Post-Newtonian limits and gravitational radiation from the superstring effective action

NUCLEAR PHYSICS B ELSEVIER

Nuclear

Physics B 423 (1994) 305—326

Post-Newtonian limits and gravitational radiation from the superstring effective action Alexandros A. Kehagias 1 Institute of Theoretical Physics, University of Nijmegen, Toernooiveld 1, 6525 Nijmegen, The Netherlands Received 25 January 1994; accepted 3 March 1994

Abstract We consider the gravitational sector of the superstring effective action with axion-matter couplings. The field equations are developed in the post-Newtonian scheme and approximate solutions for a spinning point mass and a cosmic string are presented. Furthermore, assuming vanishing axion mass and vanishing potential for the dilaton, we consider the gravitational radiation in the leading O(a’°) order. We find that the total luminosity of a radiative source has monopole and dipole components besides the standard quadmpole one. These components may possibly be checked in binary systems.

1. Introduction After the first attempts to formulate a consistent quantum theory of gravity [1], it became apparent, and strongly supplied by the non-renormalizability of general relativity [21, that the effective action of such a theory would involve, besides the Einstein tenn, also terms in higher powers in the curvature [31.These terms will determine the shortrange (high-frequency) behavior of the theory while, at large distances, the theory will be classical gravity as described by the (low-frequency) Einstein term. However, the incorporation of such high-power curvature terms in the action does not lead to the resolution of the renormalization problem. The reason is that the equations one obtains are of order higher than two. As a result, the corresponding theory has, inevitably, indefinite energy at the classical level and indefinite metric Hubert space with ghosts at the quantum level. ‘Supported by CEC under contract ERBCHBGCT920197. 0550-3213/94/$07.00 © 1994 Elsevier Science B.V. All rights reserved SSDJ 0550-3213(94)00123-V

A.A. Kehagias/Nuclear Physics B 423 (1994) 305—326

306

Today, string theory [41 seems to be the best candidate for a consistent formulation of quantum gravity and, in addition, it has the prospects to unify all interactions. However, the greatest challenge for string theory, as well as of any theory, is to make contact with experiment, and even more, to make predictions. On the other hand, all particle properties are originated from the particular solution of the string world sheet and it is difficult, in view also of the absence of any classification, to extract results and to make predictions. In this case, one may look for common features of the solutions and such a feature is the presence of tensor excitations (metric). Thus, gravitation is a universal aspect of any closed critical string theory and string-generated gravity is, potentially, a promising region for tests and confrontation of string theory with experimental and observational data. Graviton excitations in any closed critical string theory are members of a multiplet which contains, besides the graviton, also the massless modes of the scalar dilaton and the Kaib—Ramond antisymmetric two-form field [4]. The leading term in the effective action, in a conformal frame, consists of the Einstein term and the kinetic terms for the other fields. In the next-to-leading order in the a’ expansion and at the string tree level, the universal part of the effective action in any four-dimensional heterotic superstring model takes the form [6—8]

i=

f

d4xvT11~~

—

~

—

—

6e_2H,pH~

F~WF~~) + Ae

~HKApJ~E~~L +

where e~””°is the totally antisymmetric symbol (with R2

80123 =

~matter)

,

(1)

+ 1) and

—

i~’Po~4p/LP

GB

~

/LVm

2

2

is the Gauss—Bonnet tensor. This term has previously been considered in higherdimensional pure gravity theories [9] and has also been employed in the Kaluza—Klein framework [10]. The new feature in the action (1) is that we have incorporated5y~fr the coupling of the antisymmetric field toit the axial vector J~ i/iy with, in our notation, A = K/i2 three [7]. form (More~ correctly, is the expectation value of the axial vector that ~ should be coupled.) Moreover, we have collectively denoted by £matter all field contributions that will serve as the source for gravity, since the variation of this term will provide the “effective” energy—momentum tensor. The antisymmetric H,~~~-field is given by =

where ~

~

+

~

—

(3)

w~),

is the antisymmetric component of the graviton multiplet and =

~

+~

+ ~

Moreover, the Lorentz and gauge Chern—Simons three-forms ~ Tr denoting trace over suppressed Lorentz and gauge indices, by

w3T

are given, with

A.A. Kehagias/Nuclear Physics B 423 (1994) 305—326

3’~=Tr(wAR ~wAwAw), w w3T=Tr(AAF_~AAAAA),

307

(4)

with A the gauge field and w the spin connection. Usually, in the absence of sources, one may proceed further and write, at least locally, the three-form with components ~ as the dual of an exact one-form, namely as H 1app

=

—~e/~V~K~9a,

(5)

where a is the axion field. Solutions for the axion [11,13,14] and the dilaton [15] have largely been studied, the former in particular in a Kerr—Newmann background. However, when sources are present, it is not possible to express the H-field as in Eq. (5) although it may always be written as the dual of a non-exact one-form. The major reason for studying the string-generated gravity is that, as we have mentioned, the gravitational sector is a generic feature of any string theory and thus its study will improve our understanding in a model-independent way. Moreover, keeping in mind that general relativity is an experimentally tested theory [16], perhaps there is also room for similar tests in string theory [17]. This is why an extensive study of the theory is needed, which will lead, hopefully, to predictions of effects not present in general relativity. For such a purpose, one does not need to go through all the details of the theory. All predictions of general relativity, for example, have been extracted in some approximation scheme, namely, in the post-Newtonian or the weak-field approximation. In the following two sections, we are dealing with the post-Newtonian approximation method which is appropriate for slow-moving bound systems. We also examine in sect. 4 the gravitational radiation in the weak-field approximation.

2. The post-Newtonian approximation

The action (1) leads to nonlinear field equations which cannot be solved in general. Exact solutions can be found in special cases and a guide to such a search, as in general relativity, will be the isometries of spacetime. However, in view of the complexity of the field equations, it is important that some systematic approximation methods be developed, so that the main properties of the system under study become apparent. There are two such methods, the post-Newtonian and the weak-field approximation [18,20]. Here we will deal with the former case which is based on the assumption of slowmoving gravitational bound systems and let us recall some well-established results of this method based mainly in Ref. [18]. We assume that there exists an expansion of the metric in inverse powers of the speed of light for slow-moving gravitational bound systems. Since this speed has been fixed here to one, the post-Newtonian approximation corresponds to an expansion of the metric tensor in powers of a small quantity which may be taken to be a typical

308

A.A. Kehagias/Nuclear Physics B 423 (1994) 305—326

velocity of the system. Thus we expect an expansion of the form2 goo=_1~(2)goo+(4)goo~..., ,_~~(2)

~(4)

g, 1

g~1

—

g~1

. . .

~go1 + (S)g0~k...

=

(6)

where (fl)g15~is of order Greek indices run from 0 to 3, while Latin ones from 1 to 3. The inverse metric has also an expansion which may be written as “.

g00 =—1+

‘2

g00 +

4\ 00

g

+...,

1~+” g + ‘g +..., 2’’J 4 ~J

g”=c~ g0~= (3~g0~ +

+

(7)

. . . ,

with (2) 00 g

(2)

— —--

(2) ‘3

—

j2)

g 00,

g

~

—

as one may easily verify using —

g~apg

—

This expansion of the metric gives rise to a similar expansion for the connection. In fact, since the connection is expressed in terms of the metric as F~~p =

+ 8,LLgkP

~gAK(~i,,g,~

—

we find that F-symbols with even number of zeros (e.g. 2)p~ + ~4~F~~ = ( 9A +

P00)

may be written as

. .

while those with an odd number (e.g. 3)f’~~ + =

f”10)

as

+...

(

This follows from the assumption that the expansions in powers of holds in the near zone r < t and thus spatial derivatives are of order 0 (~0) while time derivatives are of order O(1). The first terms in the above expansion for the connection may explicitly be written as =

2)p

=

~2~r~

=

~(a~(2)g~J+

00

,9J(2)g~~ —

8~(2)gJ~)

( 2)g

(3)fO 00 = 2

_~~(

00,

The conventions we adapt3AF1’~ here is —that of Ref. [191, namely, +2 signature for the metric and Riemann 9KF1’A~ + V’~ApV~KP — V~’KpF1’A,~. R~’,.,AK= +

tensor defined by

A.A. Kehagias/Nuclear Physics B 423 (1994) 305 —326

309

2~g~

(3)~i

10 =

~

+ a1~

0

3~g~ 4~g + a,~ 0~8~~ 00) 4)g~J + E9J(4)g~~ ,9~(4)gJ~)

~

=

—

—

,

—

=

~(8~~ (4)j’i 00

(8)

It is straightforward now to verify that the Riemann tensor, as defined in footnote 2, has an expansion in even powers in for components with an even numberemploying of zeros 0mnt). Explicitly, (e.g. (6)—(8), R°m0n) and odd powers in for odd ones (e.g. R Eqs. we infind (2)

1 ROmOk

(2)

g

~~~9tnak

00, .~(8~~~(2)g

2~Rlmnk = ~

—

~3~R0mnk =

~

—

2)g~ ömên 10 o/a~~ ~afla,(2)g~~ + —

~4~ROmOk= _~a,2(2)g~~+ .~(4)

+ !.~ (2)

gik

+ ô~3fl(2)g~~) +

~8kD,2g~~,

~9~,9,(3)gj~~

~ (2)

2mk g00 68flg goo—4m 8Iflg~k g00~ + a~(2~g 2)g,fl~). (9) +4 00( 1 ,9~( From the above expressions we may calculate the Ricci tensor but we will first use the freedom to make coordinate transformations. So the calculation can be done in the so-called harmonic coordinate system which is the one satisfying the harmonic condition —

=

g~LVfk

0.

This condition must be satisfied to all orders in the post-Newtonian scheme. As a result, to second and third order we get ~(2) ~

~

(2)

~(2)

—

g 00

+

m

g,,,,,

+~

—

—

gmm

~

—

3~g 2o~~ = 0. 0~

(10)

These conditions simplify considerably the expression for Ricci tensor which can now be written as ~ (2)

(11)

2(2) nm 2 2(3) g,,,,, 0n 2 g (3),~ —..!v~ 0~, _! —

2~4~g

2~2~g

= ~a,

00

2

—

~V

~ 8(2)

2)gooak(2)gOo

00 —

~~nzkam(

g~

1 ~ i g®. We are now in a position to expand the Einstein tensor ~

= ~

—

(14)

310

A.A. Kehagias/Nuclear Physics B 423 (1994) 305—326

as 3~G~W +~

+ ~

G,.LP = ~

where Goo, Gmn are of even and G momentum tensor in powers of T/LV

=

+

0~of odd order. We may also expand the energy— as

+ ~

+...

which corresponds to the /3 = u/c expansion in special relativity. As a result, the Einstein field equations are now reduced to the system 2)T~~ (16) ,

= K(n

which can be solved step by step iteratively. The success of this method is due to the fact that once the metric has been calculated to say nth order, it can also be calculated to the next (n + 1) order. Thus, the approximation proceeds, the complexity of the equations increase at each step but, in principle, it can be carried on indefinitely. Finally, when we apply the post-Newtonian approximation to the case of a spherical symmetric object, we find that the first few terms are actually the first terms in the expansion of the Schwarzschild metric. However, there is no proof that the solution we get solving the equations in the post-Newtonian scheme converges to the Schwarzschild one. The important feature of the post-Newtonian scheme is its general applicability and its ability to give a first approximation where the exact results are not known. 3. The

first

post-Newtonian limit

The first post-Newtonian limit of the theory is the determination of the metric tensor to order O(~)for goo, 0(2) for gjj and O(~)for go~ [18,20]. Thus the metric may be written in the first post-Newtonian limit as goo=—l —2U+h gjj

6), 00+0(

=8,~ + h~

1+0(~)

5), go1=h~+0( where we assume that h

(17)

00, h~1and h, are of order O(~),0(2) and 0(~), respectively. Moreover, the Ricci tensor in Eqs. (11) —(15) may be written, to this approximation, as 2R 2U, 00=V 2Rnm = ~V2hnm,

2hn,

3R 0~= =

4V 8 2U 1

—

~V2hoo 2(VU)2 —

—

hniôn8iU.

(18)

It should be noted that in general relativity, as well as in the theory we discuss here, terms of order O(2~~+1)for goo and g,~,,, and O(e2~~)for gon are absent. This is because these terms satisfy the homogeneous equations

A.A. Kehagias/Nuclear Physics B 423 (1994) 305—326 =

311

2(2h1+I)g~~ = V2~2”~g

V

= 0~ 0

(19)

and the only regular solutions of the above equations are (2n+!)

—

(2n+1)

(2n+1)

~

—

(2n+1)

,.

(2n)

~

—

(2n)

g00~ gmn g~~tJ g0~ gO,, t The fact that these solutions are unspecified functions of time means that they can be eliminated by a suitable choice of gauge. However, there is also another reason for the absence of these terms. Under a time-reversal transformation, the goo, gjj components do not change sign while go~do. Similarly, the expansion parameter also changes its sign and thus from time-reversal symmetry, ignoring gravitational bremsstrahlung, the expansion in Eqs. (6), (17) follows immediately. Furthermore, as we have seen, the spatial derivatives are one order lower that the time derivatives and this in turn means that this approximation scheme is valid within the near zone r < t. The absence of terms of order O(2~1) for goo and g,,,,, and O(2~~)for gç,,, guarantee that the solutions of Eq. (16) can be matched to solutions in the far zone r > t which satisfy outgoing radiation conditions at infinity [21]. Let us now turn to the action in Eq. (1). Denoting covariant differentiation by a semicolon and ignoring gauge field contributions, the field equations which emerge from the variation of the action are 2T,~~ ~ ~ ~ + K g00

—

—

.‘

—

—

—

—

RL~g~~ — 2R~p~y~

_2Rpp(P;/ + 2R’~P;pcrgp.v+ 2R~~ ~ ~

p.pv~:r

;

a .‘

2

—~/1~r’,~ Dd, ‘.

—

;/L

d~ ;P

2R~pcP;v(P;°~

2R;,;°2R”°~p~o.g,w

+ 2R,p~o~P;U

2R,pi~u~~P;~) +~ + 12K2 (e2~

—

H,l(ARI~pKA)p

+ 12K2 ( e2’

~

KfbHVKARP,LKA) ;p

+AK2H~~0J~eKApy + A K2 H’~J~eKAp,L+ g/LVAKHKA~J~6 ~

\

~ o-

~AK2(e2~1’~l~

=

va~/;p

&~“°~J5R~ \ o-

~

—

12~Ke2~H~pKH/~ (22)

8~~

= j~A3K

~

(23)

312

A.A. Kehagias/Nuclear Physics B 423 (1994) 305—326

Eqs. (21)— (23) above are in agreement with those, by various field redefinitions and omissions, in Refs. [11,15,22]. For convenience sake, let us rewrite the field equations (21) in the form 28~8~~ + K2(T~,,— ~ (H/LKPHV~~ —

~

(W~~— ~ggpW)

—

=K

~

~g,L,,H,~H’~)

+ 12K2 (e2~ IOl~H/sI~~)tRPP,,~A) ;p + 12K2 (e_2~ K~PHV!(ARP,.LKA)p +AK2 H’~”J~eKAp,, + AK2HKAPJp~SKAp/2

—2g~,,AK2 HKApJ~s1~~0~ + AK2 ~

;p

+AK2 (e2~ K~gAvc X0~J~RP~aaK ) ;p,

(24)

where we have used the shorthand notation /~r’

—

D

~

—~/~K~’

‘)D

f

e ~ +2R~(e”~’~);puggp+ 2Rgp LI e’~”~ g,~,,,

“~IPP\

—

);j~ —

2R

‘° —

‘~

‘~‘°“

“

~

2Rgpp~(e’~);~).

(25)

W is the trace of the W/LJ~tensor and it is given by W=2R~~’(ef010l’);,~p _RDef0IOI’.

(26)

In order to proceed, we need also an expansion for the antisymmetric Kalb—Ramond field as well as for the dilaton cP. The former can be expressed as 5~B

B 0~=

+ ~

B,3 =

0~

+

+...,

+ .

.

.

.

(27)

This particular expansion originated from the fact that B0, is a vector and B,~is the dual of a vector. Thus, under parity transformations, B0, changes its sign corresponding to an odd expansion in e, while B,1 do not and corresponds to an even expansion. Moreover, the field strength ~ in this approximation, will be given by = 811~B~~1,

(28)

since the Chern—Simons term in Eq. (3) is, at least, of order 0(~).

In the same way, we may easily verify using Eq. (22) that the dilaton field has also an expansion around zero of the form =

+

O(~),

(29)

where the first term (2)~ is absent in view of the lack of matter sources for the dilaton. Let us now turn to the tensor ~ defined above. The expansion of this tensor will look like 6~W,LV + ~ +

w,~,.= ~

A.A. Kehagias/Nuclear Physics B 423 (1994) 305—326

313

as follows from Eqs. (9), (17), (18), (29) and the definition (25). Explicitly, we find that its first non-vanishing component is + (30)

2~R”8m8n~4~’P — 2(2)ROmOhi8m8n(4)~).

2~

Thus, although W~,.will contribute to the gravitational field, this contribution will start operating only at the post-Newtonian limit since the leading term in the 8-expansion of W,~,,.is of order ~6• Similarly, the Gauss—Bonnet term (2) is given to this approximation by R~iB= 8(dmökU)(ômôkU) 88m(8kUôm8kU

—

8(V2U)2 (31)

omUV2U).

—

Using Eqs. (17), (18), (27)—(31) and units written as

K2 =

8i,-, the field equations may now be

V2U=41r~0)T°°,

(32)

V2h,,=1617~1)Th1°,

(33)

mn

~ ~°~T°° mn

= —28,2U

—

(34

‘

2V2U2

—

[8(8fl,8kU)(8m8kU) = ~8,,J?nOmI

(35)

8irG(~2)T00+ ~2~T”) —

8(V2U)2]

—

12~KH~vAH~”~,

(36) (37) (38)

We observe that the first three equations above which specify the metric are identical to the post-Newtonian evaluation of the field equations in general relativity. Thus, there is no way to distinguish, to this approximation, Einstein and superstring gravity. The system of equations (32)—(38) form a consistent system and if the energy— momentum tensor is given, we can specify the graviton, the dilaton and the axion field to the first post-Newtonian limit. We will illustrate this below in two specific examples, namely, we will try to find the fields of a spinning spherical symmetric mass as well as of an infinitely long cosmic string. To begin with the former case, let us note that the energy—momentum tensor for the spinning point mass is given by

(l)T°fl =

6~Sm8k6~3~(r)

,

(39) (40)

where m=fd3x(0)T°°~

(41)

314

A.A. Kehagias/Nuclear Physics B 423 (1994) 305—326

(42)

SnCnmkfd3XXm(1)T

are the mass and the spin s density is given by =

=

(sr) of the point particle, respectively. Moreover, the spin

3~(r).

s,,c5~

The energy—momentum tensor (39), (40) specifies the metric to be (0) 00

I ~Ix—x’I (x,t)dx I

U(x,t)=—

~‘~T°”(x’t)d~x

hn(X~t)~4f

2m2

hoo(x,t)=—-—~—+O(r r

)

rn

(43)

r =

28nmk

.

(45)

Thus, the metric for a spinning point mass in spherical coordinates with the direction S as the symmetry axis may be written as ds2 =

/

—

(~i—

2m —

2m2~ + —i—) dt2

+ (1 +

—

Sjxk 4c,Jk—~--dtdx’

(dr2 + r2d02 + r2sin2Od~2).

which is, as expected, the first post-Newtonian limit of the Kerr metric. Let us now turn to the antisymmetric two-form field. In view of the invariance of ~ in Eq. (28) under the gauge transformation B,.,~—p B,.~+ 8,.,A~— 8~A,.,,

(46)

where A,~,are the components of a one-form, we may fix the gauge freedom by imposing the Coulomb condition 8,,B°~=VB=0.

(47)

In this particular gauge, Eqs. (37), (38) reduce to V2B=~VxS, HnmiO,

(48) (49)

where we have used the notation B= (~3~B 0~) and the fact that the axial current is given by J~= (0, S) with S the spin density. One may now easily verify that the solution to Eq. (48) may be written as

A.A. Kehagias/Nuclear Physics B 423 (1994) 305—326 ~

315 (50)

8ir r Moreover, as follows from Eq. (49), the space-space components of the antisymmetric two-form field are constants and thus 2~Bmn0 (51) Bmn~ by a gauge transformation. Finally, one has to verify for consistency reasons that the solutions (43)—(45), (50) satisfy the gauge conditions (10), (47) as well. It is not difficult to prove that these conditions are indeed satisfied, (47) automatically and (10) by virtue of V•

h=0,

where his the vector with components (h 1,h2,h3). The post-Newtonian scheme can applied to infinitely extended objects as well. For example, let us examine the case of an infinitely long cosmic string in the z-direction which is characterized by the equation of state (52)

T°°+T~=0.

Moreover, we will assume that it is rotating so that its energy—momentum tensor may be written as = ,~(2)(r)

(0)TZZ =

~°~T’~=0, (1)TOl1~i8ô(2)(r)

(53)

,

where i,j = 1,2 and ~.r and a- are the mass and the spin density, respectively. The Einstein equations (16) are written to lowest order as 2~Roo=K(~°~T ~ 00+~(—~°~T00+~°~T22)) =

—~Ko~f(—~°~T00 + K

(~°~T~2 — ~(~°~T00 +

(ZZ)Tzz))

(54) 2 =

Employing Eqs. (11)—(13), (17), (53) and v2U=0, =

11,

7~=0,

2~ (r)

2h, = —8i ,

V

K

—16ir,a8~2~(r)8

2h V

8ir we get

1a-81~

The solution of this system is

(55)

316

A.A. Kehagias/Nuclear Physics B 423 (1994) 305—326 U(p)

=const.,

h,1

=

—8,a~ln

h55

=

const.,

(~) (56)

h(r)=—4a-~—~~

p For the calculation of h,(r) the ansatz 3c9 h(r) = & 1A(r) + ~9,K(r). has been employed. As a result, the metric in this approximation can be written as 2 = —dt2 + dz2 + (1 4irln(p/po)) (dp2 + p2d~2)+ ~dtd~. ds This is the exact metric [12], in the a- = 0 limit, of a cosmic string with deficit angle —

i~ =ir(1 +4/L).

Eq. (48) now for the antisymmetric field can be written as V2B,(r)

=

and employing an ansatz similar to (57), we find that B,(r) =

—&~—~.

4ir

(57)

p

The corresponding ~ field is zero and the solution (58) is the counterpart of the solution in Ref. [13] in the cylindrical symmetric case. 4. Radiation Another approximation method in general relativity is the weak-field approximation where small perturbations (graviton modes) of the Minkowski metric [18,20] are considered. The field equations are reduced then to the wave equation with source the energy—momentum tensor and thus, in this case, propagating gravitational modes emerge. Today, the analysis of almost 20 years observational data from the binary pulsar PRS 1913+16 [23]seems to prove the existence of gravitational radiation [24]and it remains a direct detection of gravitational waves (by means of a gravitational antenna) [251.We will examine here wave and radiation aspects of the superstring gravity as follows from Eqs. (21)—(23). To begin with, let us recall that in the wave analysis it is convenient to introduce the quasi-orthonormal null-tetrad base [26] (e, + e~),

A.A. Kehagias/Nuclear Physics B 423 (1994) 305—326

(e, —

=

m= in=

317

(e~+ tey), (e5

.~=

—

ie~) ,

(58)

where the tetrad vectors satisfy the relation m

= —k. £ =

1.

Let us consider for example the electromagnetic field. The measurable quantity is the field strength F,.,,,. For plane waves travelling in the +z direction, F,,,,, is a function of the”retarded” time u = t z. The components of the field strength in the null-tetrad base are Fe,,, Fpq, where p, q = k, m, )~i.The Bianchi identity for F,.,,, is written as 8qF~p+ t9pFq~+ 8Fpq = 0, —

or 0,

8iFpq

where 8,,,, 8~ denote derivatives to the p, £ direction. Thus, the components Fpq are constants and there exist three independent components F,.,, which correspond to the three possible polarizations of the photon. If, moreover, F,,,,, satisfies the Maxwell equations =

0,

then in the basis (59) these equations are written as =

0

and thus F,.,,,, like Fpq, is constant (non-propagating). As a result, the Maxwell equations specify only the F,.,,, component and there are finally two independent modes F,.,,,, F~g to carry the electromagnetic energy. These components are the two polarizations modes of the photon. One may now proceed along the same lines with the gravitational field. The measurable quantity here is the Riemann tensor. There is an elegant way to analyze the polarization of gravitational waves and it is provided by the Newman—Pernose (NP) formalism [26]. For the Riemann tensor, there are six algebraically independent components which correspond to the six possible polarizations of the gravitational waves. These are, in the NP formalism, the two real functions ~l’2(u) and ~22(u) and the two complex functions ~P’ 3(u),~P4(u). These functions are actually the components = =

=

318

A.A. Kehagias/Nuclear Physics B 423 (1994) 305—326

(59) of the Riemann tensor in the null-tetrad base (59). To find the independent polarization modes for the graviton, one considers the vacuum field equations (no matter sources) and keeps only linear terms in the fields. One may then verify that the equations for the graviton is exactly the same as in general relativity and lead to ~‘2P3~’220,

~

Thus, there exist two independent modes corresponding to the two possible polarizations of the graviton. This is not the case in alternative theories for gravity. In Brans-Dicke theory for example, in addition to ~P4* 0, we also have ‘P22 # 0 [271. Let us now proceed 0) with the gravitational radiation of the effective action order. Here the theory contains thesuperstring graviton, the dilaton and in the dominant 0( a’ the antisymmetric two-form field, with Hg,~pgiven in Eq. (28), and the field equations may be written as R,.,,,

— ~g,.,,,R=K2T,,,,+ ~

—

+K2(8~8,,~

—

~g,.,,8~P8’~P) + AK2H~”~J~,eKAp,,

+ g,.wAK2 HKApJ~8”~°, ~A8K(v’~e’”J~,,e’°”~)

=

=

~

(60)

(61)

,

+ A~KHKApJ~e

~.

(62)

We will evaluate the above field equations in the weak-field approximation and we will neglect the dilaton field in view of its K2 coupling to its sources in Eq. (63). In particular, we will consider small perturbations h,,,,, around the vacuum Minkowski metric ~j,.,,,,.We may define a tensor 0,.,,,, by a I ,,A —i.

1”

~“2flP”

A’

so that the field equations in the gauge (63) and in the linearized limit can be written as na

,~

,,,,,——

K~

t,.,,, is quadratic in

,,,,,+t~,,, .

)

the fields and it is given by

= —--~(R~,,,,, — ~

+ 36(H,.,,KAH,,’~ —

~,,,,HKApH”~)

(65)

A.A. Kehagias/Nuclear Physics B 423 (1994) 305—326

319

with ~ the part of the Ricci tensor quadratic in h,.,,,. Similarly, the equations for the antisymmetric two-form field may be written as L1B,,~+

511~eK,,p,.L.

+ 8”8,,B~,.,. = ~A8KJ

~

The gauge freedom in the definition of H,.,,,,~in Eq. (28) allows one to consider a particular gauge system. Thus, we may choose the Lorentz gauge (66)

in which the B-field satisfies the wave equation, LJB,,~=

(67)

~A8”~J5”8K,,p,L.

We are interested now in plane-wave solutions in the radiation zone of the form i(kx—~t)

—

B,.,,,

__

*

—i(kx—wt)

~68

= 81,,,~t(q.x_~,t) +~

(69)

where e1,,, is symmetric and E,.,,, is antisymmetric. Moreover, k,, = (w, k) and q,.~ = (wq, q) are the null wave four-vectors for the graviton and the antisymmetric field, respectively, so that (70) The gauge conditions (64), (67) are written for the solutions (69), (70) as (71)

k’~e,.,,,=0,

2,.,,,=0.

(72)

q’ Employing the expressions (69)—(73) in (66) and integrating over spacetime, we may write (t,,,,,)

K2~

=

—

~Je~2) + 72q,.,qpE*KA~.

(73)

The brackets denote average over several wavelengths or over spacetime regions large compared to the characteristic wavelength. For the graviton, as we have seen, there are two independent modes which are actually the transverse-traceless components ~ = of the metric, defined in terms of the projection operator =

—

fin 1,

with n’

=

x’/r, as = p/,,pklp/

—

~Pkl0P”.

(74)

The antisymmetric two-form field has a total number of six components. However, they are not all independent since we have imposed the gauge condition (67). This

320

A.A. Kehagias/Nuclear Physics B 423 (1994) 305—326

condition reduces the number of independent components to three. Moreover, there is an additional freedom since we can perform gauge transformations with flIL—IL —

+

~

~~~i(q.x_wqi)

~*

Under such transformation, the amplitudes e~,,,,change accordingly to = 8,~ +

q,.,,, — q,,,.,.

However, there exist only two independent q/Lf,.,

=

~,,

since the gauge condition implies

0

and, furthermore, Ag is defined up to a divergence of a scalar. Thus, finally there exists one independent component which is actually the axion field. We define the latter as a(x,t)

—8~Jkn’B’.

(75)

We may now express (t,,,,) in terms of the truly dynamical degrees, namely, the transverse-traceless part 0~and the axion a. In terms of these, we may write Eq. (74) as = ~

~(8g0~8,,0~)

+

(76)

36(8ga8,,a)

and finally we have to evaluate the quantities in the brackets. For this, let us recall some well-known results from radiation theory. Let us suppose that is a tensor field of any order satisfying the equation ~i

L14(x,t)

= —l6irr(x,t),

where r(x, t) is the source of the field. The solution of the above equation with outgoing radiation condition at infinity may be written as ~(x, t)

=

4J

T(t

—

Ix x’k x’)d3x, Ix—xI

Far from the source, in the radiation zone (r >> R, R solution can be expressed as ~(x,t)

=

~

fr(t

—

(77)

—

r +n~x’,x’)d3x’ +O(R/r)2.

is “size” of

the source), this

(78)

Furthermore, for sufficiently slow moving sources (that is, sources within the wave zone, R < A/2rr = wavelength << r), we may express (78) as 4 1 am çb(x,t) =_~—~~fr(t_r,x’)(n.xt)md3x~. m=0

(79)

A.A. Kehagias/Nuclear Physics B 423 (1994) 305—326

321

Employing these results in our case, we may write the solution for the transverse and traceless components 0~j~ as 1-,~.(x,t)

0~

=

4~ ~ ~ —

-_~——

J(T~1 + t’3)y~(t — r,x’) (n x~)md3xF,

(80)

1

where (T”+t’3)rr is defined in analogy to Eq. (75). As one may see from Eq. (66), the contribution of the axion field to comes from the B, 3 components of the antisymmetric two-form field. However, from Eqs. (27), (51) it follows that B,1

=

Moreover, we have seen in sect. 3 that the theory as described by Eq. (21) has the same post-Newtonian limit as Einstein gravity. As a result, t’j~and consequently ~ is the same as in general relativity and the first term in Eq. (77) is the familiar general relativistic contribution. The expression for the traceless and transverse components of 0”~in Eq. (81) is actually the gravitational multipole expansion and it starts with the quadrupole m = 2 term, since there are no monopole m = 0 and dipole m = 1 terms in view of the conservation of mass and momentum and angular momentum, respectively. We can also calculate the antisymmetric two-form field which may be written, in general, as 3x’. d Its space components may be expressed, in the radiation zone, as A

~

8rrJI B~’°(x,t) = ——

x’)

(81)

A i 8~ B’~(x,t)=—————~—— /(ek~308kJ~(t_r,xI) 8rrr m! 8t~j xrOuut(8,J~(t_r,x~))(n.x~)md3x/

(82)

and thus the axion field will be given by the multipole expansion ~

_nk8tJks(t_r,xF))

x(n.x~)md3x~.

(83)

Denoting the time derivative by an overdot, the first terms of this expansion may be written as a(x, t)

=

—‘~-—n~m~(t — r) + —~—n~n 1rh~1(t — r) + —~..n~n1n,.m~~,.(t — r) 41?-r 4irr 4irr —--~--n~~,(t—r) — —~----n’n~~~1(t—r), 4irr 4irr

where

(84)

322

A.A. Kehagias/Nuclear Physics B 423 (1994) 305—326

mi(t_r)=J8J~(t_r~x’)d3x’. —

=f

r)

5(t

8j’J

—

(85)

r,x’)x~d3x’,

3x’

m,

=fa;J05(t_ r,x’)x~,x~d

11(t—r)

(86)

0

(87)

and (88)

si(t_r)=fJ~(t_r~x’)d3x’~

— r)

=f

J~(t— r,x’)x~,d3x’.

(89)

There exists a conservation law which follows from the gauge condition (64) and may be expressed as ô”(Tgp+tp.p)

0.

As a result, we may define the four-momentum p~ = /(T~0 +

and thus dPg

=

_]c(T~J+t”~)n~dS=

where the integration is over a constant-time closed hypersurface with no sources. The luminosity distribution, that is the power per unit solid angle emitted in the direction n’, is given now by =

(90)

r2n’(t°’).

Using the fact that, for q5(x, t) as given by Eq. (79), we have =

—n~8,~ + 0(R/r)2,

we may write (t 01) = —~-~nj ~(8t0~r8t0~.r) 2 =

and thus, with =

~—

—

36n1(8ta8,a),

(91)

8ir, we may express Eq. (91) as

K

~(8t0~r8t0~r)

+ 36r2(8,a8,a).

(92)

A.A. Kehagias/Nuclear Physics B 423 (1994) 305—326

323

Taking into account only the first moments, we can write for the luminosity distribution dL dfl

2

/dL\

2

9A

9A2

9A2

2

+~—~((n,.n

2

1fiz~1) :1 +~—~((n~n1n,.m~1i)

(93)

+ ~—~j((n~n1n,. s~1~) ).

As a result, the first few terms of the total luminosity, with A L = Lgr

+ ~(th,Q~1th1)

th~1Q,1~,.fn~~) + ~(‘fliJk

+

= K/I

2, are written as

QijUmn ~mn)

Q,.~,. s,.~)+...

+

(

More generally, we may write 1 (.n) ~•;~(m

L = Lgr + ~

1I~2°. ~ (.n) m

~

+~

Q

~

s,,,

QiIi2...i~

2,,)

where, in general, ~ ~

= =

=

ff

(95)

3 ,

8~J~x2x~ ...

xd~x’

Jx~x~... x,,d3 x’3,

(n + 1)!!

(~~2

(96)

.

5i,i 4

~ + cyclic permutations)

(97)

and (.n) stands for the nth-order time derivative. We have also used the shorthand

notation Lgr for the gravitational contribution to the total luminosity. This contribution starts with the quadrupole term and an expression for Lgr can be found in Ref. [28]. As one may see in Eq. (95), there are monopole, as well as dipole contributions to the total luminosity. However, since the axial current is a space-like vector, its time component Jo is proportional to the speed of the source. As a result, the dominant contribution will be the dipole one while the monopole component will be significant only for relativistic (or mildly relativistic) moving sources like binary pulsars. Note

that, as follows from Eqs. (96) and (97) it is the spin-density distribution that specifies the luminosity pattern and not the spin density itself. Unfortunately, in view of the lack of spin-polarized star models, one cannot make any estimate of the ratio L Lgr/L. —

324

5.

A.A. Kehagias/Nuclear Physics B 423 (1994) 305—326

Conclusions

We have examined here the superstring gravity as follows from the effective action (1). Gauge field contributions have been ignored since we are interested in the pure gravitational sector of the theory, namely in the effective theory for the graviton multiplet. Thus we have only considered graviton, dilaton and antisymmetric two-form (axion) modes which are the components of this multiplet. In superstring compactification there are many potential axion fields and in fact many mechanisms for the breaking of the associated Peccei—Quinn symmetry which leads to massive axions. The same holds also for the dilaton where a mass term [29], as well as a convex potential [30], is possible. Moreover, one may expect that non-perturbative effects will provide additional terms for the dilaton [31]. However, we have considered here the case where the axion and the dilaton are massless and survive in the lowenergy theory. It is for this case that we have developed the field equations to the postNewtonian limit and we found that the present theory is identical to Einstein gravity in this approximation. Thus, one will not expect deviations or corrections to the classical solar-system tests of general relativity since these tests are post-Newtonian ones. (See, however, Ref. [17]). On the other hand, in the weak-field approximation superstring gravity, contrary to general relativity, predicts all possible types of radiation. Namely, there exist monopole, dipole, quadrupole etc. contributions in the total luminosity. In Maxwell theory for example, the absence of monopole radiation is ultimately related to charge conservation and the absence of monopole and dipole radiation in general relativity, as we have already mentioned, to mass, momentum and angular momentum conservations. Thus, the absence of certain types of radiation in a theory is always due to some conservation law. However, in the present case, there is no such conservation law for J~,,= (J,~,J5) and thus all types of radiation are present. Of course, in a broken Peccei—Quinn symmetry, the axion field is massive and thus there is no axionic contribution to the luminosity of the source. In that case, the total luminosity is given by the quadrupole formula. Finally, it should be noted that, although the solar-system tests of general relativity are impressively accurate (with fractional accuracy < i0~ [5,161), they have a serious drawback in that they are weak-field tests and do not indicate the theory of gravity in strong fields like that of neutron stars or black holes. One may then expect that such systems can provide a direct testing ground for alternative gravitational theories [32]. The most promising system for such strong-field tests are the binary pulsars. In particular, the first discovered binary pulsar PRS 1913+16 [23] provides the first experimental evidence for gravitational radiation [24]. Although this system is generally accepted to be described by general relativity with gravitational damping according to the quadrupole formula, it is rather peculiar (two neutron stars with nearly equal masses) [331. There also exist other systems where potentially the quadrupole formula may be checked [28,34]. In such a case, all types of radiation must be detected from, say, a neutron star where the strong magnetic fields possibly generate a non-zero spin density.

A.A. Kehagias/Nuclear Physics B 423 (1994) 305—326

325

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