Manuscript XVIII-B@2 · XVIII-B-v2.tex · section · 27ed4f4b

Polynomial tensor structures

Polynomial tensor structures · lines 334–389

XVIII-B@2#eq:P0, XVIII-B@2#eq:P2, XVIII-B@2#eq:Q0, XVIII-B@2#eq:Q2, XVIII-B@2#eq:E

For $s = p^{2} \neq = 0$ , define

θ_{ab} (p) = η_{ab} - \frac{p _{a} p _{b}}{s},

(P_{0})_{ab c d} (P_{2})_{ab c d} = \frac{1}{3} θ_{ab} θ_{c d}, = \frac{1}{2} (θ_{a c} θ_{b d} + θ_{a d} θ_{b c}) - \frac{1}{3} θ_{ab} θ_{c d} .

These are the scalar and traceless symmetric projectors on the three-dimensional transverse space for timelike $p$ .

To avoid any division by $p^{2}$ in the final kernel, put

L_{ab} (p) = s η_{ab} - p_{a} p_{b}

and define the polynomial tensors

(Q_{0})_{ab c d} (Q_{2})_{ab c d} = \frac{1}{3} L_{ab} L_{c d} = s^{2} (P_{0})_{ab c d}, = \frac{1}{2} (L_{a c} L_{b d} + L_{a d} L_{b c}) - \frac{1}{3} L_{ab} L_{c d} = s^{2} (P_{2})_{ab c d} .

The quadratic polynomial tensor is

E_{ab c d} = s [\frac{1}{2} (η_{a c} η_{b d} + η_{a d} η_{b c}) - η_{ab} η_{c d}] - \frac{1}{2} (η_{a c} p_{b} p_{d} + η_{a d} p_{b} p_{c} + η_{b c} p_{a} p_{d} + η_{b d} p_{a} p_{c}) + η_{ab} p_{c} p_{d} + η_{c d} p_{a} p_{b} .

For $s \neq = 0$ ,

E = s (P_{2} - 2 P_{0}) .

Equation (eq:E), not this rational representation, defines $E$ also on $s = 0$ .

Displayed formulas in this passage

XVIII-B@2#eq:P0 · line 341
$(P_{0})_{ab c d} (P_{2})_{ab c d} = \frac{1}{3} θ_{ab} θ_{c d}, = \frac{1}{2} (θ_{a c} θ_{b d} + θ_{a d} θ_{b c}) - \frac{1}{3} θ_{ab} θ_{c d} .$
XVIII-B@2#eq:P2 · line 341
$(P_{0})_{ab c d} (P_{2})_{ab c d} = \frac{1}{3} θ_{ab} θ_{c d}, = \frac{1}{2} (θ_{a c} θ_{b d} + θ_{a d} θ_{b c}) - \frac{1}{3} θ_{ab} θ_{c d} .$
XVIII-B@2#eq:Q0 · line 358
$(Q_{0})_{ab c d} (Q_{2})_{ab c d} = \frac{1}{3} L_{ab} L_{c d} = s^{2} (P_{0})_{ab c d}, = \frac{1}{2} (L_{a c} L_{b d} + L_{a d} L_{b c}) - \frac{1}{3} L_{ab} L_{c d} = s^{2} (P_{2})_{ab c d} .$
XVIII-B@2#eq:Q2 · line 358
$(Q_{0})_{ab c d} (Q_{2})_{ab c d} = \frac{1}{3} L_{ab} L_{c d} = s^{2} (P_{0})_{ab c d}, = \frac{1}{2} (L_{a c} L_{b d} + L_{a d} L_{b c}) - \frac{1}{3} L_{ab} L_{c d} = s^{2} (P_{2})_{ab c d} .$
XVIII-B@2#eq:E · line 369
$E_{ab c d} = s [\frac{1}{2} (η_{a c} η_{b d} + η_{a d} η_{b c}) - η_{ab} η_{c d}] - \frac{1}{2} (η_{a c} p_{b} p_{d} + η_{a d} p_{b} p_{c} + η_{b c} p_{a} p_{d} + η_{b d} p_{a} p_{c}) + η_{ab} p_{c} p_{d} + η_{c d} p_{a} p_{b} .$
unlabeled · line 338
$θ_{ab} (p) = η_{ab} - \frac{p _{a} p _{b}}{s},$
unlabeled · line 354
$L_{ab} (p) = s η_{ab} - p_{a} p_{b}$
unlabeled · line 384
$E = s (P_{2} - 2 P_{0}) .$