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