# Time dilation in FLRW metric?

1. Jan 23, 2014

### Vincentius

Hi, could anyone help me out?

The FLRW metric in spherical coordinates is:

$\;\;$ ds2 = dt2 - a(t)2(dr2 + r22) $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$ (1)

I am considering a similar metric of the format:

$\;\;$ ds2 = $\frac{1}{a(t')^{2}}$dt'2 - a(t')2(dr2 + r22) $\;\;\;\;\;\;\;$ (2)

Are (1) and (2) equivalent? Is it just a matter of substituting dt'/a for dt in (1)?

Would the new time coordinate simply be t'=$\int$a(t)dt ?

Is (2) a known parametrization? Has it a name? What kind of time would t' represent?

The background of my question is that the FLRW metric (1) does not reflect time dilation, as e.g. in the Schwarzschild metric, while I would expect time dilation to go along with expansion of space. In the Schwarzschild metric, t is coordinate time. In the FLRW metric t is proper time already. The parallel with the Schwarzschild metric (when using isotropic coordinates) suggests a metric of format (2), or something alike.

Thanks for any help!

2. Jan 23, 2014

### pervect

Staff Emeritus
I don't recall seeing anything exactly like what you describe. But it reminds me of something that's the opposite of what you did.

If you make dt' = dt / a(t) (rather than dt' = a(t) dt) - you have conformal time. This is usually written as $d\eta$ rather than dt'. When you have flat space-slices (a spatially flat metric) then the space-time metric becomes

$$ds^2 = a(\eta)^2 \left( -d\eta^2 + dr^2 + r^2 d\Omega^2 \right)$$

which is a conformal factor multiplying a flat minkowskii metric.

If your spatial slices arent flat, you wind up with something like:
http://www.tapir.caltech.edu/~chirata/ph217/lec02.pdf

$$ds^2 = a(\eta)^2 \left[ -d\eta^2 + d\chi^2 + f(\chi) \left( d\theta^2 + sin^2 \theta \, d\phi^2 \right) \right]$$

As far as time dilation goes, the fact that changing coordinates via a simple algebraic substitution changes it should be a strong hint that it (time dilation) doesn't have any physical significance by itself because its value depends on the arbitrary choice of coordinates.

3. Jan 23, 2014

### Vincentius

The change to conformal time is not what I was aiming at. FLRW seems to avoid time dilation. I suppose a probable reason for that is that in cosmology there is no such a clear time reference as coordinate time in the Schwarzschild spacetime is, while, on the other hand, there is a clear spatial reference in FLRW to relate proper distance to, i.e. the comoving frame (times the scale factor). This, however, does not mean there is no time dilation. I agree that it does not matter to the comoving observer, he will still get around 75 years old, but cosmological time dilation would affect remote observation, like e.g. redshift, so is physically relevant in my opinion.

4. Jan 24, 2014

### pervect

Staff Emeritus
Do you think the Milne metric:
http://en.wikipedia.org/wiki/Milne_model

$ds^2 = dt^2-t^2(d \chi ^2+\sinh^2{\chi} d\Omega^2)$

"seems to avoid time dilation"?

Would you say that the flat Minkowskii metric it's equivalent to "seems to avoid time dilation"?

The equivalance can be derived by considering the Minkowskii metric

$ds^2 = d \tau^2 - dr^2$

substituting the variables

$\tau = t \, \cosh \chi$ and $r = t \, \sinh \chi$

(ref: Physical Foundations of Cosmology, through google books)

applying the chain rule d(a*b) = b da + a db

$d\tau = \cosh \chi \, dt + t \sinh \chi \, d\chi$
$dr = \sinh \chi \, dt + t \cosh \chi \, d\chi$

then a bit of algebra shows $ds^2 = d\tau^2 - dr^2 = dt^2 - t^2 d \chi^2$

5. Jan 24, 2014

### Vincentius

Quite right Pervect. There is velocity time dilation in the FLRW metric (though not at rest in the comoving frame, hence cosmologically less relevant). But I actually meant gravitational time dilation (sorry, my mistake).

If we consider expansion of the universe not as actual motion but as expansion of space itself, then one would expect (gravitational) dilation of time to go along with that. I suppose, this relates to the question of the evolution of the cosmic potential, which I can not find a conclusive answer to. Not even a proper treatment (does anyone?).

