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Constraints on the non-minimal coupling of electromagnetic field from astrophysical observations

Non-minimal coupling between the Riemann curvature and the electromagnetic field appears as quantum corrections when gravity is coupled to the standard model of particle physics. The non-minimal coupling is expected to be dominant in the strong-gravity regions such as near black-holes or the early Universe. With better instruments planned shortly, electromagnetic fields are an important source of astrophysical observations to test general relativity in strong gravity regimes. However, to precisely test general relativity in strong gravity using electromagnetic fields, it is \emph{imperative} to obtain constraints on the non-minimal coupling parameter to electromagnetic fields. As a step in this direction, we calculate the deflection angle of non-minimally coupled electromagnetic fields in the vicinity of a dynamical, spherically symmetric black hole described by the Sultana-Dyer metric. We compare the deflection angle of the photon modes for the Sultana-Dyer black hole with the Schwarzschild black hole. We show that the difference in the deflection angle for the Schwarzschild black hole is always negative, while for Sultana-Dyer is always positive. Thus, our analysis points out that the two black holes provide a distinct signature irrespective of the black hole mass. We discuss the implications of the results for future astrophysical observations.


I. INTRODUCTION
The idea of the minimal connection between the metric and matter underpins general relativity. To obtain equations of matter fields that govern physical phenomena in a gravitational field, we have to resort to the Principle of Equivalence [1,2]. The principle demands that the laws of physics in a gravitational field reduce to those of Special Relativity in the infinitesimally inertial frames. These frames are characterized by the property that the metric can be approximated by the Minkowski metric and neglecting the connection coefficients.
In flat space-time, such properties hold exactly throughout the space-time.
In the presence of a gravitational field, they can be satisfied to any desired degree of accuracy by choosing a reference frame in free fall and restricting attention to a sufficiently small subset of space-time. The condition of reducing to the laws of special relativity in the infinitesimally inertial frames puts strong restrictions on the coupling of matter to gravity.
Usually, one uses the principle of minimal coupling: the matter does not couple directly to curvature such that, at any event, we can always transform away from the influence of a gravitational field on matter [1][2][3][4]. This is an expression of the strong equivalence principle. More specifically, this means that the partial derivatives of the energy-momentum tensor are locally promoted to covariant ones (∂ µ T µν → ∇ µ T µν ), resulting in a covariant conservation equation. This assumption is required to derive the field equations [3]. The (strong) equivalence principle has been tested accurately in the solar system or weak gravity limit [5,6].
However, over the last two decades, it has been realized that while the minimal coupling is a good approximation; this need not be the scenario in strong gravity and cosmological distances. The energy-momentum tensor can no longer be conserved covariantly, resulting in an additional force in the geodesics equation that can account for the dark matter influence on galaxies [7]. It has been shown in literature that the presence of non-minimal coupling of the inflaton field ensures flatness of scalar potential in the Einstein frame at large values of the Higgs field [8,9].
Theoretically, it has been seen that any scheme unifying fundamental interactions with gravity leads to effective actions where non-minimal couplings to the geometry are present.
Higher-order curvature invariants are present in general. They emerge as loop corrections in high-curvature regimes, and extensions to general relativity are necessary [10,11]. Recently, Ruhdorfer et al did a systematic analysis and obtained an effective field theory of gravity coupled to the Standard Model of particle physics [12]. They showed that (i) the first gravity operators appear at mass dimension 6 in the series expansion, and these operators only couple to the standard model Bosons, (ii) no new gravity operators appear at mass dimension 7, (iii) in mass dimension 8 the standard model Fermions appear, and (iii) coupling between the scalar (Higgs) field and the standard model gauge Bosons appear only at mass dimension 8.
Interestingly, the terms obtained by these authors are similar to the non-minimal coupling terms to electromagnetic fields proposed by Prasanna a long time ago [13].
One cannot fix the value of the non-minimal coupling constant to the electromagnetic (EM) fields theoretically within the standard model. As mentioned earlier, matter fields with minimal coupling are well tested in weak gravity limit and not in strong gravity regions such as near black holes [14][15][16][17]. With better instruments planned shortly, EM fields are an important source of astrophysical observations to test General Relativity (GR) in strong gravity regimes. However, to precisely test GR in strong gravity using EM fields, it is imperative to obtain constraints on the non-minimal coupling parameter (λ) to EM fields.
Prasanna studied the non-minimal coupling of EM fields in curved space-times [13]. Various authors have proposed different models of non-minimal coupling (See Refs. [18][19][20][21][22]). The effects of vacuum polarization and signature of photon propagation due to non-minimal coupling were first studied considering a Schwarzschild background in Ref. [23]. Similar work for a static black hole with non-zero electric charge described by the Reissner-Nordström metric and for a neutral rotating black hole described by Kerr metric has been done respectively in Refs. [24,25]. For other related works, see Refs. [26,27].
In this work, we study the propagation of photons that are non-minimally coupled to gravity. To do so, we calculate the deflection angle of a photon in the vicinity of a dynamical, spherically symmetric black hole described by the Sultana Dyer metric. To our knowledge, the detailed evaluation of the deflection angle in a black-hole space-time to constrain the non-minimal coupling parameter is new. In Ref. [28], the authors used photon dispersion relation and computed the photon's arrival time in Schwarzschild space-time. Considering signals from radar ranging past the Sun, the authors found λ ∼ 1.1 × 10 20 cm 2 , which is about three orders of magnitude more stringent than the one obtained in Ref. [29]. We show that Sultana-Dyer solar-mass black hole improves the bound on λ.
Using the Eikonal approximation, we obtain the dispersion relations for the two modes of polarization of electromagnetic fields in Schwarzschild and Sultana-Dyer space-times. We show that the two dispersion relations corresponding to the two polarizations differ from Schwarzschild and Kerr [cf. Eq. (29)]. In the case of Schwarzschild and Kerr, the two dispersions are quadratic, and the modifications to the dispersion relation occur in the p (3) component of the momentum. However, in the case of Sultana-Dyer, the modification to the dispersion relation [cf. Eq. (35)] occurs in all the components of the momentum. Further, one of the dispersion relations is of cubic order.
We show that the deflection angle for the Sultana-Dyer black hole is inversely proportional to β 0 (which is related to horizon radius r H ). However, in the case of Schwarzschild, the deflection angle is proportional to β 0 . Further, we show that the difference in the deflection angle for the Schwarzschild black hole is always negative, while for Sultana-Dyer is always positive. Thus, our analysis points out that the two black holes provide a distinct signature with the deflection angle irrespective of the black-hole mass.
In Sec. (II), we discuss the non-minimal coupling model of the electromagnetic fields with gravity and briefly discuss the procedure to obtain the photon dispersion relation in arbitrary background. In Sec. (III), we briefly discuss the properties of the Sultana-Dyer black hole and the conserved quantities in this space-times. In Sec. (IV), we obtain the modified dispersion relations in the local inertial frame for the Schwarzschild and Sultana-Dyer black holes. In Sec. (V), we evaluate the difference in the deflection angles for the two polarization modes at the radius of photon sphere for the Schwarzschild and Sultana-Dyer black holes. We also obtain the constraint on non-minimal coupling constant λ. In Sec.
(VI), we obtain the quantitative difference in the deflection angle for Sultana-Dyer black hole from Schwarzschild. In Sec. (VII), we conclude by discussing the implications of our work for future observations. Appendices (A) -(F) contain detailed calculations.
In this work, we use (−, +, +, +) signature for the 4-D space-time metric. Greek alphabets denote the 4-dimensional space-time coordinates, and Latin alphabets denote the 3-dimensional spatial coordinates.

