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 1997͒ 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. 3͒. 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 singularity͒ 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 ͑9͒, 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 space͒, but coupled by an unusual radius function—the radial distance between two spheres with ra- r2 dial coordinates r1 and r2 is not r2Ϫr1 , but ͐ drͱgrr, 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 ␰ ͑ellipsoids͒ and of constant ␪ ͑hyperboloids͒ 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- ds2ϭdx2ϩdy2ϩdz2ϭa2͑sinh2 ␰ϩsin2 ␪͒͑d␰2ϩd␪2͒ 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 ϩa cosh ␰ cos ␪ d␾ . ͑3͒ 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 g␣␤ϭ␩␣␤ , where ␩␣␤ 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 curvature͒ in ob- hole one has to expect the occurrence of nonvanishing off- late spheroidal coordinates read diagonal metric components ͑in a coordinate basis͒, coupling t the angular coordinate with time. ␻˜ ϭdt, Explicitly, for describing the spatial part of the space– ␻˜ ␰ϭa⌺ d␰, time, we will use oblate spheroidal ͑orthogonal͒ coordinates ͑4͒ ␰, ␪, and ␾.5 Their coordinate surfaces can be expressed in ␻˜ ␪ϭa⌺ d␪, Cartesian coordinates x,y,z by ␻˜ ␾ϭa cosh ␰ cos ␪ d␾, 2 2 2 x ϩy z 2 2 ϩ ϭ1, where the abbreviation ⌺ϭͱsinh ␰ϩsin ␪ was used. a2 cosh2 ␰ a2 sinh2 ␰ As a first attempt, one would be inclined to use an ansatz similar to the Schwarzschild solution, i.e., one would multi- x2ϩy 2 z2 ply the basis 1-forms ␻˜ t and ␻˜ ␰ of the flat space–time by Ϫ ϭ1, ͑1͒ a2 cos2 ␪ a2 sin2 ␪ two unknown functions, exp f and exp g. Obviously, this will only lead to the Schwarzschild solution itself, expressed in yϪtan ␾xϭ0, 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 ␤ ␻˜ tϪsinh ␤ ␻˜␾ and describes ellipsoids of revolution (␰ϭconst.), the second, cosh ␤ ␻˜␾Ϫsinh ␤ ␻˜ t instead of ␻˜ t and ␻˜ ␾, where ␤ hyberboloids of revolution (␪ϭconst.), and the third, axial ϭ␤(␰,␪) is a coordinate-dependent Lorentz transformation planes (␾ϭconst.). 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- ␻˜ tϭe f ͓cosh ␤ dtϪsinh ␤ a cosh ␰ cos ␪ d␾͔, soids. By solving Eqs. ͑1͒ for x, y, and z, one finds ␻˜ ␰ϭega⌺ d␰, ͑5͒ ␪ xϭa cosh ␰ cos ␪ cos ␾, ␻˜ ϭa⌺ d␪, ␻˜ ␾ϭcosh ␤ a cosh ␰ cos ␪ d␾Ϫsinh ␤ dt, yϭa cosht␰ cos ␪ sin ␾, ͑2͒ which include the three unknown functions f , g, and ␤. zϭa sinh ␰ sin ␪, 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 ␰ and ␪. If the coordinate surfaces defined

898 Am. J. Phys., Vol. 65, No. 9, September 1997 Jo¨rg Enderlein 898 by the basis 1-forms ͑5͒ are completely unrelated to the in- For our basis one-forms as defined by Eq. ͑5͒, 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 Ϫ1 2 ␾ g Ϫ1 sor and the Ricci tensor. It will therefore be assumed in the ␻˜ ␰ϭ␻˜ ͑a⌺e ͒ ͓Ϫsinh ␤ tanh ␰ϩ f Ј͔Ϫ␻˜ ͑a⌺e ͒ present paper that f and g depend only on ␰ and not on ␪. ϫ cosh f cosh ␤ sinh ␤ tanh ␰ϩsinh f ␤ , ͑10a͒ This singles out the ellipsoidal coordinate surfaces for the ͓ ,␰͔ space–time structure being sought. t t Ϫ1 2 ␾ Ϫ1 ␻˜ ␪ϭ␻˜ ͑a⌺͒ sinh ␤ tan ␪ϩ␻˜ ͑a⌺͒

