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?

 

[sardar]

ach

Dear Zach,

Thank you for bringing up this interesting topic. The concept of local flatness at r = 0 for the interior spacetime of a star is indeed a fascinating one. As you mentioned, Schutz argues that enforcing local flatness leads to the conclusion that the radial metric coefficient g_rr is 1 at the origin. This makes sense, as local flatness requires the spacetime to be flat in small regions, and in flat spacetime, the metric coefficient g_rr is equal to 1.

However, as you pointed out, this analysis does not hold for the time metric coefficient g_tt. This is because time dilation is a crucial aspect of gravity, and a non-trivial g_tt is necessary to account for the effects of time dilation on the interior of the star. In fact, the fact that g_tt is not equal to 1 at the origin is a direct consequence of the presence of a gravitational potential, which causes time to pass at a different rate in different regions of spacetime.

As for your question about whether there is a way to argue for a specific value of g_tt at the origin, it is important to note that the value of g_tt is not a constant in general. It is a function of the gravitational potential, and thus can vary depending on the specific interior solution being analyzed. However, it is possible to determine the value of g_tt at the origin by solving the Einstein field equations for a specific interior solution. This would provide a more accurate and specific value for g_tt at r = 0.

In conclusion, the fact that g_rr is 1 at the origin in a locally flat spacetime is indeed remarkable, and it highlights the importance of considering both the radial and time metric coefficients when analyzing the interior spacetime of a star. I hope this provides some helpful insights and tips for further exploration. Keep up the good work!

Best,