Manuscript I-B-VP@3 · I-B-VP-v3.tex · section · 93c04eb4

N3: full tensor residuals

Retained Friedrichs computational reports › N3: full tensor residuals · lines 3411–3476

The source describes symbolic second differentiation followed by quadrature, for

α \in {1 - \frac{1}{2 π}, 1 - \frac{2}{2 π}, \frac{1}{3}}, β = 0,

and $(α, β) = (1 - 1/ (2 π), 0.3)$ , with $ξ = 0, 1/6, 1/4$ and $r = 1, 2$ , giving 24 runs.

$α, β$	$ξ$	$1440 π^{2} r^{4} (T^{t}_{t}, T^{r}_{r}, T^{φ}_{φ}, T^{z}_{z})$	rel.\ diff.	$tr T r^{4}$
$1 - 1/ (2 π), 0$	$1/6$	$(1.000488680294, \cdot, - 3.001466040882, \cdot)$	$4.6 1 0^{- 31}$	$- 2.5 1 0^{- 34}$
$1 - 1/ (2 π), 0$	$0$	$(- 7.28723785, 5.14435194, - 15.4330558, \cdot)$	$1.8 1 0^{- 31}$	$- 1.74942 1 0^{- 3}$
$1 - 2 / (2 π), 0$	$1/6$	$(1.773131450488, \cdot, - 5.319394351463, \cdot)$	$1.9 1 0^{- 31}$	$- 1.9 1 0^{- 34}$
$1/ 3, 0$	$1/6$	$(8, 8, - 24, 8)$	$2.3 1 0^{- 31}$	$- 3.9 1 0^{- 34}$
$1/ 3, 0$	$1/4$	$(28, - 2, 6, 28)$	$9.8 1 0^{- 32}$	$4.22172 1 0^{- 3}$
$1 - 1/ (2 π), 0.3$	$1/6$	$(- 3.292315687470, \cdot, 9.876947062410, \cdot)$	$6.2 1 0^{- 31}$	$1.9 1 0^{- 34}$

A dot means equality to the time component.

The reported maxima are:

relative tensor difference off-diagonal magnitude dimensionless radial conservation residual on-shell coincidence residual 6.2 \times 1 0^{- 31} 1.4 \times 1 0^{- 41} 1.5 \times 1 0^{- 33} 7.7 \times 1 0^{- 34}

The conformal trace was reported of order $1 0^{- 34}$ , and the $r = 1, 2$ values reportedly obeyed $r^{- 4}$ scaling.

The original prose stated the untwisted real trace for every nonconformal run. That is not the correct comparator for a twisted complex run. The latter must use

(ξ - \frac{1}{6}) \frac{P _{β}}{2 π ^{2} r ^{4}} .

The retained untwisted example $- 1.74942137814895 \times 1 0^{- 3}$ at $α = 1 - 1/ (2 π)$ , $ξ = 0$ , is compatible with the real trace formula.

Displayed formulas in this passage

unlabeled · line 3415
$α \in {1 - \frac{1}{2 π}, 1 - \frac{2}{2 π}, \frac{1}{3}}, β = 0,$
unlabeled · line 3455
$relative tensor difference off-diagonal magnitude dimensionless radial conservation residual on-shell coincidence residual 6.2 \times 1 0^{- 31} 1.4 \times 1 0^{- 41} 1.5 \times 1 0^{- 33} 7.7 \times 1 0^{- 34}$
unlabeled · line 3469
$(ξ - \frac{1}{6}) \frac{P _{β}}{2 π ^{2} r ^{4}} .$