All mass produces time dilation, and a reduced velocity of light isotropically, and for distant mass the effect is proportional to the inverse radius. In addition there is a radial reduction in coordinate velocity, but this is direction and does not add in the same way as the time dilation component. If mass is approximately distributed evenly, then the amount of mass in a differential radius increases proportional to R^2, so distant matter dominates the effect unless one is very close to a particular gravitational source. So how much would light speed increase if the known distant mass, including dark matter, were removed? I could do crude calculation, but there might be many factors I wouldn't think of. I'm looking for an authoritative paper. For example, if light speed would quadruple without this mass, relative to some even more distant observer (since we would not notice the difference, being local observers), then one could say the fraction attributable to the distant mass was something like (1 - 1/4) = 3/4, just as an approximate gauge.
The answer is that the question doesn't make sense, because, in curved spacetime, relative velocity depends on the path.
The starting statements are, essentially, wrong, because there isn't any such notion as an isotropic reduction of the speed of light; nor is there any radial dependence, as stated.
Once more, it's not useful to confuse space and spacetime. Time dilation makes sense in spacetime, not space and expresses the fact that the geodesic-in spacetime-maximizes the proper time.
@Stam, try reading about light bending or Shapiro delay. If you don't understand something that doesn't mean it doesn't make sense. If you don't have the answer, don't criticize the question.
BTW, in the late 1990s, measurements found space to be flat out to nearly 14 billion light years.
The apparent change is only the coordinate value assigned by the remote observer, locally it is unaffected. For matter distant from us, the speed is c relative to matter in that distant region. This is covered in Ned Wright's tutorial on page 3:
http://www.astro.ucla.edu/~wright/cosmo_03.htm
You can see it easily in this diagram where the light cones tilt away from the observer at the centre, the speed of light relative to us is then asymmetric.
http://www.astro.ucla.edu/~wright/cosmo200.gif
@George, your answer is the most thoughtful so far. My question might be too hypothetical to attract a good answer. Let me pose to you a less hypothetical version. Instead of asking about distant matter, I take the reference frame of the distant matter and consider several circumstances:
1. Light near a black hole is going coordinate speed near zero. Take away the black hole and it moves c again in the distant reference.
2. Light in the dense center of a galaxy but not especially close to a black hole will be moving almost c in the distant reference. Take away the galaxy, and it moves closer to c in the distant reference.
3. Reverse #1. Take your reference near the event horizon along with the light. Take away the distant matter. No change. It is a local measurement and by definition you still measure c. But an observer elsewhere is likely to measure no change as well, because the velocity of that light is dominated by the local metric of the BH. Do you follow and agree?
4. Continue to take your reference near the event horizon, so that your reference frame is dominated by that metric and changes relatively little. Now you are measuring the coordinate velocity of light some distance away, but still in your galaxy (very very patiently, using radar or some such method). Take away the distant matter. It was dominating the speed of light at that location, so you will presumably observe the light speed up. But how much? That represents the contribution the distant matter was making, vs just the local galaxy.
I have not constructed a perfect reference, as it is itself somewhat affected. But I have avoided the entirely hypothetical situation of the observer being outside the universe. Does that help clarify this overly complex question?
I can also ask the question a different way. The gravitational constant tells us how much mass affects other mass, and incidentally the speed of light which is easy to measure. Using the field equation, or the FRW equation, one should be able to tell if the amount of mass in the universe is correct, too much, or missing.
This question is an inversion of the decades long quest to find out the shape of the universe by measuring the amount of mass, which is difficult with dark matter and who knows what lurking about. Since the late 90s the shape is known to be flat. This produces various problems in the equations mentioned.
But a back of the envelop method, presumably valid in flat space as a "check" on the answer, would be a linear sum of gravitational potential from distant mass and the associated relative light speed.
All and GD, RS: I suggest the book by the well-known Melvin Schwartz, "Principles of Electrodynamics", of 1972. Electromagnetism is deeply affected by SR, E and B are not independent, and must have the same units (already have, in SI CGS). The velocity of EM waves depend only on local constants of the medium. There is no cosmology in that.
To repeat partly, read review below:
Unlike most textbooks on electromagnetic theory, which treat electricity, magnetism, Coulomb's law and Faraday's law as almost independent subjects within the framework of the theory, this well-written text takes a relativistic point of view in which electric and magnetic fields are really different aspects of the same physical quantity. Suitable for advanced undergraduates and graduate students, this volume offers a superb exposition of the essential unity of electromagnetism in its natural, relativistic framework while demonstrating the powerful constraint of relativistic invariance. ... The rest in online, not to copy-and-paste what anyone can find.
@Ed, thanks for your input, but we are not on the same page. I am looking at GR effects on coordinate velocity. That is not a local measure. The local constants are indeed constant locally as is the local velocity of light. But time dilation is a gravitational phenomena, and affects the velocity in non-local coordinates, causing light bending, Shapiro delay, etc. If this area is really unfamiliar to you, I've compiled some helpful information here: https://www.quora.com/Which-is-the-best-book-to-understand-the-general-relativity/answer/Robert-Shuler-4
RS: Your reply is noted but is incorrect, including your "mind-reading" of what I may be familiar with. It is an ad hominem attack by definition (be careful!), and quoting quora is not academic, in addition (as you recognize) the poor worded question in RG. If you want to focus on time dilation as a gravitational phenomena, it is not exclusively so. Perhaps a question more focused, will help get quality answers.
Ed, I think the problem is that you're doing a bit of 'mind reading' instead of following the question as asked, it's not about EM at all.
Robert, you are asking the question based on a certain mental starting point which is not quite in concordance with the way GR works which makes it difficult to answer.
RS: George, your answer is the most thoughtful so far.
It tries to get you to reappraise your viewpoint, did you study Ned Wright's diagram? Do you see how the cones tilt and follow the implication of that?
RS: My question might be too hypothetical to attract a good answer.
No, it's not hypothetical at all, it's the basis of cosmological redshift.
RS: Let me pose to you a less hypothetical version. Instead of asking about distant matter, I take the reference frame of the distant matter and consider several circumstances:
1. Light near a black hole is going coordinate speed near zero. Take away the black hole and it moves c again in the distant reference.
Right, that is because it is going at c in both cases but when the BH is present, the axes are rotated towards the BH so projecting that speed onto the distant axes gives a lower value. Locally nothing is different. Note here we are talking about the simple Schwarzschild Metric with a single mass in an otherwise empty universe.
2. Light in the dense center of a galaxy but not especially close to a black hole will be moving almost c in the distant reference. Take away the galaxy, and it moves closer to c in the distant reference.
Same reason though the matter is more widely distributed in the galaxy rather than concentrated in a single BH. The "distant" observer must be far outside the galaxy in the asymptotically flat space.
3. Reverse #1. Take your reference near the event horizon along with the light. Take away the distant matter. No change. It is a local measurement and by definition you still measure c. But an observer elsewhere is likely to measure no change as well, because the velocity of that light is dominated by the local metric of the BH. Do you follow and agree?
No, I don't follow what you mean by "Reverse #1". Either the BH is present or not, but there is no matter near the distant observer to be removed.
@George, thanks for your extended discussion. I am intending to stay completely within GR, but if I try to compute compute it myself I'd like to use the simpler math of a Schwarzschild metric rather than a cosmological metric. If there is a paper, whatever it might use I would accept.
So instead of referring to distant mass in a cosmological context, let me try a different approach.
Two masses, M1 nearby at distance r1 from a point of interest p, and M2>>M1 very far away at r2>>r1. The coordinate observer's radius is robs>>r2>>r1 so that the observer can use Schwarzschild coordinates as his own for either system.
Since r2>>r1 the point of interest along with all of system 1 can be considered to be embedded in system 2, and the time dilation factor dτ/dt is approximately
sqrt(1-2GM2/r2c^2)sqrt(1-2GM1/r1c^2).
To avoid too much abstraction, suppose that the first term is .99 and the second term is .9. The coordinate speed of light at p is then .891c.
If you remove the distant mass M2, it goes to .9c, approximately a 1% increase.
If you remove the nearer mass M1, it goes to .99c, a 10% increase.
In this case M1 at r1 account for 10% of the reduction of light velocity from the coordinate background, and M2 at r2 account for 1%.
Now if translating this to cosmology, there cannot be an observer at anything greater than r2. Humor me for a moment and let's place a hypothetical observer at that hypothetical distance >>r2. We have no problem from curvature since the observable universe is flat, but we are ignoring the expansion and thus the fact such an observer would not be able to actually observe anything. That's where a cosmological formulation would be required and I don't know either how to do that or what value to stick in for mass of the universe (there are various ambiguous values available). I don't "need" to know such things. I'd be happy to find a paper that had already calculated it. I do not care to make and defend my own calculation. But here goes the simple version . . . in the abstract, not with real numbers.
In the case where only M2 is present, the contribution of M2 was said to be 1%. If I took the mass of the universe between 13 and 14 billion light years and labeled that M2, and set r2 to the average distance 13.5 billion light years, what contribution would I get?
Further, if I continued with mass shells at 12 to13, 11 to 12, etc. what total contribution would it sum up to?
And finally, has anyone studied anything like this and written a paper on it? Or if not, what are the relevant most similar references to start with?
Hi Robert,
RS: I'd like to use the simpler math of a Schwarzschild metric rather than a cosmological metric.
You can't do that, they are fundamentally different, but we can look at what you say.
RS: Two masses, M1 nearby at distance r1 from a point of interest p, and M2>>M1 very far away at r2>>r1. The coordinate observer's radius is robs>>r2>>r1 ...
That paints the picture: p ... M1 ... M2 ... observer
RS: ... so that the observer can use Schwarzschild coordinates as his own for either system.
No, Schwarzschild coordinates apply only when there is a single body. There is no analytic solution for two bodies, however your approach as an approximation isn't too bad.
RS: the time dilation factor dτ/dt is approximately
dτ/dt = sqrt(1-2GM2/r2c^2)sqrt(1-2GM1/r1c^2)
OK, I suppose you are defining dt as the time for an observer at infinity and dτ as the time for the observer mentioned above.
RS: To avoid too much abstraction, suppose that the first term is .99 and the second term is .9. The coordinate speed of light at p is then .891c. If you remove the distant mass M2, it goes to .9c, approximately a 1% increase. If you remove the nearer mass M1, it goes to .99c, a 10% increase.
OK.
RS: Now if translating this to cosmology, there cannot be an observer at anything greater than r2.
Why not? Lets say r1 = 10 light years and r2= 10,000 light years. That is still well within our galaxy, what's the problem if the observer is 1 million light years away? That's less than half way to Andromeda and satisfies robs >> r2 >> r1.
RS: Humor me for a moment and let's place a hypothetical observer at that hypothetical distance >>r2.
As above I see no problem with that.
RS: We have no problem from curvature since the observable universe is flat, but we are ignoring the expansion and thus the fact such an observer would not be able to actually observe anything.
None of that makes any sense apart from that we have no expansion because you are using the Schwarzschild Metric which doesn't have expansion. It is asymptotic to a flat, infinite spacetime.
RS: That's where a cosmological formulation would be required and I don't know either how to do that or what value to stick in for mass of the universe ...
In the Schwarzschild metric, the only mass is the spherical body, the rest of the universe is empty. The cosmological solution is the Robertson-Walker metric and in that, there is a uniform mass density but no unique mass.
RS: I'd be happy to find a paper that had already calculated it.
You won't find anything, as it stands the question makes no sense.
RS: And finally, has anyone studied anything like this and written a paper on it?
The equation you've used above gives the time dilation as a function gravitational potential which is fine for the Schwarzschild solution but doesn't work for cosmology, the Copernican Principle on which it is built would require the potential to be the same everywhere so there is no time dilation for the reason you are describing, there is only the effect of expansion (which you ignored).
This is related and may give you some other starting points:
http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html
George, thank you for careful comments on my hypothetical situation. I left out two things.
The first is that I was using time dilation as a gauge for the coordinate speed of light. It is of course only the tangential speed. The radial speed is slower. It might be necessary to figure that for my ultimate question, bet we can get within a factor of 2 this way.
The second is that I failed to explain how I was promoting it to a cosmological model. Place r2 ~= 13.5 billion light years. Now you see why the observer cannot actually be beyond that, and still observe.
Dear Robert,
RS: The first is that I was using time dilation as a gauge for the coordinate speed of light.
You can't really do that, the rate at which a clock ticks when at rest but near to a large mass is nothing to do with the rate at which a distant object recedes from it.
RS: The second is that I failed to explain how I was promoting it to a cosmological model. Place r2 ~= 13.5 billion light years. Now you see why the observer cannot actually be beyond that, and still observe.
The most distant galaxy observed to date is GN-z11 at a redshift of 11.09. It is now at a distance of 32.2 billion light years (BLY), it was 1.34 BLY from us when the light was emitted and was receding by 4.3 light years per year at that time.
No light emitted from more than 5.85 BLY away will ever reach us, corresponding to a redshift of z=1.6.
The point is that you cannot ignore expansion. The "teardrop" diagram in Bed Wright's tutorial should make it cleared than my words, please study that carefully.
Dear George,
what does this mean?? The Universe is older than 14 billion years??
No, it means distances expanded during the time that the light was travelling, 13.38 billion years.
Hello George,
I ran across some of the strange cosmology numbers for distance you mention just last night. Like Stefano, I'm a bit surprised, but not that the objects are now much further, that's obvious. I swear I have seen many explanations by serious physicists in the past that had different numbers.
In any case, I'd like to find a paper that has made this calculation. That was what I asked. We only got into finding a back of the envelop way to make it because no one could find a paper. That is an omission and an opportunity for someone.
Just BTW, you mentioned something about lack of correlation between clocks and recession velocity. I never used recession velocity for anything in my back of the envelop attempt. Only gravitational potential. Perhaps a different number for radius is in order. But you have to admit, if the light is 14 billion years old, and changes in gravitational fields travel at the speed of light, then 14 billion light years is a reasonable guess for a non-cosmologist as to the distance to use in a calculation of gravitational potential for those objects felt at our present location.
Robert
Hi Robert,
RS: In any case, I'd like to find a paper that has made this calculation. That was what I asked.
I didn't get that from your question but the answer is simple, you need the Friedmann Equations and the equation of state for each component of the contents. You'll find that in all decent introductory textbooks on cosmology. For that reason, you probably won't find any papers on the topic beyond Friedmann's original.
When dark energy is included, there isn't an analytical solution so you have to integrate numerically. There is a standard program published called "Cosmocalc" and you can find the source code if you search for that. You can find the equations solved in a variety of standard packages like Maniacs and Maroon as well.
- Badges
- Science topic