Rotational Transformation Between Schwarzschild Metric And Kerr Metric Ling Jun Wang Department of Physics, Geology and Astronomy University of Tennessee at Chattanooga Chattanooga, TN 37403 U.S.A. Abstract: Transformation between Schwarzschild metric [1] and Kerr metric [2] is obtained for weak field at large distance (r >> GM) with low angular velocity (a ω << r 2 ). It has been found that the relativistic rotational transformation exists within a domain r < c/ω. The transformation is dependent on the space-time coordinates, and locally consistent with the Lorentz transformation. The rotational transformation obtained in this article is a direct result of Einstein s field equation. The rotational time dilation and the angle contraction are obtained to be locally consistent with the translational time dilation and length contraction of special relativity. Key-words: general relativity, Schwarzschild, Kerr, rotational transformation, time dilation, angle contraction. 1 Introduction The special theory of relativity describes the physics of inertial systems. The dynamics is based on Lorentz transformation, from which the concepts such as time dilation, length contraction, velocity addition, mass-energy relationship, and the transformation of electromagnetic fields can be derived. It can be said that Lorentz transformation is the mathematical foundation of the special theory of relativity. The transformation between rotating frames, however, does not exist. The situation is rather perplexing, considering the fact that rotational transformation can be routinely done in classical mechanics. After all, the principle of general relativity states that The law of physics must be of such nature that they apply to systems of reference in any kind of motion [3]. If we trust that the universe is isotropic, there is no reason for us to consider a system with certain orientation any better than others. The study of rotating reference systems played an important part in the origin of general relativity theory [4,5]. The rotational motion of the planets of the solar system helped to fix the constants in Einstein s field equation. Since then experiments were designed to test the general theory of relativity. Many of these experiments, such as the experiments to test transverse Doppler effect by Hay et al. [6], the Doppler shift experiment in circular orbit by Chempeney and Moon [7], and the circumnavigating cesium clock experiment by Hafele and Keating [8], involved rotational motion. In the design and analysis of these experiments, some classical relationships of rotation were taken for granted without much explanation. The rotational motion will continue to play important role in the experimental tests of relativity for two reasons. First, the particles accelerated by synchrotrons and the equipment sent in the space are all in rotational motion; Second, the rotational motion is the only practical way to maintain an acceleration of a non-inertial system for a sustainable time period. It is therefore of both theoretical and practical importance to have a thorough understanding of the rotational 1
behavior of a coordinate system in relativistic sense. It is generally believed that the non inertial coordinate systems, including rotational systems, can be treated in the general theory of relativity. Such treatment has to be derived from Einstein s field equation and its solutions. The two wellknown solutions are the Schwarzschild solution for non-rotating spherical field and the Kerr solution for rotating spherical field [2,9]. These solutions give metric tensors or the space-time intervals of the corresponding fields. The spacetime intervals, however, do not reveal full information of the coordinate transformation. To see this, let us consider an example with two inertial systems S and S. The coordinate transformation is given by the Lorentz transformation: t = γ ( t + (v/c 2 ) x ) x = γ ( v t + x ) (1) The space-time intervals are ds 2 = (c t) 2 x 2 (2) ds 2 = (c t ) 2 x 2 (3) Eq.(2) and (3) do not uniquely lead to Eq.(1). However, Eq.(1) does guarantee ds 2 = ds 2 (4) The fact that the space-time intervals do not reveal full information of the coordinate transformation can also be seen from the classical rotational transformation described in Euler equations: x = x cosφ y sinφ y = x sinφ y cosφ (5) Again we have ds 2 = dx 2 +dy 2 = dx 2 +dy 2 = ds 2. From Eqs.(1) and (5) we see that the interval can remain invariant under different transformations. It is clear that although the Kerr solution describes the field of a rotating mass, it alone does not give the rotational transformation between the rotating coordinate systems. In this article we will obtain an explicit rotational transformation consistent with the Schwarzschild solution and the Kerr solution. 2 The Rotational Transformation Classically, the transformation between coordinate systems in constant rotation is given by: t = t φ = ω t + φ (6) or, dt = dt dφ = ω dt + dφ (7) where ω is the angular velocity of relative rotation. The translational counterpart of Eq. (6) is the Galilean transformation t = t x = v t + x (8) where v is the linear velocity of relative motion. In relativity, Eq.(8) is replaced by the Lorentz transformation, Eq.(1). We expect the classical rotational transformation to be modified in a similar fashion. Namely, we expect the relativistic rotational transformation to have the general form or, t = p t + q φ φ = α (ω t + φ ) (9) dt = p dt + q dφ dφ = α (ω dt + dφ ) (10) The constant p, q and α have to be so chosen that the transformation is consistent with the Schwarzschild metric and the Kerr metric, the two solutions to Einstein s field equation. 2
3 Rotational Transformation Between Schwarzschild Metric and Kerr Metric Let us consider the gravitational field of a rotating mass with spherical symmetry as shown in Figure 1: y y ds 2 = dt 2 (ρ 2 / ) dr 2 ρ 2 dθ 2 (r 2 + a 2 ) sin 2 θ dφ 2 (2GMr /ρ 2 )( dt a sin 2 θ dφ ) 2 (14) with ρ 2 = r 2 + a 2 cos 2 θ (15) and = r 2 2GM r + a 2 (16) M ω x x where a is the angular momentum per unit mass: a = I ω / M = k ω (17) k = I / M (18) Figure 1 The coordinate system (x,y,z), or (r,θ,φ), is fixed with the mass M, which is rotating in the coordinate system (x,y,z ), or (r,θ,φ ). The gravitational field of the mass M can be measured in either of the two systems. In the natural unit system (h = c = 1), the gravitational field of a non-rotating isotropic spherical mass M is given by the Schwarzschild metric ds 2 = g µυ dx µ dx υ = (1-2GM/r) dt 2 (1-2GM/r) 1 dr 2 r 2 dθ 2 - r 2 sin 2 θ dφ 2 (11) with dx µ = (dt, dr, dθ, dφ) (12) and 1-2GM/r 0 0 0 0-1-2GM/r 0 0 g µυ = 0 0 - r 2 0 (13) 0 0 0 - r 2 sin 2 θ where we have used an approximation for large distance (r >> GM): (1-2GM/r) = 1 + 2GM/r. In system (t, r, θ, φ ), the mass M and its rest reference frame (t, r, θ, φ) are rotating with an angular velocity ω. The gravitational field surrounding the rotating mass is given by the Kerr metric [8,9]: I in Eqs. (17) and (18) is the moment of inertia. In e.g.s. unit system, a = I ω / (Mc) = k ω / c. For a homogeneous solid sphere, k = 0.6 R 2. The mass is assumed to be the rest mass on the grounds that the rotation is slow and no particle of the mass is moving at relativistic velocity. We will consider the field at large distance where r >>GM and r >>a, and the Kerr metric reduces to ds 2 = (1-2GM/r )dt 2 (1+2GM/r ) dr 2 r 2 dθ 2 r 2 sin 2 θ dφ 2 + (4GMa/r )sin 2 θ dφ dt = g λρ dx λ dx ρ (19) with dx λ = (dt, dr, dθ, dφ ) (20) and 1-2GM/r 0 0 2GMa sin 2 θ /r 0-1-2GM/r 0 0 g λρ = 0 0 - r 2 0 2GMa sin 2 θ /r 0 0 - r 2 sin 2 θ (21) We now seek a rotational coordinate transformation t t r r θ = A θ (22) φ φ or, x µ = A µλ x λ (23) 3
where A µλ are the elements of the 4x4 matrix A. Substituting Eq. (23) into Eq. (11) gives the space-time interval in terms of dx λ : ds 2 = g µυ A µλ A υρ dx λ dx ρ (24) Since the space-time interval is unique in a certain system, Eqs.(19) and (24) have to be identical. Namely, the uniqueness of space-time demands: g λρ = g µυ A µλ A υρ = A T λµ g µυ A υρ (25) In matrix form, [g λρ ] = A T [g µυ ] A (26) where A T is the transpose of A. Note that the r and θ components in [g µυ ] and [g λρ ] are identical in both the Schwarzschild metric and the Kerr metric. The transformation for r and θ is simply the identity matrix. This is not surprising at all since we are looking for a transformation of spatial rotation, which should keep r and θ unchanged. The transformation (26) reduces to a two dimensional transformation between (t,φ) and (t, φ ). For the reasons stated in the previous section, this transformation should assume the form of Eqs. (9) and (10), i.e., we look for a 2x2 matrix p q A = (27) α ω α which satisfies Eq.(26). To determine the constants p, q and α, we substitute Eqs.(13), (21) and (27) into Eq.(26) and obtain: g 00 p 2 + g 33 α 2 ω 2 = g 00 (28) g 00 pq + g 33 α 2 ω = g 03 (29) g 00 q 2 + g 33 α 2 = g 33 (30) Solving Eqs.(28)-(30) yields g 00 g 33 - g 03 2 α = (31) /\ g 33 (g 33 ω 2-2 g 03 ω + g 00 ) g 00 - g 03 ω p = (32) /\ g 00 (g 33 ω 2-2 g 03 ω + g 00 ) g 03 - g 33 ω q = (33) /\ g 00 (g 33 ω 2-2 g 03 ω + g 00 ) α/ -q/ A -1 = (34) -αω/ p/ where = α(p qω), and g 00 α/ = (35) /\ g 33 ω 2-2 g 03 ω + g 00 g 00 ω 2 αω/ = (36) /\ g 33 ω 2-2 g 03 ω + g 00 (g 03 - g 00 ω) /\ g 33 q/ = /\ (g 33 ω 2-2g 03 ω +g 00 )( g 00 g 33 - g 032 ) (37) (g 00 - g 03 ω) /\ g 33 p/ = /\ (g 33 ω 2-2g 03 ω +g 00 )( g 00 g 33 - g 032 ) (38) To obtain α, p and q in terms of ω and the coordinates, we substitute into the above expressions the matrix elements of Eqs.(13) and (21), keeping in mind that r = r and θ = θ. We have 4
α = (1 r 2 ω 2 sin 2 θ) 1/2 (39) p = (1 r 2 ω 2 sin 2 θ - 2GM/r) 1/2 (40) q = r 2 ω sin 2 θ (1 r 2 ω 2 sin 2 θ) 1/2 (41) Again we assume r >> GM and r 2 >> (aω) in obtaining Eqs. (39) (41). Note that (rω) is not an infinitesimal quantity. In c.g.s. unit system, (rω) should be replaced by rω/c = v/c, a quantity close to unity when the linear velocity of the rotating point at large distance is close to the speed of light. If we consider the case of weak field: GM/r << 1 - r 2 ω 2 sin 2 θ (42) then Eq.(40) can be approximated to be p = (1 r 2 ω 2 sin 2 θ) 1/2 = α (43) and the transformation matrix A becomes 1 r 2 ω sin 2 θ A = α (44) ω 1 The complete rotational transformation is t = α ( t + r 2 ω sin 2 θ φ ) φ = α ( ω t + φ ) (45) r = r θ = θ In c.g.s. unit system, the first equation in Eq.(45) should read t = α [ t + (r/c) 2 ω sin 2 θ φ ] (46) and Eq.(39) should read α = (1 (r ω/c) 2 sin 2 θ ) 1/2 (47) The inverse transformation is 1 -r 2 ω sin 2 θ A -1 = α (48) -ω 1 Eq.(48) shows that the inverse transformation can be obtained simply by replacing ω with - ω in the transformation matrix (44), if the condition (42) is satisfied. 