Lorentz-covariant perturbation theory for relativistic gravitational bremsstrahlung
We formulate Lorentz-covariant classical perturbation theory to deal with relativistic bremsstrahlung under gravitational scattering111This is English translation of the Moscow State University preprint issued in Russian in 1980 GGM0 (). The authors are grateful to Pavel Spirin for providing an electronic version for Fig. 1. This submission was supported by the RFBR grant 08-02-01398-a.. Our approach is a version of the fast motion approximation scheme, the main novelty being the use of the momentum space representation. Using it we calculate in a closed form the spectrum of scalar, electromagnetic and gravitational radiation. Our results for the total emitted energy agree with those by Thorne and Kovacs. We also explain why the method of equivalent gravitons fails to produce the correct result for the spectral-angular distribution of emitted radiation under gravitational scattering, contrary to the case of Weizsäcker-Williams approximation in quantum electrodynamics.
pacs:04.20.Jb, 04.65.+e, 98.80.-k
Gravitational radiation by non-relativistic and quasi-relativistic systems is low-multipole and can be easily calculated using the quadrupole formula of General Relativity with higher multipole corrections. With increasing velocities, the contribution from higher multipoles becomes dominant, so one needs another technique. The most adequate approach is the method of post-linear expansions which was discussed in early 60-ies most notably by Bertotti B (), Bertotti and Plebanski BP (), Havas and Goldberg HG (); Goldberg () (see also H (); R (); K (); SH (); In (); An (); Ros (); CS (); WeGo ()). We have developed a momentum space version of this approach GG1 () which is applied here to gravitational bremsstrahlung. Although technically different, our calculations essentially overlap and agree with those by Thorne and collaborators CT (); KT3 (); KT4 (). The results also agree with an alternative calculations by Peters Pe1 (); Pe2 (); Pe3 () based on the linear perturbation theory on Schwarzschild background. They disagree, however, with calculations based on “equivalent gravitons” method MaNu (), and we explain the origin of this disagreement.
Other approaches to relativistic bremsstrahlung problem are worth to be mentioned. One, suggested by D’Eath DE (), is based on replacing the boosted Schwarzschild metric by the impulsive gravitational wave. Another, due to Smarr Sm1 (); Sm2 (), appeals to calculation of the radiation amplitude in the low-frequency region. This overlaps with quantum calculation of the cross-section in the Born approximation Barker ().
To calculate the leading order gravitational radiation in relativistic collisions of particles interacting predominantly through the non-gravitational forces it is enough to use the linearized gravity on Minkowski background. In the case of gravitational interaction we need at least the next post-linear order. If one interprets the second order gravitational potentials in terms of Minkowski space coordinates, one finds that the source of gravitational radiation becomes non-local due to contribution of the first order gravitational stresses (similarly for non-gravitational radiation from particles interacting by gravity). This non-locality leads to destructive interference of high frequency part of radiation, so the spectrum will be different from that of the electromagnetic bremsstrahlung.
Ii Field equations in quasilinear form
Consider a system of point particles, interacting by non-gravitational fields (scalar or massless vector ) and moving in a self-consistent gravitational field described by the metric . The action can be presented as where is the sum of particle actions including non-gravitational interaction terms
and are scalar and Maxwell field actions
and the gravitational lagrangian is taken in the two-gamma form:
Assuming gravitational field to be negligible at spatial infinity, we choose asymptotically Minkowskian metric in this region and introduce the (non-tensor) metric deviation variable
Then for one has , but are not necessarily small everywhere. By convention, the indices of the quantities and will be raised and lowered by the Minkowski metric , while the indices of the true tensors are operated with the metric .
Introducing an antisymmetric tensor density
one can present Einstein equations in a divergence form
where is the total matter stress-tensor, and is Einstein’s canonical pseudotensor
Maxwell equations can be written in a similar form
where the vector-current is
Finally, the scalar wave equation reads
with the scalar current
Particle equations of motion generically read
All the above equations are exact and can be regarded as a system defining the particle motion and evolution of the scalar, vector and gravity fields and in a self-consistent way. However, since the notion of delta-functions is not defined in the full non-linear general relativity, we can deal with point particles only perturbatively, expanding all the quantities in formal series in the gravitational coupling . For this one has to pass first to quasilinear form of the field equations. For Einstein equations we single out the linear part of the -tensor:
where joins terms linear in :
while denotes non-linear in terms. Gravitational equations can be now presented as
where in all the non-linear terms are collected. To calculate gravitational radiation one needs only terms quadratic in .
Maxwell equations can be rewritten similarly:
where an effective “gravitational” vector current is given by
In the scalar case we obtain similarly:
This looks as the flat space wave equation for the spin zero field, with an important difference, however, that the “source” depends explicitly on .
Now we can further simplify the quasilinear equations (which are still exact in all orders in ) imposing the gauge conditions
which are consistent with the field equations by virtue of the identities:
In this gauge Einstein and Maxwell equations read
Iii Scalar bremsstrahlung under gravitational scattering
Consider two point masses and , one of which () carries the scalar charge . Particles interact via gravity and the systems emits both gravitational and scalar waves. In this section we calculate scalar radiation. The action reads
where , dot denotes differentiation with respect to the interval , and the metric signature is mostly minus.
The full system of equations describing the collision consists of Einstein equations, scalar field equation and particle equations as given in the previous section. The total loss of the four-momentum during the collision can be presented as
is the energy-momentum tensor of the massless field , and integration is performed over the sphere of infinite radius. The non-zero contribution comes from the terms in , which fall off at infinity as . Also, without changing the integral, one can add to the integrand the total derivative over time . Then, applying the Gauss theorem we can transform the Eq. (23) to the following form:
To exclude an infinite self-energy part it is enough to substitute as the right hand side of the Eq. (18), while as – the -odd part of the retarded potential
where is the sign function and the Fourier transforms are defined as
and similarly for . We obtain:
This expression is analogous to the usual one in electrodynamics, differing from it by presence of the non-local current which we will call the stress current.
To find and we will solve the Einstein equation, the particle equations and the scalar field equation (18) expanding and in powers of the gravitational constant . The actual expansion parameter in the ultrarelativistic collision problem will be the ratio of the gravitational radius of one of the particle to the impact parameter. One can show that approximation is valid if the particle scattering angle is small with respect to the radiation angle We ()
where – the relative velocity of the colliding particles, – the impact parameter.
We parameterize the particles world lines as
with – is the four-vector which in the rest frame of the second particle takes the form . Here and below we use brackets to denote scalar products with respect to Minkowski metric. We choose the initial conditions
so that is the four-momentum of the particle before the collision, and is the correction due to the gravitational interaction.
In the lowest order in gravitational interaction the correction to the space-time metric due to the second particle reads
Substituting this into the equation of motion of the particle we find
To calculate the Fourier-transform of we use the expression for in the lowest order (zero order in )
The Fourier-transforms and can be computed as follows. Using the integrals
where are the Macdonald functions. Lorentz-invariant integrals (III) an be conveniently computed in the rest frame of the second particle . Then two integrations from four are performed using the delta-functions, while the remaining two-dimensional integral in the plane orthogonal to is computed using the polar coordinates. The angular integral is standard, and the last one is done by contour integration.
Integration over in the expression for can be done using the Feynman parameterization. We obtain
Consider the case of non-relativistic velocities. For , then the integral in (37) is easily done and we obtain
where . From (38) and (39) it is clear that , so in the lowest in approximation radiation is entirely determined by the local current (38). Substituting it into the Eq. (28), after some simple transformations we find
One can see that for small velocities the characteristic radiation frequency is inverse to the effective time of collision .
The total energy loss during the collision can be obtained integrating (40) over angles and the frequency:
In the ultrarelativistic case the effective spread of the stress current is of the order of . So it can be expected that for the wavelengthes the source with act as point-like. Indeed, for the argument of the Macdonald functions (III) and (37) is small for all values of parameters, and with account for the leading terms we obtain
In the frequency region the contributions of the local and the non-local currents are of the same order. In this case for the spectral distribution of the total emitted energy we obtain:
For relatively high frequencies the integral in (44) is formed at . So approximately
The expressions (43), (45) and (46) together describe the behavior of the spectral curve. It follows, in particular, that in the spectral distribution there is a maximum around the frequency . The total energy loss during the collision is obtained integrating (44) over the frequency:
where is the Catalan constant.
Let us compare these results with the case of the electromagnetic interaction in Minkowski space, when the source term in the equation for does not contain the stress current . Suppose that the colliding particles are electrically charged and neglect gravitational interaction with respect to electromagnetic one. Then as the source in (28) one has to use the Fourier-transform of the trace of the particles energy-momentum tensor. After similar calculations we obtain:
In the ultrarelativistic case the spectral-angular distribution is dominated by the second term in (48). In the rest frame of the second particle we find in the leading order in :
where . Using the asymptotic expansions for the Macdonald functions for small and large arguments, from (49) we find for :
while for high frequencies
Note, that for the local source case our methods gives the energy loss without restrictions on the relative velocity of collision. Indeed, substituting (48) into (28) and integrating over frequencies and angles we obtain
Thus we see that there is substantial difference between the spectrum of the bremsstrahlung from gravitational scattering and that in the case of electromagnetic interaction. In the first case there is a maximum at , while the spectral distribution (49) is monotonous function of the frequency. For gravitational interaction the exponential cut off corresponds to the frequency , and not to as in the case of the electromagnetic scattering. Finally, the total energy loss at gravitational scattering (47) is times less that the corresponding quantity in the electromagnetic case (52) for the same scattering angle, i.e. under the condition .
These properties can be qualitatively explained by the presence of the non-local (in terms of the flat space-time picture) stress-current source in the equation for the radiated field in the case of gravitational scattering. This current has and effective transverse dimension of the order of and longitudinal of the order of ( times smaller due to the Lorentz contraction). For large wavelengthes () the source non-locality is insignificant and the low frequency limit is the same as for the electromagnetic interaction case, when there is no non-local term at all. For radiation from the most distant elements of the source exhibit a destructive interference for the angles close to , which leads to the gap in the spectrum. Finally, for the conditions for destructive interference are fulfilled for the forward direction, in which the most of the energy is emitted. This leads to substantial decrease of the radiation.
Iv Electromagnetic bremsstrahlung under gravitational scattering
The case of the electromagnetic interaction is rather similar. Let the particle carry the electric charge . Using analysis of the Sec.2 we can present Maxwell equations as follows:
where the stress-current is
Imposing the flat space Lorentz gauge on the four-potential :
we cast Maxwell equations into the form convenient for iterative solution:
It is convenient to choose two linearly independent polarization vectors as
They satisfy the following conditions:
and in the rest frame of the second particle read: where and are unit vectors along and .
The expression for the four-momentum loss due to electromagnetic interaction with polarization can be derived analogously to the Eq. (28) and reads:
where . As in the scalar case, one has to retain in only terms falling off asymptotically as . In this approximation
For small relative velocity radiation is generated predominantly by the local current , since . In this case
Integrating over frequencies and angles we get
For ultrarelativistic collisions in the low frequency range contribution of the non-local stress current is relatively small, , and we have:
For in the leading in approximation the spectral distribution of the radiated energy is given by
In (68) we performed summation over polarizations.
Using (68) one can show that for the spectral distribution behaves as follows:
while for the frequencies
For the total energy loss we obtain:
In the case of both particles electrically charged with large charge to mass ratio in geometric units one can neglect gravitational interaction and the bremsstrahlung problem is simplified considerably. Then the stress-current , and the radiation amplitude is fully given by the local current. Consider for simplicity the case . Then the Fourier-transform of the current is given by