Exploring Time and Space in a Supermassive Hollow Sphere

[Hornbein]

Suppose there were a supermassive, superdense hollow sphere. Inside of the sphere, would time move more slowly relative to outside? Would objects inside the sphere be contracted relative to outside?

I did this calculation once about the center of a neutron star. (Of course, it isn't hollow.) Time was contracted significantly, but it depended strongly on the radius of the star. This is not know precisely so it wasn't possible to make an accurate estimate. But as you can see, if there is contraction then this will tend to build on itself. More contraction => more density => more gravity => more contraction and so on until everything collapses.

 

[WannabeNewton]

Suppose there were a supermassive, superdense hollow sphere. Inside of the sphere, would time move more slowly relative to outside?

Yes if you are referring to gravitational time dilation.

Would objects inside the sphere be contracted relative to outside?

What does that even mean? Lorentz contraction is a local kinematic effect. It is only defined between two different observers at the event(s) at which their world-lines intersect. The statement "Lorentz contraction of objects inside the sphere relative to the outside" is meaningless.

 

[A.T.]

Would objects inside the sphere be contracted relative to outside?

How would you compare them? You can bring clocks together, and compare the accumulated proper times. But how to compare lengths at a distance?

 

[pervect]

re: "How would you compare them"

I imagine one could take the ratio of coordinate length to proper length, just as one takes the ratio of coordinate time to proper time and calls it "time dilation".

I'm not sure it's a good idea to encourage that sort of thinking though. So while I'll mention the idea, I'm not sure I want to propound it seriously.

My real thinking is more along the lines that "time dilation" is routinely interpreted in the context of an absolute time, which is unfortunate, and we don't really need to repeat the mistake with respect to distance. Though people seem eager to, for reasons of treating time and space symmetrically.

 

[PeterDonis]

My real thinking is more along the lines that "time dilation" is routinely interpreted in the context of an absolute time, which is unfortunate

I agree that it's unfortunate because it so often seems to lead to misconceptions. But in the particular scenario under discussion, there is a sense of "time" that is intrinsic to the scenario, because the spacetime as a whole has a timelike Killing vector field, and therefore it has an invariant notion of a "static" observer, namely, an observer following an orbit of the timelike KVF. Static observers inside and outside the shell can exchange light signals and verify that (a) they are at rest relative to each other, and (b) the proper time interval between two successive round-trip light signals is smaller for the observer inside the shell than for the one outside. This allows "time dilation" to be given an invariant meaning, *in this particular case*.

Of course this definition does not generalize to non-stationary spacetimes, which is why one has to use it with extreme caution. But it does allow a better (IMO) answer to be given to the question of why no similar comparison can be made for lengths in this scenario. That is simply because the spacetime has no spacelike KVF that can be used to define an invariant notion of "length contraction" the way we defined an invariant notion of "time dilation" above. So in this particular respect, time and space *do* work differently.

 

[pervect]

I agree that it's unfortunate because it so often seems to lead to misconceptions. But in the particular scenario under discussion, there is a sense of "time" that is intrinsic to the scenario, because the spacetime as a whole has a timelike Killing vector field, and therefore it has an invariant notion of a "static" observer, namely, an observer following an orbit of the timelike KVF. Static observers inside and outside the shell can exchange light signals and verify that (a) they are at rest relative to each other, and (b) the proper time interval between two successive round-trip light signals is smaller for the observer inside the shell than for the one outside. This allows "time dilation" to be given an invariant meaning, *in this particular case*.

In an attempt to paraphrase your argument concisely, are you are saying that if we consider time dilation written in the typical form as $\sqrt{\left| g_{00} \right|}$ it appears at first glance to be coordinate dependent. However, it can be written in coordinate independent fashion in any stationary space-time by letting $\xi^a$ be a timelike Killing vector associated with said space-time and considering the magnitude of said vector $\sqrt{ \left| \xi_a \xi^a \right| }$.

If we consider the Schwarzschild geometry as a specific example, there ARE space-like Killing vectors, but they typically represent rotational symmetries, not translational symmetries, thus they can't readily be interpreted as "length contraction / dilation" - at least I can't think of any way to interpret them thus.

I'm not sure how to express this in less technical language at the moment.

 

[A.T.]

re: "How would you compare them"

I imagine one could take the ratio of coordinate length to proper length, just as one takes the ratio of coordinate time to proper time and calls it "time dilation".

That would yield no "length contraction" within the cavity, right?

 

[PeterDonis]

In an attempt to paraphrase your argument concisely, are you are saying that if we consider time dilation written in the typical form as $\sqrt{\left| g_{00} \right|}$ it appears at first glance to be coordinate dependent. However, it can be written in coordinate independent fashion in any stationary space-time by letting $\xi^a$ be a timelike Killing vector associated with said space-time and considering the magnitude of said vector $\sqrt{ \left| \xi_a \xi^a \right| }$.

Yes.

If we consider the Schwarzschild geometry as a specific example, there ARE space-like Killing vectors, but they typically represent rotational symmetries, not translational symmetries, thus they can't readily be interpreted as "length contraction / dilation" - at least I can't think of any way to interpret them thus.

Neither can I. One way to express this is that the magnitude of the timelike KVF is constant along orbits of all of the spacelike KVFs, so the spacelike KVFs can only be used to "move between" points that have the same time dilation factor; they can't be used to compare quantities at points with different time dilation factors.