ENERGY SPECTRUM OF THE KLEIN-GORDON EQUATION IN SCHWARZSCHILD AND KERR FIELDS A. B. Gaina and I. M. Ternov UDC 531.51:524:8; 531.51.51; 52-1/-8 We e x a m i n e t h e influence of relativistic gravitational effects and rotation of a central body on the structure of the quasidiscrete energy spectrum of a spinless particle in the field of Schwarzschild and Kerr black holes. In [1-3] quasiconnected (resonance) states arising under the finite motion (E < pc =) of massive particles of various spin in Schwarzschild and Kerr fields of black holes were considered. It was shown that under fulfillment of the condition 0~4<<I['+" I/9~, for $-- O, hc ~1/2, for s > 0 (1) and arbitrary rotation of a central body (a<-~GM/c2), the spectrum of quasiconnected states is well approximated by a nonrelativistic hydrogen-like spectrum. In relation (i), ~, s, and l are the mass, spin, and orbital quantum number of a particle; M and = are the mass and rotation parameter of a black hole. However, it is obvious that the relativistic effects that cause damping of the levels of the "black-hole atom" must simultaneously bring about a more complex picture of the spectrum. In approximation (i), post-Newtonian relativistic effects as well as the rotation of a central body create, as will be shown below, the fine structure of the spectrum of the "black-hole atom." In principle, such effects could appear in theprocesses of radiation and interaction of such an atom with radiation. In the present article we calculate the structure of the energy spectrum of a spinless particle in Schwarzschild and Kerr fields. In particular, we obtain quantum analogues of the classical effects of shift of the perihelion of orbit in a central field and precession of the orbital plane in the field of a rotating body. On the quantum level the splitting of levels of a hydrogen-like spectrum corresponds to this. The general method is based on the procedure of separation of variables in the Klein-Gordon equation and the application of perturbation theory. i. We consider the motion of a scalar particle with mass p in a Schwarzschild field: g~ = diag (eL -- d, - - r ~-, - - r 2 sin 0), e" = - - e - ~ = 1 - - Ro/R, (2) where R G is the gravitational radius: RG = 2M. (Here and in the following we use the system of units c = fi = G = i.) The Klein--Gordon equation has the form i 1/ -- g o OX" V ( - - gg,~'~ ~ ~-L ~(13 -----O , w h e r e x ~- (t, r, O, ?). (3) For stationary states with given energy a, orbital momentum I and projection of momentum m on the z axis (@ = 0), passing through the coordinate origin, we have e~,~,,, = e - i ~ ' qF (r, 0, ?) = e -i~t 1 R~,, ( r ) Y}" (0, ~). /- Introducing a new radial function F = ] / ~ R , (4) we obtain for it an equation of Schr~dinger type: *Condition (i) for electrons means M << I0 x7 g. In this connection, the gravitational radius R G = 2GM/c = is significantly smaller than the Compton wavelength of the particles, i.e., we are speaking of microscopic black holes. Kishinev Politechnical Institute. M. V. Lomonosov Moscow State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. i0, Pp. 71-76, October, 1988. Original article submitted March 3, 1986. 830 0038-5697/88/3110-0830512.50 9 Plenum Publishing Corporation " 4p~ (p + ' R~) ~- (5) Here a new coordinate p = r -- RG is introduced; for p >> Rr the terms on the right-hand side of radial equation ~p-2(p + RG)-I and ~_2 (P + RG) -= can be discarded to obtain an approximate solution in this region. Under fulfillment of (I), the term eiR~/p~ as well can be considered a perturbation. If we localize the particle at some initial moment of time by means of the Coulomb function, reducing the left-hand side of (5) to zero, then calculation of the perturbation from the right-hand side, in agreement with [i], leads to small damping. Disregarding the damping and thus passing to an approximation of the discrete spectrum, we shall consider the right-hand side of (5) as a perturbation giving rise to the fine structure of the spectrum of energy states. Thus, in the null approximation the radial waves of the function are, with use of the Laguerre polynomials (see [5]): --9 0 2/~1 F.