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Conformal Einstein tensor transformation

Statement

Let $(M, g, N, o)$ be the smooth oriented, time-oriented Lorentzian four-manifold of Oriented Lorentzian fields, observers and coordinate conventions, with the curvature convention and tensors of The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors. Let $Ω : M \to (0, \infty)$ be smooth and dimensionless, and put $\tilde{g} = Ω^{2} g$ with the same orientation. Use the metric-selected connection and its tensor derivative of Metric compatibility, torsion freedom and the metric-selected connection class and Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields. Define the smooth one-form and its contractions by

$u : = Ω^{- 1} d Ω, H_{a b} : = (\nabla_{a} u)_{b}, θ : = g^{a b} H_{a b}, s : = g^{a b} u_{a} u_{b} .$

Then $H_{a b} = H_{b a}$ and

${\overset{}{Ric}}_{a b} = {Ric}_{a b} - 2 H_{a b} - g_{a b} θ + 2 u_{a} u_{b} - 2 g_{a b} s,$

$\tilde{S} = Ω^{- 2} (S - 6 θ - 6 s),$

${\tilde{G}}_{a b} = G_{a b} - 2 H_{a b} + 2 u_{a} u_{b} + 2 g_{a b} θ + g_{a b} s .$

All curvature conventions are those stated in The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors. These are mathematical transformation identities, not an assertion that the Einstein equation is conformally invariant.

Facts & Assumptions

Given: A smooth oriented Lorentzian four-manifold, smooth positive $Ω$ , and the metric-selected connections of $g$ and $\tilde{g}$ . The inherited geometric setup carries ${A C}_{ω}$ as in [A1].

[F1]

The geometry is smooth, oriented and time-oriented, with nondegenerate Lorentz metric of signature $(-, +, +, +)$ ; its smooth real forms and metric volume form have the definitions of Oriented Lorentzian fields, observers and coordinate conventions.

[F2]

Curvature is $R (X, Y) Z = \nabla_{X} \nabla_{Y} Z - \nabla_{Y} \nabla_{X} Z - \nabla_{[X, Y]} Z$ , with $R^{a}_{b c d}$ , ${Ric}_{b d} = g^{a c} R_{a b c d}$ , $S = g^{b d} {Ric}_{b d}$ , and $G_{a b} = {Ric}_{a b} - \frac{1}{2} S g_{a b}$ . (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors)

[F3]

For either smooth Lorentz metric $h = g$ or $h = \tilde{g}$ , the metric-selected connection is affine and torsion-free and has Christoffel formula $Γ [h]^{a}_{b c} = \frac{1}{2} h^{a d} (\partial_{b} h_{d c} + \partial_{c} h_{b d} - \partial_{d} h_{b c})$ . (Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields)

[F4]

For a type $(1, 2)$ tensor $C$ , the tensor derivative has components $(\nabla_{\partial_{c}} C)^{a}_{d b} = \partial_{c} C^{a}_{d b} + Γ^{a}_{e c} C^{e}_{d b} - Γ^{e}_{d c} C^{a}_{e b} - Γ^{e}_{b c} C^{a}_{d e}$ ; it commutes with contractions and annihilates $g$ and $g^{- 1}$ . (Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields)

[F5]

The exterior derivative sends a smooth $k$ -form to a smooth $(k + 1)$ -form, has coordinate formula $(d β)_{a_{0} \dots a_{k}} = (k + 1) \partial_{[a_{0}} β_{a_{1} \dots a_{k}]}$ , and obeys $d^{2} = 0$ and the graded product rule. (Lorentzian exterior calculus and Hodge star)

[F6]

For $x > 0$ and $α \in R$ , $(x^{α})^{'} = α x^{α - 1}$ ; the chain rule holds for total derivatives. (Continuity and derivatives of positive-base real powers, The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ )

[F7]

For each finite $k$ , finite componentwise sums and products of $C^{k}$ Euclidean maps and compositions of composable $C^{k}$ Euclidean maps are $C^{k}$ . Applying this at every finite $k$ in smooth manifold charts shows that finite sums, products and compositions of smooth maps are smooth. ( $C^{k}$ Euclidean maps are closed under componentwise algebra and composition)

[F8]

For either metric-selected connection, with the curvature convention of [F2], the coordinate curvature formula is $R^{a}_{b c d} = \partial_{c} Γ^{a}_{d b} - \partial_{d} Γ^{a}_{c b} + Γ^{a}_{c e} Γ^{e}_{d b} - Γ^{a}_{d e} Γ^{e}_{c b}$ , with summation over $e$ . (Tensoriality of the curvature, its coordinate formula and its point-normal metric formula)

[A1]

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

Proof

1.1givenF1F3F5F6F7A1algebra

In a chart, $\partial_{a} {\tilde{g}}_{b c} = Ω^{2} (\partial_{a} g_{b c} + 2 u_{a} g_{b c})$ and ${\tilde{g}}^{a d} = Ω^{- 2} g^{a d}$ . Substituting these into [F3] cancels the common $Ω^{2}$ and gives ${\tilde{Γ}}^{a}_{b c} - Γ^{a}_{b c} = C^{a}_{b c} : = δ_{b}^{a} u_{c} + δ_{c}^{a} u_{b} - g_{b c} u^{a}$ , where $u^{a} = g^{a b} u_{b}$ . Iterating the derivative formula in [F6] gives $D^{m} (r^{- 1}) = (- 1)^{m} m! r^{- m - 1}$ for every integer $m \geq 0$ and $r > 0$ ; all these derivatives are continuous, so $r \mapsto r^{- 1}$ is smooth on $(0, \infty)$ . Its composition with $Ω$ is smooth by [F7], and $d Ω$ is a smooth one-form by [F5], so $u$ and the displayed components $C^{a}_{b c}$ are smooth. Finally, define $A (X, Y) : = {\tilde{\nabla}}_{X} Y - \nabla_{X} Y$ for smooth local vector fields. The affine-connection rules in [F3] give $A (f X, Y) = f A (X, Y)$ and $A (X, f Y) = f A (X, Y)$ for every smooth function $f$ , while its components are the smooth differences $C^{a}_{b c}$ ; hence $A$ is a smooth tensor of type $(1, 2)$ with those components.

2.1F3F5step 1.1algebra

By [F5], $0 = d (1) = d (Ω^{- 1} Ω) = Ω^{- 1} d Ω + Ω d (Ω^{- 1})$ , so $d (Ω^{- 1}) = - Ω^{- 2} d Ω$ . Therefore $d u = d (Ω^{- 1}) \land d Ω + Ω^{- 1} d^{2} Ω = - Ω^{- 2} d Ω \land d Ω = 0$ . Torsion freedom in [F3] gives $(d u)_{a b} = \nabla_{a} u_{b} - \nabla_{b} u_{a}$ , hence $H_{a b} = H_{b a}$ .

2.2F2F4F8step 1.1algebra

Apply the coordinate curvature formula [F8] to $g$ and $\tilde{g}$ and subtract, using $\tilde{Γ} = Γ + C$ from step 1.1. Since $C$ is a tensor, [F4] gives $(\nabla_{c} C)^{a}_{d b} - (\nabla_{d} C)^{a}_{c b} = \partial_{c} C^{a}_{d b} - \partial_{d} C^{a}_{c b} + Γ^{a}_{c e} C^{e}_{d b} - Γ^{a}_{d e} C^{e}_{c b} + Γ^{e}_{d b} C^{a}_{c e} - Γ^{e}_{c b} C^{a}_{d e}$ ; the two covariant-derivative terms acting on the lower curvature-direction slots $d, c$ cancel because $Γ^{e}_{c d} = Γ^{e}_{d c}$ , while the lower $b$ -slot contributions are the last two displayed terms. These are exactly the partial-derivative and mixed $Γ C$ terms in the subtraction, and the remaining terms are $C^{a}_{c e} C^{e}_{d b} - C^{a}_{d e} C^{e}_{c b}$ . Therefore ${\tilde{R}}^{a}_{b c d} - R^{a}_{b c d} = \nabla_{c} C^{a}_{d b} - \nabla_{d} C^{a}_{c b} + C^{a}_{c e} C^{e}_{d b} - C^{a}_{d e} C^{e}_{c b}$ .

3.1F2F4step 2.1step 2.2algebra

Contracting $a$ with $c$ as in [F2], the derivative terms are $\nabla_{a} C^{a}_{d b} - \nabla_{d} C^{a}_{a b}$ . The definition of $C$ gives $C^{a}_{a b} = 4 u_{b}$ and, using $\nabla g = 0$ from [F4], $\nabla_{a} C^{a}_{d b} = H_{d b} + H_{b d} - g_{d b} θ = 2 H_{b d} - g_{d b} θ$ by step 2.1. Thus their difference is $- 2 H_{b d} - g_{b d} θ$ .

4.1step 1.1step 2.2step 3.1algebra

The first quadratic contraction is $C^{a}_{a e} C^{e}_{d b} = 4 u_{e} (δ_{d}^{e} u_{b} + δ_{b}^{e} u_{d} - g_{d b} u^{e}) = 8 u_{b} u_{d} - 4 g_{b d} s$ . Expanding the second contraction into its nine terms gives $C^{a}_{d e} C^{e}_{a b} = u_{d} u_{b} + u_{d} u_{b} - g_{d b} s + 4 u_{d} u_{b} + u_{d} u_{b} - u_{d} u_{b} - u_{d} u_{b} - g_{d b} s + u_{d} u_{b} = 6 u_{b} u_{d} - 2 g_{b d} s$ . Their difference is $2 u_{b} u_{d} - 2 g_{b d} s$ . Combining this with step 3.1 proves the Ricci transformation in the Statement.

5.1F2step 4.1algebra

Contracting the Ricci transformation with ${\tilde{g}}^{a b} = Ω^{- 2} g^{a b}$ gives $\tilde{S} = Ω^{- 2} (S - 2 θ - 4 θ + 2 s - 8 s) = Ω^{- 2} (S - 6 θ - 6 s)$ .

6.1F2step 4.1step 5.1algebra∎

By definition, ${\tilde{G}}_{a b} = {\overset{}{Ric}}_{a b} - \frac{1}{2} \tilde{S} {\tilde{g}}_{a b}$ . Substituting the Ricci formula from step 4.1 and $\tilde{S} {\tilde{g}}_{a b} = (S - 6 θ - 6 s) g_{a b}$ from step 5.1 cancels the scalar-curvature terms and leaves $G_{a b} - 2 H_{a b} + 2 u_{a} u_{b} + 2 g_{a b} θ + g_{a b} s$ . This is the claimed Einstein-tensor formula.

Remarks

The curvature sign and Ricci contraction were taken from the inspected Lorentzian supplier The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors. The conformal identities above were derived with that convention; no Riemannian curvature formula or sign convention is imported.

Conformal Einstein tensor transformation

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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