Radiation damping in conformally flat universes

ANNAIS

OF PHYSICS:

47, 166172 (1968)

Radiation

Damping

in Conformally J. M.

School of Mathematics,

Flat Universes

HOBBS

University of Newcastle upon Tyne, England

The equations of motion of a charged particle moving in a general Riemannian space, derived by Hobbs [Ann. Phys. (N. Y.) 47, 141 (1968)], are applied to conformally flat spaces. For these spaces the awkward integral term appearing in the equations is shown to vanish identically. In consequence the “tail” associated with an electromagnetic wave is constant and induced solely by the gauge corresponding to the vector potential. A subset of these spaces, which includes those with the de Sitter line element, is found in which the principle of equivalence is satisfied for a charged particle.

INTRODUCTION

The principle of equivalence is one of two experimentally established principles which form the basis for the general theory of relativity. In its simplest form it states that a gravitational force cannot be distinguished from an inertial force by any experiment which is conducted on a purely local basis. While this principle is certainly valid to a high degree of precision, and may even be valid with absolute precision for neutral matter, there is some question as to whether it can be absolutely valid for matter which carries an electrical charge. We have already seen in Hobbs (Z), hereafter referred to as I, that electro-gravitic bremsstrahlung does occur in a general Riemannian space-time. This was reflected by the occurrence of two terms in the equation of motion. The first term, which involved the Ricci tensor, was indicative of the fact that a gravitational field can be readily distinguished from an inertial field by experiments carried out over an extended region; that is, by experiments which measure field gradients. The second term, involving an integral over the whole past history of the particle, had its origin in a more subtle phenomenon associated with the failure of Huygens’ principle in curved space-time. In this paper we investigate the principle of equivalence from a cosmological standpoint in which we restrict our considerations to spaces with conformally flat line elements. The conformal invariance of Maxwell’s equations then imposes a restriction on the function v,,‘, , defined in I, which when applied to the vector potential produces the following result: Although the vector potential has a “tail” in its propagation, this “tail” is effectively constant and can be easily removed by a simple gauge transformation. We therefore conclude that the “tail” associated 166

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II

with the vector potential is induced only by the gauge corresponding to it, and as a result the field must always be propagated with a sharp wave front. Let us now turn to the equations of motion. Substituting the restricted function u,‘, we find that we are left with the usual Dirac equation plus curvature induced terms involving the Ricci tensor. These additional terms are of first order in the derivatives of the particle position and can therefore be considered as a direct modification of the incident field. Furthermore, when the incident field vanishes, the presence of these terms generally prevents the physical solution of the equations of motion being that of geodesic motion. We do find, however, that there are notable exceptions to this genera1 rule, including cosmological models with line elements of the de Sitter type.

I. THE

CONFORMALLY

FLAT-SPACE

CONDITION

Throughout the present work we shall comply with the terminology and notation of I. For ease of working, however, the equations and definitions will here be given in their tensorial form. Consider a particle with coordinates ZQ, mass m, charge e and proper time T given by

(1) where T,+ is the flat-space metric tensor and e2Ris the conformal factor. We define the vector potential at a point X, due to this particle, to be A,@)

= 4n-e I G,f,(x, z) dz”,

(2)

where G,,, is the bi-vector Green’s function solution of the wave equation gv’o’Gp’a.“‘o~ + R”,:G,/, = -~-1’2&~,&~z,

,

(3)

which, by (I, 2.27), takes the form G,,, = (87r--l(~P/~&~

6(u) - u,/J+u)>.

(4)

Let us now introduce a scalar gauge parameter, /1, defined by A = 4rre

i

#(cc, z).~ dz” = 4ne(#(x,

ZJ -

#(x, z,)),

(5)

where # is an arbitrary b&scalar and z1 and z2 represent the end points of integration. Arbitrary changes in gauge of the vector potential can then be represented by A;

= A,, - A.,, .

69

168

HOBBS

Using (2) we can convert this into the following Green’s function equation qa

= G,*, - ljkptor .

(7)

Our next step is to introduce the b&scalar Green’s function given by G(x, z) = (87~)-~{d~/~ 6(u) - vO(--u)}, which is a solution of the inhomogeneous wave equation I Ig” = G.,,,, = -g-1’2@z, ,

(8)

(9)

and choose the particular form for the b&scalar #

KG4 = @#9 qx, z), where @ is another arbitrary bi-scalar. Combining Eqs. (4), (7), (8), and (10) we see that the coefficient of the Heaviside step function in the Green’s function Cf, is Vp’a - @.pL .

(11)

We do know, however, in view of the conformal invariance of Maxwell’s equations, that it is possible in conformally flat spaces to choose a gauge such that the vector potential takes the flat space form. In this case the corresponding Green’s function will contain no Heaviside step function. This means that in conformally flat spaces it is possible to choose a bi-scalar 43, as given in (1 l), satisfying the relation VP’” = @.c’OL*

(12)

(The validity of this condition has, in fact, been checked by the somewhat longer process of using the flat-space vector potential to determine the equations of motion of a particle in conformally flat spaces.) II. THE VECTOR

POTENTIAL

Consider now the vector potential for conformally flat spaces in which we have the bi-tensor v,,, satisfying the condition (12). Introducing the bi-scalar @ we have for the advanced and retarded potentials

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169

Equation (13) shows us that the effect of the integral term is to introduce additional solutions of the wave equation, not only on the null cone at x but also solutions off the null cone represented by @.,(x, z(& co)). Clearly this term is effectively constant with respect to the particle position and therefore the induced “tail” in the propagation of the vector potential is also constant. This point can be shown more clearly by considering the gauge transformation for the retarded solution A;:

= ‘4; + A,.,, )

(14)

where the gauge parameter is chosen to satisfy A, = 43(x,

z(-co)).

(15)

The new retarded solution will therefore take the form (16)

all of whose solutions lie on the null propagated with no tail, however, it the Lorentz gauge condition. Finally (13) the field quantities will always this is nothing new since it is implicit equations. III.

THE

cone at x. This vector potential is therefore must be remembered that it will not satisfy we note that, because of the special form of be clear cut in their propagation, however in the conformal invariance of Maxwell’s

EQUATIONS

OF

MOTION

Referring to (I, 5.28) and using the condition given in equation (12) we see that the equations of motion of a charged particle in conformally flat spaces can be written in the form

where

inspection of (17) shows that in general, when the incident field vanishes, the physical solution of (17) is prevented from being that of geodesic motion by the inclusion of the terms involving the Ricci tensor. There are, however, spaces of considerable interest in which we have the quantity given in (18) vanishing identically. The necessary and sufficient condition for this to be the case is, from (18),

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HOBBS

In these spaces the equations of motion (2.16) reduce to the Dirac form

We therefore conclude that in the absence of an incident field the physical solution of the equations of motion is P = 0, that is, geodesic motion. Thus, only in spaces in which the conformal factor satisfies (2.18) can we say that, in a state of free fall, both an uncharged and a charged particle follow a geodesic path. The question now arises as to which cosmological models possess these unusual attributes? It can be easily seen that the most general solution of (19) takes the form e* = (u,F + c)-I,

(21)

where a, is an arbitrary constant four-vector and c is an arbitrary constant. The imposition of spatial isotropy effectively reduces the number of possibilities in (21) to one. This solution is eR = (Ht)-l,

(22)

which leads to the line element dam = (Ht)-‘q+

dzb dza,

(23)

where His the present value of Hubble’s constant. This can be written in the more convenient Robertson-Walker form dT2 = c2 dt2 - R2(t) du2,

where du2 is the metric of a three-dimensional

(24)

space of constant curvature and

R(t) = eHt.

(25)

This is, of course, the metric of the de Sitter universe. There are two cosmological models possessing this type of line element, and we shall deal with these individually. CASE

(a)

THE

DE SITTER

MODEL

This is a stationary model of the universe, suggested by Ehrenfest and investigated by de Sitter. For the line element dr2 = g,+.,de dZa,

the Einstein field equations, with the cosmological R, - g&R

- A) = -KT,

(26)

term, take the following form ,

(27)

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171

where Tw is the stress-energy tensor for the system. For the general line element (24), the field equations (27) take the form Tpy = (p + p/c”) &+‘ 6,y - pgp”,

(28)

where Kpc2 = --h + 3(k + R’2~-2)lR2, Kp = h - 2R”/Rc2 - (k + R’2/~2)/R2,

(29) (30)

the prime indicating differentiation with respect to t, p being the density of matter and p the pressure. For the de Sitter model, where k = 0 and R is given by (25), we find from (29) and (30) Kpc2 = --x + 3H2/c2,

Kp = x - 3H2/c2,

(31)

and on requiring that neither p nor p be negative, it is evident that they must both vanish and so p = 0,

P = 0,

h = 3H2/c2.

(32)

We are thus dealing with that approximation in which the effect of matter on the underlying metric is neglected; the nebulae are to be considered as test particles whose existence has no effect on the structure of the Universe. In the absence of matter it is not surprising that both charged and uncharged particles in a state of free fall follow a geodesic path. Our analysis therefore confirms an expected result. CASE (b) THE STEADY-STATE

MODEL

In 1948 Bondi and Gold (2) proposed a cosmological model which does not rely on Einstein’s equations. In their theory they do not postulate a specific link between matter and geometry in the form of a mathematical relation between the stress tensor and the Ricci tensor, but instead postulate a constant energy density p and demand that an observer at any point in space-time find the same Hubble recession. All points in space-time thus have to be equivalent. The above two requirements preclude the possibility of an energy conservation law. In fact, to keep the matter density p constant and to compensate for the expansion of the universe, it is necessary to assume that matter is created at a constant rate throughout space. The postulates of homogeneity and equivalence of all world points completely determine the metric. It is, in fact, of the RobertsonWalker type, and furthermore, because of the assumed constant rate of expansion of the Universe it must also be of the de Sitter type (24) with R(t) given by (25).

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The question now arises as to the interpretation of our result that free-falling charged and uncharged particles follow a geodesic path. In this context two possibilities occur. Either electro-gravitic bremsstrahlung averages out to zero for a charged point particle, or the creation field, which we have not taken into account, effects the equations of motion. The first case implies the introduction of a creation field which depends only on the end points of broken world lines-for example, the one defined by Hoyle and Narlikar (3) in terms of the scalar Green’s function

C(x) = f-l J G,, L/P,

(33)

where f is a coupling constant. With this definition of the creation field the principle of equivalence for charged particles would be satisfied. In this case, therefore, the steady-state model is a model of particular interest. Finally we consider the second possibility. A creation field introduced in the manner of (33) would now be required to depend on all points along the broken world line, not only on the end points. This would have the effect of altering Maxwell’s equations and therefore lead to a corresponding modification of the equations of motion. We shall not pursue this situation further, although it is noteworthy that a creation field of the form (33) only effects Maxwell’s equations to the extent of a gauge transformation. This, however, would not be the case for the introduction of a creation field of the latter type.

ACKNOWLEDGMENT 1 wish to thank Dr. C. Gilbert for his helpful advice during the progress of this work. RECEIVED: August 24, 1967

REFERENCES J. M. HOBBS, Ann, Phys. (N. Y.) 47, 141 (1968)-preceding paper, this issue, hereafter referred to as I. 2. H. BONDI AND T. GOLD, Monthly Notes Roy. Astron. Sot. 108, 252 (1948). 3. F. HOYLE ANLI J. V. NARLIKAR, Proc. Roy. Sot. (London) A282, 178 (1964). 1.