From a90feb7d36f8dd45e3b384abe9de541ebc67d0cc Mon Sep 17 00:00:00 2001
From: Luc <luc@bijl.us>
Date: Sun, 5 Jan 2025 20:06:39 +0100
Subject: [PATCH] update
 physics/relativistic-mechanics/schwarzschild-geometry.md: added spherical
 symmetry and exterior solution sections

---
 .../schwarzschild-geometry.md                 | 46 ++++++++++++++++++-
 1 file changed, 45 insertions(+), 1 deletion(-)

diff --git a/docs/en/physics/relativistic-mechanics/schwarzschild-geometry.md b/docs/en/physics/relativistic-mechanics/schwarzschild-geometry.md
index d56d45a..a9bbd8b 100644
--- a/docs/en/physics/relativistic-mechanics/schwarzschild-geometry.md
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@@ -1 +1,45 @@
-# Schwarzschild geometry
\ No newline at end of file
+# Schwarzschild geometry
+
+## Spherical symmetry
+
+A metric that is time-reversal and time-translation invariant is said to be **static**.
+
+> *Lemma 1*: a static, spherically symmetric metric tensor $\bm{g}: \Gamma(\mathrm{TM}) \times \Gamma(\mathrm{TM}) \to F$ must be of the form
+>
+> $$
+>   \bm{g} = A(r) dr \otimes dr + r^2 (\sin^2 (\varphi) d\theta \otimes d\theta + d\varphi \otimes d\varphi) - B(r) dt \otimes dt,
+> $$
+>
+> for all $(r, \theta, \varphi, t) \in \mathbb{R}^4$ with $A,B: r \mapsto A(r),B(r)$. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+Reducing the determination of the metric to only two functions $A$ and $B$.
+
+## Exterior solution
+
+Outside of the mass distribution the energy-momentum tensor vanishes, so we can impose $\mathbf{W} = \mathbf{0}$. Then, by imposing the weak field limit we have the following. 
+
+> *Principle 1*: a metric outside a static, spherically symmetric mass distribution is described by the **Schwarzschild metric** 
+>
+> $$
+>   \bm{g} = \Big(1 - \frac{2 G M}{c^2 r}\Big)^{-1} dr \otimes dr + r^2 (\sin^2 (\varphi) d\theta \otimes d\theta + d\varphi \otimes d\varphi) - c^2  \Big(1 - \frac{2 G M}{c^2 r} \Big) dt \otimes dt,
+> $$
+>
+> for all $(r, \theta, \varphi, t) \in \mathbb{R}^4$ with $G$ the gravitational constant and $M$ the mass of the spherically symmetric mass distribution.
+
+??? note "*Derivation*:"
+
+    Will be added later.
+
+Notice that for $r_s = \frac{2 G M}{c^2}$ the metric with these coordinates is not defined. This radius is called the **Schwarzschild radius**. 
+
+> *Theorem 1 (Birkhoff's theorem)*: the Schwarzschild metric is the only spherically symmetric solution, outside a spherical mass distribution.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+Note that static is automatically implied by spherical symmetry. An important consequence of the theorem is that a purely radially pulsating star cannot emit gravitational radiation, because outside of this star such gravitational radiation would amount to a time-dependent spherically symmetric spacetime geometry in (approximate) vacuum, which, according to the Birkhoff’s theorem, cannot be consistent with Einstein’s field equations.
\ No newline at end of file