Effect of a cosmological constant on propagation of vacuum polarized photons in stationary spacetimes

Consideration of vacuum polarization in quantum electrodynamics may affect the momentum dispersion relation for photons for a non-trivial background, due to appearance of curvature dependent terms in the effective action. We investigate the effect of a positive cosmological constant $\Lambda$ on this at one loop order for stationary $\Lambda$-vacuum spacetimes. To the best of our knowledge, so far it has been shown that $\Lambda$ affects the propagation in a time dependent black hole spacetime. Here we consider the static de Sitter cosmic string and the Kerr-de Sitter spacetime to show that there can be some non-vanishing effect due to $\Lambda$ for physical polarizations. Consistency of these results with the polarization sum rule is discussed.


A. Introduction
If we consider quantum electrodynamics in curved spacetime, the effective action for the photon contains from the loop integrals some finite part which depends on both the field strength as well as the background curvature. Such finite parts originate from the curvature dependence of the propagator in curved spacetime. In that case the photon equation of motion gets modified and consequently the dispersion relation for null geodesics may no longer hold due to the gravitational tidal forces, as was first shown in [1].
The velocity shifts of photons were investigated in various black hole backgrounds [3,5,7,8]. For various aspects associated with this phenomenon including the preservation of unitarity and causality, we refer our reader to [2,6,9].
The chief concern of this brief report is to investigate the role of a positive cosmological constant Λ in this phenomenon. The physical motivation of this study comes from the observation of accelerated expansion of our universe [19,20]. It was shown in [1] that there is no modification of photon's speed in any maximally symmetric space like de Sitter. In [8] various stationary and time dependent black holes with Λ were considered and was shown that although the velocity of the vacuum polarized photons gets modified, Λ contributes to this only in the time dependent case. Therefore, it is interesting to see whether we may find some non-trivial effect due to Λ in some physically interesting stationary spacetimes.
In particular, we shall consider below two Λ > 0 spacetimes -the static Nielsen-Olesen de Sitter cosmic string [21,22] and the Kerr-de Sitter [23] to show that we may have non-vanishing effects due to Λ, for physical or transverse polarizations.
B. Calculation of the velocity shifts The basic calculational scheme uses the 1-loop corrected effective equation of motion for photons in a weakly curved space for QED [1,6,8], where m e is the electron mass and a = − 5α 720π , b = 26α 720π and c = − 2α 720π , and α is the fine structure constant. Using eikonal approximation Eq. (1) can be cast in the leading order [1], where k a is the photon momentum, k 2 = k a k a and a b is the polarization vector, a · k = 0. We put above R ab = Λg ab for Λ-vacuum Einstein spaces. The above equation is the leading departure from the null geodesic approximation. We shall solve Eq. (2) for physical or transverse polarizations. Let us start with the static cylindrical Nielsen-Olesen cosmic string spacetimes [21,22], the exterior of which is given by where δ is a constant representing the conical singularity. The inside core metric for the Nielsen-Olesen configuration near the axis looks formally the same as above, with δ = 1, and Λ replaced with an effective cosmological constant Λ ′ = Λ + 2πλη 4 [22], where λ and η are respectively the coupling constant for quartic self interaction and the expectation value of the complex scalar field in its false vacuum. The above spacetime has a curvature singularity at ρ = π √ 3Λ [21]. Since Eq. (2) is valid only in the weak curvature limit, the following calculations will not hold near that singularity.
We choose orthonormal basis : Following e.g. [8], we next define tensors : With the help of these tensors, we express the components of the Riemann tensor as (see e.g. Chapter 6 of [24], for expressions in generic stationary axisymmetric spacetimes), where Since the conical singularity term δ in Eq. (3) does not contribute to the curvature, it is clear that the above expressions remain formally unchanged near the axis of the string, with Λ replaced with Λ ′ = Λ + 2πλη 4 .
Also, we define the individual momenta through U µν ab : We contract Eq. (2) with a vector field v b , where v b at a time corresponds to any one of the above momenta and using Eq.s (4), (5) we find We start by considering the radial photon motion, k ≡ {k 0 , 0, k 2 , 0} in Eq.s (5) and The two physical polarization vectors for radial photons correspond to the azimuthal and axial directions. For radial photons we have l ≡ {0, k 0 , 0, 0} in Eq. (5). This means a · l = 0 and we take v = l in Eq. (6). Also we have in this case p = − k2 k0 l and l · m = l · n = l · q = l · r = 0. Putting these all in together, we get Putting in the expressions of E and F (see after Eq. (4)), we get, since a · l = 0, which clearly shows that the null geodesic dispersion relation is violated due to Λ via quantum effect, and since c = − 2α 720π , it is superluminal.
The other physical polarization for radial photons is along the z-direction. From Eq. (5), in this case a · n = 0, q = k2 k0 n and we take v = n in Eq. (6). Also we have n · l = n · m = n · p = n · r = 0. Putting these all in together, we solve Eq. (6) as earlier to get which is subluminal. Next, let us consider the azimuthal or orbital photons, k ≡ {k 0 , k 1 , 0, 0}, and k 2 = k 2 0 − k 2 1 = −l 2 , from Eq.s (5). The two physical polarization states correspond to the radial and the axial directions. For polarization along the radial direction, we have a · m = 0, and we take v = m in Eq. (6). We have m · l = m · n = m · q = m · r = 0, and p = k1 k0 m. Then proceding as earlier, we evaluate Eq. (6) to find Likewise, for the polarization along the z-direction for the orbital photons, we have a · n = 0, and r = − k1 k0 n and the other inner products of n with the vectors (l, m, p, q) vanish. Accordingly, taking v = n in Eq. (6), we find Finally, we come to the photons moving in the axial or z-directions, k ≡ {k 0 , 0, 0, k 3 }, and k 2 = k 2 0 − k 2 3 = −n 2 , in Eq. (5). For physical polarization along the radial di-rection, we have a·m = 0 and q = − k3 k0 m. Then choosing v = m in Eq. (6), and noting m · p = m · n = m · l = m · r = 0, we proceed as earlier to get k 2 = 0, i.e. the null geodesic dispersion relation. For polarization along φ, we choose v = l, also giving k 2 = 0.
For a Nielsen-Olesen cosmic string spacetime, as we mentioned earlier, sufficiently near the axis, Λ can be replaced with Λ ′ = Λ + 2πλη 4 , which is due to the false vacuum of the complex scalar field. Then sufficiently inside the core, the nontrivial results of Eq.s (8)-(11) correspond to replacing Λ with Λ ′ .
Let us now come to the Kerr-de Sitter spacetime [23]. We shall work in a region far away from the Schwarzschild radius, where we can only retain term linear in the rotation parameter a. Also, owing to the observed tiny value of Λ, we can safely ignore Λa 2 terms for practical purposes. This gives where f (r) = 1 − 2M r − Λr 2

3
, and ω(r) = a 2M r 3 + Λ 3 . This can be thought of as the leading axisymmetric deformation of the Schwarzschild-de Sitter spacetime due to rotation.
For polarization along the polar direction for the radial photons, a·n = 0, and q = k2 k0 n in Eq. (5). We take v = n in Eq. (14). Procceding as above we get k 2 = 0. We note here that retaining generic rotation gives a velocity shift of at least O(a 2 ) [5] for a Kerr black hole, but we are working only with terms linear in the rotation parameter.
For orbital photons, k ≡ {k 0 , k 1 , 0, 0}, k 2 = −l 2 in Eq. (5). For polarization along radial direction, we have a · m = 0, and p = k1 k0 m. We take v = m in Eq. (14). Using m · m = −k 2 0 and p · m = −k 0 k 1 , Eq. (14) reduces to Since we are working at one-loop order, whenever the momenta are multiplied with the coefficients b or c, we can take the null geodesic dispersion relation. This means for the last term in the above equation, we can take k 0 = ±k 1 , where +(−) sign corresponds to prograde (retrograde) orbits respectively. Then we find using the explicit expressions for A, B and C (see below Eq. (13)), where +(−) sign corresponds to prograde (retrograde) orbits respectively. For orbital photons with polarization along θ, we have a · n = 0, r = − k1 k0 n in Eq. (5) and we take v = n in Eq. (14). Following similar method as earlier we find Finally, we come to photons moving along the polar angle, k ≡ {k 0 , 0, 0, k 3 }, and k 2 = −n 2 . We also have in Eq. (5), q = − k3 k0 m and r = k3 k0 l. For polarization along the radial direction, we have a · m = 0, and we take v = m in Eq. (14). Likewise, for polarization along azimuthal direction, we set v = l in Eq. (14). Then proceeding as earlier we find This completes the calculation part for velocity shifts.
We note that if we set M = 0 in Eq.s (17)- (19), we recover the vanishing result of the empty de Sitter space. If we set Λ = 0, we recover the result of the Schwarzschild spacetime [1].
Let us now check the consistency of our results with the sum rule over the physical polarizations. This states that, the averaged sum of the photon's velocity shift for physical polarization equals − (4b+8c) m 2 e R ab k a k b [6]. We put R ab = Λg ab . At one loop order we may take k 2 = 0 on the right hand side. This means on the average there is no velocity shift for photons in Λ-vacuum spacetimes. This corresponds to the earlier identically vanishing results [1,8]. For our present case any of our non-vanishing results comes like 1 ± ǫ for any set of two physical polarizations (Eq.s (8)- (11), and Eq.s (17)- (19)). Hence our result is consistent with the polarization sum rule.
It is clear that for any other Λ-vacuum spaces, the velocity shift, if it exists, should always be of the form 1 ± ǫ, for any set of two physical polarizations.
Let us summarize the results now. We have considered two stationary Λ-vacuum spacetimes -the de Sitter cylindrical cosmic string and the Kerr-de Sitter. For the cosmic string spacetime, we have found shift for radial and orbital photons (Eq.s (8)-(11)), but no shift for axial photons. The vanishing of the velocity shift along the axis can be understood as the Lorentz symmetry in the 't − z' plane of the cosmic string spacetime (Eq. (3)).
For the Kerr-de Sitter universe, we have only retained terms linear in the rotation parameter a. For the azimuthal photons, we have found non-vanishing contribution from Λ (Eq.s (17), (18).
We note that for the two different spacetimes, Λ contributes to the velocity shift in two very different ways. However, all of them are consistent with the polarization sum rule.