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:
We want to find a four-dimensional Lorentzian manifold such that it is Ricci flat. We ask that it admits two Killing vector fields which commute () and whose normal distribution 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 . 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
where and are the Killing vector fields and the functions are functions of and only. It suffices to solve for the five unknown functions and show that the orbits of 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 , we write for the full contraction against itself. In other words, we write for where is, by an abuse of notation, the inner product induced on the tensor bundle by the Lorentzian metric. In particular, 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 . We write for its dual one-form. Killing’s equation implies that is anti-symmetric, so the two form
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
This implies that 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 which vanishes for a solution of the Einstein vacuum equation.
As a Maxwell field, the form has a natural electromagnetic decomposition
where * is the Hodge dual operator, which we can write in co-ordinates where is the volume form (or the Levi-Civita symbol in an orthonormal frame). When is not a null-vector, we claim that, as in the case of standard Maxwell theory on Minkowski space (where is replaced by the time co-ordinate derivative), we can reconstitute from the electromagnetic components using the following algebraic identity
Now, let us examine the magnetic part . It is also called the twist of the Killing vector field . Observe that by Frobenius’ theorem, the normal bundle to the vector field is integrable if and only if . In other words, the twist tells us whether is hypersurface-orthogonal. As seen above, is exact: it arises from the potential . We claim that since is Killing, and since that the space is Ricci-flat, also has a potential. Observe that . So is a closed two-form. Now use the Cartan relation where is a form and a vector field by taking and . Since is Killing, and is geometric, we must have that . Therefore we conclude that
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 such that . This is called the Ernst potential.
So why do we care about the Ernst two-form? Recall our ansatz 1.2, we know that . Now, observe that the dual one-form to the Killing vector field is given by , and so using that and are independent of and , we have
On the other hand, taking the orientation
we can compute using 2.4 an expression for in terms of and . Observe that and should both be independent of and , so that , . Therefore
and so, by observing that the inverse metric is given by
we can write
which implies that
(Continued in Part 2)