Mach's principle, GR and the nature of space

[PeterDonis]

on top of that axisymmetric (has axial KVF $\psi^{a}$)

This wouldn't actually be true of the isolated Earth-Moon case, or indeed of any case involving two isolated bodies orbiting their mutual center of mass. You would need something like a rotating fluid.

 

[WannabeNewton]

Sorry if this is a dumb question, does the anti-symmetrization here $\epsilon_{abcd}\nabla^{c}\psi^{d}$ mean that the covariant derivative can be replaced with partial derivatives ?

Not in general, no. When you evaluate the expression in a coordinate basis what you end up calculating is $\epsilon_{\mu\nu\alpha\beta}g^{\alpha \sigma}g^{\beta \gamma}\nabla_{\sigma}\psi_{\gamma}$; the christoffel symbols are certainly important in the calculation. For example the Komar mass is defined in an extremely similar way as $M = -\frac{1}{8\pi}\int _{S}\epsilon_{abcd}\nabla^{c}\xi^{d}$. In the coordinates adapted to the time-like KVF $\xi^{a}$, its coordinate representation is given by $\xi^{\mu} = \delta^{\mu}_{t}$ hence $\nabla^{\nu}\xi^{\mu} = g^{\nu\alpha}\Gamma _{\alpha t}^{\mu}$. If you had on the other hand just used partials then $\partial^{\nu}\xi^{\mu} = g^{\nu\alpha}\partial_{\alpha}\delta^{\mu}_{t} = 0$ which is of course not a general result.

Presumably the Minkowski space-time in the cylindrical chart has the necessary KVFs, but zero ang. momentum ?

Yes the so defined angular momentum is coordinate independent as you can see (one can show that Riemann and Lebsgue integrals of differential forms over smooth manifolds are coordinate independent). In fact one can show using Stoke's theorem that $J = \frac{1}{16\pi}\int _{S}\epsilon_{abcd}\nabla^{c}\psi^{d} = -\int _{\Sigma}T_{ab}n^{a}\psi^{b}dV$ where $\Sigma$ is a space-like hypersurface chosen so that $\psi^{a}$ is tangent to $\Sigma$, $S = \partial \Sigma$, and $n^{a}$ is the unit outward normal to $\Sigma$. Now for any static axisymmetric space-time we have that the time-like KVF $\xi^{a}$ is normal to some family of space-like hypersurfaces $\Sigma$ and that $\xi^{a}\psi_{a} = 0$ hence $\psi^{a}$ lies tangent to such $\Sigma$.

Choosing $n^{a} = \alpha \xi^{a}$ as our outward normal then, in the coordinates adapted to the KVFs we have $T_{\mu\nu}\xi^{a}\psi^{b} = T_{0\phi}$. Now we know that in these coordinates, $g_{0\phi} = 0$ and that $R_{0\phi} = 0$ hence by Einstein's equations $T_{0\phi} = 0$ thus $J = 0$. This is a covariant result so it holds for all coordinate systems. This includes Minkowski space-time and Schwarzschild space-time.

On the other hand, for stationary (but not necessarily static!) axisymmetric space-times, $J \neq 0$ in general. For example $J = Ma$ for the Kerr space-time of a completely isolated rotating spherically symmetric body; again $J$ is an unambiguous quantity.

 

[Mentz114]

Thanks for a complete answer, which will take me some time to verify. Right now, I don't believe you

 

[Nabeshin]

But such a situation would not be a solution of the Einstein Field Equation: there is no solution of the EFE that has the Earth and Moon both remaining at rest relative to each other for all time, without moving. At least, not if the Earth or the Moon has nonzero mass.

Yes, GR is decidedly non-Machian from this point of view.

Edit: In a fully Machian theory, the two bodies would likely just fall together at some (rapid) rate under their mutual gravitational pull. So they're not truly stationary. Regardless, GR doesn't describe this (the rate would be much much bigger than a GW-merger timescale).

 

[Nabeshin]

If the space-time is indeed asymptotically flat, and on top of that axisymmetric (has axial KVF $\psi^{a}$) we have an absolute invariant of the space-time that corresponds to (in most cases) the total angular momentum of the space-time, given by $J = \frac{1}{16\pi}\int _{S}\epsilon_{abcd}\nabla^{c}\psi^{d}$ where $S$ is a topological 2-sphere taken in the vacuum region; this is akin to the Komar mass. It is, as mentioned, an absolute quantity and can certainly distinguish truly isolated static configurations from truly isolated non-static ones (static meaning the space-time has a hypersurface orthogonal time-like KVF) without reference.

One can also define such an invariant corresponding to total angular momentum for non-axisymmetric (but still asymptotically flat) space-times but it is much less trivial.

Do you know how this relates to the calculation of the Chern-Pontryagin invariant? I was given the impression the CP invariant also tells you about the 'spin' in a spacetime (region). However, I don't entirely understand it outside the trivial case of kerr, and know it (oddly enough) produces zero for the godel universe.

 

[WannabeNewton]

Do you know how this relates to the calculation of the Chern-Pontryagin invariant? I was given the impression the CP invariant also tells you about the 'spin' in a spacetime (region). However, I don't entirely understand it outside the trivial case of kerr, and know it (oddly enough) produces zero for the godel universe.

The only thing I know about the CP invariant is that it is related to the winding number of an instanton. The rest is beyond me however, sorry to say

You might try ch22 of Frankel's "Geometry of Physics".