Is there a known parametrization for time dilation in the FLRW metric?

[Vincentius]

Hi, could anyone help me out?


The FLRW metric in spherical coordinates is:


\;\; ds² = dt² - a(t)²(dr² + r²dΩ²) \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; (1)


I am considering a similar metric of the format:

\;\; ds² = \frac{1}{a(t')^{2}}dt'² - a(t')²(dr² + r²dΩ²) \;\;\;\;\;\;\; (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'=\inta(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!

 

[pervect]

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

<br /> ds^2 = a(\eta)^2 \left( -d\eta^2 + dr^2 + r^2 d\Omega^2 \right)<br />

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

<br /> 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]<br />

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.

 

[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.

 

[pervect]

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 $ds^2 = d\tau^2 - dr^2 = dt^2 - t^2 d \chi^2$

 

[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?

 

[pervect]

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.

 

[PeterDonis]

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

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.

 

[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.

 

[PeterDonis]

I am comfortable with the idea that gravitational potential is ambiguous, depending on the observer

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.

And if it is an unresolved question

It's not. See below.

The sun clearly produces a potential over here.

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.

The closest next star also, I suppose.

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

It is likely not the usual Newtonian sum, but it should add up to something and have physical meaning.

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.

The FLRW metric may not be suitable

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.

but how can the physical concept of gravitational potential not exist in the cosmos.

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.

I believe the FLRW metric actually does reflect this (via a(t)) for the spatial part.

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.

 

[pervect]

Pervect, you were talking about velocity time dilation, not gravitational dilation, right?

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.

 

[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 exert 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.

 

[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 :-)

 

[PeterDonis]

However, this doesn't imply these two stars do not exert 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.

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.

I can't see why this does not apply to n-stars and to the cosmos as a whole.

See above.

The question is what makes the cosmos non stationary?

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.

The fact that more mass is steadily piling up within the moving particle horizon seems quite relevant to me.

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.

 

[PeterDonis]

it makes sense to consider gravity as source of expansion of space itself.

"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.)

This could happen on the cosmic scale via the increase of mass appearing inside the particle horizon.

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.

 

[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.

 

[PeterDonis]

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).

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.

The potential can be calculated (in principle) for the present status of the mass, as we always do.

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.

expansion is caused by the extra matter appearing within the moving horizon.

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.

I even question if gravitational attraction affects expansion at all. I don't know.

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).

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?

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

(This is in fact much like your argument about there is no place to go to, but now in the opposite 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.

The "motion" or "metric change" views are frequently mixed up in literature, which does not really help clarification IMO.

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.

Gravitational attraction falls in the first category.

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.

 

[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.

 

[PeterDonis]

I suppose this is the right place to discuss interpretation of GR, right?

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.

What I consider a major misconception is that the presence of the cosmic masses is irrelevant to whatever local system one considers.

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

the gravitational field of the distant stars is effectively zero everywhere.

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.

Above that, the cosmic masses do not appear explicitly in any GR equation.

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

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.

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.

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

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.

The standard interpretation boils down to a Newtonian interpretation (via the well known isolated inertially expanding small sphere of mass)

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

disregards the presence of the cosmic mass outside of the sphere.

No, it doesn't. See above.

Then for sure one needs something extra to explain acceleration of expansion.

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.

 

[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.

 

[PeterDonis]

I still can't see where the distant cosmic masses appear explicitly in the equations.

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 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).

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.

The cosmic masses also seem to appear implicitly via the presence of absolute space in GR.

There is no "absolute space" in GR.

This absolute element has bothered Einstein for most of his life.

Do you have any specific references?

 

[Vincentius]

I am afraid Peter we disagree on everything :-)

The cosmic masses do not show up in the "local" GR metrics: Minkowski, Schwarzschild, etc. The background of these metrics (i.e. Minkowski space) is supposed to be empty. Again, the cosmic masses appear to be irrelevant, since whenever we apply the Schwarzschild metric (e.g. in the anomalous perihelion precession) it is always in a real universe filled with matter. The huge potenial these masses exert on the local bodies (Mercury, the sun) is flat, but should be as relevant as the potential of the sun and Mercury (this is what Schroedinger's paper is about). Therefore, the flat "empty" metric = the flat cosmic background, IMO. Then it is also easy to understand why the gyroscope direction is fixed relative to the stars. What other explanation is there for that fact?

