Null multipole particles as sources of pp-waves

J. B. Griffiths^{∗}

Department of Mathematical Sciences

Loughborough University Loughborough, Leics. LE11 3TU, U.K.

and J. Podolský^{†}

Department of Theoretical Physics,

Faculty of Mathematics and Physics, Charles University, V Holešovičkách 2, 18000 Prague 8, Czech Republic.

October 10, 1997

Abstract

We describe a class of impulsive gravitational pp-waves generated by null particles, each of which have an arbitrary multipole structure. The nature of each point source is described for all possible modes.

PACS class 04.20.Jb, 04.30.Nk

Keywords: Impulsive gravitational waves, pp-waves.

∗E–mail: J.B.Griffiths@Lboro.ac.uk †E–mail: Podolsky@mbox.troja.mff.cuni.cz

The widely known class of pp-waves [1] (plane-fronted gravitational waves with parallel rays) is defined by the property that the space-times admit a covariantly constant null vector field. The metric can be written in the form

ds^{2} = 2dudr + 2H(u, ζ, ζ̄)du^{2} − dζdζ̄  (1)

where the complex coordinate ζ spans the plane wave surfaces. The field equations for aligned null radiation reduce to the two-dimensional Poisson equation

ΔH = 4H_{ζζ̄} = 8πT_{uu}.  (2)

In the vacuum case, the general solution can be written in the form H = f(u, ζ) + f̄(u, ζ̄) where f is an arbitrary function of u and ζ, holomorphic in ζ.

Of particular interest is the case given by Aichelburg and Sexl [2] in which f = μ logζ δ(u) where μ is a real constant. Introducing polar coordinates ζ = ρe^{iϕ}, we see that, in this case,

H = 2μ logρ δ(u).  (3)

Substituting this into (2) indicates that

T_{uu} = \frac{1}{2} μ δ(ρ) δ(u)

where δ(ρ) represents the two-dimensional delta function. Thus, it can be seen that this solution describes an impulsive gravitational wave generated by a single null particle located at the origin of polar coordinates in the wave surface. It was originally obtained by boosting a Schwarzschild black hole to the speed of light while its mass is reduced to zero in an appropriate way [2]. Some more complicated solutions have also been obtained [3]–[6] by boosting other members of the Kerr–Newman family of solutions in given directions.

We consider here a more general class of impulsive waves for which H = H(ρ, ϕ)δ(u), or f = f(ζ)δ(u), and T_{uu} = J(ρ, ϕ)δ(u). First we rewrite equation (2) in the form

\frac{1}{ρ} \frac{∂}{∂ρ} (ρ\frac{∂H}{∂ρ} ) + \frac{1}{ρ^{2}} \frac{∂^{2}H}{∂ϕ^{2}} = 8πJ(ρ, ϕ).  (4)

We then consider solutions that can be separated in the form H_{m} = h(ρ)e^{−imϕ} where m is a non-negative integer. For situations which are vacuum everywhere, except possibly at the origin, (4) becomes h′′ + \frac{1}{ρ}h′ − \frac{m^{2}}{ρ^{2}}h = 0 for which solutions are given by

h = \{ \begin{matrix}a_{0} − b_{0} logρ  for m = 0 \\ a_{m}ρ^{m} + b_{m}ρ^{−m}  for m ≥ 1 \end{matrix}

where a_{m} and b_{m} are constants. The components a_{0} and a_{1}ρ can be removed by a simple transformation, while the a_{2}ρ^{2} component describes a plane gravitational wave of arbitrary Figure 1: These pictures illustrate the monopole, dipole and quadrupole modes showing the dependence of the functions H_{0}, H_{1} and H_{2} near the singular point representing the source of the impulsive waves.

polarization. The component −b_{0} logρ is the solution (3) above with −b_{0} = 2μ which describes the impulsive gravitational wave generated by a single monopole particle. The components a_{m}ρ^{m} for m > 2 are unbounded at infinity and will not be considered further here.

In this letter, we concentrate on the remaining asymptotically flat cases in which h = b_{m}ρ^{−m} for m ≥ 1. These correspond to f(ζ) = \frac{1}{2}β_{m}ζ^{−m} where β_{m} is a complex constant. Putting β_{m} = b_{m}e^{imϕ_{m}}, the arbitrary real constants b_{m} and ϕ_{m} describe the amplitude and phase of each component. These can then be written in the form

H_{m} = b_{m}ρ^{−m} cos[m(ϕ − ϕ_{m})].

It can be seen that each of these components describes a multipole solution of order m, with the monopole solution H_{0} given by (3). The multipole character of the first three of these modes is clearly illustrated in Fig. 1. These terms each have the appropriate ϕ dependence. Further, by substituting into (4), it can be shown that the source of the m^{th} mode is proportional to the m^{th} derivative of the delta function δ(ρ) with respect to ρ.

In order to prove this statement, let us consider a sequence h_{m}(ρ) satisfying the recurrence relation h_{m+1} = −c_{m}h′_{m} where c_{0} = 1 and c_{m} = \frac{1}{m} for m ≥ 1. We then define another sequence J_{m}(ρ) by the relation

h_{m}′′ + \frac{1}{ρ}h′_{m} − \frac{m^{2}}{ρ^{2}}h_{m} = 8πJ_{m}.  (5)

By differentiating (5) we obtain the recurrence relation for J_{m}:

J_{m+1} = −c_{m}J′_{m} − \frac{m}{4πρ^{2}} (h_{m+1} − \frac{1}{ρ}h_{m}) .

Thus J_{1} = −J′_{0}, and J_{m+1} = −\frac{1}{m}J′_{m} for m ≥ 1 provided h_{m+1} = \frac{1}{ρ}h_{m}. The required conditions are satisfied for the sequence of solutions h_{m} = ρ^{−m}, with h_{0} = − logρ, introduced above. Then, since we know that J_{0} = −\frac{1}{4}δ(ρ), we obtain that J_{m} = −\frac{1}{4} \frac{(−1)^{m}}{(m−1)!}δ^{(m)}(ρ), and this may be considered to represent the source of the m^{th} mode.

The general solution for a single particle with an arbitrary multipole structure can thus be written as a sum of all the above components:

H = (−b_{0} logρ + ∑_{m=1}^{∞} b_{m}ρ^{−m} cos[m(ϕ − ϕ_{m})])δ(u)

in which the source in the impulsive wavefront u = 0 is given by

J = −\frac{b_{0}}{4}δ(ρ) − ∑_{m=1}^{∞} \frac{b_{m}}{4} \frac{(−1)^{m}}{(m − 1)!}δ^{(m)}(ρ) cos[m(ϕ − ϕ_{m})].

Finally, we may observe that since (2) is linear, it follows that solutions can be constructed which contain an arbitrary number of such arbitrary multipole particles distributed arbitrarily over the wave surface. In addition, solutions are not restricted to one impulsive wave in the same space-time, and so further solutions with exotic sources may be constructed.

Acknowledgments

JP was supported by a visiting fellowship from the Royal Society and, in part, by the grant GACR-202/96/0206 of the Czech Republic and the grant GAUK-230/96 of the Charles University.

References

[1] Kramer, D., Stephani, H., MacCallum, M. A. H., and Herlt, E. (1980) Exact solutions of Einstein’s field equations, Cambridge University Press, §21.5.

[2] Aichelburg, P. C.. and Sexl, R. U. (1971). Gen. Rel. Grav., 2, 303–312.

[3] Ferrari, V. and Pendenza, P. (1990). Gen. Rel. Grav., 22, 1105–1117.

[4] Loustó, C. O. and Sánchez, N. (1992). Nucl. Phys. B, 383, 377–394.

[5] Balasin, H. and Nachbagauer, H. (1995). Class. Quantum Grav., 12, 707–713.

[6] Balasin, H. and Nachbagauer, H. (1996). Class. Quantum Grav., 13, 731–737.

Figure caption

These pictures illustrate the monopole, dipole and quadrupole modes showing the dependence of the functions H_{0}, H_{1} and H_{2} near the singular point representing the source of the impulsive waves.