~ (p) = C.t (22.~) ~+~e ~ L._~_~ (2~p), where Q n = ~ p ~ - - ~ , (6) determined by the condition of boundedness of the wave function for p ~ 0: 1 -q-- l + ( 2 ~} / ~- -_ _~ )~ ~ --n:=O, . -- o --I (7) From this 2n ~ As i s o b v i o u s , 8 n4 § .... 9 ~ = i n t h e n u l l a p p r o x i m a t i o n we c h o s e , n , n= l~l~n,. (8) t h e s p e c t r u m i s d e g e n e r a t e i n t h e number ; . We obtain the normalization coefficients Cn; from normalization condition oo co f g~176 ]/ - - gd3x ;( ) = 1 -- ,,R~ 9 r>RG -1 R ~"(r) dr = I F~,; (?) d,< (9) r RG Hence o el. = -n n (n + l)! (n- (z0) Z-- 1)! In first order perturbation theory it is not difficult to find ~oi~= (11) < ~,11) > + <ttY(? > + < tl;'? ) > , where <IU~ I ) > =~-"RS_<? < w~2)> /~n3(lj-1:2)' 2~OMi (I ~to) t t ( z ~ I/2)Ro( / > \ pz (9 ,'-- Ro) n ~ (l ,-+-1/2) In addition, it is not difficult to convince oneself that for I # 0 the term W(~) = R~/ g 4p 2 (p + RG)= makes a contribution only of the next order of smallness, and for I = 0, this term as well can be omitted, since for integration in limits R o < r < ~c with functions Fn0, region r - R o ~ R G introduces a fundamental contribution. Thus, the first relativistic correction, which removes the degeneracy in the number l, is • h-~t ~ . . .2p .. 3,'. 3 n ~ (1 § 1/2) From (8) and (12) we g e t t h e r e s u l t a n t expression for the binding c l e in a Schwarzschild field up to terms of order 06 . (12) energy of a spinless parti- It is not difficult to show that in the case of quasiclassical motion with Z >> I, the correction obtained corresponds exactly to the classical effect of systematic displacement of perihelion of the orbit of the test particle. For this we write the binding energy of the particle by means of adiabatic invariants 831 I r= prdr=2= n rq- , 1o= podO=2= l--m-5 , I,=, = ~ p=d~ -= 2=m and find the change in the spatial part of the classical action function after one revolution of the particle: 2=~M 15 4 AS r + 2=L ~ ] / 2 t l - - u/p) M]/2(1_~/~)~ ' 6=(~M)-* L Here we assumed that the classical moment of the particle equals L = (10-5 1~)/2=. If we choose as the plane of motion 0 = ~-L(I~=0), then for the displacement of perihelion of the orbit after one revolution we find O( .%?= .%S~)=2~ i 6=~'~ t~ , (13) which coincides with the Einstein value (see, e.g., [4]). 2. We consider the motion of a scalar particle by Klein--Gordon equation (3) in the Kerr metric (Boyer--Lindquist coordinates) : d s ' = (1 _ 2 r dt~--~----'-dr~ r'+a"4-2Mra----'--JsinO]" v 2Mra sin ~ Od~dt, (14) A = r ~- - 2Mr § a-, - - g = ~-~ sin ~ 0. (15) X sin ~ @d~~ + - - where X,=r2 _ ~ a 2 cos ~-0, -F / X The moment J of the central body is related to parameter a by relation J =Adanz. (16) The spatial part of the wave function in a stationary state satisfies equation -- ~ g " + 2~g'~L~ + ~2 -5 ] / ~ ^ where ~, ~' = I, 2, 3 (r, 0, ?); L~ = -- io_ a~ Ox ~ 07 (r, e, ~) = O, (17) is the projection operator of the orbital momentum of a particle onto the z axis (0 = 0), which is the action integral. case Obviously, ~=~.t~(a). in the general (18) We limit ourselves further to case (i) for arbitrary rotation a<~M. According to [i], the spectrum in the null approximation is nonrelativlstic and hydrogen-like. In the next, i.e., first relativistic, approximation deviations from formula (12) are possible for fine structure at the expense of an alteration in the diagonal components of the metric Kerr tensor, and correspondingly of the geometry in the region of the barrier and hole. However, in contravariant components gt~ of the metric tensor, only the square of a enters; correspondingly, in the spatial part of the d'Alambertian, the projection of moment m enters only in its square as well, and therefore the energy of the particle can be represented in the form (_~ ) ~M ~ = 1 -- - n,l,m + A~. t m~(a ~) 2n~ ' ' ~ A~n.t.