III. CALCULATING THE RICCI TENSOR ϫ͓cosh f cosh ␤ sinh ␤ tan ␪Ϫsinh f ␤,␪͔, ͑10b͒

There follows a brief description of the calculation of the t ␰ g Ϫ1 Riemann curvature and Ricci tensor in the language of dif- ␻˜ ␾ϭ␻˜ ͑a⌺e ͒ ͓sinh f cosh ␤ sinh ␤ tanh ␰ ferential forms, following mainly Ref. 6. The Einstein sum- ␪ Ϫ1 ϩcosh f ␤, ͔ϩ␻˜ ͑a⌺͒ ͓Ϫsinh f cosh ␤ mation convention is used throughout. ␰ ␣ ϫsinh ␤ tan ␪ϩcosh f ␤ ͔, ͑10c͒ First, we have to find the connection one-forms ␻˜ ␤ ,␪ ␣ ␥ ϭ⌫␤␥␻˜ , which are defined by the first Cartan relation: ␰ ␰ 3 Ϫ1 ␻˜ ␪ϭ␻˜ ͑a⌺ ͒ cos ␪ sin ␪ ␣ ␣∧ ␤ ␣ ␥∧ ␤ ␪ 3 g Ϫ1 d␻˜ ϭϪ␻˜␤ ␻˜ ϭϪ⌫␤␥␻˜ ␻˜ . ͑6͒ Ϫ␻˜ ͑a⌺ e ͒ cosh ␰ sinh ␰, ͑10d͒

Additionally, because the metric components in our basis are ␰ t g Ϫ1 constant, the connection one-forms are antisymmetric ͓see ␻˜ ␾ϭϪ␻˜ ͑a⌺e ͒ ͓cosh f cosh ␤ sinh ␤ tanh ␰ Ref. 6, Eq. ͑14.31b͔͒: ␾ g Ϫ1 2 ϩsinh f ␤,␰͔Ϫ␻˜ ͑a⌺e ͒ cosh ␤ tanh ␰, ͑10e͒

␻˜ ␣␤ϭϪ␻˜␤␣ . ͑7͒ ␪ t Ϫ1 ␻˜ ␾ϭ␻˜ ͑a⌺͒ ͓cosh f cosh ␤ sinh ␤ tan ␪Ϫsinh f ␤,␪͔ Relation Eq. ͑6͒ is of no direct use for the calculation of the ␾ Ϫ1 2 ␣ ϩ␻˜ ͑a⌺͒ cosh ␤ tan ␪. ͑10f͒ connection one-forms. But the new symbols c␤␥ antisym- metric in ␤ and ␥ defined by Here, a prime denotes differentiation by ␰, and ␤,␰ and ␤,␪ denote the two partial derivatives of ␤. ␣ 1 ␣ ␤∧ ␥ Next, one calculates the Riemann curvature 2-form, which d␻˜ ϭϪ 2c␤␥ ␻˜ ␻˜ ͑8͒ is defined by the second Cartan relation: are easy to calculate by applying the exterior derivative to our basis one-forms. Comparing Eq. ͑8͒ with Eq. ͑6͒ one can ˜␣ 1 ␣ ␥∧ ␦ ␣ ␣∧ ␥ R␤ϵ 2R␤␥␦␻˜ ␻˜ ϭd␻˜␤ϩ␻˜␥ ␻˜␤ . ͑11͒ see that ⌫␣␥␤Ϫ⌫␣␤␥ϭc␤␥␣ . By applying a cyclic permuta- The R␣ are the components of the Riemann tensor. Insert- tion of the indices to ⌫␣␥␤Ϫ⌫␣␤␥ϭc␤␥␣ , and then adding/ ␤␥␦ subtracting the resulting equations, one finds that ing Eqs. ͑10͒ into the last equation, one obtains the following nonvanishing and distinct components of the Riemann ten- ␣ ␣␭ 1 ␣␭ ␥ ␻˜ ␤ϭg ␻˜ ␭␤ϭ 2g ͑c␭␤␥ϩc␭␥␤Ϫc␤␥␭͒␻˜ . ͑9͒ sor:

t 2 2g 2 Ϫ1 2 2 2 2 2 2 R␰t␰ϭ͓a e ⌺ ͔ ͕͑sinh f Ϫsinh 2f ͒͑cosh ␤ sinh ␤ tanh ␰ϩ␤,␰͒ϩ͑sinh 2f Ϫ4 sinh f ͒cosh ␤ sinh ␤ Ϫ2 2 2 2g Ϫ2 2 2 ϫtanh ␰ ␤,␰Ϫ⌺ sinh ␤ sin ␪ ͑e Ϫ1͒ϩ⌺ cosh ␰ sinh ␰ f ЈϪ f Ј ϩsinh ␤ tanh ␰ ͑2 f ЈϪgЈ͒ϩ f ЈgЈϪ f Љ͖, ͑12a͒

t 2 g 2 Ϫ1 2 2 2 2 R␰t␪ϭ͓2a e ⌺ ͔ ͕2͑sinh 2f Ϫsinh f ͓͒cosh ␤ sinh ␤ tanh ␰ tan ␪Ϫ␤,␰␤,␪͔ϩ͑sinh 2f Ϫ4 sinh f ͒ Ϫ2 2 ϫcosh ␤ sinh ␤͓tanh ␰ ␤,␪Ϫtan ␪␤,␰͔ϩ2⌺ sinh ␤͑cosh ␰ sinh ␰ tan ␪Ϫtanh ␰ cos ␪ sin ␪͒ Ϫsinh2␤͑1Ϫsinh2 ␤͒tanh ␰ tan ␪ϩ2͑⌺Ϫ2 cos ␪ sin ␪Ϫsinh2 ␤ tan ␪͒f Ј͖, ͑12b͒

t 2 2g 2 Ϫ1 Ϫ2 2 2g R␰␰␾ϭ͓a e ⌺ ͔ ͕cosh f ͓⌺ cosh ␤ sinh ␤ sin ␪ ͑e Ϫ1͒Ϫcosh ␤ sinh ␤ tanh ␰ ͑ f ЈϪgЈ͒Ϫ2 f Ј␤,␰͔ 2 2 2 Ϫ2 2g ϩsinh f ͓cosh ␤ sinh ␤͑cosh ␤ϩsinh ␤͒tanh ␰Ϫ2 cosh ␤ sinh ␤ tanh ␰ f ЈϪ⌺ e cos ␪ sin ␪␤,␪ Ϫ2 ϩ͑⌺ cosh ␰ sinh ␰Ϫtanh ␰Ϫ f ЈϩgЈ͒␤,␰Ϫ␤,␰␰͔͖, ͑12c͒

t 2 g 2 Ϫ1 Ϫ2 R␰␪␾ϭ͓a e ⌺ ͔ ͕cosh f ͓⌺ cosh ␤ sinh ␤͑cos ␪ sin ␪ tanh␰Ϫcosh ␰ sinh ␰ tan ␪͒ϩcosh ␤ sinh ␤ 2 2 ϫtan ␪͑tanh ␰ϩ f Ј͒Ϫ f Ј␤,␪͔Ϫsinh f ͓cosh ␤ sinh ␤͑cosh ␤ϩsinh ␤͒tanh ␰ tan ␪Ϫ͑cosh ␤ sinh ␤ tan ␪Ϫ␤,␪͒f Ј Ϫ2 2 Ϫ2 2 Ϫ͑⌺ cosh ␰ sinh ␰ϩsinh ␤ tanh ␰͒␤,␪Ϫ͑⌺ cosh ␪ sin ␪ϩcosh ␤ tan ␪͒␤,␰ϩ␤,␰␪͔͖, ͑12d͒

t 2 2 Ϫ1 2 2 2 2 2 2 R␪t␪ϭ͓a ⌺ ͔ ͕͑sinh f Ϫsinh 2f ͓͒cosh ␤ sinh ␤ tan ␪ϩ␤,␪͔ϩ͑4 sinh f Ϫsinh 2f ͒cosh ␤ sinh ␤ tan ␪␤,␪ ϩ⌺Ϫ2 sinh2 ␤ sinh2 ␰͑eϪ2gϪ1͒ϪeϪ2g⌺Ϫ2 cosh ␰ sinh ␰ f Ј͖, ͑12e͒

