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Scattering of massless particles and spacetime shifts
NUCLEAR
P H VS I C S B
Nuclear Physics B 385 (1992) 604—622 North-Holland
_________________
Scattering of massless particles and spacetime shifts Valeria Ferrari International Centerfor Relativistic Astrophysics (ICRA), Dipartimento di Fisica “G. Marconi Universita’ di Roma, Rome, Italy
Maurizio Martellini
*
Sezione INFN, Universita’ di Roma, Rome, Italy Received 11 May 1992 Accepted for publication 19 June 1992
We compute the differential cross section of the scattering of two massless particles, in the vicinity of a Cauchy horizon generated by the collision of two plane gravitational waves. One of the two particles, and the associated impulsive wave, is assumed to be part of the curved background, the second is a test particle. We show that for large momentum transfer I q I, the cross section is independent of I q I.
1. Introduction In the classical theory of general relativity a massless particle must necessarily be accompanied by a gravitational shock wave. This can be seen in a variety of ways, for example by boosting the Schwarzschild solution along an arbitrary direction, and taking the limit when the velocity of the boost tends to the speed of light and the mass of the black hole tends to zero in some appropriate manner. As Aichelburg and Sexl [1] have shown, the resulting metric is an exact solution of Einstein’s equations which describes a massless point-like particle, reminiscent of the black-hole singularity, and an impulsive gravitational wave to which the gravitational field of the black hole reduces in this extreme limit. Another interesting algorithm to generate the same solution is the “scissor-and-paste” procedure introduced by Penrose [21,consisting in the following operations: cut the spacetime in two pieces along a null surface, shift one with respect to the other by a given amount, and glue them together by a suitable coordinate transformation. This method, which has also been used by Dray and ‘t Hooft [3] to generate a solution describing a Schwarzschild black hole with a massless particle “sitting” on Permanent address: Dipartimento di Fisica, Universita’ di Milano, Milan, and Sezione INFN, Pavia, Italy. *
0550-3213/92/$05.00 © 1992
—
Elsevier Science Publishers B.V. All rights reserved
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the horizon, will be described (and used) in detail in the following sections. Here we only want to remark that, according to this procedure, massless particles can be understood as shifts in space and time. Unlike the classical theory of electromagnetism, where only waves are allowed to exist, in the classical theory of general relativity there exist exact solutions which describe both massless particles and their associated gravitational waves. It seems, therefore, that the “double nature” of particles and waves which is exhibited by some phenomena in nature, and which has been unified by the theory of quantum mechanics, is in some sense already contemplated in general relativity, although at a classical level. This is a remarkable fact whose implications are not yet understood. Under these premises, it would be extremely interesting to find an exact solution of Einstein’s equations describing the collision of two such particle waves. It would certainly improve our understanding of the reasons why gravity seems to resist to any attempt of quantization. Moreover, it would describe the gravitational interaction of particles whose energy is dominated by kinetic energy rather than rest mass, and these effects might possibly be observed in real processes of scattering of high-energy particles. Unfortunately it seems exceedingly difficult to find such solution, but a number of things can be done to approach, by successive levels of approximation, the fully relativistic situation. For example, as a first approximation ‘t Hooft [4] has considered the scattering of a massless test particle with the gravitational field of a particle wave traveling in flat spacetime. In this case the differential cross section is shown to depend on the inverse fourth power of the exchanged momentum q the same behaviour as the Rutherford cross section. It is interesting to note that the same dependency on q has also been found by Veneziano [51in the case of the scattering of two strings in flat space-time. As a next level of approximation, one might ask how the scattering of massless particles takes place when it occurs in a curved background, assuming that one of the two particles and the associated shock wave belongs to the exact background solution, and the second is a test particle. For example, we might choose a background solution describing the interaction of gravitational plane waves. Sofar, all attempts to add null particles on the incoming wavefronts and to follow their successive interaction in terms of an exact solution have been unsuccessful. However, null particles can be added to a colliding-wave background in the following way. It is known that the interaction of pure gravitational plane waves does not necessarily produce an all-embracing curvature singularity. A variety of exact solutions have been found [6—91in which pure gravitational waves interact and, after a finite amount of time, generate a Cauchy horizon in place of the curvature singularity. In this case the space-time can be extended across the horizon by a suitable coordinate transformation. As we shall show in sect. 2, by using the scissor-and-past procedure, massless particles can be added on the =
—
,
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horizon of these colliding-wave solutions, in a way similar to that used by Dray and ‘t Hooft to add a massless particle on the horizon of a Schwarzschild black hole. The resulting exact solutions provide an interesting model to explore the properties of scattering processes of null particles in a curved background, and, in particular, in the vicinity of a horizon. We would like to stress that in our solution the two pure gravitational colliding waves and one of the two particles constitute the background, while the other scattering particle has to be understood as a test field given by the solution of a massless wave equation in this background. In principle we might solve a relativistic wave equation for massless fields (for example the Klein—Gordon equation) and compute the cross section of the scattering of this wave by the particle wave present in the curved background. It is clear that the low-energy behaviour of the cross section would depend on the detailed structure of the particular background we select through an effective potential, and on the type of massless field under examination. However, the high-energy behaviour, which is essentially governed by the eikonal approximation, is independent on the kinematic of the scattered fields. A motivation for this assumption is that in the high-energy limit (wavelength of the massless particle tending to zero) the incident test particle is probing the “microscopic” point-like graviton structure of the background by the exchange of virtual gravitons. A similar situation arises when a high-energy light particle scatters on a proton in a parton model of hadrons. The result of our investigation is the following. We find that in the high-energy (eikonal) limit, and for high exchanged momentum q I, the cross section is independent of the exchanged momentum. The behaviour is therefore completely different from the case analysed by ‘t Hooft, in which a test particle scatters on a single particle-wave in flat space-time, and from the Veneziano amplitude in string theory. In their case the cross section exhibits a behaviour which is characteristic of scattering by a potential barrier, while in our case it is elastic scattering by the “point-like” structure of the background. 2. Space-time shifts and massless particles Impulsive pure gravitational plane waves are the most elementary wave-like solutions of Einstein’s equations. The idea of interpreting these waves as shifts in space and time is originally due to Penrose, and it is known as the “scissor-andpaste” procedure. It operates in the following way. Consider a flat space-time ds2=2du dv—dx2—dy2,
(1)
where u and v are null coordinates u=~/~(t-z),
v=~/~(t+z).
(2)
/
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Cut the space-time along the surface u 0, and assume that for u coordinate u is replaced by a shifted coordinate, =
u—v+e(u)f(x,
>
y),
0 the
(3)
where e is the Heaviside step function, and f=f(x, y) is a function which is left unspecified for the moment. For u > 0 the metric becomes ds2
=
2 du[dv
+
e(u)f, dx’]
—
dx2
—
dy2,
i
=
1, 2.
(4)
If we now introduce the following set of new coordinates i2
=
u,
i~ v =
+
e(i2)f,
~ =xt,
(5)
d~2 d92,
(6)
the metric (4) takes the form ds2
=
2 di2[di~
—
ö(i2)f di2]
—
—
which is flat everywhere except on the null surface i2 u 0. Therefore, the transformation (5) “glues” together the two parts of flat spacetime, u <0 and u > 0, that have been shifted with respect to each other, by introducing an impulsive “disturbance” traveling along u 0 at the speed of light. The solution (6) has been studied by several authors, starting from Brinkman in 1923 [10], and it represents a wave traveling in flat space-time along the positive z-direction. The only non-vanishing component of the Ricci tensor is =
=
=
R~
=
~(â)~f,
(7)
where ~ is the laplacian operator in the transverse coordinates ~ and quently, the metric (6) represents a pure gravitational impulsive wave if —‘
9.
f=~—9~,
Conse-
(8)
or an impulsive wave associated to a distribution of massless particles if ~
(9)
For example, the metric for a massless particle traveling in flat space-time along the positive z-direction, is obtained by solving the equation z1f=32~rp~2(~, 9), where p is a constant, and we are are adopting the convection G
(10) =
c
=
1. The
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solution in terms of the original coordinates (u, v, x, [11]) is
f=
y)
(originally due to Bonnor
2 +y2 —4p lo~(x ~2
(11)
where C is a constant scale wavelength. The stress—energy tensor that correspond to the solution (11) is T~~=4p~2(x, y)~(u).
(12)
Thus, the constant p does represent the momentum of the massless particle located at x y 0. In this case the metric agrees with the solution obtained by Aichelburg and Sexl (cf. ref. [3]) by boosting the Schwarzschild solution in the limit v c, and it will be referred to as a “particle—wave” solution, because it describes an impulsive gravitational wave and the massless particle which is its source. =
=
—‘
3. Massless particles in a curved background As anticipated in sect. 1 we do not know an exact solution which describes the scattering and the gravitational interaction of two particle waves. The purpose of the present investigation is to construct a model which mimics as closely as possible this interaction. In this view we shall discuss how and where it is possible to add one or more massless particles to a background of two interacting pure gravitational plane waves, by using the scissor and past procedure. Let us consider for example the simplest solution describing the collision of two impulsive gravitational waves with collinear polarization, the Khan—Penrose [121 solution. Before the collision, the two incoming waves travel in flat space-time, and it is not difficult to add to the shift due to the pure impulsive wave (eq. (8)), a further shift due to the assumed presence of a massless particle (eq. (11)). This is equivalent to add to the solution of eq. (10) a solution of the associated homogeneous equation. However, when we try to construct the solution in the region where the two particle waves interact, the scissor-and-paste procedure is not of much use, essentially because we are not able to solve the problem as an initial-value problem, and we do not know that type of source will be produced after the impact. Since we are dealing with massless particles, the shift must be applied along a null surface, and, apart from the wavefronts, no other “privileged” null surfaces exist in the Khan—Penrose solution. However, the procedure can be successfully applied in a different context. It is known that when pure gravitational plane waves collide and interact, a curvature singularity appears as a consequence of a process of mutual focusing.
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The Khan—Penrose solution belongs to this class of solutions. There exist, however, other classes of solutions in which the interaction of pure gravitational plane waves produces a Cauchy horizon in place of the curvature singularity. In this case the solution can be extended across the horizon by a suitable coordinate transformation, and the picture of the space-time can be completed. Thus, in these solutions there exist another “privileged” null surface on which we can try to put a null particle by using the shift method. In order to do that, we have selected one of the simplest solutions with horizon [81, in which the region where the two waves interact is locally isometric to a part of the Schwarzschild solution. The metric in the interaction region is ds2 =E(1
+
)2(
1_~2
—
1_~2)
—
(~~)
dx~—(1
+
~2)(1
~)2
dy2, (13)
where sj is the time elapsed from the instant of collision ~ 0), and ,a cos 0 measures the distance along the direction of propagation (js 0 when the two waves collide). E is a constant associated to the energy carried by the incoming waves. In fact, before the collision the only non-vanishing Weyl scalars are respectively =
=
=
1
3
1
3
1114=__
E 2(1+u)3
E 2(1+v)3
(14)
where u and 1) are dimensionless outgoing and ingoing null coordinates. In geometric unities, ~I’()and ~[‘4 have dimensions 12, i.e. that of an energy per unit volume, and the dimensions of E follow as a consequence. The local isometry of this solution with the Schwarzschild spacetime is immediately manifested if we perform the following coordinate transformation: r
—
m
~.t=cos0
m y=mçb,
x=t,
E=m2.
(15)
The metric (13) becomes ds2=
—
i)
dr2—
—
i) dt2—r2(d02+sin2O d~2),
which is the Schwarzschild metric inside the horizon (r
<
2m). The surface sj
(16) =
1,
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that is, the horizon produced after the collision of the two incoming waves, is thus mapped onto the horizon of the Schwarzschild black hole r 2m. In order to introduce a massless particle by performing a shift of the horizon, we need to extend the metric (13) by a suitable set of new coordinates, and the isometry with the Schwarzschild solution suggests the following generalization of the Kruskal coordinates (for details see ref. [8]) =
T=
—f~(i
X= ~/~(1 T2 —X2
=
=
)1/2
—
~(1
—
—
~)1/2
4osh(x/4V~), e~’~”c
e~1~~4 sinh(x/4V~),
~) e~’~’~”2,
-tanh(x/4fE~).
(17)
The metric (13) becomes ds2
1~~2[dT2 dX2]
32E_e
=
—
—
1+?7
E~
+
)2
d~2 (1 —
—
~2)(1
+
)2
dy2.
1—/2 (18)
or, introducing the null coordinates U=T—X, ds2
=
(19)
V=T+X,
32E_e1~~2dU dv— E ~ + ~ 1+ij l/.L
d~2 (1 —
—
~2)(1
+
~)2
dy2. (20)
If we replace ~ and ~ by the expressions given in eqs. (15), the metric (20) becomes the Schwarzschild solution in Kruskal coordinates i~,
ds2
=
32m3
eT/2m
dU dv— r2[d02
+
sin20 d~2].
(21)
The shift can be performed either on the metric in the form (20), or, alternatively, in the form (21). We shall use the form (21) in order to make the comparison with the Schwarzschild case easier. We would like to remark that the choice of the minus sign in the definition of T in eqs. (17) is not standard, and it has been done for a physical reason: in the region of interaction the arrow of time is fixed by the coordinate which is in fact the time elapsed from the instant of collision; we require that in the region of i~,
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//:Z~Z1~~ZZZ~::\\\ Fig. 1. The space-time diagram for the extended solution. The dashed region A represents the region where the incoming waves interact. Regions A and B are isometric as well as regions C and D.
interaction the time and the coordinate T, that must describe the time in all the accessible space-time following sj 1, both point in the same direction, in the sense that an increase of correspond to an increase of T. The two-dimensional diagram illustrating the extended space-time is shown in fig. 1. The quadrant T2 > X2, T < 0 (the dashed region A) is isometric to the solution (18). It represents the region of interaction of the two colliding waves, and it corresponds to the “white-hole” region in the standard Kruskal diagram for black holes. We have removed the region “before” sj 0, corresponding to the spacetime preceeding the collision, because that region must be covered by the usual Penrose extension [121in terms of null coordinates u and v defined as i~
=
i~
=
=
ui/i
~=ui/1_v2
—
+
vi/i~u2,
_vi/1_u2.
Since the metric (18) is unchanged by the transformation T T, the quadrant T2 >X2, T> 0 (region B in fig. 1) is isometric to the former. In the extended region B, there is a curvature singularity on 1 (T2 X2 1). The regions C and D are isometric to the Schwarzschild solution outside the horizon (see ref. [81for details), but is should be remembered that the isometry is only local, because in the colliding-wave solutions both Killing vectors have open orbits. The structure of the space-time is now easily investigated. For example, null test particles starting in region A (our interaction region) follow 45°lines in the —‘
—
=
—
=
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Kruskal coordinates, and, since we have inverted the arrow of time with respect to the standard Kruskal coordinates for the Schwarzschild black hole, every photon starting in region A leaves this region, crosses the horizon and escapes to infinity in region C or D without hitting the singularity that is present in region B. However, a photon starting in region C, for example, can escape to infinity or fall into the singularity in a finite time. Finally, we may note that
~)
(i e~’~~2
uv= (1
=
e~2m,
—
(22)
and therefore the horizon s~ 1 is mapped onto the null surfaces U 0 and V 0. We are now in a position to add a massless particle on the horizon produced by the collision of two pure gravitational plane waves. We first replace the coordinates U and V by dimensional coordinates =
=
U-’mU,
=
V-*mV.
(23)
We cut the space-time along the surface U 0 (a cut along V 0 can be performed in complete analogy), shift, for U> 0, the coordinate V by the amount =
=
v-v+e(U)f(~,~), with
f
(24)
unspecified for the moment, and glue the two parts of space-time U<0,
U>0,
(25)
with the coordinate transformation
ü=U,
~‘=V+ø(U)f,
/i=~,
~=4.
(26)
As a result, the metric (20) takes the form ds2
=
~e~2m
dU[d~— ~(U)f dO]
—
r2[d~2 + sin2~d~2j,
(27)
where now r r(U, 1/), and f has to be understood as function of the new coordinates. This metric belongs to the general class =
ds2=2A(O, l~)dL2[d~2—~(U)fdO] —g(U, J2~h~
1,
(28)
dx’ 1 dx where 16m A(U, V)
=
_____e_r/2m,
g(U, V)
=
r2.
(29)
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613
As shown in ref. [31,it represents a massless particle traveling in the U-direction, and and located at ~i 0, provided the function f satisfies the following equation: =
=
—
where p is a constant,
~
1]f= 32irp(gA)
~)
(,062(12
(30)
is the laplacian associated to the two-metric d2
(ds2) 2
1
=
~2
2 +
(1
— ~2)
d~
,
(31)
and (gA) Iu=o=2m
e1 =25E e1.
(32)
The meaning of the constant p is the following. The stress—energy tensor for the massless particle traveling along the U-direction is Tab
=
4p~2(ii,
)~(U)~~.
(33)
We are working in unities c G 1, therefore [Tab] [12]. U has the dimension of length (and [~(U)] [i~]); since ~i and ~ are dimensionless coordinates, it follows that p is the momentum of the particle (a length in geometric unities) divided by the square of a scale length A: =
=
=
=
momentum (34)
A2
The equation relevant to our problem finally is
~),
[~1~—1]f=2~rkp~2(,i,
(35)
k=29m2e’=29Ee1.
(36)
where
Eq. (35) should be compared with eq. (10). The difference between the two is the term —f on the left-hand side, and the factor k on the right-hand side, due to the fact that we are now operating a shift in a curved background. The explicit form of the equation for r is [(i_
~2)~2
—
2jth~+ 1
~
2a4 —
lIf(JL, ~)
2(~,~), =
2~kp6
(37)
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where we have dispensed with the use of the ‘hat’ on the coordinates. The solution can be found by assuming a factorized form for the function f: f(IL,
4)
~
=
e~’,
(38)
where a is an arbitrary constant. With this assumption eq. (37) becomes ~ 1(1 ~2)32
—
2~
+
1
~2
—
1Jfa(~) e~
=
~
(39)
The previous equations holds in the sense of distributions. Due to the presence of the ~(~) the only contribution to the solution comes from a 0, and eq. (39) becomes =
[(1 ~
—
1]fo(,a) =2~rkp~(~s),
(40)
The solution can be found in terms of a series of Legendre polynomials: f(~t,y)
=f~)
=
—kp ~ 1=0
~
~ +
+
~ P,(~z).
(41)
)
~.
Eq. (41) corresponds to the solution found by Dray and ‘t Hooft in ref. [3]. In that case it represents a massless particle sitting on the horizon of a Schwarzschild black hole at 0 0. In the next section we shall scatter a massless test particle on the particle in the background, and we shall compute the scattering cross section. The results we shall obtain will be applicable both to our solution, and to the aforementioned Dray—’t Hooft solution. =
4. Scattering of massless particles in a curved background In the previous section we have found an exact solution of Einstein’s equations which describes a massless particle located at (~ 0, 4 0) on the horizon produced by the interaction of pure gravitational plane waves. It is clear that the solution is, in some sense, “artificial”, because we do not know where does the particle come from: it simply happens to be there when the horizon is produced at the time 1 after the instant of collision. However, the solution provides a model to study the problem of scattering of massless particles in a situation in which at least one of the particles belongs to a curved background. In the following analysis, this particle will be indicated with the label ‘2’. We shall now send a massless test particle, indicated with the label ‘1’, against the particle wave in the background, assuming, for example, that the particle 1 =
i~ =
=
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Scattering of massless particles
travels along V 0 with an arbitrary momentum p~ p~, and a momentum p ~ in the transverse plane ~ y). In the eikonal approximation the wave function of the particle 1 will be =
=
~I’~=exp(i[p1x1 _p~V]),
(42)
where we have now assumed that c h 1. If the particle 2 were not present in the background, the test particle 1 would cross the line U 0 in the (U, V) diagram shown in fig. 1, and proceed straight along the line V 0. The impact with the particle 2 forces the particle 1 to suffer a shift in the V-direction =
=
=
=
~1V=f,
(43)
where f is given by eq. (41). Afterwards, the particle will proceed along a straight line, with unaltered momentum (as it follows from the geodesic equations). The wave function after the impact will be t’~=exp(i[p~~x1_p~(V+f)])=exp(_ip~f)1I1~’).
(44)
S12_exp(ip1~~f),
(45)
The factor
can therefore be interpreted as the reflection coefficient, or the two-body scattering matrix. It should be remembered that the function f given by eqs. (41) and (36) contains of the particle in the background 2), giventhe by information eq. (34), andabout aboutthe the momentum energy density of the background of colliding ~( waves, E (or the mass of the black hole m) expressed in the new unities. We shall now compute the scattering amplitude
A=(p~ISi
2Ip~)=Nf
(2ir)
~exp[i(p1~—p0~1)x]S12,
(46)
where N is a normalization constant. We shall assume that the inner product in the Hilbert space is normalized to ~
(47)
~
By introducing the following variables q
=
—pm,
~
2,
-
=
s
(exchanged momentum) (Mandelstam variables)
=~~~(2),
—q 0O
P 1(0)[1+’]
f=E 1+1(1+1) P1(g),
f= —kp~f,
(48)
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the scattering amplitude, up to the irrelevant normalization constant, is A
=
f dO exp[iof’7]
In appendix A we show that if the function integral form oo
-
f[cos(0)]
=
0
exp[iksf[~z]].
(49)
f defined in eq. (48) is written in the
cos(zyT) dz Vcoshz—coso
(50)
,
the integral can be evaluated, and the solution is given in terms of the Legendre functions Pa(cos 0): f[cos(o)J =CP0,(—cos 0) —iDPa(cos 0),
(51)
where C and D are constants given in appendix A, and a = + ~ evaluate the scattering amplitude (49) we shall now assume that —
-~
In order to
ks<<1.
(52)
From eq. (36) it Mfollows that, since we are now using the unities c = h 2,where G = M1~ 01 is the Planck mass, eq. (52) is equivalent to 3sm2 1, G namely
=
<<
1, and (53)
M~ s*z<—~-, in the case of scattering near the Schwarzschild horizon, or
(54)
M~ s*z*z—~— E
(55)
in the case of a scattering near the horizon produced by the collision of gravitational plane waves. Thus, the energy of the particles must be much smaller than the sixth power of the Planck mass, rescaled with the energy of the background E (or the square of the mass of the black hole). We are therefore in a regime where strong quantum effects can be neglected (we shall add some further comments on this point in sect. 5). Under these conditions, eq. (49) becomes A={f dO exp[i0~~]+iksf dOexp[iOV~fl[CPa(—cos
0) —iDPa(cos (56)
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617
The scattering amplitude can be evaluated explicitly only in the limit of large momentum transfer:
f~7>>i.
(57)
In this case, as shown in appendix B, the result is (58) and the corresponding differential scattering cross section g
2p~’~ Jn ~ p~ outj1 d out F ,~(l) I
n-i =
(ks)2{{~(fl
~ (n
n=1
m=0
[3
+
(2m
+ 1)21)
(
D—C (2n)!
2
[3+2m+12])(~[C+_nnDj)I
2
}
~
(59)
This equation should be compared with the result obtained by ‘t Hooft in ref. [4]. In that case he considered the scattering of a massless test particle by a single particle wave traveling in flat space-time, and he found the following cross section: ‘[p—~p~I d2p~~=4G2(s2/t2)d2p~,
(60)
that is, apart from the factor s2, the Rutherford cross section. Therefore, the process is essentially a scattering by a potential, and for large momentum transfer the cross section goes to zero. In our case the same scattering process occurs in curved space-time in the vicinity of a horizon, and one of the two scattering particles, together with its shock wave, belongs to the curved background. Since C and D are constants eq. (59) shows that for large momentum transfer the cross section is independent on the exchanged momentum.
5. Concluding remarks We have shown that when the scattering of massless particles takes place in the vicinity of the horizon generated by the collision of two pure gravitational waves, for large momentum transfer the cross section is independent of t. The same result holds if the scattering process occurs in the vicinity of a Schwarzschild black hole, when the particle wave that belongs to the background is sitting on that horizon.
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This is due to the isometry existing between the Schwarzschild solution and the solution for colliding waves that we have used in sect. 3, and to the shift procedure used to add massless particles to a background that can be applied to both situations with similar results. This result suggests that a constant cross section is a generic outcome of scattering processes of massless particles in the vicinity of horizons. It should be remarked that in the standard eikonal approach to scattering problems, both particles are usually assumed to be test particles moving in a given linearized background. In this way, the non-linearity of their gravitational interaction is neglected. In our approach we still do not compute the full non-linear interaction; however, one of the two particles, together with its wave, is part of the curved background, and the eikonal approximation is applied only to the incident test particle. It is interesting to note that the result established in sect. 4 holds in the limit defined by eqs. (54) and (55). Let us consider first the case when the background solution represents a Schwarzschild black hole plus a wave particle on the horizon. If the black hole is a macroscopical object, then the condition M~ S ~
m
imposes a severe constraint on the energy of the scattering particles. In the case of our solution, two plane gravitational waves collide and generate a “white-hole” region (region A in fig. 1), and a particle wave is put on the resulting horizon. We might think to this solution as to a particle wave “dressed” by a gravitational field originated by the gravitational interaction of the incoming waves whose intensity is %Ii~= m. If we assume that the energy of the colliding waves is comparable with the Planck mass then eq. (55), M~ S <<
E
suggests that the energy of the scattering particles may be allowed to be remarkably large, compatibly with the semiclassical approximation. Appendix A THE EVALUATION OF THE FUNCTION
f
We shall now evaluate explicitely the integral
f[cos
~ cos(z~/J)dz
01= 0
~/coshz—cos0
.
(A.1)
/
V Ferrar( M. Martellini
Scattering of massless particles
619
Since the argument of the integral is an even function of z, eq. (A.1) can be written as ~ exp(iz~/I)dz f[coso]=~f . (A.2) —
Vcosh z
—~
—
cos 0
Transforming cosh z into its exponential form, and putting u
=
ez,
(A.3)
eq. (A.2) becomes u’V’~du
-
f[cosO]=~f
(A.4)
,
~/~/u2
—~
+
1—2 cos 0
or, alternatively u’~~”d2u
~
-
-~
i/i
(A.5)
+ 2uv + u2
where v = —cos 0,
a
=
~.
(A.6)
This integral can be expressed in terms of the Legendre functions P,~’ (cf. ref. [13], p. 160, eq. (33)) P~(L’)= F(1
2~F(1_2~)(v2_1)~2 ~)F(—~ v)F(v + 1) —
—
—~
Re(~+v)<0,
f du(1
+
2uv
-1/2
+ u2)~
~
Re(~—v)<1.
In our case ~s = 0 and = ia, and the conditions are satisfied. If we rewrite the integral (A.5) in the following form i-’
2f[v]
=
—
—
(_1)1~l/2fu’~1~2~(1 2u’v
+ u~2)~/2du’
± fu~o
du,
—
where we have replaced u f[v]
~-
=
—*
1/2)(1
+ 2uv + u2)
t/2
(A.7)
u’ in the first integral, it is easy to show that
CPt/ 2±~~(~) iDPi/2+~~( —v). —
(A.8)
620
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where
exp(_~/fl
‘IT
— ,
C= +
D=
,
exp(~/~)
1
+
exp(2~T)
(A.9)
and we have used the property 1/2-i~/~7~( v)
~
=
P
(A.10)
1/2+i~/~7~( LI),
Appendix B THE EVALUATION OF THE SCATTERING AMPLITUDE
We shall now evaluate explicitely the scattering amplitude given in eq. (56): A =N{f dO e~°~ +iksl],
(B.1)
where I=
f dO e’~~[CPl/2+~~( —cos 0)
—
1DP_I/2+~~( +cos
0)1,
(B.2)
and p = We shall assume that (B.3) therefore the following results will be true for large momentum transfer. Under this condition the first integral in the scattering amplitude is negligible
f dO e~°~ =
—
~
(e’~~ 1)
0.
—
(B.4)
The Legendre functions P1/2+~~(+cos 0) can be expressed as a power series (cf. ref. [13], p. 174, eq. (5)) P1/2+1~(cos0)
=
1
+ ~ n=i
[n
0
[4~2 +
(2j
+
1)21
2
[sin~o]2nJ,
[4~2+
(21+
1)2]
2
[cos~o]2tlj;
J~~
Pt/ 2±~P(—cos0)
=
1
+
~ [n_i n=1 )°
(B.5)
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621
consequently I can be written as
=
n~
[‘h’ [~(21 +
+ 1)2]
~ (2n)! [Df~do
+iCf dO eb0~[cos~O]2n]=
e°~[sin~0]
I
~{n_1
(2n)!~~C121. (B.7)
I~and 12 can be evaluated explicitely in terms of the beta functions F(x)F(y) (B.8)
B(x,y)= F(x+y) and of the hypergeometric functions F. For example, I~is
=
22n+t
{exp(i~[~
—
~])B[~
XF[—2n, ~[~7 —n, +exp(i~[n
+
—n, i]
—n
~])B[~
+
—n, 2n
1, —i] +
1]F[0, ~
—n,
f~7+n + 1,
_i]} (B.9)
In the limit of large momentum transfer I~reduces to Ii=~{1+i(_1y(~}.
Similarly for
‘2
we
(B.10)
find ((2n)! 12
2
Substituting the expressions of I~and finally find
~ The series clearly converges.
22~~ +
‘2
~J.
(B.11)
in eq. (B.7) and then in eq. (B.1), we
(B.12)
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Scattering of massless particles
References [1] P.C. Aichelburg and R.U. SexI, Gen. Rel. Gray. 2 (1971) 303 [21 R. Penrose, Rev. Mod. Phys. 37 (1965) 215 [3] T. Dray and G. ‘t Hooft, NucI. Phys. B253 (1985) 173 [41G. ‘t Hooft, Phys. Lett. B198 (1987) 61 [5] G. Veneziano, Nuovo Cimento A57 (1968) 190 [6] V. Ferrari and J. Ibañez, Gen. Rel. Gray. 19 (1987) 405 [7] V. Ferrari, J. Ibañez and M. Bruni, Phys. Rev. D36 (1987) 1053 [8] V. Ferrari and J. Ibañez, Proc. R. Soc. London A417 (1988) 417 L91 S. Chandrasekhar and B.C. Xanthopoulos, Proc. R. Soc. London A408 (1986) 175 [10] H.W. Brinkman, Proc. Nat. Acad. Sci. (USA) 9 (1923) 1 [11] W.B. Bonnor, Commun. Math. Phys. 13 (1969) 163 [121 K.A. Khan and R. Penrose, Nature 229 (1971) 185 [131 H. Bateman, High transcedental functions, Vol. 1 (McGraw-Hill, New York, 1953) p. 193
P H VS I C S B
Nuclear Physics B 385 (1992) 604—622 North-Holland
_________________
Scattering of massless particles and spacetime shifts Valeria Ferrari International Centerfor Relativistic Astrophysics (ICRA), Dipartimento di Fisica “G. Marconi Universita’ di Roma, Rome, Italy
Maurizio Martellini
*
Sezione INFN, Universita’ di Roma, Rome, Italy Received 11 May 1992 Accepted for publication 19 June 1992
We compute the differential cross section of the scattering of two massless particles, in the vicinity of a Cauchy horizon generated by the collision of two plane gravitational waves. One of the two particles, and the associated impulsive wave, is assumed to be part of the curved background, the second is a test particle. We show that for large momentum transfer I q I, the cross section is independent of I q I.
1. Introduction In the classical theory of general relativity a massless particle must necessarily be accompanied by a gravitational shock wave. This can be seen in a variety of ways, for example by boosting the Schwarzschild solution along an arbitrary direction, and taking the limit when the velocity of the boost tends to the speed of light and the mass of the black hole tends to zero in some appropriate manner. As Aichelburg and Sexl [1] have shown, the resulting metric is an exact solution of Einstein’s equations which describes a massless point-like particle, reminiscent of the black-hole singularity, and an impulsive gravitational wave to which the gravitational field of the black hole reduces in this extreme limit. Another interesting algorithm to generate the same solution is the “scissor-and-paste” procedure introduced by Penrose [21,consisting in the following operations: cut the spacetime in two pieces along a null surface, shift one with respect to the other by a given amount, and glue them together by a suitable coordinate transformation. This method, which has also been used by Dray and ‘t Hooft [3] to generate a solution describing a Schwarzschild black hole with a massless particle “sitting” on Permanent address: Dipartimento di Fisica, Universita’ di Milano, Milan, and Sezione INFN, Pavia, Italy. *
0550-3213/92/$05.00 © 1992
—
Elsevier Science Publishers B.V. All rights reserved
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Scattering of massless particles
605
the horizon, will be described (and used) in detail in the following sections. Here we only want to remark that, according to this procedure, massless particles can be understood as shifts in space and time. Unlike the classical theory of electromagnetism, where only waves are allowed to exist, in the classical theory of general relativity there exist exact solutions which describe both massless particles and their associated gravitational waves. It seems, therefore, that the “double nature” of particles and waves which is exhibited by some phenomena in nature, and which has been unified by the theory of quantum mechanics, is in some sense already contemplated in general relativity, although at a classical level. This is a remarkable fact whose implications are not yet understood. Under these premises, it would be extremely interesting to find an exact solution of Einstein’s equations describing the collision of two such particle waves. It would certainly improve our understanding of the reasons why gravity seems to resist to any attempt of quantization. Moreover, it would describe the gravitational interaction of particles whose energy is dominated by kinetic energy rather than rest mass, and these effects might possibly be observed in real processes of scattering of high-energy particles. Unfortunately it seems exceedingly difficult to find such solution, but a number of things can be done to approach, by successive levels of approximation, the fully relativistic situation. For example, as a first approximation ‘t Hooft [4] has considered the scattering of a massless test particle with the gravitational field of a particle wave traveling in flat spacetime. In this case the differential cross section is shown to depend on the inverse fourth power of the exchanged momentum q the same behaviour as the Rutherford cross section. It is interesting to note that the same dependency on q has also been found by Veneziano [51in the case of the scattering of two strings in flat space-time. As a next level of approximation, one might ask how the scattering of massless particles takes place when it occurs in a curved background, assuming that one of the two particles and the associated shock wave belongs to the exact background solution, and the second is a test particle. For example, we might choose a background solution describing the interaction of gravitational plane waves. Sofar, all attempts to add null particles on the incoming wavefronts and to follow their successive interaction in terms of an exact solution have been unsuccessful. However, null particles can be added to a colliding-wave background in the following way. It is known that the interaction of pure gravitational plane waves does not necessarily produce an all-embracing curvature singularity. A variety of exact solutions have been found [6—91in which pure gravitational waves interact and, after a finite amount of time, generate a Cauchy horizon in place of the curvature singularity. In this case the space-time can be extended across the horizon by a suitable coordinate transformation. As we shall show in sect. 2, by using the scissor-and-past procedure, massless particles can be added on the =
—
,
606
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Scattering of massless particles
horizon of these colliding-wave solutions, in a way similar to that used by Dray and ‘t Hooft to add a massless particle on the horizon of a Schwarzschild black hole. The resulting exact solutions provide an interesting model to explore the properties of scattering processes of null particles in a curved background, and, in particular, in the vicinity of a horizon. We would like to stress that in our solution the two pure gravitational colliding waves and one of the two particles constitute the background, while the other scattering particle has to be understood as a test field given by the solution of a massless wave equation in this background. In principle we might solve a relativistic wave equation for massless fields (for example the Klein—Gordon equation) and compute the cross section of the scattering of this wave by the particle wave present in the curved background. It is clear that the low-energy behaviour of the cross section would depend on the detailed structure of the particular background we select through an effective potential, and on the type of massless field under examination. However, the high-energy behaviour, which is essentially governed by the eikonal approximation, is independent on the kinematic of the scattered fields. A motivation for this assumption is that in the high-energy limit (wavelength of the massless particle tending to zero) the incident test particle is probing the “microscopic” point-like graviton structure of the background by the exchange of virtual gravitons. A similar situation arises when a high-energy light particle scatters on a proton in a parton model of hadrons. The result of our investigation is the following. We find that in the high-energy (eikonal) limit, and for high exchanged momentum q I, the cross section is independent of the exchanged momentum. The behaviour is therefore completely different from the case analysed by ‘t Hooft, in which a test particle scatters on a single particle-wave in flat space-time, and from the Veneziano amplitude in string theory. In their case the cross section exhibits a behaviour which is characteristic of scattering by a potential barrier, while in our case it is elastic scattering by the “point-like” structure of the background. 2. Space-time shifts and massless particles Impulsive pure gravitational plane waves are the most elementary wave-like solutions of Einstein’s equations. The idea of interpreting these waves as shifts in space and time is originally due to Penrose, and it is known as the “scissor-andpaste” procedure. It operates in the following way. Consider a flat space-time ds2=2du dv—dx2—dy2,
(1)
where u and v are null coordinates u=~/~(t-z),
v=~/~(t+z).
(2)
/
V Ferrari, M. Martellini
Scattering of massless particles
607
Cut the space-time along the surface u 0, and assume that for u coordinate u is replaced by a shifted coordinate, =
u—v+e(u)f(x,
>
y),
0 the
(3)
where e is the Heaviside step function, and f=f(x, y) is a function which is left unspecified for the moment. For u > 0 the metric becomes ds2
=
2 du[dv
+
e(u)f, dx’]
—
dx2
—
dy2,
i
=
1, 2.
(4)
If we now introduce the following set of new coordinates i2
=
u,
i~ v =
+
e(i2)f,
~ =xt,
(5)
d~2 d92,
(6)
the metric (4) takes the form ds2
=
2 di2[di~
—
ö(i2)f di2]
—
—
which is flat everywhere except on the null surface i2 u 0. Therefore, the transformation (5) “glues” together the two parts of flat spacetime, u <0 and u > 0, that have been shifted with respect to each other, by introducing an impulsive “disturbance” traveling along u 0 at the speed of light. The solution (6) has been studied by several authors, starting from Brinkman in 1923 [10], and it represents a wave traveling in flat space-time along the positive z-direction. The only non-vanishing component of the Ricci tensor is =
=
=
R~
=
~(â)~f,
(7)
where ~ is the laplacian operator in the transverse coordinates ~ and quently, the metric (6) represents a pure gravitational impulsive wave if —‘
9.
f=~—9~,
Conse-
(8)
or an impulsive wave associated to a distribution of massless particles if ~
(9)
For example, the metric for a massless particle traveling in flat space-time along the positive z-direction, is obtained by solving the equation z1f=32~rp~2(~, 9), where p is a constant, and we are are adopting the convection G
(10) =
c
=
1. The
608
V Ferrar4 M. Martellini
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Scattering of massless particles
solution in terms of the original coordinates (u, v, x, [11]) is
f=
y)
(originally due to Bonnor
2 +y2 —4p lo~(x ~2
(11)
where C is a constant scale wavelength. The stress—energy tensor that correspond to the solution (11) is T~~=4p~2(x, y)~(u).
(12)
Thus, the constant p does represent the momentum of the massless particle located at x y 0. In this case the metric agrees with the solution obtained by Aichelburg and Sexl (cf. ref. [3]) by boosting the Schwarzschild solution in the limit v c, and it will be referred to as a “particle—wave” solution, because it describes an impulsive gravitational wave and the massless particle which is its source. =
=
—‘
3. Massless particles in a curved background As anticipated in sect. 1 we do not know an exact solution which describes the scattering and the gravitational interaction of two particle waves. The purpose of the present investigation is to construct a model which mimics as closely as possible this interaction. In this view we shall discuss how and where it is possible to add one or more massless particles to a background of two interacting pure gravitational plane waves, by using the scissor and past procedure. Let us consider for example the simplest solution describing the collision of two impulsive gravitational waves with collinear polarization, the Khan—Penrose [121 solution. Before the collision, the two incoming waves travel in flat space-time, and it is not difficult to add to the shift due to the pure impulsive wave (eq. (8)), a further shift due to the assumed presence of a massless particle (eq. (11)). This is equivalent to add to the solution of eq. (10) a solution of the associated homogeneous equation. However, when we try to construct the solution in the region where the two particle waves interact, the scissor-and-paste procedure is not of much use, essentially because we are not able to solve the problem as an initial-value problem, and we do not know that type of source will be produced after the impact. Since we are dealing with massless particles, the shift must be applied along a null surface, and, apart from the wavefronts, no other “privileged” null surfaces exist in the Khan—Penrose solution. However, the procedure can be successfully applied in a different context. It is known that when pure gravitational plane waves collide and interact, a curvature singularity appears as a consequence of a process of mutual focusing.
V Ferrari, M. Martellini
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Scattering of massless particles
609
The Khan—Penrose solution belongs to this class of solutions. There exist, however, other classes of solutions in which the interaction of pure gravitational plane waves produces a Cauchy horizon in place of the curvature singularity. In this case the solution can be extended across the horizon by a suitable coordinate transformation, and the picture of the space-time can be completed. Thus, in these solutions there exist another “privileged” null surface on which we can try to put a null particle by using the shift method. In order to do that, we have selected one of the simplest solutions with horizon [81, in which the region where the two waves interact is locally isometric to a part of the Schwarzschild solution. The metric in the interaction region is ds2 =E(1
+
)2(
1_~2
—
1_~2)
—
(~~)
dx~—(1
+
~2)(1
~)2
dy2, (13)
where sj is the time elapsed from the instant of collision ~ 0), and ,a cos 0 measures the distance along the direction of propagation (js 0 when the two waves collide). E is a constant associated to the energy carried by the incoming waves. In fact, before the collision the only non-vanishing Weyl scalars are respectively =
=
=
1
3
1
3
1114=__
E 2(1+u)3
E 2(1+v)3
(14)
where u and 1) are dimensionless outgoing and ingoing null coordinates. In geometric unities, ~I’()and ~[‘4 have dimensions 12, i.e. that of an energy per unit volume, and the dimensions of E follow as a consequence. The local isometry of this solution with the Schwarzschild spacetime is immediately manifested if we perform the following coordinate transformation: r
—
m
~.t=cos0
m y=mçb,
x=t,
E=m2.
(15)
The metric (13) becomes ds2=
—
i)
dr2—
—
i) dt2—r2(d02+sin2O d~2),
which is the Schwarzschild metric inside the horizon (r
<
2m). The surface sj
(16) =
1,
610
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Scattering of massless particles
that is, the horizon produced after the collision of the two incoming waves, is thus mapped onto the horizon of the Schwarzschild black hole r 2m. In order to introduce a massless particle by performing a shift of the horizon, we need to extend the metric (13) by a suitable set of new coordinates, and the isometry with the Schwarzschild solution suggests the following generalization of the Kruskal coordinates (for details see ref. [8]) =
T=
—f~(i
X= ~/~(1 T2 —X2
=
=
)1/2
—
~(1
—
—
~)1/2
4osh(x/4V~), e~’~”c
e~1~~4 sinh(x/4V~),
~) e~’~’~”2,
-tanh(x/4fE~).
(17)
The metric (13) becomes ds2
1~~2[dT2 dX2]
32E_e
=
—
—
1+?7
E~
+
)2
d~2 (1 —
—
~2)(1
+
)2
dy2.
1—/2 (18)
or, introducing the null coordinates U=T—X, ds2
=
(19)
V=T+X,
32E_e1~~2dU dv— E ~ + ~ 1+ij l/.L
d~2 (1 —
—
~2)(1
+
~)2
dy2. (20)
If we replace ~ and ~ by the expressions given in eqs. (15), the metric (20) becomes the Schwarzschild solution in Kruskal coordinates i~,
ds2
=
32m3
eT/2m
dU dv— r2[d02
+
sin20 d~2].
(21)
The shift can be performed either on the metric in the form (20), or, alternatively, in the form (21). We shall use the form (21) in order to make the comparison with the Schwarzschild case easier. We would like to remark that the choice of the minus sign in the definition of T in eqs. (17) is not standard, and it has been done for a physical reason: in the region of interaction the arrow of time is fixed by the coordinate which is in fact the time elapsed from the instant of collision; we require that in the region of i~,
V Ferrari, M. Martellini
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611
//:Z~Z1~~ZZZ~::\\\ Fig. 1. The space-time diagram for the extended solution. The dashed region A represents the region where the incoming waves interact. Regions A and B are isometric as well as regions C and D.
interaction the time and the coordinate T, that must describe the time in all the accessible space-time following sj 1, both point in the same direction, in the sense that an increase of correspond to an increase of T. The two-dimensional diagram illustrating the extended space-time is shown in fig. 1. The quadrant T2 > X2, T < 0 (the dashed region A) is isometric to the solution (18). It represents the region of interaction of the two colliding waves, and it corresponds to the “white-hole” region in the standard Kruskal diagram for black holes. We have removed the region “before” sj 0, corresponding to the spacetime preceeding the collision, because that region must be covered by the usual Penrose extension [121in terms of null coordinates u and v defined as i~
=
i~
=
=
ui/i
~=ui/1_v2
—
+
vi/i~u2,
_vi/1_u2.
Since the metric (18) is unchanged by the transformation T T, the quadrant T2 >X2, T> 0 (region B in fig. 1) is isometric to the former. In the extended region B, there is a curvature singularity on 1 (T2 X2 1). The regions C and D are isometric to the Schwarzschild solution outside the horizon (see ref. [81for details), but is should be remembered that the isometry is only local, because in the colliding-wave solutions both Killing vectors have open orbits. The structure of the space-time is now easily investigated. For example, null test particles starting in region A (our interaction region) follow 45°lines in the —‘
—
=
—
=
612
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Scattering of massless particles
Kruskal coordinates, and, since we have inverted the arrow of time with respect to the standard Kruskal coordinates for the Schwarzschild black hole, every photon starting in region A leaves this region, crosses the horizon and escapes to infinity in region C or D without hitting the singularity that is present in region B. However, a photon starting in region C, for example, can escape to infinity or fall into the singularity in a finite time. Finally, we may note that
~)
(i e~’~~2
uv= (1
=
e~2m,
—
(22)
and therefore the horizon s~ 1 is mapped onto the null surfaces U 0 and V 0. We are now in a position to add a massless particle on the horizon produced by the collision of two pure gravitational plane waves. We first replace the coordinates U and V by dimensional coordinates =
=
U-’mU,
=
V-*mV.
(23)
We cut the space-time along the surface U 0 (a cut along V 0 can be performed in complete analogy), shift, for U> 0, the coordinate V by the amount =
=
v-v+e(U)f(~,~), with
f
(24)
unspecified for the moment, and glue the two parts of space-time U<0,
U>0,
(25)
with the coordinate transformation
ü=U,
~‘=V+ø(U)f,
/i=~,
~=4.
(26)
As a result, the metric (20) takes the form ds2
=
~e~2m
dU[d~— ~(U)f dO]
—
r2[d~2 + sin2~d~2j,
(27)
where now r r(U, 1/), and f has to be understood as function of the new coordinates. This metric belongs to the general class =
ds2=2A(O, l~)dL2[d~2—~(U)fdO] —g(U, J2~h~
1,
(28)
dx’ 1 dx where 16m A(U, V)
=
_____e_r/2m,
g(U, V)
=
r2.
(29)
/
V Ferrar4 M. Martellini
Scattering of massless particles
613
As shown in ref. [31,it represents a massless particle traveling in the U-direction, and and located at ~i 0, provided the function f satisfies the following equation: =
=
—
where p is a constant,
~
1]f= 32irp(gA)
~)
(,062(12
(30)
is the laplacian associated to the two-metric d2
(ds2) 2
1
=
~2
2 +
(1
— ~2)
d~
,
(31)
and (gA) Iu=o=2m
e1 =25E e1.
(32)
The meaning of the constant p is the following. The stress—energy tensor for the massless particle traveling along the U-direction is Tab
=
4p~2(ii,
)~(U)~~.
(33)
We are working in unities c G 1, therefore [Tab] [12]. U has the dimension of length (and [~(U)] [i~]); since ~i and ~ are dimensionless coordinates, it follows that p is the momentum of the particle (a length in geometric unities) divided by the square of a scale length A: =
=
=
=
momentum (34)
A2
The equation relevant to our problem finally is
~),
[~1~—1]f=2~rkp~2(,i,
(35)
k=29m2e’=29Ee1.
(36)
where
Eq. (35) should be compared with eq. (10). The difference between the two is the term —f on the left-hand side, and the factor k on the right-hand side, due to the fact that we are now operating a shift in a curved background. The explicit form of the equation for r is [(i_
~2)~2
—
2jth~+ 1
~
2a4 —
lIf(JL, ~)
2(~,~), =
2~kp6
(37)
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V Ferrari, M. Martellini
/
Scattering of massless particles
where we have dispensed with the use of the ‘hat’ on the coordinates. The solution can be found by assuming a factorized form for the function f: f(IL,
4)
~
=
e~’,
(38)
where a is an arbitrary constant. With this assumption eq. (37) becomes ~ 1(1 ~2)32
—
2~
+
1
~2
—
1Jfa(~) e~
=
~
(39)
The previous equations holds in the sense of distributions. Due to the presence of the ~(~) the only contribution to the solution comes from a 0, and eq. (39) becomes =
[(1 ~
—
1]fo(,a) =2~rkp~(~s),
(40)
The solution can be found in terms of a series of Legendre polynomials: f(~t,y)
=f~)
=
—kp ~ 1=0
~
~ +
+
~ P,(~z).
(41)
)
~.
Eq. (41) corresponds to the solution found by Dray and ‘t Hooft in ref. [3]. In that case it represents a massless particle sitting on the horizon of a Schwarzschild black hole at 0 0. In the next section we shall scatter a massless test particle on the particle in the background, and we shall compute the scattering cross section. The results we shall obtain will be applicable both to our solution, and to the aforementioned Dray—’t Hooft solution. =
4. Scattering of massless particles in a curved background In the previous section we have found an exact solution of Einstein’s equations which describes a massless particle located at (~ 0, 4 0) on the horizon produced by the interaction of pure gravitational plane waves. It is clear that the solution is, in some sense, “artificial”, because we do not know where does the particle come from: it simply happens to be there when the horizon is produced at the time 1 after the instant of collision. However, the solution provides a model to study the problem of scattering of massless particles in a situation in which at least one of the particles belongs to a curved background. In the following analysis, this particle will be indicated with the label ‘2’. We shall now send a massless test particle, indicated with the label ‘1’, against the particle wave in the background, assuming, for example, that the particle 1 =
i~ =
=
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V Ferrar4 M. Martellini
615
Scattering of massless particles
travels along V 0 with an arbitrary momentum p~ p~, and a momentum p ~ in the transverse plane ~ y). In the eikonal approximation the wave function of the particle 1 will be =
=
~I’~=exp(i[p1x1 _p~V]),
(42)
where we have now assumed that c h 1. If the particle 2 were not present in the background, the test particle 1 would cross the line U 0 in the (U, V) diagram shown in fig. 1, and proceed straight along the line V 0. The impact with the particle 2 forces the particle 1 to suffer a shift in the V-direction =
=
=
=
~1V=f,
(43)
where f is given by eq. (41). Afterwards, the particle will proceed along a straight line, with unaltered momentum (as it follows from the geodesic equations). The wave function after the impact will be t’~=exp(i[p~~x1_p~(V+f)])=exp(_ip~f)1I1~’).
(44)
S12_exp(ip1~~f),
(45)
The factor
can therefore be interpreted as the reflection coefficient, or the two-body scattering matrix. It should be remembered that the function f given by eqs. (41) and (36) contains of the particle in the background 2), giventhe by information eq. (34), andabout aboutthe the momentum energy density of the background of colliding ~( waves, E (or the mass of the black hole m) expressed in the new unities. We shall now compute the scattering amplitude
A=(p~ISi
2Ip~)=Nf
(2ir)
~exp[i(p1~—p0~1)x]S12,
(46)
where N is a normalization constant. We shall assume that the inner product in the Hilbert space is normalized to ~
(47)
~
By introducing the following variables q
=
—pm,
~
2,
-
=
s
(exchanged momentum) (Mandelstam variables)
=~~~(2),
—q 0O
P 1(0)[1+’]
f=E 1+1(1+1) P1(g),
f= —kp~f,
(48)
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Scattering of massless particles
the scattering amplitude, up to the irrelevant normalization constant, is A
=
f dO exp[iof’7]
In appendix A we show that if the function integral form oo
-
f[cos(0)]
=
0
exp[iksf[~z]].
(49)
f defined in eq. (48) is written in the
cos(zyT) dz Vcoshz—coso
(50)
,
the integral can be evaluated, and the solution is given in terms of the Legendre functions Pa(cos 0): f[cos(o)J =CP0,(—cos 0) —iDPa(cos 0),
(51)
where C and D are constants given in appendix A, and a = + ~ evaluate the scattering amplitude (49) we shall now assume that —
-~
In order to
ks<<1.
(52)
From eq. (36) it Mfollows that, since we are now using the unities c = h 2,where G = M1~ 01 is the Planck mass, eq. (52) is equivalent to 3sm2 1, G namely
=
<<
1, and (53)
M~ s*z<—~-, in the case of scattering near the Schwarzschild horizon, or
(54)
M~ s*z*z—~— E
(55)
in the case of a scattering near the horizon produced by the collision of gravitational plane waves. Thus, the energy of the particles must be much smaller than the sixth power of the Planck mass, rescaled with the energy of the background E (or the square of the mass of the black hole). We are therefore in a regime where strong quantum effects can be neglected (we shall add some further comments on this point in sect. 5). Under these conditions, eq. (49) becomes A={f dO exp[i0~~]+iksf dOexp[iOV~fl[CPa(—cos
0) —iDPa(cos (56)
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Scattering of massless particles
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The scattering amplitude can be evaluated explicitly only in the limit of large momentum transfer:
f~7>>i.
(57)
In this case, as shown in appendix B, the result is (58) and the corresponding differential scattering cross section g
2p~’~ Jn ~ p~ outj1 d out F ,~(l) I
n-i =
(ks)2{{~(fl
~ (n
n=1
m=0
[3
+
(2m
+ 1)21)
(
D—C (2n)!
2
[3+2m+12])(~[C+_nnDj)I
2
}
~
(59)
This equation should be compared with the result obtained by ‘t Hooft in ref. [4]. In that case he considered the scattering of a massless test particle by a single particle wave traveling in flat space-time, and he found the following cross section: ‘[p—~p~I d2p~~=4G2(s2/t2)d2p~,
(60)
that is, apart from the factor s2, the Rutherford cross section. Therefore, the process is essentially a scattering by a potential, and for large momentum transfer the cross section goes to zero. In our case the same scattering process occurs in curved space-time in the vicinity of a horizon, and one of the two scattering particles, together with its shock wave, belongs to the curved background. Since C and D are constants eq. (59) shows that for large momentum transfer the cross section is independent on the exchanged momentum.
5. Concluding remarks We have shown that when the scattering of massless particles takes place in the vicinity of the horizon generated by the collision of two pure gravitational waves, for large momentum transfer the cross section is independent of t. The same result holds if the scattering process occurs in the vicinity of a Schwarzschild black hole, when the particle wave that belongs to the background is sitting on that horizon.
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V Ferrari, M. Martellini
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Scattering of massless particles
This is due to the isometry existing between the Schwarzschild solution and the solution for colliding waves that we have used in sect. 3, and to the shift procedure used to add massless particles to a background that can be applied to both situations with similar results. This result suggests that a constant cross section is a generic outcome of scattering processes of massless particles in the vicinity of horizons. It should be remarked that in the standard eikonal approach to scattering problems, both particles are usually assumed to be test particles moving in a given linearized background. In this way, the non-linearity of their gravitational interaction is neglected. In our approach we still do not compute the full non-linear interaction; however, one of the two particles, together with its wave, is part of the curved background, and the eikonal approximation is applied only to the incident test particle. It is interesting to note that the result established in sect. 4 holds in the limit defined by eqs. (54) and (55). Let us consider first the case when the background solution represents a Schwarzschild black hole plus a wave particle on the horizon. If the black hole is a macroscopical object, then the condition M~ S ~
m
imposes a severe constraint on the energy of the scattering particles. In the case of our solution, two plane gravitational waves collide and generate a “white-hole” region (region A in fig. 1), and a particle wave is put on the resulting horizon. We might think to this solution as to a particle wave “dressed” by a gravitational field originated by the gravitational interaction of the incoming waves whose intensity is %Ii~= m. If we assume that the energy of the colliding waves is comparable with the Planck mass then eq. (55), M~ S <<
E
suggests that the energy of the scattering particles may be allowed to be remarkably large, compatibly with the semiclassical approximation. Appendix A THE EVALUATION OF THE FUNCTION
f
We shall now evaluate explicitely the integral
f[cos
~ cos(z~/J)dz
01= 0
~/coshz—cos0
.
(A.1)
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V Ferrar( M. Martellini
Scattering of massless particles
619
Since the argument of the integral is an even function of z, eq. (A.1) can be written as ~ exp(iz~/I)dz f[coso]=~f . (A.2) —
Vcosh z
—~
—
cos 0
Transforming cosh z into its exponential form, and putting u
=
ez,
(A.3)
eq. (A.2) becomes u’V’~du
-
f[cosO]=~f
(A.4)
,
~/~/u2
—~
+
1—2 cos 0
or, alternatively u’~~”d2u
~
-
-~
i/i
(A.5)
+ 2uv + u2
where v = —cos 0,
a
=
~.
(A.6)
This integral can be expressed in terms of the Legendre functions P,~’ (cf. ref. [13], p. 160, eq. (33)) P~(L’)= F(1
2~F(1_2~)(v2_1)~2 ~)F(—~ v)F(v + 1) —
—
—~
Re(~+v)<0,
f du(1
+
2uv
-1/2
+ u2)~
~
Re(~—v)<1.
In our case ~s = 0 and = ia, and the conditions are satisfied. If we rewrite the integral (A.5) in the following form i-’
2f[v]
=
—
—
(_1)1~l/2fu’~1~2~(1 2u’v
+ u~2)~/2du’
± fu~o
du,
—
where we have replaced u f[v]
~-
=
—*
1/2)(1
+ 2uv + u2)
t/2
(A.7)
u’ in the first integral, it is easy to show that
CPt/ 2±~~(~) iDPi/2+~~( —v). —
(A.8)
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Scattering of massless particles
where
exp(_~/fl
‘IT
— ,
C= +
D=
,
exp(~/~)
1
+
exp(2~T)
(A.9)
and we have used the property 1/2-i~/~7~( v)
~
=
P
(A.10)
1/2+i~/~7~( LI),
Appendix B THE EVALUATION OF THE SCATTERING AMPLITUDE
We shall now evaluate explicitely the scattering amplitude given in eq. (56): A =N{f dO e~°~ +iksl],
(B.1)
where I=
f dO e’~~[CPl/2+~~( —cos 0)
—
1DP_I/2+~~( +cos
0)1,
(B.2)
and p = We shall assume that (B.3) therefore the following results will be true for large momentum transfer. Under this condition the first integral in the scattering amplitude is negligible
f dO e~°~ =
—
~
(e’~~ 1)
0.
—
(B.4)
The Legendre functions P1/2+~~(+cos 0) can be expressed as a power series (cf. ref. [13], p. 174, eq. (5)) P1/2+1~(cos0)
=
1
+ ~ n=i
[n
0
[4~2 +
(2j
+
1)21
2
[sin~o]2nJ,
[4~2+
(21+
1)2]
2
[cos~o]2tlj;
J~~
Pt/ 2±~P(—cos0)
=
1
+
~ [n_i n=1 )°
(B.5)
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Scattering of massless particles
621
consequently I can be written as
=
n~
[‘h’ [~(21 +
+ 1)2]
~ (2n)! [Df~do
+iCf dO eb0~[cos~O]2n]=
e°~[sin~0]
I
~{n_1
(2n)!~~C121. (B.7)
I~and 12 can be evaluated explicitely in terms of the beta functions F(x)F(y) (B.8)
B(x,y)= F(x+y) and of the hypergeometric functions F. For example, I~is
=
22n+t
{exp(i~[~
—
~])B[~
XF[—2n, ~[~7 —n, +exp(i~[n
+
—n, i]
—n
~])B[~
+
—n, 2n
1, —i] +
1]F[0, ~
—n,
f~7+n + 1,
_i]} (B.9)
In the limit of large momentum transfer I~reduces to Ii=~{1+i(_1y(~}.
Similarly for
‘2
we
(B.10)
find ((2n)! 12
2
Substituting the expressions of I~and finally find
~ The series clearly converges.
22~~ +
‘2
~J.
(B.11)
in eq. (B.7) and then in eq. (B.1), we
(B.12)
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Scattering of massless particles
References [1] P.C. Aichelburg and R.U. SexI, Gen. Rel. Gray. 2 (1971) 303 [21 R. Penrose, Rev. Mod. Phys. 37 (1965) 215 [3] T. Dray and G. ‘t Hooft, NucI. Phys. B253 (1985) 173 [41G. ‘t Hooft, Phys. Lett. B198 (1987) 61 [5] G. Veneziano, Nuovo Cimento A57 (1968) 190 [6] V. Ferrari and J. Ibañez, Gen. Rel. Gray. 19 (1987) 405 [7] V. Ferrari, J. Ibañez and M. Bruni, Phys. Rev. D36 (1987) 1053 [8] V. Ferrari and J. Ibañez, Proc. R. Soc. London A417 (1988) 417 L91 S. Chandrasekhar and B.C. Xanthopoulos, Proc. R. Soc. London A408 (1986) 175 [10] H.W. Brinkman, Proc. Nat. Acad. Sci. (USA) 9 (1923) 1 [11] W.B. Bonnor, Commun. Math. Phys. 13 (1969) 163 [121 K.A. Khan and R. Penrose, Nature 229 (1971) 185 [131 H. Bateman, High transcedental functions, Vol. 1 (McGraw-Hill, New York, 1953) p. 193