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Constant coupled electrovac rescaling

Statement

Let $g, F, j, Λ, c, G, μ_{0}, κ$ be as in SI and geometrized Einstein–Maxwell parameters and the four-dimensional Lorentzian curvature convention of The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors. Suppose they satisfy the formal sourced Einstein–Maxwell equations

$G_{a b} (g) + Λ g_{a b} = \frac{κ}{μ_{0}} Q_{a b} (g, F), d F = 0, d (⋆_{g} F) = μ_{0} ι_{j} {vol}_{g},$

where $Q$ is the quadratic tensor of Exact geometric electrovac normalization. For a positive dimensionless constant $λ$ , define

$g^{'} : = λ^{2} g, F^{'} : = λ F, Λ^{'} : = λ^{- 2} Λ, j^{'} : = λ^{- 3} j .$

Then $(g^{'}, F^{'}, j^{'}, Λ^{'})$ satisfies the same equations with the same numerical constants $c, G, μ_{0}$ . In particular, a source-free electrovac solution ( $j = 0$ ) maps to a source-free electrovac solution. No transformation law for an additional matter stress tensor or matter action is asserted.

Facts & Assumptions

Given: The smooth Lorentzian geometry, parameter dictionary and quadratic tensor of SI and geometrized Einstein–Maxwell parameters, Exact geometric electrovac normalization and The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors. The inherited geometric setup carries ${A C}_{ω}$ as in [A1].

[F1]

For $F = α F$ , the Einstein equation above is equivalent to $G_{a b} + Λ g_{a b} = 2 Q_{a b} (g, F)$ . (Exact geometric electrovac normalization)

[F2]

Under a smooth positive conformal factor $Ω$ , the Maxwell pair $(F, j)$ maps to $(F, Ω^{- 4} j)$ for $\tilde{g} = Ω^{2} g$ . (Four-dimensional conformal Maxwell covariance)

[F3]

The metric-selected connection has the Christoffel formula of Metric compatibility, torsion freedom and the metric-selected connection class, and curvature, Ricci, scalar and Einstein tensors have the definitions of The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors.

[A1]

The inherited choice assumption is Countable Choice ${A C}_{ω}$ , not full Choice. (The Axiom of Countable Choice ( ${A C}_{ω}$ ))

Proof

1.1givenF3A1algebra

Since $λ$ is constant, $(g^{'})^{a b} = λ^{- 2} g^{a b}$ and $\partial_{c} g_{a b}^{'} = λ^{2} \partial_{c} g_{a b}$ . Substitution in the Christoffel formula [F3] gives $Γ^{'} [g^{'}]^{a}_{b c} = Γ [g]^{a}_{b c}$ ; hence the two curvature operators and their lower-index Ricci tensors agree by [F3]. Tracing with $(g^{'})^{- 1}$ gives $S [g^{'}] = λ^{- 2} S [g]$ , so $G_{a b} (g^{'}) = G_{a b} (g)$ .

1.2givenF1algebra

In the definition of $Q$ , $F_{a c}^{'} = λ F_{a c}$ while $F_{b}^{'}^{c} = (g^{'})^{c d} F_{b d}^{'} = λ^{- 1} F_{b}^{c}$ , so the first quadratic term is unchanged. Also $F_{c d}^{'} F^{' c d} = λ^{- 2} F_{c d} F^{c d}$ and $g_{a b}^{'} = λ^{2} g_{a b}$ , so the trace term is unchanged. Therefore $Q_{a b} (g^{'}, F^{'}) = Q_{a b} (g, F)$ .

2.1F1step 1.1step 1.2algebra

We have $Λ^{'} g_{a b}^{'} = λ^{- 2} Λ λ^{2} g_{a b} = Λ g_{a b}$ . Together with steps 1.1 and 1.2, both sides of the Einstein equation are unchanged. Equivalently, $F^{'} = α F^{'} = λ F$ and the normalized equation in [F1] has the same quadratic tensor under $(g, F) \mapsto (g^{'}, F^{'})$ .

3.1F2algebra∎

Apply [F2] with the constant conformal factor $Ω = λ$ : $(g, F, j)$ maps to $(g^{'}, F, λ^{- 4} j)$ . The differential-form Maxwell equations are linear in the pair $(F, j)$ , so multiplying both entries of this solution by $λ$ gives $(g^{'}, F^{'}, j^{'}) = (g^{'}, λ F, λ^{- 3} j)$ , which also solves them. This proves the sourced Maxwell transformation; when $j = 0$ , the transformed source remains zero.

Remarks

The scaling is an exact identity for the displayed equations. Because lengths and the cosmological parameter are rescaled, it does not say that a fixed apparatus or fixed dimensionful matter data are unchanged. A matter model must supply its own transformation law before any coupled matter invariance can be claimed.

Constant coupled electrovac rescaling

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources