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At the bottom of the Wikipedia article https://en.wikipedia.org/wiki/Lema%C3%AEtre_coordinates for the Schwarzschild metric, there appears
" The Lemaître coordinate system is synchronous, that is, the global time coordinate of the metric defines the proper time of co-moving observers.
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Can somebody give me a symbolic/geometric/intrinsic statement of synchronus/co-moving observers?
I would prefer an "intrinsic" definition. I have found they are less open to "interpretation"/hand-waving. The use of "co" throws me off in curved space-time.

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You can find the general theory of synchronous coordinate systems in Landau-Lifschitz Vol 2, The Classical Theory of Fields chapt. 97. However, definitions are quite straightforward: a synchronous coordinate system has a metric of the form

$$ ds^2 = dt^2 + g_{ij}dx^i dx^j \qquad i, j = 1,2,3, $$

that is $\ g_{00} = 1, \ g_{0k}= 0. $

In these reference systems time-like lines are geodetics, and this means that a curve of the form

$$ x^i = \text {const}, \qquad i = 1,2,3 $$

is the trajectory of a free falling, non-interacting particle: this is a comoving observer. Now, I don't know what you mean by intrinsic: this depends on our particular coordinate system, so it isn't intrinsic.

However, you can always find such a system solving the Hamilton-Jacobi equation

$$ g^{ij}\frac{d\tau} {dx^i} \frac{d\tau} {dx^j} = 1 $$

and the metric has a nice expression. In cosmology the comoving system is characterized by the property that the CBR is isotropic.

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  • $\begingroup$ I take this to mean that, if I have two test particles 1,2 at a point p with different velocities v_1,v_2 and respectively following their geodesic paths g_1(s),g_2(s) . The paths will have the same t(s) coordinate t=s ? i.e. proper time on all geodesics equals the t coordinate. Since you answered this so easily; I might be back with using something like this on the I-II boundary of the Kerr Metric where O'Neil says there is no timelike Killing field across the boundary. Not quite there yet though. $\endgroup$
    – rrogers
    Dec 5 '16 at 22:14
  • $\begingroup$ My mistake and confusion arose because I took "co" to mean two; which it sometimes does in English. My bad. $\endgroup$
    – rrogers
    Dec 5 '16 at 22:28
  • $\begingroup$ Hi rrogers, you are welcome to ask anything, I'm glad to help if I can. $\endgroup$
    – Luca
    Dec 6 '16 at 14:52
  • $\begingroup$ Yes, co-moving actually refers to the metric of the space-like 3-space you live in: co-moving observers just let themselves be led about by the variations of the space-like metric $\endgroup$
    – Luca
    Dec 6 '16 at 15:16
  • $\begingroup$ So for a primitive coordinate proof I examine the Christoffell symbol geodesic equation terms and show that dt/ds =k , k for reparametrization, for all input vectors and d^2(t)/ds^2 =0 ? On an even more basic level I can prove that this coordinate system exists by testing for matrix congruence; counting positive and negative eigenvalues. I like being able to phrase things in terms of matrices, at the end, because I trust embodied computer algorithms more for matrices than any other algorithms (including parsing which should be foolproof!). I hope these comments are not too elementary. $\endgroup$
    – rrogers
    Dec 6 '16 at 19:00

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