Do electromagnetic waves harbour gravitational waves?

Brian Bramson

Abstract

In linearized, Einstein–Maxwell theory on flat spacetime, an oscillating electric dipole is the source of a spin-2 field. Within this approximation to general relativity, it is shown that electromagnetic waves harbour gravitational waves.

Keywords:

1. Introduction

An oscillating electromagnetic dipole generates electromagnetic waves. Does it also radiate gravitationally? After all, the dipole produces a Maxwell field whose energy–momentum tensor, according to general relativity, serves as a source of gravitation via Einstein's field equations. Over 80 years ago, Reissner & Nordstrøm discovered that Maxwell's field contributes gravitationally. They showed that solving the coupled Einstein–Maxwell equations for an idealized, ‘point’ particle yields a gravitational field that depends on the particle's mass and electric charge. (See, for example, Hawking & Ellis (1973) and Misner et al. (1973). The Kerr–Newman solution described by Chinnapared et al. (1965) allows for the additional presence of spin.)

This paper considers an approximation to general relativity and the full Einstein–Maxwell equations. A Maxwell field is taken to be the source of a spin-2 field on a background Minkowski spacetime. The coupled equations are linear in the spin-2 field and quadratic in the Maxwell field. (This defines the meaning of the term ‘linearized Einstein–Maxwell’.) The resulting equations are solved for an idealized, oscillating, electric, point dipole. Not only are electromagnetic waves produced but also spin-2 waves.

The spinor formalism described by Penrose & Rindler (1987) is used throughout this paper, §2 providing a brief outline. The advantage is one of economy. The Maxwell field, with six (real) components in the tensor formalism, is represented by three (complex) components in the spinor formalism while the spin-2 field, with 10 components in the tensor formalism, has five in the spinor formalism. However, the tensors and spinors appearing in this paper are viewed ab initio as geometric objects rather than as sets of components, the abstract index convention of Penrose (1968) being employed.

The Maxwell field envisaged in this paper is that considered previously by Janis & Newman (1965). It is generated by an oscillating, electric dipole whose centre of mass is fixed. Thus the geometric configuration will involve a uniquely defined, timelike geodesic Embedded Image relative to which things are referred. The assumption of no incoming radiation leads to the employment of standard, retarded, null coordinates tied to Embedded Image, together with a null tetrad and associated spinor dyad, as described in §3.

Section 4 reviews the correspondence between geometric objects in Minkowski spacetime and functions on the two-dimensional, unit sphere Embedded Image. Spin-weighted functions are then reviewed together with some properties of eth, a useful differential operator tangent to Embedded Image. Finally, the results of some key sphere integrals are presented.

Maxwell's equations are solved in §5 for an idealized, point, electric dipole. The solution involves the dipole's time derivatives from the zeroth to the second and is non-singular everywhere except on the timelike geodesic Embedded Image. The result agrees with that found by Janis & Newman (1965). In addition, it is noticed that the Maxwell field may be generated from a solution to the scalar wave equation.

The linearized Einstein–Maxwell equations (i.e. linear in the spin-2 field) are presented in §6 and solved in a background Minkowski spacetime using the formalism of Newman & Penrose (1962). The solution for the spin-2 field involves the electric dipole's time derivatives from the zeroth to the fifth, is non-singular everywhere except on Embedded Image and possesses radiative parts. Strictly speaking, no energy is radiated by spin-2 waves in flat spacetime. However, by regarding the spin-2 field as an approximation to a gravitational field, an estimate of the power output in gravitational waves is presented in §7.

Throughout, units are chosen for which the speed of light c=1.

2. Spinor formalism

The following is informal and by no means comprehensive; but it will form a useful reference for what follows, in particular with regard to signs and other conventions. Essentially, Penrose & Rindler (1987) are followed.

Vectors Embedded Image and their duals Embedded Image will be written using lower-case, Roman indices, the scalar product Embedded Image being Lorentz invariant. Spinors Embedded Image and dual spinors Embedded Image will be identified via upper-case, Roman indices, the scalar product Embedded Image being invariant under the covering group SL(2,C) of the (proper, orthochronous) Lorentz group. Spin space has two complex dimensions. Complex conjugate spinors Embedded Image and their duals Embedded Image will be written with primed indices.

A vector Embedded Image may be written in spinor notation Embedded Image; this displays its membership of the tensor product of spin space and its complex conjugate. If Embedded Image is null, it takes the form Embedded Image and if, in addition, Embedded Image is real Embedded Image is the complex conjugate Embedded Image of Embedded Image.

The Minkowski metric Embedded Image and its inverse Embedded Image are chosen to have signatures (+−−−) and spinor equivalentsEmbedded Image(2.1)where the epsilon objects are skew symmetric in their indices and whereEmbedded Image(2.2)Spinor indices are raised and lowered according toEmbedded Image(2.3)and similarly for primed indices. Only symmetric spinors matter because, for example,Embedded Image(2.4)Related identities includeEmbedded Image(2.5)A real Maxwell field Embedded Image (skew-symmetric in its indices) defines a symmetric spinor Embedded Image according toEmbedded Image(2.6)The dual of the Maxwell field amounts toEmbedded Image(2.7)the alternating symbol Embedded Image, skew in all its indices, chosen so that in any proper, orthochronous, Minowski, orthonormal tetrad Embedded Image,Embedded Image(2.8)The complex self-dual and anti self-dual fields Embedded Image take the forms,Embedded Image(2.9)Maxwell's equations in the presence of a source Embedded Image (a 4-current) form a pair,Embedded Image(2.10)where Embedded Image is the standard derivative operator in Minkowski spacetime. EquivalentlyEmbedded Image(2.11)or, in the spinor formalism,Embedded Image(2.12)At this stage, it is worth stating that the electromagnetic, energy–momentum tensor Embedded Imageab takes a particularly simple form,Embedded Image(2.13)To end this section, consider Einstein–Maxwell theory in general relativity and note that, by virtue of Einstein's field equations, the Riemann curvature tensor (responsible for tidal forces) is given by (see eqns. (4.6.1), (4.6.3) and (5.2.6) of Penrose & Rindler 1987)Embedded Image(2.14)where G is Newton's constant.

3. Retarded null coordinates, tetrad and spinor dyad

Imagine a source following a timelike geodesic Embedded Image in Minkowski spacetime Embedded Image and radiating waves. To describe this process in detail it will make sense to use a system of coordinates tied to Embedded Image. In the absence of incoming waves it proves convenient to adopt retarded null coordinates.

Figure 1 depicts the geodesic Embedded Image on which has been placed an arbitrary origin O. A point X in Embedded Image defines a pair of points R(etarded) and A(dvanced) on Embedded Image, which are the respective intersections of the past and future light cones of X with Embedded Image. The position vector of X from O is given byEmbedded Image(3.1)Here, ta is the unit, timelike, future-pointing tangent vector to Embedded Image and represents Embedded Image's velocity, while u measures the proper time lapsed from O to R. ℓa, defined away from Embedded Image, is the outgoing, future-pointing, null vector normalized to unity against Embedded Image and parallel to the displacement from R to X. r is the standard radial distance in Embedded Image's rest frame. (If Embedded Image were to be written for the displacement from R to X, r would be Embedded Image.)

Figure 1

Retarded null coordinates based on a timelike geodesic Embedded Image: xa=uta+rℓa.

The line element is given byEmbedded Image(3.2)the precise form of the last term depending on how one labels the null rays leaving Embedded Image, i.e. the generators of the future light cones u=const. For example, with regard to a constant, proper, orthochronous, orthonormal tetrad whose timelike member is Embedded Image, the components of Embedded Image take the formEmbedded Image(3.3)where θ and ϕ are standard angular coordinates. In this case,Embedded Image(3.4)(However, the results of this paper are independent of choice of labelling.) Away from Embedded Image, it proves useful to define a second, future-pointing, null vector parallel to the displacement from X to A according toEmbedded Image(3.5)Next, choose a normalized spinor dyad Embedded Image such thatEmbedded Image(3.6)and use this to extend Embedded Image and Embedded Image to a null tetrad Embedded Image by settingEmbedded Image(3.7)Following convention, writeEmbedded Image(3.8)The metric takes the formEmbedded Image(3.9)and Embedded Image's velocity has the following properties:Embedded Image(3.10)The gradient of equation (3.1) for xa yields the following useful expressions:Embedded Image(3.11)Embedded Image(3.12)Embedded Image(3.13)which, in turn, imply thatEmbedded Image(3.14)Embedded Image(3.15)Embedded Image(3.16)Embedded Image(3.17)the last of these using equation (3.5). Next, the phase freedom in the spinor dyad is reduced by demanding thatEmbedded Image(3.18)the remaining freedom taking the form of a real phase independent of u and r:Embedded Image(3.19)The behaviour of the spinor dyad under the action of δ and Embedded Image is given byEmbedded Image(3.20)the precise details of the spin coefficient β depending on the method for labelling the generators of the light cones emanating from Embedded Image. Equations (3.18) and (3.20) imply thatEmbedded Image(3.21)Embedded Image(3.22)Useful results are the commutation relationsEmbedded Image(3.23)and properties of β's derivatives,Embedded Image(3.24)With the (θ, ϕ) labelling employed in equation (3.4), explicit expressions for δ and β that satisfy equations (3.23) and (3.24) are given byEmbedded Image(3.25)

4. Some functions on the unit sphere

Start with the fixed, timelike, future pointing, unit vector ta and the future pointing, null vector a normalized to unity against ta and ranging over the unit sphere Embedded Image. Let xa and ya be arbitrary vectors orthogonal to ta (unrelated to the xa of equation (3.1)) and define the functions on Embedded ImageEmbedded Image(4.1)With the association (3.3), X and Y are linear combinations of the associated Legendre functions Embedded Image with =1 and m=0, ±1. For completeness, the vector za orthogonal to ta, xa and ya,Embedded Image(4.2)defines another =1 function on Embedded Image,Embedded Image(4.3)Next, given the three metric,Embedded Image(4.4)define the symmetric, transverse, trace-free tensorEmbedded Image(4.5)and the corresponding function on Embedded ImageEmbedded Image(4.6)Embedded Image comprises linear combinations of the Embedded Image with =2 and m=0, ±1, ±2.

Next consider spin-weighted functions on Embedded Image. Any quantity η (indices suppressed) that under the phase transformation (3.19) behaves according toEmbedded Image(4.7)where s is half-integral, is said to have spin-weight s. In that case the operators Embedded Image (eth) and Embedded Image are defined byEmbedded Image(4.8)In eth notation,Embedded Image(4.9)From this it follows that, given ka orthogonal to ta and Embedded Image,Embedded Image(4.10)Of interest is the application of Embedded Image and Embedded Image to products of omicrons and iotas. Let w(p, q) be the symmetrized outer product of p omicrons and q iotas,Embedded Image(4.11)Then suppressing indices,Embedded Image(4.12)Further, writing w=w(p, q),Embedded Image(4.13)In fact, Embedded Image is the spin-weight of w. Generally, if η has spin-weight s,Embedded Image(4.14)This section ends with some useful sphere integrals:Embedded Image(4.15)(To prove these, note that they can only involve ta and gab.) Next, let ka be a unit vector orthogonal to ta as above. Then, in standard spherical coordinates,Embedded Image(4.16)It follows that, if n is a non-negative integer,Embedded Image(4.17)Further,Embedded Image(4.18)Embedded Image(4.19)

5. Electric Maxwell dipole

Consider an electric dipole confined to a timelike geodesic Embedded Image with velocity ta in Minkowski spacetime. There are two ways to derive the field: the first uses the formalism of Newman & Penrose (1962) and amounts to that presented by Janis & Newman (1965); the second generates the field from a solution to the scalar wave equation and provides an application of Penrose's spin-raising techniques.

Decomposing the Maxwell field into its spinor parts according to equation (2.6), Maxwell's equation (2.12) away from sources becomeEmbedded Image(5.1)Using the spinor dyad established in §3, writeEmbedded Image(5.2)Then Maxwell's equations becomeEmbedded Image(5.3)Recalling from §3 thatEmbedded Image(5.4)seek a solution generated by an electric dipole lying on the geodesic Embedded Image. So, let ka be a real, unit, spacelike vector orthogonal to ta and writeEmbedded Image(5.5)Next, let z(u) be any smooth, real function of time and, for typographical expediency, writeEmbedded Image(5.6)Then an exact solution to Maxwell's equation (5.3) for r>0 (i.e. away from Embedded Image) and with dipole moment zka is given by (Janis & Newman 1965)Embedded Image(5.7)It turns out that this may be generated from the scalar fieldEmbedded Image(5.8)Away from Embedded Image, this satisfies the wave equation. To see this, note that its first and second derivatives are given byEmbedded Image(5.9)Embedded Image(5.10)From this, it follows thatEmbedded Image(5.11)Furthermore, if Embedded Image is any constant symmetric spinor, the symmetric spinor fieldEmbedded Image(5.12)satisfies Maxwell's equations. In passing, the complex fields,Embedded Image(5.13)play the rôles of left-handed potential and left-handed Hertz potential for the Maxwell field:Embedded Image(5.14)Embedded Image(5.15)Embedded Image(5.16)WriteEmbedded Image(5.17)ThenEmbedded Image(5.18)It is not hard to see thatEmbedded Image(5.19)whence the components of the Maxwell field are given byEmbedded Image(5.20)In particular, chooseEmbedded Image(5.21)where ka is real, unit and orthogonal to ta. ThenEmbedded Image(5.22)in which caseEmbedded Image(5.23)

6. The linearized Einstein–Maxwell equations

The equations to be solved in a background Minkowski spacetime away from the timelike geodesic Embedded Image, are (see eqn. (5.2.7) of Penrose & Rindler 1987)Embedded Image(6.1)where Embedded Image and Embedded Image define the Maxwell field for the electric dipole lying on Embedded Image and where Embedded Image is the spinor equivalent of the spin-2 field. The five complex components of the spin-2 field are given byEmbedded Image(6.2)On using equation (5.3), the linearized Einstein–Maxwell equation (6.1) yields the following for solution (see, e.g., Exton et al. 1969):Embedded Image(6.3)Embedded Image(6.4)Embedded Image(6.5)Embedded Image(6.6)Embedded Image(6.7)Embedded Image(6.8)Embedded Image(6.9)Embedded Image(6.10)To proceed, recall the definitions of Δ and D from equation (5.4), use the expressions (5.7) for the oscillating, electric dipole and employ the expression (4.10). It turns out that a particular solution to equations (6.3)–(6.10) may be found by choosingEmbedded Image(6.11)ψ0 is chosen to vanish because a solution is sought that is driven solely by the Maxwell field. It implies that the gravitational quadrupole moment vanishes. Allowing ψ0 to be non-zero would result in the addition of a solution to the free spin-2 equations.

Given the vanishing of ψ0, equation (6.3) is solved for ψ1 to yieldEmbedded Image(6.12)where Embedded Image may be found in equation (4.10). S takes the formEmbedded Image(6.13)and is a pure =1 contribution. Equation (6.4) serves to constrain sa according toEmbedded Image(6.14)In fact, a particular solution may be found by setting ja=0. Thus, chooseEmbedded Image(6.15)Solving equation (6.5) for ψ2 yieldsEmbedded Image(6.16)The μ term is pure =0 while the Embedded Image term is =2. Equation (6.6) serves to constrain μ and yieldsEmbedded Image(6.17)It turns out, from equation (6.19) below, that the function m(u) may not be set equal to zero. Thus, ψ2 is given by the sum of =0 and =2 parts:Embedded Image(6.18)Equation (6.7) for ψ3 likewise contains an =0 part and an =2 part. The former yieldsEmbedded Image(6.19)which provides the reason why m(u) may not be set zero. In fact, m(u) is the Bondi–Sachs mass of the system (see equation (7.4) below) and equation (6.19) expresses the fact that m(u) decreases as electromagnetic energy is radiated away. The =2 part of equation (6.7) yieldsEmbedded Image(6.20)The expressions (6.18) and (6.20) for ψ2 and ψ3 now satisfy equation (6.8) exactly. Finally, solving equation (6.9) for ψ4 yieldsEmbedded Image(6.21)Equation (6.10) is then satisfied exactly. The entire solution for the spin-2 field is given by equations (6.11), (6.15) and (6.18)–(6.21). Note that, if the dipole source is monochromatic with frequency ω, the spin-2 field, being quadratic in the zi, will have frequency 2ω.

To end this section consider the Bondi–Sachs News function N defined implicitly and uniquely byEmbedded Image(6.22)and given here byEmbedded Image(6.23)In the full theory N governs the gravitational power radiated (see, e.g., Exton et al. 1969).

7. Power output

Standard theory shows that the electromagnetic power output is governed by the asymptotic behaviour of the field, in particular by the coefficient of r−1 in the expression for ϕ2. WritingEmbedded Image(7.1)the momentum flux Embedded Image is given byEmbedded Image(7.2)Indeed, if ψ2 is expanded,Embedded Image(7.3)the system's Bondi–Sachs momentum is defined byEmbedded Image(7.4)whose rate of change, on using equation (6.7), is given byEmbedded Image(7.5)(An alternative and more elegant approach involves rescaling the metric so that ‘infinity’ is regarded as a finite, null boundary to the spacetime.) For the oscillating electric dipole, equations (6.18) and (7.4) yieldEmbedded Image(7.6)while equations (5.7) and (7.2) yieldEmbedded Image(7.7)Strictly speaking, in linearized Einstein–Maxwell theory, no energy is radiated by spin-2 waves. There is indeed a spin-2 radiation field, behaving like 1/r for large r, but the right-hand side of equation (7.5) does not involve the spin-2 field.

In the full theory, the radiation of gravitational waves is well defined for asymptotically flat spacetimes, the power output being governed by the News function N. The momentum flux Embedded Image is then given byEmbedded Image(7.8)An estimate of Embedded Imagea may be obtained by using the expression (6.23) derived for N in linearized Einstein–Maxwell theory:Embedded Image(7.9)Embedded Image(7.10)Thus,Embedded Image(7.11)The power radiated gravitationally as presented in equation (7.11) may be compared with that radiated electromagnetically as depicted in equation (7.7). For a monochromatic source (Embedded Image), it is clear that the power output in gravitational waves is proportional to the square of the electromagnetic power output (compare page 978 of Misner et al. 1973). However, the factor of proportionality (of the order of Embedded Image) is very small. The two power outputs are comparable only when energy is radiated at the rate of around 1052 W; this corresponds to the energy conversion of one Planck mass (10−8 kg) in around the Planck time (10−43 s).

8. Conclusion

Do electromagnetic waves harbour gravitational waves? The ideal approach would be to solve the coupled Einstein–Maxwell equations for an asymptotically flat spacetime given a suitable oscillating Maxwell source. The Maxwell field, by Einstein's field equations, would in turn become a source for a gravitational field and this ought to possess wave-like properties.

The purpose of this paper has been to present a preliminary analysis in linearized Einstein–Maxwell theory on a background Minkowski spacetime with the Maxwell stress tensor serving as a source of the linearized spin-2 field. The Maxwell field was chosen to satisfy Maxwell's equations everywhere except on a given timelike geodesic Embedded Image. The source of the field was considered to be confined to Embedded Image and the chosen solution corresponded to that of an electric dipole. A solution to the linearized Einstein–Maxwell equations was then obtained away from Embedded Image and it was shown to contain spin-2 waves. Thus, the electromagnetic waves generated by an oscillating electric dipole harbour spin-2 waves in linear theory.

For a monochromatic source, the spin-2 wave frequency is twice that of the frequency of the electric dipole. Strictly speaking, in linear theory, no energy is radiated by spin-2 waves. However, by regarding the spin-2 field as an approximation to a gravitational field in the full theory, an estimate of the power output in gravitational waves was obtained. This turns out to be proportional to the square of electromagnetic power output. However, the factor of proportionality, of the order of Embedded Image is very small. Embedded Image is one Planck unit of power, around 1052 W.

Two extensions have been proposed by a referee. The first would be to generalize the dipole source to allow for a magnetic component. The second would use the present analysis as the first step in an iterative solution to the full Einstein–Maxwell equations.

Acknowledgments

I am grateful to Timothy Field for reviewing the original version of this paper, to colleagues at QinetiQ for much encouragement and to two referees whose constructive comments have added clarity to the text.

Footnotes

    • Received October 24, 2005.
    • Accepted January 6, 2006.

References

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