Einstein's ring can be seen around massive galaxies: If the light-bending was due to atmospherical effects, this should be impossible.

JES

Yes that is true. However, the question is quantitative and it concerns Einstein rings reported on Sun masses and how much the observed effect is due gravitational bending of light and not contributed to other effects which can have a similar outcome.

Well, I think the gravitational effect can't be denied, but, you are right in the sense that denser plasma should give a lower velocity to the photons and therefore a bend should appear. If this can give reis to Einstein's ring around a star is not evident. I don't know the dependence of the plasma density versus radius.

JES

A salient and unique fact regarding light bending per GR is that it is achromatic. This is proven as radio photons emitted from nearby stars and very distant quasars are bent by the *same* amount as their optical photons. A medium will produce slightly different frequency-dependent photon lightspeeds producing chromatic effects. Attempts to explain photon bending by a medium surrounding a mass like a star or an active galaxy must assume some very special properties to do this in the very different astrophysical environments of these objects. Occam's razor very strongly favors GR as the best explanation of photon trajectories in space curved by masses, not a special medium surrounding them.

Good point! So if the bending was more due Snell refraction then Einstein ring would appear not white but slightly colored?

However, as far as i know the frequency is not affected by the the refractive index of the medium therefore this criterion I believe cannot be used.

In frames of GR stars act n the trajectories of light (bending of -space). Moreover, space-time acts on the frequency of the light. These conclusions are results of the solution of null geodesic lines equations describing 4-trajectories of photons. The spatial equations describe the form of heirs trajectories, the equation fourth equation gives the expression for the frequency f photons in the given space-time. So, we can calculate Doppler effect for every space-time.

So Einstein Rings are Doppler shifted? Thus are chromatically distorted?

Larissa Borissova : " So, we can calculate Doppler effect for every space-time. "

We are unable to calculate the Doppler effect because we are subject to this effect ourselves. We are not able to detect changes in our motion vector or determine these changes over time. The same applies to the observed objects. We are unable to detect the changes in the Doppler effect to which we are subject. The propagation of changes in the Doppler effect is limited by the speed of light. The Doppler effect changes dynamically over time and does not act synchronously on matter in the universe. This statement seems highly illogical in the context of observing distant galaxies.

I am now confused. Seems to me that there is no any rigorous analysis of why Einstein rings around stars (not galaxies) are 100% GR phenomena and should be contributed exclusively due to gravity. If this theory is not falsifiable it is not correct. I'm not convinced yet that a single even hypermassive star is capable of producing a gravitational Einstein Ring optical effect around it. This should only refer to the mass of galaxies, clusters and black holes. Is there no formula in GR that predicts the amount of curl introduced on a light beam skimming the surface of an astronomical body per mass?

To add to my prior comment, here is a reference on observed gravitational bending of radio waves by the Sun, affirming consistency with GR:

https://inspirehep.net/literature/818782

To clarify astronomers' understanding regarding Einstein rings: since standard GR predicts and observations find that gravitational lensing is achromatic, there are no color distortions in a lensed image (but there are shape distortions, i.e. a galaxy can be stretched into an arc or a ring). Bending is independent of wavelength, so blue photons are bent by the same amount as red photons and as radio photons. A chromatic effect on would lead to smeared optical images, like a spectrum, blue on one end and red on the opposite, that do not appear in any observed lensed images. Several Einstein rings are observed at radio wavelengths since the background galaxy is a strong radio emitter.

Adding a few other observed facts asked or mentioned by others above regarding gravitational lensing: redshifts of lensing galaxies are always smaller than lensed galaxies (a few of which are Einstein rings). On average, lensed background galaxies have about twice the redshift, and thus are about twice as far away, as the foreground lensing galaxy (one example: https://academic.oup.com/mnras/article/343/2/L29/1043278). Background galaxies can be multiply imaged by a foreground galaxy, and these multiple images have the *same* redshift (two examples: https://hubblesite.org/contents/media/images/2006/23/1931-Image.html?news=true and https://arxiv.org/abs/1811.02505). Note Figure 1 in the latter reference: two background galaxies each have distinct interior structure that than be seen in the lensed images (labeled 1.1, 1.2, 1.3 and 2.2, 2.2, 2.3). Multiple images result from relative distances and non-alignment of the background galaxy with the centroid of the foreground one, and the mass distribution in the latter.

There is also at least one supernova (called supernova Refsdal) in one background galaxy that was imaged at multiple and overlapping times by a foreground galaxy cluster because of differing photon path lengths and resulting light travel times. Each of these time delayed images has the same light curve, spectrum and redshift within observational uncertainties (https://arxiv.org/abs/1512.05734 ), strongly indicating it is the *same* object.

