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

Willie Wai Yeung Wong

wongwwy@math.msu.edu

Michigan State University

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

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

- XLA: Xinliang An,

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": 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

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

- Assume:
- Stationary + axi-symmetric
- Asymptotically flat + regular event horizon

- Conclude: Kerr-Newman

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

- Appearance of "additional symmetries"

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

- 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 \[g = g^M = g^Q + r^2 g^F.\]

- \(\mathrm{Ric}^M\) 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 \(T\)

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.

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

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

**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.

**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:

...

...

- 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.
- The solution is a Lorentzian product with \(r\) being constant. \(Q\) is maximally symmetric, and the Yang-Mills/Maxwell field is covariantly constant.
- \(g^Q(dr,dr)\) is generically non-zero, and the solution is a generalized Reissner-Nordstrom(-(A)dS) solution where we allow non-spherical fibers \(F\).

- 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.

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"

"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.

- Killing vector field generates
__Ernst two form__ __Assume__alignment between Ernst two form, Maxwell field, and Weyl curvature- Obtain local rigidity

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

- 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)

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

- 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

- \(P_+ = \overline{P_-}\)
- 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

- Let \(H\) be the Faraday tensor, and \(\mathcal{H}:= P_+ H\).
- \(T_{ab} = \mathcal{H}_{ac} \overline{\mathcal{H}}_b{}^c\)
- Maxwell's equation \(\iff d\mathcal{H} = 0 \iff \mathrm{div}~\mathcal{H} = 0\)

- \(W\): Weyl curvature
- \(E\): traceless Ricci component
- \(S\): scalar curvature component
- In \(\Lambda\)-electrovac: \(S \iff \Lambda\) and \(E \iff T\).
- \(P_+ W = W P_+\), \(P_+S = SP_+\)
- \(P_+ E = E P_-\).

- \(\nabla\xi\) is antisymmetric
- \(\nabla^2\xi = \mathrm{Riem}(\xi)\) (Jacobi equation)
- In Einstein vacuum: \(F = d\xi\) solves free Maxwell's equation

"Ernst two form"

- 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

- \(\sigma:M\to\mathbb{C}\), 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

- 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 \(P_+H\)

**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.

- \(\Lambda\)-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).

- 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.

- Worked invariantly using natural, geometric scalars

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?

- \(\mathcal{H}= P_+H\): 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

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

- Definition of $\sigma,\Xi$ as potentials
- Maxwell's equation
- Jacobi equation for \(\nabla^2\xi\)
- Second Bianchi identity for the Weyl field

- 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.)

- 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 \(\xi\).