# Local rigidity in general relativity

2022 Conference on Geometric Analysis and Hyperbolic PDE — Online — 2022.12.12

## Local rigidity in general relativity

Willie Wai Yeung Wong
wongwwy@math.msu.edu
Michigan State University

Slides available :   https://slides-n-notes.qnlw.info

## Credits page

• Co-authors:
• XLA: Xinliang An, Nat'l University Singapore
• MCR: Manuel Catacora Rios, Michigan State University

## Einstein's equations

Einstein's equations on a Lorentzian manifold $$(M,g)$$ $\mathsf{G}_{ab} = \mathrm{Ric}_{ab} - \frac12 S g_{ab} = T_{ab} - \tilde{\Lambda} g_{ab}$

• $T_{ab}$: stress-energy tensor
• $\tilde{\Lambda}$: cosmological constant
• $\mathrm{Ric}_{ab}$: Ricci curvature
• $S$: scalar curvature

## Local rigidity

• "Local": results can be derived on small open neighborhoods; does not depend on global topology / asymptotics / boundary conditions.
• "Rigidity":
• Assume certain geometric properties / ansatz

## "Not local" rigidity

Example: black-hole uniqueness. ($$\tilde{\Lambda} = 0$$, Einstein-Maxwell)

• Assume:
• Stationary + axi-symmetric
• Asymptotically flat + regular event horizon
• Conclude: Kerr-Newman

## Local rigidity

#### Examples

1. Gauss-Codazzi / Initial data constraint
2. Birkhoff's Theorem (Einstein-Maxwell): spherically symmetric implies static.

## Overview

1. Generalized Birkhoff's Theorem (joint with XLA).
2. Creating extra Killing vector field from curvature alignments. (joint with MCR)

## Warped product ansatz

• Topologically: $$M = Q \times F$$, $$\dim Q = 2$$, $$\dim F \geq 2$$
• $$\dim Q = 2$$ to mimic / generalize spherical symmetry
• Metrically: $$g^M$$ is pseudo-Riemannian; there exists a pseudo-Riemannian metric $$g^Q$$ on $$Q$$ and a pseudo-Riemannian metric $$g^F$$ on $$F$$, and a warping function $$r:Q\to \mathbb{R}_+$$ such that $g = g^M = g^Q + r^2 g^F.$

## Block diagonality of Einstein tensor

• $$\mathrm{Ric}^M$$ and $$\mathsf{G}^M$$ are block diagonal. $\mathsf{G}^M = \begin{pmatrix} \underbrace{\mathsf{G}^Q}_{=0} + G_1(r,g^Q,\dim F,S^F) \\ & \mathsf{G}^F + G_2(r,g^Q,\dim F, S^Q) \cdot g^F \end{pmatrix}$
• Puts some restrictions on compatible stress-energy tensors $$T$$

## Generalized Birkhoff #1 (isotropy)

Using that the $$F$$ component of $$\mathsf{G}^M$$ is $$\mathsf{G}^F + G_2\cdot g^F$$, we see immediately

Proposition. Given $$M = Q\times_r F$$ a pseudo-Riemannian warped product. If the $$F$$ block of $$\mathsf{G}^M$$ is pure-trace, then it is equal to $$\lambda \cdot g^F$$ for some $$\lambda:Q \to \mathbb{R}$$. If additionally $$\dim F \geq 3$$ then $$F$$ is Einstein.

## More about $$G_1$$

• $$G_1 = \frac12 \mathrm{tr}^{Q} G_1 \cdot g^Q + \widehat{G}_1$$
• The trace free part $$\widehat{G}_1$$ is proportional to the trace-free part of the Hessian $$\nabla^2 r$$ on $$Q$$.

## Generalized Kodama vector field

$K = (* dr)^\sharp$
• $$dr$$ is one form on $$Q$$
• $$*$$ is Hodge operator on $$Q$$
• $$\sharp$$ is with respect to $$g^Q$$.
• Note: automatically $$K(r)$$ = 0.

## Generalized Birkhoff #2 (local rigidity)

Proposition. Given $$M = Q\times_r F$$ a pseudo-Riemannian warped product, with $$\dim Q = 2$$. If $$G_1$$ is pure trace, then either the warping function $$r$$ is constant, or the Kodama vector field is Killing.

Proof: $$\mathcal{L}_K g^Q \propto$$ trace-free part of Hessian of $$r$$.

Remark: $$\widehat{G}_1 = 0$$ is automatically true for Einstein-vacuum, Einstein-Maxwell, and Einstein-Yang-Mills.

## Generalized Birkhoff #3 (integration)

Theorem [XLA, W '17]. Given $$M = Q\times_r F$$ a pseudo-Riemannian warped product, solving either Einstein-vacuum, Einstein-Maxwell, or Einstein-Yang-Mills, with or without cosmological constant. Then (1) $$M$$ admits a non-trivial Killing vector field tangent to $$Q$$ and (2) $$F$$ is Einstein (and hence has constant scalar curvature).

...

...

Moreover, the metric can be explicitly integrated. With $g^Q(dr,dr) = \frac{S^{F}}{n(n-1)} - \frac{2m}{r^{n-1}} - \frac{2\tilde{\Lambda}}{n(n+1)}r^2 + \frac{q^2}{n(n-1)r^{2n-2}}$ (here $$n := \dim F$$, and $$m,q$$ are constants of integration)

And exactly one of the three following cases hold:

...

...

1. The solution is a plane-wave vacuum solution with $$\tilde{\Lambda}= m = 0$$; $$F$$ is Ricci-flat, $$Q$$ is isometric to Minkowski space, and $$r$$ is an optical function.
2. The solution is a Lorentzian product with $$r$$ being constant. $$Q$$ is maximally symmetric, and the Yang-Mills/Maxwell field is covariantly constant.
3. $$g^Q(dr,dr)$$ is generically non-zero, and the solution is a generalized Reissner-Nordstrom(-(A)dS) solution where we allow non-spherical fibers $$F$$.

## Remarks

• First two propositions are implicitly known to many people, at least in the context of studying when $$M$$ is a pseudo-Riemannian Einstein manifold.
• Our main contribution is the addition of matter field.

## Setting

Focus on 4-dimensional Lorentzian manifolds solving $\mathrm{Ric}_{ab} = T_{ab} + \Lambda g_{ab}$ with $$T_{ab}$$ the electro-magnetic stress-energy.

"$$\Lambda$$-electrovac"

## Motivating question

"How can we identify, locally, the explicit Kerr-NUT-AdS family of solutions when given a solution to the $$\Lambda$$-electrovac equations with one Killing vector field?"

Implicitly we care more about the case with the K.v.f. is the pseudostationary one.

The "global" analogue is the black hole uniqueness problem.

## Rough description of results

• Killing vector field generates Ernst two form
• Assume alignment between Ernst two form, Maxwell field, and Weyl curvature
• Obtain local rigidity

## Prior works

• Marc Mars (CQG 1999) (CQG 2000) vacuum
• W (AHP 2009) electrovac
• Marc Mars, José Senovilla (AHP 2015) $$\Lambda$$-vacuum

## Prior applications to black hole uniqueness

• Alexandru Ionescu, Sergiu Klainerman (Inventiones 2009)
• Spyros Alexaski, Alexandru Ionescu, Sergiu Klainerman (CMP 2010)
• W (PhD Thesis)
• Pin Yu (PhD Thesis)
• W, Pin Yu (CMP 2014)

## Prior applications to black hole uniqueness

1. Combine with Carleman estimates to force global alignment from some "seed".
2. Use "failure to align" as a measure of distance to Kerr-NUT-AdS family and prove "small data rigidity".

## Self-duality

• Let $$*$$ denote the Hodge operator acting on two-forms. $* H_{ab} = \frac12 \epsilon_{abcd} H^{cd}$
• 4D Lorentzian: $$** = -1$$.
• $$\bigwedge^2(T^*M) \otimes \mathbb{C} = \Lambda_+M \oplus \Lambda_-M$$
• $$\Lambda_\pm M$$ are eigenspaces of $$*$$ with eigenvalues $$\pm i$$ respectively.
• Write $$P_{\pm}$$ the corresponding projection operators

## Self-duality

• $$P_+ = \overline{P_-}$$
• Let $$\mathcal{X} = P_+ \mathcal{X}$$, and $$\mathcal{Y} = P_+ \mathcal{Y}$$, then
• $$\mathcal{X}_{ab} \overline{\mathcal{Y}}^{ab} = 0$$
• $$\mathcal{X}_{ab} \mathcal{X}^{cb}$$ is pure trace
• $$\mathcal{X}_{ab} \overline{\mathcal{X}}^{cb}$$ is a real symmetric tensor

## Self-duality

Simplifies statements and proofs in $$\Lambda$$-electrovac systems. Example:
• Let $$H$$ be the Faraday tensor, and $$\mathcal{H}:= P_+ H$$.
• $$T_{ab} = \mathcal{H}_{ac} \overline{\mathcal{H}}_b{}^c$$
• Maxwell's equation $$\iff d\mathcal{H} = 0 \iff \mathrm{div}~\mathcal{H} = 0$$

## Ricci decomposition

$\mathrm{Riem} = W + E + S$
• $$W$$: Weyl curvature
• $$E$$: traceless Ricci component
• $$S$$: scalar curvature component
• In $$\Lambda$$-electrovac: $$S \iff \Lambda$$ and $$E \iff T$$.
• $$P_+ W = W P_+$$, $$P_+S = SP_+$$
• $$P_+ E = E P_-$$.

## Killing vector field $\xi$

• $$\nabla\xi$$ is antisymmetric
• $$\nabla^2\xi = \mathrm{Riem}(\xi)$$ (Jacobi equation)
• In Einstein vacuum: $$F = d\xi$$ solves free Maxwell's equation
"Ernst two form"

## Mars (2000)

• Vacuum: $$E = S = 0$$
• $$\mathcal{F}:= P_+ F$$, $$\mathcal{C}:= P_+ W$$
• Theorem If $$\mathcal{C}$$ is proportional to the trace-free part of $$\mathcal{F}\otimes \mathcal{F}$$, then $$M$$ has an additional Killing vector field.
• Basic algebraic step of the proof: constructing Ernst potential

## Ernst potential

• $$\sigma:M\to\mathbb{C}$$, with $$\nabla\sigma = \iota_\xi \mathcal{F}$$
• Existence: if $$\zeta$$ is vector field, $$H$$ is two form, and
• $$\mathcal{L}_\xi H = 0 = \mathcal{L}_\xi (* H)$$
• $$H$$ solves Maxwell's equations
then a potential for $$P_+H$$ exists.

## W (2009)

• Electrovac: $$E \neq 0 = S$$ $$\implies$$ $$\mathrm{div} F = \mathrm{Ric}(\xi) \neq 0$$
• But $$\mathcal{F}:= P_+ F + 4 \bar{\Xi} P_+H$$ solves Maxwell's equation
• $$H$$: Faraday tensor of electrovac solution
• $$\Xi$$: potential for $$P_+H$$
So use this for the role of Ernst two form.
• Theorem If $$\mathcal{C}$$ is proportional to the trace-free part of $$\mathcal{F}\otimes \mathcal{F}$$, and $$P_+H$$ is proportional fo $$\mathcal{F}$$, then $$M$$ has an additional Killing vector field.

## Mars-Senovilla (2015)

• $$\Lambda$$-vacuum: $$E = 0 \neq S$$ $$\implies \mathrm{div} F = \Lambda \xi \neq 0$$
• But $$d(P_+F) = dF + d*F = \Lambda (*\xi) \implies \iota_\xi d(P_+F)=0$$
• So Ernst potential can still be defined!
• Same Theorem as Mars (2000).

## MCR, W (2022 Work in progress)

• Due to Ricci decomposition, the two approaches of W (2009) and Mars-Senovilla (2015) can be combined.
• Theorem If $$\mathcal{C}$$ is proportional to the trace-free part of $$\mathcal{F}\otimes \mathcal{F}$$ (here $\mathcal{F} = P_+F + 4 \bar{\Xi} P_+H$), and $$P_+H$$ is proportional fo $$\mathcal{F}$$, then $$M$$ has an additional Killing vector field.

• Mars (2000) and W (2009):
• Heavy use of tetrad formalism
• Assumed explicit form of proportionality factors
• Obtained the second Killing vector field as a by product of integrating Einstein's equations.

• Mars-Senovilla (2015) and MCR-W:
• Worked invariantly using natural, geometric scalars
• Mars-Senovilla still used some tetrad-based computations
• Assumed the existence of proportionality factors, derived their form as a consequence
• Natural geometric description of the second K.v.f.

## Remarks on applications

A short-coming of Ionescu-Klainerman (2009) and W (PhD Thesis) is that the requirement of explicit proportionality factors means that certain scalars are required to take explicit values along the event horizon, making them "conditional" results.

Next step: using the new formulation without explicit factors (perhaps combined with the Alexakis-Ionescu-Klainerman local Hawking theorem), can this be upgraded?

## Outline of proof

Basic geometric quantities:
• $$\mathcal{H}= P_+H$$: self dual Faraday tensor
• $$\xi$$: hypothesized Killing vector field
• $$\Xi$$: scalar electromagnetic potential $$d\Xi = \iota_\xi \mathcal{H}$$
• $$\mathcal{F} = P_+(d\xi) + 4 \bar{\Xi} \mathcal{H}$$: modified Ernst two form
• $$\sigma$$: Modified Ernst potential
• $$\mathcal{C} = P_+ W$$: self dual Weyl curvature

## Outline of proof

Hypotheses: There exists complex scalar functions $D, Q$ such that
• $$\mathcal{H}_{ab} = D \mathcal{F}_{ab}$$
• $$\mathcal{C}_{abcd} = Q(\mathcal{F}_{ab}\mathcal{F}_{cd} - \frac13 \mathcal{F}^2 (P_+)_{abcd})$$
• ($$\mathcal{F}^2 = \mathcal{F}_{mn}\mathcal{F}^{mn}$$)

## Outline of proof

Step 1: Derive differential relations between unknown scalars $\sigma, \Xi, D, Q, \mathcal{F}^2$
• Definition of $\sigma,\Xi$ as potentials
• Maxwell's equation
• Jacobi equation for $$\nabla^2\xi$$
• Second Bianchi identity for the Weyl field

## Outline of proof

Step 2: Integrate differential relations
• Let $$J = \dfrac{Q \mathcal{F}^2 - 4\Lambda}{(1 - 4 D \bar{\Xi}) \sqrt{\mathcal{F}^2}}$$
• The scalars $$\sigma, \Xi, D, Q, \mathcal{F}^2$$ can be solved to be rational functions of $$J$$
• (In Mars (2000) and Wong (2009), the expressions for $$D$$ and $$Q$$ are assumed, not derived.)

## Outline of proof

Step 3: Solve Killing's equation
• Let $$\eta^a = \mathcal{F}^{ab} \overline{\mathcal{F}}_{bc}\xi^c$$
• Show that $\nabla_a \eta_b + \nabla_b\eta_a = A_a \eta_b + A_b \eta_a + B_a \xi_a + B_b \xi_a.$
• Show, using steps 1 and 2 that $$A,B$$ can be integrated, and so there exists a second Killing vector field in the span of $$\eta$$ and $$\xi$$.