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Article of through of Gravitational Waves

Charles H.-T. Wang * and Melania Mieczkowska

Department of , University of Aberdeen, King’s College, Aberdeen AB24 3UE, UK * Correspondence: [email protected]

Abstract: Zero-point fluctuations are a universal consequence of theory. fluctua- tions of electromagnetic field have provided crucial evidence and guidance for QED as a successful quantum field theory with a defining gauge symmetry through the , , and spontaneous emission. In an accelerated frame, the thermalisation of the zero-point electromagnetic field gives rise to the Unruh effect linked to the Hawking effect of a via the equivalence principle. This principle is the basis of general covariance, the symmetry of general relativity as the classical theory of gravity. If exists, fluctuations of the gravita- tional field should also to the and dissertation of general forms of and matter. Here we present a novel theoretical effect involving the spontaneous emission of soft by as they bend around a heavy mass and discuss its observational prospects. Our analytic and numerical investigations suggest that the gravitational bending of starlight predicted by classical general relativity should also be accompanied by the emission of gravitational waves. This in turn redshifts the light causing a loss of its energy somewhat analogous to the bremsstrahlung of   by a heavier charged . It is suggested that this new effect may be important for a combined astronomical source of intense gravity and high- such as X-ray binaries Citation: Wang, C.H.-T.; and that the proposed LISA mission may be potentially sensitive to the resulting sub-Hz stochastic Mieczkowska, M. Bremsstrahlung of gravitational waves. Light through Spontaneous Emission of Gravitational Waves. Symmetry Keywords: quantum gravity; quantum gravity phenomenology; gauge symmetry; quantum vacuum; 2021, 13, 852. https://doi.org/ spacetime fluctuations; gravitational decoherence; gravitational bremsstrahlung; gravitational waves; 10.3390/sym13050852 gravitational ; X-ray binary

Academic Editors: Sergei D. Odintsov and Ignatios Antoniadis

Received: 14 April 2021 1. Introduction Accepted: 8 May 2021 The quantum vacuum is a remarkable consequence of the quantum field theory (QFT). Published: 11 May 2021 To be sure, the (QED) as the first successful QFT has received crucial guidance and support through its quantum vacuum effects including the Lamb Publisher’s Note: MDPI stays neutral shift, Casimir effect, and spontaneous emission. with regard to jurisdictional claims in Although the physical reality of the quantum vacuum seems to contradict the void published maps and institutional affil- classical vacuum, it in fact forges essential links between classical and . iations. The general agreement between the classical emission rate and quantum spontaneous emission rate of electromagnetic (EM) dipole have been well-known at atomic scales (see e.g., [1]). Such an agreement is also clear in the classical radiation and the quantum spontaneous emission of the Landau levels [2], in the context of detecting Copyright: © 2021 by the authors. Unruh radiation as a quantum vacuum effect in non-inertial frames [3]. Licensee MDPI, Basel, Switzerland. At present, a fully quantised theory of gravity is still to be reached (for some recent This article is an open access article developments, see e.g., [4–6] and references therein). Nevertheless, the effective QFT distributed under the terms and for linearised general relativity is expected to yield satisfactory physical descriptions at conditions of the Creative Commons sufficiently lower than the Planck scale [7–9]. Indeed, the spontaneous emission Attribution (CC BY) license (https:// rate of gravitons for a nonrelativistic bound system due to the zero-point fluctuations creativecommons.org/licenses/by/ of spacetime in linearised quantum gravity has been recently shown [10,11] to agree 4.0/).

Symmetry 2021, 13, 852. https://doi.org/10.3390/sym13050852 https://www.mdpi.com/journal/symmetry Symmetry 2021, 13, 852 2 of 16

with the quadrupole formula of gravitational wave radiation in general relativity [12]. The preservation of the local translational symmetry of linearised gravity is crucial in the theoretical steps of establishing this agreement through the gauge invariant Dirac quantisation technique [8]. Based on this development, the next challenge would be the spontaneous emission of gravitons from a relativistic and unbound system, which we will address in this paper. The possible gravitational radiation by photons has long been a subject of interest and has been considered by a many researchers with various approaches [13–21]. The obtained size of the effect has generally been quite small. We therefore seek an amplified effect in the astronomical context involving the deflec- tion of starlight by a celestial body or distribution of mass. We show that soft gravitons are spontaneously emitted resulting in scattering modes of incident photons to decay into lower energy scattering modes in the fashion of the bremsstrahlung of electrons by ions [22–25]. Our preliminary estimates of such effects suggest they may be important for high frequency photons deflected by a compact heavy mass. Under weak gravity, the polarisations of light subject to gravitational bending are expected to be negligible. Therefore, as a first approximation, the effect of of is neglected similar to neglecting spins in standard descriptions of the bremsstrahlung of electrons. Throughout this paper, we adopt units in which the c equals unity, unless otherwise stated. We also write log10 = log and use Greek indices µ, ν ... = 0, 1, 2, 3 and Latin indices i, j ... = 1, 2, 3 for spacetime (t, x, y, z) and spatial (x, y, z) coordinates, respectively.

2. Light Modelled as Massless Scalar in a Weak Central Gravitational Field As alluded to in the introduction section, in what follows, we shall model photons as massless scalar with a linearised metric

gµν = ηµν + hµν (1)

where ηµν = diag(−1, +1, +1, +1) is the Minkowski metric and hµν is the metric perturba- tion arising from a spherical gravitational source with mass M? so that

h00 = h11 = h22 = h33 = −2Φ (2)

in terms of the Newtonian potential [12]

GM? Φ = − (3) r p with r = x2 + y2 + z2 and the gravitational constant G. Figure1 illustrates the schematic physical and geometrical configurations under consideration. Treating the above source as a gravitational lens, its effective is given approximately by n = 1 − 2Φ [26]. This gives rise to the approximate dispersion relation

ω = (1 + 2Φ)p (4)

for a real massless scalar field φ having a frequency ω and wave vector p with wave number p = |p|. We will continue to denote wave vectors associated with the scalar field by p and q and wave vectors associated with the gravitons by k, unless otherwise stated. Symmetry 2021, 13, 852 3 of 16

To capture the salient physical effects carried by the light frequency and to simplify our technical derivations, we consider the wave number of the scalar to peak around some fixed value p◦. Then, it follows from Equation (4) to leading contributions, that

ω2 = p2 + v(r) (5)

where

GM? v(r) = −4p2 . (6) ◦ r

Figure 1. An illustration of the key physical and geometrical features of an astronomical configuration for the gravitational bremsstrahlung of light involving an X-ray binary and a possible detection concept with LISA. Here the mass of the compact object is assumed to be dominant for simplicity.

Using Equation (6), the Lagrangian density of the scalar field

1 L = − p−g g ∇µφ∇νφ 2 µν reduces to 1 1 L = − ηµνφ φ − v(r)φ2 (7) 2 ,µ ,ν 2 with v(r) as the effective external scalar potential [27,28]. The stress–energy tensor of the scalar field is given by

1 1 Tµν = (ηµαηνβ + ηµβηνα − ηµνηαβ)φ φ − ηµνvφ2. (8) 2 ,α ,β 2 The field equation follows as

µν η ∂µ∂νφ − vφ = 0. (9)

To solve this field equation, one naturally invokes the separable ansatz

φ(r, t) = ψ(r)e−iωt (10) Symmetry 2021, 13, 852 4 of 16

so that the general solutions are the real parts of the linear combinations of Equation (10). Substituting the above ansatz into Equation (9), we see that this field equation is equiva- lent to

−∇2ψ + vψ = ω2ψ (11)

taking the form of a time-independent Schrödinger equation. It follows that the solutions to the field Equation (9) representing the deflection of light with an incident wave vector p and frequency ω = p can be obtained from the solutions of the Schrödinger Equation (11) describing a scattering problem involving a Coulomb-type, i.e., 1/r, central potential as the “scattering wave functions” of the form

∞ l 1 l m m∗ ψp(r) = ∑ ∑ 4π i wl(η, ρ)Yl (rˆ)Yl (pˆ ) (12) ρ l=0 m=−l

m where pˆ = p/p, r = (x, y, z), Yl (r) are spherical harmonics, and wl(η, ρ) are the Coulomb wave functions satisfy the wave equation (see e.g., [29])

n d2 h 2η l(l + 1) io + 1 − − w (η, ρ) = 0 (13) dρ2 ρ ρ2 l

using the dimensionless variable ρ = p r and dimensionless parameter η = −ν/p with

2 ν = 2 GM? p◦. (14)

By virtue of the orthogonality of wl(η, ρ), we can choose the normalisation of wl(η, ρ) so that ψp(r) satisfy the following orthonormality Z 3 ∗ 0 d r ψp0 (r)ψp(r) = δ(p − p ), (15) Z 3 ∗ 0 0 d p ψp(r )ψp(r) = δ(r − r ). (16)

It is useful to introduce the “momentum representation” scattering wave functions [30] ψq(p) of the above “position representation” scattering wave functions ψq(r) given by

Z 1 3 −ip·r ψq(p) = d r e ψq(r), (17) (2π)3/2 Z 1 3 ip·r ψq(r) = d p e ψq(p). (18) (2π)3/2

The orthogonality of ψq(p) follows immediately from Equations (15)–(18) to be Z 3 ∗ 0 0 d q ψq(p )ψq(p) = δ(p − p ), (19) Z 3 ∗ 0 d q ψp0 (q)ψp(q) = δ(p − p ). (20)

For a weak interaction with the central potential where |η|  1, the first order Born approximation yields ν ψp(q) = δ(p − q) + (21) π2(p2 − q2)|p − q|2 Symmetry 2021, 13, 852 5 of 16

Note that the first term of Equation (21) corresponds to the first term of Equation (22), which represents the incident (asymptotically) free particle. This term does not contribute to Equation (31) under Markov approximation of the gravitational master equation as discussed in Reference [9]. The corresponding asymptotic scattering of the position r is given by

ipr 1 h ip·r e i ψp(r) = e + f (θ) (22) (2π)3/2 r

with the scattering amplitude ν f (θ) = (23) 2p2 sin2(θ/2)

where θ = ](r, p) is the scattering angle. Just as with the Coulomb potential, so does the infinitely long range of the Newtonian potential imply a divergent total scattering cross section. However, to account for the realistic limited dominance of this potential due to other influences beyond a range distance Rrng = 1/e, which can be conveniently incorporated by modifying the Newtonian potential with an additional exponential-decay factor of e−er as a long-range regularisation. On the other hand, the finite extension with a radius R? = 1/δ of the gravity source means the need for a compensating short-range potential within this radius. These considerations lead to the following phenomenological Yukawa regularisation

1 e−er e−δr → − (24) r r r with e and δ as long and short range regularisation parameters, respectively. Accordingly, we find the regularised scattering wave function to be ν ψq(p) = δ(p − q) + π2[p2 − (q + ie)2][|p − q|2 + e2] ν − (25) π2[p2 − (q + iδ)2][|p − q|2 + δ2]

with p◦  δ > e.

3. Quantisation of the Scalar Field in the Regularised Potential

Using scattering wave function ψp(r) derived in the preceding section, we can now perform the so-called second quantisation of the scalar field φ into a quantum field in the as follows s Z h i 3 h¯ −iωpt † ∗ iωpt φ(r, t) = d p apψp(r)e + apψp(r)e (26) 2ωp

where h¯ is the reduced and the creation and annihilation operators ap and † ap satisfy the standard nontrivial canonical commutation relation

 †  0 ap, ap0 = δ(p − p ). (27)

The associated field momentum is given by π = ∂φ/∂t = φ˙, which satisfies the equal time field commutation relation

[φ(r, t), π(r0, t)] = ih¯ δ(r − r0)

following from Equations (16) and (27). Symmetry 2021, 13, 852 6 of 16

Substituting Equation (18) into Equation (26), we can write φ in terms of the momen- tum representations of the scattering wave functions as follows s Z 3 3 h¯ h ip·r −iqt † ∗ −ip·r iqti φ(r, t) = d q d p aqψq(p)e e + a ψ (p)e e . (28) 2(2π)3q q q

The coupling of φ to the metric fluctuations due to low energy quantum gravity in addition to the metric perturbation Equation (2) due to the lensing mass M? is through the TT transverse-traceless (TT) part of its stress–energy tensor Equation (8) to be τij := Tij [8] given by

τij(r, t) = Pijklφ,k(r, t)φ,l(r, t), (29)

with the Fourier transform Z 3 −ik·r τij(k, t) = d r τij(r, t) e Z 3 −ik·r = d r e Pijkl(k)φ,k(r, t)φ,l(r, t) (30)

where Pijkl is the TT projection operator [8,31]. † † Using Equation (28) and applying normal orders and neglecting apaq and apaq terms through the rotating wave approximation [32], we see that Equation (30) becomes

Z h¯ 3 3 0 3 00 pk pl τij(k, t) = Pijkl(k) d p d p d p 2 pp0 p00

h † ∗ −i(p0−p00)t † ∗ i(p0−p00)ti ap00 ap0 ψp0 (p)ψp00 (p − k)e + ap0 ap00 ψp0 (p)ψp00 (p + k)e . (31)

From Equation (31) we see that

† τij(−k, t) = τij(k, t) (32)

as a useful property for later derivations.

4. Coupling to the Gravitational Quantum Vacuum To induce the spontaneous emission of the photon lensed by the regularised Newto- nian potential, we now include an additional TT gravitational wave-like metric perturbation TT hµν into Equation (1), which carries spacetime fluctuations at zero temperature [8]. The photon is assumed to travel a sufficiently long distance and time past the gravitational lens (see justifications below) so that we can neglect any memory effects in its statistical interactions with spacetime fluctuations. Additionally, we consider the energy scale to be low enough for the self interaction of the photon to be negligible. This to the Markov quantum master equation

Z 3 Z ∞ κ d k −iks † ρ˙(t) = − ds e [τij(k, t), τij(k, t − s)ρ(t)] + H.c. (33) h¯ 2(2π)3k 0

in the , where κ = 8πG, τij(k, t) is given by Equation (31), and H.c. denotes the Hermitian conjugate of a previous term, for the reduced density operator, i.e., , ρ(t) of the photon by averaging, i.e., tracing, out the degrees of freedom in the noisy gravitational environment [8,10]. Symmetry 2021, 13, 852 7 of 16

It is convenient to express Equation (31) in the form Z 3 0 3 00 h 0 00 −i(p0−p00)t † 0 00 i(p0−p00)ti τij(k, t) = d p d p τij(k, p , p )e + τij(−k, p , p )e (34)

in terms of Z 0 00 h¯ 3 pk pl † ∗ τij(k, p , p ) = Pijkl(k) d p a 00 ap0 ψp0 (p)ψ 00 (p − k) (35) 2 pp0 p00 p p

From Equations (34) and (32) we also have Z † 3 0 3 00 h 0 00 −i(q0−q00)t † 0 00 i(q0−q00)ti τij(k, t) = d q d q τij(−k, q , q )e + τij(k, q , q )e (36)

Substituting Equation (34) into Equation (33) we have

Z 3 3 0 3 00 Z ∞ κ d k d p d p −i(k−p0+p00)s −i(p0−p00)t † 0 00 ρ˙(t) = − ds e e [τij(k, t), τij(k, p , p )ρ(t)] + H.c. h¯ 2(2π)3k 0 Z 3 3 0 3 00 Z ∞ κ d k d p d p −i(k+p0−p00)s i(p0−p00)t † † 0 00 − ds e e [τij(k, t), τij(−k, p , p )ρ(t)] + H.c. . (37) h¯ 2(2π)3k 0

We then apply the Sokhotski–Plemelj theorem

Z ∞ 1 ds e−ies = πδ(e) − iP (38) 0 e where P is the Cauchy principal value. Since this Cauchy principal value contributes to a renormalised energy that can be absorbed in physical energies [32], we can neglect it. The remaining part of Equation (37) on account of Equation (36) is

3 3 0 3 00 3 0 3 00 πκ Z d k d p d p d q d q 0 00 ρ˙(t) = − δ(k − (p0 − p00))e−i[k+(q −q )]t h¯ 2(2π)3k  0 00 0 00 × [τij(−k, q , q ), τij(k, p , p )ρ(t)] + H.c. 3 3 0 3 00 3 0 3 00 πκ Z d k d p d p d q d q 0 00 − δ(k − (p0 − p00))e−i[k−(q −q )]t h¯ 2(2π)3k  † 0 00 0 00 × [τij(k, q , q ), τij(k, p , p )ρ(t)] + H.c. 3 3 0 3 00 3 0 3 00 πκ Z d k d p d p d q d q 0 00 − δ(k + (p0 − p00))e−i[k+(q −q )]t h¯ 2(2π)3k  0 00 † 0 00 × [τij(−k, q , q ), τij(−k, p , p )ρ(t)] + H.c. 3 3 0 3 00 3 0 3 00 πκ Z d k d p d p d q d q 0 00 − δ(k + (p0 − p00))e−i[k−(q −q )]t h¯ 2(2π)3k  † 0 00 † 0 00 × [τij(k, q , q ), τij(−k, p , p )ρ(t)] + H.c. . (39)

Due to the weakness of interactions, the density matrix will evolve only slightly over time so that

ρ(t) = ρ0 + ∆ρ(t) (40)

where ρ0 = ρ(t = 0) is the initial density matrix and the components of ∆ρ(t) are small compared to the components of ρ0. Symmetry 2021, 13, 852 8 of 16

Then Equation (39) yields

3 3 0 3 00 3 0 3 00 πκ Z d k d p d p d q d q Z t 0 00 ∆ρ(t) = − δ(k − (p0 − p00)) ds e−i[k+(q −q )]s h¯ 2(2π)3k 0

n 0 00 0 00 o × [τij(−k, q , q ), τij(k, p , p )ρ0] + H.c.

3 3 0 3 00 3 0 3 00 πκ Z d k d p d p d q d q Z t 0 00 − δ(k − (p0 − p00)) ds e−i[k−(q −q )]s h¯ 2(2π)3k 0

n † 0 00 0 00 o × [τij(k, q , q ), τij(k, p , p )ρ0] + H.c.

3 3 0 3 00 3 0 3 00 πκ Z d k d p d p d q d q Z t 0 00 − δ(k + (p0 − p00)) ds e−i[k+(q −q )]s h¯ 2(2π)3k 0

n 0 00 † 0 00 o × [τij(−k, q , q ), τij(−k, p , p )ρ0] + H.c.

3 3 0 3 00 3 0 3 00 πκ Z d k d p d p d q d q Z t 0 00 − δ(k + (p0 − p00)) ds e−i[k−(q −q )]s h¯ 2(2π)3k 0

n † 0 00 † 0 00 o × [τij(k, q , q ), τij(−k, p , p )ρ0] + H.c. . (41)

For t → ∞, physically corresponding to t greater than the effective interaction time of the system, we can again apply the Sokhotski–Plemelj theorem Equation (38) to Equa- tion (41) and neglect the Cauchy principal value terms to obtain

∆ρ := ∆ρ(t → ∞) πκ Z d3k d3 p0 d3 p00 d3q0 d3q00 = − δ(k + (q0 − q00)) δ(k − (p0 − p00)) h¯ 2(2π)3k

n 0 00 0 00 o × [τij(−k, q , q ), τij(k, p , p )ρ0] + H.c.

πκ Z d3k d3 p0 d3 p00 d3q0 d3q00 − δ(k − (q0 − q00)) δ(k − (p0 − p00)) h¯ 2(2π)3k

n † 0 00 0 00 o × [τij(k, q , q ), τij(k, p , p )ρ0] + H.c.

πκ Z d3k d3 p0 d3 p00 d3q0 d3q00 − δ(k + (q0 − q00)) δ(k + (p0 − p00)) h¯ 2(2π)3k

n 0 00 † 0 00 o × [τij(−k, q , q ), τij(−k, p , p )ρ0] + H.c.

πκ Z d3k d3 p0 d3 p00 d3q0 d3q00 − δ(k − (q0 − q00)) δ(k + (p0 − p00)) h¯ 2(2π)3k

n † 0 00 † 0 00 o × [τij(k, q , q ), τij(−k, p , p )ρ0] + H.c. . (42)

More precisely, the limit t → ∞, means t exceeds the effective interaction duration between the photon and the lensing mass with an effective force range ∼ 1/e, and this is satisfied if the photon is measured at a large distance with t  1/e. Considering the time integrals in Equation (41), the validity of the foregoing limit is further justified through the numerical simulations described towards the end of this paper where k and |p − q| are found to peak around e. Symmetry 2021, 13, 852 9 of 16

Adopting the Born approximation (25) and keeping up to the first orders in ν, we see that Equation (35) becomes

0 (0) 0 (1) 0 τij(k, p, p ) = τij (k, p, p ) + ν τij (k, p, p ) (43)

in terms of ( ) (0) 0 h¯ Pijkl k pk pl 0 † τ (k, p, p ) = δ(p − p − k) a 0 ap (44) ij 2 ppp0 p ( ) 0 0 (1) 0 h¯ Pijkl k h pk pl pk pl i † τij (k, p, p ) = p 0 + 0 ap0 ap 2π2 pp0[|p − p0 − k|2 + e2] |p − k|2 − (p − ie)2 |p + k|2 − (p + ie)2 −(e → δ) (45)

satisfying

(0)† 0 (0) 0 τij (k, p, p ) = τij (−k, p , p) (46)

(1)† 0 (1) 0 τij (k, p, p ) = τij (−k, p , p) (47)

which are consistent with Equation (32). Substituting Equations (43) into Equation (42) and repetitively using the same argu- ment leading to free particles suffering no Markovian gravitational decoherence, we obtain the 2nd order (in ν) asymptotic change of the density matrix

2 Gν2 Z 1 ∆ρ = − d3k d3 p d3 p0 d3q d3q0 δ(k + (q − q0)) δ(k − (p − p0)) πh¯ k (1) 0 (1) 0 × [τij (−k, q, q ), τij (k, p, p )ρ0] + H.c. (48)

On account of Equations (14) and (47), Equation (48) takes a more explicit form as follows

Z Pijkl(k) ∆ρ = −ζ d3k d3 p d3 p0 d3q d3q0 δ(k − p + p0) δ(k + q − q0) kppp0qq0 0 0 n 1 h pi pj pi pj i o × + − (e → δ) [|p − p0 − k|2 + e2] |p − k|2 − (p0 − ie)2 |p0 + k|2 − (p + ie)2

n 1 h q q q0 q0 i o × k l + k l − (e → δ) [|q − q0 + k|2 + e2] |q + k|2 − (q0 − ie)2 |q0 − k|2 − (q + ie)2  † †  × aq0 aq , ap0 ap ρ0 + H.c. (49)

where

2¯h G3 M2 p4 ζ = ? ◦ (50) π5 is a dimensionless parameter. Symmetry 2021, 13, 852 10 of 16

The action of Equation (49) can be obtained from its action on an initial basis matrix element of the form

† ρ = |p ihq | = a |0ih0|aq . (51) 0 0 0 p0 0 Substituting this form (51) into Equation (49) and using the relation

 † †  0 0 0 0 aq0 aq , ap0 ap ρ0 = δ(p − p0)δ(q − p )|q ihq0| − δ(p − p0)δ(q − q0)|p ihq|

obtained from Equation (27), we can usefully write the resulting ∆ρ as

∆ρ = ∆$\ + ∆$[ + H.c. (52)

where Z ( ) | ih | \ 3 3 3 Pijkl k p q0 ∆$ = −ζ d k d p d q δ(k + q − p0) δ(k + q − p) √ kq pp0

n 1 h p i p j qiqj i o × 0 0 + − ( → ) 2 2 2 2 2 2 e δ |p0 − q − k| + e |p0 − k| − (q − ie) |q + k| − (p0 + ie) n 1 h q q p p i o × k l + k l − (e → δ) (53) |q − p + k|2 + e2 |q + k|2 − (p − ie)2 |p − k|2 − (q + ie)2

does not lose energy and Z ∆$[ = ζ d3k k

Z n δ(k + p − p ) h p0i p0j pi pj i o × P (k) d3 p 0 + − (e → δ) |pi mnij p 2 2 | − |2 − ( − )2 | + |2 − ( + )2 p0kp [|p0 − p − k| + e ] p0 k p ie p k p0 ie Z n δ(k + q − q ) h p p q q i o × P (k) d3q 0 0k 0l + k l − (e → δ) hq| (54) mnkl p 2 2 | − |2 + ( + )2 | + |2 − ( − )2 q0kq [|q0 − q − k| + e ] q0 k q ie q k q0 ie is responsible for dissipating photon energy by emitting bremsstrahlung gravitons. The setup above allows us to consider a wave packet profile Z |ψi = d3 p ψ(p)|pi (55)

as an initial normalised state with Z d3 p |ψ(p)|2 = 1 (56)

and a mean wave number vector p◦ so that Z 3 2 d p p |ψ(p)| = p◦. (57)

This can be used to construct an initial density matrix Z Z 3 3 ∗ ρ0 = |ψihψ| = d p d q ψ(p)ψ (q) |pihq|. (58)

Then from Equations (58) and (54), we have Z Z [ 3 3  ∆$ = ζ d k |pijihpij| = ζ d k |p+ihp+| + |p×ihp×| (59) Symmetry 2021, 13, 852 11 of 16

where

|piji = |piji(k) Z Z 3 0 3 0 0 1 = Pijkl(k) d p d p ψ(p ) δ(k + p − p ) pp0kp n 1 h p0 p0 p p i o × k l + k l − (e → δ) |pi |p0 − p − k|2 + e2 |p0 − k|2 − (p − ie)2 |p + k|2 − (p0 + ie)2 1 Z Z = 3 0 | i 3 d p dΩp Aij p (60) p◦ p0>k p=p0−k

with the dimensionless TT amplitude

0 Aij = Aij(k, p , Ωp) 3 3/2 0 p◦ p := Pijkl(k) ψ(p ) pp0k n 1 h p0 p0 p p i o × k l + k l − (e → δ) . |p0 − p − k|2 + e2 |p0 − k|2 − (p − ie)2 |p + k|2 − (p0 + ie)2 p=p0−k

5. Gravitational Bremsstrahlung with a Single Momentum Initial State For simplicity, we now restrict to single momentum initial state, deferring the more general wave-packet initial states to a future investigation. Such a state is obtained by setting q0 = p0 = p◦ in Equation (51) so that Equation (54) takes the form Z Z [ 3 3  ∆$ = ζ d k |pijihpij| = ζ d k |p+ihp+| + |p×ihp×| (61) k

where Z −3/2 |piji = |piji(k) = p◦ dΩp Aij|pi (62)

p=p◦−k

with the dimensionless TT amplitude

Aij = Aij(k, Ωp) 3/2 p◦ p Pijkl(k) n 1 h p p p p i o = √ 0k 0l + k l − ( → ) : 2 2 2 2 2 2 e δ . k |p◦ − p − k| + e |p◦ − k| − (p − ie) |p + k| − (p◦ + ie) p=p◦−k

There are two orthogonal parts of Aij:

+ × Aij = Aij + Aij

corresponding to the two (+ and ×) polarisations of the gravitational waves [31], so that the gravitational wave square amplitude decomposes accordingly as

2 ∗ + +∗ × ×∗ |A| := Aij Aij = Aij Aij + Aij Aij . (63)

For numerical evaluations, we can conveniently choose p◦ = p◦zˆ, then we have

p p3/2 n 1 h p2δ δ p p i o = ◦√ ◦ 3k 3l + k l − ( → ) Aij 2 2 2 2 2 2 e δ . k |p◦ − p − k| + e |p◦ − k| − (p − ie) |p + k| − (p◦ + ie) p=p◦−k Symmetry 2021, 13, 852 12 of 16

For example, in the limit of a point source of gravity with effective radius 1/δ → 0, Equation (63) yields

2 3  4 2 p◦ p p◦P3333(k) |A|e = k [|p − p − k|2 + e2]2 2 2 2 ◦ |p◦ − k| − (p − ie) 2 2 4  2 p◦ p P33ij(k)pˆi pˆj p Pijkl(k) pˆi pˆj pˆk pˆl +   + (64) < (| − |2 − ( + )2)(| + |2 − ( + )2) 2 2 2 p◦ k p ie p k p◦ ie |p + k| − (p◦ + ie) p=p◦−k

where

2 2 2 2 2 2 2 2 2 |p◦ − k| − (p − ie) = (|p◦ − k| − p + e ) + 4e p ,  2 2 2 2  2 2 2 2 2 2 2 < (|p◦ − k| − (p + ie) )(|p + k| − (p◦ + ie) ) = (|p◦ − k| − p + e )(|p + k| − p◦ + e ) − 4e p◦ p, 2 2 2 2 2 2 2 2 2 |p + k| − (p◦ + ie) = (|p + k| − p◦ + e ) + 4e p◦,

and 1 P (k) = (1 − kˆ2)2, 3333 2 3 1 P (k)pˆ pˆ = pˆ2 − 2(kˆ pˆ )kˆ pˆ + (kˆ pˆ )2(kˆ2 + 1) + kˆ2 − 1, 33ij i j 3 i i 3 3 2 i i 3 3  ˆ 22 Pijkl(k)pˆi pˆj pˆk pˆl = 1 − (ki pˆi) .

2 2 It is also useful to introduce |A|δ := |A|e→δ. For simplicity, let us adopt as a reasonable first order-of-magnitude estimate of the gravitational wave square amplitude for a source of gravity with an effective range 1/e and radius 1/δ to be

2 2 2 |A|e,δ := |A|e − |A|δ. (65)

The total dissipated outgoing energy corresponding to Equations (52) and (61) then follows as 2ζ Z Z = 3 | |2 ( ) ∆E 3 d k dΩp hk¯ A e,δ k, Ωp p◦ 2ζh¯ Z p◦ Z Z = 3 | |2 ( ) 3 dk k dΩk dΩp A e,δ k, Ωp . (66) p◦ 0

To obtain an expression in term of dimensionless quantities, we express p◦, p, k, e as dimensionless quantities in units of p◦. Then, by replacing p◦ → p◦ p◦, p → p◦ p, k → p◦k, e → p◦e, δ → p◦δ and using Equation (50) with the mass–energy of the gravity source E? = M?, we arrive at the fractional energy loss µ := ∆E/E◦ = ∆E/(hp¯ ◦) given by

4  S S  µ = h¯ G3E2 p4 e − δ π5 ? ◦ e δ 4 E2E4  S S  = ? ◦ e − δ 5 6 (67) π EP e δ

where EP is the Planck energy and

Z 1 Z Z 3 2 Se = e dk k dΩk dΩp |A|e(k, Ωk, Ωp) 0 Z 1 Z Z Z Z = dk dθk dφk dθp dφp Fe(k, θk, φk, θp, φp) (68) 0 Symmetry 2021, 13, 852 13 of 16

2 with |A|e obtained from Equation (64) for p◦ → 1 and

Fe(k, θk, φk, θp, φp) ek2 p3 sin θ sin θ  1 (1 − kˆ2)2 = k p 2 3 2 2 2 2 2 2 2 2 2 4 [|p◦ − p − k| + e ] (|p◦ − k| − p ) + 2e (|p◦ − k| + p ) + e h i 2 2 ˆ ˆ 1  ˆ 2 ˆ2 ˆ2  4 22 2 p pˆ3 − 2(ki pˆi)k3 pˆ3 + 2 (ki pˆi) (k3 + 1) + k3 − 1 p 1 − (kˆ pˆ )  + + i i 2 2 2 2 2 2 2 2 2 2 4 . (69) (|p◦ − k| − p + e )(|p + k| − 1 + e ) − 4e p (|p + k| − 1) + 2e (|p + k| + 1) + e p=1−k

We also have analogous constructions for Sδ and Fδ. Numerical evaluations of Fe and Fδ as functions of (k, θk, φk, θp, φp) show sharp but finite peaks around small k, θk and θp for small e, δ. This leads to finite numerical integrations with Se ≈ Sδ ≈ 100 for e, δ  1. Therefore, for a gravity source with a characteristic radius 1/δ and effective range 1/e in units of 1/p◦, from Equation (67) we have

E2E4  1 1  = ? ◦ − µ 6 . (70) EP e δ approximately, corresponding to the rough spread of the photon impact parameter to be from the surface of the gravitational lens to one radius from the surface. 2 4 6 The energy loss rate above can be interpreted µ ≈ Γ τ where Γ = E?E◦/EP is the effective emission transition rate by the photon, and τ = 1/e − 1/δ is the effective interaction time between the photon and the lensing mass M? with the corresponding effective interaction range Rrng = 1/e. See Figure1. This is consistent with the long travel time or distance assumption stated in Section4. In full physical units, Equation (70) becomes

3 h¯ G   2 5 µ = R − R? M ω (71) c12 rng ? ◦ where the speed of light c has now been reinstated.

6. Conclusions and Discussion Based on the gravitational quantum vacuum, which has recently been shown to lead to gravitational decoherence [8] and gravitational spontaneous radiation that recovers the well-established quadrupole radiation [10], in this paper we have presented, to our knowledge, the first approach to the spontaneous bremsstrahlung of light due to the combined effects of gravitational lensing and spacetime fluctuations. Our present work yields a new quantum gravitational mechanism whereby starlight emits soft gravitons and becomes partially redshifted. This effect may contribute to the stochastic gravitational wave background [11,33,34]. We also note that while the term (53) for the outgoing light does not undergo photon to graviton energy conversion, it exhibits a type of recoherence of photons [35]. Our work naturally raises the prospect of potential detection of the released stochastic gravitational waves. Addressing this question in detail requires a further investigation, which is currently underway by the authors. It is however of interest at the present stage to envisage a plausible observation scenario. To this end, let us take the well studied strong X-ray binary Cygnus X-1 [36] because this system has both a large gravitational field and a strong X-ray source similar to that illustrated in Figure1. However, both the compact object/black hole and the companion supergiant star in the Cygnus X-1 binary system are massive, with a total mass M? ≈ 40M and an orbiting radius ≈ 20R , which we will take as the effective R?. This would make Rrng to be in the region of 100R and so from the discussions in Section5, the spontaneously emitted stochastic gravitational waves would have a mean in the same region having a mean frequency f ∼ 0.01 Hz. Using these parameters, the fractional energy loss µ in Equation (71) is plotted in Figure2 Symmetry 2021, 13, 852 14 of 16

with the effective gravity range parameter d := (Rrng − R?)/R? chosen to be 0 ≤ d ≤ 4, corresponding to the rough spread of the photon impact parameter to be from the surface of the gravitational lens to Rrng ≈ 100R discussed above. The detection of sub-Hz gravitational waves is a unique strength of the proposed LISA mission, which is expected to reach a corresponding characteristic strain sensitivity close to h ∼ 10−22 [37], as shown in Figure3. 37 −1 From the X-ray luminosity of Cygnus X-1 LX ≈ 4 × 10 erg s [38,39] and its distance D ≈ 6100 light years to the Earth, one gets the arriving X-ray energy flux to be 2 −7 −2 −1 SX = LX/(4πD ) ≈ 10 erg cm s . Let us suppose that the total gravitational wave luminosity of Cygnus X-1 arises from LGW = µeffLX, in terms of an effective photon-to- graviton energy transfer rate 0 < µeff < 1. We can then estimate the characteristic strain h 2 2 2 of the gravitational waves with energy density expression UGW = c ω h /(32πG) where 2 2 2 h = h+ + h× with ω = 2π f and the energy flux SGW = c UGW so that SGW = µeffSX. −10 −2 −1 For example, if µeff = 0.1% and f = 0.01 Hz, then SGW ≈ 10 erg cm s yielding h ≈ 10−22.

Figure 2. The fractional energy loss µ given by Equation (71) as a function of the photon frequency f◦ (Hz) in logarithmic scale along the horizontal axis evaluated with a chosen range of the effective gravity range parameter d = (Rrng − R?)/R? for the X-ray binary Cygnus X-1. Above, log µ is plotted in (a) as a projected height onto the (log f◦, d) plane and in (b) as a line for three representative values of d. It is obvious that as the photon frequency f◦ increases from the optical spectrum, the 17 photon-to-graviton energy conversion “chips in” at around f◦ = 3 × 10 Hz at the start of the soft X-ray spectrum, so that µ ∼ 0.01 for d = 4 corresponding to Rrng ∼ 100R . Although here µ appears to carry on increasing with f◦, the validity of Equation (71) is strictly limited to small µ due to the Born approximation with weak interactions we have assumed in Equation (21) used to derive µ in Equation (71). Nonetheless, the ascending trend of µ with the photon frequency f◦ suggests that starting from the soft X-ray , the gravitational bremsstrahlung of light in the vicinity of Cygnus X-1 may be important.

As shown in Figure3, we choose moderate effective transfer rate values 0.01% ≤ µeff ≤ 1% −5 −3 across the gravitational wave frequencies 10 Hz ≤ f ≤ 10 Hz. Indeed, for µeff & 10 at around f ∼ 0.01 Hz, the characteristic strain of the bremsstrahlung gravitational waves from Cygnus X-1 could be above the sensitivity level of LISA and hence potentially de- tectable. Furthermore, the buildup of similar sources could also contribute to an overall stochastic gravitational wave background. Based on the initial results and estimates re- Symmetry 2021, 13, 852 15 of 16

ported in this work, we plan to analyse and quantify the properties of the bremsstrahlung gravitational waves from intense astronomical sources of light and gravity in a more realistic setting for future publication.

Figure 3. Estimated characteristic strain h of the gravitational waves with effective photon-to-graviton

energy transfer rate µeff for Cygnus X-1 with 0.01% ≤ µeff ≤ 1% against the gravitational wave frequency f in Hz compared with the LISA characteristic strain sensitivity curve [37].

Author Contributions: Conceptualisation, C.H.-T.W.; methodology, C.H.-T.W. and M.M.; software, C.H.-T.W. and M.M.; validation, C.H.-T.W. and M.M.; formal analysis, C.H.-T.W.; investigation, C.H.-T.W.; resources, C.H.-T.W. and M.M.; data curation, C.H.-T.W. and M.M.; writing—original draft preparation, C.H.-T.W.; writing—review and editing, C.H.-T.W. and M.M.; visualisation, C.H.-T.W. and M.M.; supervision, C.H.-T.W.; project administration, C.H.-T.W.; funding acquisition, C.H.-T.W. and M.M. Both authors have read and agreed to the published version of the manuscript. Funding: C.W. was funded by the Cruickshank Trust and M.M. was funded by the Student Awards Agency Scotland (SAAS). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: The data presented in this study are included in Figures2 and3. Acknowledgments: C.W. wishes to thank John S. Reid for many years of stimulating discussions on the topics and background of this work. Conflicts of Interest: The authors declare no conflict of interest.

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