Mass density of the cosmos does appear in the Friedmann equations, allright. But it considers only the local effect of masses attracting each other (i.e. the Newtonian analogy of a local sphere) and assuming that that is everywhere the same. But it remains a local mechanism. The appearance of new masses inside the moving particle horizon is nowhere in the Friedmann model. Again, it apparently doesn't matter what happens outside the local sphere.

To my knowledge, Newtonian physics (i.e. absolute space) arises in the weak field limit of the Schwarzschild metric, i.e. in Minkowsky spacetime. Then, how can there not be absolute space in GR? Again, this conceptual problem is overcome if one realizes that the "empty" and absolute Minkowski background is nothing but the cosmic background. Then motions are all relative to something, i.e. the cosmic masses and then absolute space has gone.

Such an understanding would take away many issues in a century of confusing GR interpretation. Like e.g. the twin paradox. I hear you say: it is not a paradox! Actually it isn't, IF one involves the cosmic masses, which determine who is accelerating and who is not. Without that frame it is a paradox: both twins can claim the other accelerates. Without the cosmic background, it is the celebrated inertial frame whose existence is an impossible postulate in empty space, like absolute space, unobservable.

I remember a quote by old Einstein, approximately: "it took Newton a lot of effort to get absolute space in. But it takes much more effort to get it out again". But I could not find it back. Anyway, this is about Einstein's failed attempts to get Mach's principle in and so absolute space out. This is also why Einstein for quite some time insisted that the cosmological constant must have a matterial origin. I agree with that view.

 

[PeterDonis]

The cosmic masses do not show up in the "local" GR metrics: Minkowski, Schwarzschild, etc.

To the extent this is true (see below), that's because those are local; they are only supposed to describe a local region of the universe, not the universe as a whole.

The background of these metrics (i.e. Minkowski space) is supposed to be empty.

No, it's supposed to be asymptotically flat. But I described in an earlier post how the rest of the matter in the universe can physically produce this boundary condition, assuming (as is true to a good approximation on average) that it is spherically symmetric about the local region. You have not addressed that point at all.

Again, the cosmic masses appear to be irrelevant, since whenever we apply the Schwarzschild metric (e.g. in the anomalous perihelion precession) it is always in a real universe filled with matter.

Matter which is spherically symmetrically distributed about the local region in which we are applying the Schwarzschild metric; which means, as I've already said, that the effect of all that matter is asymptotic flatness in the local region.

The huge potential these masses exert on the local bodies

There is no such potential in GR. If you think there is, please show your work. Otherwise you are not talking about GR, you are talking about your own speculative theory, which you said you didn't want to do. If you want to talk about GR, you can't just keep claiming there is a potential in situations where GR says the concept of "potential" is not well-defined.

(this is what Schroedinger's paper is about)

Once again, is that paper about GR? Or is it about quantum mechanics? AFAIK Schrodinger only did work in QM, not GR. I can't find the paper online so I can't read it to check.

the flat "empty" metric = the flat cosmic background, IMO.

No, the asymptotically flat local metric = the effect of the spherically symmetric rest of the matter in the universe.

Then it is also easy to understand why the gyroscope direction is fixed relative to the stars. What other explanation is there for that fact?

The fact that a spherically symmetric matter distribution surrounding an isolated empty region produces a flat metric within that region.

Mass density of the cosmos does appear in the Friedmann equations, allright. But it considers only the local effect of masses attracting each other (i.e. the Newtonian analogy of a local sphere) and assuming that that is everywhere the same.

If by "local" you simply mean that every tensor equation is local, yes, this is true. But that's true of *every* solution in GR, not just the FRW solution. GR is a tensor theory; tensor equations describe local physics. Global properties arise from evaluating local physics at each event in a region of spacetime.

The appearance of new masses inside the moving particle horizon is nowhere in the Friedmann model.

It most certainly is. You said it yourself, in what I quoted just above: the mass-energy in the Friedmann equations is everywhere the same. "Everywhere" includes both inside and outside the particle horizon; new masses "appear" inside the horizon because the horizon's spatial location changes with time, not because the masses themselves "move". All of this is already contained in the FRW model.

Again, it apparently doesn't matter what happens outside the local sphere.

You appear to have some significant misunderstandings of how the FRW model works. See above.

To my knowledge, Newtonian physics (i.e. absolute space) arises in the weak field limit of the Schwarzschild metric

Yes. More precisely, the weak field, slow motion limit (all velocities much less than c in a coordinate chart in which the central mass is at rest).

i.e. in Minkowsky spacetime.

