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For a long time I had basically taken for granted the usual interpretation of one-parameter families of orthogonal space-like hypersurfaces relativized to a time-like congruence as "global simultaneity surfaces" (c.f. MTW section 27.3), and left it at that. See also p.24 of this paper: http://arxiv.org/pdf/gr-qc/0311058v4.pdf and p.5 of this paper: http://arxiv.org/pdf/gr-qc/0506127.pdf But a past discussion with PeterDonis about operational and mathematical definitions of "synchronizable reference frames" given in the text "General Relativity for Mathematicians"-Sachs and Wu has me confused about some aspects of the surface-level identification of hypersurface orthogonality with "global simultaneity", all relative to a time-like congruence.

More specifically, consider a time-like congruence with 4-velocity field ##\xi^{\mu}##. If the congruence is irrotational, that is, ##\xi_{[\gamma}\nabla_{\mu}\xi_{\nu]} = 0##, then there exists a one-parameter family of space-like hypersurfaces ##\Sigma_t## everywhere orthogonal to ##\xi^{\mu}## in which case we can write ##\xi^{\mu} = h \nabla^{\mu} t## for some positive scalar field ##h## (technically ##\xi_{[\gamma}\nabla_{\mu}\xi_{\nu]} = 0## only guarantees such a foliation locally in general, but we can consider cases wherein it holds globally for simplicity).

Sachs and Wu define a "synchronizable reference frame" to be a time-like congruence for which ##\xi^{\mu} = h \nabla^{\mu} t## (section 2.3). If ##h = 1## identically then they call this a "proper time synchronizable reference frame"; obviously this implies that ##\nabla_{[\mu}\xi_{\nu]} = 0## which itself holds if and only if the congruence is irrotationalandgeodesic. Physically, clocks at rest in a "proper time synchronizable reference frame" are freely falling and so undergo no gravitational time dilation whereas clocks at rest in a "synchronizable reference frame" will, in general, undergo gravitational time dilation of different factors due to different proper accelerations.

In section 5.3 they then give these definitions operational meaning (see attachments below). However I have a couple of problems with their discussion of clock synchronization throughout ##\xi^{\mu}##, assuming that ##\nabla_{[\mu}\xi_{\nu]} = 0##.

(1) Doesn't the congruence need to be rigid, that is, ##\nabla_{(\mu}\xi_{\nu)} = 0##? Take, for example, the Painleve congruence in Schwarzschild space-time; this congruence is both geodesic and irrotational but it is not rigid. Because of gravitational tidal forces exerted by the self-gravitating source, two Painleve observers that are infinitesimally separated radially will have a non-vanishing relative radial velocity between them. Therefore they can't synchronize their clocks because their clocks tick at different rates due to their relative Lorentz factor.

(2) Say that ##\nabla_{(\mu}\xi_{\nu)} = 0## does hold. Sachs and Wu's discussion of clock synchronization throughout ##\xi^{\mu}## makes no explicit use of ##\xi_{[\gamma}\nabla_{\mu}\xi_{\nu]} = 0##. Where then in their operational method of synchronizingallthe clocks following orbits of ##\xi^{\mu}## do they implicitly make use of the irrotationality of the congruence? More precisely, they take two clocks A and B infinitesimally separated and Einstein synchronize them (call this "local Einstein synchronization"). Then B is locally Einstein synchronized with a third clock C and so on. By doing this we get a space-like hypersurface ##t = \text{const}## which is everywhere orthogonal to ##\xi^{\mu}## and represents the set of events that are simultaneous with respect toallthe clocks in this congruence. But clearly this only works if all the clocks in the congruence are synchronized with one another. In other words if clock A is locally Einstein synchronized with clock B and clock B is locally Einstein synchronized with clock C then we need clock A to be synchronized with clock C i.e. we require the synchronization to be transitive, which Sachs and Wu don't seem to mention at all.

If we consider the family of clocks laid out around a rotating ring in flat space-time, then two infinitesimally separated clocks on the ring can be locally Einstein synchronized but as we know all the clocks on the ring cannot be synchronized with one another i.e. the synchronization fails to be transitive. All the clocks on the ring experience the same time dilation factor and ##\nabla_{(\mu}\xi_{\nu)} = 0## but ##\xi_{[\gamma}\nabla_{\mu}\xi_{\nu]} \neq 0##.

So how does one prove that ##\xi_{[\gamma}\nabla_{\mu}\xi_{\nu]} = 0 \Rightarrow## local Einstein synchronization is transitive? Otherwise I don't even see how this is a clock synchronization procedure throughout ##\xi^{\mu}## and hence I don't see how the identification of hypersurface orthogonality with "global simultaneity" can be interpreted as the set of events that are simultaneous with respect toallthe clocks in this congruence.

Thanks in advance!

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# Global simultaneity surfaces

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