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I was studying the book of Hartle on general relativity. In chapter 9, "The Geometry Outside a Spherical Star", he suddenly introduces a metric named Schwarzschild metric and then goes on describing the geometry it produces. I did not quite get how exactly this was a metric generated by a spherical start. There must be some methodology of arriving at this metric. In non-relativistic Newtonian limit, I know how to show $g_{00} = 1+2\phi/c^2$, but for general case, I did not find anything useful.

What is the systematic logical sequence behind this result?

Qmechanic
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user157588
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    The intuitive "ansatz" approach (see @G. Smith's answer) is the usual way taught in text books. If you crave for a "systematic logical" approach (invoking symmetry groups or algebraic structure/Petrov type), see the book: http://content.schweitzer-online.de/static/catalog_manager/live/media_files/representation/zd_std_orig__zd_schw_orig/002/190/827/9780521467025_foreword_pdf_1.pdf – MadMax Jul 31 '20 at 17:31
  • There are some sites in the web with the derivation. Ex:https://web.stanford.edu/~oas/SI/SRGR/notes/SchwarzschildSolution.pdf, It is a little tedious, but it is good to know how GR works. – Claudio Saspinski Jul 31 '20 at 17:35
  • This is derived in my Phys.SE answer here. – Qmechanic Jul 31 '20 at 17:42

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A typical method is to make the ansatz that the spherically-symmetric metric has the form

$$ds^2=-A(r)dt^2+B(r)dr^2+r^2(d\theta^2+\sin^2{\theta}d\phi^2)$$

and determine which functions $A(r)$ and $B(r)$ make the metric satisfy the Einstein field equations, which in vacuum are $R_{\mu\nu}=0$ everywhere (except perhaps at some singularity).

This ansatz lets you reduce partial differential equations to ordinary differential equations.

You should try doing this yourself, by hand! Calculate the Christoffel symbols, the Riemann tensor, and the Ricci tensor in terms of $A$ and $B$ and their first and second $r$-derivatives. Then set Ricci to zero and solve for $A$ and $B$. You will have a great sense of satisfaction in solving Einstein’s field equations for a Schwarzschild black hole.

It turns out that this metric describes not just black holes but also the vacuum outside an uncollapsed and non-rotating star. But it is easiest to first think about the all-vacuum black hole case.

G. Smith
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