How to derive the Kerr metric by cheating quite a bit. Part 1

by Willie Wong

This came from a lecture I gave to MAT 451 at Princeton University on April 23, 2009. I have originally written this up as a LaTeX document; since I won’t be publishing this in any conventional way (and indeed, the material covered is rather unconventional), I figure I’ll use this as an experiment for the first posts on this blog. The original is a 10 page paper, which is why I am splitting this into several installments.

MAT 451 is a senior level mathematics course in which the instructor has great leeway in deciding what to teach. This year my thesis advisor was in charge, and focused the discussion on mathematical aspects of general relativity. This first post will, therefore, be rather on the technical side: the reader is assumed to have familiarity with basic pseudo-Riemannian geometry and with various aspects of general relativity. I will, however, be happy to answer any questions left in the comments.

The Kerr metric I refer to is, of course, the rotating black hole solution in Einstein’s theory of general relativity. For a brief history surrounding its discovery, see Dautcourt’s survey “Race for the Kerr field”.

1. Introduction and the first ansatz

In this note we give a heuristic derivation of the Kerr metric, in a way quite significantly different from the classical methods. This is in no way a formal write-up, so for a more rigorous derivation, and for references, please see the wonderful article by Roberto Bergamini and Stefano Viaggiu, “A novel derivation for Kerr metric in Papapetrou gauge,” Class. Quantum Grav. 21 4567–4573 (2004). The method described herein is inspired by Marc Mars’ paper “A spacetime characterization of the Kerr metric,” Class. Quantum Grav. 16 2507–2523 (1999), and also by my 2009 PhD dissertation.

The question we seek to answer is: find a solution of the Einstein vacuum equations that is stationary and axially symmetric. However, to actually answer the question, we will need to impose very significant, and not ab initio justifiable, constraints. It is only with great hindsight (that we know the solution we seek already) that the constraints seem natural. On the other hand, we will try to argue that these constraints are not completely wild guesses: by following a particular coherent chain of thought, it may have been possible to have obtained the Kerr metric through this method with no prior knowledge of the metric.

The principal argument here is thus: we first consider the Ernst potential on a stationary solution to the Einstein equations. Next we compute the Ernst potential, the Ernst two-form, and the Weyl curvature of the Schwarzschild metric. By observing that the Schwarzschild solution is algebraically special, we make the ansatz that we want to search for a similarly algebraically special solution. We find that through some cosmic co-incidences, there exist additional solutions beside the Schwarzschild metric that satisfy the algebraic condition. And through a calculation we (almost) show that this is the Kerr metric.

First we begin by quantifying the type of solutions we are actually looking for. The presentation here is standard, and the assumptions given here are made from first principles and form the common starting point of every “derivation” of the Kerr metric. In particular, we define what a solution is and what the symmetry conditions means. We seek to answer the following problem:

Problem 1.1
We want to find a four-dimensional Lorentzian manifold (M,g_{ab}) such that it is Ricci flat. We ask that it admits two Killing vector fields \tau^a,\eta^a which commute ([\tau,\eta] = 0) and whose normal distribution \{\tau,\eta\}^\perp \subset TM is space-like and integrable.

It is clear that from the formulation above, our solution manifold can be ruled by two transverse foliations: one by the integrated normal distribution, one by the tangential distribution to \tau,\eta. Each of the foliation is two-dimensional, so the induced metric on it can be diagonalized (at least locally). Therefore immediately the formulation of the problem admits the following ansatz

Equation 1.2
\displaystyle ds^2 = -S dt^2 + 2Q dt d\phi + R d\phi^2 + V dr^2 + W d\theta^2

where \partial_t = \tau and \partial_\phi = \eta are the Killing vector fields and the functions S,Q,R,V,W are functions of r and \theta only. It suffices to solve for the five unknown functions and show that the orbits of \eta are closed. Unfortunately, if one writes down the Ricci-flat condition for the above ansatz, one gets a monstrous set of equations that takes tens of pages to be written down.

A quick note: in the following, for a tensor quantity X_{abc\cdots d}, we write X^2 = X_{abc\cdots d}X_{lmn\cdots o}g^{al}g^{bm}g^{cn}\cdots g^{do} for the full contraction against itself. In other words, we write X^2 for g(X,X) where g(\cdot,\cdot) is, by an abuse of notation, the inner product induced on the tensor bundle by the Lorentzian metric. In particular, X^2 is a scalar that can be of arbitrary sign.

2. The algebraic alignment condition

To further simplify the equations, we want to impose additional constraints. Here we give a line of (perhaps questionable) reasoning that leads to certain algebraic constraints.

The Ernst two-form and Ernst potential

Consider the Killing vector field \tau^a. We write \tau_a for its dual one-form. Killing’s equation implies that \nabla_a\tau_b is anti-symmetric, so the two form

Equation 2.1
F_{ab} = (d\tau)_{ab} = \nabla_a\tau_b - \nabla_b\tau_a = 2 \nabla_a\tau_b

is defined. This two-form is called the Ernst two-form. As is well known, the second covariant derivatives of a Killing vector field is given by the Riemann curvature tensor

Equation 2.2
\nabla_cF_{ab} = 2\nabla_c\nabla_a\tau_b = 2 R_{dcab}\tau^d

This implies that F_{ab} is a Maxwell field: it is curl free (one can see this from either the fact that the double exterior derivative of a form vanishes, or the first Bianchi identity [the two are equivalent]), and has its divergence given by \nabla^aF_{ab} = -2 R_{db}\tau^d which vanishes for a solution of the Einstein vacuum equation.

As a Maxwell field, the form F_{ab} has a natural electromagnetic decomposition

Equation 2.3
E_a := F_{ab}\tau^b = 2(\nabla_a\tau_b)\tau^b = \nabla_a\tau^2 ~,\quad B_a = ({}^*F)_{ab}\tau^b

where * is the Hodge dual operator, which we can write in co-ordinates ({}^*F)_{ab} = \frac12 \epsilon_{abcd}F^{cd} where \epsilon_{abcd} is the volume form (or the Levi-Civita symbol in an orthonormal frame). When \tau^a is not a null-vector, we claim that, as in the case of standard Maxwell theory on Minkowski space (where \tau is replaced by the time co-ordinate derivative), we can reconstitute F_{ab} from the electromagnetic components using the following algebraic identity

Equation 2.4
\tau^2 F_{ab} = E_a\tau_b - E_b\tau_a - \epsilon_{abcd}B^c\tau^d

Now, let us examine the magnetic part B_a. It is also called the twist of the Killing vector field \tau^a. Observe that by Frobenius’ theorem, the normal bundle to the vector field \tau^a is integrable if and only if B_a = 0. In other words, the twist tells us whether \tau^a is hypersurface-orthogonal. As seen above, E_a is exact: it arises from the potential \tau^2. We claim that since \tau is Killing, and since that the space is Ricci-flat, B_a also has a potential. Observe that d{}^*F = -{}^*\delta {}^*F = 0. So {}^*F is a closed two-form. Now use the Cartan relation \mathcal{L}_X\omega = d(i_X\omega) + i_X(d\omega) where \omega is a form and X a vector field by taking X = \tau and \omega = {}^*F. Since \tau is Killing, and F is geometric, we must have that \mathcal{L}_\tau{}^*F = 0. Therefore we conclude that

Equation 2.4′
d(i_\tau{}^*F) = 0 = dB

Now if we assume our space-time is simply connected (or let’s say we look at a simply connected domain), we can define, up to a constant, a real-valued scalar \Theta such that d\Theta = B. This \Theta is called the Ernst potential.

So why do we care about the Ernst two-form? Recall our ansatz 1.2, we know that \tau^2 = -S. Now, observe that the dual one-form to the Killing vector field \tau is given by \tau^\flat = -S dt + Q d\phi, and so using that S and Q are independent of t and \phi, we have

Equation 2.5
\displaystyle F = \partial_rS dt\wedge dr + \partial_\theta S dt \wedge d\theta - \partial_r Q d\phi\wedge dr - \partial_\theta Q d\phi\wedge d\theta

On the other hand, taking the orientation

Equation 2.6
\displaystyle dvol = \left( (SR + Q^2)VW \right)^{\frac{1}{2}} dt\wedge d\phi \wedge dr \wedge d\theta

we can compute using 2.4 an expression for F in terms of E and B. Observe that \tau^2 and \Theta should both be independent of t and \phi, so that B = \partial_r\Theta dr + \partial_\theta\Theta d\theta, \quad E = -\partial_rS dr - \partial_\theta S d\theta. Therefore

Equation 2.7
\begin{array}{rl}  E\wedge \tau^\flat & = -S \partial_r S dt\wedge dr - S\partial_\theta S dt\wedge d\theta + Q\partial_rS d\phi \wedge dr + Q\partial_\theta S d\phi\wedge d\theta \\  B\wedge \tau^\flat & = S\partial_r\Theta dt\wedge dr + S\partial_\theta\Theta dt\wedge d\theta - Q\partial_r\Theta d\phi\wedge dr - Q\partial_\theta\Theta d\phi\wedge d\theta \end{array}

and so, by observing that the inverse metric is given by

\displaystyle - \frac{R}{SR+Q^2}(\partial_t)^2 + \frac{2Q}{SR+Q^2}\partial_t\partial_\phi + \frac{S}{SR+Q^2}(\partial_\phi)^2 + \frac{1}{V}(\partial_r)^2 + \frac{1}{W}(\partial_\theta)^2

we can write

Equation 2.8
\displaystyle -S F = -S \partial_r S dt\wedge dr - S\partial_\theta S dt\wedge d\theta + Q\partial_rS d\phi \wedge dr + Q\partial_\theta S d\phi\wedge d\theta - \left( (SR + Q^2)VW \right)^{\frac{1}{2}}( \frac{\partial_r\Theta}{V} d\phi\wedge d\theta - \frac{\partial_\theta\Theta}{W} d\phi \wedge dr )

which implies that

Equation 2.9
\begin{array}{rl} \displaystyle S \partial_r Q &= Q\partial_r S +  \sqrt{(SR+Q^2)VW} \frac{\partial_\theta\Theta}{W} \\ \displaystyle S \partial_\theta Q &= Q\partial_\theta S - \sqrt{(SR+Q^2)VW}\frac{\partial_r\Theta}{V} \end{array}

(Continued in Part 2)

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