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The proper time in Kruskal coordinates is given by $$d\tau^2=\frac{4}{e^rr}(dT^2-dX^2)$$ (where r is measured in units of Schwarzschild radii). Since at $T=+/-\infty$, $dT^2-dX^2$ goes to zero for an observer at constant $r$ (the hyperbolas of constant $r$ at $T=+/-\infty$ become light-like), does this not imply that the clock of an observer at rest in a gravitational field does not tick at $T=-\infty$, then ticks at an increasing rate up to $T=0$, then slows to zero again as $T$ goes to $\infty$? Therefore, shouldn't this mean that during the expansion of the Universe, clocks at rest in gravitational fields tick at increasing rates over time until a point of full expansion and then begin to tick slower again as they observe the Universe to collapse (but given that the expansion/contraction of the Universe is governed by the FRW metric, how do these two metrics relate?)?

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  • $\begingroup$ "Therefore, shouldn't this mean that during the expansion of the Universe" - aren't the Kruskal coordinates used for the static (at least outside the horizon) Schwarzschild geometry? $\endgroup$ – Alfred Centauri Sep 4 '15 at 1:54
  • $\begingroup$ Well, that's where I get confused because when I look at the visual depiction of the metric (en.wikipedia.org/wiki/Kruskal%E2%80%93Szekeres_coordinates#/…), it seems to suggest that for an observer at constant r, the proper distance and times go to zero at t or T = +/- infinity, so does this not mean that there is a time dependance? How else can that be interpreted? It seems to me that the value of the coordinate time (as opposed to the interval) is just as significant as the value of the radius at which the observer is at rest. $\endgroup$ – Chris L. Sep 4 '15 at 2:10
  • $\begingroup$ From the diagram, it seems that an observer at rest in a gravitational field would see space expand and then contract regardless of the mass content of the Universe (because the metric describes empty space outside a spherically symmetric body). Looks like length contraction and time dilation from special relativity. If the equivalence principle holds, it seems as if being at rest in a gravitational filed will accelerate the observer from V=-c at t=-infinity to V=c at t=infinity, where the velocity is characterized by the observed length contraction and time dilation of empty space. $\endgroup$ – Chris L. Sep 4 '15 at 2:21
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The proper time in Kruskal coordinates is given by $$d\tau^2=\frac{4}{e^rr}(dT^2-dX^2)$$ (where r is measured in units of Schwarzschild radii).

Sure, and you can pick whatever well formed coordinates you want near an event. It doesn't change the physics.

Since at $T=+/-\infty$, $dT^2-dX^2$ goes to zero for an observer at constant $r$ (the hyperbolas of constant $r$ at $T=+/-\infty$ become light-like),

This is not true. At every $T$ the curve of constant $r,$ $\theta,$ and $\phi$ is timelike, not lightlike. A surface of constant $r$ has a tangent in the $T$ direction at $T=0$ then it asymptotically approaches the $T=X$ surface which means the constant $r$ surface has $dT>dX$ at every single $T.$ In fact $T=e^{r/2}\sinh(t/2r)\sqrt{r-1}$ and $X=e^{r/2}\cosh(t/2r)\sqrt{r-1}.$ So $dT^2-dX^2$ is proportional to $X^2-T^2$ which is a constant.

does this not imply that the clock of an observer at rest in a gravitational field does not tick at $T=-\infty$, then ticks at an increasing rate up to $T=0$, then slows to zero again as $T$ goes to $\infty$?

No, a clock at rest (constant $r,$ $\theta,$ and $\phi$) ticked and infinite number of times before $T=0$ and ticks an infinite number of times after $T=0.$

If you want to compute how how many ticks you get per change in $T$ you can compute that. But this is unrelated to whether the curve asymptotically approaches a lightlike curve.

Therefore, shouldn't this mean that during the expansion of the Universe, clocks at rest in gravitational fields tick at increasing rates over time until a point of full expansion and then begin to tick slower again as they observe the Universe to collapse (but given that the expansion/contraction of the Universe is governed by the FRW metric, how do these two metrics relate?)?

Coordinates have nothing to do with anything. You could take Minkowski space and replace t with $\sinh(t).$ Then $x,y,z,$ and $t'=\sinh(t/t_0)$ are perfectly fine coordinates. You can write the flat Minkowski metric in the $x,y,z,$ and $t'=\sinh(t/t_0)$ coordinate system and then clocks tick slower per $t'$ than they do per $t$ but the physics didn't change. And the curve of constant $x,y,$ and $z$ doesn't asymptotically approach a lightlike curve.

So that had nothing to do with anything. The amount of ticks of clock per coordinate time decreased for tines away from $t=0$ was just because we used an increasingly stretched of coordinate then. But we are free to use what well formed coordinates we want.

Don't confuse coordinates with physics.

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