Do electromagnetic waves harbour gravitational waves?

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 relative to which things are referred. The assumption of no incoming radiation leads to the employment of standard, retarded, null coordinates tied to , 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 . Spin-weighted functions are then reviewed together with some properties of eth, a useful differential operator tangent to . 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 . 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 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 and their duals will be written using lower-case, Roman indices, the scalar product being Lorentz invariant. Spinors and dual spinors will be identified via upper-case, Roman indices, the scalar product being invariant under the covering group SL(2,C) of the (proper, orthochronous) Lorentz group. Spin space has two complex dimensions. Complex conjugate spinors and their duals will be written with primed indices.

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

The Minkowski metric and its inverse are chosen to have signatures (+−−−) and spinor equivalents(2.1)where the epsilon objects are skew symmetric in their indices and where(2.2)Spinor indices are raised and lowered according to(2.3)and similarly for primed indices. Only symmetric spinors matter because, for example,(2.4)Related identities include(2.5)A real Maxwell field (skew-symmetric in its indices) defines a symmetric spinor according to(2.6)The dual of the Maxwell field amounts to(2.7)the alternating symbol , skew in all its indices, chosen so that in any proper, orthochronous, Minowski, orthonormal tetrad ,(2.8)The complex self-dual and anti self-dual fields take the forms,(2.9)Maxwell's equations in the presence of a source (a 4-current) form a pair,(2.10)where is the standard derivative operator in Minkowski spacetime. Equivalently(2.11)or, in the spinor formalism,(2.12)At this stage, it is worth stating that the electromagnetic, energy–momentum tensor ^ab takes a particularly simple form,(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)(2.14)where G is Newton's constant.

3. Retarded null coordinates, tetrad and spinor dyad

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

Figure 1 depicts the geodesic on which has been placed an arbitrary origin O. A point X in defines a pair of points R(etarded) and A(dvanced) on , which are the respective intersections of the past and future light cones of X with . The position vector of X from O is given by(3.1)Here, t^a is the unit, timelike, future-pointing tangent vector to and represents 's velocity, while u measures the proper time lapsed from O to R. ℓ^a, defined away from , is the outgoing, future-pointing, null vector normalized to unity against and parallel to the displacement from R to X. r is the standard radial distance in 's rest frame. (If were to be written for the displacement from R to X, r would be .)

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Figure 1

Retarded null coordinates based on a timelike geodesic : x^a=ut^a+rℓ^a.

The line element is given by(3.2)the precise form of the last term depending on how one labels the null rays leaving , 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 , the components of take the form(3.3)where θ and ϕ are standard angular coordinates. In this case,(3.4)(However, the results of this paper are independent of choice of labelling.) Away from , it proves useful to define a second, future-pointing, null vector parallel to the displacement from X to A according to(3.5)Next, choose a normalized spinor dyad such that(3.6)and use this to extend and to a null tetrad by setting(3.7)Following convention, write(3.8)The metric takes the form(3.9)and 's velocity has the following properties:(3.10)The gradient of equation (3.1) for x^a yields the following useful expressions:(3.11)(3.12)(3.13)which, in turn, imply that(3.14)(3.15)(3.16)(3.17)the last of these using equation (3.5). Next, the phase freedom in the spinor dyad is reduced by demanding that(3.18)the remaining freedom taking the form of a real phase independent of u and r:(3.19)The behaviour of the spinor dyad under the action of δ and is given by(3.20)the precise details of the spin coefficient β depending on the method for labelling the generators of the light cones emanating from . Equations (3.18) and (3.20) imply that(3.21)(3.22)Useful results are the commutation relations(3.23)and properties of β's derivatives,(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 by(3.25)

4. Some functions on the unit sphere

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

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

5. Electric Maxwell dipole

Consider an electric dipole confined to a timelike geodesic with velocity t^a 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 become(5.1)Using the spinor dyad established in §3, write(5.2)Then Maxwell's equations become(5.3)Recalling from §3 that(5.4)seek a solution generated by an electric dipole lying on the geodesic . So, let k^a be a real, unit, spacelike vector orthogonal to t^a and write(5.5)Next, let z(u) be any smooth, real function of time and, for typographical expediency, write(5.6)Then an exact solution to Maxwell's equation (5.3) for r>0 (i.e. away from ) and with dipole moment zk^a is given by (Janis & Newman 1965)(5.7)It turns out that this may be generated from the scalar field(5.8)Away from , this satisfies the wave equation. To see this, note that its first and second derivatives are given by(5.9)(5.10)From this, it follows that(5.11)Furthermore, if is any constant symmetric spinor, the symmetric spinor field(5.12)satisfies Maxwell's equations. In passing, the complex fields,(5.13)play the rôles of left-handed potential and left-handed Hertz potential for the Maxwell field:(5.14)(5.15)(5.16)Write(5.17)Then(5.18)It is not hard to see that(5.19)whence the components of the Maxwell field are given by(5.20)In particular, choose(5.21)where k^a is real, unit and orthogonal to t^a. Then(5.22)in which case(5.23)

6. The linearized Einstein–Maxwell equations

The equations to be solved in a background Minkowski spacetime away from the timelike geodesic , are (see eqn. (5.2.7) of Penrose & Rindler 1987)(6.1)where and define the Maxwell field for the electric dipole lying on and where is the spinor equivalent of the spin-2 field. The five complex components of the spin-2 field are given by(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):(6.3)(6.4)(6.5)(6.6)(6.7)(6.8)(6.9)(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 choosing(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 yield(6.12)where may be found in equation (4.10). S takes the form(6.13)and is a pure ℓ=1 contribution. Equation (6.4) serves to constrain s^a according to(6.14)In fact, a particular solution may be found by setting j^a=0. Thus, choose(6.15)Solving equation (6.5) for ψ₂ yields(6.16)The μ term is pure ℓ=0 while the term is ℓ=2. Equation (6.6) serves to constrain μ and yields(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 ℓ=0 and ℓ=2 parts:(6.18)Equation (6.7) for ψ₃ likewise contains an ℓ=0 part and an ℓ=2 part. The former yields(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 ℓ=2 part of equation (6.7) yields(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 yields(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 z_i, will have frequency 2ω.

To end this section consider the Bondi–Sachs News function N defined implicitly and uniquely by(6.22)and given here by(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. Writing(7.1)the momentum flux is given by(7.2)Indeed, if ψ₂ is expanded,(7.3)the system's Bondi–Sachs momentum is defined by(7.4)whose rate of change, on using equation (6.7), is given by(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) yield(7.6)while equations (5.7) and (7.2) yield(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 is then given by(7.8)An estimate of _a may be obtained by using the expression (6.23) derived for N in linearized Einstein–Maxwell theory:(7.9)(7.10)Thus,(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 (), 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 ) 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 . The source of the field was considered to be confined to and the chosen solution corresponded to that of an electric dipole. A solution to the linearized Einstein–Maxwell equations was then obtained away from 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 is very small. 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.