From 88930aca6ec667b749f036bb81e4b30e180aa607 Mon Sep 17 00:00:00 2001 From: Luc Date: Mon, 19 Aug 2024 22:04:05 +0200 Subject: [PATCH] Added first part of relativistic formalism. --- README.md | 4 +- config/en/mkdocs.yaml | 2 +- .../relativistic-formalism.md | 75 ++++++++++++++++++- 3 files changed, 75 insertions(+), 6 deletions(-) diff --git a/README.md b/README.md index 2313ca2..81b654b 100644 --- a/README.md +++ b/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). \ No newline at end of file +This is the repository containing the mathematics and physics wiki for [wiki.bijl.us](https://wiki.bijl.us). \ No newline at end of file diff --git a/config/en/mkdocs.yaml b/config/en/mkdocs.yaml index ef58daf..ca7f835 100755 --- a/config/en/mkdocs.yaml +++ b/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 diff --git a/docs/en/physics/relativistic-mechanics/relativistic-formalism.md b/docs/en/physics/relativistic-mechanics/relativistic-formalism.md index 63fee34..c7f8483 100644 --- a/docs/en/physics/relativistic-mechanics/relativistic-formalism.md +++ b/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 \ No newline at end of file