# Time dilation in FLRW metric?

PeterDonis
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"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.

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
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2019 Award
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).

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.

lets 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?

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PeterDonis
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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.

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 text books lol

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
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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.

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PeterDonis
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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).

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
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2019 Award
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.

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