899 Am. J. Phys., Vol. 65, No. 9, September 1997 Jo¨rg Enderlein 899 t 2 g 2 Ϫ1 Ϫ2 R␪␰␾ϭ͓a e ⌺ ͔ ͕cosh f ͓⌺ cosh ␤ sinh ␤͑cos ␪ sin ␪ tanh ␰Ϫcosh␰ sinh ␰ tan ␪͒ϩcosh ␤ sinh ␤ tanh ␰ 2 2 ϫtan ␪Ϫ f Ј␤,␪͔Ϫsinh f ͓cosh ␤ sinh ␤͑cosh ␤ϩsinh ␤͒tanh ␰ tan ␪Ϫcosh ␤ sinh ␤ tanh ␰ f Ј Ϫ2 2 Ϫ2 2 Ϫ͑⌺ cosh ␰ sinh ␰Ϫcosh ␤ tanh ␰͒␤,␪Ϫ͑⌺ cos ␪ sin ␪Ϫsinh ␤ tan ␪͒␤,␰ϩ␤,␰␪͔͖, ͑12f͒ t 2 2 Ϫ1 Ϫ2 2 Ϫ2g 2 2 2 R␪␪␾ϭ͓a ⌺ ͔ ͕⌺ cosh f cosh ␤ sinh ␤ sinh ␰͑1Ϫe ͒ϩsinh f ͓cosh ␤ sinh ␤͑cosh ␤ϩsinh ␤͒tan ␪ Ϫ2 Ϫ2g Ϫ2 ϩ͑⌺ cos ␪ sin ␪ϩtan ␪͒␤,␪Ϫe ⌺ cosh ␰ sinh ␰ ␤,␰Ϫ␤,␪␪͔͖, ͑12g͒ t 2 2g 2 Ϫ1 2g 2 2 2 2g 2 R␾t␾ϭ͓a e ⌺ ͔ ͕sinh 2f cosh ␤ sinh ␤͑e tan ␪␤,␪Ϫtanh ␰ ␤,␰͒Ϫsinh f ͓cosh ␤ sinh ␤͑e tan ␪ 2 2g 2 2 2 ϩtanh ␰͒ϩe ␤,␪ϩ␤,␰͔Ϫcosh ␤ tanh ␰ f Ј͖, ͑12h͒ t 2 g 2 Ϫ1 2 2 R␾␰␪ϭ͓a e ⌺ ͔ ͕sinh f ͓␤,␪ f ЈϪ͑cosh ␤ϩsinh ␤͒͑tanh ␰ ␤,␪ϩtan ␪␤,␰͔͒Ϫcosh f cosh ␤ sinh ␤ tan ␪ f Ј͖, ͑12i͒

␰ 2 2g 6 Ϫ1 2 2 2 2 2g 2 R␪␰␪ϭ͓a e ⌺ ͔ ͕͑sinh ␰ cos ␪Ϫcosh ␰ sin ␪͒͑1Ϫe ͒ϩ⌺ cosh ␰ sinh ␰ gЈ͖, ͑12j͒ ␰ 2 2g 2 Ϫ1 2 2 2 2 2 2 R␾␰␾ϭ͓a e ⌺ ͔ ͕͑sinh 2f ϩsinh f ͒͑cosh ␤ sinh ␤ tanh ␰ϩ␤,␰͒ϩ͑4 sinh f ϩsinh 2f ͒ Ϫ2 2 2 2g 2 ϫcosh ␤ sinh ␤ tanh ␰ ␤,␰ϩ⌺ cosh ␤ sin ␪͑e Ϫ1͒ϩcosh ␤ tanh ␰ gЈ͖, ͑12k͒ ␰ 2 g 2 Ϫ1 2 2 R␾␪␾ϭ͓2a e ⌺ ͔ ͕͑4 sinh f ϩsinh 2f ͒cosh ␤ sinh ␤͑tanh ␰ ␤,␪Ϫtan ␪␤,␰͒Ϫ2͑sinh f ϩ2 sinh 2f ͒ 2 2 Ϫ2 2 ϫ͓cosh ␤ sinh ␤ tanh ␰ tan ␪Ϫ␤,␰␤,␪͔ϩ2⌺ cosh ␤͑cos ␪ sin ␪ tanh ␰Ϫcosh ␰ sinh ␰ tan ␪͒ ϩcosh2 ␤͑cosh2 ␤ϩ1͒tanh ␰ tan ␪͖, ͑12l͒ ␪ 2 2 Ϫ1 2 2 2 2 2 2 R␾␪␾ϭ͓a ⌺ ͔ ͕͑sinh f ϩsinh 2f ͓͒cosh ␤ sinh ␤ tan ␪ϩ␤,␪͔Ϫ͑4 sinh f ϩsinh 2f ͒cosh ␤ sinh ␤ tan ␪␤,␪ ϩ⌺Ϫ2 cosh2 ␤ sinh2 ␰͑1ϪeϪ2g͖͒. ͑12m͒

