update physics/relativistic-mechanics/relativistic-formalism: added curvature, energy momentum and field equations sections

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

@@ -69,8 +69,10 @@ 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 
 
@@ -78,4 +80,61 @@ $$
     \frac{4^2 (4^2 - 1)}{12} = 20.
 $$
 
-## 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.
+
+## 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.