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A Heuristic Way of Obtaining the Kerr Metric

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A Heuristic Way of Obtaining the Kerr Metric A heuristic way of obtaining the Kerr metric Jo¨rg Enderlein CST-1, MS M888, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 ~Received 29 August 1996; accepted 11 April 1997! An intuitive, straightforward way of finding the metric of a rotating black hole is presented, based on the algebra of differential forms. The representation obtained for the metric displays a simplicity which is not obvious in the usual Boyer–Lindquist coordinates. © 1997 American Association of Physics Teachers. I. INTRODUCTION techniques were developed, like the spinor technique of Pen- rose and Newman,2 or Ba¨cklund transformation techniques The formulation of general relativity by Albert Einstein in ~for a review see, for example, Ref. 3!. Despite their great 1915 was one of the greatest advances of modern physics. It success in treating the Einstein equations, these methods are describes the dependence of the structure of space–time on technically complicated and are known mainly to the special- the distribution of matter, and the converse effect of this ists working in the field. Taking into account the great physi- space–time structure on matter distribution. Despite the cal importance of the Kerr solution, it is desirable to have a overwhelming clarity of its foundation and the elegance of more straightforward way of finding it from the vacuum Ein- its basic equations, it has proved to be very difficult to find stein equations. Although a straightforward but nonetheless exact analytical solutions of the Einstein equations. More- general way for finding the Kerr solution can be found in the over, of all the exact solutions which are known, only a classic work, The Mathematical Theory of Black Holes,byS. limited class seem to have a real physical meaning. Among Chandrasekhar,4 we will present here a more heuristic way them are the famous solutions of Schwarzschild and Kerr for of finding this solution, revealing its simplicity and elegance black holes, and the Friedman solution for cosmology. Al- by using an alternative presentation of its metric. Techni- though the ‘‘simple’’ solution for a static, spherically sym- cally, all calculations will be presented in the language of metric black hole ~static vacuum solution with spherical differential forms. symmetry and central singularity! was found by Schwarzs- child shortly after Einstein’s publication of his equations, 1 II. A HEURISTIC GUESS FOR THE METRIC OF A nearly 48 years were to elapse before Kerr discovered the ROTATING BLACK HOLE vacuum solution for the stationary axisymmetric rotating black hole. What we are looking for is the metric of a stationary, Today, there exists a wealth of literature about solving the axially symmetric solution of the vacuum Einstein equations. Einstein equations and about their solutions. Powerful new The term ‘‘stationary’’ implies that there are no dependen- 897 Am. J. Phys. 65 ~9!, September 1997 © 1997 American Association of Physics Teachers 897 cies of the space–time structure on time t. Axial symmetry implies that there is no dependency on the coordinate of revolution in axially symmetric coordinates. In constructing a guess for the metric, we will first try to choose an appro- priate coordinate presentation of the flat space–time metric. This is then modified by the introduction of additional un- known functions, whose form is subsequently determined by imposing the vacuum Einstein equations. The aim is to look for a simple guess for the metric of a rotating black hole. To do this, one can consider the known Schwarzschild solution of a static black hole and try to generalize it to a possible metric of a rotating black hole. The Schwarzschild space– time can be thought of as consisting of two-dimensional co- ordinate surfaces with the 2-metric of spheres ~in Euclidean three-dimensional space!, but coupled by an unusual radius function—the radial distance between two spheres with ra- r2 dial coordinates r1 and r2 is not r22r1 , but * drAgrr, r1 Fig. 1. Schematic representation of the oblate spheroidal coordinates. Sur- where grr is the rr component of the metric. Exactly this, faces of constant j ~ellipsoids! and of constant u ~hyperboloids! are shown. together with a radius-dependent metric component gtt , causes a nonvanishing curvature of the space–time. What could be a possible generalization of this space–time struc- ture giving that of a rotating black hole? From a pure tech- ds25dx21dy21dz25a2~sinh2 j1sin2 u!