All other nonvanishing components of the Riemann tensor ␰ Ϫ2 Ϫg 2 2 R␪ϭ͑a⌺͒ e ͕͑cosh ␤ sinh ␤ϩ1͒tan ␪ tanh ␰ can be found by employing the symmetry properties of ␣ ␣␮ Ϫsinh2 ␤ tan ␪ f ЈϪ␤ ␤ ϩ⌺Ϫ2͓cos ␪ sin ␪͑tanh ␰ R␤␥␦ϭg R␮␤␥␦ : ,␰ ,␪ 2 R␣␤␥␦ϭR␥␦␣␤ϭϪR␤␣␥␦ϭϪR␣␤␦␥ . ͑13͒ ϩ f Ј͒Ϫcosh ␰ sinh ␰ tan ␪͔ϩ2 sinh f ͓␤,␰␤,␪ Finally, by contracting the Riemann tensor, one finds the Ϫcosh2 ␤ sinh2 ␤ tan ␪ tanh ␰͔ components of the Ricci tensor: ϩsinh 2f cosh ␤ sinh ␤͓tanh ␰ ␤,␪Ϫtan ␪␤,␰͔͖, ␣ ␥␣ R␤ϭR ␥␤ . ͑14͒ ͑15d͒ For our space–time metric, the Ricci tensor components are: ␪ Ϫ2 Ϫ2 2 Ϫ2g t g Ϫ2 2g 2 2 R␪ϭ͑a⌺͒ ͕⌺ sinh ␰ ͑1Ϫe ͒ R␫ϭ͑a⌺e ͒ ͕͑1Ϫe ͒sinh ␤ϩsinh ␤ tanh ␰ ͑ f ЈϪgЈ͒ Ϫ2 Ϫ4 2 2 2 2 Ϫ cosh sinh f Ϫg ϩ cosh sin Ϫtanh ␰ f ЈϪ f Ј ϩ f ЈgЈϪ f ЉϪsinh 2f ͓cosh ␤ ⌺ ␰ ␰ ͑ Ј Ј͒ ⌺ ͓ ␰ ␪ 2 2 Ϫ2g 2 2g 2 2 2g 2 2 Ϫsinh ␰ cos ␪͔͑1Ϫe ͒ ϫsinh ␤͑e tan ␪ϩtanh ␰͒ϩe ␤,␪ϩ␤,␰͔ 2 2 2 2 2 2 2g ϩ2 sinh f ͓cosh ␤ sinh ␤ tan ␪ϩ␤,␪͔ ϩ4 sinh f cosh ␤ sinh ␤͑e tan ␪␤,␪Ϫtanh ␰ ␤,␰͖͒, ͑15a͒ Ϫ2 sinh 2f cosh ␤ sinh ␤ tan ␪␤,␪͖, ͑15e͒ Rt ϭ͑a⌺eg͒Ϫ2͕cosh f ͓cosh ␤ sinh ␤͑1Ϫe2gϩ f ЈϪgЈ͒ ␾ ␾ g Ϫ2 2 2g R␾ϭ͑a⌺e ͒ ͕cosh ␤͓e Ϫ1Ϫtanh ␰ ͑ f ЈϪgЈ͔͒ ϩ2 f Ј␤,␰͔ϩsinh f ͓cosh ␤ sinh ␤͑2 tanh ␰ f Ј ϩ4 sinh2 f cosh ␤ sinh ␤ tanh ␰ ␤ Ϫe2g tan ␪␤ 2g 2 2 ͓ ,␰ ,␪͔ Ϫ͑e tan ␪ϩtanh ␰͒cosh 2␤͒ϩ␤,␰͑ f ЈϪgЈ͒ 2 2 2 2g 2 2 2g ϩsinh 2f͓cosh ␤ sinh ␤͑tanh ␰ϩe tan ␪͒ϩ␤,␰ ϩe ͑␤,␪␪Ϫtan ␪␤,␪͒ϩ␤,␰␰ϩtanh ␰ ␤,␰͔͖, ͑15b͒ ϩe2g␤2 ͔͖. ͑15f͒ ␰ g Ϫ2 2 2 ,␪ R␰ϭ͑a⌺e ͒ ͕2 sinh ␤ tanh ␰ f ЈϪ f Ј ϩtanh ␰ gЈϩ f ЈgЈ Ϫ2 2g 2 2 2 Ϫ f ЉϪ1ϩ⌺ ͓e ͑2 sin ␪Ϫcos ␪͒Ϫcosh ␰ Although the recipe for calculating the Ricci tensor is rela- tively simple, the resulting algebraic calculations may be ϩcosh ␰ sinh ␰ ͑ f ЈϩgЈ͔͒ϩ2⌺Ϫ4͓e2g cos2 ␪ sin2 ␪ time-consuming, and it is recommended that one uses sym- 2 2 ϩcosh ␰ sinh ␰͔ϩ2 sinh f cosh ␤ sinh ␤ tanh ␰ ␤,␰ bolic programs for the calculations, like MAPLE or MATH- EMATICA. All calculations in the present paper were done 2 2 2 2 2 7 ϩ2 sinh f ͓cosh ␤ sinh ␤ tanh ␰ϩ␤,␰͔͖, ͑15c͒ with the help of MATHEMATICA.

