Manuscript XX-C@1 · XX-C.tex · section · 7a8cab37

Translation-invariant causal kernel

An intrinsic inverse-distance obstruction for two rods › Translation-invariant causal kernel · lines 1411–1459

XX-C@1#eq:jump

Use the orthonormal coframe $d t, a (t) d x^{i}$ , and write

h_{ij} = a^{- 2} h_{ij} .

In these field components the rod-source matrix is time independent:

(\calCz)^{ia} = \frac{1}{2} δ^{ia} δ^{j k} h_{j k} - h_{ia} - \frac{p}{A} δ^{ia} φ .

Spatial translation invariance and uniqueness make the causal kernel translation invariant:

E_{B} (t, x; t^{'}, y) = e (t, t^{'}; x - y) .

Its restriction to fixed $t, t^{'}$ exists as a spatial distribution: null wavefront covectors have nonzero spatial parts and do not meet the conormal bundle of that time restriction.

For fixed $t, t^{'}$ , the distribution $e$ is supported in the ball

∣ z ∣ \leq ∣ \int_{t^{'}}^{t} a (u)^{- 1} d u ∣ .

Let

m (t, t^{'}) = \int_{R^{3}} e (t, t^{'}; z) d^{3} z .

This is a well-defined finite-dimensional matrix, smooth in the two times. The Cauchy jump normalization gives

m (t^{'}, t^{'}) = 0, \partial_{t} m (t, t^{'})_{t = t^{'}} = - a (t^{'})^{- 3} A^{- 1} .

The minus sign comes from the principal differential part $- \partial_{t}^{2}$ of $□_{g ˉ}$ .

Displayed formulas in this passage

XX-C@1#eq:jump · line 1451
$m (t^{'}, t^{'}) = 0, \partial_{t} m (t, t^{'})_{t = t^{'}} = - a (t^{'})^{- 3} A^{- 1} .$
unlabeled · line 1415
$h_{ij} = a^{- 2} h_{ij} .$
unlabeled · line 1420
$(\calCz)^{ia} = \frac{1}{2} δ^{ia} δ^{j k} h_{j k} - h_{ia} - \frac{p}{A} δ^{ia} φ .$
unlabeled · line 1429
$E_{B} (t, x; t^{'}, y) = e (t, t^{'}; x - y) .$
unlabeled · line 1440
$∣ z ∣ \leq ∣ \int_{t^{'}}^{t} a (u)^{- 1} d u ∣ .$
unlabeled · line 1445
$m (t, t^{'}) = \int_{R^{3}} e (t, t^{'}; z) d^{3} z .$