This may be a lot to digest, but the subject is a rich observational one, with many instructive images, and sophisticated data analyses.

Dear Emmanouil,

General Relativity works quite well to explain the bending of EM waves. The Shapiro Delay is an example where EM waves bending is present but also a considerable delay effect is present (200 microseconds between Earth and Venus at the opposite sides of the Sun) and beautifully predicted by GR.

The Shapiro Delay experiment is also important because it excludes phenomena of possible refraction of the Sun's corona, since the signals were made travel quite far from the surface of the SUN, unlike the sun deflection from starlight which may be subject of such an ambiguity.

Although you have to be careful because the curved trajectory of EM wave is always in euclidean space and such increased path length is very small, not at all sufficient to account for the delay of 200micro-sec.

On one hand the curvature of space-time, as good tool of calculation, is able to account for such a delay.

On the other hand, in Euclidean space where the phenomenon occurs, you can explain the delay of EM waves across a gravitational field only with a Variable speed of light. As a matter of fact it is what happens for real.

Preprint SHAPIRO TIME-DELAY, Curved 4D space-time or Variable speed of light

The light spreads in the [ace-time along null geodesic lines for which ds^2 = 0. It means that the light spreads momentary in the space-time and with the velocity 300.000 km/sec in the 3-space. The bending of light trajectories consists from 2 parts: bending of 3-trajectories; 2) changing of the frequency of the light. The results can be obtained as solutions of null geodesic equations which ape applied to 4-dimensional equation of null geodesic. Un other words, it is necessary to solve the equation of null geodesic lines relative to 4-dimensinal wave vector K^\alpha = (\omega/c)dx^\alpha/d\sigma, where \alpha = 0,1,2,3, d\sigma is 3-dimensional spatial interval. These equations are obtained by Zelmanov in terms of physically observed values. These equations can be applied for calculations of trajectories of light and its frequency in every space-time, including cosmological. So we can calculate Doppler-effect. Analogical problem can be solved for mass particles moving with v 0. The equations of motion in these case are written relative to 4-dimensional energy-impulse vector P^\alpha = m_0(dx^\alpha/ds) where m_0 is the mass at rest. We obtain the value of relativistic mass m = m_0/(1 - V^2/c^2)^1/2, where m is relativistic mass and 3-trajectories. .

General Relativity is a good tool of calculation nothing more, same as it is the space-time in SR with its due limitations.

GR was born in the mind of Einstein as a possible "theory of everything" but eventually it was clear that although GR gave predictions of new phenomena (time dilation and light delay and bending) it was not complete. GR stemmed from one tiny problem (the rosetta trajectory of Mercury) when in 1915 men were not even aware to be part of a galaxy. We cannot be surprised if it is not able to give account to other important phenomena related to gravitational dynamics (DE and DM effects) which are active at galactic scale. GW waves are provided by perturbations of the linearised "metric" of EFE but the reason of such variation is something external to it.

Thank You All for your excellent answers.

Einstein rings around a star are *not* resolvable with any current telescope. However, gravitational microlensing permits the brightness increase of a background star by a foreground star associated with rings to be detected. The probability of this alignment with Earth is about 10^-6 to 10^-5 in our galaxy, so the only way to detect it is by observing a very large number of stars, which led to the MACHO, OGLE and similar projects. This was initiated by Bodan Paczyński and nicely summarized by him: https://arxiv.org/abs/astro-ph/9604011 .

I can say the same: stars bend 3-trajectories in the 3-space and change frequency of photons. Photons fly along 4-dimensional trajectories where ds^2 = 0 in fact momentary). We perceive theirs 3-dimensional projections in the 3-space as 3dimensional lines alng them photons move with the light velocity. It is to take into account that the light spreads in the space-time.

Dear Larissa Borissova ,

>

in a Four dimensional warped space-time yes, their trajectory is found under that condition, where the speed of light is constant.

In Euclidean space which is where their trajectory physically occurs the speed of light is not constant but is a function of the gravitational potential.

The speed of light is constant c= 300000 km/cec inside 3-space where our 3-dimensional bodies are. But the light spreads in the space-time along 4-dimensional trajectories ds^2 = 0. Until we'"ll think that c is maximal velocity of light, we"ll cannot use GR, because this theory is created for 4-dimensinal space (space-time). Value c is maximal only in 3-space, but our main task is the study of time. light ray is bending in 3-space, ut the frequency of photon an be calculated exactly only in space-time. we can take into account not only motion in 3-space but and changes of time caused by different forces, for example by gravitational force and rotation (see Hafele-Keating experiments). They show that the temp of cloks depends on gravitational field and rotation of planet. The cause of temporal bending is just the change of course of time caused by gravity and rotation. And the time is the fourth coordinate. The problem of bending of light rays is in space-time but not only is 3-space.