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ELEMENTARY PARTICLE PHYSICS AND FIELD THEORY GRAVITATION FIELD IN THE VAIDYA PROBLEM ALL@WING SEPARATION OF VARIABLES IN THE HAMILTON--JACOBI EQUATION Vo G~ Bagrov, V. V. Obukhov~ and A~ V. Shapovalov UDC 530~12:531o51 I. INTRODUCTION The problem of solving the Einstein equations with an energy-momentum tensor of matter in the form T~j = 4~ Q~:~J' k~# = 0 is known in the literature as the Vaidya problem. This problem was first investigated in [i, 2]. In particular, Einstein equations averaged over the space--time domain that describe the gravitation field in the presence of high-frequency, low-amplitude radiation (gravita- tional or electromagnet) that is propagated in a background of curved space-time reduce to the Vaidya problem. High-frequency electromagnetic radiation is described by light rays in the geometric optics approximation, Investigation of light ray propagation is an effective means of studying the structure of space-time and of physical processes occurring therein. Another important means of analyzing GRT fields is investigation of the motion of test particles. In this connection, the solution of the Vaidya problem is of interest in cases when the equations of motion of test particles can be integrated effectively, A solution of the Vaidya problem is considered in this paper under the condition that the Hamilton--Jacobi equation for a test charge averaged over the space--time domain, is integrated in a zero approximation by separation of variables by using isotropic full sets of motion integrals. 2. FUNDAMENTAL EQUATIONS In order to establish the influence of the high-frequency electromagnetic field on the averaged motion of the test charge, we consider the fundamental considerations resulting in the Vaidya problem. We write the self-consistent system of Einstein-Maxwell equations (with the cosmological term A) in the form gavgFj~ = 0. (2) Here, i, j, ~ = 0, I, 2, 3; gij is the metric tensor of the signature (%, , ~ --)~ gi~gK~ = 6 i is the Kronecker delta, Rij is the Ricci tensor, V. is the operator of the cow,riant J j ~J . derivative with respect to the variable x relative to the tensor giji Fij is the electro- magnetic field tensor given by the potential Ai by the formula Fij = Aj, i -- Ai~ j; the comma denotes the partial derivative, Tij(~) (i/4~)[g13FmlFml/4 ~ FIkF3 ~ is the e._ectro~ magnetic field energy-momentum tensor given ty the tensor~~ij;,• = const, The Hamilton-- Jacobi equation for the test charge in an external field Ai is written in the form #3 (s, ~+A~) (S j+~j)--~ = 0. (3) We assume that the electromagnetic field in a certain coordinate system is the super- position of a slowly varying field A i and a low-amplitude high-frequency field Bi: V. V. Kuibyshev Tomsk State University. Instituteof Heavy-Current Electronics, Tomsk Division of the Siberian Branch of the Academy of Sciences of the USSR. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. i0, pp. 3-8, October, 1986. Original article submitted August 29, 1984o i0038-5697/86/2910-0775512.50 1987 Plenum Publishing Corporation 775

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Page 1: Gravitation field in the Vaidya problem allowing separation of variables in the Hamilton-Jacobi equation

ELEMENTARY PARTICLE PHYSICS AND FIELD THEORY

GRAVITATION FIELD IN THE VAIDYA PROBLEM ALL@WING SEPARATION

OF VARIABLES IN THE HAMILTON--JACOBI EQUATION

Vo G~ Bagrov, V. V. Obukhov~ and A~ V. Shapovalov UDC 530~12:531o51

I. INTRODUCTION

The problem of solving the Einstein equations with an energy-momentum tensor of matter in the form

T~j = 4~ Q~:~J' k~# = 0

is known in the literature as the Vaidya problem. This problem was first investigated in [i, 2]. In particular, Einstein equations averaged over the space--time domain that describe the gravitation field in the presence of high-frequency, low-amplitude radiation (gravita- tional or electromagnet) that is propagated in a background of curved space-time reduce to the Vaidya problem.

High-frequency electromagnetic radiation is described by light rays in the geometric optics approximation, Investigation of light ray propagation is an effective means of studying the structure of space-time and of physical processes occurring therein. Another important means of analyzing GRT fields is investigation of the motion of test particles. In this connection, the solution of the Vaidya problem is of interest in cases when the equations of motion of test particles can be integrated effectively,

A solution of the Vaidya problem is considered in this paper under the condition that the Hamilton--Jacobi equation for a test charge averaged over the space--time domain, is integrated in a zero approximation by separation of variables by using isotropic full sets of motion integrals.

2. FUNDAMENTAL EQUATIONS

In order to establish the influence of the high-frequency electromagnetic field on the averaged motion of the test charge, we consider the fundamental considerations resulting in the Vaidya problem.

We write the self-consistent system of Einstein-Maxwell equations (with the cosmological term A) in the form

gavgFj~ = 0. (2)

Here, i, j, ~ = 0, I, 2, 3; gij is the metric tensor of the signature (%, , ~ --)~ gi~gK~ = 6 i is the Kronecker delta, Rij is the Ricci tensor, V. is the operator of the cow,riant J j ~J .

derivative with respect to the variable x relative to the tensor giji Fij is the electro- magnetic field tensor given by the potential Ai by the formula Fij = Aj, i -- Ai~ j; the comma denotes the partial derivative, Tij(~) (i/4~)[g13FmlFml/4 ~ FIkF3 ~ is the e._ectro~ magnetic field energy-momentum tensor given ty the tensor~~ij;,• = const, The Hamilton-- Jacobi equation for the test charge in an external field Ai is written in the form

#3 (s, ~+A~) (S j + ~ j ) - - ~ = 0. (3)

We a s s u m e t h a t t h e e l e c t r o m a g n e t i c f i e l d i n a c e r t a i n c o o r d i n a t e s y s t e m i s t h e s u p e r - position of a slowly varying field A i and a low-amplitude high-frequency field Bi:

V. V. Kuibyshev Tomsk State University. Instituteof Heavy-Current Electronics, Tomsk Division of the Siberian Branch of the Academy of Sciences of the USSR. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. i0, pp. 3-8, October, 1986. Original article submitted August 29, 1984o

i0038-5697/86/2910-0775512.50 �9 1987 Plenum Publishing Corporation 775

Page 2: Gravitation field in the Vaidya problem allowing separation of variables in the Hamilton-Jacobi equation

A~ = Ai+B~. (4)

Questions associated with the concept of the "slowness" of the variation of the field Ai and the "rapidity" of the variation of the field B i are discussed in the known paper of Isaacson [3]. Appropriate discussions can also be found in Sec. 1.3 in [4] and [5]. We briefly recall the fundamentals. The characteristic parameters are the distances L and I in which the fields A i and Bi, respectively, vary substantially. In a space--time domain of dimension d

l << d <<.L (5)

the electromagnetic waves are propagated as plane waves in a background of almost plane space-time. Selected as the smallness parameter is e = I/L, e << ~, it is hence assumed that

Ar = O(1), A~,j.= O(1), B~ = O(e), B~,j = O(1). (6)

The electromagnetic field tensor has the form

f~j=.o(1), h i= 0(1).

In principle, the presence of the high-frequency electromagnetic field can cause the appearance of a high-frequency gravitational field (gravitational waves). Then the space matrix is represented in the form gij = Yij + hij. Here Yij is a smoothly varying metric of the background space, and hij is the low-amplitude high-frequency gravitational wave induced by the electromagnetic wave. By using (i), it is easy to show that hij has an order in c not less than 0(~2). For this reason, as well as because of the linearity in hij (with slowly-varying coefficients) of the highest terms in e in (i) and (3), the induced gravita- tional wave hij does not enter into the average equations (i) and (3) if it is limited to the highest terms in e therein. On the other hand, the estimates presented in [6] show that the energy of the induced gravitational wave is quite small compared with the electromagnetic wave causing it because of the smallness of the coupling constant, even in the presence of resonance conditions. Takingthese circumstances into account, we do not take account of the induced gravitational wave hij and we consider gij a smoothly varying metric (varying substantially within distances on the order of L).