No. The weak-field, slow motion limit of the Schwarzschild solution is Newtonian gravity. Newtonian gravity is not Minkowski spacetime. Newtonian gravity is an *approximate* theory (key word: see further comments below); it is not GR.

Then, how can there not be absolute space in GR?

Because Newtonian gravity is not GR. Newtonian gravity is, as above, an approximation: the weak field, slow motion approximation to the Schwarzschild solution. GR explains why Newtonian gravity, as an approximate theory, is a good approximation in the weak field, slow motion limit; but GR does *not* say that Newtonian gravity, with its absolute space, is "correct". If all fields are weak and all motion is slow compared to c, Newtonian gravity works simply because all the relativistic effects that show that there is *not* really absolute space are too small to matter. That's not the same as saying those effects magically disappear because we're using Newtonian gravity to get approximate answers. The correct theory is still GR, a relativistic theory with no absolute space in it.

I won't bother commenting on the rest of your post, since it should be evident from what I've said to this point that you are either not talking about GR, or misunderstanding what GR says.

 

[Mordred]

I am afraid Peter we disagree on everything :-)

The cosmic masses do not show up in the "local" GR metrics: Minkowski, Schwarzschild, etc. The background of these metrics (i.e. Minkowski space) is supposed to be empty. Again, the cosmic masses appear to be irrelevant, since whenever we apply the Schwarzschild metric (e.g. in the anomalous perihelion precession) it is always in a real universe filled with matter. The huge potenial these masses exert on the local bodies (Mercury, the sun) is flat, but should be as relevant as the potential of the sun and Mercury (this is what Schroedinger's paper is about). Therefore, the flat "empty" metric = the flat cosmic background, IMO. Then it is also easy to understand why the gyroscope direction is fixed relative to the stars. What other explanation is there for that fact?

Mass density of the cosmos does appear in the Friedmann equations, allright. But it considers only the local effect of masses attracting each other (i.e. the Newtonian analogy of a local sphere) and assuming that that is everywhere the same. But it remains a local mechanism. The appearance of new masses inside the moving particle horizon is nowhere in the Friedmann model. Again, it apparently doesn't matter what happens outside the local sphere.

To my knowledge, Newtonian physics (i.e. absolute space) arises in the weak field limit of the Schwarzschild metric, i.e. in Minkowsky spacetime. Then, how can there not be absolute space in GR? Again, this conceptual problem is overcome if one realizes that the "empty" and absolute Minkowski background is nothing but the cosmic background. Then motions are all relative to something, i.e. the cosmic masses and then absolute space has gone.

Such an understanding would take away many issues in a century of confusing GR interpretation. Like e.g. the twin paradox. I hear you say: it is not a paradox! Actually it isn't, IF one involves the cosmic masses, which determine who is accelerating and who is not. Without that frame it is a paradox: both twins can claim the other accelerates. Without the cosmic background, it is the celebrated inertial frame whose existence is an impossible postulate in empty space, like absolute space, unobservable.

I remember a quote by old Einstein, approximately: "it took Newton a lot of effort to get absolute space in. But it takes much more effort to get it out again". But I could not find it back. Anyway, this is about Einstein's failed attempts to get Mach's principle in and so absolute space out. This is also why Einstein for quite some time insisted that the cosmological constant must have a matterial origin. I agree with that view.

PeterDonis is providing accurate answers, take any time slice in the universes history. Maintain the homogeneous and isotropic nature of the universe at sufficient scale where this is true (above 100 Mpc).

The FLRW metric requires that homogeneous and isotropic nature to work. You cannot use the FLRW metric to descibe local anistrophies.

At any time slice in a homogeneous and isotropic universe there is no gravitational potential difference to cause a time dilation other than local effects such as stars BH's etc. However these are only small, local regions compared to the universe as a whole.

at any time slice in our universes history the universe as a whole is considered homogeneous and isotrophic. So how can a universal time dilation occur?

 

[PeterDonis]

At any time slice in a homogeneous and isotropic universe there is no gravitational potential difference

A better way to state this would be to say that the concept of "gravitational potential" is not well-defined for the universe as a whole. It is only well-defined in a stationary spacetime, and the universe is not stationary.

at any time slice in our universes history the universe as a whole is considered homogeneous and isotrophic.

This isn't quite correct as you state it, because there is only one set of time slices in which the universe appears homogeneous and isotropic; these are the "comoving" time slices that are used to generate the standard FRW coordinate chart. But an observer that is moving relative to the "comoving" observer (who is at rest in the FRW chart) will *not* see the universe as homogeneous and isotropic, and his "time slices" will reflect that.

 

[Mordred]

ah yea I gotcha its been a while, took a break from my self studies.

" the standard physical interpretation of the t coordinate in the FLRW model is to identify it with the proper time of a standard clock at rest in the comoving system"

Is the quoted statement still valid or accurate? its derived from this related paper

On the physical basis of cosmic time

http://arxiv.org/pdf/0805.1947v1.pdf

its an interesting article that describes numerous problems in defining time with physical processes or measuring rods etc. Never did hear of any counter arguments to this paper, not to say they don't exist.

 

[PeterDonis]

"the standard physical interpretation of the t coordinate in the FLRW model is to identify it with the proper time of a standard clock at rest in the comoving system"

Is the quoted statement still valid or accurate?

Yes.

 

[Vincentius]

It is proper time indeed. And IMO it slows down, while space expands (these go together, just as in the Schwarzschild metric). Picture what happens if one increases the central mass in the Schwarzschild spacetime. The increase of total mass within the expanding particle horizon could generate the same effect.

At any time slice in a homogeneous and isotropic universe there is no gravitational potential difference to cause a time dilation other than local effects such as stars BH's etc. However these are only small, local regions compared to the universe as a whole.

at any time slice in our universes history the universe as a whole is considered homogeneous and isotrophic. So how can a universal time dilation occur?

The point is that gravitational interaction through space is not within a time slice, but across time. The gravitational effect of distant matter on us today represents the past. So there is a gradient over space when there is a gradient over time, which there likely is by the moving particle horizon.

 

[PeterDonis]

And IMO it slows down, while space expands (these go together, just as in the Schwarzschild metric).

"Slows down" by what criterion? Certainly not the FRW metric; in that metric, coordinate time is the same as proper time for comoving observers. Once again, if you want to talk about GR, and not your personal speculations, you need to talk about what GR actually says.

Picture what happens if one increases the central mass in the Schwarzschild spacetime. The increase of total mass within the expanding particle horizon could generate the same effect.

Whether it "could" or not in some hypothetical model, it doesn't in the actual model GR uses, the FRW model.

The point is that gravitational interaction through space is not within a time slice, but across time. The gravitational effect of distant matter on us today represents the past.

This is true; a more precise way of saying it is to say that the spacetime curvature observed at any event in spacetime can be explained entirely by the presence of mass-energy in the past light cone of that event. This is true for FRW spacetime, since it's true of any spacetime in GR; that is, it's true of any spacetime that is a solution of the Einstein Field Equation.

However, this...

So there is a gradient over space when there is a gradient over time

...does not follow from the above; FRW spacetime is a counterexample. In the standard "comoving" coordinate chart, there is a gradient of spacetime curvature with time (because the scale factor changes with time), but not with space (because the scale factor is the same everywhere in space).

 

[Mordred]

Ok I may be missing something here, I am by no means an expert in either cosmology or SR/GR. So feel free to correct any errors I may make, its how I learn.

let's look at the flight path of a photon emitted from the CMB in a Milne universe.

at the time of emission the average density of the universe is higher than it is today, however their is no localized gravity well, so the photon flight path is still determined by the Universe geometry (which is close to flat). see here for geometry effects on lightpaths http://cosmology101.wikidot.com/universe-geometry
move ahead in time a billion years. the average density of the universe is lower but once again their is no localized gravity well, so the flight path remains unchanged. repeat every billion years yields the same conditions other than the average densities, but at any point in time there is still no localized gravity well to alter the straight flight path of the photon.

So where would you expect to see a time dilation or am I missing something?

 

[PeterDonis]

lets look at the flight path of a photon emitted from the CMB in a Milne universe.

The Milne universe is empty (no matter-energy present), so it's not a good example. You may be confusing the Milne universe with the critical density FRW universe, which has flat spatial slices (so the spatial geometry is easy to conceptualize), but nonzero energy density (since the energy density has to be just at the critical value such that the expansion rate of the universe goes to zero as the time goes to infinity).

at the time of emission the average density of the universe is higher than it is today

Not in the Milne universe, it isn't; it's zero for all time in the Milne universe. The statement is true for a critical density FRW universe.

however their is no localized gravity well, so the photon flight path is still determined by the Universe geometry (which is close to flat).

