140 lines
No EOL
5.5 KiB
Markdown
140 lines
No EOL
5.5 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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??? note "*Proof*:"
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Will be added later.
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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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## Curvature
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> *Definition 1*: let $\mathbf{W}: \Gamma(\mathrm{TM}) \times \Gamma(\mathrm{TM}) \to F$ denote the **Ricci tensor** which is defined as
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>
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> $$
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> \begin{align*}
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> \mathbf{W} &= \frac{1}{2} R_{ijk}^k dx^i \vee dx^j,\\
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> &= \frac{1}{2} W_{ij} dx^i \vee dx^j,
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> \end{align*}
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> $$
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>
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> 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.
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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.
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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**.
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> *Definition 2*: let $\mathbf{G}: \Gamma(\mathrm{TM}) \times \Gamma(\mathrm{TM}) \to F$ denote the **Einstein tensor** which is defined as
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>
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> $$
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> \mathbf{G} = \mathbf{W} - \frac{1}{2} W \bm{g},
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> $$
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>
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> with $\mathbf{W}$ the Ricci tensor, $\bm{g}$ the metric tensor and $W$ the Ricci scalar.
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The **Einstein tensor** encapsulates the curvature of the manifold while satisfying the posed conditions (Lovelock's theorem). Such as the following proposition.
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> *Proposition 2*: the Einstein tensor $\mathbf{G}: \Gamma(\mathrm{TM}) \times \Gamma(\mathrm{TM}) \to F$ has the following properties
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>
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> 1. $\mathbf{G} = G_{|ij|} dx^i \vee dx^j$,
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> 2. $D_i \mathbf{G} = 0$.
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??? note "*Proof*:"
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Will be added later.
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## Energy and momentum
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> *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,
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>
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> 1. $\mathbf{T} = T^{|ij|} \partial_i \vee \partial_j \in \bigvee^2(\mathrm{TM})$,
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> 2. $D_i \mathbf{T} = 0$.
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Property 1. is a result of the zero torsion axiom and property 2. is the demand of conservation of energy and momentum.
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The **energy momentum tensor** describes the matter distribution at each event in spacetime. It acts as a *source* term.
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## Einstein field equations
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> *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
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>
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> $$
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> \mathbf{G} + \Lambda \bm{g} = \kappa \mathbf{T},
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> $$
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>
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> with $\kappa = \frac{8 \pi G}{c^4}$ and $\Lambda, G$ the cosmological and gravitational constants respectively.
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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. |