Boyer-Lindquist coordinates

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Boyer-Lindquist coordinates are a description of a rotating (“Kerr”) black hole. They are a generalisation of Schwarzschild coordinates. They are the simplest coordinates for calculations, generally speaking, because the line element has just one cross-term in these coordinates. They were derived by Robert Boyer and Richard Lindquist, and published in a 1967 paper.

Metric and line element

With coordinates (t,r,θ,φ) the metric line element is:

ds^2=-\Big(1-\frac{2Mr}{\rho^2}\Big)dt^2-\frac{4Mra\sin^2\theta}{\rho^2}dt\,d\phi+\frac{\rho^2}{\Delta}dr^2+\rho^2\,d\theta^2+\Big(r^2+a^2+\frac{2Mra^2\sin^2\theta}{\rho^2}\Big)\sin^2\theta\,d\phi^2

where \rho^2\equiv r^2+a^2\cos^2\theta (also written Σ by e.g. Frolov & Novikov), and \Delta\equiv r^2+a^2-2Mr (called a “discriminant” by Carter §4.3 in Wiltshire et al) are standard notation. Every source I have checked uses an equivalent expression to this one, apart from the original paper by Boyer & Lindquist (§2) which has a different sign for the coefficient of dt dφ, presumably due to Kerr’s original a, the sign of which was quickly changed by the community. The above metric is used by O’Neill §2.1, Visser §1.5, Teukolsky §2, and others listed shortly. With some algebra, the coefficient of d\phi^2 may be rearranged to (r^2+a^2)^2-a^2\Delta\sin^2\theta, so the version given by Frolov & Novikov §3.2.1 is equivalent.

Carter (§4.3 in Wiltshire et al) gives the equivalent expression:

ds^2=-dt^2+\frac{2Mr}{\rho^2}(dt-a\sin^2\theta d\phi)^2+\rho^2\Big(\frac{dr^2}{\Delta}+d\theta^2\Big)+(r^2+a^2)\sin^2\theta\,d\phi^2

O’Neill (§2.6) has, in addition to our first-listed version:

ds^2=-\frac{\Delta}{\rho^2}(dt-a\sin^2\theta\,d\phi)^2+\frac{\rho^2}{\Delta}dr^2+\rho^2d\theta^2+\frac{\sin^2\theta}{\rho^2}((r^2+a^2)d\phi-a\,dt)^2

Chandrasekhar (§54) gives the following expression, which is changed to our metric signature convention:

ds^2=-\frac{\rho^2\Delta}{\Sigma_\textrm{Chandra}^2}dt^2+\frac{\Sigma_\textrm{Chandra}^2}{\rho^2}\Big(d\phi-\frac{2aMr}{\Sigma_\textrm{Chandra}^2}dt\Big)^2\sin^2\theta+\frac{\rho^2}{\Delta}dr^2+\rho^2d\theta^2

which also turns out to be equivalent. He uses \delta\equiv\sin^2\theta (§53) but a different definition of \Sigma: \Sigma_\textrm{Chandra}^2\equiv(r^2+a^2)^2-a^2\Delta\sin^2\theta (§54; also Frolov & Novikov label this A), for which we gave an identity above. [Check, earlier text reads: which we can show is also equal to (r^2+a^2)\rho^2+2Mra^2\sin^2\theta].

To write the components of the symmetric metric tensor, don’t forget to halve the off-diagonal terms, and so Chandrasekhar §54 is also equivalent:

g_{\mu\nu}=\begin{pmatrix} -\Big(1-\frac{2Mr}{\rho^2}\Big) & 0 & 0 & -\frac{2Mra\sin^2\theta}{\rho^2} \\ 0 & \frac{\rho^2}{\Delta} & 0 & 0 \\ 0 & 0 & \rho^2 & 0 \\ -\frac{2Mra\sin^2\theta}{\rho^2} & 0 & 0 & \Big(r^2+a^2+\frac{2Mra^2\sin^2\theta}{\rho^2}\Big)\sin^2\theta \end{pmatrix}

Inverse metric

The components of the inverse metric are found by inverting the above matrix. Notice it is in a “block diagonal” form (at least, if we were to rearrange the order of coordinates), so we merely have to invert each 2×2 block. This is trivial for the r-θ block; for the t-φ block the determinant is… and so

g^{\mu\nu}=\begin{pmatrix} -\frac{1}{\Delta}\Big(r^2+a^2+\frac{2Mra^2\sin^2\theta}{\rho^2}\Big) & 0 & 0 & -\frac{2Mra}{\rho^2\Delta} \\ 0 & \frac{\Delta}{\rho^2} & 0 & 0 \\ 0 & 0 & \frac{1}{\rho^2} & 0 \\ -\frac{2Mra}{\rho^2\Delta} & 0 & 0 & \frac{\Delta-a^2\sin^2\theta}{\rho^2\Delta\sin^2\theta} \end{pmatrix}

This is given by Frolov & Novikov §D.1.

