Conservation of the volume form

[relativityfan]

hi,

is it true that for any valid spacetime in general relativity that:
if time dilated/contracted by k, space is contracted/dilated by k?

what is the mathematical explanation for this?

therefore, if one knows the time dilation for a point in spacetime, then one knows automatically the length contraction?

for the Kerr metric, the problem is that at the event horizon of a black hole space is infinitely contracted but time does not seem infinitely dilated...

 

[Mentz114]

hi,

is it true that for any valid spacetime in general relativity that:
if time dilated/contracted by k, space is contracted/dilated by k?

There are solutions of the field equations where this is not true.

Dust solutions usually have g₀₀=-1, for instance. It isn't true generally for Weyl vacuums either.

 

[relativityfan]

really?
but the time dilation does not occur only for g_00, the determinant of the metric must be evaluated.
could you provide or link to a full example of such metric tensor, that is physically valid?

 

[Mentz114]

arxiv.org/pdf/gr-qc/0004016

 

[DrGreg]

is it true that for any valid spacetime in general relativity that:
if time dilated/contracted by k, space is contracted/dilated by k?

I'm assuming you are referring to so-called "gravitational" dilation, and not the "special-relativistic" time dilation/length contraction due to motion.

Well, there is simple counter-example to your proposition: the "Rindler metric"

ds^2 = \frac{a^2 x^2}{c^2} dt^2 - dx^2 - dy^2 - dz^2​

in which there is "gravitational" time dilation but no change in spatial distance. If you are not familiar with Rindler coordinates, they are coordinates relative to a rigidly accelerating rocket in flat spacetime (i.e. no gravity apart from the "artificial" gravity experienced by the rocket's occupants).

(The Rindler metric is just the Minkowski metric expressed relative to another coordinate system.)

This shows that gravitational time dilation depends on your choice of coordinates and is not an intrinsic (i.e. frame-independent) property of a spacetime.

 

[relativityfan]

thank you,
well the coordinate t in this Rindler metric is not time but a combination of space and time

the time dilation is of course not frame independant, but if you change the coordinates SYSTEM and not the spacetime position/frame of the observer, the lorentz factor must remain the same

sorry, but I have still not any convincing example.
for me the gravitationnal time dilation is just a consequence of special relativity: its exactly the same phenomena