update physics/relativistic-mechanics/relativistic-formalism.md: renamed field equation section

docs/en/physics/relativistic-mechanics/relativistic-formalism.md

@@ -69,13 +69,72 @@ Similarly, we have the following.
 > $$
 >   \mathbf{R} = \frac{1}{8} R^i_{jkl} (\partial_i \wedge dx^j) \vee (dx^k \wedge dx^l).
 > $$
->
+
-> such that $R^i_{jkl}$ has a dimension of 
+??? note "*Proof*:"
- 
+
+    Will be added later.
+
 Such that $R^i_{jkl}$ has a dimension of 
 
 $$
     \frac{4^2 (4^2 - 1)}{12} = 20.
 $$
 
-## Axioms of Einstein
+## Curvature
+
+> *Definition 1*: let $\mathbf{W}: \Gamma(\mathrm{TM}) \times \Gamma(\mathrm{TM}) \to F$ denote the **Ricci tensor** which is defined as 
+>
+> $$
+> \begin{align*} 
+>     \mathbf{W} &= \frac{1}{2} R_{ijk}^k dx^i \vee dx^j,\\
+>                 &= \frac{1}{2} W_{ij} dx^i \vee dx^j,
+> \end{align*}
+> $$
+>
+> with $R_{ijk}^k$ the contracted Riemann holor and let $W$ be the **Ricci scalar** be defined as $W = W_{ij} g^{ij}$ with $g^{ij}$ the dual metric holor.
+
+The Ricci tensor and scalar are normally denoted by the symbol $R$ but this would impose confusion with the curvature tensor, therefore it has been chosen to assign symbol $W$ to the Ricci tensor and scalar.
+
+The **Ricci tensor** is a contraction (simplification) of the Riemann curvature tensor. It provides a way to summarize the curvature of a manifold by focusing on how volumes change in the presence of curvature. The **Ricci scalar** summarizes the curvature information contained in the **Ricci tensor**.
+
+> *Definition 2*: let $\mathbf{G}: \Gamma(\mathrm{TM}) \times \Gamma(\mathrm{TM}) \to F$ denote the **Einstein tensor** which is defined as
+>
+> $$
+>   \mathbf{G} = \mathbf{W} - \frac{1}{2} W \bm{g},
+> $$
+>
+> with $\mathbf{W}$ the Ricci tensor, $\bm{g}$ the metric tensor and $W$ the Ricci scalar.
+
+The **Einstein tensor** encapsulates the curvature of the manifold while satisfying the posed conditions (Lovelock's theorem). Such as the following proposition.
+
+> *Proposition 2*: the Einstein tensor $\mathbf{G}: \Gamma(\mathrm{TM}) \times \Gamma(\mathrm{TM}) \to F$ has the following properties
+>
+> 1. $\mathbf{G} = G_{|ij|} dx^i \vee dx^j$,
+> 2. $D_i \mathbf{G} = 0$.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+## Energy and momentum
+
+> *Definition 3*: let $\mathbf{T}: \Gamma(\mathrm{T^*M}) \times \Gamma(\mathrm{T^*M}) \to F$ denote the **energy momentum tensor** which is defined by the following properties,
+>
+> 1. $\mathbf{T} = T^{|ij|} \partial_i \vee \partial_j \in \bigvee^2(\mathrm{TM})$,
+> 2. $D_i \mathbf{T} = 0$.
+
+Property 1. is a result of the zero torsion axiom and property 2. is the demand of conservation of energy and momentum.
+
+The **energy momentum tensor** describes the matter distribution at each event in spacetime. It acts as a *source* term.
+
+## Einstein field equations
+
+> *Axiom 2*: the Einstein tensor $\mathbf{G}: \Gamma(\mathrm{TM}) \times \Gamma(\mathrm{TM}) \to F$ relates to the energy momentum tensor $\mathbf{T}: \Gamma(\mathrm{T^*M}) \times \Gamma(\mathrm{T^*M}) \to F$ by
+>
+> $$
+>   \mathbf{G} + \Lambda \bm{g} = \kappa \mathbf{T},
+> $$
+>
+> with $\kappa = \frac{8 \pi G}{c^4}$ and $\Lambda, G$ the cosmological and gravitational constants respectively.
+
+This equation (these equations) relate the geometry of spacetime to the distribution of matter within it. For a given $\mathbf{T}$ the system of equations can solve for $\bm{g}$ and vice versa.

docs/en/physics/relativistic-mechanics/schwarzschild-geometry.md

@@ -1 +1,45 @@
-# Schwarzschild geometry
+# 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.