~ (a) (19) ~ The energy A~ (a) = A ~ 7 ms (a 2) gives the fine structure. For a << M, the dependence o n m 2 and a 2 disappears, and ~is'fine structure is defined by expression (12). This is a case of a Kerr metric linearized in a, when separation of variables can be carried out to the end by means of spherical harmonics. Deviations from formula (12) take place when a + M. It is possible, however, to show strictly that in this connection the number m = enters only in terms of order ~6M"a2mm/n6. l =. Therefore it is possible to assume that Am (=) = Amn, l (a =) for 2~M<<I+I/2 832 a~.~M. (20) Calculation of this splitting in an extremal Kerr field (a + M) is difficult, and we shall not carry it out. It is more interesting to consider the interaction of a particle with the nondiagonal components g~,~, making a contribution A~ (~) = A ~ ~ m(a). Calculation of this splitting can be carried out independently of the calculation o~t~e fine structure A~ (=). In first order perturbation theory we get r~ // n3l(l+l/2)(l + 1) (21) Thus, an additional interaction between the moment of the central body and the orbit of the particle has a quasimagnetic character and completely takes away the degeneracy of the levels of the spectrum in magnitude as well as in direction of the projection of the moment of the particle. Each level of the spectrum of a particle with quantum numbers n and Z splits into 22 + i sublevels in correspondence with the possible number of projections of the orbital momentum of the particle. In this sense, the considered splitting recalls the normal Zeeman splitting observed in relatively weak magnetic fields, if spin effects are not considered (see [5]). We note that in the considered case the splitting (21) leads to hyperfine structure, since by order of magnitude A~3, ~M ]m{~l. ~ l+ I l It is not difficult to show that in the quasiclassical limit I >> i, the Lense--Thirring effect of the systematic shift of perihelion of the orbit and precession of the orbital plane (if this plane is not equatorial) in the field of a rotating body corresponds to the considered splitting [4]. 3. We showed that relativistic gravitational effects in Schwarzschild and Kerr Fields give rise to a complex structure of the spectrum of the quasiconnected states of massive particles. Under fulfillment of condition (i), they create fine and hyperfine splitting of the levels of the nonrelativistic hydrogen-like spectrum, and the indicated splitting significantly exceeds the width of the levels caused by capture of particles into a black hole (see [i]). The splitting of levels 2p and 2s in a Schwarzschild field ~c~ 4 \ hc / = ~ \ M ~ L / (22) is a characteristic relativistic effect for a scalar particle. Such a magnitude of splitting is retained also in a Kerr field in the case of weak rotation (a << GM/c2), however, it changes under passage to extremal rotation (a + GM/c2). It is interesting to note that this effect is three times greater than the analogous effect for the Kleirr-Gordon equation in a Coulomb electric field in plane space (for effective charge ~M/M~L + Z~; see [5]). The interaction of the orbit of a particle with the angular momentum of a black hole leads to splitting of the 2p level of a spinless particle into three sublevels: 2p(+l), 2p(~ 2p (-~), differing by energy Af.,p paCf G~Mi' _ and, in correspondence with _ = 1 ( a '['1~M'I ~ ) (23) [i], level 2p(+I) is superradiant. In conclusion, we note that in the case of a macroscopic central body, formulas (12), (19), and (21) are unsuitable for states of a particle with orbital momentum somewhat exceeding the critical orbital momentum near the center of a stable orbit. In these cases, the relativistic effects and the effects connected with rotation of the central body become the determining effects. We note as well that in [6, 7] an attempt was undertaken to calculate the energy spectrum of the Kleirr-Gordon and Dirac equations. However, in these articles the particle was localized near the event horizon of a black hole, where the quasidiscrete spectrum is absent. This obviously led to incorrect results. LITERATURE CITED i. 2. I. M. Fiz., I.M. 56-62 Ternov, V. R. Khalilov, G. A. Chizhov, and A. B. Gaina, Izv. Vyssh. Uchebn. Zaved., No. 9, 109-114 (1978). Ternov, A. B. Gaina, and G. A. Chizhov, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 8, (1980). 833 3. 4. 5. 6. 7. D. V. Gal'tsov, G. V. Pomerantseva, and G. A. Chizhov, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 8, 81-85 (1984). L. D. Landau and E. M. Lifshits, Theory of Fields [in Russian], Nauka, Moscow (1973). A. A. Sokolov, I. M. Ternov, and V. Ch. Zhukovskii, Quantum Mechanics [in Russian], Nauka, Moscow (1979). L. H. Ford, Phys. Rev. D, 12, 2963 (1975). B. R. lyer and A. Kumar, Pramana, ~, 500 (1977). DYNAMICS OF THE SYMMETRIES OF A PHYSICAL SPACE AND THEORY OF EQUIVARIANT BORDISMS A. K. Guts and G. B. Fishkina UDC 530 9 The origin of the homogeneity and isotropy of the universe is analyzed on the basis of a theory of equivariant bordisms. A real physical space or, more precisely, its geometry has several symmetries (translation, rotation, and reflection) 9 The general theory of relativity has generated the idea of a geometry of a space which varies in time 9 One can thus speak in terms of the dynamics of spatial symmetries. Since a space is merely "a shadow of a four-dimensional space--time," the extension of this assertion of Minkowski to the symmetries of a space should mean that the spatial symmetries are only one of the manifestations of the symmetries of the space--time. In other words, if G is the symmetry group of the space-time then the restriction of its application in spacelike cross sections constitutes spatial symmetries. This approach to the problem of the symmetries of a space makes it necessary to appeal to the mathematical theory of G manifolds, i.e., manifolds with the action of group G [I]. It is pertinent to recall here that any Riemannian manifold with symmetry group (isometry group) G is a G manifold. At the same time, the question of the dynamics of spatial symmetries is solved on the basis of the theory of equivariant bordisms or G bordisms [2, 3]. We show below how to use these theories to solve the question of the acquisition by a space of such important properties as homogeneity and isotropy in the course of the evolution of the universe. i. We denote by G a group and by M n an n-dimensional, smooth, oriented manifold 9 We denote by G -~ Diff+(M n) a homomorphism, where Diff+(M n) is the group of orientation-conserving diffeomorphisms of the manifold M n. One then says that M n is a G manifold and that G acts on M n. We denote the image of an element ~ 6 G under O-+Dif[_(M '~)by g, and we write g(x) for the image of a point x ~ M rL when element g acts on it. If G is a Lie group, we also require that the mapping G x M n . M n, (g, x) + g(x) from our definition of the action be smooth, We denote a G manifold M n with a fixed action of group G by <G, Mn>. Two G manifolds <G, Mn> and <G, Mn> are equivalent if there exists an orientation-conserving diffeomorphism ~:M?~M,, which is an equivariant mapping, i.e , if we have~(glx))=g(~(x)) for all fI:O, x 6 M'#. Let us assume that <G, Mn> is an orientable closed, i.e., compact without an edge, Gmanifold. It is bordantly zero if one can find acompact, oriented G manifold <G, V n+l > for which <G, ~vn+l> is equivalent to <G, Mn>. Here ~Vn+l is the boundary of manifold V ~+l. For two G manifolds <G, Mn> and <G, Mn> there exists an ordinary unconnected union < O, M ~n, U ~ 4 ~n >. Let us assume --<G, Mn> = <G, --Mn>,where--Mn is the manifold M n with the opposite orientation. The pair <G, Mn> is bordant with respect to the pair <G, Mn> if the unconnected union < O. ~ ! (--M~) > *is bordantly zero. The relation of equivariant bordantness is the relation of equivalency on the set of all closed and oriented n-dimensional manifolds [2] 9 We denote the class of bordisms of G manifold <G, Mn> by [G, M n], and we denote the set of all such classes by ~n(G). The unconnected union of G manifolds induces the structure of an Abelian group in ~n(G). If we omit the Omsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. i0, pp. 76-79, October, 1988. Original article submitted March i0, 1986. 834 0038-5697/88/3110-0834512.50 91989 Plenum Publishing Corporation