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Regauging an explicit harmonic-gauge plane-wave amplitude to transverse-traceless form

Example

Work in the inertial chart of the flat model with the wave covector $k^{a} = ω (1, 0, 0, 1)$ , $ω = 1$ , and let $X$ be the space of symmetric amplitudes of Plane-wave perturbations of the flat linearization: planar profiles, wave covectors, amplitudes, TT form and the linearized field equations clause (c). Consider the amplitude

$X = (\begin{array}{cccc} - 2 & - 2 & 0 & 0 \\ - 2 & 1 & 0 & 2 \\ 0 & 0 & - 1 & 0 \\ 0 & 2 & 0 & 2 \end{array}),$

whose components in the orthonormal basis $(e_{0}, e_{1}, e_{2}, e_{3})$ are displayed in the order $(0, 1, 2, 3)$ . Then:

$X$ lies in $V_{0} = {X : k^{a} {\overset{ˉ}{X}}_{a b} = 0}$ and is not TT: $X (e_{0}, e_{0}) = - 2 \neq 0$ and its trace is $tr X = - X_{00} + X_{11} + X_{22} + X_{33} = 2 + 1 - 1 + 2 = 4 \neq 0$ .
The unique $Z$ of Harmonic-gauge plane-wave amplitudes split uniquely into a pure plane-wave gauge part and a transverse-traceless representative, with a two-dimensional space of polarizations clause (c) with $X_{a b}^{'} : = X_{a b} - k_{a} Z_{b} - k_{b} Z_{a}$ TT is $Z = (1, 2, 0, 1)$ , and $X^{'} = (\begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}),$ so the amplitudes of the TT representative are $A_{+} = 1$ , $A_{\times} = 0$ .
With $ξ_{b} = Z_{b} \sin (k \cdot x)$ the perturbation $h_{a b} = X_{a b} \cos (k \cdot x)$ satisfies $h_{a b}^{'} : = h_{a b} - \partial_{a} ξ_{b} - \partial_{b} ξ_{a} = X_{a b}^{'} \cos (k \cdot x)$ , which is a TT plane wave in harmonic gauge.

Verification

Given: The inertial chart, $k^{a} = (1, 0, 0, 1)$ , $k_{a} = (- 1, 0, 0, 1)$ , the amplitude $X$ above and the vector $Z = (1, 2, 0, 1)$ .

[L1] A symmetric amplitude $X$ is in $V_{0}$ exactly when $k^{a} {\overset{ˉ}{X}}_{a b} = 0$ for every $b$ . For $X \in V_{0}$ and $ω \neq 0$ in the adapted frame, its unique TT representative is $X_{a b}^{'} = X_{a b} - k_{a} Z_{b} - k_{b} Z_{a}$ , where $Z_{0} = - X_{00} / (2 ω)$ , $Z_{1} = - X_{01} / ω$ , $Z_{2} = - X_{02} / ω$ , $Z_{3} = Z_{0} - X_{03} / ω$ (Harmonic-gauge plane-wave amplitudes split uniquely into a pure plane-wave gauge part and a transverse-traceless representative, with a two-dimensional space of polarizations clauses (b)-(c)).

[L2] For $h_{a b} = X_{a b} \cos (k \cdot x)$ and $ξ_{b} = Z_{b} \sin (k \cdot x)$ one has $\partial_{a} ξ_{b} = k_{a} Z_{b} \cos (k \cdot x)$ and $h^{'} = X^{'} \cos (k \cdot x)$ with $X^{'} = X - k Z - k Z$ (Harmonic-gauge plane-wave amplitudes split uniquely into a pure plane-wave gauge part and a transverse-traceless representative, with a two-dimensional space of polarizations clause (d), The derivatives of sine and cosine are cosine and minus sine, The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , 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, Plane-wave perturbations of the flat linearization: planar profiles, wave covectors, amplitudes, TT form and the linearized field equations clauses (c)-(d)).

1.1L1algebra

(Membership in $V_{0}$ and failure of TT.) The trace is $tr X = η^{a b} X_{a b} = - X_{00} + X_{11} + X_{22} + X_{33} = 2 + 1 - 1 + 2 = 4$ , so ${\overset{ˉ}{X}}_{a b} = X_{a b} - 2 η_{a b}$ has entries ${\overset{ˉ}{X}}_{00} = - 2 + 2 = 0$ , ${\overset{ˉ}{X}}_{01} = - 2$ , ${\overset{ˉ}{X}}_{03} = 0$ , ${\overset{ˉ}{X}}_{11} = 1 - 2 = - 1$ , ${\overset{ˉ}{X}}_{13} = 2$ , ${\overset{ˉ}{X}}_{22} = - 1 - 2 = - 3$ , ${\overset{ˉ}{X}}_{33} = 2 - 2 = 0$ and ${\overset{ˉ}{X}}_{12} = {\overset{ˉ}{X}}_{23} = 0$ . With $k^{0} = 1$ , $k^{3} = 1$ , $k^{1} = k^{2} = 0$ , the four contractions are $k^{a} {\overset{ˉ}{X}}_{a 0} = {\overset{ˉ}{X}}_{00} + {\overset{ˉ}{X}}_{30} = 0$ , $k^{a} {\overset{ˉ}{X}}_{a 1} = {\overset{ˉ}{X}}_{01} + {\overset{ˉ}{X}}_{31} = - 2 + 2 = 0$ , $k^{a} {\overset{ˉ}{X}}_{a 2} = 0$ , $k^{a} {\overset{ˉ}{X}}_{a 3} = {\overset{ˉ}{X}}_{03} + {\overset{ˉ}{X}}_{33} = 0$ ; hence $X \in V_{0}$ . It is not TT because $X (e_{0}, e_{0}) = X_{00} = - 2 \neq 0$ and $tr X = 4 \neq 0$ . This is clause 1.

2.1L1step 1.1algebra

(The TT representative.) Since $X \in V_{0}$ by step 1.1, [L1] with $ω = 1$ gives the unique $Z$ as $Z_{0} = - X_{00} / 2 = 1$ , $Z_{1} = - X_{01} = 2$ , $Z_{2} = - X_{02} = 0$ , $Z_{3} = Z_{0} - X_{03} = 1$ , which is the displayed $Z$ . Forming $k_{a} Z_{b} + k_{b} Z_{a}$ with $k_{a} = (- 1, 0, 0, 1)$ gives the matrix with entries $(k Z + Z k)_{00} = - 2$ , $(k Z + Z k)_{01} = - 2$ , $(k Z + Z k)_{03} = 0$ , $(k Z + Z k)_{11} = 0$ , $(k Z + Z k)_{13} = 2$ , $(k Z + Z k)_{22} = 0$ , $(k Z + Z k)_{23} = 0$ , $(k Z + Z k)_{33} = 2$ , and subtracting from $X$ gives the displayed $X^{'}$ , whose only nonzero entries are $X_{11}^{'} = 1$ and $X_{22}^{'} = - 1$ ; hence $A_{+} = X_{11}^{'} = 1$ and $A_{\times} = X_{12}^{'} = 0$ . This is clause 2.

3.1L2step 2.1algebra∎

(The perturbation.) By [L2] the gauge-transformed perturbation of the plane wave $h_{a b} = X_{a b} \cos (k \cdot x)$ is $h^{'} = X^{'} \cos (k \cdot x)$ , and $X^{'} \in T \subseteq V_{0}$ is TT and in harmonic gauge in the sense of Plane-wave perturbations of the flat linearization: planar profiles, wave covectors, amplitudes, TT form and the linearized field equations clause (c). This is clause 3.

Remarks

Uniqueness. The example exhibits the unique solution of the linear system of Harmonic-gauge plane-wave amplitudes split uniquely into a pure plane-wave gauge part and a transverse-traceless representative, with a two-dimensional space of polarizations clause (c): any other $Z$ would give a different tensor whose $0$ -components would not vanish, and the lemma's uniqueness statement forbids two TT representatives in the same class.
What the amplitudes are. The pair $(A_{+}, A_{\times}) = (1, 0)$ is the gauge-invariant first-order content of this class up to the frame choice: a screen rotation mixes the pair as in The screen and spatial projectors are orthogonal projections of rank two and three, and the TT projection is the orthogonal projection onto the two-dimensional transverse-traceless space clause (d), and the class, not the coordinate pair, is the intrinsic object.

Regauging an explicit harmonic-gauge plane-wave amplitude to transverse-traceless form

Example

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources