Photon-graviton pair conversion

Andreas Källberg, Gert Brodin and Mattias Marklund

Department of Physics, Umeå University, SE-901 87 Umeå, Sweden

Email: gert.brodin@physics.umu.se and mattias.marklund@physics.umu.se

Received 5 August 2005
Published 28 December 2005

PACS numbers: 04.20.Cv, 04.30.Nk, 95.30.Cq

The interaction between electromagnetic waves (EMWs) and gravitational waves (GWs) has been considered by many authors using approximate methods [1-5] or by deriving exact solutions [6]. The latter solutions are usually found on a flat vacuum background, in which case the simplest nonlinear interaction processes involving three waves are excluded due to energy-momentum conservation. In the present letter we will consider four-wave interaction on a Minkowski background, the lowest order process allowing for energy exchange between photons and gravitons. Using the classical Einstein-Maxwell system, we adopt a calculation scheme inspired by quantum mechanics [7-12]. Assuming that two GWs interact with two EMWs, the only combination leading to energy exchange, the coupled system of equations is solved perturbatively, taking into account terms up to third order in the amplitudes. The four-wave interaction coefficients (giving the quantum mechanical transition rates) are calculated in the centre of mass system for arbitrary polarizations and scattering angle. Thus, the local energy transfer rate between GWs and EMWs is found, i.e. the probability of photoproduction by two gravitons or that of the inverse process.

Although the timescale for coherent GW-EMW pair conversion is shorter than the incoherent one, the large energy densities in the early universe makes it interesting to consider incoherent phenomena involving multiple conversion processes. Using the analogy between the interaction coefficients and the scattering matrix elements, the differential cross section for graviton-photon pair conversion is calculated. In particular, it is found that a thermalization between photons and gravitons due to pair conversion requires a temperature T → T_p, where T_p is the Planck temperature.

In this letter, we use metric signature ( -   +   +   + ), tetrad indices a, b,  ...  =  0, 1, 2, 3, α, β,  ...  =  1, 2, 3, and coordinate indices μ, ν,  ...  =  0, 1, 2, 3. The velocity of light in vacuum is c  =  1.

We use the formalism developed in [1, 2, 4, 13], introducing an observer four-velocity, V^a, and decomposing the electromagnetic field into an electric and a magnetic part. Introducing an orthonormal frame {e_a  =  e^μ_a∂_μ}, where e₀  =  V  =  V^a e_a, (i.e. V^a  =  δ^a₀), the Maxwell field equations can be written in terms of the electric and magnetic fields with the inclusion of effective charges and currents originating from the gravitational field (see e.g. [1, 2, 4]).

From the field equations one may derive the following inhomogeneous wave equations:

where and where the three-vector notation E ≡ (E^α)  =  (E¹, E², E³) etc and ∇ ≡ (e₁, e₂, e₃) from [1, 2, 4] is used. The C^a_bc are the commutation functions, i.e. [e_a, e_b]  =  C^c_ab e_c.

We consider plane waves with weakly space and time dependent amplitudes, i.e. E  =  E(x^μ)exp[ik_μ x^μ]  +  c.c. and |∂_μ E(x^ν)| |k_μ||E(x^ν)|, where c.c. denotes the complex conjugate. In a flat vacuum background, energy-momentum conservation excludes three wave processes leading to energy exchange between photons and gravitons. Furthermore, the only four-wave combination converting energy between the gravitational and electromagnetic degrees of freedom involves two EMWs and two GWs. We denote the gravitational and electromagnetic waves by h_A, h_B and E_A, E_B respectively. The matching conditions for resonant four-wave interaction are [14] , corresponding to energy-momentum conservation. In the centre of mass system, all frequencies are equal and denoted by ω. Thus, in this system we have , , fulfilling the matching conditions, and corresponding to pairwise counterpropagating waves1^Note1. We consider terms up to cubic order in the amplitudes, and note that the corresponding effects can be grouped into four categories:

We will only include the energy conversion terms from point (iv) above, and the final interaction equations will then be of the form [14]

and permutations thereof allowed by the wave matching. Here Φ_1,2 and Φ_3,4 represent mutually counterpropagating wave fields, (i.e. E_A,B and h_A,B respectively), ≡ ∂²_t  -  ∂²_x  -  ∂²_y  -  ∂²_z, and the coefficients M depend on the scattering angle, the polarization of the waves and on the frequency, ω.

We introduce perturbations of the Minkowski metric, g_μν  =  η_μν  +  h_μν. The perturbed energy momentum tensor T_ab should then satisfy Einstein's equations G_ab  =  κT_ab, where G_ab is the perturbation to the Einstein tensor due to the metric perturbation.

In order to determine the nonlinear gravitational response to the gravitational wave perturbations, we first consider the vacuum case. The perturbed metric is

where the h⁽¹⁾_μν are wave perturbations to first order in amplitude and the h⁽²⁾_μν are the second-order response to these perturbations. For the linear wave perturbations we choose the transverse and traceless gauge [15], such that the only nonzero components are h⁽¹⁾₁₁  =   - h⁽¹⁾₂₂ ≡ h ₊ (t, z) and h⁽¹⁾₁₂  =  h⁽¹⁾₂₁ ≡ h_×(t, z). To second order, the vacuum Einstein equations take the form R⁽¹⁾_ab  +  R⁽²⁾_ab  =  0, where R⁽¹⁾_ab is linear in h _+ ,× and R⁽²⁾_ab is quadratic in h _+ ,× and linear in h⁽²⁾_μν. From the second-order vacuum Einstein equations, we see that we only need the following components of h⁽²⁾_μν, corresponding to a choice of gauge: h⁽²⁾₀₀, h⁽²⁾₀₃, h⁽²⁾₁₁, h⁽²⁾₂₂ and h⁽²⁾₃₃. For the metric (4) we choose the tetrad basis e^(vac)_a  =  e^μ(vac)_a∂_μ  =  (e^μ(1)_a  +  e^μ(2)_a)∂_μ where

In this basis the only nonzero components of R⁽¹⁾_ab are R⁽¹⁾₁₁  =   - R⁽¹⁾₂₂  =  1/2(∂²_t  -  ∂²_z)h ₊  and R⁽¹⁾₁₂  =  1/2(∂²_t  -  ∂²_z)h_×. The nonzero second-order components are R⁽²⁾₀₀, R⁽²⁾₀₃, R⁽²⁾₁₁, R⁽²⁾₂₂ and R⁽²⁾₃₃ where R⁽²⁾_ab  =  R⁽²⁾_ab(∂²h⁽²⁾_ab, ∂²h² ₊ , ∂²h²_×). The second-order nonlinear response terms h⁽²⁾_μν are related to the linear wave perturbations by R⁽²⁾_ab  =  0, which gives h⁽²⁾₁₁  =  h⁽²⁾₂₂. Furthermore, adding R⁽²⁾₀₀ to R⁽²⁾₃₃, using the approximation ∂_t ≈ ±∂_z on h _+ ,×, we get

For two (counterpropagating) gravitational waves h_A and h_B, we have h ₊   =  h_A +   +  h_B +  and h_×  =  h_A×  +  h_B×, where the h_A and h_B are given by h_A  =  h_A(t, z)exp[i(k_hz  -  ωt)]  +  c.c. and h_B  =  h_B(t, z)exp[i( - k_hz  -  ωt)]  +  c.c.. From the form of the evolution equations (3), we see that the only terms contributing to the four-wave interaction are terms proportional to h_Ah_B (and h_A h_B), i.e. terms exp[ - 2iωt] (and exp[2iωt]). By examining R⁽²⁾_ab we see that the only components containing terms with this purely temporal dependence are h⁽²⁾₁₁, h⁽²⁾₂₂ and h⁽²⁾₃₃. Inserting the ansatz for h_A and h_B into equation (9), we get h⁽²⁾₁₁  =  (h² ₊   +  h²_×)/4, and the equation R⁽²⁾₀₀  =  0 then gives h⁽²⁾₃₃  =   - h⁽²⁾₁₁. These second-order quantities will enter the evolution equations (1) and (2), and contribute to the interaction coefficients in (3).

The directly resonant terms described above are not the only contribution to the coupling coefficients. Let E  =  E_A  +  E_B  =  (E_A1  +  E_B1) e₁  +  (E_A2  +  E_B2) e₂  +  (E_A3  +  E_B3) e₃ and similarly for B. Furthermore, let E_A1  =  E_A1(t, x, z)exp[i(k_Exx  +  k_Ezz  -  ωt)]  +  c.c., E_B1  =  E_B1(t, x, z)exp[i( - k_Exx  -  k_Ezz  -  ωt)]  +  c.c. etc. Inserting this into the nonlinear wave equations for the electromagnetic fields, the largest terms on the right-hand side of equations (1) and (2) are terms of the form Eh, Bh etc, which induces oscillations at frequencies and wavenumbers that are non-resonant (nr), i.e. ω_nr ≠ k_nr. These nr fields may combine with fields of appropriate frequency and wavenumber in (1) and the corresponding evolution equations for h_A, h_B, making the resulting terms resonant with the original wave perturbation, thus contributing to the coupling coefficients in (3).

In order to take these contributions into account we need to modify our ansatz for the electromagnetic field components, to include the nr fields. For this purpose, we use the lowest order approximation ∂_tB  =   - ∇ × E, where ∇  =  (∂_x, ∂_y, ∂_z), in all second-order terms in equation (1). We thus make the ansatz E^(tot)₁  =  E_A1  +  E_B1  +  E^(nr)₁ etc for the total electromagnetic field. Inserting this into equations (1)-(2), including terms that will contribute to the four-wave coupling, we see that the terms linear in h_A,B take the form E^(nr)  =  c₁ E_A h_A  +  c₂ E_A h_B  +  c₃ E_B h_A  +  c₄ E_B h_B  +  c.c., where the c_i are coefficients determined by the field geometry. Thus, we find that the nr fields will contain oscillating factors exp[i(k_E  -  k_h)]  +  exp[i(k_E  +  k_h)]  +  c.c.. Applying this together with the previous expressions for the original perturbations, noting that for the given field geometry we may use k_Ez  =  ωcos θ, k_Ex  =  ωsin θ and k_h  =  ω, one obtains explicit expressions for the nr electric and magnetic field components. These expressions are greatly simplified when expressing the nr field components in terms of the linear polarization states for the original waves. We thus define the two independent linear electromagnetic polarization states according to E_A1  =   - cos θE_A + , E_A2  =  E_A×, E_A3  =  sin θE_A + , and E_B1  =  cos θE_B + , E_B2  =  E_B×, E_B3  =   - sin θE_B + . The parts of the nr field components contributing to the four-wave interaction take the form

plus an analogous term with spatial dependence exp[i(k_E  +  k_h)].

Terms proportional to E_AE_B and E_A E_B oscillates as exp( - 2iωt) and exp(2iωt), respectively. These terms induce non-resonant GW fields h^(nr), with the same temporal variation, through Einstein's equations. Nonlinearities in the Einstein tensor of the form h^(nr)h_A,B are then resonant with the original perturbations, and will contribute to the coefficients in the evolution equations for the GW amplitudes.

Calculating the part of the energy momentum tensor varying as exp(±2iωt), we make the ansatz E_A1  =  E_A1(t, x, z)exp[i(k_Exx  +  k_Ezz  -  ωt)]  +  c.c. etc. We have not included the non-resonant fields calculated above in the ansatz, since these fields do not appear to second order. The nonzero components of the energy momentum tensor varying as exp(±2iωt) are the purely spatial components of T_ab.

Since the energy-momentum tensor induces gravitational fields with the same temporal variation, exp(±2iωt), we assume a general metric perturbation with this time dependence. We can separate out terms in Einstein's equations describing the GWs and the nonlinear response to these, and terms describing the metric response to the energy momentum tensor of the EMWs, and therefore we can calculate the nr gravitational fields separately. We use the gauge h^(nr)_0ν  =  0, and we choose a general tetrad (i.e. a basis where all h^(nr)_αβ are included), such that the extra terms corresponding to the nr fields basis may be added to the tetrad (5)-(8). We only need to include terms up to in the Einstein tensor, since higher order terms in h^(nr)_μν will be of at least fourth order in the electromagnetic amplitude. The nonzero components of the Einstein tensor to this order are the purely spatial components of G^(nr)_ab. By letting h^(nr)_αβ → h^(nr)_αβexp[ - 2iωt]  +  c.c. one obtains explicit expressions for the non-resonant gravitational fields from Einstein's equations.

We now have the ingredients needed to calculate the evolution equations for the amplitudes of the waves and explicitly give the coefficients in (3). We expand our metric ansatz (4), in order to include the non-resonant GW fields, as g_μν  =  η_μν  +  h^TT_μν  +  h⁽²⁾_μν  +  h^(nr)_μν, and include the driven non-resonant part in our ansatz for the electromagnetic fields. The final tetrad basis is given by e_a  =  (e^μ(vac)_a  +  e^μ(nr)_a)∂_μ, where

and where we put h⁽²⁾₀₀  =  h⁽²⁾₀₃  =  0 in e^μ(vac)_a. Using this basis, and inserting explicit expressions for all fields into equation (1), one obtains equations, involving several timescales, for the field components. Taking an average, over several wavelengths and periods, of the equations, terms resonant with the original perturbation are obtained [14]. Expressing the equations using the linear polarization states, we finally obtain

from equation (1), where H_I ≡ h_A +  h_B +   -  h_A× h_B×, H_II ≡ h_A +  h_B×  +  h_A× h_B + , M ₊  ≡ ω²(1  +  α²)/2, M_× ≡ ω²α and α ≡ cos θ.

The evolution equations for the gravitational wave amplitudes are G₁₁  -  G₂₂  =  κ(T₁₁  -  T₂₂) for h ₊  and G₁₂  +  G₂₁  =  κ(T₁₂  +  T₂₁) for h_×. Inserting explicit expressions for all fields, we obtain the following equations:

where E_I ≡ E_A +  E_B +   +  E_A× E_B×, E_II ≡ E_A +  E_B×  -  E_A× E_B + , m ₊  ≡ κ(1  +  α²) and m_× ≡ 2κα. The coupled system of equations (12)-(15) describes the evolution of the slowly varying wave amplitudes.

Introducing the energy densities of the waves and for the gravitational and electromagnetic wave, respectively, we note that the total wave energy, , is conserved in the interaction described by equations (12)-(15).

In order to illustrate the physics of (12)-(15), we assume long pulses, i.e. →  - 2iω∂_t. For simplicity, we consider the case where all waves have ` + ' polarization, and assume that the initial amplitudes and relative phases of the wave envelopes are fixed in such a way that E_A +   =  E_B +  ≡ E and h_A +   =  h_B +  ≡ h, where and . We insert this into equations (12) and (14), and re-express the equations as evolution equations for the energy densities and , and the relative phases of the waves. Then, for the special choices of initial values (_h  -  _E)|_t = 0  =  ±π/4, one obtains very simple analytical solutions for the ratio of electromagnetic and gravitational wave energy densities. The results in this case are in the case when (_h  -  _E)(0)  =   - π/4 and for (_h  -  _E)(0)  =  π/4, where .

Thus we see that the characteristic timescale for graviton-photon conversion is rather slow. In particular, as far as the interaction between four long pulses is concerned, there is only time for large energy conversion if the waves also contain enough energy to significantly modify the background curvature (cf point (iii) above). However, we note that the GW-EMW conversion takes place in a spacetime that is locally flat, and a given pulse (EMW or GW) can interact with other pulses (or even individual quantas) consecutively, leading to significant conversion after multiple processes. Thus, we here emphasize that the coefficients in equations (12)-(15) can be used also to describe incoherent interaction involving multiple processes. Writing the equations in terms of the vector potential, A, rather than the electric field and making a normalization , we note that and in (12)-(15). The coefficients and now correspond to the scattering amplitude matrix elements, from which the cross section can be calculated straightforwardly. Following [16], we find that the differential cross section for graviton-photon conversion averaged over all polarization states is

where L_P is the Planck length.

The process of graviton-photon conversion may lead to thermalization between gravitons and photons in the early universe. For thermalization to occur before expansion slows down the conversion rate too much, the collision frequency must fulfil ν  =  σn > H, where H is the Hubble parameter and n is the number density. Considering a photon gas at temperature T in the radiation dominated era, using H ~ 1/t, the above condition can be written as , where T_P is the Planck temperature and is the effective number of degrees of freedom. Thus thermalization, in case this is still a meaningful concept when T ~ T_P, must be described within a quantum theory of gravity. The classical photon-graviton conversion process nevertheless puts some limits on the background levels of gravitational radiation, since it makes a relic GW spectrum much above the thermal level unlikely.

We note, however, that the timescale for coherent interaction is shorter than that for incoherent single particle interactions by a factor of the order k²L²_P, and thus physical scenarios with significant GW-EMW conversion in astrophysics are still possible within a classical framework. Due to the large energy densities required, however, such processes are likely to take place in significantly curved backgrounds, for which a description lies outside the scope of this letter.

A comparison of the cross section (16) to that obtained by using Feynman diagrams would be of much interest, but as far as we know, the latter result is yet to be deduced. We note, however, that an interaction Lagrangian of the form , as used in, e.g., [11] in linear quantum gravity, will not be sufficient here, since the effect considered in this letter is quadratic in the gravitational wave amplitudes.

Acknowledgment

Andreas Källberg, who was supported by the Swedish National Graduate School of Space Technology, was tragically killed in an accident shortly after this work was completed.

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Notes

Note1

 Apart from processes leading to GW-EM energy conversion, we can also have pure scattering processes where the EMW and GW energies are conserved separately, i.e. one photon and one graviton scatters into another photon-graviton pair. This process is not considered here.