Coordinate transformation

From Kerr coordinates:

dt_{BL}=dt-\frac{2Mr}{\Delta}dr, d\phi_{BL}=d\phi+\frac{a}{\Delta}dr (Boyer & Lindquist, §2)

Compare Frolov & Novikov §D.7

[There are two types of Kerr coordinates, based on ingoing/outgoing principal null directions. (O’Neill §2.5) O’Neill gives ingoing version]

Visser §1.5 gives dt_{BL}=dt-\frac{2Mr}{\Delta}dr, d\phi_{BL}=-d\phi-\frac{a}{\Delta}dr

Transformation with Kerr coordinates (“K”):
dv=dt+\frac{r^2+a^2}{\Delta}dr (same in Carter §4.3)
d\phi_K=d\phi_{BL}+\frac{a}{\Delta}dr (Teukolsky §2, but opposite sign of \phi_{BL} in Carter §4.3)
Also given in Kerr \S2.6, but different.

First integrals

These are especially useful for timelike geodesics.

\rho^2\diff{\theta}{\lambda} = \sqrt{\Theta}

\rho^2\diff{r}{\lambda} = \sqrt{R}

\rho^2\diff{\phi}{\lambda} = -(aE-L_z/\sin^2\theta)+(a/\Delta)P

\rho^2\diff{t}{\lambda} = -a(aE\sin^2\theta-L_z)+(r^2+a^2)P/\Delta

Here

\Theta = \mathcal L-\cos^2\theta[a^2(\mu^2-E^2)+L_z^2/\sin^2\theta]

P = E(r^2+a^2)-L_za-eQr

R = P^2-\Delta[\mu^2r^2+(L_z-aE)^2+\mathcal L]

(MTW §33.5)

Properties

Coordinate singularity at Δ=0.

When a→0, Schwarzschild coordinates result.

Canonical vector fields V^\mu=(r^2+a^2,0,0,a), W^\mu=(a\sin^2\theta,0,0,1). Then identities \fvec V\cdot\fvec\partial_\phi=\Delta a\sin^2\theta, \fvec V\cdot\fvec\partial_t=-\Delta, \fvec W\cdot\fvec\partial_\phi=(r^2+a^2)\sin^2\theta, \fvec W\cdot\fvec\partial_t=-a\sin^2\theta, \fvec V\cdot\fvec V=-\Delta\rho^2, \fvec W\cdot\fvec W=\rho^2\sin^2\theta, \fvec V\cdot\fvec W=0 (O’Neill §2.1).

More properties of B-L coordinates: g_{tt}g_{\phi\phi}-g_{t\phi}^2=-\Delta\sin^2\theta, \det(g_{\mu\nu})=-\rho^4\sin^2\theta (in Frolov for instance).

Hartle \S15.2

Principal null directions. These are null geodesics, and can be thought of as photons directly approaching/receding from the black hole, at least at large r (Floyd p.52 says B-L adapted to the “outgoing” n-congruence). In these coordinates: (O’Neill §2.5)

\left(\frac{r^2+a^2}{\Delta},\pm 1,0,\frac{a}{\Delta}\right)

There is a “price to be paid for the algebraic simplicity that has made it the most widely known expression for the Kerr solution” — singular where Δ=0 (Carter §4.3 in Wiltshire et al).

Singular. For null geodesic at least, “as such a curve approaches a horizon, so Δ→0, it exhibits the infinite spiraling and slowing that signals the failure of Boyer-Linquist coordinates.” (O’Neill §2.5)

Null tetrad (Frolov & Novikov §D.6).

Christoffel symbols and curvature tensors

These are given, for Boyer-Lindquist coordinates, in Frolov & Novikov §D.2, Mueller §2.14.1, and possibly O’Neill §2.

