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

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Luc Bijl 2025-01-05 18:20:17 +01:00
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docs/en/physics/relativistic-mechanics/relativistic-formalism.md
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@ -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.