D. Radu and D. Tatomir- Gravitational scattering of massive particles on a background with axial symmetry

8/3/2019 D. Radu and D. Tatomir- Gravitational scattering of massive particles on a background with axial symmetry

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Gravitational scattering of massive

particles on a background w ith axialsymmetry

D. Radu and D. Tatomir

Abstract: Using the S-matrix formalism and the Feynman diagram technique, thegravitational scattering of the minimally and non-minimally coupled scalar, spinor, vector,spin-vector, and spin-2 massive particles, in a background described by KerrNewmangeometry is studied for any value of the scattering angle. We find that the differentialcross sections of the scalar, spinor, and vector particles in the backward direction andultrarelativistic case are finite and consequently the backscattered particles must have the

opposite helicity, whereas for the spin-vector and spin-2 particles in the same case, thedifferential cross sections are clearly infinite. It has been shown, for the particular case whenthe angular momentum of the scatterer vanishes (i.e., for the Schwarzschild geometry) and inthe small-angle approximation and ultrarelativistic limit as well, the differential cross sectionsare all of the same type, i.e., in this special limit case the gravitational particle scattering isspin independent.

PACS Nos. 03.80+r, 11.80-m, 03.70+k

Resume : Nous utilisons le formalisme de la matrice S et la technique des diagrammes deFeynman pour tudier la diffusion gravitationnelle tous les angles, en couplage minimalou non, de particules massives scalaires, spinnorielles, vectorielles, spin-vectorielles et despin 2 sur un fond dcrit par une gomtrie de KerrNewman. Nous trouvons, pour les troispremiers types de particules, que les sections efficaces diffrentielles en direction arrire dansla limite ultrarelativiste sont finies et que, par consquent, les particules doivent tre dhlicitoppose. Par contre, la section efficace est clairement infinie pour les deux derniers types departicules. Il a dj t dmontr que, dans la double approximation des petits angles et desvitesses ultrarelativistes et lorsque le moment angulaire du diffuseur est nul (gomtrie deSchwarzschild), alors toutes les sections efficaces sont du mme type. Clairement dans ce caslimite, la diffusion gravitationnelle des particules est indpendante du spin.

1. Introduction

The problem of the scattering of particles of various spin values by a gravitating body that generates

an axisymmetric gravitational field has been considered in many studies (see, for example, refs. 1 and

2). In the present paper, we develop an unified approach for the description of scattering processes for

the massive scalar, spinor, vector, spin-vector (RaritaSchwinger), and spin-2 particles in the frame

of quantum gravity (QG). The concrete results (scattering cross sections and the basic characteristics

of the process of interaction between the massive spin-0, 1/2, and spin-1 particles and the stationary

gravitational field described by KerrNewman geometry) generalize all the results previously given inrefs. 3 and 46, in the long-wavelength, weak-field limit (wavelength corresponding to the incident

particles size of body/radius of scatterer gravitational radius/mass of scatterer). They also refer

Received December 2, 1996. Accepted July 2, 1997.

D. Radu and D. Tatomir. Department of Theoretical Physics Al.I. Cuza University Blvd. Copou, 11, Iasi,R-6600, Romania. Telephone: +40-32-144760; FAX: +40-32-213330; e-mail: [email protected].

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2 Can. J. Phys. Vol. 76, 1998

to the interaction process implying the massive spin-3/2 and spin-2 particles and the above-mentioned

gravitational-field processes, which according to our knowledge, have not been studied so far in such

a context. Besides, the massive spin-2 field is the subject matter of some recent papers in the field

[7]; its treatment in the present paper has as its reason the clearing-up of at least a particular domainin which this field is implied.

A study on almost the same subject but concerning massless spin-0, 1, and spin-2 particles was un-

dertaken by De Logi and Kovacs Jr. in ref. 1 using Feynmans perturbative method (based upon Feyn-

man diagrams). The Feynman technique was first used in connection with quantum-electrodynamical

processes and its efficiency as a problem-solving tool soon led to its widespread use in many aspects

of quantum interactions, including QG [8]. The results obtained by De Logi and Kovacs, in the limit

of small scattering angles, are in agreement with those obtained by Peters [2] who used a method

based on Greens function formalism; they also showed that their method may be successfully used

in studying gravitational scattering processes for massive or half-integer spin particles this is what

we do in this paper for massive spin-0, 1/2, 1, 3/2, and spin-2 particles. Using a different method

to that of De Logi and Kovacs, and other authors, to determine the interaction Lagrangians between

the gravitational field and the matter fields under consideration, we will use the principle of minimal

coupling in QG, that is equivalent to considering the Lagrangians of the matter fields studied in curvedspace-time. In fact, this implies replacing the usual derivatives in the expressions of the Lagrangians

with the corresponding covariant derivatives. As we will see, this fact determines the appearance of

some correction terms in the expressions of the interaction Lagrangians even in the first-order ap-

proximation. For the massive scalar field we also used a nonminimal coupling introduced by means

of the scalar curvature U.We have to stress that unlike the majority of authors who usually use Guptas interaction La-

grangian (in the so-called Gupta coupling formalism; see, for example ref. 9):

OJxswdlqw @ 4

5kW

we obtain the interaction Lagrangians using the above-mentioned method in the form:

Olqw @ OJxswdlqw . Ofrrulqw +,

where Ofruulqw +, is the first-order correction in the coupling constant +,. This correction term is dueto the contribution brought by the covariant derivative of the corresponding field functions. In the

first-order approximation it is proportional to the usual first-order derivatives of the weak gravitational

field given in terms of k (see relation (4)). For instance, in the case of the massive spin-2 fieldthis correction term is

Ofruulqw +, @ +k> . k> k>,+*>* . **>,where *> is the usual derivative of the tensor field function. Also we mention that some extra-background to our approach has been provided in Chap. 14 of the ref. 10.

Our paper comprises four sections and two appendices. The second section offers the general

working frame, and the calculus and notation conventions as well. The third section is dedicated

to the study of the scattering processes of the massive spin-0, 1/2, 1, 3/2, and spin-2 particles in

the slightly curved gravitational stationary background described by KerrNewman geometry, while

the last section contains a final overview on the results presented in the paper as well as some

concluding remarks. Appendix A contains the expressions for the matrix elements of the creation and

annihilation matter field operators, as well as the expressions for the first-order vertices corresponding

to the gravitational interactions of the massive spin-0, 1/2, 1, 3/2, and spin-2 fields, whereas Appendix

B contains the explicit expressions for the coefficient functions that appear in the relations giving

both the differential and total (integral) cross sections for the processes studied in this paper.

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Radu and Tatomir 3

2. General conventions and notations

We denote the metric tensor, the Minkowski tensor (diag(.4> 4> 4> 4)), and the tensor of theweak gravitational field by

j

,

, and|

, respectively. Following Feynman [8] and because

we require that m| m 4 everywhere, we expand the gravitational field about the flat Minkowskibackground

sjj @ | , where j @ ghw+j, and @s

49J (in natural units, J beingthe Newton constant).

In BoyerLindquist coordinates, the KerrNewman geometry is described by the line element

gv5 @[gu5

. g5

. +u5 . d5,vlq5 g!5 f5 gw5 . 5PJuS +d vlq5 g! f gw,5

(1)[ u5 . d5 frv5 > u5 5P Ju . d5

For large and very large u, the line element becomes the line element of a flat space-time. Thus,if in the far field one passes from the metric (1) to Cartesian coordinates: { @ u vlq frv !> | @

u vlq vlq !> } @ u frv (where ! denotes the angle !), then one obtains

gv5 @

4 5PJ

u. R

4

u6

gw5 .

7%mnoJ

Vn{o

u6. R

4

u6

gw g{m

4 .5PJ

u

mn .

gravitational radiation terms

that die out as R4u

g{m g{n> +m>n>o @ 4> 6, (2)Here 7@ P@ is the angular momentum and P is the mass of the body that creates the gravitationalfield, so that

j33 @ 4 5PJu

> j3m @ jm3 @5P J

u6%mnodn{o> jmn @

4 .

5PJ

u

mn > +m>n>o @ 4> 6, (3)

Taking into account the relations

j @ . k> k @ | 45

|> > | @ | (4)

we find

krr @ 5PJu

> krm @ kmr @ 5%mnoPJ

u6dn{o> kmn @ 5P J

umn > +m>n>o @ 4> 6, (5)

and the Fourier transform for | is given by

|rm+^, @ |mr+^, @ l|7m+^, @ l|m7+^, @ 4+5,6@5

P

5

m^

m5

%mnodnto

|rr+^, @ |77+^, @ 4+5,6@5

P

m^m5 > |mn+^, @ 3> +m>n>o @ 4> 6, (6)

If the angular momentum per unit mass @ vanishes in the above expressions, we recover the linearized

Schwarzschild geometry. In the following, we will not specify what set of values take the running

indices, because our convention establishes that components of 3-vectors are labeled by Roman in-

dices, while components of 4-vectors carry Greek indices. Also, a comma denotes the usual derivative,

whereas semicolon denotes the covariant derivative with respect to the space-time coordinates.

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4 Can. J. Phys. Vol. 76, 1998

3. Gravitational scattering of the massive scalar, spinor, vector,spin-vector, and spin-2 particles

The problem of the scattering of massive particles of various spins propagating in a slightly curved

space-time may be treated by quantizing both the gravitational background and the scattered field.

In a such scenario, the two fields couple according to the Feynman vertex rules. However, since

our interest is restricted to a gravitational background geometry generated by energy-momentum

distributions that are not appreciably affected by the scattering process, we may replace the virtual

graviton by an external gravitational field, i.e., we may use the external field approximation for a

stationary gravitational field described by KerrNewman geometry [1].

In this paper we shall limit ourselves to interactions proportional to 5 (single graviton exchange),calculating the scattering cross sections in the first Born approximation.

As one knows, the matter field theories in a fixed curved background are in general nonrenormal-

izable. It is not even clear, so far, to what extent the semiclassical approximation, i.e., quantized matter

in a classical background geometry, can provide reliable results. However, the processes studied in

this paper prove to be finite, that is, the concrete calculus can be done to the end without obtaining

divergent terms, at least in the first order of the Born approximation.To obtain the first-order Lagrangians that express the interaction between the gravitational field

and the massive scalar, spinor, vector, spin-vector, and tensor fields, we use the principle of minimal

coupling; for the massive scalar field we also considered a nonminimal coupling introduced by means

of scalar curvature U. According to this principle, for scalar, spinor, vector, and tensor fields we mustadd to the expression of the gravitational-field Lagrangian, the massive scalar-, spinor-, vector-, and

tensor-field Lagrangians written in curved space [11, 12]:

Ovfdodu Op>3 @sj j*>*> +p5 . eU,** (7)

Ovslqru Op>4@5 @ l5sj ##> #># sjp## (8)

Oyhfwru Op>4 @ sj45

jj J J . p5jEE

(9)

Owhqvru Op>5 @sjjj+j*>*> p5**, (10)

It is easy to see that for the scalar and vector fields we considered the KleinFock formalism

and the Proca formalism, respectively. In relation (7) U is the curvature scalar and e is an arbitraryconstant; the particular value for this constant depends on the method used to obtain the general

scalar-field equation, or on other considerations. For example, e @ 4@7 in the case in which theKleinFock equation is obtained by squaring the Dirac equation in curved space-time, e @ 4@6to obtain the KleinFock equation by the special Feynman summation (over histories) in special

relativity, and e @ 4@9 for the conformally invariant massless scalar theory. In the above relationsJ @ E>

E> is the tensor of the massive vector field, E> is the covariant derivative of the

vector field function, and *> is the covariant derivative of the tensor-field function that satisfiesthe constraints: +

sj*,> @ 3 and * @ 3.In the case of the RaritaSchwinger field, besides the principle of minimal coupling we also use

the vierbein formalism, so that the Lagrangian of this system can be written as follows (see ref. 13):

Ovslqyhfwru Op>6@5 @sjj

l

5+#

#G

# #$G#, . p##

@ 4Op>6@5 .5Op>6@5 (11)

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Radu and Tatomir 5

where

4Op>6@5 @sjj

l

5+#>

# ##>, . p##and

5Op>6@5 @l

;

sjj#+> . >,##> being the usual derivative of the RaritaSchwinger function. Obviously there is a big differencein studying the interaction of massless and massive particles, that means a system described by a

Lagrangian containing no mass term in comparison to one that contains such a term [14]. The presence

of a mass term in the Lagrangian seriously affects the physical content of the problem. Even if a

gravitino is considered at the moment to have no mass, it can have a nonzero rest mass due to the

cosmological constant, as shown in ref. 15. Concerning the RaritaSchwinger field we used the mass

term proposed in ref. 16, that means lp## , a term that for reasons of simplicity has been

written under the more convenient form pj## , (this fact being perfectly possible because of

the following constraints: # @ 3, C# @ 3> [16] ).The above expressions for 4Op>6@5 and

5Op>6@5 were been obtained by inserting the expressionfor the covariant derivative of the spin-vectors:

$G# @ #> #> ##G @ #> # (12)where are the FockIvanenko spin coefficients of the affine connection; they have the followingexpression:

@4

7>

(13)

where are the generalized Dirac matrices:

@ O+,a+,> @ O+,a+, (14)

a+, being the usual Dirac matrices in 5 hyperbolic representation of Minkowski space-time. Theexpression for > is

> @ > (15)In relation (14) O+, and O+, are the vierbein coefficients satisfying the following constraints:

O+,O+, @ j > O+,O+, @ j (16)

Since all our considerations refer only to the first-order approximation, we give below the linearized

relations for the quantities that appear in the calculus [12]:

j @ k . R+5, (17)

j @ . k . R+5

, (18)

@4

5

k> . k> k>

. R+5, (19)

@ a 45

ak . R+5, (20)

@ a .4

5ak

. R+

5, (21)

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6 Can. J. Phys. Vol. 76, 1998

where

k @ | 45

|> k @ |

4

5|> | @ |

(22)

It is very simple to show that the 5Op>6@5 term in (11) gives no contribution in the first-orderapproximation. Indeed, we have

5O+4,p>6@5+, @

l

49#a

aa#+k> k>, @

l

49#a

+aa aa,#k> @ 3

where the well-known anticommutation relations:

ia> aj @ aa . aa @ 5 (23)have been used. Taking into account the previous considerations, the first-order interaction Lagrangian

between the weak gravitational and the massive RaritaSchwinger fields reads

O+4,+lqw,p>6@5+, @

l

7+#>a#

#a#>,kl

5+#>a

##a#>,|p##|(24)

Finally, for the spinor field, as is easy to see from relation (8), we consider the usual massive Dirac

Lagrangian, while for the massive spin-2 field we use the simplest Lagrangian proposed by Schwinger

[17].

Passing to the flat complex Minkowski space-time:

{3 @ w> {m $ {m > {7 @ lw> aC $ lC> $ > C CC{

$ CC{

(25)

the first-order interaction Lagrangians between the gravitational and massive scalar, spinor, vector,

spin-vector, and tensor fields are, respectively,

O+4,+lqw,p>3+, @

*>*>| .

4

5**+p5| e|>,

+59=4,

O+4,+lqw,p>4@5+, @

4

7

##> #>#

v 45

p##| +59=5,

O+4,+lqw,p>4+, @

JJx . p

5EE|

+59=6,

O+4,+lqw,p>6@5+, @

4

7

#># ##>

| 47

p##| +59=7,

O+4,+lqw,p>5+, @ *>*>| 5

*>*> . p

5**

| .

*>*>

.4

5p5**

| +k> . k> k>,+*>* . **>, +59=8,

where

v @ | .4

5| (27)

x @ | 47

| (28)

J @ E> E> (29)We have also taken advantage of the RaritaSchwinger field equation #> @ p# and its

adjoint. According to standard quantum field theory (QFT), the parts of the (26.1)(26.5) Lagrangians

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Radu and Tatomir 7

Fig. 1. The (massive field) graviton (massive field)

vertex. The wavy line represents a graviton. The con-

tinuous lines represent either scalar, spinor, vector,

spin-vector, or spin-2 quanta.

Fig. 2. The spatial orientation of the angular momen-

tum D and the scattered direction S relative to the

incident direction S .

cast in normal form that describe the interaction of the massive scalar, spinor, vector, spin-vector, and

spin-2 particles with gravity, are, respectively [12],

Qk

O+4,+lqw,p>3+{,

l@

k*+,> +{,*

+.,> +{,|

h{w +{,

.4

5*+,+{,*+.,+{,

p5|h{w+{, e|h{w>+{,

+63=4,

Qk

O+4,+lqw,p>4@5+{,

l@ 4

7

#+,

+{,#+.,> +{, #

+,

> +{,#+.,+{,

vh{w +{,

45

p#+,

+{,#+.,+{,|h{w+{, +63=5,

Qk

O+4,+lqw,p>4+{,

l@

J+, +{,J

+., +{,x

h{w +{, . p

5E+, +{,E+., +{,|

h{w +{,

+63=6,

Qk

O+4,+lqw,p>6@5+{,

l@

4

7

#+,

>+{,#+., +{, #

+,

+{,#+.,> +{,

|h{w +{,

4

7

p#+,

+{,#+., +{,|

h{w+{, +63=7,

Qk

O+4,+lqw,p>5+{,

l@ *+,>+{,*+.,>+{,|h{w +{, 5

*+,>+{,*

+.,>+{, . p

5*+, +{,*+., +{,

|h{w +{, .

*+,>+{,*+.,>+{, .

4

5p5*+, +{,*

+., +{,

|h{w+{,

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8 Can. J. Phys. Vol. 76, 1998

The processes are described by the Feynman-type diagram in Fig. 1, where s and +u, and alsos3 and +v, are the four-momenta and polarization indices (u> v @ 4 to 5v . 4, where v is the spin ofthe particles) of the initial and final particles, respectively, and t is the four-momentum of the virtual

graviton. The spatial orientation of the angular momentum @ and the scattered direction R

relativeto the incident direction R are shown in Fig. 2.

Using the S-matrix formalism, we deduce the Feynman rules for diagrams in the external gravi-

tational field, described by KerrNewman geometry, which allows us to calculate the matrix element

ks3mVmsl in the approximation. Thus we find that;A?A=

|h{wmn +^, @ 3

|h{wm7 +^, @ |h{w7m +^, @

45

4+5,6 @ 5

Pmqm5 +@ ^,m

|h{w77 +^, @ |h{w+^, @ 4

+5,6 @ 5Pmqm5

(31)

;A?A=

vh{wmn +^, @45

4+5,6 @ 5

Pmqm5

mn

vh{wm7 +^, @ vh{w7m +^, @

45

4+5,6 @ 5

Pmqm 5 +@ ^,m

vh{w

77 +^, @6

5

4

+5,6 @ 5

P

mqm5

(32)

;A?A=

xh{wmn +^, @ 47 4+5,6 @ 5 Pmqm 5 mnxh{wm7 +^, @ x

h{w7m +^, @

45

4+5,6 @ 5

Pmqm 5

+@ ^,mxh{w77 +^, @

67

4+5,6 @ 5

Pmqm 5

(33)

;A?A=

kh{wmn +^, @ 45 4+5,6 @ 5 Pmqm5 mnkh{wm7 +^, @ k

h{w7m +^, @

45

4+5,6 @ 5

Pmqm5

+@ ^,mkh{w77 +^, @

45

4+5,6 @ 5

Pmqm5

(34)

We specify that any derivation with respect to { is equivalent to the appearance of a supple-mentary factor lt. Taking into account the above relations, the matrix elements in the external field

approximation corresponding to the diagram in Fig. 1 can be expressed as follows:

V+,s3 s ks3mV+,msl @ I+,+s3> s,+t3, (35)

where the superscript symbol (*) takes the place of 0, 1/2, 1, 3/2, and 2, depending on which field is

being considered (i.e., the scalar, spinor, vector, spin-vector, and tensor field, respectively). In relation

(35), t3 @ s33 s3 @ 3 states the energy conservation law, and the expressions for I+,+s3> s, aregiven by

I+3,+s3> s, @l5P

5s3+5,5mR Rm5 T+3,+s3> s, +69=4,

I+4@5,+s3> s, @5pP

7s3+5,5

mR

R

m5

T+4@5,+s3> s, +69=5,

I+4,+s3> s, @l5P

5s3+5,5mR Rm5 T+4,+s3> s, +69=6,

I+6@5,+s3> s, @5pP

s3+5,5mR Rm5 T+6@5,+s3> s, +69=7,

I+5,+s3> s, @l5P

5s3+5,5mR Rm5 T+5,+s3> s, +69=8,

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Radu and Tatomir 9

where s3 (and s33 ) is the energy of the initial (and final) particles, whereas T

+,+s3> s, are given by(see Appendix A):

T+3,+s3> s, @ ss3 |h{w +^, e5 |h{w>+^, . p5

5+6:=4,

T+4@5,+s3> s, @ x+v,+R,w+4@5,x+u,+R, +6:=5,

T+4,+s3> s, @k

h+u, +R,h+v, +R

,s3s h+u, +R,h+v, +R,s3s h+u, +R,h+v, +R,s3s.h+u, +R,h

+v, +R

,s3sl

xh{w +^, . p5h+u, +R,h

+v, +R

,|h{w +^, +6:=6,

T+6@5,+s3> s, @ x+v, +R,w+6@5,x+u, +R, +6:=7,

T+5,+s3> s, @ h+v, +R,h+u, +R,

s3s|

h{w +^, +s3s .

4

5p5,|h{w+^,

. 5h+v,

+R

,h

+u,

+R,s3

s . p5

|h{w

+^, .k

tkh{w

+^, . tkh{w

+^, tkh{w

+^,l

ks3h

+v, +R

,h+u, +R, sh+v, +R,h+u, +R,l

+6:=8,

where

w+4@5, @ +s . s3, v

h{w +^, 5lp|h{w+^, (38)

w+6@5, @4

7

+s . s

3, |

h{w +^, lp|h{w+^,

(39)

|h{w+^, @ +5,6@5m^m5P

|h{w+^, @ 4 (40)

|h{w +^, @ +5,6@5 m^m5

P|h{w +^, (41)

vh{w +^, @ +5,6@5 m^m5

Pvh{w +^, (42)

xh{w +^, @ +5,6@5 m^m5

Pxh{w +^, (43)

In the above relations, x+u,+R,>

x+u,+R, @ +x+u,,|+R,7

> h+u, +R,> x

+u, +R,>

kx+u, +R, @ +x

+u, ,|+R,7

l>

and h+u, +R, on the one hand,

and x+v,+R,>

x+v,+R, @ +x+v,,|+R,7

> h+v, +R,> x

+v, +R,

kx+v, +R, @ +x

+v, ,|+R,7l

> and

h+v,

+R

, on the other hand, are the polarization spinors, vectors, spin-vectors, and tensors of the initialand final particles, respectively. We denote the characteristic quantities the mass, the four-momentum,

and the energy by the common notations p, s, and s3 for all five fields, respectively, and the daggerrepresents the Hermitian conjugation.

The differential cross section is given by the well-known expression

g @ +5,5

-[i=vs=

mI+s3> s,m5.

l=vs=

s53 g (44)

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where g @ vlq g g!, being the scattering angle. To evaluate the differential cross sections we

must find the expressions for

-[i=vs=

mI+s3> s,m5.

l=vs

. Thus, we have

-[i=vs

mI+,+s3> s,m5.

l=vs

@

-[i=vs

F+,mT+,+s3> s,m5.

l=vs

@ F+,4

5v . 4

[sro

mT+,+s3> s,m5 (45)

F+, being the corresponding coefficients that are very simply to identify, v is the spin for the

each field, and[sro

mT+,+s3> s,m5 are the corresponding polarization sums (because they have long

expressions we prefer not to give them here). For the spinor and spin-vector massive fields these

sums are in fact two corresponding traces over the spinorial indices, i.e.,

[sro

mT+4@5,+s3> s,m5 @5

[u>v@4

mx+v,+R,w+4@5,x+u,+R,m5

@ Wuk

7+w+4@5,,|7S

+4@5,+., +R

,w+4@5,S+4@5+., +R,

l(46)

[sro

mT+6@5,+s3> s,m5 @7[

u>v@4

mx+v, +R,w+6@5,x+u, +R,m5

@ Wuk

7+w+6@5,,|7S

+6@5,+., +R

,w+6@5,S+6@5+., +R,

l(47)

where S+4@5,+., and S

+6@5,+., are the covariant projection operators for the positive-energy spinors and

spin-vectors, respectively. To evaluate the polarization sums we take into account the fact that the

polarization spinors, vectors, spin-vectors, and tensors satisfy the following relations [17, 18]:

S+4@5+., +R, @

5[u@4

x+u,+R,x+u,+R, @s . lp

5lp(48)

+R, @6[

n@4

h+n, +R,h+n, +R, @ g (49)

S+6@5,+.,+R, @

7[u@4

x+u, +R,x+u, +R,

@s . lp

5lp

4

6 .

l

6p+s s, . 5

6p5ss

(50)

and

>+R, @8[

n@4

h+n, +R,h+n, +R, @

4

5+gg . gg, 4

6gg (51)

respectively, where g @ .s

s

p5.

After laborious calculus for the differential cross sections of the massive scalar, spinor, vector,

spin-vector, and spin-2 particles, one obtains the following expressions, respectively:

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Radu and Tatomir 11

g+3, @ +JP,5g

vlq7 5

%4 . y5

5y5

5 5e 4 . y

5

y5vlq5

5. 7e5 vlq7

5

&

. +JP,5g

vlq7 5

d5s53 vlq5 vlq5 vlq5 ! +85=4,

g+4@5, @ +JP,5g

vlq7 5

%4 . y5

5y5

5 6 . y

5

7y5vlq5

5

&. +JP,5d5s53^i4+, . i5+y> ,vlq

5

. i6+,vlq5 frv ! . i7+,frv5 ! . i8+y> ,vlq

5 frv5 !`g +85=5,

g+4, @ +JP,5g

vlq7 5

%4 . y5

5y5

5 5

6vlq5

5

5

y5 vlq5

5

&. +JP,5d5s53

i4+y> , . i5+y> ,vlq5 . i6+y> ,vlq5 frv ! . i7+,vlq5 frv5 !g +85=6,g6@5 @ +JP,5

g

vlq7

5+

4 . y5

5y5 5

469y7+4

y5,5%+48 74y

5 . 8y7 . 54y9,y5 vlq5

5

.7+6 9y5 8y7,y7 vlq7 5

. ;+6 . y5,y9 vlq9

5

. +JP,5d5s53

k

i4+y> , . i5+y> ,vlq5 . i6+y> ,vlq5 frv ! . i7+y> ,frv

5

5frv5 !

lg +85=7,

g+5, @ +JP,5g

vlq7 5

+4 . y5

5y5

5.

5

8

4

6y+4 y5,5

^45;y43 vlq43

5 97y5+4 . 8y5,vlq;

5

. 7y9+:8y7 . 83y5 86,vlq9 5

. 7y7+56y9 454y7 . 588y5 :4,vlq7 5

y5+64y; 85;y9 . 4447y7 :9;y5 . 484, vlq5 5

. 48+y43 46y; . 67y9

67y7

. 46y5

4, vlq5 5. +JP,5d5s53^i4+y> , . i5+y> ,frv5 . i6+y> ,vlq5 frv ! . i7+y> ,vlq

5 frv5 !`g +85=8,

where we denoted the ratiomRms

3

by y. The corresponding expressions for il+y> ,> l @ 4> 8 appearingin the above relations are given in Appendix B. Allowing @ to vanish (linearized Schwarzschild

geometry) one recovers the results we have already obtained in refs. 19 and 20, i.e.,

g+3,Vfkz @ +JP,

5 g

vlq7 5

%4 . y5

5y5

5 5e 4 . y

5

y5vlq5

5. 7e5 vlq7

5

&+86=4,

g+4@5,Vfkz @ +JP,

5 g

vlq7 5

%4 . y5

5y5

5 6 . y

5

7y5vlq5

5

&+86=5,

g+4,Vfkz @

+JP,5

vlq7 5

%4 . y5

5y5

5 5

6vlq5

5

5

y5 vlq5

5

&+86=6,

g+6@5,Vfkz @

+JP,5

vlq7 5

+4 . y5

5y5

5 4

69y7+4 y5,5 ^+48 74y5 . 8y7 . 54y9,y5 vlq5

5

.7+6 9y5 8y7,y7 vlq7 5

. ;+6 . y5,y9 vlq9

5`

g +86=7,

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12 Can. J. Phys. Vol. 76, 1998

Fig. 3. Variation of integral (total) cross section

j| L |

Ec k with respect to the both ratio ' S

R

3

and angle k, for massive scalar particles in the caseof minimal coupling (K ' f).


j| L |

Ec k with respect to both the ratio ' S

R

3

and angle k, for massive scalar particles in the caseof nonminimal coupling (K ' *e).

g+5,Vfkz @

+JP,5

vlq7 5

+4 . y5

5y5

5.

5

8

4

6y+4 y5,5

45;y43 vlq43

5 97y5+4 . 8y5,vlq;

5

. 7y9+:8y7 . 83y5 86,vlq9 5

. 7y7+56y9 454y7 . 588y5 :4,vlq7 5

y5+64y;

85;y9 . 4447y7 :9;y5 . 484, vlq5 5

. 48+y43 46y; . 67y9 67y7 . 46y5 4,

vlq5 5

g +86=8,

If, in addition, we ask for the angle to take small and/or very small values then,, the differentialcross sections are all1 of this form

g+,Vfkz @ +JP,

5 g

vlq7 5

4 . y5

5y5

5 gUxwk= (54)

i.e., they are differential cross sections of the Rutherford type. As we can see from relations (53.1)

(53.5), the expression for gUxwk= is contained in these relations as the first term. Since this term (i.e.,gUxwk=) is the exact differential cross section for the minimally coupled massive scalar particles (forinstance, the scalar mesons) we can interpret the second term in relations (53.2)(53.5) as being the

spin contribution of the massive spinor, vector, spin-vector, and spin-2 particles, respectively.

It is worthwhile pointing out that for the same conditions as above (i.e., @ @ 3 ), in the caseof minimal coupling +e @ 3, for the massive scalar field that because of the u4 dependence of

the gravitational potential (53.1) reduces to the usual 4@ vlq7 5 Rutherford-type cross section. Fornonminimal coupling +e 9@ 3,, the cross section still exhibits the Rutherford-type angular dependence,but only for 4. This feature of the differential cross section for massive scalar particles alsoappears for the massless scalar field, which is not surprising, because it is the scalar curvature Uthat gives rise to e-dependent terms in the differential cross section. If U is nonzero only along the

1 Moreover, for the massive scalar field this result is valid for any value of the scattering angle, if the coupling with gravity

is a minimal one (i.e., K ' f)

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Radu and Tatomir 13


j| L |

Ec k with respect to both the ratio ' S

R

3

and angle k, for massive spinor particles.

Fig. 6. Variation of integral (total) cross section with

respect to both the ratio ' S

R

3

and angle k, formassive vector particles.

Fig. 7. Variation of integral (total) cross-section with

respect to the both ratio ' S

R

3

and angle k, formassive spin-vector particles.

Fig. 8. Variation of integral (total) cross section with

respect to both the ratio ' S

R

3

and angle k, formassive spin-2 particles.

world line of the scatterer, we see that for large impact parameters (i.e., small scattering angles) the

scalar curvature cannot significantly contribute to the differential cross section, in agreement with

ref. 1. Also, (52.1) shows that the effect of angular momentum, for the massive scalar field only, is

to add a positive semidefinite term to g+3,Vfkz, which for small scattering angles is negligible with

respect to g+3,Vfkz. This can be easily understood by recalling that for large impact parameters the

u4 dependence of the gravitational potential k33 dominates the u5 dependence of the magnetic-

type gravitational field k3m , which, in fact, is the source of the angular-momentum term [1]. Alsowe mention that, although we have considered an exact solution with U @ 3 as a background, fromquantum point of view and taking into consideration the first-order contributions of the gravitons,

one gets a purely quantum fluctuation of U given by the formula

sjU @ 45

|>

which gives emphasis to the nontrivial e-coupling term

4

5e|>+{,*

+{,*+{,

in the model Lagrangian (see the relation (26.1)).

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14 Can. J. Phys. Vol. 76, 1998

Fig. 9. Variation of *?dj| L |

Ec ko with respect to

both the ratio '

S

R

3

and angle k, for massive scalarparticles in the case of minimal coupling (K ' f).

Fig. 10. Variation of *?dj| L |

Ec ko with respect to

both the ratio '

S

R

3

and angle k, for massivescalar particles in the case of nonminimal coupling

(K ' *e).

3.1. The backward-scattering limit caseThe method we have used to determine the differential cross sections allows us to study the backward-

scattering limit case, since the results obtained (concerning gg ) are valid for any value of the scattering

angle. The results obtained in this limit ( $ ) differ substantially from one case to another, thisfeature shows the strong dependence of the backward differential cross section on the spin value

of the particles. Thus, for the massive scalar field scattering in the backward direction is finite and

independent of the angular momentum @:

g

g

+3,@

@ +JP,5

%7e5 .

4 . y5

5y5

5 5e+4 . y

5,

y5

&(55)

and in the ultrarelativistic (UR) limit, it coincides with that obtained by De Logi and Kovacs for

massless scalar particles:

g

g

+3,

@

y @ 4

@ +JP,5+4 5e,5 (56)

The massive spin-1/2 and spin-1 particles also have finite values for the backward differential

cross sections, but unlike the scalar-particle case, they depend on the angular momentum per unit

mass squared @5 and also on the angles and !. It is interesting to point out that this angulardependence of the backward differential cross sections still appears for the UR limit case, when

g

g

+4@5,

@

y @ 4

@ +JP,5@5s53

>>!, gg+y>>>!,, which means we determine the total diffusion cross sectionsas -indefinite integrals of the differential cross sections:

+y>>, @ ]+,

g

g+y>>>!,g @ ]

+,

g vlq ]5

3

i+y>>>!,g! (59)

then, using the LeibnizNewton formula we determine the integral scattering cross sections for the

five processes in the following manner:

wrw+y> , @ +y>>,@

+y>>@%4

(60)

where % is a very small positive number. The exact numerical estimations used % @ 439. Rigorouslyspeaking the graphs obtained give only qualitative dependencies. It is clear that in the % $ 3 limit

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16 Can. J. Phys. Vol. 76, 1998


j| L |

Ec k with respect to the angle k, for massivescalar particles in the UR case ( ' ), for both min-imal (K ' f) and nonminimal (K ' *e) couplings.


j| L |

Ec k with respect to the angle k for massivespinor particles in the UR case ( ' ).

Fig. 17. Variation of integral (total) cross sectionwith respect to the angle k, for massive vector parti-cles in the UR case ( ' ).

Fig. 18. Variation of integral (total) cross sectionwith respect to the angle k, for massive spin-vectorparticles in the quasi-UR case ( ' fbbb).

there are diverging expressions obtained for wrw+y> , in all five cases, the reason for this beingexplained above. The calculus gives the following expressions:

+3,+y>>, @ 7+@5s53 vlq5 5e5,frv +4 . y

5,5

y7frvhf5

5

. 49y5@5s53 vlq

5 e+4 . y5,y5

oq+vlq

5, +94=4,

4@5+y>>, @

7@5s53+6 . 5y

5 vlq5 :frv5,frv 49

@5s53+4 . 5y5 vlq5 6frv5,

frv5 y7

+4 . y5,5frvhf5 5

5y5

6 . y5+4 46@5s53 . 7@5s53 frv5, oq+vlq 5 , +94=5,+4,+y>>, @

9

; . @5s53+7 46y5 . 45 frv 5 . 4


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Radu and Tatomir 17

Fig. 19. Variation of integral (total) cross section with respect to the angle k, for massive spin-2 particles in thequasi-UR case ( ' fbbb).

+6@5,+y>>, @

765+4 y5, 9i4+y> ,frv . 9i5+y> ,frv5 . @

5s53i6+y> ,frv6

.6

5i7+y> ,frv7,

y

7

693+4 y5,@5s53 frv

5

5frv8

+4 y5,5

y7frvhf5

5 5

, oq+vlq

5, +94=7,

+5,+y>>, @

,frv . 43i5+y> ,frv5 .8

6y5i6+y> ,frv6

.8

5y7i7+y> ,frv7 . y

9i8+y> ,frv8

+4 . y

5,5

y7frvhf5

5

;6y5+4 y5, i9+y> ,oq+vlq 5 , +94=8,

where the corresponding functions il+y> ,> l @ 4> 9 are given in Appendix B, and the graphs forwrw+y> , @ +y>> @ , +y>> @ % @ 439, are given in Figs. 38. Due to the very rapidvariation of wrw+y> , in the range of the very small values of y, we use a logarithmic scale in Figs.914 that emphasizes the dependence of the integral diffusion sections on small and very small values

of the ratio mRm@s3 y. Also, since the variation of the integral scattering cross section with respectto the angle covers a values domain which is far greater than that corresponding to the variationwith respect to the parameter y, from Figs. 38 we cannot clearly see how the -dependence of thesecross sections looks. This is why we prefer to present the dependence of the integral scattering cross

sections on the angle (in the most interesting case, namely, for the massive spin-0, 1/2, and spin-1particles in the UR case, and for the massive spin-3/2 and spin-2 particles in the quasi-UR case)

separately, in Figs. 1519. We have to say that in this section (beginning with relation (61.1)) we didnot considered the factor (GM)5, which, as a constant, does not affect the shape of the surfaces and

curves, and for convenience in exact numerical calculations we considered @5s53 @ 4. Therefore, thenumerical values appearing in all the figures have only qualitative and not quantitative significance.

4. Summary and conclusions

We have developed an unified approach to the description of scattering processes for the massive

scalar, spinor, vector, spin-vector (RaritaSchwinger), and spin-2 particles in the frame of QG. The

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18 Can. J. Phys. Vol. 76, 1998

basic method we used here is that offered by the principle of minimal coupling in QG. For the

massive scalar field we also used nonminimal couplings. The calculations were made in the first-

order approximation but we also took into account the contribution given by the covariant derivative

according to the principle of minimal coupling in QG. Consequently, a greater generality of resultswas produced that referred to the value scale of the scattering angle, which thus was not forced to take

only small values. This fact allowed us to approach the special case of the backward-scattering limit.

The differential cross sections for the weak-field gravitational scattering of scalar, spinor, vector, spin-

vector, and spin-2 particles were calculated using Feynman perturbative methods. All the expressions

giving the differential cross sections are of the form:g

g$

Nhuu

@

g

g$

Vfkz

. terms which are proportional to @5s53

and, except for the massive scalar field, they exhibit a Rutherford-type angular dependence for @ @ 3and 4. For the massive scalar field, if@ @ 3 , the differential scattering cross section exhibits aRutherford-type angular dependence in the case of minimal coupling, and for nonminimal coupling

(introduced by scalar curvature U) this type of angular dependence appears only for

4. Also,for the massive scalar field, the effect of the angular momentum @ is to add a positive semidefinite

term togg$

Vfkz

, whereas, for the other massive fields, this effect is more complicated due to the

nonzero value of the spin v. We also studied the backward-scattering limit case and have obtainedthe result that the differential cross sections of the scalar, spinor, and vector particles in the backward

direction as well as for the UR case are finite, and consequently the backscattered particles must have

the opposite helicity, whereas for spin-vector and spin-2 particles, in the same case, the differential

cross sections are clearly infinite. We also determined the integral scattering cross sections of the five

types of particles as -indefinite integrals of the differential cross sections and we represented thequalitative dependencies of wrw+y> , on both the ratio mRm@s3 y and angle graphically. Thesegraphs permit a clear observation of the UR divergence of the massive spin-3/2 and spin-2 particles

cross sections.

Note that ifp is set equal to zero in (9) the final result for the scattering cross section is invariant

under a change (a gauge transformation) of the formE $ E . Ciwhere i is an arbitrary scalar function (in fact, taking into account relation (29), if p @ 3 in (9)then besides a factor of 1/2 one recovers the electromagnetic field Lagrangian). Unlike this, in the

corresponding spin-2 field case (i.e., setting p @ 3 in (10)) the final result for the scattering crosssection (corresponding to gravitons) is not invariant under the analogous gauge transformation, which

in this instance is of the following form:

* $ * . Ci . Ciwhere i is an arbitrary vector function. This fact can be understood by taking into account that, ingeneral, the Feynman-diagram formalism preserves the gauge invariance as long as all the diagrams

of the same order in the coupling constant are included. The external field approximation that we

used here serves to simplify the algebra, but the effect of the omitted diagrams is to lose the gauge

invariance of the scattering amplitude.

Finally, it is worthwhile pointing out that for @ @ 3, in the small-angle approximation, thedifferential scattering cross sections of massive spin-0, 1/2, 1, 3/2, and spin-2 particles have the

same form, and in UR case they coincide with those corresponding to the massless scalar particles,

neutrinos, photons, gravitinos, and gravitons, i.e., the gravitational particle scattering in this limit

case is spin-independent [21, 22] in agreement with other authors results (see, for example, ref. 23)

obtained by different means in few particular cases.

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Radu and Tatomir 19

Acknowledgments

The authors thank Professor I. Gottlieb for many stimulating discussions and useful suggestions. We

are also grateful to Professor I. Merches for a pointer regarding our literary style.

References

1. W. De Logi and S.J. Kovacs Jr. Phys. Rev. D: Part Fields, 16, 237 (1977).

2. P.C. Peters. Phys. Rev. D: Part. Fields, 13, 775 (1976).

3. G.W. Gibbons. Commun. Math. Phys. 44, 245 (1975).

4. D. Radu and D. Tatomir. Nuovo Cimento Soc. Ital. Fiz. B, 111(12), 1465 (1996).

5. D. Tatomir and D. Radu. Rom. J. Phys. 39(7&8), 383 (1994).

6. D. Tatomir, D. Radu, and O. Mihalache. Rom. J. Phys. 39(5&6), 491 (1994).

7. S. Hamamoto. Preprints Nos. Toyama-84, hep-th /9509144 (1995); Toyama-90, hep-th /9605084 (1996).

8. R.P. Feynman. Acta Phys. Pol. 24, 697 (1963).

9. D. Boccaletti, V. De Sabbata, P. Fortini, and C. Gualdi. Nuovo Cimento Soc. Ital. Fiz. B, 11, 289 (1972).

10. M.D. Scadron. Advanced quantum theory and its applications through Feynman diagrams. Springer-

Verlag, Berlin, Germany. 1991.

11. D. Tatomir. Analele Stiintifice ale Univ. Al.I. Cuza Iasi (n.s.), Sect. Ib, Phys. Tom. XXIV. 1978. pp. 91.

12. D. Tatomir. Ph.D. thesis, Al I. Cuza University, Iasi, Romania. 1981.

13. H. Cohen. Nuovo Cimento Soc. Ital. Fiz. A, 52, 1242 (1967).

14. D. Tatomir, D. Radu, and O. Mihalache. Studia Univ. Babes-Bolyai Cluj-Napoca, Physica, XXXVIII, 2.

1993. pp.31.

15. S. Deser and B. Zumino. Phys. Rev. Lett. 38, 1433 (1977); E. Cremmer, S. Ferrara, L.Girardello, and A.

van Proeyen. Phys. Lett. 116B, 231 (1982); Nucl. Phys. B212, 413 (1983).

16. S. Deser, J.H. Kay, and K.S. Stelle. Phys. Rev. D: Part. Fields, 16, 2448 (1977).

17. J. Schwinger. Particles, sources and fields. Addison-Wesley, U.S.A. 1970. pp. 219.

18. D. Lurie. Particles and fields. John Wiley and Sons, New York. 1968. pp. 38&51.

19. D. Tatomir, D. Radu, and O. Mihalache. Analele Universitatii Timisoara, Seria Stiinte Fizice, Vol. XXX,

pp. 35, 49 (1993).

20. D. Tatomir and D. Radu. Analele Univ. Timisoara, Seria Stiinte Fizice, Vol. XXX, pp. 43 (1993).

21. D. Tatomir. In Abstracts of contributed papers for the discussion groups. Jena, July 14 19, 1980.

22. D. Tatomir. In The 10th International Conference in G.R.G. Padova. July 49, 1983. pp. 140.

23. K. Lotze. Acta Phys. Pol. B, 9, 665, 677 (1978).

Appendix A

To construct the S-matrix elementss3mV+,ms for the transition s j$ s3 in the external field approxi-

mation, we need to know, on the one hand, the matrix elements of the creation and annihilation matter

field operators between the state that contains one particle (with a given four-momentum and a fixed

polarization state) and the vacuum state, and, on the other hand, the expressions for the first-order

vertices corresponding to the interaction processes of the massive spin-0, 1/2, 1, 3/2, and spin-2

particles with the external gravitational field. In the following we enlist the matrix elements for the

each field under consideration. They are

*+,+R,> *+,+R, $ 4+5,6@5

s5s3

> *+,> +R,> *+,> +R, $ ls+5,6@5s5s3 +D4,

#+d,+.,+R, $ 4+5,6@5

up

s3x+d,+R,> #

+d,+,+R, $ 4

+5,6@5

up

s3x+d,+R, +D5,

#+d,+.,> +R, $ls

+5,6@5

up

s3x+d,+R,> #

+d,+.,

> +R, $ ls

+5,6@5

up

s3x+d,+R, +D6,

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20 Can. J. Phys. Vol. 76, 1998

E+d,+, +R,> E+d,+, +R, $

h+d, +R,

+5,6@5s

5s3+D7,

J+d,+, +R,> J+d,+, +R, $

lk

h+d, +R,s h+d, +R,sl+5,6@5

s5s3

+D8,

#+d,+., +R, $x+d, +R,

+5,6@5s

5s3> #

+d,+,

+R, $x+d, +R,

+5,6@5s

5s3+D9,

#+d,+.,> +R, $lsx

+d, +R,

+5,6@5s

5s3> #

+d,+,

> +R, $ lsx

+d, +R,

+5,6@5s

5s3+D:,

*+d,+, +R,> *+d,+, +R, $

h+d, +R,

+5,6@5s

5s3> *

+d,+,> +R,> *

+d,+,> +R, $

lsh+d, +R,

+5,6@5s

5s3+D;,

and for the first-order vertex:

l+5,7+R3

R

^,+t

3,.

In the above expressions forq

incidentemergent particles we have

q+d,$+u,>+,$+.,>R$R+d,$+v,>+,$+,>R$R3 .

Appendix B

Below we list the explicit expressions appearing in relations (52.2)(52.5), (61.4), and (61.5).

In relation (52.2)

i4+, @4

7vlq5 frvhf5

5+E4=4,

i5+y> , @4

7frvhf5

5+< . 9frv . frv5 y5 vlq5 , +E4=5,

i6+, @45

vlq +E4=6,

i7+, @ , @ 47

frvhf5

5+ , @7

6

6 . 5y5 5vlq5

5+4 y5 frv5

5,

;

6y5 7

6+E5=4,

i5+y> , @ 7frvhf5 5

;6

y5 76

i4+y> , +E5=5,

i6+y> , @7

6vlq +4 y5 vlq5

5, +E5=6,

i7+, @ 76

frw5

5+E5=7,

In relation (52.4)

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Radu and Tatomir 21

i4+y> , @4

69frvhf5

5

;5 y5+45< 9;y5 . 6y7, . 5 frv +64 :3y5 . 5:y7, y5 frv5

+4 , @4

69frvhf5

5

;5 y5+45< 9;y5 . 6y7, . 5 frv +64 :3y5 . 5:y7, y5 frv5

+4< 47y5 y7, . 5y7 frv6 +7 . y5 frv , +4 y5,5 i4+y> , +E6=5,i6+y> , @

4

4;vlq +8 9y5 . 6y7 7y5 frv . 5y7 frv5 ,+4 y5,5 +E6=6,

i7+y> , @ 5 , @4

78frvhf5

5^5:3 9


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22 Can. J. Phys. Vol. 76, 1998

i5+y> , @ 7y5+6 . y5, . @5s53^+9: 488y5 . ;7y7 . 7y9, . +:5 479y5 . 99y7, frv +8 5 , @ 43 67y5 . 5:y7 . , @ 5 . 8y5 . y7 . +5 . 9y5,frv y5+4 . y5,frv5 +E8=7,

i8+y> , @ 48 59y5 54y7 477y5@5s53+4 y5,vlq5 +E8=8,In relation (61.5)

i4+y> , @ 5749 43349y5 . 43:57y7 8685y9

Documents

D. Radu and D. Tatomir- Gravitational scattering of massive particles on a background with axial symmetry