~dj21du2! nical point of view, replacing the spheres by rotational ellip- 2 2 2 2 soids is the simplest thing one can try. Both types of surfaces 1a cosh j cos u df . ~3! are described by simple second-order algebraic equations. Of Next, the basis 1-forms in the above coordinates for the flat course, there is no direct physical justification for such a space–time have to be chosen. It is convenient for the sub- guess, and the only way to prove its validity will be to solve sequent calculations to choose them in such a way that the the Einstein equations and to find a noncontradictory solu- metric acquires the diagonal form gab5hab , where hab tion. However, simply replacing the spherical coordinate sur- denotes the Minkowski flat space–time metric. Thus the ba- faces is not sufficient for a guess metric. For a rotating black sis one-forms of the flat space–time ~zero curvature! in ob- hole one has to expect the occurrence of nonvanishing off- late spheroidal coordinates read diagonal metric components ~in a coordinate basis!, coupling t the angular coordinate with time. v˜ 5dt, Explicitly, for describing the spatial part of the space– v˜ j5aS dj, time, we will use oblate spheroidal ~orthogonal! coordinates ~4! j, u, and f.5 Their coordinate surfaces can be expressed in v˜ u5aS du, Cartesian coordinates x,y,z by v˜ f5a cosh j cos u df, 2 2 2 x 1y z 2 2 1 51, where the abbreviation S5Asinh j1sin u was used. a2 cosh2 j a2 sinh2 j As a first attempt, one would be inclined to use an ansatz similar to the Schwarzschild solution, i.e., one would multi- x21y 2 z2 ply the basis 1-forms v˜ t and v˜ j of the flat space–time by 2 51, ~1! a2 cos2 u a2 sin2 u two unknown functions, exp f and exp g. Obviously, this will only lead to the Schwarzschild solution itself, expressed in y2tan fx50, quite unfortunate coordinates. Recalling that one is looking for a rotating black hole solution, one could try to use the where a is a positive constant. The first of these equations Lorentz transformed basis 1-forms cosh b v˜ t2sinh b v˜f and describes ellipsoids of revolution (j5const.), the second, cosh b v˜f2sinh b v˜ t instead of v˜ t and v˜ f, where b hyberboloids of revolution (u5const.), and the third, axial 5b(j,u) is a coordinate-dependent Lorentz transformation planes (f5const.). Some coordinate surfaces of this system parameter. Then one arrives at the following set of basis are depicted in Fig. 1. The constant a is an arbitrary param- 1-forms: eter, defining the location of the common foci of the ellip- v˜ t5e f @cosh b dt2sinh b a cosh j cos u df#, soids. By solving Eqs. ~1! for x, y, and z, one finds v˜ j5egaS dj, ~5! u x5a cosh j cos u cos f, v˜ 5aS du, v˜ f5cosh b a cosh j cos u df2sinh b dt, y5a coshtj cos u sin f, ~2! which include the three unknown functions f , g, and b. z5a sinh j sin u, The use of such an ansatz for finding a stationary axially symmetric vacuum solution entails no essential loss of gen- so that the Euclidean line element in these coordinates is erality, as long as all three unknown functions are assumed given by to depend on both j and u. If the coordinate surfaces defined 898 Am. J. Phys., Vol. 65, No. 9, September 1997 Jo¨rg Enderlein 898 by the basis 1-forms ~5! are completely unrelated to the in- For our basis one-forms as defined by Eq. ~5!, the connection trinsic character of the final solution, then this ansatz leads to one-forms explicitly read over-complicated expressions for the Riemann curvature ten- t t g 21 2 f g 21 sor and the Ricci tensor. It will therefore be assumed in the v˜ j5v˜ ~aSe ! @2sinh b tanh j1 f 8#2v˜ ~aSe ! present paper that f and g depend only on j and not on u. 3 cosh f cosh b sinh b tanh j1sinh f b , ~10a! This singles out the ellipsoidal coordinate surfaces for the @ ,j# space–time structure being sought. t t 21 2 f 21 v˜ u5v˜ ~aS! sinh b tan u1v˜ ~aS! III. CALCULATING THE RICCI TENSOR 3@cosh f cosh b sinh b tan u2sinh f b,u#, ~10b! There follows a brief description of the calculation of the t j g 21 Riemann curvature and Ricci tensor in the language of dif- v˜ f5v˜ ~aSe ! @sinh f cosh b sinh b tanh j ferential forms, following mainly Ref. 6. The Einstein sum- u 21 1cosh f b, #1v˜ ~aS! @2sinh f cosh b mation convention is used throughout. j a 3sinh b tan u1cosh f b #, ~10c! First, we have to find the connection one-forms v˜ b ,u a g 5Gbgv˜ , which are defined by the first Cartan relation: j j 3 21 v˜ u5v˜ ~aS ! cos u sin u a a∧ b a g∧ b u 3 g 21 dv˜ 52v˜b v˜ 52Gbgv˜ v˜ . ~6! 2v˜ ~aS e ! cosh j sinh j, ~10d! Additionally, because the metric components in our basis are j t g 21 constant, the connection one-forms are antisymmetric @see v˜ f52v˜ ~aSe ! @cosh f cosh b sinh b tanh j Ref.
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