Expansion of the particle horizon (more mass coming in) suggests increase of the potential, hence, expansion of proper distance (relative to comoving distance) AND gravitational dilation of time. This is why I would expect gravitational time dilation to appear in the spacetime metric of the universe. So, I am looking for such a metric, similar to (2) in my original post. Something alike does appear in the optical analogy of GR, the "Polarizable Vacuum" theory (e.g. contributions of Dicke and Putthof), but not in mainstream GR, as far as I know. Any hints?

6. Jan 24, 2014

### pervect

Staff Emeritus
My view, in a nutshell, is that time dilation is simply the ratio of the coordinate time difference to the proper time. So I expect time dilation to vary with a change of coordinates, the concept of time dilation itself is explicitly dependent on one's choice of coordinates.

Therefore I don't really see the problem you're trying to solve. I thought maybe some of the examples I gave would help you resolve your confusion, but apparently it didn't.

You also asked some questions about how the metric transformed, I hope that that the example of transforming it using algebra was of help. It's also possible to transform the metric using tensor transformation rules, but I find the algebraic approach simpler to describe.

7. Jan 24, 2014

### Staff: Mentor

That's because there isn't one; the concept of "gravitational potential" only makes sense in a stationary spacetime, and FRW spacetime is not stationary.

8. Jan 24, 2014

### Vincentius

Pervect, you were talking about velocity time dilation, not gravitational dilation, right? Thanks for pointing out anyway. The problem I am trying to solve is finding out if the cosmological constant has indeed a material origin, as Einstein tried to hold for quite some time, but which conjecture he dropped after acknowledging de empty solution of de Sitter. Potential rules the metric. So does the cosmic potential, I suppose.

Peter: I try to grasp this. I am comfortable with the idea that gravitational potential is ambiguous, depending on the observer, as pretty much everything in GR. And if it is an unresolved question, I can live with that. But, I have a hard time understanding that gravitational potential makes no sense in the cosmos. The sun clearly produces a potential over here. The closest next star also, I suppose. And so on. It is likely not the usual Newtonian sum, but it should add up to something and have physical meaning. The FLRW metric may not be suitable, but how can the physical concept of gravitational potential not exist in the cosmos. And if it does, I would expect the metric to reflect this. I believe the FLRW metric actually does reflect this (via a(t)) for the spatial part. But this is not the case for time.

9. Jan 24, 2014

### Staff: Mentor

Actually, in a stationary spacetime, it isn't. More precisely, a "gravitational potential" can be defined that is not observer-dependent; it is a Lorentz scalar that has an invariant value at every event. When you see "gravitational potential" talked about in GR, that's usually what is meant.

However, this definition only works in a stationary spacetime. See further comments below.

It's not. See below.

The sun is an isolated gravitating mass, so the spacetime around it can be approximated as a stationary spacetime, in which a potential can be defined. But this "potential" only makes sense when defined relative to the stationary patch of spacetime centered on the sun. It does not make sense otherwise.

In the stationary patch of spacetime centered on that star, yes. But not otherwise.

To the extent that we can define a stationary patch of spacetime including both stars, and provided gravity is weak enough everywhere in that stationary patch, then yes, you can add the two potentials generated by each star to get a total potential at a given point in the stationary patch of spacetime. (If gravity isn't weak enough, you can't add potentials linearly, but to the extent you can define a stationary patch of spacetime containing both stars, you can still define a potential.) But this potential will still only have meaning within the stationary patch of spacetime, not otherwise.

It's not a question of "metric" in the sense of which coordinate chart we use. Whether or not a spacetime is stationary is an invariant property of the spacetime; it's the same regardless of our choice of coordinates.

Because the cosmos as a whole is not stationary. It just happens that there are a lot of isolated gravitating systems in the cosmos such that a stationary patch of spacetime *can* be defined around them, and within any such stationary patch, a gravitational potential can be defined. But there's no way to combine all these potentials to get a potential for the universe as a whole, because the universe as a whole is not stationary.

The a(t) in the FRW metric has nothing to do with gravitational potential. The fact that it's a function of t is a direct reflection of the fact that the cosmos as a whole is not stationary.

10. Jan 24, 2014

### pervect

Staff Emeritus
I believe the same formula works for either sort of time dilation. For instance, gravitational time dilation on the Earth is also the ratio of coordinate clocks to proper time (for instance, time as kept by local atomic clocks).

There are well known examples (such as Einstein's elevator) where in one coordinate system (the accelerating rocket frame) there is gravity but no motion, so all time dilation is due to gravity. In another coordinate system (an inertial frame), there is no gravity (but there is motion), so there is no gravitational time dilation but there is velocity time dilation. So the distinction between the two is also observer and coordinate dependent, it's a part of GR's general diffeomorphism invariance.

11. Jan 24, 2014

### Vincentius

I am afraid I must disagree Peter. Again, the potentials of two stars may not exactly add up. So, the Newtonian definition is not quite right. Agreed sofar. However, this doesn't imply these two stars do not excert a gravitating force together anymore (on whatever particle in their environment). So it takes energy to get away from the two stars? Hence, the concept of potential remains valid. I can't see why this does not apply to n-stars and to the cosmos as a whole. Again, I do see it is a complicated question.

The question is what makes the cosmos non stationary? The fact that more mass is steadily piling up within the moving particle horizon seems quite relevant to me.

12. Jan 24, 2014

### Vincentius

Ok, Pervect, I am aware of the parallel. Expansion of the universe is considered as metric change, not motion (main stream cosmology, but I can imagine an equivalent interpretation in motion). As source of metric change we have gravity and energy. Since dark energy is hypothetical so far, it makes sense to consider gravity as source of expansion of space itself. This would appear when the cosmic potential increases, similar to what would happen if the mass in the Schwarzschild metric increases, distances increase and clocks slow down. This could happen on the cosmic scale via the increase of mass appearing inside the particle horizon. At least, my hypothesis :-)

13. Jan 24, 2014

### Staff: Mentor

Once again, that's because the two stars can be modeled as being isolated in a stationary spacetime. However, I do see that I emphasized the "stationary" part before but not the "isolated" part; the latter is important too.

For example, the stars being an isolated system is implicit in your statement that it takes energy to "get away" from the stars. Get away to where? To infinity, where the "potential" is zero. But in the cosmos as a whole, there is no "infinity"; there is nowhere to "escape" to, because the mass-energy in the cosmos is everywhere. So there's no way to measure the "potential" at a location in the cosmos, because there's no infinity relative to which you can measure it.

See above.

Consider a stationary isolated gravitating body like the Sun. (What I'm going to say applies to a stationary isolated system with multiple bodies too, but it's simpler to see with a single body.) It has a definite center of mass, which is the "center"--the spatial origin--of the isolated system. Because there is a unique center of mass, it is possible to have a symmetric configuration of all the matter around that center of mass, and that symmetric configuration can be in a stationary equilibrium. (If the mass is not rotating, the equilibrium will be static--nothing will move at all. But if the mass is rotating, even though its parts are in motion, the motion is stationary--its parameters won't change with time.)

The cosmos as a whole, however, does not have a unique center of mass; on a large scale, its mass-energy is spread evenly throughout the entire cosmos. Such a configuration cannot be in a stationary equilibrium. Mathematically, this is a consequence of the Einstein Field Equation, but intuitively, it should be evident from the fact that there is no center of mass, so there is no center around which to form a symmetric configuration that can be in equilibrium.

I'm not sure what you mean by "piling up". The average density of mass-energy in the universe is decreasing as it expands. The total amount of mass-energy inside the spatial volume within our cosmological horizon is increasing, but the spatial volume itself is increasing even faster. The density of mass-energy is the key parameter affecting the dynamics.

14. Jan 24, 2014

### Staff: Mentor

"Gravity" isn't a source of anything; it's an effect. The universe is expanding because it started out expanding in the Big Bang; in other words, it's due to initial conditions, not to "gravity" causing expansion of space. The effect of the mass-energy in the universe (ordinary matter and radiation, not dark energy) is to slow down the expansion, not speed it up. (That's why dark energy is hypothesized, btw: there's no other form of stress-energy we know of that could cause the expansion to speed up. "Gravity" won't do it, because, as I said above, "gravity" is an effect, not a source.)

As I noted in my previous post, the key parameter is density of mass-energy, not total mass; and the density of mass-energy is decreasing. But if it were increasing, the effect would be a increasing tendency for the expansion to slow down.