II. THE MODEL
The most general model of non-minimal coupling of EM fields with gravity consists coupling of EM field tensor with the Riemann and Ricci tensors as well as with the Ricci scalar.

4
The model is described by the following action: where, The above action is CPT invariant and invariant under general coordinate transformations; however, it violates Einstein's equivalence principle [28]. In this work, we shall focus on a simpler model of non-minimal coupling, which we obtain by setting, q 1 = q 2 = 0 in Eq. (2) and the action reduces to: The reason for this choice is to compare the constraints obtained by earlier authors for time-independent spherically symmetric black holes [28] with the Sultana-Dyer black hole.
Varying the action (3) with respect to the gauge field A µ , we get: Using the Bianchi identity and symmetries of the Riemann curvature tensor in Eq. (4), we obtain the following relation: The RHS of the above equation is a coordinate-dependent quantity and lead to a modified dispersion relation. In the case of Maxwell's theory, we will have the same dispersion relation (p µ p µ = 0) for both polarization modes. The first term in the RHS modifies the kinetic term, while the second term acts as a mass term. As the coupling terms are linearly related to EM fields, the non-minimal coupling leads to mixing of the two polarization modes.
For static black holes, like Schwarzschild (de Sitter) or Kerr (de Sitter) metric, that satisfy R µν ∝ g µν , the second term in the RHS of the above expression vanishes. However, for black holes in an expanding universe, like Sultana-Dyer metric, R µν = Λg µν . Hence, we retain the last term in the RHS of the above expression.

