Colliding Plane Fronted Waves and a GravitoElectromagnetic Searchlight Peter A. Hogan1and Dirk Puetzfeld1 2y 1School of Physics University College Dublin Beleld Dublin 4 Ireland

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Colliding Plane Fronted Waves and a Gravito–Electromagnetic Searchlight

Peter A. Hogan1, ∗and Dirk Puetzfeld1, 2, †

1School of Physics, University College Dublin, Belﬁeld, Dublin 4, Ireland

2University of Bremen, Center of Applied Space Technology and Microgravity (ZARM), 28359 Bremen, Germany

(Dated: January 2, 2023)

We present a formulation of Einstein–Maxwell vacuum ﬁelds due to plane fronted electromagnetic

waves sharing their wave fronts with gravitational waves. This is based on a recent geometrical

reconstruction of plane fronted wave ﬁelds by the authors which clearly identiﬁes the cases in which

the wave fronts collide or do not collide. In the former case our construction suggests an explicit

example of a searchlight beam, accompanied by gravitational radiation, which sweeps across the

sky. This gravito–electromagnetic searchlight and its properties are described in detail.

PACS numbers: 04.20.-q; 04.20.Jb; 04.20.Cv

Keywords: Classical general relativity; Exact solutions; Fundamental problems and general formalism

I. INTRODUCTION

We describe a formulation of Einstein–Maxwell vac-

uum ﬁelds of plane fronted electromagnetic waves sharing

their wave fronts with gravitational waves. This follows

our recent reconstruction [1,2] of plane fronted waves

which enables a clear distinction between plane waves

with colliding wave fronts (Kundt [3] waves) and waves

with non–colliding wave fronts (pp–waves [4,5]). Our

new formulation suggests a simple example for which the

gravitational and electromagnetic radiation ﬁelds van-

ish if the waves are not colliding. This is an example

of a searchlight beam, accompanied by gravitational ra-

diation, sweeping across the sky. We refer to it as a

gravito–electromagnetic searchlight. Such waves are the

asymptotic limit of spherical waves emitted by an isolated

source [6]. The wave fronts are colliding or not colliding

depending upon the motion of the source. The paper

is organized as follows: in section II we give a detailed

description of our formulation of plane fronted waves in

the context of Einstein–Maxwell theory. This is followed

in section III by the gravito–electromagnetic searchlight

which is an explicit solution of the Einstein–Maxwell

equations which exploits variables which arise naturally

in our geometrical construction of plane fronted waves.

The solution is also described in coordinates closely asso-

ciated with rectangular Cartesians and time as this makes

the solution more surveyable. This leads to further prop-

erties of the solution described in section IV by making

use of tensor ﬁelds on a background Minkowskian space

time. The paper ends with a discussion of our results in

section V.

II. EINSTEIN–MAXWELL VACUUM FIELDS

We use units for which the gravitational constant

G= 1 and the speed of light in a vacuum c= 1 and

∗peter.hogan@ucd.ie

†dirk@puetzfeld.org;http://puetzfeld.org

we choose the sign conventions of Synge [7]. Thus in a

local coordinate system {xi}we obtain from the Riemann

curvature tensor components Rijkm the Ricci tensor com-

ponents Rjk =gim Rijkm. The Ricci scalar R=gjk Rjk,

the Einstein tensor components Gjk =Rjk −1

2gjk Rand

the Einstein–Maxwell vacuum ﬁeld equations

Gjk =−κ Ejk with Ejk =Fjp Fkp−1

4gjk Fmp Fmp ,

(1)

with the Maxwell tensor Fij =−Fji satisfying Maxwell’s

vacuum ﬁeld equations

Fij ;j= 0 with Fij;k+Fki;j+Fjk;i= 0 .(2)

In (1)κ= 8 π,Ejk =Ekj with Ejj= 0 is the electromag-

netic energy–momentum tensor, indices are raised with

gij and lowered with gij and gij is deﬁned by gij gjk =δi

and gij =gji are the components of the metric tensor.

The semicolon denotes covariant diﬀerentiation with re-

spect to the Riemannian connection calculated with the

metric tensor gij . We shall denote by Aithe compo-

nents of a 4–potential from which the components of the

Maxwell tensor can be obtained via

Fij =Aj;i−Ai;j=Aj,i −Ai,j ,(3)

with the comma denoting partial diﬀerentiation with re-

spect to the coordinates xi. With Fij given by (3) the

second equation in (2) is satisﬁed. The ﬁrst equation in

(2) can be written in terms of the 4–potential as ([7], p.

357)

gjk Ai;j;k+Rij Aj= 0 provided Ai;i= 0 .(4)

This relies on the Ricci identities in Synge’s form

Ai;j;k−Ai;k;j=ApRpijk ,(5)

which reveals his convention for the deﬁnition of the Rie-

mann curvature tensor. Making use of this, following

the substitution of (3) into the ﬁrst of Maxwell’s equa-

tions (2), results in (4). As a ﬁnal preliminary we note

arXiv:2210.14968v2 [gr-qc] 29 Dec 2022

the components of the Weyl conformal curvature tensor

Cijkm are given by

Cijkm =Rijkm +1

2gik Rjm +gjm Rik

−gim Rjk −gjk Rim

6R(gim gjk −gik gjm).(6)

We now consider a novel form for the line element

of the space–time model for plane fronted gravitational

waves sharing their wave fronts with electromagnetic

waves which has been constructed in [1,2]:

ds2=gij dxidxj= 2 |dζ−β(u)v du|2+2 q du (dv+H du),

(7)

with

q(ζ, ¯

ζ, u) = β(u)¯

ζ+¯

β(u)ζ+1

2γ(u).(8)

The coordinate ζis complex with complex conjugate ¯

while u, v are two real coordinates. The function β(u)

is an arbitrary complex–valued function of the coordi-

nate uwith complex conjugate denoted by ¯

β(u), γ(u)

is an arbitrary real–valued function of u, and H(ζ, ¯

ζ, u)

is a real–valued function to be determined by the ﬁeld

equations. The hypersurfaces u= constant are null and

generated by the shear–free, expansion–free, geodesic in-

tegral curves of the null vector ﬁeld ∂/∂v. It is demon-

strated explicitly in [1,2] that these null hypersurfaces

intersect, and thus the wave fronts collide, if and only if

β(u)6= 0. Labelling the coordinates xi= (ζ, ¯

ζ, v, u) for

i= 1,2,3,4 we ﬁnd that the only non–vanishing compo-

nents of the Riemann curvature tensor are

R1424 =−q Hζ¯

ζ, R1414 =−q Hζζ , R2424 =−q H¯

ζ¯

ζ,

(9)

with the subscripts here, and in the sequel, denoting

partial derivatives. The Ricci tensor has only one non–

vanishing component

R44 = 2 q Hζ¯

ζ.(10)

Consequently the Ricci scalar vanishes and the non–

vanishing components of the Weyl conformal curvature

tensor are

C1414 =−q Hζζ and C2424 =−q H¯

ζ¯

ζ.(11)

Next we consider a 4–potential on the space–time with

line element (7) given by the 1–form

A=f(ζ, ¯

ζ, u)du , (12)

with fa real–valued function of its arguments. This leads

to a candidate for Maxwell ﬁeld Fij having non–vanishing

components

F14 =fζand F24 =f¯

ζ,(13)

with the subscripts denoting partial derivatives as before.

Now Maxwell’s vacuum ﬁeld equations (2) are satisﬁed

provided fsatisﬁes

fζ¯

ζ= 0 ,(14)

and thus fζis an analytic function of ζwith an arbi-

trary dependence on u. The electromagnetic energy–

momentum tensor in (1) has one non–vanishing compo-

nent

E44 = 2 fζf¯

ζ.(15)

With (10) and (15) the Einstein–Maxwell ﬁeld equations

(1) provide the following single equation

Hζ¯

ζ=−κ q−1fζf¯

ζ.(16)

In order to survey what we have here it is useful to intro-

duce the null tetrad ki, li, mi,¯migiven in the coordinates

xi= (ζ, ¯

ζ, v, u) for i= 1,2,3,4 by

ki=δi

3, mi=δi

2,¯mi=δi

li=q−1β v δi

1+q−1¯

β v δi

2−q−1H δi

3+q−1δi

4,(17)

and the three complex bivectors

Lij = ¯milj−¯mjli,(18)

Mij =mi¯mj−¯mimj+kilj−kjli,(19)

Nij =mikj−mjki.(20)

Each of these bivectors is self–dual in the sense that

∗Lij =i Lij ,∗Mij =i Mij and ∗Nij =i N ij with

i=√−1 and ∗Lij =1

2ηijkm Lkm etc. with ηijkm =

√−g ijkm,g= det(gij ) and ijkm the totally skewsym-

metric Levi–Civita permutation symbol in four dimen-

sions with 1234 = 1. Any self–dual complex bivector

can be expressed as a linear combination of these three

self–dual bivectors. In particular we have from (13)

Fij −i∗Fij = 2 q−1fζNij .(21)

The covariant vector ﬁeld kiand the complex bivector

Nij satisfy the important equations, in the current con-

text,

ki;j=−q−1¯

β mikj−q−1β¯mikj,(22)

given in [1,2], and

Nij;k=−q−1β Mij kk,(23)

so that if β= 0 then kiand Nij are covariantly constant.

Using (23) we conﬁrm that (21) satisﬁes Maxwell’s equa-

tions in the form

(Fij −i∗Fij );j=−2q−1fζ¯

ζki= 0 .(24)

It also follows from (21) that

(Fij −i∗Fij )kj= 0 ,(25)

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CollidingPlaneFrontedWavesandaGravito{ElectromagneticSearchlightPeterA.Hogan1,andDirkPuetzfeld1,2,y1SchoolofPhysics,UniversityCollegeDublin,Beleld,Dublin4,Ireland2UniversityofBremen,CenterofAppliedSpaceTechnologyandMicrogravity(ZARM),28359Bremen,Germany(Dated:January2,2023)WepresentaformulationofE...

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Colliding Plane Fronted Waves and a GravitoElectromagnetic Searchlight Peter A. Hogan1and Dirk Puetzfeld1 2y 1School of Physics University College Dublin Beleld Dublin 4 Ireland

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