arcs that are easily larger than detectable points.

No. I want to say that the sizes of stars are many orders of magnitude smaller than our best achievable resolution, so there is no hope of measuring the stretch. If the gravitational lensing is there, then the stars would still be circular. The stretch is smaller than the point spread function of the telescope, so all stars appear circular independent of the lensing is there or not. To give an analogy, it would be the same if you would try to infer something about the shape of the light bulb of a car which is 5 km from you. You have no way to do that with your eyes - you just see the light but not its shape. In both cases (stars and the car light bulb) you have no information about the shape of the light source, so you can't rule out any hypothesis (lensing) based on these data.

>> With lensing theory, some stars will appear to be circular ring because lensing is circularly symmetrical. The stretching I am talking about is that the lensing will cause point source appear stretched to arcs or rings. If a galaxy has distinguishable stars, all stars will appear stretched when lensed. See the horseshoe galaxy typically used as an example of gravitational lensing. But there really isn't typical stars been stretched, but a rather fuzzy galaxy.

I think you mix two sources of light here. Before you were talking about lensing of images of individual stars. My calculations prove that there is no way to detect the stretch because the angular sizes are too small. Now you are showing a lensed image of an entire galaxy (the blue ring). You can't distinguish individual stars there. I am not sure what you want to infer from this. From my point of view this image is the proof that lensing is real - otherwise how would you produce such a ring?

>> Another example is the HE 0435-1223 at https://cerncourier.com/a/gravitational-lens-challenges-cosmic-expansion/

The four stars are not stretched, but still dots. I consider the lensing theory invalid in this case.

These are not stars. These are the images of a lensed background galaxy. And again, the resolution of this image is not good enough to detect the stretch. Imagine 4 arcs which are each around 0.1x0.5 arcsec on the sky. Then you have a telescope with a resolution of 1.5 arcsec. Then all these arcs will appear as circular, symmetric objects. You can't rule out the lensing theory using such data.

> As of the deflection by solar atmosphere, it is so easy to create the deflection with a density parameters and match the measured deflection. As there is no direct measurements to verify the density of the solar atmosphere, thus the theory is arbitrary and easy to match any numbers the lensing theory can give. I am not interesting in writing down such man made parameters.

All right, do you know any paper which demonstrates this? Or can you do such calculations yourself? Without this this alternative theory has no grounds. I understand that you are not interested in deriving this, but then you don't have arguments that your theory is correct. If you claim a discovery you need to provide a proof, not just say that you are not interested in demonstrating that it works.

> Nevertheless, I believe the lensing needs more proof because it is not just about the Sun, but about many many black holes and galaxies that are in the universe everywhere.

Yes! More proofs is always good. Up to know we have :

1) lensing by the Sun

2) Lensing by individual galaxies (the ring you you have just sent)

3) Lensing by clusters of galaxies (the arcs you send a few days ago)

4) Microlensing events (brightening of stars being lensed by another star)

All these data cannot be explained by anything else but lensing, so I would say that this is extremely convincing. But if you have an idea how to test this further or devise an alternative theory, it would be perfect.

> The lensing theory causes so much trouble because it means all the images taken by any telescope are just the artifacts of the results of numerous visible or invisible lensing products. How can you tell or prove which image is lensed and which is not, and how can you believe what astronomers talked about is from lensed measurements and non-lensed measurements.

That's correct. A lot of images are lensed to some extend. But this is not so problematic - you can easily verify that lensing is minimal if the alignment of the lens and the source along the line of sight is not nearly perfect (within arcsec even for galaxy lenses). Hence, you can safely assume that most of the images are not lensed at all, if there is nothing massive along the line of sight.

Found an example of the Einstein Cross (which I mis-remembered as a "star") in several places. This one shows changes in the lensing due to the motion of the foreground galaxy relative to the background quasar: http://hyperphysics.phy-astr.gsu.edu/hbase/Astro/eincros.html

Hi Michal Michalowski ,

Let's talk about the simplest case to avoid the possibilities that we are talking about different things.

There is a star very far away, it will appear as a dot (point source) no matter how powerful the telescope is. Now there is a point black hole stands between the star and the observer. Based on your computation, how will the far star look like to the observer if it is lensed:

(1) becoming one dot at a different location from its unlensed position?

(2) becoming two or three or four or five .... dots at different locations?

(3) becoming an arc or circle around the black hole?

The link given by you and also Courtney Seligman obviously match (2). Now the question is: isn't lensing rotationally symmetric? If the theory has rotational symmetry, shouldn't the dot be stretched into an arc as in (3).

If (2) is insisted, how come one source can become two or three or four or five dots without stretching because of the rotational symmetry? How those locations are determined? I don't think lensing can come up with any good theory that can first specify the magic number of dots that should show up and then the specific locations that the dots should appear. The explanation may come up with some data fitting similar to the every easy solar atmosphere density parameters.

For the solar atmosphere density parameters, just use the required bending and then write down the required density distribution. It is always possible, like middle school question. But this is not a theory, but a data fitting.

Hi Hong Du

Good idea, that will simplify the discussion. First to be precise, such a star (unlensed) would not appear as a dot, but as a disk, for example a 2D Gaussian, because we can't have a telescope with infinite resolution.

Now, all three of your options are correct, it just depends on the orientation:

* if the star, BH (lens) and observer are exactly on the same line then option (3) is valid - a star will be stretched into a perfect ring. Its size (Einstein radius) depends only on the mass of the lens (BH) and the distances, see the equation under "Putting θS = 0 and solving for θ1 gives" in

https://en.wikipedia.org/wiki/Einstein_radius

Note that this is the only rotationally-symmetric case. Lensing is not rotationally symmetric in general because the source may be shifted away from the line of sight towards the lens.

* if the star is shifted slightly away from a perfect alignment, then we will see two images - option (2). One towards the lens will be inside the Einstein radius, and the other will be outside the Einstein radius. To get more images you need a more complex case - the lens and/or the source cannot be point-sources.

* Finally if the stars is shifted away from a line-of-sight towards the lens by an angle larger than a critical value, we would only see a single image, displaced compared to its unlensed position - option (1).

In all of these cases the image will be stretched. As I showed before, in order to detect this stretch you need good enough resolution, which is impossible if your source is too far away and/or your lens is not massive enough.

I am not sure why you say that lensing cannot "come up with any good theory that can first specify the magic number of dots that should show up and then the specific locations that the dots should appear". This is derived only from General Relativity, nothing else is assumed. You take the mass of the lens , the distance, the alignment and work from GR to predict these observables. As I said above, this results in two images for a simple case, see for example slide 17 and 22 in this lecture:

https://lweb.cfa.harvard.edu/~dfabricant/huchra/ay202/lectures/lecture12.pdf

You can take GR and calculate more complex solutions precisely.

Solar atmosphere - it is far from a middle school exercise. Most astronomers accept the gravitational lensing explanation for the bending, so I think it would be an extraordinary achievement to show that another theory is equally good - somebody would have already done this if it was that simple. Of course if you take just one star, you can always explain it with a well chosen refraction index. However, I suspect this will fail because you have constraints at several different angular distances from the Sun, so in order to explain them all at the same time with solar atmospheric refraction, you would need an extremely unlikely, discontinuous model of the atmosphere.

My point here is that before anybody accept such explanation it needs to be shown that it works (predicts correct angles) and is consistent with other data (e.g. what we know about the density and optical properties of the solar vicinity).

Hi Michal Michalowski ,

Thanks a lot for the link at https://lweb.cfa.harvard.edu/~dfabricant/huchra/ay202/lectures/lecture12.pdf.

(1) For understanding of not perfect point image, I thoroughly understand that because I did research on CCD images and their point spread function (PSF) that ranges over several orders of magnitude, and I can predict that the radiometric calibration of CCD camera can be off by a few percent if not deconvoluted by the PSF.

Now I am going to cite some key phrases for the harvard lecture note: Einstein Ring, Magnification (gravitational lensing preserves surface brightness but changes the apparent solid angle of the source =>magnification.

So point star shall not remain point star, it must be stretched, into ring or whatever shape that satisfy the lensing equation, which is continuous, not a point like condition that only very specific angles can satisfy. That is what I have meant by saying stretching. If I don't see stretching, I will suspect why only a few angles will satisfy the lensing equation. There must be special conditions that requires extra equations to satisfy, whether it is because the mass distribution of the black hole or whatever, which is not easy the characterize because there is no measurement data.

Now lets examine the Einstein Ring, check the attached image from the lecture note with two bright false-colored dots in the top left image about Hewitt+1987 MG1131+0456: they are not lensed or stretched as the circle, in addition one is inside the ring, one is outside the ring, so how we can say the two dots are lensed? You can declare that those two dots are not lensed because they are located in front of the lensing black hole, but look at the image below it (bottom left with cutoff), do you think it lacks the stretching effect. I see an obvious blob that doesn't get stretched like the ring.

Hi Hong Du

"So point star shall not remain point star, it must be stretched, into ring or whatever shape that satisfy the lensing equation, which is continuous, not a point like condition that only very specific angles can satisfy."

In reality there and no point stars all of them have finite, non-zero dimensions. We only talk about point sources in a context of a given telescope, meaning that the source is smaller than the PSF i.e. you can't measure its size or shape (only brightness).

So the star has some dimension and it is stretched by lensing. But if its unlensed angular size is 0.6 micro arcsec, then even if it is streched by a factor of 1000 (extreme lensing, never seen, as far as I know) into 600x0.6 micro arcsec , it would still appear as a circular image if your resolution is 1 arcsec, i.e. 1000 times worse than this size. This is because the image you detect is the true stretched image convolved with the PSF. Hence, there is no way you can see a stretch of a star with the current technology, so you can't use the lack of the stretch as evidence against lensing.

The colour images you refer to are real examples of lensing of galaxies by galaxies (not by black holes!). Hence both the sources and the lenses are not point-like objects - they are extended and they have some irregular structures. This complicates the lensing effect. On top of that, again we have the effect of imperfect resolution. Hence, the image is not a regular ring which you might expect. I have two immediate options why these little structures are circular:

1) the same story as before - maybe they are images of regions which are bright but small. Then the stretch would make them elongated, but if the resolution is poor then their images may still be smaller than the PSF and appear circular.

2) maybe this objects are in the foreground.

I have no idea about the details of these two particular objects, but if you want to rule out any hypothesis (GR and/or lensing) you need much stronger evidence, than just two images with component which we may think should look differently (probably because we don't understand the details). There are a lot of strong evidence supporting lensing, so the evidence to rule it out must be equally strong. In this case I provided two alternative explanations, so your evidence is weak.

Broadening the discussion a little bit - if you don't think the lensing is real, then what theory you propose to explain all the instances which are interpreted by gravitational lensing - rings, Einstein crosses, arcs behind galaxy clusters, microlensing events, etc?

Hi Michal Michalowski ,

Thanks for your patient reasoning.

When you say "But if its unlensed angular size is 0.6 micro arcsec, then even if it is streched by a factor of 1000 (extreme lensing, never seen, as far as I know) into 600x0.6 micro arcsec , it would still appear as a circular image if your resolution is 1 arcsec, i.e. 1000 times worse than this size." you are still talking about stars that are lensed but not visibly appear to be lensed. I believe in this case, nobody would cite these stars as lensed stars because they are still look like point stars, there is no reason that any will think a point star is lensed or stretched.

For all the images that are cited to be lensed, they are either circular rings spanning over multiple pixels or multiple stars at different locations. The circular rings are all much greater the resolution of the imagers (or the PSF). This is the stretching I have been referring to, not the theoretical stretching none detectable due to resolution.

I am showing the attached lensed_Quasar, they shows four bright stars that are called been lensed. Even you can say that the lensing black hole or galaxy has mass distribution to move one star images to four locations, isn't there be some stretching because the lensing should be analytical not digital or some sharp delta functions, right? Besides, I can immediately say that the images are composite images from different cameras because the blue and amber colors are having different diffraction patters of the camera aperture, and PSF are not processed when you see the patterns around bright stars.

I am also showing another picture from the same image to emphasize the effects.

I have my own theory, that is what we see in optical images are mostly true images without lensing except for some deflected images that are deflected by gases (or electron distribution) of stars, which is rare because space is mostly empty. The ring structures are everywhere as we see ring galaxies everywhere, they are truely not lensed because you can see stars in them. Such as the cartwheel galaxy that I attached. No stars are arced.

Plus: I especially don't like the concept of brightening effect by lensing. This is simply asking for trouble because if we still believe in energy conservation, then the brightening of star image in some areas will mean weakening in other areas, probably not far away from the observed area. This further destroy a lot of results that astronomers rely on to calculate things around the lensed areas.

Here I am attaching probably the best lensing image in terms of stretching patter for the lensed Smiley Cheshire Cat image. See the two green arrows pointing to the stretched (lensed) stars or galaxies or whatever. They do look like true lensed image because they are stretched into arc segments satisfying lensing equation. But when you look at the red arrows, you will see totally unlensed (unstretched) stars. They appear to be in the same lensed structure. How the lensing can selectively stretch some objects while sparing those stars? You can say that the stars are located in front of the lensing black hole or whatever mass. But look, these unlensed stars all perfectly belong to the ring including the characteristic bluck color. Moreover, the bottom left arrow pointing to a unstretched galactic structure that might have a great number of unstretched stars that look smaller than the resolution of the image..

So if we believe in lensing, then there are obvious problems and we only get a lot of puzzles like I mentioned above without progress and there is no alternative. If we give up lensing, then we have new physics to think about instead of getting stuck at some theory that may not be suitable for such ringed observations. Why don't we accept that the rings are actually ringed galaxies? Why they cannot be ringed galaxies, just because there exists a lensing theory?

Hi Hong Du

"I believe in this case, nobody would cite these stars as lensed stars because they are still look like point stars, there is no reason that any will think a point star is lensed or stretched."

I had the impression that it was you who claimed that these stars should be stretched by lensing, and you used this as an argument against lensing. Hence, I made the points that yes, these stars should be strechted, but this effect is too small to be measurable with current instrumentation, so the lack of the stretch cannot be used to rule out the lensing explanation.

"For all the images that are cited to be lensed, they are either circular rings spanning over multiple pixels or multiple stars at different locations. The circular rings are all much greater the resolution of the imagers (or the PSF). This is the stretching I have been referring to, not the theoretical stretching none detectable due to resolution."

These are not multiple stars, but multiple images of entire galaxies (quasars in that case, but this could happen to non-quasar galaxies as well). Indeed, such lensing images are bigger, because the lens is always an entire galaxy - a mass of the order of 10^12 solar masses. At the beginning of the discussion you were expecting such stretching signatures caused by the black hole in the centre of our Galaxy. It has the mass of 4e6, so the stretch will be approximately million times smaller. That's why it is not possible to measure this.

Anyway, I am not sure what your argument about the arcs and rings are. I view them as the evidence of lensing. Are you implying that they are inconsistent with lensing?

"Even you can say that the lensing black hole or galaxy has mass distribution to move one star images to four locations, isn't there be some stretching because the lensing should be analytical not digital or some sharp delta functions, right?"

Yes, there is stretching, but it is smaller than the PSF, so we can't see the elongation. (Note again that these 4 images are of a galaxy not an individual star).

"I have my own theory, that is what we see in optical images are mostly true images without lensing"

Do you mean that these four images are just four separate objects? This is inconsistent with the data. First, in all such cases the multiple images have the same properties - the same colours, the same emission lines, the same redshifts, the same spectral shapes, the same variability (when one source changes brightness, the other ones do exactly the same after some delay stemming from geometrical effects). That behaviour would be extremely unlikely - you can't have 4 identical objects not related to each other changing its properties together.

"The ring structures are everywhere as we see ring galaxies everywhere, they are truely not lensed because you can see stars in them. Such as the cartwheel galaxy that I attached. No stars are arced."

You can't see individual stars in them because they are so distant that our spatial resolution is not good enough. You can't use this as the evidence against lensing.

The cartwheel galaxy is a nice example - you are right that it is not lensed. We now this, because we measured the redshift of the core in the middle and the ring and they are the same - they are at the same distance. However, all other arcs and rings we are discussing have much higher redshifts than the central galaxies, so they are not physically related structures. Your theory cannot explain them.

"I especially don't like the concept of brightening effect by lensing. This is simply asking for trouble because if we still believe in energy conservation, then the brightening of star image in some areas will mean weakening in other areas, probably not far away from the observed area. This further destroy a lot of results that astronomers rely on to calculate things around the lensed areas."

There is no violation of the energy conservation. If you take a normal lens and you look at a light source it will appear brighter, but not because the energy is produced (which you seem to imply), but because more light rays are bend towards the observer. Optical lenses do not violate energy conservation. The same happens to gravitational lensing. The light source emits the same amount of energy when it is lensed or not. When it is not lensed only one ray is registered by the observer (going straight). If there is lensing, then multiple rays are bend into the filed of view of the observer, so the source appears brighter. The energy conservation is fulfilled.

"But when you look at the red arrows, you will see totally unlensed (unstretched) stars. They appear to be in the same lensed structure. How the lensing can selectively stretch some objects while sparing those stars?"

The same as before - they are bright structures in the background galaxy that are smaller than the PSF. They are stretched, but given their small size they still appear point-like because the resolution is not good enough. We would need to have a better telescope to resolve them and measure their shapes.

"the bottom left arrow pointing to a unstretched galactic structure that might have a great number of unstretched stars that look smaller than the resolution of the image.."

This structure appears to me as very stretched from top-left to bottom right - around the lens galaxy, so all is fine here. Did I misunderstood your point?

"Why don't we accept that the rings are actually ringed galaxies? Why they cannot be ringed galaxies, just because there exists a lensing theory?"

We can't accept this theory because it is inconsistent with the data. For most of such rings (except of the cartwheel galaxy and a few rare others), the redshift of the ring is much higher than the redshift of the middle galaxy, so they can't be at the same distance - they are not parts of the same galaxy. Then if you say that they are just two galaxies (one normal and one ring) at different redshifts (chance association), then for each of such chance alignments we should see thousands of such ring galaxies not aligned with closer ones (because in your theory there is no reason why they should only appear only in the background of other galaxies). We have not seen such galaxies at all, which invalidates your theory.

Forgot to mention one more evidence supporting lensing - supernova Refsdal lensed into 5 images. After the initial four images were recorded it was predicted that the fifth image would appear again because some of the light rays take more time to reach us and therefore there is a delay. Indeed such image of this SN appeared , as predicted. How could that be explained with anything but lensing? That we had 5 identical supernovae exploding by chance exactly where and when GR/lensing predicts them to explode? This is impossible.

https://en.wikipedia.org/wiki/SN_Refsdal

"The supernova Refsdal reappeared punctually at the predicted position between mid-November 2015 and December 11, 2015[6] (with the exact date being uncertain by approximately one month which is the interval between two consecutive Hubble observations),[7] in excellent agreement with the blind model predictions made before the reappearance was observed."

By the way the images of the SN, i.e. four yellow dots here:

http://news.berkeley.edu/wp-content/uploads/2015/02/einsteincross.jpg

do not appear stretched because again - the size of such explosion is much smaller than the resolution, so we can't see the stretch.

Hi Michal Michalowski ,

Thanks again for your patient and detailed discussion. You wrote "I had the impression that it was you who claimed that these stars should be stretched by lensing, and you used this as an argument against lensing." Sorry that I couldn't said things clearer.

See the attached image. For any stars, no matter how small (or far) they appear in the image, when the lensing equation is satisfied, θE represents a circle, a 2*pi arc, not a point. Even the star is a point as you mentioned in micro arcsecs, but a 2*pi arc is definitely larger than PSF because it spans a circle, way larger than the image pixel, larger than many galaxies in the same photo. So Einstein lensing predicts that a point star will show up like a ring, a red circle as I overlapped on the attached image. And the ring structures are so representatively shown as Einstein lensing ring.

My point is that, no, the lensing doesn't work, because any point structures in the Einstein ring should be stretched equally because they satisfy θE based on its location. But why they are not stretching? See the three structures I am pointing to in the attached image. Notice that the two upper bright structures are definitely larger than PSF, but they are not stretched, moreover, one structure is inside the ring, the other is outside the ring. They simply cannot be from the lensing effect. The lowest ring structure has a blob that shows no sign of stretching. Don't they also follow the θE equation?

You can say that those structures do not belong to the lensed source. If it can be said that way, why can't I say that the ring is not caused by lensing but is an actual ring galaxy at far distance.

The best test for validating the lensing would be observe the ring over many years, and see if they change shape as the lensing condition changes. The lensing condition won't maintain all the time. If a whole Einstein ring disappears, then it would be the best evidence.

The predicted appearance of lensed objects at https://en.wikipedia.org/wiki/SN_Refsdal seems strange to me, why they are not showing up in a ring arrangement that is typical for lensing equation, the predicted locations of the lensing is beyond my understanding of gravitational lensing.

massless ''interact'' with gravity as well as massive objects due to the deformation of space-time geometry by energy concentrations. this is GR interpretation of geometry

Sorry about the attached picture missing the unstretched objects in the previous post. Now is included here with three unstretched objects.

Also, I have attached a zoomed-in image of the Smiley ring to show where I think the ring is not stretched, I believe it is a normal galaxy structure, referring to your comment "This structure appears to me as very stretched from top-left to bottom right - around the lens galaxy, so all is fine here."

Hi Michal Michalowski ,

you wrote "We can't accept this theory because it is inconsistent with the data. For most of such rings (except of the cartwheel galaxy and a few rare others), the redshift of the ring is much higher than the redshift of the middle galaxy, so they can't be at the same distance - they are not parts of the same galaxy."

You mentioned a very interesting phenomena that I am actually looking for observational evidence: the redshift of the ring is much higher than the redshift of the middle galaxy. Does it apply to all ring galaxies? Do you see less redshift in the stars in the rings and more in the core? Based on my theories, the stars in the rings and stars in the core of a ring galaxy should have different redshifts. Can you please refer some papers to me?

Thanks a lot.

Hong

Hi Hinnerk Albert ,

That is the typical Einstein's space-time concept. I am developing a new theory that doesn't need it, but I still get gravitational time-dilation which agree with Einstein's result (ignoring higher order differences). My theory has to give up the space-time warping, and doesn't support gravitational lensing.

Hi Hong Du

"So Einstein lensing predicts that a point star will show up like a ring"

Not always! An object is stretched into an Einstein ring only if it is perfectly aligned along the line-of-sight to the lens. The structures you are referring to are not perfectly aligned, so, consistently with GR, they do not appear as point sources.

"The lensing doesn't work, because any point structures in the Einstein ring should be stretched equally because they satisfy θE based on its location."

Again this is wrong. Lensing is not the same for all the sources behind the lens. The lensing influence depends on the sizes of the sources and their locations. Hence a very small bright region within a lensed galaxy will be lensed differently than the overall distribution of light in this galaxy.

"Notice that the two upper bright structures are definitely larger than PSF, but they are not stretched, moreover, one structure is inside the ring, the other is outside the ring. They simply cannot be from the lensing effect. The lowest ring structure has a blob that shows no sign of stretching. Don't they also follow the θE equation?"

Yes, this is perfectly consistent with the lensing theory. The ring is a part of the distant galaxy which is aligned perfectly on the line observer-lens-source. That's why it is lensed into a ring. The two images are likely images of another structure which is not aligned perfectly, so we see only two images of it (instead of a ring). Lensing predicts that one image would be inside one outside the Einstein radius, so all is fine. Do you have the name of this lensing system? Then I could look up the details if you want.

"You can say that those structures do not belong to the lensed source. If it can be said that way, why can't I say that the ring is not caused by lensing but is an actual ring galaxy at far distance."

I say that both the ring and two blobs are lensed. They are just different sources. The ring cannot be just a "ring galaxy", because it has a redshift much higher than the lens (I will verify this if you give me the system name). As I said earlier, if lensing didn't work, once in a while we could see such rare chance alignment of rings and foreground galaxies, but then we should see a lot more "ring galaxies" not aligned with foreground galaxies. We have not seen them, so this invalidates your argument.

"The predicted appearance of lensed objects at https://en.wikipedia.org/wiki/SN_Refsdal seems strange to me, why they are not showing up in a ring arrangement that is typical for lensing equation"

This is not a perfect ring, because the supernova is not perfectly aligned with the lens. Moreover, the lens itself is not a simple point source but a galaxy within a cluster, so the lensing signature will be more complex.

" the predicted locations of the lensing is beyond my understanding of gravitational lensing."

I wouldn't bring an ad personam argument myself, but since these are your words, then you may reflect on this - if you are criticising a theory which you don't understand, then it is unlikely that you are right. It could be more successful to first understanding a theory and then reject it.

"Also, I have attached a zoomed-in image of the Smiley ring to show where I think the ring is not stretched, I believe it is a normal galaxy structure"

Again, to me this structure is stretched - there are many blurred "lines" going from top-left to the bottom-right. There are no lines going in the perpendicular direction.

"You mentioned a very interesting phenomena that I am actually looking for observational evidence: the redshift of the ring is much higher than the redshift of the middle galaxy. Does it apply to all ring galaxies? Do you see less redshift in the stars in the rings and more in the core? Based on my theories, the stars in the rings and stars in the core of a ring galaxy should have different redshifts. Can you please refer some papers to me?"

Yes, the redshifts of the rings are higher than the redshifts of the rings. It applies to all such systems which are identified as lenses. There are a few rare examples where the rings are at the same redshift as the core (like the Cartwheel galaxy), which means that they are not lensed, just two structures of the same galaxy. There are no examples of a ring with lower redshift than the core. Your theory would predict the existence of such systems as well. This is not seen, so this goes against your theory. Moreover, your theory predicts many more rings without cores, and this is not seen either.

There are hundreds of papers on this. You can for example look at this database of lensing systems:

https://lweb.cfa.harvard.edu/castles/

zs is the redshift of the background lensed galaxy (source) and zl is the redshift of the lens - always lower.

Thank you Michal Michalowski for providing the link to database of lensing systems.

For sure I will take some time to learn the lensing equations and decide if my perception makes sense or not because I don't just take for granted any theory easily including my own, I go back and forth between different theories by testing them under different conditions, and I compare results from different theories. I need more analysis to determine which way to go instead of just taking a theory as final.

Thank you for your help, and now the lensing object that has two distinct unlensed objects is Hewitt+1987 MG1131+0456. And hopefully some redshift or spectral information can be used to determine the two objects.

Also I think you might be interested, so I attach the screenshot of the appearance of stretched star when it passes behind a point mass. It is taken from https://www.astro.umd.edu/~richard/ASTR680/Clusters_Lec2_3_Astro680_2019.pdf. You can see that even the star is a dot, it will be stretched into one or two arcs definitely distinguishable as the arcs will appear so much larger taking up segments of the sky, it doesn't matter if the star is a point or not. When the alignment is not perfect, the lensing makes two arcs, when the alignment is perfect (straight line), it is a ring.

Hi Michal Michalowski ,

As shown in the attached image in my previous post, (a) when a far star approaches from behind a black hole (or lensing galaxy) on the left, it first has a small phantom image on the right of the black hole like a black dot. Hmm, this is already a problem to me because there will definitely be many celestial bodies behind a lensing object that are close, they will all tend to create phantom images around the lensing object. Had we noticed them or not? Now black holes will no longer be black because there are always some stars that are behind, and project some phantom images around the black holes. They are no longer invisible. I am not sure if what we see is true or false. (b) as the star move closer, we see two phantom images in arcs. You can see the arcs are much larger than the star as the size of the star is no longer relevant now, the arcs are much longer in the rotational direction than the image resolution. Suppose the star size is microsec, the arcs could be thousands of microsecs maybe. Now there is another issue here, when the stars are lensed like that, the energy will be spread out thousands of times. If the star is a true point star, you won't be able to see it, let a lone the magnification. Remember the radiance theorem? If an image is magnified, it's radiance must decrease. (c) as the star move further closer aligning to the lensing object, the two phantom images connect, forming a near circle until (d) it is exactly behind the lensing object to form a perfect ring. Therefore, anything next to the star should do the same as one of (a)(b)(c)(d), anything that are geometrically close to the ring should behave similar: they must be stretched like the point star. If you see a star that is not stretched in the ring, that star must be unrelated to the star in terms of distance.

Now, there is another problem related to such lensing, are we able to see the star itself directly except the phantom images? The answer is probably no, because as you will say, they don't satisfy the lensing equation, and the light is deflected out of the sight. Alright, I believe most of the stars behind the lensing object but close to the lensing objects in sight should not satisfy the lensing equation. But this is a huge blow to astronomers because that means any lensing objects are blinding numerous stars behind it that don't satisfy the lensing equation. All we see around the lensing objects should only be things that are in front of the lensing object. Don't we see things behind the supermassive black holes (SMBH) at the center of Milky Way where we live? Don't we see any stars been lensed by our SMBH? Are there any stars in Milky Way out of the millions of stars that satisfy the lensing equation and show as rings, just one or two? We don't need to analyze how the remote rings or crosses are lensed, why don't we look at the center of our galaxy to look for proof?

All right, let's go through all the scales involved to understand for which lens-source separation the lensing should be noticeable (and hence lensing theory can be tested) and for which it is so small that it is not measurable (so with such objects the lensing theory cannot be tested). I will go through all the relevant examples we have discussed so far. It is important to note two things:

1) strong lensing (multiple images, arcs, rings etc) only happens if the the source is separated on the sky from the lens by an angle comparable to the Einstein radius. If it is further away the stretch is very small. Note that in your schematic plot the source is actually always within the Einstein radius, so the stretch is big.

2) The maximum strength of lensing is when the lens is in the middle between the source and observer. In other configurations we get smaller Einstein radii and weaker lensing.

Assumptions:

1) Both the lens and the source are very small ("point-sources")

2) I will use the Einstein radius equation which is written on your schematic plot; it is the convenient equation under "Substituting for the constants gives" in

https://en.wikipedia.org/wiki/Einstein_radius

Situation 1:

Lensing caused by the BH in the centre of our Galaxy (M=4e6 MSun) of a star on the orbit around the BH. We have started our discussion with this. The maximum distance of these stars from the BH is around 20 000 AU (most of them are closer which would make the lensing signal weaker). We have

DL = 8kpc = 8e-6 Gpc: the observer-lens distance

DLS = 20 000 AU = 9.6962723e-11 Gpc: the lens-source distance

DS = 8kpc + 20 000 AU = 8e-6 Gpc: the observer-source distance

The Einstein radius is then: 0.007 arcsec. This is the size of the ring if the star would be aligned perfectly with the BH. All stars which we can say that are 20 000 AU behind the BH are at least 0.1-1 arcsec (14-140 Einstein radii) away from the BH on the sky. At such large separations we do not expect any measurable stretch, because the stretch will be smaller than 0.007 arcsec, whereas the resolution of these data (PSF) is 0.07 arcsec. Even if a star aligns perfectly the 0.007 arcsec ring would still be blurred and be seen as a perfect circle with a radius of 0.07 arcsec.

Conclusion - you cannot rule out lensing with the data we have around the galaxy centre, because the expected stretch is much lower than the resolution. We need to wait for better optical data to do this.

Situation 2:

Lensing by a star similar to the Sun 1 kpc away (nearby) of an object 2 kpc away (for maximum lensing signal):

DL=1 kpc = 1e-6 Gpc

DLS= 1 kpc = 1e-6 Gpc

DS = 2 kpc = 2e-6 Gpc

This gives the Einstein radius of 0.002 arcsec, so also not measurable. We can see brightening of such background star, but not its stretch. This phenomena is called microlensing.

Situation 3:

Lensing of a distant galaxy by another galaxy. Let's take the mass of the lens to be 1e11 MSun - not so big galaxy and the distances 1/2 Gpc. With this we get the Einstein radius of 0.6". This is perfectly measurable and indeed this is what we see as arcs/rings around lens galaxies which you sent during the discussion.

Situation 4:

Now let's only change the lens and assume that this is a cluster of galaxies with a mass of 1e15 MSun. Then we get the Einstein radius of 63 arcsec. Indeed this is the scale of elongated arcs we also discussed for the Abell cluster.

If you would like to analyse more examples, let me know.

These were of course simple and unrealistic scenarios. Usually neither lens nor the source are compact and they usually are not rotationally symmetric. That's a reason why the rings are not perfect, the arcs are wiggly and some parts of sources are lensed more than the other.

In conclusion I would say that all the data I described above are perfectly consistent with the prediction from GR. In situations for which GR predicts a stretch too small to be measured, we indeed don't see the stretch. When GR predicts a measurable stretch we indeed detect it.

I will now answer your questions.

MG1131+0456 - as I expected this is a lensing of a complex source, so the result is also complex. See Fig. 8 of this paper, which I also attach here:

Article Multifrequency radio images of the Einstein ring gravitation...

The source is a radio galaxy, which consist of a central core (crosses on the upper panel) and two extended radio lobes (jets; circles and squares on this panel). One part of the upper lobe is aligned with a caustic line (a diamond shape line), so it is streched into a ring. This is shown on the lower panel which shows how the model looks on the sky. The core is away from the caustic, so it is lensed into two images on each side of the lens - these are the smaller sources one inside and one outside the ring. Hence the geometry of this object can be explained with GR with a radio galaxy geometry which we see in an unlensed form for many other examples.

"this is already a problem to me because there will definitely be many celestial bodies behind a lensing object that are close, they will all tend to create phantom images around the lensing object. Had we noticed them or not? Now black holes will no longer be black because there are always some stars that are behind, and project some phantom images around the black holes."

There is no problem here. To see a ring, the alignment would need to be perfect. As I shown above, for the BH in the centre of our Galaxy this would need to be within 0.007 arcsec. There are no such stars perfectly aligned because we only see a few tens of those. Stars which are farther away (which in principle could be aligned) cannot be seen because there is too much dust in the centre of our Galaxy.

"Now there is another issue here, when the stars are lensed like that, the energy will be spread out thousands of times. If the star is a true point star, you won't be able to see it, let a lone the magnification. Remember the radiance theorem? If an image is magnified, it's radiance must decrease."

Could you describe this problem a bit more, because I don't understand what is wrong. As I see it, the total energy of a lensed star doesn't change at all. The only effect is that instead of one light ray coming to the observer, we see several, so the source appear brighter. This also mean that for an observer located elsewhere in the Universe this star would appear dimmer. The energy is conserved, it is just re-directed. This is exactly the same mechanism as an optical lens - it doesn't create energy, just focus it to one point.

I am not sure what you mean that we won't be able to see a true point star. First, there are no point stars, every one of them has finite dimension. Second, however small they are, we will still see them if they are bright enough.

I have never heard about the radiance theorem (unless by a different name) and can't find a good description, so could you explain to me how this applies here? I see that this connects somehow to light going through media with different refraction indices, so this doesn't seem to apply here. For all of our discussion and calculations we assume that the light is travelling in vacuum and this is enough to agree with the data perfectly.

"If you see a star that is not stretched in the ring, that star must be unrelated to the star in terms of distance."

Yes, that's one of the option, but the second is that this star is behind the lens, but it is away from the lens on the sky by more than several Einstein radii. Then we wouldn't see the stretch either.

"I believe most of the stars behind the lensing object but close to the lensing objects in sight should not satisfy the lensing equation."

Again, we see no star within 0.007" from the BH in our Galaxy (our resolution is too poor too look for them), so "close to the lensing object" is something which we don't have technical capability to see. We will surely do this test in the future when we have such capability.

"But this is a huge blow to astronomers because that means any lensing objects are blinding numerous stars behind it that don't satisfy the lensing equation."

No, it's not a problem. Lensing is extremely rare, only the objects perfectly aligned are lensed. As I shown above, GR predicts that basically no star in our Galaxy should appear to be lensed by the BH - either they are too far away from the Einstein radius, or they are too far away from us to be seen.

"Don't we see things behind the supermassive black holes (SMBH) at the center of Milky Way where we live?"

Slightly behind the BH? Yes, we see the stars orbiting the BH. However, we don't see stars significantly behind the BH, because there is too much dust there and distant stars are to dim to see them.

"Don't we see any stars been lensed by our SMBH?"

No, because the Einstein radius is too small. You would need a star on the other side of our Galaxy to have a hope to see lensing, but such stars are not observable because they are too dim.

"Are there any stars in Milky Way out of the millions of stars that satisfy the lensing equation and show as rings, just one or two?"

There are many stars which were shown to be lensed, but not by the BH but by either some small stars, or brown dwarfs or planets. See many examples mentioned here:

https://en.wikipedia.org/wiki/Gravitational_microlensing

If you need more details on this phenomena, let me know.

However, note that the Einstein radius in such cases is of the order of 0.001 arcsec (Situation 2 above). This is much smaller than the resolution of our telescopes, so no rings can be seen. We need to have much better telescope to see them and for now can only see brightening.

"We don't need to analyze how the remote rings or crosses are lensed, why don't we look at the center of our galaxy to look for proof?"

Because the Einstein ring there is too small to be detected by our telescopes.

Hi Michal Michalowski ,

With what you calculated, there is a lensing angle of 0.007" (arcsec) for a star that is orbiting 20 000 AU away from the BH at GC and right behind the BH. Look at the UCLA image of the GC, I would say that the resolution is ~0.05". Sure enough such Einstein ring will not be visible. But there are plenty of stars behind BH that are not orbiting the BH that can be 100 times or 10000 times farther from the BH (that is only 0.01 kpc and 1 kpc). They are expected to have 0.07" ring or 0.7" ring, well within the resolution. If they are right behind BH, we should have arcs that are visible, right? We don't see any, see the attached UCLA image.

You may say they are not right behind the BH, say there is significant θs, then use θ1=θs+θE2/θ1, we have θ1=-sqrt(θs+4θE2)/2-θs/2 and θ1=sqrt(θs+4θE2)/2+θs/2. They are all arcs which appear to be somewhere around 0.07" to 0.7" away from the BH. Again, I don't see such arcs around the BH in the UCLA image either.

If we trust lensing, then we ruled out any possible stars right behind BH ranging from GC all the way to the end of Milky way. Any stars close to the 0.07" and 0.7" are also ruled out. All the stars we see in the attached UCLA image are just stars not satisfying the lensing equations and in front of BH. Do you think it is reasonable?

No, there are two mistakes in your reasoning. First, in order to rule out lensing you would need to find such star which is expected to be lensed, but it is not. Then you have ruled out lensing. Before you find such star you don't have the argument. These observations are difficult, so you can't say that the fact that we don't see these stars means that the lensing theory doesn't work. It is the observational limitation, not problems with lensing. You say that we should see arcs, but you haven't demonstrated that there is even a single star for which this should happen. At the very least you would need to calculate the surface density of stars in the far-side of the Galaxy and show that there should be some stars within 0.7" of the BH.

Second, there is actually very simple explanation that we don't see very many stars significantly behind the BH. This region is extremely dusty, so the farther you look at, the more severe dust attenuation is. Hence, it is progressively more difficult to see any stars in the background. This is actually the reason why these observations are done at near-IR not the optical.

In other words - the data you presented in the image show that there is no star 0.7" away from the BH which is in the far background (all stars getting close to the BH are on the orbit around it and are very close to it). In these data no assumption was made on GR, these are just data. Hence, I am not sure how you try to use these data to rule out lensing. Even if we reject lensing, there is still no star 0.7" away from the BH, so I don't see the argument here.

We can talk about why we don't see such stars, but whatever we agree on has nothing to do with lensing. Either the dust attenuation is too severe, or the number density of stars is too low to have any bright star within 0.7", even if we include everything all the way to the far side. Either way, this can't be used to test lensing.

All right, I made the calculations of the number density of stars and how many are expected within 0.7" by chance . I assume that there are 1e11 stars in the Milky Way and that they are distributed in a disk with 16 kpc radius and 1 kpc thickness. Hence the volume number density of stars is:

nden = 1e11 / (PI*16^2 * 1) = 1.24340e+08 stars / kpc^3

We are looking towards the centre of a galaxy, so observing a star within alpha=0.7" means that it must be located somewhere in a cone with a tip located at the Earth (8 kpc from the Galaxy centre) and an opening angle of 0.7". The hight of this cone is

h= 8 kpc + 16 kpc = 24 kpc.

The radius of the base of this cone evaluated at the far edge of the Galaxy is:

r = h * tan (alpha/2) = 4.07244e-05 kpc

The volume of this cone is

V = 1/3 * r^2 * h = 4.16820e-08 kpc^2

Hence the number of stars within this cone will be:

N = V * nden ~5.

So there are 5 random stars within 0.7" of the BH (plus of course those which are there not by chance, but because of the gravity of the BH; as I shown earlier, they cannot be lensed). Now 1e11 are all the stars - luminous and faint. There are a lot more faint stars than luminous stars, so we can safely conclude that all of these 5 are faint so they are not detectable towards the BH, because they are too far away (far side of the Galaxy) and there is too much dust. Hence it is not strange that we don't see any background star 0.7" away from the BH.

But I want to stress once more - just the fact that we don't see such stars is not an argument against lensing. You can rule out lensing only if you do see such star and it is not lensed.

And I also demonstrate that we could only see the very brightest stars in this image, so it is certain than neither of those random 5 are that bright and hence are not visible. The faintest of the stars around the BH has a Kband apparent magnitude (~flux) of m=18.5, see the "List of stars" in

https://en.wikipedia.org/wiki/Sagittarius_A*_cluster

I assume that this is the limiting magnitude of that image. Hence a star at the far edge of the galaxy (dist_pc = 24 000 pc away from us) in order to be visible would need to have an absolute magnitude (~luminosity) of

M = m - 5*(log10(dist_pc) - 1 ) = 1.6 mag.

The Sun has the Kband absolute magnitude of 5, so such star would need to be 3.5 magnitude or 25 times brighter than the Sun. There are such stars, but they are extremely rare, so you wouldn't get such star by chance in a volume in which you expect only 5 stars in total.

Hi Michal Michalowski , thank you for very detailed calculations. Now we all agree that in the 1”x1” range, there’s no visible lensed image. That’s a great conclusion. As of dust, that is true also. I remember that the attached image is either radio frequency or infrared, they go farther than visible light, so if stars show up in the image that is 8 kpc away from the sun, then there should be stars visible 16 kpc away unless the dust distribution got thicker beyond the GC. Remember that the lensing do have to be right behind the BH, we can do jus expand the field of view, and there are still lensing that should be partial arcs. But no mentioning either. I am not saying that just because the Milky Way shows no lensing then lensing doesn’t exist. It should at least cast some doubts.

So this forces us to look somewhere else where there is good lensing where far stars are right behind the black hole. There are Einstein rings. Now you should agree that no matter how small the star appears to be, they should be circles, if not big arcs. So if you see a point star in a lensing ring, you will think it must not satisfy the lensing equation. One or two of such point stars are ok, but many is unacceptable, and a blob simply cannot work. You already proved how hard it is to have something right at a specific location even when stars are dense sideviewing Milky Way.

BTW, the UCLA image of 1”x1” field of view already has 10 stars counted and tracked with their orbits, not mentioning others behind which covers larger volumes beyond GC. Maybe GC has higher star density than other parts elsewhere or maybe only for stars near GC only.

Hi Hong Du

Yes, I agree that no matter how small a star is, it should be lensed into a visible arc if it is aligned with a lens. My point is that given how many stars are in the Milky Way, the chance of finding even one of such star is zero.

Do I understand correctly that you still imply that some of the stars we see towards the Galactic centre should be lensed? I get it from your sentence "One or two of such point stars are ok, but many is unacceptable". The point is that *all* of these "many" stars are orbiting the black hole, so, as I showed earlier, the lensing signature for all of them is negligible. You need stars much farther away from the BH in order for them to be lensed. I calculated that only 5 stars (of any brightness) are expected given the number density of stars in the galaxy and strictly none of them should be bright enough to be detected (because bright stars are extremely rare). Hence, the lack of such distant stars is very much expected independent of whether lensing works or not. Hence their lack cannot be used to rule out lensing. Moreover, I even haven't included dust into this consideration of 5 stars - the more distant a star is, the more dust is in front of it, so they would need to be even brighter (and hence rarer) to be detected.

But it seems to me that this reasoning doesn't convince you, so I will try to calculate how many such bright stars should be there behind the BH for which we can hope hope to measure the lensing signature.

As before let's assume the limiting K-band magnitude of 18.5. For example, at a distance of 8 kpc from the BH (16 kpc to us) this corresponds to an absolute magnitude of

M = m - 5*(log10(dist_pc) - 1 ) = 2.5 mag.

We need to correct this for dust. Towards the Galactic centre the dust attenuation at K-band is 2.4 mag:

https://ui.adsabs.harvard.edu/abs/2011ApJ...737...73F/abstract

Here I am making an absolutely optimistic assumption that the attenuation measured for stars near the BH is the same as for a star 8 kpc behind it, so I am effectively assuming that the is no dust behind the BH. If I correctly assumed that there is dust also behind the BH, then the number of stars I derive at the end would be even lower (because they would need to be even more luminous to be detected).

Coming back to our star, in order for it to be detected it would need to have the dust-corrected absolute magnitude of

M = 2.5 - 2.4 = 0.1 mag.

This is 5 mag, so a factor of 100 brighter than the Sun. So you see that only extreme giants could be visible there. Let's calculate how many of them exist. The luminosity function of stars (=how many stars of a given luminosity exist) shows that stars with a K-band absolute magnitude around 0 are 100 times less frequent than those with an absolute magnitude of 5 (like the Sun), see Fig 4 in

http://adsabs.harvard.edu/pdf/1982ApJ...255..181M

Hence, out of our 5 stars within the cone 0.7" from the BH, we expect only 0.05 to be bright enough the be detectable. Since this number is less than 1, the most probable option is that there is not a single star within 0.7" from the BH including all stars all the way towards the Galaxy edge which is bright enough to be detected (apart from those who are very close to the BH because of its strong gravity - there number density of stars is much higher than the Galaxy average I am using). We could have been lucky to get one such star and measure the lensing, but apparently we are not lucky. Hence, the fact that we don't see any lensed stars there doesn't cast any doubts on lensing.

You mention that the solution is to expand the field of view to see more and more stars, but this not the case. At the maximum distance from the BH (16 kpc - the Galaxy edge), the Einstein radius is 1.64". Hence we can zoom out only by a factor of 2, getting 4 times more stars (i.e. only 0.2 stars bright enough to be detected, still less than 1), but that's it. We can't zoom out more, because at larger angular distances from the BH even the farthest stars on the other side of the Galaxy are not expected to be lensed strongly.

I hope this will convince you that the fact that none of the stars around the BH is visibly lensed is because they are all too close to the BH to be lensed, whereas farther stars are so rare than we don't expect even one of them to align within 2" with the BH. Hence we do not have any evidence against lensing here.

Hi Michal Michalowski , thank you for providing many more stars that is within 1" of the view of BH and the star brightness analysis. I just attached the image of the infrared image to show the stars.

Remember I calculated that if the star is only 0.01 kpc and 1 kpc (instead of the orbital distance in the 20,000 AU range) beyond the BH at GC, it should have a lensing circle that extends 0.07" and 0.7" , respectively. Since the more than 10 stars orbiting the BH are already 8 kpc away from us, therefore adding 0.01 kpc or 1 kpc should not change the magnitude too much using the same dust attenuation and 1/r2. Look at the stars in the attached images, there are really numerous dim stars, should some of them be 0.01 kpc away from the BH and show 0.07" circles or arcs? or even 1 kpc away from the BH and show 0.7" circles?

I cannot believe that all the dim stars we see in the image are either in front of the BH (shouldn't them be brighter?) or less than 0.01 kpc away from the BH (all orbiting around BH busily in space in the 20,000 AU range).

So I am not convinced still unless you can prove that stars 0.01 kpc behind the BH is not detectable or forcefully rule out that there is no way that stars can exist behind BH, even around its neighborhood, and disagree with stars in the image I attached.

Regardless of what I said above, we need to review the Smiley Cheshire Cat image that is typically used as lensed image. You should now understand my concerns about those point stars and galactic structures I pointed out with red arrows. Once a star satisfy the lensing equation, especially the circular ones, the stars must look circular, or at least arcs. Now I have pointed out multiple stars on the arcs and a structure on the arcs that do not appear lensed. This is definitely a unacceptable situation. If the lensing theory were developed myself, I would not stop there, I would find out reasons to see why multiple stars and even structures in the circle but show no sign of stretching. This is basically an obvious failure of the theory. You cannot have multiple stars located at the same lensing circle but not affected by the lensing. You can say that those stars are coincidentally being there. Ok, just check other lensed images, you still can find point stars and structures that are not lensed for the rings. Are they all coincidental?

I would rather accept the ring, the stars and structures are really what they are without lensing rather than believing they are coincidental.

Hi Hong Du

"Since the more than 10 stars orbiting the BH are already 8 kpc away from us, therefore adding 0.01 kpc or 1 kpc should not change the magnitude too much using the same dust attenuation and 1/r2"

Its not the point that we couldn't see such luminous stars 1 kpc behind the BH. The point is that 1 kpc behind the BH, the stellar density is too low for even a single star to be there. The number density of stars around the BH is indeed high, so there are stars there, but 1 kpc behind we are in the average place in the Galaxy, so we don't expect even a single star in such a small volume.

By the way 0.01 kpc is not enough for our purpose - the Einstain radius will be 0.07", so comparable with the resolution of this image, so it will not be possible to see the arcs. You need to have the resolution at least several times smaller than the size of the object (arcs) in order to have any hope of measuring its shape.

"I cannot believe that all the dim stars we see in the image are either in front of the BH (shouldn't them be brighter?) or less than 0.01 kpc away from the BH (all orbiting around BH busily in space in the 20,000 AU range)."

I think it is the most reasonable to say that the person who proposes the argument shows that it works. You claim that some of these stars should be lensed, so you need to provide the evidence for that. If you find which of these stars are behind the BH and close enough on sky to be lensed, then we can discuss that. If you can't find any, then you don't have an argument.

Remember that more nearby stars don't need to appear bright in the sky - if they are just small stars (which are a lot more numerous, so more probable to find), then they would still be faint even if they are closer.

I also don't understand why you show a zoom-out image around the BH. As I said before only stars within 1-2" (depending on the distance) can be strongly lensed, so it doesn't matter how many stars we see several or even tens arcsec away - they will not be lensed anyway.

"So I am not convinced still unless you can prove that stars 0.01 kpc behind the BH is not detectable or forcefully rule out that there is no way that stars can exist behind BH, even around its neighborhood, and disagree with stars in the image I attached."

0.01 kpc is not enough, because the stretch of 0.07" cannot be detected with the current data. But sure, I can prove that there shouldn't be any stars within 1 kpc from the BH (because they are too rare), or any other distance you pick. The radii of the cone with alpha=1" opening angle evaluated at h=8 kpc and H=9 kpc are:

r=h*tan(alpha/2) = 1.35748e-05 kpc

R=H*tan(alpha/2) = 1.52716e-05 kpc

Hence the volume of such cone extending from the BH to 1 kpc behind the BH is:

V = 1/3 * PI*R^2*H - 1/3 * PI*r^2*h = 2.44231e-10 kpc^3

With the density of stars nden= 1.24340e+08 stars / kpc^3 I calculated above, this means that in this cone we expect

V*nden = 0.03 stars

This is less than 1, so we expect no stars 1 kpc from the BH which is within 1 " from it. Not because it cannot be seen there, but because the density of stars away from the Galaxy centre is too low for any stars to happen to be in such a small volume.

Now I even should make a correction that not all stars are bright enough to be detected in this image, and bring this number down, but there is not point - even without this we expect zero stars 1 kpc behind the BH within 1" from it.

I can keep making these calculations for other parameters, but the conclusion wouldn't change. If a star is to be lensed by the BH, it needs to be very close to the line of sight of the BH, and this volume is too small for any star to be randomly there.

Cheshire Cat (I love the name!) - well, earlier in the discussion you claimed that the ring with two point sources is the evidence against lensing. I gave you the proof that it can be explained by a simple lensing configuration in which a lens is a radio galaxy withe compact core and two extended radio jet lobes. Now you show an image which is extremely complicated lensing arrangement - we have at least 5 galaxies in the group which act as lenses and at least two sources at z=1 and z=1.4 which are lensed, see for example this paper

https://ui.adsabs.harvard.edu/abs/2009MNRAS.392..104B/abstract

Lensing is not regular - the stretch depends on the position of the source, on its distance and on its intrinsic shape. Hence in a complicated system like that it is at least possible to have one structure extremely elongated (ring) and one more compact (the point sources). Maybe there are at different redshifts? Maybe one is crossing the critical line (at which the stretch is maximum so we see the ring) and the other is outside of it so is not stretched? Who knows. The point is that if you want to rule out the theory you need to have a clean argument. You would need to show that what lensing predicts for this object is inconsistent with the data. You don't show the lensing prediction, just a very simplified view on lensing based on our feeling on how it works for point sources and point lenses. In this system neither is true. Hence, to use it against lensing you would need to provide a lens model which is robust and show that it cannot predict the what we see with data.

OK, I have found a lens model for the Cheshire Cat, see the attached figure which is from Fig. 2 bottom left in this paper (it is called SDSS J1038+4849 there):

https://ui.adsabs.harvard.edu/abs/2020ApJS..247...12S/abstract

The system is even more complicated that what I inferred from the previous paper. There are more sources and the critical line is very irregular. The point sources (galaxy C) you are concerned about are multiple images of a galaxy at z=2.8. The ring at which they are located is also part of this galaxy. As I said before, the ring is stretched because it is an image of the part of the galaxy which lies exactly on the critical line. The point sources compe from a part which is away from the critical line so it is lensed into four images.

The authors summarised the lensing model for this system by "The model is well constrained and is classified in category A." Your claim was that this system is inconsistent with the lensing theory. However, they have found a lensing model which fits perfectly the data, so your claim is wrong. You can't rule out lensing using this system because it is consistent with the GR prediction.

If you want to look for evidence against lensing I would look for simpler, cleaner systems. When you look at messy complicated systems you can get all sort of complicated images, so you will never be able to use it as a test case for lensing.

And I have a related challenge for your theory. If you assume that there are no lensing, then all these rings and arc we see are intrinsic shapes of background galaxies (is this correct in your theory?). For the Cat we have an elongated, curved structures 20 arcsec long or so. For cluster images we viewed before we have arcs that are 100 arcsec long or so. These structures are much further away from the central structures (the eyes of the cat or the cluster of galaxies), so they are not physically related and in your theory they are just chance projections. If that's the case, then why do all these structures align with these less distant massive structures and wrap around them on the sky? If they are not physically related in any way, as your theory predicts (correct me if I am wrong here), then we should see a lot of such structures with no visible central structures (just arcs no lensing galaxies) and some of the arcs should curve away from the massive structures not around them. We don't see such lonely rings/arcs, which is a weakness of your theory.

Hi Michal Michalowski ,

Thanks again for the followup posts. I am showing the zoomed-out infrared image around GC to show that how many stars are visible in the field of view. I don't worry how many stars should theoretically show up, what I am interested in is how many stars we can see in the image. The numerous stars are true distinguishable stars, and we cannot all rule them out as in front of BH or cannot be in the range 0.01 to 1 kpc behind the BH. It will be a tough theory to rule all of them out. And the lensing equation doesn't require the stars must be exactly behind the BH to show the rings and arcs.

For stars that are few arcsecs away from the BH, θs is around 0" to 1", the lensing equation can be expanded as a Taylor theory because θs is a small term compared with θE . These stars will appear to be arcs at θE+θs/2+θs2/8/θE. You see that they should also appear near θE which is around the intended range of 0.07" to 0.7". There are just so many chances of getting a ring or arc at least given that many stars showing in the image around 1", but none showed up in the image. Isn't this already casting doubts unless you truly believe the conclusion that all the many stars bright or dim in the image are proven to be not behind BH in the range of 0.01 to 1 kpc. Well this is a great conclusion by itself because it takes out statistics about the stars near GC. If you apply such conclusion to all images of galaxies or black holes, you will get more such conclusions ruling out stars and galaxies near or behind them.

Thank you Michal Michalowski for the colorful lensing image for cat. Now you see, in order to explain the lensing effect, people have to put in more and more parameters including various sources, double lenses, and various data to fit what they want to believe. This is data fitting, not good sense of science. Of course you can arbitrarily throw in many parameters to make analysis the same as what you believe. But doing so is only going to hide the real physics.

The lensing cases I posted here are the most popular and supposedly the most convincing lensing examples that I could find. There is simply no perfectly good lensing images. I would love to see you prove me wrong. That is why I am here to ask people to provide good and simple lensing images so that I can believe in the lensing. But no. Even for the almost perfect cat lensing example, people have to come up with various reasons and parameters and multiple lensing or lensed objects. I just need simple examples. Shouldn't there be enough of them in our sky if lensing is a true theory? Aren't black holes far enough to be considered point source for lensing? While it is just so hard to find one single star that is is 0.01 to 1 kpc behind BH of our own Milky Way, yet, there are two lenses with complicated mass distributions to explain its morphology in a single image of 30"x30" of sky.

I just need a couple of perfect lensing images. May be there is no one at all or do we have to accept a theory that has no good examples?

As of my own theory, I am not so confident, so I cannot just let people know that my theory is going to explain what we see. But I cannot be as certain as the lensing theory has been. I cannot throw in parameters and fit the observations. But I do have some really good clues, and I need to find some simple observations (which I already have found some) to support my theory and at the same time I need good lensing examples to disapprove my theory so I know where I got wrong.

Those rings and stars for the cat lensing example still look like a ring galaxy to me, although you have pointed out they have different properties. But ring galaxies are a matter of fact, and you cannot force ring galaxies to appear exactly what you expected them to look, as has been shown in various confirmed ring galaxy images.

So, lensing theory is still a big question mark to me. But I do appreciate that you brought in so many insightful perspectives.

Hi Hong Du

"I don't worry how many stars should theoretically show up, what I am interested in is how many stars we can see in the image."

An image 10" across misses the point that only stars within 1-2" can be lensed. My calculations show that within such a small volume (opening angle of 1") we expect no stars by chance. Before you prove otherwise (identify a star that is in the background), there is no argument against lensing. The point is that if you select a random 1"x1" piece of the sky you will rarely find even a single bright star there. This is actually very good for us, because otherwise extra-galactic astronomy would be nearly impossible.

"There are just so many chances of getting a ring or arc at least given that many stars showing in the image around 1", but none showed up in the image."

I have already showed that we expect on star within 1" of the BH in the background, so no, there are no chances of a star showing a ring. Do you think I made a mistake in these calculations?

Isn't this already casting doubts unless you truly believe the conclusion that all the many stars bright or dim in the image are proven to be not behind BH in the range of 0.01 to 1 kpc.

Why shouldn't I believed in this conclusion? Sure, these are approximate calculations, but the order of magnitude should be all right. Even if I am off by a factor of 10, then the fact that we don't see any lensed star in the background is perfectly consistent with the expectation, so there is no argument against lensing. You need to have strong arguments in order to rule out any theory, this is too weak.

Well this is a great conclusion by itself because it takes out statistics about the stars near GC. If you apply such conclusion to all images of galaxies or black holes, you will get more such conclusions ruling out stars and galaxies near or behind them.

Could you explain this a bit more? We cannot apply these calculations for any other galaxies, because for them our resolution does not allow us to resolve the Einstein rings of any BH in them.

"Now you see, in order to explain the lensing effect, people have to put in more and more parameters including various sources, double lenses, and various data to fit what they want to believe. This is data fitting, not good sense of science. Of course you can arbitrarily throw in many parameters to make analysis the same as what you believe. But doing so is only going to hide the real physics."

You have selected as an example one of the most complex lensing example (the Cat), so I am not sure why you are complaining that a lot of parameters are needed to explain it. You call it bad science, but you can clearly see multiple arcs at different radii and of different colours (i.e. different galaxies being lensed) and multiple lensing galaxies, so how can the model be simple in such a case?

I am not trying to say that this image is a proof of lensing, but equally, this image cannot be used as an evidence against it (which is what you are trying to do, as far as I understand). If a given theory can explain the data, then these data cannot be used to rule the theory out. You would need a dataset which cannot be explained by the theory to rule it out.

"There is simply no perfectly good lensing images. I would love to see you prove me wrong."

What do you mean by "good lensing image"? From my point of view all images with arcs and rings are good demonstration of lensing. Do you mean simple? If yes, I strongly disagree with your reasoning - the fact that the Universe is not simple doesn't mean that it works in a different way that you expect. Though there are some lensing images much simpler than the Cat, for example:

https://upload.wikimedia.org/wikipedia/commons/1/11/A_Horseshoe_Einstein_Ring_from_Hubble.JPG

https://www.almaobservatory.org/en/press-releases/dwarf-dark-galaxy-hidden-in-alma-gravitational-lens-image/

Are you looking for anything specific to perform your tests on this?

"Even for the almost perfect cat lensing example, people have to come up with various reasons and parameters and multiple lensing or lensed objects."

The cat lensing is far for a regular image (is the what you mean by "perfect"?). It is full of separate arcs and multiple sources. There are also visible multiple lensing galaxies. Hence, people don't come up with multiple lensing, but just explain what we directly see on the image.

"Aren't black holes far enough to be considered point source for lensing?"

There are no BH far enough and massive enough to produce a good lensing image. Look at my calculations above:

1) a stellar-mass BH produces an Einstein ring orders of magnitude smaller than our best resolution;

2) the supermassive BH in the centre of our Galaxy would produce a 1-2" Einstein ring (detectable), but the number density of stars is too low for any background star to align within this value;

3) a supermassive BH in other galaxies produce an Einstein ring orders of magnitude smaller than our best resolution.

There are no other BH which we can use. We do see lensing, but produced by entire galaxies, because their masses (and hence Einstein radii) are millions of times larger than those of even most massive BHs in the Universe. But then the lensing gets complicated - the lenses are not point sources. Moreover, most of the lensed sources themselves are not point sources either - they are also galaxies, so their different parts are lensed differently.

While it is just so hard to find one single star that is is 0.01 to 1 kpc behind BH of our own Milky Way, yet, there are two lenses with complicated mass distributions to explain its morphology in a single image of 30"x30" of sky.

This is because the mass of the supermassive BH in the Galaxy centre has a mass of 4e6 solar masses, whereas a typical galaxy has a mass of 1e11-1e12 and a cluster can reach 1e14-1e15. Hence, the Einstein radii produced by the latter are millions times bigger and hence so much easier to find.