Killing tensor

Frolov & Novikov (§D.3, from setting the charge Q=0 for the Kerr geometry) give the rank-2 Killing tensor corresponding to Carter’s constant as:

K_{\mu\nu}=\begin{pmatrix} a^2\Big(1-\frac{2Mr\cos^2\theta}{\rho^2}\Big) & 0 & 0 & -\frac{a\sin^2\theta}{\rho^2}(\Delta a^2\cos^2\theta+r^2(r^2+a^2)) \\ 0 & -\frac{a^2\cos^2\theta\rho^2}{\Delta} & 0 & 0 \\ 0 & 0 & r^2\rho^2 & 0 \\ -\frac{a\sin^2\theta}{\rho^2}(\Delta a^2\cos^2\theta+r^2(r^2+a^2)) & 0 & 0 & \frac{\sin^2\theta}{\rho^2}(r^2(r^2+a^2)^2+\frac{1}{4}\Delta a^4\sin^22\theta) \end{pmatrix}

Then K_{\mu\nu}u^\mu u^\nu=\mathcal K, Carter’s constant, which is conserved along a geodesic. (Note: I use [quantity] per unit mass of the particle. Many authors use momentum instead of velocity in the above equation). This tensor is also the “square” of a Killing-Yano tensor f which is antisymmetric: K_{\mu\nu}=f_{\mu\rho}f^\rho_\nu:

f_{\mu\nu}=\begin{pmatrix} 0 & -a\cos\theta & ar\sin\theta & 0 \\ a\cos\theta & 0 & 0 & -a^2\cos\theta\sin^2\theta \\ -ar\sin\theta & 0 & 0 & r(r^2+a^2)\sin\theta \\ 0 & a^2\cos\theta\sin^2\theta & -r(r^2+a^2)\sin\theta & 0 \end{pmatrix}

[This is formally identical to the Kerr coordinates case, apart from a couple of minus signs; also recall that φ‘s are defined differently].

Jezierski & Łukasik (2006, §4) give the Killing-Yano tensor in the form:

f=r\sin\theta\,d\theta\wedge[(r^2+a^2)d\phi-a\,dt]+a\cos\theta\,dr\wedge(dt-a\sin^2\theta\,d\phi)

which is equivalent. Carter (§4.5 in Wiltshire et al) gives an expression in terms of a canonical tetrad:

f_{\mu\nu}=2a\cos\theta\fvec e^{\hat 1}_{[\mu}\fvec e^{\hat 0}_{\nu]}+2r\fvec e^{\hat 2}_{[\mu}\fvec e^{\hat 3}_{\nu]}

which also turns out to be equivalent. We can also define a “total angular momentum vector per unit mass” (compare Carter) by \mathcal J_\mu=f_{\mu\nu}u^\mu. This vector propagates parallely, and its square is the Carter constant: \mathcal K=\mathcal J_\mu\mathcal J^\nu (check u / p mixing…)  [Care: I am using … per unit mass, whereas many authors don’t divide by the mass]

History

Kerr published the discovery of the rotating black hole solution in 1963. Later, “In Papapetrou (1966) [http://adsabs.harvard.edu/abs/1966AnIHP…4…83P] there is a very elegant treatment of stationary axisymmetric Einstein spaces. He shows that if there is a real non-singular axis of rotation then the coordinates can be chosen so that there is only one off-diagonal component of the metric. We call such a metric quasi-diagonalizeable.” (Kerr §2.6 in Wiltshire et al 2009).

Boyer and Lindquist published their coordinates in 1967. Sadly, Boyer was killed 2 weeks after the editor received the submitted paper. Kerr claims that he and Ray Sachs also discovered this solution, but did not consider it: “Having derived this canonical form, we studied the metric for at least ten minutes and then decided that we had no idea how to introduce a reasonable source into a metric of this form, and probably would never have.” (ibid.) But they did not publish it, so Kerr appropriately credits Boyer & Lindquist.

The sign convention of Kerr’s original rotation parameter a was quickly changed by the community. Kerr credits this (in §2.5 of Wiltshire et al) to Boyer (see e.g. the comparison with Lense-Thirring precession in Boyer & Lindquist §2). Boyer & Lindquist termed their coordinates “S” coordinates, because they generalise Schwarzschild-Droste coordinates for a non-rotating black hole.

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