The Milne universe is not "close to flat"; it's exactly flat. (Put another way, the Milne universe is just a region of Minkowski spacetime, with an unusual coordinate chart on it.) That's why photon flight paths in the Milne universe are straight lines; spacetime is flat because the energy density is zero.

So where would you expect to see a time dilation or am I missing something?

Yes, you are, even aside from what I pointed out above. In the Milne universe, "comoving" observers are observers moving outward from the "Big Bang" (which is just an event at the origin of a Minkowski coordinate chart in which the Milne universe is the "upper wedge", the interior of the future light cone of the origin) at all possible velocities (i.e, all possible timelike worldlines through the origin). Since these observers are obviously in relative motion with respect to a global inertial frame, you would expect to see "time dilation" between them.

However, the reason all this works is that, as above, the spacetime is still flat, so "time dilation" obviously works in it, since it's just Minkowski spacetime. For an FRW spacetime with nonzero energy density (including, but not limited to, the critical density model I referred to above), spacetime is no longer flat, nor is it stationary, so there is no way to define "time dilation" or "gravitational potential" in the usual fashion.

 

[Mordred]

ah gotcha, thanks for the clarification on the Milne universe. I was under the understanding that it was a matter removed model only. Not sure where I picked that mistake up from, it certainly wasn't in any of my textbooks lol

 

[Vincentius]

Peter: if something is different from the FRW metric that doesn't mean it is in conflict with GR. Your statement is in effect: there is no gravitational time dilation, because it is not in the FRW metric. That doesn't mean it does not exist. This is precisely the point a raised in my original post. I think it's not there, because we simply don't know (yet) how to calculate the cosmic potential and how it evolves. But I believe their are some clues, as I indicated before, which could help us moving forward on this, all within GR context. FRW model is not free from interrogation as long as we have no acceptable answer to what dark energy is supposed to be.

 

[PeterDonis]

if something is different from the FRW metric that doesn't mean it is in conflict with GR. Your statement is in effect: there is no gravitational time dilation, because it is not in the FRW metric. That doesn't mean it does not exist.

Ok, then show me another metric that correctly describes the universe as a whole but also allows a meaningful "gravitational time dilation" to be defined. Just waving your hands and saying "cosmic potential" doesn't mean any such metric consistent with GR does exist.

I think it's not there, because we simply don't know (yet) how to calculate the cosmic potential and how it evolves. But I believe their are some clues, as I indicated before, which could help us moving forward on this, all within GR context.

None of the clues you have mentioned so far are "within GR context"; they are either speculations on alternative theories to GR that didn't pan out, or dealing with other theoretical frameworks altogether (such as quantum mechanics).

FRW model is not free from interrogation as long as we have no acceptable answer to what dark energy is supposed to be.

Modeling dark energy in the FRW model is easy: it's a positive cosmological constant. If that's not enough for you because we don't know what microphysics produces a positive cosmological constant, that's not a problem with the FRW model; *any* large-scale model that produced the same predictions would be open to the same objection.

Also, I don't see what dark energy, or the observations that lead us to incude it in the standard cosmological model, have to do with any "cosmic potential". None of the suggestions you have made about how that might be the case were valid alternatives to the standard FRW model with a positive cosmological constant; they were misunderstandings of how the FRW model works.

 

[PeterDonis]

The increase of total mass within the expanding particle horizon

Btw, I haven't commented on this before: according to the best current model of the universe (the one with a positive cosmological constant, aka "dark energy"), the total mass within our particle horizon is *decreasing*, not increasing. That's because the expansion of the universe is accelerating, and the effect of the accelerating expansion (which moves matter outside the particle horizon) outweighs the effect of the increasing age of the universe (which increases the distance to the particle horizon).

 

[Mordred]

One quick question on a com-moving observer...

if an observer is in relative motion to the fundamental observer and to the fundamental observer the universe is homogeneous and isotropic. Am I correct in thinking that the com-moving observer would not see the universe as isotropic?

 

[PeterDonis]

if an observer is in relative motion to the fundamental observer and to the fundamental observer the universe is homogeneous and isotropic. Am I correct in thinking that the com-moving observer would not see the universe as isotropic?

What you are calling the "fundamental observer" is what is usually called a "comoving observer". Those observers see the universe as homogeneous and isotropic. An observer in relative motion to the comoving observer in his vicinity will *not* see the universe as homogeneous and isotropic.

 

[Mordred]

thanks that's what I thought but wanted to make sure