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Static supertranslation shear without Bondi news

Physical scope

A formal boundary-data example with an exact flat seed and finite shear action; it distinguishes shear from news and does not claim a full transformed vacuum metric or observation.

Example

Let $q_{A B}$ be the unit round metric on $S^{2}$ , $D_{A}$ its Levi-Civita derivative, and $d Ω$ its area form. On $I = R$ and $r > r_{0} > 0$ , with $r_{0}$ a length, the exact flat Bondi metric is $g_{0} = - d u^{2} - 2 d u d r + r^{2} q_{A B} d x^{A} d x^{B}$ ; its shear, news and mass aspect are $C_{A B} = N_{A B} = m = 0$ . Let $m_{B} (u) = \frac{1}{4 π} \int_{S^{2}} m (u, x) d Ω$ . Let $Y (x) = \frac{1}{2} (3 x_{3}^{2} - 1)$ on $S^{2}$ , set $f = a Y$ for a nonzero length $a$ , and use the finite active boundary action $C_{A B}^{(f)} (u, x) = C_{A B} (u + f (x), x) - 2 D_{A} D_{B} f + q_{A B} D^{C} D_{C} f$ , $N_{A B}^{(f)} (u, x) = N_{A B} (u + f (x), x)$ . It gives $C_{A B}^{(f)} = - 2 a T_{A B}$ , where $T_{A B} = D_{A} D_{B} Y - \frac{1}{2} q_{A B} D^{2} Y$ , and $N_{A B}^{(f)} = 0$ . Treat these as formal boundary data with $m = 0$ and impose the local coefficient constraint $\partial_{u} m = \frac{1}{4} D_{A} D_{B} N^{A B} - \frac{1}{8} N_{A B} N^{A B}$ . Then $m_{B} = 0$ , $d m_{B} / d u = 0$ , and $Δ C_{A B} = 0$ on every retarded-time interval, where $Δ C_{A B} = C_{A B} (u_{2}, x) - C_{A B} (u_{1}, x)$ for $u_{1} < u_{2}$ . No full finite-transformed vacuum metric is asserted. A nonzero shear value alone therefore does not imply nonzero news or a nonzero shear difference.

Example references

A smooth vacuum Bondi expansion

Bondi supertranslations and their action on the shear

Bondi mass-aspect constraint coefficient

Facts & Assumptions

Given: The exact flat Bondi end and boundary data in the Example, the finite active boundary shear action of Bondi supertranslations and their action on the shear, and the stipulated local coefficient constraint whose vacuum-expansion version is established in Bondi mass-aspect constraint coefficient.

[F1]

The angular metric $q$ is positive definite, $D$ is its Levi-Civita derivative, and the news is $N_{A B} = \partial_{u} C_{A B}$ (A smooth vacuum Bondi expansion).

[F2]

In the fixed Bondi frame the finite active boundary action is $C_{A B}^{(f)} (u, x) = C_{A B} (u + f (x), x) - 2 D_{A} D_{B} f + q_{A B} D^{2} f$ and $N_{A B}^{(f)} (u, x) = N_{A B} (u + f (x), x)$ on its stated domain (Bondi supertranslations and their action on the shear).

[F3]

The local mass-aspect equation $\partial_{u} m = \frac{1}{4} D_{A} D_{B} N^{A B} - \frac{1}{8} N_{A B} N^{A B}$ is imposed on the formal boundary data; it is the coefficient proved under the stronger smooth-vacuum hypotheses of Bondi mass-aspect constraint coefficient.

[A1]

Countable Choice ${A C}_{ω}$ is the inherited smooth Lorentzian sphere-bundle convention; full Choice is not used (The Axiom of Countable Choice ( ${A C}_{ω}$ )).

Verification

Given: [F1]–[F3] and [A1].

1.1A1F1givenalgebra

The restriction $Y = \frac{1}{2} (3 x_{3}^{2} - 1)$ is smooth on the sphere because it is the restriction of a polynomial. Put $T_{A B} : = D_{A} D_{B} Y - \frac{1}{2} q_{A B} D^{2} Y$ . Its trace is zero. At $θ = π / 2$ , $D_{θ} D_{θ} Y = 3$ , $D_{ϕ} D_{ϕ} Y = 0$ and $D^{2} Y = 3$ , so $T_{θ θ} = 3 / 2$ ; hence $T$ is nonzero.

2.1F1F2step 1.1algebra

With $t = u + r$ , the metric $g_{0} = - d u^{2} - 2 d u d r + r^{2} q$ becomes $- d t^{2} + d r^{2} + r^{2} q$ , the Minkowski metric in spherical spatial coordinates. Thus its Bondi data are exactly $C_{A B} = N_{A B} = m = 0$ . Since $f = a Y$ is independent of $u$ , [F2] gives $C_{A B}^{(f)} = - 2 a D_{A} D_{B} Y + a q_{A B} D^{2} Y = - 2 a T_{A B}$ and $N_{A B}^{(f)} = 0$ . We use only these as formal boundary data; we do not assert that they are the coefficients of a finite transformed vacuum metric.

3.1F3step 2.1algebra∎

Set $m (u, x) = 0$ for the formal data. With $N_{A B} = 0$ , [F3] is satisfied and $m_{B} (u) = \frac{1}{4 π} \int_{S^{2}} m d Ω = 0$ , so $d m_{B} / d u = 0$ . The shear $C_{A B}^{(f)} = - 2 a T_{A B}$ is independent of $u$ , hence $Δ C_{A B} = 0$ on every interval. These conclusions follow directly from the stipulated data and constraint; neither vacuum mass-loss nor the memory proposition is invoked.

Static supertranslation shear without Bondi news

Example

Example references

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources