Gravitational radiation in massless-particle collisions
The angular and frequency characteristics of the gravitational radiation emitted in collisions of massless particles is studied perturbatively in the context of classical General Relativity for small values of the ratio of the Schwarzschild radius over the impact parameter. The particles are described with their trajectories, while the contribution of the leading nonlinear terms of the gravitational action is also taken into account. The old quantum results are reproduced in the zero frequency limit . The radiation efficiency outside a narrow cone of angle in the forward and backward directions with respect to the initial particle trajectories is given by and is dominated by radiation with characteristic frequency .
- 1 Introduction
- 2 Notation – Equations of motion
- 3 Radiation amplitude
- 4 Characteristics of the emitted radiation
- 5 Conclusions – Discussion
- A Conventions
- B Computation of momentum integrals
- C Asymptotic expansion of higher-frequency integrals
The problem of gravitational radiation in particle collisions has a long history and has been studied in a variety of approaches and approximations. The interested reader may find a long and rather comprehensive list of relevant references in GST (), where the emitted gravitational energy, as well as its angular and frequency distributions in ultra-relativistic massive-particle collisions were computed. The condition imposed in GST () that the radiation field should be much smaller than the zeroth order flat metric restricted the region of validity of our approach to impact parameters much greater than the inverse mass of the colliding particles, and made our conclusions not applicable to the massless case.
However, the problem of gravitational radiation in massless-particle collisions is worth studying in its own right and has attracted the interest of many authors in the past as well as very recently. Apart from its obvious relevance in the context of TeV-scale gravity models with large extra dimensions GKST (), it is very important in relation to the structure of string theory and the issue of black-hole formation in ultra-planckian collisions ACV (), Dvali (). Nevertheless, to the best of our knowledge, complete understanding of all facets of the problem is still lacking. The emission of radiation in the form of soft gravitons was computed in Weinberg () in the context of quantum field theory, but in that computation the contribution of the non-linear graviton self-couplings i.e. the stress part of the energy-momentum tensor, was argued to be negligible. The result of the quantum computation for low-frequency graviton emission was reproduced by a purely classical computation in Wbook (), due entirely to the colliding particles and leaving out the contribution of the stress part of the energy-momentum tensor.
In the pioneering work DEath () or its recent generalization to arbitrary dimensions Herdeiro (), the special case of collisions with vanishing impact parameter was studied, with emphasis on the contribution to the radiation of the stress part of the energy momentum tensor, leaving out the part related to the colliding particles themselves. In a more recent attempt Taliotis () the metric was computed to second order, but no computation of the radiation characteristics was presented, apart from an estimate of the emitted energy based essentially on dimensional analysis. More recently, a new approach was put forward for the computation of the characteristics of the emitted radiation GV (), based on the Fraunhofer approximation of radiation theory. However, this method cannot be trusted at very low frequencies and, furthermore, it ignores the non-linear terms of the gravitational action, which are expected to be important in the high frequency regime. Thus, we believe it is fair to conclude, that the issue of the frequency and angular characteristics as well as the efficiency of gravitational radiation in ultra-relativistic particle collisions is not completely settled yet.
The purpose of the present paper is to extend the method used in GST () to the study of gravitational radiation in collisions of massless particles with center-of-mass energy and impact parameter . The formal limit (or equivalently for the Lorentz factor) of the massive case leads to nonsensical answers for the radiation efficiency, i.e. the ratio of the radiated to the available energy, the characteristic radiation frequency , or the characteristic emission angle . The whole set-up of the computation in the massive case is special to that case and, consequently, does not allow to extract safe conclusions related to massless-particle collisions. In particular, the massive case computation was performed in the lab frame, the choice of polarization tensors was special to the lab frame, while, being interested in ultra-relativistic collisions, we organized the computation of the energy-momentum source in a power series of the Lorentz factor . Here, we shall deal directly with massless collisions in the center-of-mass frame and correct the above inadequacies of our previous results. We shall study classically the gravitational radiation in the collision of massless particles using the same perturbative approach as in GST (). The scattered particles will be described by their classical trajectories, eliminating potential ambiguities in the separation of the radiation field from the field of the colliding particles, inherent in other approaches. Furthermore, at the level of our approximation we shall take into account the contribution of the cubic terms of the gravitational action to the radiation source, which will be shown to be essential for the consistency of our approach. Finally, the efficiency outside a narrow cone in the forward and backward directions will be obtained as a function of the only available dimensionless quantity , formed out of the four parameters , relevant to the problem at hand.
The rest of this paper is organized as follows: In Section 2 we describe the model, our notation, the equations of motion and the perturbative scheme in our approach. This is followed by the computation in Section 3 of the total radiation amplitude, i.e. the sum of the local and stress part. Section 4 focuses on the study of the angular and frequency characteristics of the emitted radiation in the most important regimes of the emission-anglefrequency plane. . Furthermore, in a separate subsection we compare the results of this paper to previous work and verify that they are compatible in their common regime of validity. Our conclusions are summarized in Section 5, while in three Appendices the interested reader may find the details of several steps of the computations and the proofs of basic formulae, used in the main text.
2 Notation – Equations of motion
The action describing the two massless particles and their gravitational interaction reads
where is the einbein of the trajectory in terms of the corresponding affine parameter , and the summation is over the two particles. We will be using unprimed and primed symbols to denote quantities related to the two particles.
Variation of the einbeine gives for each particle the constraint
Using the reparametrization invariance we can choose constant. Furthermore, we can use the remaining freedom of rescalings to set on the particle trajectories. Finally, we can shift the affine parameters to set at the positions of closest approach of the two particles. Before the collision the particle positions are at negative and . They “collide” when they are at .
For identical colliding particles in the center-of-mass frame we can choose and, consequently, . With the gauge choice , the two einbeine are finally determined by the condition
with the energy of each colliding particle.
Thus, the particles move on null geodesics, while variation of leads to the particle equation of motion:
and similarly for . At zeroth order in the gravitational interaction, the space-time is flat and the particles move on straight lines with constant velocities, i.e.111The upper left index on a symbol labels its order in our perturbation scheme.
The particle energy-momentum is defined by , i.e. for each particle
At zeroth order, in particular, it is given by
and is the source of the first correction of the gravitational field. Given that is traceless, the perturbation satisfies for each particle separately the equation
whose solution in Fourier space is
where , , while and . Since the particle momenta satisfy , the consistency conditions are also satisfied to this order.
In coordinate representation they are
where is the dimensional Fourier transform of :
with and is the position and impact vector, respectively, in the transverse plane and an arbitrary constant with dimensions of length.
Write for the metric and substitute in (2.5) to obtain for the first correction of the trajectory of the unprimed particle the equation
The interaction with the self-field of the particle has been omitted and due to the primed particle is evaluated at the location of the unprimed particle on its unperturbed trajectory.
Integrating it over , the first-order correction to velocity is given by
The integration constants are chosen and in order to satisfy the initial conditions .
Thus, the components of are
Making use of the formulae GS ()
satisfied by the distributions and their Fourier transform, respectively, we can express collectively in the following useful form
which vanish for all . Indeed, the massless particle trajectories should remain undisturbed before the collision.
Finally, we integrate (2.16) and fix the integration constants so that is regular and satisfies . We end up with
or, equivalently, in components
From these it is straightforward to reproduce the leading order expressions of the two well-known facts about the geodesics in an Aichelburg-Sexl metric, namely
The time delay at the moment of shock equal
The refraction caused by the gravitational interaction by an angle
in the direction of the center of gravity.
Clearly, similar expressions to the above are obtained for the primed particle trajectory. For the perturbation , in particular, we have
To summarize: We have obtained the first order corrections of the gravitational field, sourced by the straight zeroth-order trajectories of two colliding massless particles. It is identical with the leading term of the Aichelburg-Sexl metric describing the free particles and it can be shown to coincide with the limit of the corresponding field due to massive particles. The perturbations and of the trajectories of the colliding particles in the center-of-mass frame and with impact parameter were also computed. Finally, the known expressions Dray87 () for the time delay and the leading order in scattering angle were reproduced.
As will be shown in the next section, the arbitrary scale in the expressions for and disappears, as it ought to, from physical quantities such as the gravitational wave amplitude or the frequency and angular distributions of the emitted energy.
3 Radiation amplitude
We proceed with the computation of the energy-momentum source of the gravitational radiation field. The gravitational wave source has two parts. One is the particle energy-momentum contribution, localized on the accelerated particle trajectories given in the previous section. The other is due to the non-linear self-interactions of the gravitational field spread over space-time. One should keep in mind that we are eventually interested in the computation of the emitted energy, given by (4.1). It involves projection of the energy-momentum source on the polarization tensors and imposing the mass shell condition on the emitted radiation wave-vector. Thus, whenever convenient, we shall simplify the expressions for the Fourier transform of the energy-momentum source by imposing the on-shell condition , as well as by projecting it on the two polarizations.
3.1 Local source
We start with the direct particle contribution to the source of radiation. We call it “local”, because, as mentioned above, it is localized on the particle trajectories. The first order term in the expansion of (2.6) is
where is evaluated at and is evaluated at . Its Fourier transform is
Similarly for the primed particle with replaced by .
Introducing the momentum integrals
the first-order correction to the source becomes222Terms, coming from the integration constants in (2.16) and (2.17), contain and lead to the extra terms proportional to and . With the on-shell condition the latter is equivalent to and these terms do not contribute in the subsequent integration.
Note that the integrals and contain one massless Green’s function. This is in accordance with the fact that , expressed through them, is the source of radiation from the colliding particles. and are computed in Appendix B. They are
and upon substitution into (3.3) lead to
with . Similarly
for the contribution of the primed particle, obtained from by the substitution .
Eventually, and will be contracted with the polarization vectors and , we will construct in the next section. They have zero time component and, therefore, satisfy and . Thus, one may effectively replace in the energy momentum tensor by when they are not contracted, to obtain
3.2 Non-local stress source
The contribution to the source at second-order coming from the expansion of the Einstein tensor reads GST ()
It contains products of two first-order fields. Thus, it is not localized, hence its name “non-local”. It is also called “stress”, being part of the stress tensor of the gravitational field.
Upon substitution of and of the previous section in the above expression we obtain for the Fourier transform of
in terms of the integrals
(). We use the definition , while we have omitted the terms proportional to as well as the longitudinal ones proportional to or in anticipation of the fact that they will eventually vanish, when contracted with the radiation polarization tensors. Finally, as in the case of one can effectively substitute to obtain:
Note that contain the product of two graviton Green’s functions, which signals the fact that is due to radiation from “internal graviton lines” in a Feynman graph language, through the cubic graviton interaction terms. It will be explicitly demonstrated below that in the zero frequency limit the contribution of in the emitted radiation is negligible, as argued in Weinberg (). Nevertheless, it will become clear that it contributes significantly at high frequencies and, as will be shown next, it plays an important role in the cancellation of the dependence in physical quantities.
3.3 Cancellation of the arbitrary scale
As anticipated, in this subsection we will demonstrate explicitly that the arbitrary scale disappears from the final expression of the total contribution to the source of the gravitational radiation. As will become clear below, the local and stress parts of the source each depends on , but their sum is independent and finite. According to their expressions in (3.7) and (3.8), and depend on through , while depends on through terms proportional to (with no extra factors ) in the expressions of and , evaluated in Appendix B. All these unphysical terms will be shown to cancel out and will end up with expressions (3.24) and (3.25) for the total energy-momentum source for the two polarizations separately333The reader, who is not interested in the details, may go directly to these formulae for the total source..
We proceed in steps:
1. Split with444The integrals and are singled-out, because they can be computed exactly. See Appendix B.
3. The remaining stress contribution is a linear combination of , and , which have been computed in Appendix B. Taking, as above, into account the fact that they will eventually be contracted with the polarization vectors and that we shall set in the integral for the radiation energy and momentum we are interested in, they are555Note that we use non-standard symbols for the modified Bessel functions, namely . In this notation the differentiation rule reads for any , while the zero-argument limit is for . :
Having anticipated that the dangerous terms for divergence and dependence are the ones which contain the integral of with , since according to Appendix B lead to 666The integral containing the hatted Macdonald of index , which near behaves as , diverges logarithmically at both ends of the integration region. In Appendix B it is shown that this logarithmic behavior is related to the one of (Eqns. (B.6, B.8)). Alternatively, one could regularize these divergent integrals by shifting the index of all Macdonald functions by , which makes all integrations convergent, and take the limit in the very end of the computation. it is natural to treat separately the terms in which contain , from the ones which contain or . Thus, in a suggestive notation, we split: with
Substituting the explicit form of and simplifying, we obtain
4. Consider, next . Using the formulae derived in Appendix B, i.e.
takes the form
5. Thus, the sum