15. Jan 25, 2014

### Vincentius

Thanks Peter, appreciate your input. The notion of potential can still exist in a homogeneous (observable) universe. Taking mass out to an empty universe is imaginary of course, but it can be imagined (suppose e.g. the universe outside of the observble universe is empty). But this irrelevant. The potential can be calculated (in principle) for the present status of the mass, as we always do. Also consider that potential always adds up, i.e. never cancels out like in the net force of gravity. The gradient of the cosmic potential (i.e. force) may be small on the large scale, the potential is still huge. I like to view potential as a pressure compressing the unit distance (or curve spacetime if you like). The pressure can be flat, so irrelevant energy-wise, but can still be huge and increasing and can so act on the metric.

Ok, different wording: matter (not gavity) is a source of expansion: More mass coming in => stronger potential => expansion => density goes down. And so on. Hence, expansion is caused by the extra matter appearing within the moving horizon.

I even question if gravitational attraction affects expansion at all. I don't know. Given the cosmological principle, there is pure symmetry in the universe at large scale. Why would mass go into any direction, apart from local peculiar motion? (This is in fact much like your argument about there is no place to go to, but now in the opposite direction.) The "motion" or "metric change" views are frequently mixed up in literature, which does not really help clarification IMO. Gravitational attraction falls in the first category. The mechanism I mention in the second category. It is conceivable, I believe, that both mechanisms act at the same time, but again, I have some doubts about gravitational attraction as a factor in the expansion.

16. Jan 25, 2014

### Staff: Mentor

You can construct a model like this, yes, but its dynamics won't be the same as the dynamics of our actual universe. So you can't use it to define a potential for our actual universe, only for the imaginary universe of the model.

Show your work, please. Make sure that the math you show addresses the points I raised earlier, about our actual universe being non-stationary and non-isolated. (No, your imaginary model where the universe outside our observable universe is empty won't suffice; see my comments above.)

I won't bother commenting on the rest of what you say about "potential" since there's no point unless you can address the points I've already made.

Again, show your work. This bears no relation that I can see to how the dynamics of the universe are derived using the Einstein Field Equation. If you're not using the EFE, then you're not talking about GR, you're talking about some speculative theory of your own, which is off limits here.

Well, cosmologists using GR do know: it does. As I've said before, the gravitational attraction of ordinary matter and energy acts to slow the expansion; it's straightforward to show this using the EFE (it's a homework exercise in most GR textbooks).

On average, it doesn't; the average motion of matter in the universe does not pick out any preferred direction.

No, it isn't, because the FRW model of the universe does not require a preferred direction, whereas defining a potential does require there to be an infinity somewhere for objects to escape to.

They are often mixed up in pop science presentations, yes. That's unfortunate, but if you really want to understand the physics, you shouldn't be reading pop science presentations anywhere. The textbook presentations I'm aware of (mainly the one in MTW) do a good job of treating this issue.

No; it can fall in either category. That's a key point of the FRW model: the "gravitational attraction" of ordinary matter can act to slow the expansion even though no individual piece of matter is "changing direction" due to any other, on average, so there is no preferred direction anywhere. I suggest reviewing a good textbook presentation of the FRW model; you appear to have some basic misconceptions about it.

17. Jan 26, 2014

### Vincentius

Peter: I am a big fan of GR. And I am not trying to discuss any privat little theory. We do have some serious issues, though, which I do not attribute to GR itself, rather to (yes) misconceptions in the (standard) interpretation of GR. I suppose this is the right place to discuss interpretation of GR, right? (though my original question was a bit more practical). What I consider a major misconception is that the presence of the cosmic masses is irrelevant to whatever local system one considers. I suppose the reason for this opinion is that the gravitational field of the distant stars is effectively zero everywhere. Above that, the cosmic masses do not appear explicitly in any GR equation. But, as far as the cosmic masses could be relevant, then it is likely via the cosmic potential. Schroedinger concluded the cosmic background potential must be equal to -$\frac{1}{2}$c2. Sciama came to a similar result. Hence, trying to interpret GR, this suggests the factor c2 in the various metrics is the implicit representation of the cosmic potential. Statements like it is "a pure coincidence" that our gyro compasses follow the stars, serve standard interpretation, since it does not deal with the cosmic masses. The same is apparently the case in the standard interpretion of the FLRW metric, which is (like GR) fine in itself. The standard interpretation boils down to a Newtonian interpretation (via the well known isolated inertially expanding small sphere of mass), i.e. disregards the presence of the cosmic mass outside of the sphere. Then for sure one needs something extra to explain acceleration of expansion. Recognizing that the cosmic masses do matter (the effect of the cosmic potential on the metric) is IMO a scientifically viable, if not mandatory, position in considering the question of dark energy. Gravitational potential is of primary importance in all of GR. Abandoning cosmic potential, not only effectively but even conceptually, seems a mistake to me.

