Local flatness at r = 0 for star interior spacetime

[FunkyDwarf]

Hi All,

I am interested in the discussion in section 10.5 of Schutz's First Course in GR book. Specifically, the conditions at r = 0 of a static, spherically symmetric interior star (or whatever) solution e.g. Schwarzschild interior solution.

He argues that by enforcing local flatness one finds that the radial metric coefficient g_rr is 1 at the origin. Is there some way one can argue as to what the g_tt (g_00 as he writes) would be at the origin? Analysis of specific interiors shows that it is not 1, which makes sense as otherwise there would be no time dilation and thus, to an external observer, no potential. In fact it seems to me quite remarkable that g_rr should be 1 at the origin.

Any tips?
Cheers,
Z

 

[A.T.]

He argues that by enforcing local flatness one finds that the radial metric coefficient g_rr is 1 at the origin. Is there some way one can argue as to what the g_tt (g_00 as he writes) would be at the origin?

g_rr = 1 can be understood intuitively from embedding diagrams of a 2D spatial slice. It relates the radial distance ds on the curved surface (proper radial distance), and its projection on the plane dr(Schwarzschild radial coordinate difference):

At the center the curved surface and the plane are parallel so dr = ds:

g_tt is not that simple, because it is not local. It relates the local proper time to a distant observer's time. You have to combine the interior and exterior metric to find it.

 

[PeterDonis]

In fact it seems to me quite remarkable that g_rr should be 1 at the origin.

I don't have Schutz's book so I don't know if he mentions the theorem that spacetime inside a spherically symmetric mass distribution is flat; but that theorem is enough to show that g_rr = 1 at r = 0 for a spherically symmetric massive object.

 

[FunkyDwarf]

Thanks A.T., the non-locality part is I guess what I was looking for.

PeterDonis: How can spacetime be flat (except locally) inside a generic spherical mass?