We derive the Kerr solution in a pedagogically transparent way, using physical symmetry and gauge arguments to reduce the candidate metric to just two unknowns. The resulting field equations are then easy to obtain, and solve. Separately, we transform the Kerr metric to Schwarzschild frame to exhibit its limits in that familiar setting.
Deep Dive into De/re-constructing the Kerr Metric.
We derive the Kerr solution in a pedagogically transparent way, using physical symmetry and gauge arguments to reduce the candidate metric to just two unknowns. The resulting field equations are then easy to obtain, and solve. Separately, we transform the Kerr metric to Schwarzschild frame to exhibit its limits in that familiar setting.
The two fundamental asymptotically flat, Schwarzschild (S) and Kerr (K) [1] solutions, of General Relativity, were derived almost half a century apart, due to the latter's complexity-it is still daunting when first encountered. Given K's physical importance, our aim is to provide a transparent, physically instructive, derivation. We will use the labor-saving Weyl method that obviates first wading through the full array of Einstein's equations, then inserting the desired special features of the candidate solution. Instead, we first specify the metric as extensively as possible, using physical and symmetry arguments, then get just the two relevant field equations from the correspondingly reduced Einstein action. This procedure is perfectly legitimate [2], despite appearances. Of course, the equations must still be solved; fortunately they are quite easy. While the metric ansatz, and its plausibility, of course stand on the shoulders of K, the process provides useful insight into its physics.
We will begin by reviewing the oblate spheroidal (OS) coordinates first introduced in this context by [3], then narrow to our candidate metric in this frame. The mechanics of obtaining, and solving, the reduced field equations follows. Separately, we will exhibit the S metric in OS, and that of K in ordinary S, frames. The latter in particular allows one to get a different perspective on K and its limit to S than in OS.
In OS coordinates, the optimal framework for axial symmetry, K has the form [3]
where (a, m) are (the only) constant parameters. [These coordinates are not to be confused with their Cartesian namesakes; as always, they are defined by the interval’s form and by their ranges, though the latter are the usual ones.] Deriving this metric from a simple ansatz will be our endproduct. First, consider some limits for orientation. Clearly m = 0 = a represents flat space in spherical coordinates. The parameter m is aptly named, being the usual ADM mass of the solution (1), that is the system’s total energy as measured by an observer at spatial infinity. This is obvious since there (1) coincides with the familiar asymptotic form of S,
We know, by the positive energy theorems of GR, that the vanishing of this single parameter is sufficient to imply flat spacetime. Indeed, at m = 0, (1) is the direct product of time with the textbook metric for Cartesian three-space expressed in OS coordinates. We must next face K’s off-diagonal components, with their linear rather than quadratic dependence on a, and attendant loss of reflection symmetry. Fortunately, (sub-)intervals of this type are familiar already in flat space, describing rotating systems with angular velocity a, and total ADM angular momentum J = ±a m. Note finally that the remaining, angular elements (dθ 2 , dφ 2 ) retain their flat space (OS) forms, precisely as they do in the S solution in S coordinates: these frames are in fact defined as keeping “flat” surface area. The differences between K and flat space, then, are entirely contained in the coefficients of dr 2 and (dt -a sin 2 θdφ) 2 , just as they are (without the extra rotation term) for S. [It is, however, surprising that both coefficients in (1) depend only on r and not on θ, as would be expected a priori.]
The above de-construction of the K metric provides our basis for its re-construction, starting with a metric ansatz with maximal physical and gauge information and minimal number of unknown functions. We propose
where we have replaced the angle θ by q = cos 2 θ for notational convenience. The kinematical OS factor Σ is defined in (1), and (Z, D) are the two unknowns, to be determined by the field equations. We follow the (simpler) construction of S, in the spirit of [4], keeping Σ in g rr (and its reciprocal in dt 2 ), but modify it by the unknown function D. Still following the example of S, the coefficient of the whole rotation term, including dt 2 , involves a different unknown, Z. [Time-independence, the hallmark of K, is also assumed,although it might be derivable by the methods of [4].] Pursuing the analogy still further, we take both unknowns to depend only on r; this assumption greatly simplifies the derivation by turning the field equations into ordinary differential ones1 . While one may argue that introduction of spin should not affect the spherical symmetry of the dt 2 coefficient, this is not as defensible here as for S. Rather, its virtue is pragmatic: it is too simple a guess not to be tried first, even if we didn’t know it would work.
Evaluating, and varying, the Einstein action with respect to (Z, D) is best done using an algebraic program, unlike for S, whose explicit action is simple; some irreducible, if purely mechanical, complexity remains. The resulting two equations are
It is easy to solve these two sets of equations (note that each requires separate vanishing of the q 0 and q 1 coefficients). Either set is just an ordinary first order equation for D ′ or Z ′ , an a
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