Added first part of relativistic formalism.

README.md

@@ -1,3 +1,3 @@
-# My notes
+# Mathematics and physics wiki
 
-This is the repository containing my digitalized notes for [wiki.bijl.us](https://wiki.bijl.us).
+This is the repository containing the mathematics and physics wiki for [wiki.bijl.us](https://wiki.bijl.us). 

config/en/mkdocs.yaml

@@ -156,7 +156,7 @@ nav:
       - 'Differential manifolds': mathematics/differential-geometry/differential-manifolds.md
       - 'Tangent spaces': mathematics/differential-geometry/tangent-spaces.md
       - 'Transformations': mathematics/differential-geometry/transformations.md
-      - 'Lengths and volumes': mathematics/differential-geometry/lenghts-and-volumes.md
+      - 'Lengths and volumes': mathematics/differential-geometry/lengths-and-volumes.md
       - 'Linear connections': mathematics/differential-geometry/linear-connections.md
       - 'Derivatives': mathematics/differential-geometry/derivatives.md
       - 'Torsion': mathematics/differential-geometry/torsion.md

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

@@ -4,9 +4,78 @@ From now on, we refer to space and time as spacetime.
 
 ## Fundamental assumptions
 
-> *Postulate 1*: spacetime is continuous, homogenous and isotropic.
+> *Postulate 1*: spacetime is continuous.
 
-aaa.
+Implying that there is no fundamental limit to the precision of measurements of spatial positions, velocities and time intervals. 
 
-> *Postulate 2*: there exists a [neighbourhood]() of a point in spacetime in which the axioms of [Euclidean]() geometry hold.
+> *Postulate 2*: there exists a [neighbourhood]() in spacetime in which the axioms of [Euclidean]() geometry hold.
 
+A reformulation of the postulate in the Newtonian formalism compatible with the new formulation. 
+
+> *Postulate 3*: all physical axioms have the same form in all inertial frames.
+
+This principle is dependent on the definition of an inertial frame, which in my view is not optimal. It will have to be improved.
+
+> *Principle 1*: spacetime is not instantaneous.
+
+Implying that there exists a maximum speed with which information can travel. 
+
+> *Axiom 1*: spacetime is represented by a torsion-free pseudo Riemannian manifold $M$ with 3 spacial dimensions and 1 time dimension.
+
+Torsion-free means that $\mathbf{T} = \mathbf{0}$, the [torsion tensor]() is always zero.
+
+## Lorentz transformations
+
+Will be added later.
+
+## Results from the fundamental assumptions
+
+> *Theorem 1*: let $\bm{g} \in \Gamma(\mathrm{TM})$ be the pseudo Riemannian inner product on $\mathrm{TM}$, then it follows that from [Hamilton's principle]() that the covariant derivative is equal to zero:
+>
+> $$
+>   \forall i \in \{1, 2, 3, 4\}: D_i \bm{g} = \mathbf{0},
+> $$
+>
+> which is called *metric compatibility*.
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+A linear connection $\nabla$ on a torsion-free pseudo Riemannian manifold with metric compatibility is called the **Levi-Civita connection** with its linear connection symbols denoted as the **Christoffel symbols**.
+
+> *Theorem 2*: the Christoffel symbols $\Gamma_{ij}^k$ (of a Levi-Civita connection) are covariantly symmetric 
+>
+> $$
+>   \Gamma_{ij}^k = \Gamma_{ji}^k,
+> $$
+>
+> for all $(i,j,k) \in \{1,2,3,4\}^3$, and may be given by
+>
+> $$
+>   \Gamma_{ij}^k = \frac{1}{2} g^{kl} (\partial_i g_{ij} + \partial_j g_{il} - \partial_l g_{ij}),
+> $$
+>
+> for all $\bm{g} = g_{ij} dx^i \otimes dx^j \in \Gamma(\mathrm{TM})$. 
+
+??? note "*Proof*:"
+
+    Will be added later.
+
+Similarly, we have the following.
+
+> *Proposition 1*: let $\mathbf{R}: \Gamma(\mathrm{T^*M}) \times \Gamma(\mathrm{TM})^3 \to F$ be the Riemann curvature tensor on a manifold $M$ over a field $F$, defined under the Levi-Civita connection. Then it may be decomposed by
+>
+> $$
+>   \mathbf{R} = \frac{1}{8} R^i_{jkl} (\partial_i \wedge dx^j) \vee (dx^k \wedge dx^l).
+> $$
+>
+> such that $R^i_{jkl}$ has a dimension of 
+ 
+Such that $R^i_{jkl}$ has a dimension of 
+
+$$
+    \frac{4^2 (4^2 - 1)}{12} = 20.
+$$
+
+## Axioms of Einstein