4 Discussion A. Existence of rotational transformation In order for the transformation to be physically meaningful, α has to be real. We must have r ω < c (49) This condition is not surprising at all since (r ω) is the local linear velocity. If the gravitational field is strong (GM/r is not negligible), the more stringent condition (42) instead of (49) should be imposed. B. Non uniformity of transformation Within the domain of rotational transformation r < c/ω, the transformation matrix A is dependent on the coordinates r and θ, i.e., there does not exist an uniform rotational transformation between (t,φ) and (t,φ ) for all the values of r and θ. On any cylinder coaxial with the Z-axis with given radius (r sinθ), however, the rotational transformation is uniform. C. Rotation of Z-axis (sinθ = 0) On the rotational axis, sinθ = 0. The transformation reduces to the classical rotational transformation (6), due to the fact that the linear velocity of any point on the Z-axis is zero. D. Rotation of the X-Y plane (sinθ = 1) On the X-Y plane, sinθ = 1. The rotational transformation becomes 5
t = α ( t + (r/c) 2 ω φ ) φ = α (ω t + φ ) (50) with α = (1 r 2 ω 2 sin 2 θ) 1/2 = (1 v 2 /c 2 ) 1/2 (51) Since the product (rω) is the local linear velocity v, the constant α is the familiar γ factor in the special relativity. Actually, the transformation is identical to the Lorentz transformation if (rφ) is replaced with the linear arc distance x. E. The rotational time dilation Let us look at the differential form of the transformation (50): dt = α (dt + (r/c) 2 ω dφ ) dφ = α (ω dt + dφ ) (52) Suppose there is a clock fixed in the (t,r,θ,φ ) system, dφ = 0. We have dt = α dt = (1- (rw/c) 2 ) -1/2 dt (53) which shows that the time dilation due to rotational motion is locally identical to that of special relativity. F. The angle contraction Now suppose the observer in the (t,r,θ,φ ) system measures certain angular displacement dφ. This angular displacement is measured by the observer in the (t,r,θ,φ) system to be dφ, but it has to be measured simultaneously, namely, dt must be equal zero. We therefore have dφ = dφ /α = (1- (rω/c) 2 ) dφ (54) The angle contraction is locally consistent with the Lorentz length contraction. 5 Conclusion We have obtained a relativistic rotational transformation between the Schwarzschild metric and the Kerr metric for weak field at large distance (r >> GM) with low angular velocity (a ω << r 2 ). It has been found that the relativistic rotational transformation exists within a domain r < c/ω. The transformation is dependant on the space-time coordinates, and locally consistent with the Lorentz transformation. Of particular interest are the rotational time dilation and the angle contraction derived from such rotational transformation that are also locally consistent with the time dilation and Lorentz length contraction of the theory of special relativity. Since the Schwartzschild metric and the Kerr metric are respectively the solutions to Einstein s field equation for non-rotating and rotating masses respectively, the rotational transformation thus obtained in this article is a direct result of Einstein s field equation. References: [1] Shwarzschild, Berl. Ber, 1916, pp189. [2] P. Kerr, Phys. Rev. Lett, Phys.Rev Lett., Vol.11, 1963, p237. [3] A. Einstein, The Foundation of the General Theory of Relativity in The Principle of Relativity, Methuen,London, 1923, Dover reprint. [4] L. Marder, Time and The Space-Traveler, University of Pennsylvania Press, Philadelphia, 1970. [5] H. Arzelies, Relativistic Kinematics, Pergamon, Oxford, 1966. [6] H.J. Hay, J.P. Schiffer, T.E. Cranshaw and P.A. Egelstaff, Measurement of the Red Shift in an Accelerated System using the Mossbauer Effect in Fe, Phys. Rev. Lett., Vol. 4, 1960, p165. [7] D.C. Champeney, and P.B. Moon, Absence of Doppler shift for Gamma Ray source and Detector on Same Circular Orbit, Proc. Phys. Soc., Vol. 77, 1961, p350. [8] J.C. Hafele and R.E. Keating, Relativistic Time for Terrestrial Circumnavigations, Amer. J. Phys., Vol. 40, 1972, p81. [9] R.H. Boyer and R.W. Lindguist, J. Math. Phys., Vol.8, 1967, p265. 6