Is spacetime flat inside a spherical shell? - Physics Stack Exchange most recent 30 from physics.stackexchange.com 2019-09-15T12:06:38Z https://physics.stackexchange.com/feeds/question/43626 https://creativecommons.org/licenses/by-sa/4.0/rdf https://physics.stackexchange.com/q/43626 41 Is spacetime flat inside a spherical shell? Leos Ondra https://physics.stackexchange.com/users/7786 2012-11-07T11:44:26Z 2019-04-20T04:11:42Z <p>In a perfectly symmetrical spherical hollow shell, there is a null net gravitational force according to Newton, since in his theory the force is exactly inversely proportional to the square of the distance.</p> <p>What is the result of general theory of relativity? Is the spacetime flat inside (given the fact that orbit of Mercury rotates I don't think so)? How is signal from the cavity redshifted to an observer at infinity?</p> https://physics.stackexchange.com/questions/43626/-/43640#43640 81 Answer by Qmechanic for Is spacetime flat inside a spherical shell? Qmechanic https://physics.stackexchange.com/users/2451 2012-11-07T16:11:07Z 2016-06-18T14:41:09Z <p>Here we will only answer OP's two first question(v1). Yes, <a href="http://en.wikipedia.org/wiki/Shell_theorem">Newton's Shell Theorem</a> generalizes to <a href="http://en.wikipedia.org/wiki/General_relativity">General Relativity</a> as follows. The <a href="http://en.wikipedia.org/wiki/Birkhoff%27s_theorem_%28relativity%29">Birkhoff's Theorem</a> states that a spherically symmetric solution is static, and a (not necessarily thin) vacuum shell (i.e. a region with no mass/matter) corresponds to a radial branch of the <a href="http://en.wikipedia.org/wiki/Schwarzschild_metric">Schwarzschild solution</a> </p> <p>$$\tag{1} ds^2~=~-\left(1-\frac{R}{r}\right)c^2dt^2 + \left(1-\frac{R}{r}\right)^{-1}dr^2 +r^2 d\Omega^2$$</p> <p>in some radial interval $r \in I:=[r_1, r_2]$. Here the constant $R$ is the <a href="http://en.wikipedia.org/wiki/Schwarzschild_radius">Schwarzschild radius</a>, and $d\Omega^2$ denotes the metric of the angular $2$-sphere.</p> <p>Since there is no mass $M$ at the center of OP's internal hollow region $r \in I:=[0, r_2]$, the Schwarzschild radius $R=\frac{2GM}{c^2}=0$ is zero. Hence the metric (1) in the hollow region is just flat Minkowski space in spherical coordinates. </p>