"Those rings and stars for the cat lensing example still look like a ring galaxy to me, although you have pointed out they have different properties."

No, this is not a ring galaxy. The redshifts of the arcs and multiple point sources are much higher than the redshift of the central galaxies (the eyes of the cat), Hence, they cannot be parts of the same structure.

But ring galaxies are a matter of fact, and you cannot force ring galaxies to appear exactly what you expected them to look, as has been shown in various confirmed ring galaxy images."

Could you explain this a bit more? What do I force? Indeed there exist ring galaxies. For them the redshift of the ring is the same as the redshift of the central core. But we are talking about structures for which the redshift of the ring is much higher than the central core. They are perfectly explained by lensing. In your theory they must be chance projections, so we should also see such rings not associated with central cores. This is the argument against your theory.

Let me summarise our discussion. You are looking for evidence against lensing. We can rule out lensing if we find:

1) A star that is far behind the BH in the Galaxy centre aligned within 1-2" from it, for which the GR prediction is an arc larger than 0.1" or so. We don't know such a star, so this can't be used to rule out lensing. The lack of such stars is perfectly consistent with what we know about the number of stars in the Galaxy.

2) A galaxy lensing configuration which is inconsistent with the GR prediction. For example that could be a configuration of images, which cannot be explained by GR or a system for which GR can only agree with the data with some crazy parameters that are unreasonable, for example a mass of galaxy of 1e15 solar masses. All systems we discuss (and are discussed in the literature) can be explained by lensing within GR, so this can't be used to rule out lensing.

3) A positional displacement of a star close to the surface of the Sun which is inconsistent with the GR prediction. This is not the case either, all measured displacements agree perfectly with GR (by the way the Newtonian theory predicts a value by a factor of two less than the data).

Have I missed anything? If not, then we don't have even a single dataset which is inconsistent with lensing. Just the fact that something doesn't conform to our qualitative expectation cannot be used as evidence against any theory.

Hi Michal Michalowski ,

I am not sure if your calculation about number of stars around Milky Way GC is correct or not. But obviously, the images from UCLA shows that there are more stars than you calculated.

I just found another source at http://www.astronomy.ohio-state.edu/~ryden/ast162_7/notes31.html#:~:text=Within%20a%20parsec%20of%20the,0.2%20star%20per%20cubic%20parsec.

According to that link, there are many more stars near GC than other part of the galaxy. The density is 107/pc3, or 1016/kpc3 around GC. If we are looking at 1"x1" sky around GC and 1 kpc behind BH at GC, then we would expect 1016/kpc*(5x10-6x8kpc)2x1kpc=1.6x107.

But this number doesn't make sense either, because the image from UCLA definitely shows otherwise. There are not that many stars in the picture. So I don't know what to believe.

I would rather believe what I see in the image. I would think statistically some of them behind BH between 0.01kpc to 1 kpc makes more sense than the prediction of the calculations which we don't know where the numbers actually come from, and what physics are involved in the average star density. You already see stars in the image, why not believe in what we see rather than what calculations tell you. Aren't calculations meant for explaining what we see rather than the other way around?

The solar bending experiment must rule out the atmospheric effect. Noting doing so only means the person who is doing the calculation is not very careful, or at least incomplete, not mentioning other possibilities.

For the perfect horse shoe lensing image that a lot of people are citing, let me show you the zoomed-in screenshot of the picture in the attached screenshot. You can also find the picture at https://www.nao.ac.jp/en/contents/gallery/2016/20160315-alma-full.jpg taken by ALMA which has higher resolution.

See the few dots I am pointing at with red arrows. They are not stretched or arced. You can say these are different stars at different locations because of their redshifts are different. But having numerous of them perfectly aligned with the ring is statistically impossible as you calculated that even no single star should appear in 1"x1" area behind BH at GC in a 1 kpc range within a star-studded galaxy.

Let's think it the other way: why not the perfect circular ring doesn't show a few bright sharp arcs lensed from a few brightest single stars in the lensed galaxy, don't they have stars? You may say, the lensed galaxy has no individual particularly brightest stars. Then you may also say, these dots might be imaging artifacts. Well, why not take another image of the same ring and compare over time. Then you may say, that doesn't make sense, because the stars are not supposed to be arced or stretched, see the photos of the Einstein cross of the four stars are individual stars not arced or stretched because of the lensing object has perfect mass distribution that bent the same star image to the perfect point locations. If that is the case, why there is a ring ?

The lensing theory appears too flexible to me. There are endless parameters to tweak in order to believe it. My imagination becomes a barrier accepting it. I just at least need a perfect ring with a sharp star drawing a sharp circle or arc, not some dots showing up in a ring that appears like a ring galaxy.

But don't worry about the details of stretching, lensing theory has been stretched to lens individual dots to multiple individual dots without worrying about arc or stretching at all. I have attached the screenshot, but you can check the link here at

https://www.syfy.com/syfywire/sunburst-arc-reveals-how-infant-universe-lit-up

It is about the sun burst arcs. One structure is lensed into 6 structures with no stretching for the ultraviolet light but stretching for the longer wavelengths.

Isn't this amazing? With all the discussions we had without convincing me, now the lensing theory can apply to different wavelengths in the visible wavelengths, and lensed to different locations. I am even more unconvinced after seeing the lensing theory been cited this way.

Hi Hong Du

Let me start by an important point. Many times you use the argument that a point sources (not stretched) located at a ring/arc cannot be produced by lensing. This is wrong - such configuration can easily be produced by lensing. You just need to have a source which is extended and have some structure.

In order to prove this I will provide two examples of point sources located at the ring produced numerically assuming only lensing. First example is a bit silly, but it demonstrates the point well. See the attached image of a lensed dog from

http://www.mattianegrello.com/?page_id=513

On the lensed image on the right you can see that the eyes of the dog in the lower image are not stretched at all but they are located on a ring which is stretched by more than 180 degrees. This is because the eyes are not located exactly on the caustic line, whereas the other part of the dog are. This image was produced by applying only lensing equation to the image of the dog, so this demonstrates that lensing can produce an image in which you have a point source located on an arc and they both come from the same source.

A second, more serious example is coming from my own playing around with the lens simulator:

http://pcollette.webege.com/mirage.html

You can upload any image there, set up the lens and see how the lensed image would look. Feel free to use it as well to investigate what lensing can and cannot produce.

On the image you will see that the blue complex source marked on the left is lensed into two arcs. The lower arc is clearly composed of a point-source on the left and an arc extending to the right.

Hence, I provided two images which were produced exclusively by applying lensing (there are no foreground images or different physical mechanisms in these images) and I was able to reproduce what you repeatedly show as evidence against lensing - a point source on a ring. This proves that your argument is wrong.

On the ALMA lensing ring (the galaxy is called SDP.81 in case you want to look up the details): I have just proven than point sources on a ring can be produced by lensing. However, actually these point sources are not real structures - these are just noise spikes. Its difficult to recognise because you are showing a highly processed image presumably made by an graphical artist to look nice. I attach the image of the actual data from this paper:

Article ALMA Imprint of Intergalactic Dark Structures in the Gravita...

You can see that such point sources are scattered around everywhere. The point is that on interferometric images noise is correlated in adjacent pixels, so noise manifest itself as several adjacent pixels going slightly up or down together (not like in typical CCD image where one pixel can go up and the next one can go down due to noise). Hence, one needs to be really careful when interpreting any point source which is fainter than 5 times the noise value.

Anyway, SPD.81 can perfectly be reproduced by lensing, so it cannot be used as evidence against this theory!

See attached figure with the lens model from this paper:

https://ui.adsabs.harvard.edu/abs/2020MNRAS.494.5542R/abstract

The middle column shows residuals: data minus model. They are consistent with noise, so lensing model works good, so there is no reason to use this example against lensing.

By the way, isn't this the perfect lensing example you are looking for? The source is shown on the right column, it is extended, but it is a simple regular blob. The lens is also a regular simple galaxy. It's difficult to find a simpler lensing configuration.

"The lensing theory appears too flexible to me. There are endless parameters to tweak in order to believe it. My imagination becomes a barrier accepting it."

I have a funny but accurate analogy to this. Do Maxwell equation as well appear too flexible to you to believe them? From them you can derive how light behaves when it goes through glass. When you take a regular ideal lens, you will find a simple and elegant solution. However, when you take a piece of horribly twisted glass with density changing everywhere, then you will get a very complicated solution and very complicated image if you shine a laser through it. It is the same with GR. If you take a point source and point lens you will get a simple solution and a simple image. If you take a complicated configuration you will get a complicated solution and a complicated image. It is not a weakness of Maxwell equations that twisted pieces of glass exist and it is not a weakness of GR that complicated gravitational structures exist. In both cases it is not theories that requires endless parameters but the complexity of structures we are trying to describe.

Sun Burst image - why do you expect the same lensing image at different wavelengths? It is inconsistent with what we know about galaxies (lensed or not). At ultraviolet you see young stars and in the optical - mostly older ones. They are very rarely distributed in the same way in any galaxy. I can provide many examples if you wish. If the spatial distributions of light is different at different wavelengths, then the lensed images are also different. There is not problem here.

I am not sure if your calculation about number of stars around Milky Way GC is correct or not. But obviously, the images from UCLA shows that there are more stars than you calculated.

I am only calculating the number of stars that are far behind the BH, not around the BH. The UCLA image show only stars around the BH (because the star number density is much higher than than in the background). Hence my calculations only show that we do not expect any star far behind and is consistent with this image.

According to that link, there are many more stars near GC than other part of the galaxy. The density is 107/pc3, or 1016/kpc3 around GC. If we are looking at 1"x1" sky around GC and 1 kpc behind BH at GC, then we would expect 1016/kpc*(5x10-6x8kpc)2x1kpc=1.6x107.

In the link you quoted they say:

Within a parsec of the galactic center, the estimated number density of stars is about 10 million stars per cubic parsec.

Hence you can't take the number density of stars calculated for the central parsec and apply it for a central kpc. Within the kpc the number density is much lower - similar to what I am assuming. Hence your number is wrong.

I would rather believe what I see in the image. I would think statistically some of them behind BH between 0.01kpc to 1 kpc makes more sense than the prediction of the calculations which we don't know where the numbers actually come from, and what physics are involved in the average star density.

Inverting your statement - I would rather base our inference on data not on what we think. I also don't know why you say that we don't know what my number come from. I referred to all observational papers which quote these data. Is there any of these number questionable for you? Remember, that even if we revise these numbers by a factor of 10 up we would still expect no detectable star far behind the BH, so this doesn't matter.

You already see stars in the image, why not believe in what we see rather than what calculations tell you. Aren't calculations meant for explaining what we see rather than the other way around?

That's why I am asking you to show which star is behind the BH and should be lensed. Before this can be done, the argument is very weak. Maybe there are stars which should be lensed, maybe not. We can't rule out a well-established theory based on something like that.

The solar bending experiment must rule out the atmospheric effect. Noting doing so only means the person who is doing the calculation is not very careful, or at least incomplete, not mentioning other possibilities.

Yes, I guess this is what is happening - this experiment tells us that the density of gas in the solar corona is too low to cause any measurable refraction. Again, before it is demonstrated that refraction can explain the data equally well as GR, the argument against GR is very weak. Moreover, how do you know that people making GR calculations are not careful and did not take into account refraction? Maybe they did concluding that it is not important?

Hi Michal Michalowski ,

Thank you so much for taking time to search for references and posting images. However, I still cannot be convinced. This is probably not important to you, and that is why I really appreciate your help and the enthusiasm to science.

I especially like the web tool you posted for http://pcollette.webege.com/gl.html. I attached an image using the tool. The purpose is to demonstrate the stretching I am looking for any Einstein ring. With the parameters, for a ring to span most of the circle, the lensed object cannot vary a lot in space, and they must be very much stretched. Points are not an option. For the SDP.81, you accepted multiple individual structures like Ad1, Ad2, Dq1 Bq1, Dd1 ...in the paper at Article ALMA Imprint of Intergalactic Dark Structures in the Gravita...

BTW, your mylensing.png image shows the correct stretching effect I am looking for. I can immediately recognize the stars around the circles being very obviously lensed. But I don't see any astronomical images that carry the stretching effect.

As of the light bent by solar atmosphere, I quote NASA website at https://einstein.stanford.edu/content/relativity/q793.html

"Could the displacement of star images near the sun be caused by refraction in the atmosphere of the Sun, not by general relativity?

No. Long wavelength electromagnetic radio waves are, in fact, refracted by the plasma in the solar photosphere, chromosphere and corona, but this effect can be accounted for, leaving a frequency-independent bending of the amount predicted by general relativity.

In 1974-75 a series of radio observations were made of the occultation by the sun of the quasars 3C273 and 3C279 by astronomers Fomalont and Sramek. The measurements were made at 2.7 and 8.1 gigacycles. Because refraction from the solar corona varies with the square of the observing frequency as n^2 - 1, where n is the plasma index of refraction, it is possible from a 2-frequency observation to eliminate most of the effects caused by refraction in the solar atmosphere. General relativity predicts that the 'lensing' of light by a gravitational field does NOT depend on the frequency of the light, unlike lensing of light by optical means."

You see that for long wavelengths, it is obvious that light is bent by solar atmosphere. But why not visible light? NASA didn't explain why.

But I can give the densities here: solar atmosphere density is 10-7 g/cc, while Earth's atmosphere density is 10-3g/cc. But don't forget that the Sun has a radius 100 times larger. There is no reason to think light is not bent by the solar atmosphere. The bending of light at horizon on the Earth is about 30'. You can rough estimate how much bending the solar atmosphere can do at grazing able by just simple maths.

On the other hand gravitational lensing is expected to cause bending of about 0.9".

The dog stretching image is interesting, but it doesn't help the problems. The bowl is stretched to the size of the whole body of the dog. You can do things like that with computer stretching with different proportions, but lensing has to follow the lensing equation. They are different types of stretching. Maxwell equations are great, and I can compute the light scattering patters using the Mie theory which are the Maxwell equation solutions for light scattering off spheres, and predict rings at different wavelengths. Aerosols in the atmosphere can have different shapes, but the scattering of light by aerosols still shows circular symmetry, which agrees fairly well with Mie theory and observation such as the rainbow.

If gravitational lens is true, lensing objects are typically far from us and should have point like behavior, and that is why I am looking for the stretching. You can not put in multiple lenses that satisfy lensing equation or different lensing mass distributions to explain what is observed. That is too imaginary. I do not believe that the lensing objects can have that much effect to explain what we see. Adding arbitrary parameters can explain anything with any theory.

You wrote "Sun Burst image - why do you expect the same lensing image at different wavelengths? It is inconsistent with what we know about galaxies (lensed or not). At ultraviolet you see young stars and in the optical - mostly older ones. They are very rarely distributed in the same way in any galaxy. I can provide many examples if you wish. If the spatial distributions of light is different at different wavelengths, then the lensed images are also different. There is not problem here"

You are probably talking about different things. For the same celestial body, such as a bright star or a galaxy consisting of stars with various ages, it has different wavelengths no matter what. If one of the wavelengths shows stretching, then other wavelengths must show the same amount of stretching. The images I uploaded shows that long wavelengths are arcs, but UV wavelengths are multiple dots, as the authors claim the Sun Burst image is a lensed image with one object lensed to multiple locations. That is impossible for gravitational lensing.

Large galaxy clusters contain both dark matter and normal matter. The immense gravity of all this material warps the space around the cluster, causing the light from objects located behind the cluster to be distorted and magnified. This phenomenon is called gravitational lensing. This sketch shows paths of light from a distant galaxy that is being gravitationally lensed by a foreground cluster.

Dear Hong

Yes gravitational lensing is correct. And it is almost 360 degrees around the black hole.And I have explained it to you in the 1st answer.

..

Thank you...

Hi Neeraj Meena ,

you basically re-expressed what is gravitational lensing, which I know. But I am questioning against it. You can review the previous questions, answers and reasoning, mostly between Michal Michalowski and me for the issues involved in the gravitational lensing. There are many open issues in my opinion, and these issues are shaking the validity of gravitational lensing.

Hi Michal Michalowski ,

I think it is necessary to follow up with your star number calculation claiming no star 0.01 to 1 kpc behind BH at Milky Way GC with some literature. But I have calculated there are much more than thousands of stars, and I have shown images of numerous stars around the BH. You are pretty confident I am wrong.

So I just looked up the literature, and found a term called stellar cusp, which describes the especially high density of stars near the GC. See the attached image of a paper posted at https://arxiv.org/pdf/1702.00219.pdf titled "The Stellar Cusp around the Milky Way’s central black hole". The left most image shows numerous stars around the BH at GC at lateral distance +/- 0.05 kpc.

So your calculation surely is questionable, and my claim that Einstein lensing is also questionable.

And for the horse shoe lensing image, I have attached the web tool illustration and the image of the horse shoe with possible lensing routes.

Hi Michal Michalowski , using the web tool, if the resulting image is a ring, then lensed objects must be behind the BH, and there should be no broken arcs. The horse shoe appear to be a good ring, but it has broken arcs, so it forces us to use lensed object at offsets from the BH. Obviously the G point provided by the

the attached paper screenshot shows no good lensing as expected as indicated by the yellow arrows, with yellow question marks indicating the possible images that are not show. The green arrows tries to use the web tool's lensing features in order to locate the possible BH. But obviously, Dd2 cannot lens into Aq1, Ad1 and Ad2 based on the brightness and shapes.

So the lensing theory is very shaky no matter how I look at it. In my theoretical framework, I don't need bending of space time to predict the results of special and general relativities, so why do I have to give in to lensing. I have options, but other researchers don't, so they are stuck with what they have, and forced to throw in various parameters to try to fit the observations.

Hi Hong Du

"Thank you so much for taking time to search for references and posting images. However, I still cannot be convinced. This is probably not important to you, and that is why I really appreciate your help and the enthusiasm to science."

I would like to convince you but I am discussing this for the fun of it. Moreover, I am learning myself a lot, because in order to express these arguments clearly I need to get my head around it first. Let's get into it!

On SDP.81:

"With the parameters, for a ring to span most of the circle, the lensed object cannot vary a lot in space, and they must be very much stretched. Points are not an option."

Why do you think the source needs to be a point source in order to see a nearly perfect ring? I have shown you before a figure from the paper which applies GR to SDP.81 and they produce a nearly complete ring with an extended source.

"For the SDP.81, you accepted multiple individual structures like Ad1, Ad2, Dq1 Bq1, Dd1 (...), but you are saying the dots I pointed out in the high resolution image is simply noise or artifacts. These claims don't seem convincing to me."

I know that this structures are real, whereas the "point-sources" may not because the former are very bright, whereas the fluxes of the latter are consistent with noise. Let's have a look at real data, they are publicly available here here:

https://almascience.eso.org/alma-data/science-verification

Look for "16. SDP.81", and I downloaded

Band7 -> SDP81_Band6+7_ReferenceImages.tgz

The image ends with tt0.fits. You can use a software like ds9 to open it to examine yourself. I attach the picture showing the data. I can do a proper analysis if you wish, but just by looking on how many negative peaks there are with the value of a few*10^-5 Jy, I can robustly say that the noise is at this level. Now your point sources are positive peaks again with a few*10^-5 Jy, so they are not even higher than the noise - we cannot even assume that they are real! Even the brightest blob in the southern part of the ring has 10^-4 Jy, so it is significant at most at a 2-3 sigma level - very questionable. On the other hand the blobs marked in the paper have fluxes of 4-10*10^-4 Jy, so they are 10-20 times above the noise - definitely real.

You can accept these individual Ad1, AD2 ... Dd1, but do not accept the points and consider them noise. I don't think the points I referred to with red arrows are noises. We can find another high resolution image to see if they are still there.

It's not the point if they are on another image or not (though feel free to check on band 6 or band 4 data). The point is that their fluxes are consistent with the noise in the image, so we cannot base (or reject) any theory using them.

"It is really hard to have individual lensed Ad1, Ad2, Dq1 .... objects to align up in a circle of they are off axis"

You got it wrong here, these objects are images of the same galaxy, not individual objects.

"But that galaxy will be too large considering the lensing distance (billions of light years)."

I have no idea what this claim is based on. Could you provide some calculations showing that the galaxy turns out to be too large (and what is too large)?

"If are objects very close, they must be right behind the lens, meaning they must be arcs."

No. Only objects right on the caustic line will be arcs. Those outside of it will be lensed into multiple objects.

"using the web tool, if the resulting image is a ring, then lensed objects must be behind the BH, and there should be no broken arcs."

You make a mistake in interpreting the data. You are assuming that the lens and the source are a point-like objects. In the case of SDP.81, for the lens this is 100% not true - the lens is a galaxy (not a BH) clearly extended on the optical image. Second, the source was also shown not to be point-like (the reconstruction I sent before), so your model is useless in interpreting these data.

"Obviously the G point provided by the the attached paper screenshot shows no good lensing as expected as indicated by the yellow arrows, with yellow question marks indicating the possible images that are not show."

I don't understand the point you are trying to make here. I have shown you a paper with a lensing model perfectly fitting the data for SDP.81 (residuals consistent with noise). Now you draw several straight lines and you claim that GR cannot fit the object? Well, these authors provided such good model, so your point is invalid. In order to prove that a theory doesn't fit you would need to explore the entire reasonable parameter space and find no good match. You haven't done this, whereas. they have and they have found a good model. I think you are assuming that the lens is a point-source and that's why you misinterpreted this.

On the images and the lines you have drawn - do I understand correctly that you claim that all lensed images should have a counter-image on the other side of the lens? This is only true for point-like (or spherically symmetric) lenses (the ones which are considered in the web tool I sent you). If a lens is not symmetric you may have non-symmetric images. See the attached image with a simple elliptical lens from:

https://cds.cern.ch/record/614102/files/0304438.pdf

The upper panel shows several sources and their location with respect to caustic lines and the lower shows the images. You can see that you can easily get an image without a counter-image on the other side of the lens if the source is located away from the line of symmetry, for example the green image located just outside the inner caustic.

Hi Neeraj Meena

Yes, this is what I mean. We are discussing whether there are some alternative explanations of the images which we interpret as lensing. As far as I see whatever example we discuss it always turn out to be perfectly consistent with GR whereas an alternative explanation is either lacking or severely inconsistent with the data.

On the bending near the Solar limb:

"You see that for long wavelengths, it is obvious that light is bent by solar atmosphere. But why not visible light? NASA didn't explain why."

Plasma physics is not my expertise, but I guess that the point is that radio waves scatter off the electrons of that energy, whereas optical photons have too high energy to scatter. For them only the standard refraction (change in the refractive index) can produce bending. Anyway, what they mean here is that the effect of the atmosphere can be subtracted off and there is still a displacement consistent with lensing which cannot be explained by the atmosphere.

But anyway, the key sentence from the NASA article is this one:

"General relativity predicts that the 'lensing' of light by a gravitational field does NOT depend on the frequency of the light, unlike lensing of light by optical means."

The displacement of stellar and quasar images near the Solar limb was measured at a wide range of wavelength - different optical wavelengths and even radio. Lensing predicts that the displacement should not depend on wavelength and indeed we see that it doesn't. Refraction is wavelength-dependent - light of different wavelengths bends differently (that's why a prism splits the colours). If this was what was happening here, we should see different displacements for different wavelengths. We don't, so this is the prove that refraction is not at play here.

To convince yourself about it see Fig 2 from this paper (also attached here)

https://arxiv.org/pdf/1409.7812.pdf

The parameter on the y-axis is the strength of the displacement expressed in units so that it is equal 1 if it is consistent with the prediction of GR. You see that all displacements are consistent with GR at a level of 1e-4. And here we don't have *any* free parameters to play with, because the mass and the distance of the Sun is measured from other observations and the true positions of these objects is known from observations when the Sun is far away. So do you say that refraction just happens to produce the displacement in perfect agreement with GR and for some weird reason happens not to depend on wavelength even though we know it should? This is just impossible, it is much simpler to conclude that this is a confirmation of GR.

"The dog stretching image is interesting, but it doesn't help the problems. The bowl is stretched to the size of the whole body of the dog. You can do things like that with computer stretching with different proportions, but lensing has to follow the lensing equation. They are different types of stretching."

Are you implying that the dog image was produced by some kind of "computer stretching"? No, it was produced by applying the lensing equation to it. The point is that if a given source (the bowl) is closer to the caustic line, then it is stretched more. If another is further away (eyes), then it is not stretched at all. That's how lensing works.

Adding arbitrary parameters can explain anything with any theory.

But GR is the *only* theory that explains the lensing images. Before another theory is presented, consistent with all the data, we have no reason to reject it.

You are probably talking about different things. For the same celestial body, such as a bright star or a galaxy consisting of stars with various ages, it has different wavelengths no matter what. If one of the wavelengths shows stretching, then other wavelengths must show the same amount of stretching.

This is incorrect. A galaxy usually have different distribution of light at different wavelengths. Hence, two images at different wavelengths are very likely to produce different lensing stretch. See the example of the Andromeda galaxy from

https://www.forbes.com/sites/startswithabang/2019/03/25/what-would-the-milky-way-look-like-if-you-could-see-all-of-its-light/

The visible and ultraviolet images look very differently, so if you apply lensing equation to such an image, you will produce different stretch.

"That is impossible for gravitational lensing."

On the contrary - it is very easy with lensing if the distribution of light at different wavelength is different.

Stellar cusp in the centre of the Galaxy:

Could you explain a bit more what your argument is? You have shown an image with thousands of stars many arcsec away from the BH, so none of.them is expected to be lensed. Only those on your right image could in principle be lensed if they are far away. There are just a few tens of stars there and most importantly, neither you nor anybody else have shown that even one of them is in the background, but they have been investigated for decades. Before you show one such star there is no argument against lensing.

I have one more argument supporting lensing and ruling out your alternative theory (I assume that it is not able to explain this, but feel free to demonstrate otherwise). I was only pointed out very recently that we do have a resolved image of a single star lensed by another star:

https://ui.adsabs.harvard.edu/abs/2019ApJ...871...70D/abstract

They observed a star which increased its brightness 10 times (2.5 mag), see the attached image with the light curve. It was bright enough for interferometric observations to be carried out and they indeed resolve the source into two images 2 milli-arcsec away (these two images were too small to measure their shapes), see the second image. So now we have a perfectly normal star observed for years beforehand doing nothing, then it increases its brightness by a factor of 10 and the flux evolves perfectly as GR predicts (the black line in the lightcurve plot) and it starts to be seen as a double source. After that the flux goes back to normal the star goes back to being single and then again keep doing nothing for years. I can't even image another theory which would allow a normal star to brighten, split into two pieces and then go back to normal. Isn't it this best lensing example you are looking for?

"BTW, your mylensing.png image shows the correct stretching effect I am looking for. I can immediately recognize the stars around the circles being very obviously lensed. But I don't see any astronomical images that carry the stretching effect."

That artificial image I created is of course nice, but there the tool assumes that everything on the image is behind the lens. In real astronomical image you will never see something like that because stars on this image will always be in the foreground, so will not be lensed if the lens is another galaxy.

But you haven't addressed the point I am making with the image. The blue extended source at which the arrow points is lensed into something you claim is impossible - a ring with a point-source located on it. I could do that with just a simple (though extended) source and a point-like lens, so nothing too fancy. The Universe is more complex than that, so can produce a point-source on a ring/arc equally easy.

Hong Du I think I got your point about the stellar cusp. Do you mean that since we see a lot of stars distributed 50 pc away from the BH in projection on the sky, then we should see a lot of them 50 pc behind? But at such distance the Einstein radius is 150 mili-arcsec, only 3 times larger than the resolution of the Keck image. With such data you won't be able to measure such ring - it would only be a low-significance extension, very difficult to find and interpret in such a crowded field. We need to wait for larger telescopes.

Hi Michal Michalowski ,

Thanks a gain for providing information regarding the SDP.81 at https://almascience.eso.org/alma-data/science-verification and I downloaded band 6 and band 7 but band 4 never completed downloading. I have adjusted the contrast and attached the images so as to compare the images at different frequencies. Distinct structures instead of complete arcs or significant stretch of arcs can be seen (only looking at strong signal ignoring background noise).

It is agreed that only individual visual objects not complete or significant segments of arcs are seen. Now you contribute this effect as the effect of a lensing galaxy structure instead of a BH. Now the question is why the galaxy is not visible near the center? Isn't it supposed to be closer than the lensed source and more visible? Another more important questions is: how possible that the one single lensed source being projected to a perfect ring arrangement while only appearing at some specific discrete locations? What kind of lensing galaxy mass distribution can produce such specific ring locations? How many parameters have to be put in to make multiple lensed object? Why not study the structure of the lensing galaxy and write some papers and do some measurements?

If you can image such thing can happen, I cannot. I can not arbitrarily throw in parameters to make myself believe some theory. I can only say this theory is totally incomplete instead of a mature theory because such theory can make anything happen without constraint. What is the point of such theory? You can say all the stars in the sky with the same light spectrum are from the same lensed source, because you can throw in parameters to make it happen.

For the visible light and radio wave bending by solar atmosphere: if radio wave and visible light have to be processed differently, it already shows that there exists effects more than GR can predict because GR only predicted fixed bending regardless of frequencies. Now you see from radio wave to visible light there are differences. GR is incomplete in the light bending and light traveling no matter how you look at it. GR has no idea how light or radio wave should bend or not traveling in space with particle distribution, it is out of GR's scope.

It is known that radio wave and visible light travels at different speeds through space with different particular densities, they take different times to arrive from far distance to Earth, you know GR have no idea about this. If light travels through media such as the Sun's atmosphere at speeds different from that in vacuum, it will bend more when the media has uneven density. Same thing will happen for radio waves but to a lower degree, but it is more sensitive to charge density . Even NASA says that radio wave is bent by solar atmosphere. No matter what, GR cannot account for all the light or radio wave bending in the solar atmosphere.

If we cannot measure the solar atmosphere density accurately, we cannot determine how much is from atmospheric bending and how much is from GR. The conclusion is incomplete in any sense for both GR and atmospheric bending. I have already shown that with present value of solar atmosphere density, the bending of visible light is already comparable to GR bending. That is a big uncertainty that must be ruled out before GR bending can be evaluated.

Your perception that different wavelengths are lensed differently is not what GR can predict. Lensing equation doesn't have that, GR doesn't have it. You can say that the attenuation of different wavelengths is different, but you cannot say that they are lensed differently. But this is on top of GR, and even this cannot explain why one object can be lensed to multiple locations unless you tweak the mass distribution on top of it, which can make anything possible.

For the stellar cusp thing, my calculation shows any star between 0.01 kpc to 1 kpc behind the black hole at GC gives rings or arcs between 0.07" to 0.7", the UCLA image resolution is much better than 0.1".

The missing Einstein ring is a mystery. Unless you think the stars you see are already lensed because lensed stars can remain stars, and anything can happen with proper mass distribution even it is in the galaxy. But no ring at this time.

The micro lensing event you referred to at Article First Resolution of Microlensed Images

The curve matching is perfect, but what is the x-axis and y-axis?

"Hi Hong Du

I'm back here! First on SDP81:

"It is agreed that only individual visual objects not complete or significant segments of arcs are seen."

Yes, though check if this granularity is not again because of noise. Is the difference in flux between bright regions and fainter adjacent regions in the arc larger than 3*the noise level? To do this properly you can plot the flux of the pixels in the middle of the arc as a function of the position along the arc. This will be a wiggly line, so then you can see if the wiggles are consistent with noise or are real.

"Now you contribute this effect as the effect of a lensing galaxy structure instead of a BH."

I would say this is due to the structure in the lensed background galaxy, not the lens, but some structure in the lens can also play a role. Note than lensing by the supermassive BH in the centre of the lens is irrelevant - its mass is at most a few percent of the mass of the galaxy, so the lensing signal from the BH is negligible.

"Now the question is why the galaxy is not visible near the center? Isn't it supposed to be closer than the lensed source and more visible?"

The image with the arc is taken at the millimetre wavelengths. The lens is an elliptical galaxy with (almost?) no dust, so it doesn't emit in the millimetre. However, this galaxy is clearly visible in the optical - it is the blue blob which is visible in the first combined image of this object you sent some time ago. It is also shown in a few papers I send around. The redshift of this galaxy has been measured and is much lower than the redshift of the arc.

"Another more important questions is: how possible that the one single lensed source being projected to a perfect ring arrangement while only appearing at some specific discrete locations? What kind of lensing galaxy mass distribution can produce such specific ring locations?"

By "single lensed source" you don't mean a point source? The source is a galaxy, so it is extended. How is it possible? Take a look at the papers I sent around - you just need a blob (background galaxy) lensed by another blob (the lens), aligned almost perfectly, but with a small shift. Very easy configuration.

"How many parameters have to be put in to make multiple lensed object?"

Can you have a look at the paper to get the number of free parameters yourself? I will have a guess:

1, 2) the position of the lens - not really free, because the lens is visible, so the position is fixed

3) the mass of the lens

4) the axial ratio of the the lens (assumed to be for example a single isothermal sphere or a a Gaussian)

5) the positional angle of the lens

6, 7) the position of the background galaxy

8) the axial ratio of the background galaxy (assumed to be a Gaussian)

9 the positional angle of the background galaxy

So we have 9 free parameters or even 7 if we consider the lens position to be fixed, or even 5 because parameters 4) and 5) are also not completely free, because they can be measured from the optical image. The distances are measured through spectroscopy independently of the lensed modelling. The number of datapoints is probably around hundred. We get a perfect fit, so this demonstrates the correctness (and beauty imho) of GR.

Why not study the structure of the lensing galaxy and write some papers and do some measurements?

This is done in the papers on this object and many others.

"If you can image such thing can happen, I cannot."

Our imagination has nothing to do with this. The authors took GR, model the image with 5-9 (or whatever) free parameters and hundreds of datapoints and obtained a good fit. The conclusion is that GR is consistent with this dataset.

"I can not arbitrarily throw in parameters to make myself believe some theory. I can only say this theory is totally incomplete instead of a mature theory because such theory can make anything happen without constraint. "

I am not sure what you mean here. There are 7 free parameters. But more importantly - before you show an alternative, more elegant, and simpler explanation of this dataset, there is no argument against lensing.

"You can say all the stars in the sky with the same light spectrum are from the same lensed source, because you can throw in parameters to make it happen."

Could you add more details on this, I am not sure what you mean. Stars we see in the sky are not lensed, because they are in the Milky Way, so apart from microlensing events, they are not lensed.

On the displacement of stellar positions close to the Solar Limb:

"For the visible light and radio wave bending by solar atmosphere: if radio wave and visible light have to be processed differently, it already shows that there exists effects more than GR can predict because GR only predicted fixed bending regardless of frequencies."

I am not sure what your argument is here. The situation is the following

* We have the following expectations:

1) GR predicts than bending is the same at all wavelengths

2) Refraction predicts that bending depends on wavelengths.

* We have the following data

1) The bending is exactly the same at all optical/near-infrared wavelengths and even in radio

2) The amount of bending agrees perfectly with the GR prediction down to 10^-4. The prediction is done with no free parameters, because the Solar mass and the distance to Earth is measured very precisely from other observations.

3) Nobody (correct me if I am wrong here) has presented a model of the plasma density distribution around the Sun, which would produce the star displacement exactly as we see in the data.

Hence, the conclusion must be that GR is consistent with the data, whereas the refraction is not at play here.

"Now you see from radio wave to visible light there are differences. "

That's surprising, please demonstrate that this is the case. From the plot I sent before (gamma vs time) you can see that the displacement in the visible light is exactly the same as in the radio.

"GR is incomplete in the light bending and light traveling no matter how you look at it. GR has no idea how light or radio wave should bend or not traveling in space with particle distribution, it is out of GR's scope."

Again, I am not sure I am getting your point here, could you elaborate?

"Even NASA says that radio wave is bent by solar atmosphere. "

Yes, NASA, you and I agree here - radio waves (and all other electromagnetic light) can be bent by refraction in the solar atmosphere. But the data tells us that this effect is negligible. If it was the main effect, then the bending would be different at different wavelengths. It is not.

"I have already shown that with present value of solar atmosphere density, the bending of visible light is already comparable to GR bending. "

How have you shown this? You just state that refraction can explain the bending, but you have never provided any model which predicts the correct amount of bending.

I assume that this is about the Sun Burst galaxy (the one with different lensed images at the ultraviolet and optical).

"Your perception that different wavelengths are lensed differently is not what GR can predict. Lensing equation doesn't have that, GR doesn't have it."

I completely agree with this! If (un-lensed) images of a source are *identical* at two wavelengths, then lensing will produce identical images. However, galaxies are not identical at different wavelengths - I demonstrated that with the image of the Andromeda galaxy. Basically all galaxies have different distributions of light when you look at them at different wavelengths. Hence, the lensed image at different wavelengths will also be different. You can convince yourself that this is the case by taking the image of the Andromeda galaxy (or any other galaxy) at ultraviolet and optical and process them with the online lensing tool I sent before using the same lensing parameters (the mass and the position of the lens). You will get different lensed images, not because GR works differently at different wavelengths, but because the input images are different.

"You can say that the attenuation of different wavelengths is different, but you cannot say that they are lensed differently."

Agreed, GR is independent of wavelength. What is different is the input images:

1) the distribution of light in the ultraviolet is different than in the optical (because UV is dominated by massive stars which are located at different positions than less-massive stars, which dominate the light in the optical)

2) Indeed, as you mention - the attention is different, so more dust regions will appear much fainter in the UV than in the optical.