Following [3-5], in order to obtain the average equations in the geometric optics ap- proximation, we represent the electromagnetic wave potential B i in the space--time domain of dimension d that satisfies condition (5), in the form of plane waves. Without limiting the generality, we can limit ourselves to a monochromatic wave

B. = a.e i~ + "~e-iO. (7)

Here a M is the slowly varying amplitude, and ~K is the complex conjugate to a~

a~ = 0 ( ~ ) , a~,j = 0 ( ~ ) , ~ j = O,j, ~j = 0 ( ~ - ~ ) , ~j,~ = 0 ( ~ - ~ ) . ( 8 )

Substituting (4) and (7) into (i) and retaining terms highest in E, after taking the average over a space-time domain of dimension d, we obtain the equation [3, 4]

Rij - -Agi j = 4~• +• (9)

where the q u a n t i t y Q = - ( a i a i + a i a i ) i s p r o p o r t i o n a l to the e f f e c t i v e energy d e n s i t y of the e l e c t romagne t i c wave, ~i i s the r a d i a t i o n f l u x p ropaga t ion v e c t o r [4] , and 4~Tij(F) = g iJFm~F~/4 -- FiKFj k i s the energy-momentum t enso r of the s lowly va ry ing e l e c t r o m a g n e t i c f i e l d .

We s u b s t i t u t e (4) and (7) i n t o the Maxwell equa t ion (2) and impose the Lorenz c o n d i t i o n g iJviAj = 0 on the p o t e n t i a l (4). Separa t ing the s lowly va ry ing p a r t (terms wi thou t the exponen t i a l f a c t o r e l0) and the r a p i d l y o s c i l l a t i n g p a r t (terms having the e x p o n e n t i a l f a c - to r ) in these equa t ions and t ak ing account of the order of smal lness in ~ of the q u a n t i t i e s (6) and (8) e n t e r i n g t h e r e , we o b t a i n the fo l l owing equa t ions [ 3 , 4]

W F~j = 0, (I0)

776,

Page 3: Gravitation field in the Vaidya problem allowing separation of variables in the Hamilton-Jacobi equation

IcjK l = O, ( i i )

Vj (QmJ) = 0, (12)

ai~ = O~ (13)

Taking into account that ~j = O, j, we obtain from (ii):

~lVT~i : K~Vi~] = O. (14)

To obtain the average Hamilton--Jacobi equation, we substitute (4) and (7) into (3) and we represent the desired function S in the form

S = S + R e m + R e -'~

Here S = 0(i), R + 0 as e § 0. In order to establish the order of the quantity R in the parameter , we extract the rapidly oscillating part (the terms containing the exponential factor e i0) from (3) and by equating it to zero we retain the terms highest in e in the equation~ We obtain R = iE(Pa)/PK) to the accuracy of terms on the order of 0(c3). use the notation

P i ' = S , i + A i , (Pa) = P i a i, (PK)-~-PiKq

We here

Taking into account that R = O(E2), as well as (II) and (13), and after taking the average over the space-time domain of dimension d (5) (which reduces to discarding terms containing the exponential factor e i0 in (3) in the long run), we obtain the average Hamilton--Jacobi equation in the form

gi~P*P~--Q--m2-----O" (15)

Let us note that the first term is of the order 0(i), Q = O(E=).

Let us pose the general problem of classification of the solutions of the system (9), (12), (14) under the condition that (15) allows total separation of variables in the zero approximation Q = 0 in a certain (privileged) cordinate system. An analogous problem for electrical vacuum spaces without high-frequency radiation was solved completely by the authors in [7, 8]. Consequently, this problem is actually a continuation of the papers mentioned. We, henceforth, retain the definitions and notation used in [7, 8]. In addition we introduce just one other definition, the electrical vacuum Stoeckel--Vaidya space (or the vacuum space in the absence of the slowly varying electromagnetic field Fij) of the type (n.n ~ ) will be called the solution of the average self-consistent system of Einstein--Maxwell equations (9)-(12), (14) in the presence of high-frequency electromagnetic radiation if (15) allows complete separation of variables of the type (n.n') in the zeroth approximation Q = 0.

All Stoeckel--Vaidya vacuum spaces of the type (n.l) are found in this paper~

3. SPACES OF THE TYPE (n.l)

Using the explicit form of the metric of Stoeckel spaces of the type (n.l), [7, 8], we represent the linear element in the Robinson--Trautman form by using an obvious coordinate transformation

d~ 2 = 2 (Adx o + Tdx~ + Bdx~ + Cdx3) dx o -- U~dx~ -- V2dx~. ( -16)

In this section the functions of one variable are denoted by lower-case Greek letters with a right subscript a3 = a3(x3), the constants are denoted by lower-case Latin letters without subscripts, all the remaining functions are denoted by upper-case letters, and the coordinate indices are located below. The functions A, B, C, T, U, V in (16) have the fol-

lowing form.

For spaces of the type (i.i): A =--AM/2Vo =, B = C = 0,

~J'~= - ~ / G , v ~= -ca~G, M = %Vo +~2G + ~ G ,

a = a o vo + a~ G + a, G , Vo = ~ - ~ , G = ~ - %,

v 3 : % - ~ 2 , T =A/Vo , ~, ~ = • 1.

777

Page 4: Gravitation field in the Vaidya problem allowing separation of variables in the Hamilton-Jacobi equation

For spaces of the type (2.1):

A = - - ( A / 2 ) ( % + f 2 ~ / O ) , T = - - V t = A = ~ o + ~ a ,

B = ~alO, C = O, U s = A/G, e = ~'~o + "~ + +oX,, 6 = - ( ~ o ~ + 2~o~,~ + "r~ + Po)

(the dot denotes the derivative with respect to the corresponding variable).

For spaces of the type (3.1):

A = ( 1 / 2 ) ( * ] + d ) ~ ) , B = ~ 2 , C=(Da , T = - - U ~ = - V ~ = - 1 ,

| 0 "

(17)

(18)

We select the tetrade as follows

I~ = {1, 0, 0, 0), n~ = . (A, T, B, C), m~ = (0, 0, Ull /2 , i V I V 2 ) .

Let us present the spin coefficients

~ = p = ~ = ~ = 0 , ~=- - -~ , ~ + ~ = - - ~ ,

~. = [ ( & , - T~)IU + ~ ( T~ -- C ,)l V]t2 V~T,

= = [ V(T,2 - - Be ) + 2 r v , 2 -47 i ( C , i U - - UV3 - - 2TU, s)]/4 V 2 V U T ,

={ U, o V + U V,o + i [(BT~ -- CTa)]T + C,2 - - B~ + (CB,t - - CaB) /T]} /2UV,

= {V[T ( B , o - A~) + B (An --Tie) + A (T,2 - - B e ) ] + iU [T(Aa --C,o) + C (T,o -- A,O + A ( C a - T~)]} / I /2VUT,

7 ---- {2 (An -- T.o) U V + i[BT~ -- T&z + caT - CaB -- C (T,~ -- B O]}/4VUT.

Since ~ = 0 = ~ = e = O, there follows from the Newman--Penrose equations w.o = ~I = r = ~ox = 0. Therefore, ~i agrees with the principal isotropic direction while the element (16) yields algebraically special solutions.

In our case

eOoo = tcdcjliPQ, ~,o = tcacjVmJQ, ~ o = xac'/mtmJQ,

IDIa = tCtlcjlirdQ, do21 = &xjnimJQ, eo~ = tcd:ln~nlQ,

hence <i~ i = 0. T h e r e f o r e , Ki = a l l + bmi + bmi . S i n c e Kic i = 0, t h e n b = 0. Hence , t h e vector c i is collinear to the vector I i and there is just one nonzero component of the pro- jection of the Ricci tensor deviator ~aa = e12Q.

Let us write all the nonequivalent equations of the Newman--Penrose system (excluding

those which are definitions of the quantities Pi):

~ - - ~Xt~ - - ( ~ + ~,~) - - (7 + ~ ) ~ + - ~ + (3~ + ~' + ~) ~ = 0~Qx,

D (2~ - - 7) + ~ (= - - 2,:) - - ' ~ - - 2 ~ + 2= (~ - - ~) - - ~ - - - ~ + 2~,~ - - A ----- 0,

ox - ~ - ~:~ + ( ~ - ~) ~ = o,

' ~ - - g~ + D-f - - =~--- ~ + 2=~ + =~ = 0, (19)

~r - ~ + (~ - - ~ - ~) ~ - ~ + ~ = o

( h e r e A = R/4 i s t h e c o s m o l o g i c a l c o n s t a n t ) .

I n o r d e r to s a t i s f y t he c o n d i t i o n s ( l l ) and ( 1 3 ) , we s h o u l d s e t 0~ = 1. Then t h e f i r s t equation of the system (19) permits determination of the function Q in terms of the functions A, B, C, T, U, V, which are found from the remaining equations in the system (19). Omitting the details of the computations, we present the final results (the functions of one variable in the cases following are arbitrary if the specific form is not indicated).

Type (i.i). In (16) we should set

A =- -6o (,o2+~o3)/2 Vo, U~ .... V ~ = 60Vo.

I) Vo = ehxi - -cosm;

3) Vo=x~, + x ~ ;

R----.C----A--O, T----~o,

2) Vo'= exp&;

4) V o = 1.

778

Page 5: Gravitation field in the Vaidya problem allowing separation of variables in the Hamilton-Jacobi equation

Type (2.1)o In (17) we should set

1. Oa = a~ a," 3u;] ~= A ( ~ ] + 6c2o~] - - 3c ~) + 6 x % , - - A = % -~ + c 2, % = 2 c / ~ o.

1) % = a~,~/~], o = • (a~o + b ) / ~ + 2cx, L , ~g = 1/(a~ o~+ 20~o + d); 2) c = 0 , o = IA2/~, ~o = const ;

3) c = ~c = o, e = lx~, ~ = V 3 / A / x ~ .

II. % = A = 0 , A = - - p g .

1) 9 = l/'x~, pg = a ~ 0 , ~a = bx3; 2) P, = lx3, ~ = cons t .

Type (3.1). In (18) we should set Bov = 0.

The authors are grateful to Doctor of Physical-Mathematical Sciences V. S. Smirnov for interest in the research and useful discussions of the results.

LITERATURE CITED

i. V. V. Narlikar and P. C. Vaidya, Nature, 159, No. 4045, 642 (1947). 2. P. C, Vaidya, Phys. Rev., 83, No. i, i0 (1951). 3. R. Ao Isaacson, Phys. Rev., 166~ No. 5, 1263-1271; 1272-1280 (1968). 4o V. P. Frolov, Tr. FiAN SSSR, 96, 72-180 (1977). 5o C~ Mizner, K. Torn, and J~ Wheeler, Gravitation [Russian translation], Vol. 2, Mir,

Moscow (1977), 6o Ya. B. Zei'dovich, Zh. Eksp. Teor~ Fiz., 6_55, No. 4(10), 1311 (1973). 7. V. G. Bagrov and V. V. Obukhov, Ann~ Phys., 40, No. 7, 415 (1983). 8. Vo G0 Bagrov, V. V. Obukhov, and A. V. Shapovalov, Vyssh. Uchebn. Zaved., Izv. Fiz., No.

i, 6 (1983).

EFFECT OF VIOLATING T-INVARIANCE OF THE ELECTROMAGNETIC INTERACTION

OF HADRONS IN THE PROCESSES n + p + d + y AND n + p + d + e + + e--

M. Po Rekalo UDC 539.12

T-odd correlations that are possible in the processes n + p + d + u and n + p + d § e + + e--and that arise due to the capture of thermal neutrons are determined in terms of threshold amplitudes (for the process n § p + d + y) or electromagnetic inelastic form factors (for the process n + p § d + e + + ~-)o The mechanism of these reactions is not given in concrete form, but those general properties of the electromagnetic hadronic current are used that are preserved with three-dimensional reflections~

I~ The invariance of the electromagnetic interaction of hadrons for temporal reflec- tions (T-invariance) has been studied in many different experiments. The search for T-odd asymmetry of inelastic scattering of electrons from a polarized proton target, e--+ p + e-- + X [I], the transverse polarization of deuterons in the reaction e-- + d + e-- + d [2], and the T-odd correlation in the decay Z ~ § A + e+e - [3] has yielded negative results. The search for a violation of T-invariance was done in processes with real photons: by comparing the differential cross sections of the direct and reverse reactions n + p § d + ~ and y + d + n + p [4] and by measuring the ~elative phase of multipole amplitudes for electromagnetic

Kharkov Physicotechnical Institute, Academy of Sciences of the Ukrainian SSR. Trans- lated from Izvestiya Vysshikh Uehebnykh Zavedenii, Fizika, No. I0, pp. 8-13, October, 1986.

Original article submitted August 27, 1984~

0038-5697/86/2910-0779512.50 �9 1987 Plenum Publishing Corporation 779