900 Am. J. Phys., Vol. 65, No. 9, September 1997 Jo¨rg Enderlein 900 IV. FINDING A SOLUTION OF THE VACUUM EINSTEIN EQUATIONS

For the vacuum, the right-hand side of the Einstein equations, ␣ ␣ ␣ R␤Ϫ͑1/2͒␦␤Rϭ8␲T␤ , ͑16͒ ␣ vanishes, and the equations reduce to R␤ϭ0. When considering Eqs. ͑15͒, it seems to be impossible to find a straightforward solution for the unknown functions f , g, and ␤. But one can use the fact that the unknown functions f and g do not depend on the coordinate ␪. The idea is to consider the equations R␣ϭ0 in the limits ␪ 0 and ␪ ␲/2. When taking the latter limit ␤ → → one has to be careful because of the divergence of tan ␪. Performing a series expansion of the Ricci tensor around ␪ϭ␲/2 reveals that the coefficient of the term with the highest divergence, ϳ(␲/2Ϫ␪)Ϫ2, has the form

Ϫsinh 2f sinh 2␤ 00Ϫ2 sinh f cosh 2␤ cosh ␤ sinh ␤ 000 0 . ͑17͒ 2a2 cosh2 ␰ 0 0 2 sinh2 f sinh 2␤ 0 ͩ 2 sinh f cosh 2␤ 0 0 sinh 2f sinh 2␤ ͪ Since ␤ is assumed to be a smooth function, this implies that ␤͑␰,␲/2͒ϭ0,

␤,␪͑␰,␲/2͒ϭ0, ͑18͒ and that any derivative of ␤ with respect to ␰ on the symmetry axis equals zero. This also cancels automatically the divergent term proportional to (␲/2Ϫ␪)Ϫ1. Assuming also that ␤ is everywhere differentiable and symmetric with respect to the equatorial plane, one finds the additional constraint

␤,␪͑␰,0͒ϭ0. ͑19͒ For the limits ␪ 0 and ␪ ␲/2, the derivatives of the unknown functions are distributed within the Ricci tensor as → → fЈ,gЈ,␤,␪␪ ͕f Ј, f Љ,gЈ,␤,␰͖ лл ͭ␤,␰ ,␤,␰␰ ͮ л ͕fЈ,fЉ,gЈ,␤ ͖ лл ␮ ,␰ lim R␯ ϳ ͑20͒ ␪ 0 лл͕fЈ,gЈ͖л → fЈ,gЈ,␤,␪␪ лл͕fЈ,gЈ,␤,␰͖ ͫ ͭ␤,␰ ,␤,␰␰ ͮ ͬ and as

͕f Ј, f Љ,gЈ,␤,␪͖ лл͕␤,␪␪͖ л ͕fЈ,fЉ,gЈ͖ лл ␮ lim R␯ ϳ . ͑21͒ ␪ ␲/2 лл͕fЈ,gЈ,␤,␪͖л → ͫ ͕␤,␪␪͖ лл͕fЈ,gЈ,␤,␪͖ͬ ␣ As a first step, one can try to exclude the two unknowns f Љ and ␤,␪ in the equations lim␪ ␲/2 R␤ϭ0. By examining the → t ␰ ␪ coefficients of these two functions in the components of the Ricci tensor, one finds that lim␪ ␲/2(RtϪR␰ϩR␪) will eliminate both terms: → 2 2g t ␰ ␪ ͑1Ϫsinh ␰͓͒1Ϫe ͔Ϫcosh ␰ sinh ␰ ͑3 f ЈϩgЈ͒ lim ͑RtϪR␰ϩR␪͒ϭ 2 2g 4 . ͑22͒ ␪ ␲/2 a e cosh ␰ → ␪ Moreover, the limit lim␪ 0 R␪ also does not contain both terms: → 2 2g ␪ ͑1Ϫsinh ␰͓͒1Ϫe ͔Ϫcosh ␰ sinh ␰ ͑ f ЈϪgЈ͒ lim R␪ϭ 2 2g 4 . ͑23͒ ␪ 0 a e sinh ␰ → Subtracting the numerators of Eqs. ͑22͒ and ͑23͒ yields f Ј(␰)ϭϪgЈ(␰). Taking into account that both f and g tend to zero for ␰ ϱ͑flat Minkowski space–time at infinity͒, one has f (␰)ϭϪg(␰). Substitution of this relation into Eq. ͑23͒ and integrating the→ latter leads to sinh ␰ exp͓ f ͑␰͔͒ϭexp͓Ϫg͑␰͔͒ϭ 1ϪA , ͑24͒ ͱ cosh2 ␰ with A as an integration constant.

