Local rigidity in general relativity

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
  • Obtain additional geometric properties

"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.

This talk

• Appearance of "additional symmetries"

Overview

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

Warm Up: Generalized Birkhoff's Theorem

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

• \(dr\) 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).

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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:

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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.

Curvature Alignments

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

• 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

• \(W\): 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.

Additional technical differences

• 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.

Additional technical differences

• 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

• \(\mathcal{H}= P_+H\): 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

• \(\mathcal{H}_{ab} = D \mathcal{F}_{ab}\)
• \(\mathcal{C}_{abcd} = Q(\mathcal{F}_{ab}\mathcal{F}_{cd} - \frac13 \mathcal{F}^2 (P_+)_{abcd})\)
• (\(\mathcal{F}^2 = \mathcal{F}_{mn}\mathcal{F}^{mn}\))

Outline of proof

• 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

• 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

• 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 \(\xi\).

Thank you for listening!