How to derive the Kerr metric by cheating quite a bit. Part 2

by Willie Wong

(… continued from Part 1)

The anti-self-dual fields and complexification

For ease of algebraic manipulations, often we consider the anti-self-dual versions of two-forms. Observe that on a four-dimensional Lorentzian manifold, the Hodge star operator takes two-forms to two-forms, and squares to -1. This implies that its eigenvalues can only be \pm i. So we complexify our geometry by \otimes_\mathbb{R}\mathbb{C} linearly (so in particular (X+iY)^2 = X^2 + 2i g(X,Y) - Y^2 and not the Hermitian product). It is clear that (via a little bit of linear algebra) that the space of two-forms \Lambda^2T^*M splits after complexification

Equation 2.10
\Lambda^2T^*M\otimes_\mathbb{R}\mathbb{C} = \Lambda_+ \oplus\Lambda_-

where \Lambda_\pm are spaces of complex-valued two-forms that have eigenvalues \pm i under * respectively. It is also clear that there is a natural isomorphism from \Lambda^2T^*M to each of \Lambda_\pm (they all have real dimension 6).

So instead of focusing on real-valued two-forms, we’ll focus on \Lambda_-, elements of which are called anti-self-dual two-forms. The canonical isomorphism between \Lambda^2T^*M \leftrightarrow \Lambda_- is given by

\Lambda^2T^*M \ni X_{ab} \leftrightarrow \frac12(X_{ab} + i {}^*X_{ab}) = \mathcal{X}_{ab} \in \Lambda_-

The anti-self-dual forms enjoy many marvelous algebraic properties, especially with regards to tensor products and their traces. For a list of such properties, see the paper of Mars referenced before or Chapter 2 of the my PhD dissertation. Those algebraic properties, however, will not be needed here.

In the following we denote by \mathcal{F}_{ab} := \frac12(F_{ab} + i{}^*F_{ab}) the anti-self-dual Ernst two-form. Observe that it is a closed two-form, and thus \mathcal{F}_{ab}\tau^a is also closed by the argument before, and on a simply-connected domain is given by the potential \sigma:

Equation 2.11
\sigma = \frac{S-1}{2}  - i \frac{\Theta}{2}
\nabla_b\sigma = \nabla_b( \frac{S}{2} - i\frac{\Theta}{2} ) = - \frac12 (E_b + i B_b) = \mathcal{F}_{ab}\tau^a

We call \sigma the complex Ernst potential. (The normalization that \Re\sigma = (S-1)/2 is to accommodate the physical assumption of asymptotic flatness: near “spatial infinity”, \tau^a is expected to approach a time translation, meaning that \tau^2 \to -1. The chosen normalization allows \Re\sigma \to 0 at spatial infinity as a consequence.)

Next, one can observe that the Riemann curvature tensor can be viewed as a symmetric map from \Lambda^2T^*M to itself. The Ricci decomposition of the Riemann curvature tensor into

\textrm{Riemann} = \textrm{Weyl} \oplus \textrm{traceless Ricci} \oplus \textrm{Scalar}

is a purely algebraic decomposition on the space of such maps (I gave a hand-out about this decomposition in Week 2 of the class; see also I.M. Singer and J.A. Thorpe, “The curvature of 4-dimensional Einstein spaces”, in Global Analysis: Papers in Honor of K.Kodaira). The property we will use is that for the Weyl conformal tensor, we can define its left and right Hodge duals

{}^*W_{abcd} = \frac12\epsilon_{ab}{}^{ef}W_{efcd}~;\quad W^*_{abcd} = \frac12W_{abef}\epsilon^{ef}{}_{cd}

and verify that {}^*W_{abcd} = W^*_{abcd} which is equivalent to the statement that, viewing the Weyl curvature as a map from two-forms to two-forms, it commutes with the Hodge star operator. In any case, since the Hodge dual is well-defined (the left and right actions are equal), we can define the anti-self-dual Weyl curvature as

Equation 2.12
\mathcal{C}_{abcd} := \frac12(W_{abcd} + i{}^*W_{abcd})

Principal null directions

Let X_{ab} be a real-valued two-form on our four dimensional Lorentzian manifold. Consider the eigenvalue problem for X_{ab}. In the Riemannian case, because the metric is positive definite, there
exists no nontrivial solutions to

X_{ab}r^b = \lambda r_a

since by contracting against the vector r^a, we obtain

0 = r^aX_{ab}r^b = \lambda g(r,r)

where the first equality follows from the anti-symmetry of two-forms. So either r is the zero-vector or that it is in the kernel of X. Contrast to the case where the metric is pseudo-Riemannian. The
expression above tells us that either r is a null vector, or it has eigenvalue 0.

