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Explicit screen projection of a symmetric spatial tensor: projection, transverse-traceless part, amplitudes and idempotence

Example

Work in the inertial chart of the flat model (The flat model spacetime: the Minkowski metric on R^4 and its inertial chart) with the observer $n = \partial_{0}$ and propagation direction $s = \partial_{3}$ , so that the screen is $Σ = R e_{1} \oplus R e_{2}$ and, in the basis $(e_{0}, e_{1}, e_{2}, e_{3})$ , the screen and spatial projectors of The wave screen of an observer and propagation direction, and the transverse-traceless projection of a symmetric two-tensor have the component matrices

$p = diag (0, 1, 1, 0), h = diag (0, 1, 1, 1) .$

Let $A$ be the symmetric spatial tensor with component matrix

$(A_{a b}) = (\begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 3 & 1 & 2 \\ 0 & 1 & - 1 & 0 \\ 0 & 2 & 0 & 5 \end{array}),$

so $A (n, \cdot) = 0$ . Then:

The screen projection and the trace of $A$ on the screen are $p_{a}^{c} p_{b}^{d} A_{c d} = (\begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 3 & 1 & 0 \\ 0 & 1 & - 1 & 0 \\ 0 & 0 & 0 & 0 \end{array}), p^{c d} A_{c d} = A_{11} + A_{22} = 2.$
The TT projection is $Π_{n, s} (A) = (\begin{array}{cccc} 0 & 0 & 0 & 0 \\ 0 & 2 & 1 & 0 \\ 0 & 1 & - 2 & 0 \\ 0 & 0 & 0 & 0 \end{array}),$ which is transverse ( $Π_{n, s} (A) (s, \cdot) = 0$ ), traceless ( $η^{a b} Π_{n, s} (A)_{a b} = 2 - 2 = 0$ ) and equals its own TT projection: $Π_{n, s} (Π_{n, s} (A)) = Π_{n, s} (A)$ .
Its amplitudes are $A_{+} = 2$ and $A_{\times} = 1$ , and it is the tensor $2 (e^{1} \otimes e^{1} - e^{2} \otimes e^{2}) + (e^{1} \otimes e^{2} + e^{2} \otimes e^{1})$ .

Verification

Given: The inertial chart, $n = \partial_{0}$ , $s = \partial_{3}$ , the component matrices $p$ and $h$ of the screen and spatial projectors, and the symmetric spatial tensor $A$ above.

[L1] In the adapted basis, $p^{c d} = p_{c d} = diag (0, 1, 1, 0)$ , the TT projection acts on a symmetric $A$ by $(Π_{n, s} A)_{11} = \frac{1}{2} (A_{11} - A_{22})$ , $(Π_{n, s} A)_{22} = - \frac{1}{2} (A_{11} - A_{22})$ , $(Π_{n, s} A)_{12} = (Π_{n, s} A)_{21} = A_{12}$ and all other components zero; the TT tensors are exactly the tensors with $A_{11} = - A_{22}$ , $A_{12} = A_{21}$ and all other components zero, with amplitudes $A_{+} = A_{11}$ , $A_{\times} = A_{12}$ (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 clauses (c)-(d), The wave screen of an observer and propagation direction, and the transverse-traceless projection of a symmetric two-tensor clauses (b)-(d)).

[L2] The screen projection of a spatial tensor is obtained by keeping the components with indices in ${1, 2}$ and setting the others to zero, and $p^{c d} A_{c d} = A_{11} + A_{22}$ (The wave screen of an observer and propagation direction, and the transverse-traceless projection of a symmetric two-tensor clauses (a)-(b), Linear map between vector spaces over the same field, Real and complex inner product spaces, with the inner product linear in the first argument).

1.1L2algebra

(Screen projection and trace.) In the adapted basis, $p^{a}_{b} = diag (0, 1, 1, 0)$ , so $p_{a}^{c} p_{b}^{d} A_{c d}$ keeps exactly the $11, 12, 21, 22$ entries of $A$ , namely $3, 1, 1, - 1$ , and kills the $13, 23, 33$ entries; the trace over the screen is $p^{c d} A_{c d} = A_{11} + A_{22} = 3 + (- 1) = 2$ . This is clause 1.

2.1L1step 1.1algebra

(TT projection.) By [L1] and step 1.1, $(Π_{n, s} A)_{11} = \frac{1}{2} (3 - (- 1)) = 2$ , $(Π_{n, s} A)_{22} = - 2$ , $(Π_{n, s} A)_{12} = (Π_{n, s} A)_{21} = 1$ , all other components zero, which is the displayed matrix. Its contraction with $s = \partial_{3}$ vanishes because the $3$ -row and $3$ -column are zero; its trace is $η^{a b} (Π_{n, s} A)_{a b} = 2 + (- 2) = 0$ ; and applying the same rule to $Π_{n, s} (A)$ , whose nonvanishing entries satisfy $2 = - (- 2)$ and $1 = 1$ , returns the same matrix, so $Π_{n, s}^{2} (A) = Π_{n, s} (A)$ . This is clause 2.

3.1L1step 2.1algebra∎

(Amplitudes and reconstruction.) By [L1] applied to the matrix of step 2.1, $A_{+} = 2$ , $A_{\times} = 1$ , and the tensor is $A_{+} (e^{1} \otimes e^{1} - e^{2} \otimes e^{2}) + A_{\times} (e^{1} \otimes e^{2} + e^{2} \otimes e^{1})$ . This is clause 3.

Remarks

The projection kills the $3$ -components. The longitudinal entries $A_{13} = 2$ , $A_{23} = 0$ and the trace component $A_{33} = 5$ are removed by the screen projection; the screen trace $A_{11} + A_{22} = 2$ is then removed by the half-trace subtraction, leaving the traceless screen block with amplitudes $(2, 1)$ .
No physical reading. The example computes a projection of a chosen matrix in the flat model; the tensor $A$ is not a metric perturbation and no wave, source or observation is involved.

Explicit screen projection of a symmetric spatial tensor: projection, transverse-traceless part, amplitudes and idempotence

Example

Verification

Remarks

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