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First-order curvature of the flat linearization and of TT plane waves, its gauge invariance and its contraction to the linearized Ricci tensor

Statement

Let $(M, g, N) = (R^{4}, η, \partial_{0})$ be the flat model with its inertial chart and let $(ε_{1}, h, q)$ be a flat linearized family over an open set $Ω \subseteq R^{4}$ with first-order coefficients $Γ^{(1) a}_{b c}$ as in Flat linearized perturbations: families, trace reversal, gauge transformation and harmonic gauge; write $R^{a}_{b c d} (λ)$ for the curvature components of $g (λ)$ (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors, Tensoriality of the curvature, its coordinate formula and its point-normal metric formula) and define the first-order curvature coefficient

$R^{(1) a}_{b c d} : = \partial_{λ} R^{a}_{b c d} (λ) ∣_{λ = 0}, R_{a b c d}^{(1)} : = η_{a e} R^{(1) e}_{b c d} .$

Then:

(a) General first-order formula. The coefficient is defined (the component family is differentiable in $λ$ at $0$ ) and

$R^{(1) a}_{b c d} = \partial_{c} Γ^{(1) a}_{d b} - \partial_{d} Γ^{(1) a}_{c b}, R_{a b c d}^{(1)} = \frac{1}{2} (\partial_{b} \partial_{c} h_{a d} + \partial_{a} \partial_{d} h_{b c} - \partial_{a} \partial_{c} h_{b d} - \partial_{b} \partial_{d} h_{a c}) .$

(b) Gauge invariance. For every smooth covector field $ξ$ on $Ω$ , with $h_{a b}^{'} : = h_{a b} - \partial_{a} ξ_{b} - \partial_{b} ξ_{a}$ the gauge-transformed perturbation of Flat linearized perturbations: families, trace reversal, gauge transformation and harmonic gauge clause (d), $R_{a b c d}^{(1)} (h^{'}) = R_{a b c d}^{(1)} (h)$ : the first-order curvature coefficient is unchanged by gauge transformations.

(c) Planar and TT waves. Let $F = (F_{a b})$ be a smooth planar profile with $h_{a b} (x) = F_{a b} (u (x))$ , $u = x^{0} - x^{3}$ , and let $n_{a} = \partial_{a} u$ ; then

$R_{a b c d}^{(1)} = \frac{1}{2} (n_{b} n_{c} F_{a d}^{''} + n_{a} n_{d} F_{b c}^{''} - n_{a} n_{c} F_{b d}^{''} - n_{b} n_{d} F_{a c}^{''}) .$

If the profile is spatial ( $F_{0 a} = 0$ for all $a$ ), then $R_{0 i 0 j}^{(1)} = - \frac{1}{2} F_{i j}^{''}$ and, raising indices, $R^{(1) i}_{00 j} = + \frac{1}{2} F^{''}^{i}_{j}$ . If the profile is TT with amplitudes $A_{+}, A_{\times}$ , then $F^{''}$ is supported on the screen with $F_{11}^{''} = A_{+}^{''}$ , $F_{22}^{''} = - A_{+}^{''}$ , $F_{12}^{''} = F_{21}^{''} = A_{\times}^{''}$ and all other second derivatives of the profile zero; in particular

$R_{0101}^{(1)} = - \frac{1}{2} A_{+}^{''}, R_{0102}^{(1)} = - \frac{1}{2} A_{\times}^{''}, R_{0202}^{(1)} = + \frac{1}{2} A_{+}^{''} .$

(d) Contraction to the Ricci coefficient. For every perturbation $h$ ,

$η^{a c} R_{a b c d}^{(1)} = {Ric}_{b d}^{(1)},$

the right-hand side being the first-order coefficient of Flat linearized perturbations: families, trace reversal, gauge transformation and harmonic gauge clause (b). For a TT planar perturbation both sides vanish: ${Ric}_{b d}^{(1)} = 0$ and $S^{(1)} = 0$ .

Facts & Assumptions

Given: A flat linearized family $(ε_{1}, h, q)$ over $Ω$ , a smooth covector field $ξ$ , and a smooth planar profile $F$ .

[L1]

The curvature components of the metric $g (λ)$ are given by $R^{a}_{b c d} = \partial_{c} Γ^{a}_{d b} - \partial_{d} Γ^{a}_{c b} + \sum_{e} (Γ^{a}_{c e} Γ^{e}_{d b} - Γ^{a}_{d e} Γ^{e}_{c b})$ , the curvature is a $(1, 3)$ -tensor, and ${Ric}_{b d} = g^{a e} R_{e b a d}$ , $S = g^{b d} {Ric}_{b d}$ , $G_{a b} = {Ric}_{a b} - \frac{1}{2} S g_{a b}$ . Since the background curvature vanishes at $λ = 0$ , differentiation gives ${Ric}_{b d}^{(1)} = η^{a e} R_{e b a d}^{(1)}$ , $S^{(1)} = η^{b d} {Ric}_{b d}^{(1)}$ and $G_{a b}^{(1)} = {Ric}_{a b}^{(1)} - \frac{1}{2} S^{(1)} η_{a b}$ (Tensoriality of the curvature, its coordinate formula and its point-normal metric formula, The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors, Flat linearized perturbations: families, trace reversal, gauge transformation and harmonic gauge clauses (a)-(c), The inverse metric, musical isomorphisms and index raising).

[L2]

The first-order Christoffel coefficients are $Γ^{(1) a}_{b c} = \frac{1}{2} η^{a d} (\partial_{b} h_{c d} + \partial_{c} h_{b d} - \partial_{d} h_{b c})$ , the component family $λ \mapsto Γ^{a}_{b c} (λ, p)$ is smooth with second derivative bounded on compact parameter sets, and $Γ^{a}_{b c} (0) = 0$ (First-order curvature of a flat linearized family, gauge transformations and the quadratic remainder clauses (a), (e), Flat linearized perturbations: families, trace reversal, gauge transformation and harmonic gauge clause (b)).

[L3]

Mixed partial derivatives of a $C^{2}$ function commute and derivatives of products and sums obey the usual rules (Clairaut--Schwarz theorem for continuous second partial derivatives, Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ , The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, $C^{r}$ and smooth maps between smooth manifolds).

[L4]

For a planar profile, $\partial_{c} h_{a b} = F_{a b}^{'} (u) n_{c}$ and $\partial_{d} \partial_{c} h_{a b} = F_{a b}^{''} (u) n_{d} n_{c}$ , and $□ h_{a b} = 0$ ; for a TT profile the trace, the divergence and hence the contraction $\partial_{a} \partial_{b} h^{a b}$ vanish (The phase coordinate u = x^0 - x^3 is null and profile functions are flat-harmonic and constant along its transport curves clause (c)).

[L6]

The map $h \mapsto R_{a b c d}^{(1)} (h)$ given by the formula of (a) is $R$ -linear in $h$ and its derivatives, and ${Ric}_{b d}^{(1)} = \frac{1}{2} (\partial_{c} \partial_{b} h^{c}_{d} + \partial_{c} \partial_{d} h^{c}_{b} - □ h_{b d} - \partial_{b} \partial_{d} h)$ (First-order curvature of a flat linearized family, gauge transformations and the quadratic remainder clause (b), Flat linearized perturbations: families, trace reversal, gauge transformation and harmonic gauge clause (b)).

Proof

technique · direct; differentiate the coordinate curvature formula at $\lambda=0$, substitute the first-order Christoffel symbols, then cancel the pure-gauge terms and specialize to planar profiles

1.1L1L2L3algebra

(Differentiating the curvature formula.) By [L2] the Christoffel component families are smooth in $(λ, p)$ with $Γ (0) = 0$ . Differentiating the coordinate formula of [L1] in $λ$ at $0$ , the product terms $Γ Γ$ contribute $2 Γ (0) \partial_{λ} Γ (0) = 0$ , so $R^{(1) a}_{b c d} = \partial_{c} Γ^{(1) a}_{d b} - \partial_{d} Γ^{(1) a}_{c b}$ ; this also shows that the coefficient is defined as the derivative of a differentiable component family. Substituting the formula of [L2] into $η_{a e} R^{(1) e}_{b c d}$ and using $η_{a e} Γ^{(1) e}_{d b} = Γ_{a d b}^{(1)} = \frac{1}{2} (\partial_{d} h_{a b} + \partial_{b} h_{a d} - \partial_{a} h_{d b})$ gives $R_{a b c d}^{(1)} = \frac{1}{2} \partial_{c} (\partial_{d} h_{a b} + \partial_{b} h_{a d} - \partial_{a} h_{d b}) - \frac{1}{2} \partial_{d} (\partial_{c} h_{a b} + \partial_{b} h_{a c} - \partial_{a} h_{c b}),$ and the two terms $+ \frac{1}{2} \partial_{c} \partial_{d} h_{a b}$ and $- \frac{1}{2} \partial_{d} \partial_{c} h_{a b}$ cancel by [L3], leaving the displayed formula. This proves (a).

2.1L3L6step 1.1algebra

(Gauge invariance.) Let $H_{a b} : = \partial_{a} ξ_{b} + \partial_{b} ξ_{a}$ , so that $h^{'} = h - H$ and $R^{(1)} (h^{'}) = R^{(1)} (h) - R^{(1)} (H)$ by [L6]. By the formula of step 1.1, $R_{a b c d}^{(1)} (H) = \frac{1}{2} (\partial_{b} \partial_{c} H_{a d} + \partial_{a} \partial_{d} H_{b c} - \partial_{a} \partial_{c} H_{b d} - \partial_{b} \partial_{d} H_{a c}) .$ Substituting $H_{a d} = \partial_{a} ξ_{d} + \partial_{d} ξ_{a}$ and expanding, the eight third-derivative terms group as $(\partial_{b} \partial_{c} \partial_{a} ξ_{d} - \partial_{a} \partial_{c} \partial_{b} ξ_{d}) + (\partial_{b} \partial_{c} \partial_{d} ξ_{a} - \partial_{b} \partial_{d} \partial_{c} ξ_{a}) + (\partial_{a} \partial_{d} \partial_{b} ξ_{c} - \partial_{b} \partial_{d} \partial_{a} ξ_{c}) + (\partial_{a} \partial_{d} \partial_{c} ξ_{b} - \partial_{a} \partial_{c} \partial_{d} ξ_{b}),$ and each parenthesis vanishes by [L3]. Hence $R^{(1)} (H) = 0$ and $R^{(1)} (h^{'}) = R^{(1)} (h)$ . This proves (b).

2.2L4step 1.1algebra

(Planar waves.) Let $h_{a b} = F_{a b} (u)$ ; by [L4], $\partial_{b} \partial_{c} h_{a d} = F_{a d}^{''} n_{b} n_{c}$ , and substituting into the formula of step 1.1 gives the displayed planar formula. If $F_{0 a} = 0$ , then substituting $n_{0} = 1$ , $n_{i} = 0$ for $i = 1, 2$ and $n_{3} = - 1$ gives $R_{0 i 0 j}^{(1)} = \frac{1}{2} (n_{i} n_{0} F_{0 j}^{''} + n_{0} n_{j} F_{i 0}^{''} - n_{0} n_{0} F_{i j}^{''} - n_{i} n_{j} F_{00}^{''}) = - \frac{1}{2} F_{i j}^{''}$ . For the raised component, the four-term formula of step 1.1 is antisymmetric in the first and in the last index pair, so $R_{a 00 j}^{(1)} = - R_{a 0 j 0}^{(1)} = R_{0 a j 0}^{(1)} = - R_{0 a 0 j}^{(1)} = - (- \frac{1}{2} F_{a j}^{''}) = \frac{1}{2} F_{a j}^{''}$ ; raising the first index gives $R^{(1) i}_{00 j} = η^{i a} \frac{1}{2} F_{a j}^{''} = + \frac{1}{2} F^{''}^{i}_{j}$ . If in addition the profile is TT, then $F_{i j}^{''}$ vanishes whenever $i = j = 3$ or one index equals $3$ , and $F_{11}^{''} = - F_{22}^{''} = A_{+}^{''}$ , $F_{12}^{''} = F_{21}^{''} = A_{\times}^{''}$ ; the displayed components follow. This proves (c).

2.3L1L4L6step 1.1algebra

(Ricci contraction.) Contracting the formula of step 1.1 with $η^{a c}$ term by term gives $η^{a c} R_{a b c d}^{(1)} = \frac{1}{2} (\partial_{b} \partial_{c} h^{c}_{d} + \partial_{a} \partial_{d} h^{a}_{b} - □ h_{b d} - \partial_{b} \partial_{d} h),$ which is the formula of [L6] for ${Ric}_{b d}^{(1)}$ (the first two terms are the same contraction with renamed dummy indices). For a TT planar perturbation, [L4] gives zero trace, zero divergence and $□ h_{b d} = 0$ , so every term in the Ricci formula of [L6] vanishes, independently of boundedness of the profiles. Thus ${Ric}_{b d}^{(1)} = 0$ and its contraction $S^{(1)} = 0$ by [L1]. This proves (d).

3.1step 1.1step 2.1step 2.2step 2.3∎

(Conclusion.) Steps 1.1, 2.1, 2.2 and 2.3 prove the four clauses from the coordinate curvature formula, the first-order Christoffel symbols and the commutation of partial derivatives. No field equation or profile boundedness is assumed: the vanishing statement of (d) follows directly from the null-phase and TT identities; no choice principle beyond the inherited countable choice of the cited suppliers is used.

Remarks

Physical reading. Clauses (a)-(c) identify a gauge-invariant curvature content of a linearized perturbation and compute it for plane waves; clause (d) verifies that the contraction reproduces the independently defined Ricci coefficient. The physical interpretation of $R^{(1)}$ as tidal response requires the free-fall postulate and is given in the response theorem of this page.
Sign conventions. The displayed signs are those of the curvature convention $R^{a}_{b c d} = \partial_{c} Γ^{a}_{d b} - \partial_{d} Γ^{a}_{c b} + Γ Γ$ fixed by Tensoriality of the curvature, its coordinate formula and its point-normal metric formula and The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors; changing the overall curvature sign would reverse the signs in clause (c).

First-order curvature of the flat linearization and of TT plane waves, its gauge invariance and its contraction to the linearized Ricci tensor

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources