# Is spacetime volume preserved in presence of a mass in General Theory of Relativity (GTR)?

In many publications about the warping of spacetime in presence of a mass the metric of spacetime is stretched, expanded :

By the simultaneously introduced additional dimension the overall (hyper)volume of spacetime is also expanded.

So is this common illustration in consonance with theory of general relativity or in other words is spacetime volume preserved in presence of a mass ? If not, how is GTR balanced, meaning how is the change in spacetime volume countered to achieve a closed theory, an universal equilibrium. It is a bit strange to assume that whenever a new star is born the (hyper-) volume of spacetime increases a bit and in addition this leads to symmetry problems and induces a massive philosophical problem...

• This is a difficult question to answer because the volume of all of spacetime (at least for the simplest solutions) is infinite. Also, when a star is born, the mass was already there previously. – Javier May 24 at 18:21
• not all mass was generated at big bang ( big bounce ) there is still energy-mass conversion in both directions forbes.com/sites/paulrodgers/2014/05/19/… – ralf htp May 24 at 18:29
• Indeed, but it's energy that warps spacetime, not mass. And anyway I'm not sure why it would be such a big problem for volume to change. – Javier May 24 at 18:29
• There is no conservation of spacetime volume, and physicists do not regard this fact as a philosophical problem. – G. Smith May 24 at 18:31
• Well, the determinant of the Schwarzschild metric is independent of the mass, so I suppose you could say that local spacetime volume is preserved in that situation. (The radial dimension grows and the time dimension shrinks by the same factor.) I haven’t checked whether this is true for a Kerr black hole. In a Friedman universe, spacetime volume is definitely not preserved. – G. Smith May 24 at 18:43

The question is not posed in a meaningful way, so GR doesn't answer it. Conservation laws are not statements that a certain quantity is the same if you change the conditions in a certain way. More specifically, in the context of GR, we can't compare spacetime A (without the added mass) and spacetime B (with the added mass) and define whether the same parcel of space has the same volume. There is no way to even decide what constitutes the "same" parcel of space in A and B.

• yes there is cf en.wikipedia.org/wiki/… – ralf htp May 24 at 19:32
• in addition volume preservation is critical for conservation laws in general via Liouville's theorem ( en.wikipedia.org/wiki/Liouville%27s_theorem_(Hamiltonian) ) – ralf htp May 24 at 19:39
• @ralfhtp Those links refer to conservation of the 3-volume of a certain region as it evolves in time. They're not comparing 4-volumes of different spacetimes. – Javier May 24 at 20:00
• can be expanded to 4D the underlying concept does not loose its validity. can answer tomorrow, is late here ... – ralf htp May 24 at 20:30

i did research this deeper and think i can answer this now. Thanks to all of you for the help, this is a more complex topic than i initially thought. In some sense Ben Crowell is right with his answer, however this answer is simply not satifying and in addition there is a clean mathematical way how two volumes in $$\mathbb{R^n}$$ can be compared : the Jacobian determinant ( https://en.wikipedia.org/wiki/Jacobian_matrix_and_determinant#Jacobian_determinant ).

His answer is accurate in the sense that GR does not give an answer to the problem because GR simply is not a closed theory and conservation laws simply do not hold anymore in GR cf i.e. https://link.springer.com/article/10.1007/BF01691807 and https://motls.blogspot.com/2010/08/why-and-how-energy-is-not-conserved-in.html .

And actually this is a philosophical problem and also a physical that is tried to recover with approaches like the Wheeler-DeWitt equation ( https://en.wikipedia.org/wiki/Wheeler%E2%80%93DeWitt_equation ) that introduces a global Hamiltonian and fixes the global energy to zero what is accurate from a philosophical viewpoint.

If the local spacetime volume in presence of a energy-mass is preserved depends on the used metric, like G. Smith accurately stated in the comments under the question. In the 'cosmological standard model' the Friedman universe, it is not preserved. The Friedmann universe is modeled on a perfect fluid in that only pure expansive stresses are allowed, not even shear stresses so in presence of any energy-mass spacetime volume is definitely not preserved.

i came up with the problem of preservation of spacetime volume because there is a self-field problem in electrodynamics ( including QED ) and theories of relativity ( including SR ) cf https://en.wikipedia.org/wiki/Abraham%E2%80%93Lorentz_force