We say that a null vector r^a is a principal null vector of the two-form X_{ab} if it is an eigenvector. The eigenvalue equation can be evidently re-written in the following form

Equation 2.13
r_{[c}X_{a]b}r^b = 0

Now treating the Weyl curvature as a symmetric map from two-forms to two-forms, we can also ask for the eigenvalues and eigenvectors. Naturally the eigenvectors and eigenvalues of the two-forms can now be lifted to the level of the Weyl curvature, and thus we say that a null vector r^a is a principal null vector of the Weyl curvature tensor W_{abcd} if

Equation 2.14
r^br_{[e}W_{a]bc[d}r_{f]}r^c = 0

(Observe that, in form, 2.14 is a simple generalization of 2.13.) Since r^a is a null vector, it makes no sense to try to normalize it to unit length, so we can’t find a preferred unit eigenvector. Hence it is traditional also to refer to principal null directions instead of the principal null vectors. (Furthermore, the space of null directions form a \mathbb{S}^2 bundle over the manifold, with a conformal structure induced by the Lorentz transformations [i.e. Local diffeomorphisms]. So working, at least locally, with elements in the space of null directions can be reduced to working on \mathbb{CP}^1, where a lot of algebraic tools are available. This is sort of one way to look at spinors in the 4-dimensional, Lorentzian case.)

Following is a theorem about the existence of principal null directions for two-forms and Weyl-fields (resp. spin 1 and spin 2 fields). The two-form case is classical and well-known in the physics literature. The Weyl-field case is due to Petrov.

Theorm 2.15
Let X_{ab} be a real-valued two-form, and W_{abcd} be a (0,4)-tensor satisfying all algebraic symmetries of the Weyl conformal curvature, on a four dimensional Lorentzian manifold. Then at every point X_{ab} has two (possibly coïncidental) principal null directions, and W_{abcd} has four (possibly coïncidental) principal null directions, unless the tensors vanish identically.

From the spinor point of view, the above theorem is simple to prove (see Penrose and Rindler, Spinors and space-time for example). A quick sketch: as we remarked above that the space of null-directions can be identified with \mathbb{CP}^1. A (perhaps not-so-)simple calculation verifies that under this identification 2.13 and 2.14 become a degree-two and a degree-four polynomial on \mathbb{CP}^1 respectively. By the fundamental theorem of algebra, the polynomials have two and four zeros respectively when counted with multiplicity.

We say X_{ab} is non-degenerate or non-null if the two principal null directions are distinct. Using a bit of linear algebra, one sees that this is equivalent to its anti-self-dual part having non-zero norm

Equation 2.16
\mathcal{X}^2 = \mathcal{X}_{ab}\mathcal{X}^{ab} \neq 0

Let l^a and k^a stand for future pointing vector-fields corresponding to the two distinct principal null directions, we can ask that they are normalized to have l^ak_a = -1. Then we have the re-consitution formula for a non-null two-form:

Equation 2.17
\mathcal{X}_{ab} = \frac12 \sqrt{-\mathcal{X}^2} (l_ak_b - k_al_b + i\epsilon_{abcd}l^ck^d)

where the square root is taken with respect to complex numbers, so there exists two roots; therefore up to exchanging the labels l^a and k^a, the above equation is well-defined. From 2.17 it is also clear that

Equation 2.18
\mathcal{X}^2 = -4(\mathcal{X}_{ab}k^al^b)^2

A similar statement can be had for a special type of Weyl fields. We say a Weyl field is Petrov type D if it only has two distinct principal null directions, and each of the directions has algebraic multiplicity 2. Roughly speaking, one can think such a Weyl field as a tensor product W_{abcd} \sim X_{ab}\otimes X_{cd} of non-null two-forms. (A more precise notion of this is given by the concept of a “symmetric spinor product” defined in my PhD thesis.) For a Petrov type D field, we can write down an analogous formula to 2.17, which we will omit here. Analogous to 2.18, we see that a Petrov type D field W_{abcd} is similarly characterized by its two principal null directions and the scalar \mathcal{C}_{abcd}k^al^bk^cl^d (or \mathcal{C}^2) where \mathcal{C}_{abcd} is the anti-self-dual part of W_{abcd}.