901 Am. J. Phys., Vol. 65, No. 9, September 1997 Jo¨rg Enderlein 901 It remains to find a solution for ␤͑␰,␪͒. One can seek it by ⌬ sin2 ␪ ␪ ds2ϭϪ ͓dtϪa sin2 ␪ d␾͔2ϩ ͓͑r2ϩ␳2͒d␾ close examination of the equation R␪ϭ0, which results in ␳2 ␳2 sinh ␰ 2 4 sin2 cosh2 ϩ4 4 tan cosh sinh A ␳ ͓ ␪ ␰ ⌺ ␤,␪ ␪ ␤ ␤͔ 2 Ϫadt͔2ϩ dr2ϩ␳2d␪2, ͑27͒ cosh ␰ ⌬ Ϫ͓4 sin2 ␪ cosh2 ␰Ϫ⌺4͑␤ Ϫtan ␪ cosh ␤ sinh ␤͒2͔ ,␪ where the abbreviations ⌬ϭr2Ϫ2Mrϩa2, ␳2ϭr2 sinh ␰ 2 ϩa2 cos2 ␪, and 2MϵAa were used. Equation ͑27͒ is the ϫ A ϭ0. ͑25͒ ͩ cosh2 ␰ͪ standard representation of the Kerr metric in Boyer– Lindquist coordinates for a rotating black hole with mass M This equation still looks quite complicated, but one can sat- and angular momentum SϭaM ͓see Eqs. ͑33.2–4͒ in Ref. isfy Eq. ͑25͒ by making the two square brackets vanish sepa- Ϫ1 6͔. But now, the original simplicity of the metric as dis- rately. This leads to the solution cosh ␤ϭ⌺ cosh ␰ and played in Eqs. ͑5͒ is no longer obvious. hence sinh ␤ϭ⌺Ϫ1 cos ␪, where the integration constant is determined by taking into account lim␪ ␲/2␤ϭ0. Since, for any fixed ␰, this solution obeys the nonlinear→ ordinary differ- ential equation ͑ODE͒ Eq. ͑25͒ with the boundary condition ACKNOWLEDGMENTS lim␪ ␲/2␤ϭ0, it is the only solution by the uniqueness theo- rem→ for ODEs. By directly inserting the resulting solutions of I thank W. Pat Ambrose ͑LANL͒ for kindly reading the f , g, and ␤ into the Ricci tensor, it is straightforward to manuscript. I would like to thank the referees for a very prove that they really constitute a solution of the vacuum thorough and critical reading of the manuscript, which led to Einstein equations. an improved and more readable paper. I would like to ac- knowledge the support of the German Academic Exchange Service, granting my stay with the Los Alamos National V. CONCLUSION Laboratory. Collecting all results of the last section, the resulting met- ric finally has the form sinh cosh 1R. P. Kerr, ‘‘Gravitational field of a spinning mass as an example of 2 ␰ ␰ ds ϭϪ 1ϪA 2 dt algebraically special metrics,’’ Phys. Rev. Lett. 11, 237–238 ͑1963͒. ͫ cosh ␰ͬͩ ⌺ 2 R. Penrose and W. Rindler, Spinors and Space-Time ͑Cambridge U.P., 2 Cambridge, 1984͒. cos ␪ 3 Ϫ a cosh ␰ cos ␪ d␾ D. Kramer, H. Stephani, M. MacCallum, and E. Herlt, Exact Solutions of ⌺ ͪ Einstein’s Equations ͑Cambridge U.P., Cambridge, 1981͒. 4S. Chandrasekhar, The Mathematical Theory of Black Holes ͑Oxford U.P., Ϫ1 sinh ␰ New York, 1983͒, pp. 273–318. ϩ 1ϪA a2⌺2d␰2ϩa2⌺2d␪2 5 ͫ cosh2 ␰ͬ H. Margenau and G. M. Murphy, The Mathematics of Physics and Chem- istry ͑van Nostrand, Princeton, 1956͒, 2nd ed., p. 182. 6 cosh ␰ cos ␪ 2 C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation ͑Freeman, San ϩ Ϫ Francisco, 1973͒, pp. 354–363. a cosh ␰ cos ␪ d␾ dt . ͑26͒ 7 ͩ ⌺ ⌺ ͪ The MATHEMATICA package DIFFORM.M for performing differential geom- etry algebra and the MATHEMATICA notebook KERR.MA containing all alge- By changing the variables ␰ to rϭa sinh ␰ and ␪ to ␲/2 braic calculations of the present paper can be found on the internet at Ϫ␪, one finds the equivalent form http://www.berlinet.de/ϳenderlein/Krr.

BLACK HOLES The absence of any sharp spike, or cusp, of light on sub-arc-second scales bolsters the idea that M15 probably does not have a black hole in its midst... . For me, this rather mundane development was welcome news. While teaching college astronomy courses over the years, I had resisted the temptation to endow the heart of virtually every poorly understood object in the Universe with a black hole. The ‘‘bandwagon’’ appeal among astronomers who would have black holes lurking in darkened nooks and crannies practically everywhere—in the centers of galaxies, star clusters, exploding stars, even at the core of our Sun—was unconvincing to me, especially since there is no unambiguous evidence that even one such black hole actually exists anywhere. They probably do, but they could just as well be figments of our imagination.

Eric J. Chaisson, The Hubble Wars ͑HarperCollins, New York, 1994͒, p. 299.

902 Am. J. Phys., Vol. 65, No. 9, September 1997 Jo¨rg Enderlein 902