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Setup for the question: Under the Cosmological Principle, mass and energy density can be treated as homogenous and isotropic at large scale. This Principle can be applied to any scale in a model. We consider an infinitely large volume of spacetime, or 'universe', containing mass-energy with perfect gravitational isotropy. It's absolutely flat (i.e. Euclidean in spatial dimensions x,y,z).

We know that increased density gives time dilation. We now consider two instant (dt) 'universes' having unequal 'mass' (mass-energy) densities.

My question: How is differential time dilation between these two instant 'universes' expressed as a function of density alone? Such an expression cannot contain any length or distance term.

Has anybody read about, heard of, or perhaps even derived this expression?

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  • $\begingroup$ perfect gravitational anisotropy Do you mean isotropy? $\endgroup$ Commented Jun 19 at 18:55
  • $\begingroup$ Yes of course, my bad. ISOTROPY. $\endgroup$ Commented Jun 19 at 19:32
  • $\begingroup$ Then please correct your post by editing it. Comments are not considered clarifications. $\endgroup$ Commented Jun 19 at 19:35
  • $\begingroup$ It seems like you are implying that events would proceed in slow motion for Universe A compared to Universe B if they had different mass densities (presumably as seen by some Godlike figure outside of both). This is not necessarily the case. Any observer in either universe looking only at his own local vicinity would not perceive any time dilation. $\endgroup$ Commented Jun 19 at 19:40
  • $\begingroup$ What does time dilation between universes mean? $\endgroup$ Commented Jun 20 at 0:42

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There is no such expression, because that is not a meaningful quantity to calculate.

The elapsed time for an observer is the length of their worldline. Time dilation refers to the possibility that two observers could have different worldline lengths between two agreed-upon reference times. Those reference times correspond to 3D spatial slices within 4D spacetime. Here is a quick drawing:

spacetime diagram for time dilation

Time dilation refers to how the lengths of the worldlines (blue) between the two times (black) can be different. The problem is that to construct those reference times (black) in the first place, you need a description of the spacetime lying between the two observers. And that is going to involve some notion of distance or position.

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  • $\begingroup$ Thank you Sten, I think I understand your position. Let me rephrase my question. Consider a comoving sphere of perfect fluid, with a clock at the center. The sphere gets less dense with time. Are you saying that comoving elapsed time at the clock has some sort of distance dependence? $\endgroup$ Commented Jun 19 at 19:48
  • $\begingroup$ In that scenario, you need to pick the reference times (spatial slices) between which you are measuring the clock's elapsed time. How do you want to pick them? $\endgroup$ Commented Jun 19 at 19:57
  • $\begingroup$ You have gotten right to the heart of my question. I am partial to density, as it comoves monotonically with volume or spatial slice. My initial density would be very, very close to singularity, having a corresponding spatial slice of whatever-it-is. My next spatial slice would be a million times more voluminous, so the rest mass density would be 1/1,000,000. Let's leave kinetic and light energy densities out of it for now, OK? Next, a billion times more voluminous/less dense. $\endgroup$ Commented Jun 19 at 20:41
  • $\begingroup$ What I am really after here is an analytical expression of this comoving elapsed or cosmic time. Spatial slice or density, take your pick. $\endgroup$ Commented Jun 19 at 20:42
  • $\begingroup$ You are looking for the time elapsed between some reference density values? The rate of change of the density is $d\rho/dt=-3(1+w)H\rho$, where $\rho$ is the density, $w$ is the equation of state parameter, and $H$ is the Hubble rate. Thus the time elapsed per logarithmic interval (e-fold) in $\rho$ is $(d\ln\rho/dt)^{-1}=-[3(1+w)H]^{-1}$. Note however that this does not match any conventional approach for how to define time dilation. $\endgroup$ Commented Jun 19 at 20:48

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