The Schwarzschild metric

Let us now examine the Schwarzschild metric

Equation 2.19
\displaystyle ds^2 = -(1-\frac{2M}{r}) dt^2 + (1-\frac{2M}{r})^{-1}dr^2 + r^2 (d\theta^2 + \sin^2\theta d\phi^2)

We take \tau^a = \partial_t, so F = \frac{2M}{r^2} dt\wedge dr. Now notice that B = 0, since \partial_t is hypersurface orthogonal. Therefore the complex Ernst potential is given by

Equation 2.20
\sigma = \frac{S-1}{2} = -\frac{M}{r}

Now, a direct computation (which we’ll omit here) shows that

Equation 2.21
\displaystyle \mathcal{F}^2 = - \frac{4M^2}{r^4} = -\frac{4}{M^2}\sigma^4
\displaystyle \mathcal{C}^2 = \frac{24M^2}{r^6} = \frac{24}{M^4}\sigma^6

What about the principal null directions? It is easy to see that F_{ab} is non-null and W_{abcd} is type D using spherical symmetry of the Schwarzschild metric. Simply speaking, if r^a is a vector defined at some point p such that it is a principal null direction. Let O be an element of SO(3) such that its action on (M,g) fixes the point p. Then the induced diffeomorphism is an isometric map from T_pM to itself. Consider the vector (O^*r)^a, it will necessarily then be another principal null direction. Using the algebraic classification theorem, one sees now that r^a must be fixed by the action of O. Therefore any principal null direction must be in the plane spanned by \partial_t and \partial_r. Since W_{abcd} and F_{ab} do not vanish identically (else we are in Minkowski space), the only possible vectors which can be the principal null directions are (after normalizing to k^al_a = -1 and requiring them to be future pointing)

Equation 2.22
\displaystyle k = \frac{1}{\sqrt{1-\frac{2M}{r}}}\partial_t + \sqrt{1-\frac{2M}{r}}\partial_r ~,\qquad l = \frac{1}{\sqrt{1-\frac{2M}{r}}}\partial_t - \sqrt{1-\frac{2M}{r}}\partial_r

To see that both k and l are principal null directions, we use the fact that the Schwarzschild metric also has a discrete time-reflection symmetry sending t \leftrightarrow -t. Under this change k\leftrightarrow -l. Hence the algebraic multiplicity of k and l as principal null directions must be equal. Therefore F_{ab} is non-null and W_{abcd} is type D.

Revised ansatz

In view of the special algebraic properties of the Schwarzschild metric, we revise our initial guess and ask for a solution to the following problem

Problem 2.23
Find a four-dimensional Lorentzian manifold (M,g_{ab}) such that the following conditions hold:

  1. (M,g_{ab}) is Ricci flat.
  2. It admits two Killing vector fields \tau^a,\eta^a which commute ([\tau,\eta] = 0) and whose normal distribution is space-like and integrable.
  3. The Ernst two-form is non-null; the Weyl curvature has Petrov type D; and their principal null directions are aligned.
  4. The Ernst potential satisfies the following: there exists a real valued constant M such that \mathcal{F}^2 = -\frac{4}{M^2}\sigma^4, and \mathcal{C}^2 = \frac{24}{M^4}\sigma^6.

(Recall that the real Ernst potential \Theta is defined only up to a (real) constant. The condition here should read to mean that there exists a normalization for \Theta such that the conditions described here holds. One sees that the only possible normalization in the asymptotic flat case is by assuming \Theta vanishes at spatial infinity.)

In general, of course, it is not immediately obvious that a solution to Einstein’s equation with all the listed properties exist. In practice, it suffices to try to calculate until a contradiction is found, or until a self-consistent answer emerges. Here, we use our amazing hindsight that the Kerr-metric actually satisfies the above conditions in the formulation of the above problem. Of course, we claim that the asking of the above question is not completely unreasonable in view of the properties of the Schwarzschild metric.

(Continued in Part 3)