18. Jan 26, 2014

### Staff: Mentor

Yes; but you do realize, I hope, that any claim along the lines of "the standard interpretation has serious issues" is an extraordinary claim and will require extraordinary evidence and arguments to back it up.

Who said the cosmic masses were irrelevant? See further comments below.

Remember that "gravitational field" in the usual sense is not how GR models gravity. Gravity is modeled as spacetime curvature. The correct way to state what you're trying to say here is that, from the standpoint of an isolated local system like the solar system, the curvature produced by the rest of the mass-energy in the universe is effectively zero. This is not just an assumption made for convenience; there are good physical reasons for it. See further comments below.

Yes, they do; they appear in the Friedmann equations, which are the Einstein Field Equations specialized to the standard FRW cosmological models.

None of these models have anything to do with GR; they are speculations on possible alternative theories to GR, none of which have panned out. In the standard GR model, there is no "cosmic potential", and no need for one to take proper account of the rest of the masses in the universe. See below.

No, this is wrong. The FRW metric explicitly includes the global effect of the mass-energy in the universe; the presence of nonzero stress-energy is what makes the parameter $a(t)$ in the metric a function of time (if there were zero stress-energy in the universe, $a$ would just be a constant that could be set equal to 1, to obtain the Minkowski metric).

More importantly, GR does *not* say that it is "pure coincidence" (is that a quote? where from?) that gyroscopes keep pointing at the same distant star, nor that other local inertial effects are "coincidence". What GR does say, in the FRW model of the universe as a whole, is that, from the standpoint of a local isolated system like the solar system, the rest of the mass-energy in the universe, on average, is distributed in a spherically symmetric fashion outside the boundary of the system (where the choice of "boundary" is somewhat arbitrary, but that's not an issue here). And there is a theorem in GR which says that if you have a spherically symmetric mass-energy distribution outside of some spherical boundary, that distribution produces zero spacetime curvature inside the boundary. (There is a similar theorem in Newtonian gravity that says there is zero gravitational field in an empty region inside a spherically symmetric mass distribution.) *That* is why we can treat an isolated system like the solar system as asymptotically flat, despite the presence of all the other mass-energy in the universe: because, from the standpoint of the solar system, the *effect* of all the other mass-energy in the universe is to produce, physically, an asymptotically flat boundary condition around the solar system.

I don't understand what you mean by this; it doesn't resemble any "standard interpretation" of GR that I'm aware of.

No, it doesn't. See above.

Yes; and the "something extra" *cannot* be the ordinary mass-energy we observe in the universe. Not only is that already taken into account (see above), but ordinary mass-energy *cannot* produce an accelerating expansion; that's a consequence of the Einstein Field Equation. That's why we need to postulate something like dark energy to explain the accelerating expansion.

19. Jan 27, 2014

### Vincentius

I still can't see where the distant cosmic masses appear explicitly in the equations. I can see that they appear implicitly, via the speed of light (I suggest you read Erwin Schrödinger, Ann. der Phys., 77, 325-336 (1925) to convince yourself it is not speculation). The cosmic masses also seem to appear implicitly via the presence of absolute space in GR. This absolute element has bothered Einstein for most of his life. I think it should bother us too. So if we want to get any further with this question, it makes sense to recognize the cosmic masses as the carrier of absolute space. In that line of thought, empty Minkowski space is not actually empty, but is the mathematical representation of the cosmic background. Evolution of the cosmic background then is likely to affect absolute Minkowski space (via change in the speed of light in my view). I am sure this is irrelevant to local mechanics, we will continue to measure a fixed speed of light locally. But understanding evolution of the cosmos without recognizing its connection to the notion of absolute space in GR is doomed to fail IMO.

20. Jan 27, 2014

### Staff: Mentor

Which equations are you looking at? The Friedmann equations are here:

http://en.wikipedia.org/wiki/Friedmann_equations

The $\rho$ and $p$ that appear in the equations are the average mass-energy density and pressure of the matter in the universe. (The $\Lambda$ in the equations is the cosmological constant, but we can leave that out for this discussion.)

I don't know if this paper is available online (haven't been able to find it by Googling). Is it based on GR, or quantum mechanics? GR is a classical theory, not a quantum theory.

There is no "absolute space" in GR.

Do you have any specific references?

Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook