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
Michigan State University

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

Credits page

Einstein's equations

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

Local rigidity

"Not local" rigidity

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

Local rigidity


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

This talk


  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

Block diagonality of Einstein tensor

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 \(G_1\)

Generalized Kodama vector field

\[ K = (* dr)^\sharp \]

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 \(F\).


Curvature Alignments


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


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

Prior works

Prior applications to black hole uniqueness

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".




Simplifies statements and proofs in \(\Lambda\)-electrovac systems. Example:

Ricci decomposition

\[ \mathrm{Riem} = W + E + S \]

Killing vector field $\xi$

Mars (2000)

Ernst potential

W (2009)

Mars-Senovilla (2015)

MCR, W (2022 Work in progress)

Additional technical differences

Additional technical differences

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:

Outline of proof

Hypotheses: There exists complex scalar functions $D, Q$ such that

Outline of proof

Step 1: Derive differential relations between unknown scalars \[ \sigma, \Xi, D, Q, \mathcal{F}^2 \]

Outline of proof

Step 2: Integrate differential relations

Outline of proof

Step 3: Solve Killing's equation

Thank you for listening!