A. Photon dispersion relation
To obtain the trajectory of photons, choosing a locally flat inertial frame is convenient.
Hence, we define a local inertial frame attached to the non-inertial one. Let p (µ) correspond to the photon momentum in the local inertial frame. This is related to momentum in noninertial frame (p ν ) by the relation: p ν = e ν (µ) p (µ) , where, e ν (µ) is the tetrad connecting the two -inertial and non-inertial -frames.
In the case of time-dependent black holes we consider, the variations of the curvature along the radial coordinate (R) and the time (T ) are long compared to the EM waves reduced wavelength and period, i.e.
Under the Eikonal approximation, we can locally treat the EM waves as planar and monochromatic [28]: Thus, in the local inertial frame, the Bianchi identity of the EM field tensor and Eq. (5) reduces to: Fixing the space-time index µ to the 3-space j in Eq. (8) leads to: Setting µ → j, ν → k and α = 0 -the Bianchi identity in local inertial frame (7) becomes: Combining the above two equations leads to the following: Substituting Eq. (8) into the above equation, the equation of motion in the Eikonal approximation is: where, For different space-times, the above equation of motion (12) will yield different dispersion relations and can potentially provide constraints on λ.
The constraints on λ are obtained in the literature by considering static black hole spacetimes, like Schwarzschild and Kerr. In this work, we obtain constraints on λ by considering a spherically symmetric black hole in cosmological space-time, commonly referred to as Sultana-Dyer metric [30]. To our knowledge, such an analysis has not been done, and we show that this provides a stringent constraint on the non-minimal coupling constant λ.

III. SULTANA-DYER BLACK HOLE
Sultana and Dyer obtained an exact spherically symmetric black hole solution in expanding cosmological space-time [30]: where M is mass of the non-rotating black hole and the conformal factor a(η) = (η/η 0 ) 2 .
The conformal time η is related to the comoving time t via the relation dt = dη a(η). We We list below some of the properties of the Sultana-Dyer space-time: 1. The above line-element corresponds to a black hole of mass M in a spatially flat FLRW universe with scale factor a(t) ∝ t 2/3 . This corresponds to black holes in the matter-dominated epoch.

Sultana-Dyer is sourced by two non-interacting perfect fluids -null dust and normal
dust. The stress-energy tensor is T µν = T µν = ρ o u µ u ν describes the normal dust and T (2) µν = ρ n k µ k ν describes a null dust with density ρ n and k α k α = 0.
3. r = 2GM ≡ r H remains an event horizon because the conformal transformation preserves the causal structure. The Kretschmann scalar for the line element (14) is: It implies curvature singularities occur at η = 0 and r = 0. The singularity at η = 0 is spacelike for r > r H , timelike for r < r H and null for r = r H [31]. The singularity at r = 0 is spacelike and surrounded by the event horizon.
4. The energy density of the dust is positive only in the region At the event-horizon r = r H , the energy conditions are satisfied everywhere for η < 2r H .
In other words, after this time η, those particles closest to the event horizon become superluminal [30]. As we will see, this condition translates to a constraint on λ. (14) in non-geometrized unit leads to:

Writing the line element
Substituting the following dimensionless variables: in the Sultana-Dyer metric (17), we have where H 0 is Hubble constant, In the rest of the work, we will use the dimensionless line-element ds defined in Eq. (20).
This choice is suitable because the affine parameter (τ ) and conserved quantities are dimen-

sionless.
A. Conserved quantities from symmetries of the metric Although Sultana-Dyer metric (20) is time-dependent, the space-time has conserved quantities. Besides the spherical symmetry, the metric is invariant under conformal transformations. Taking into account the conformal symmetry; the following quantity is conserved w.r.t dimensionless affine parameter τ [30,31]: Due to the azimuthal symmetry, the following quantity is conserved [30]: In Appendix. (A), we obtain the relation between the above quantities with the conserved quantities for the dimension-full coordinates. In Sec. (V), we show that the dispersion relation depends on the value ofL/Ẽ. In the rest of this work, we obtain the change in photon propagation in the Sultana-Dyer space-time due to the presence of non-minimal coupling (3). We also compare our results with Schwarzschild space-time.

IV. MODIFIED DISPERSION RELATIONS
The non-minimal coupling of the electromagnetic fields with gravity leads to modified photon dispersion relation (12). This change will be significant near strongly gravitating objects like black holes. As we show in this section, additionally, the nature of the dispersion relation is different for time-independent (like Schwarzschild, Kerr) and time-dependent black hole (like Sultana-Dyer) space-times.
To derive the photon dispersion relation we fix the photon momentum: ) -confined in the orbital plane of the gravitating object. Imposing the transverse condition we choose the electric field to be E (1) , E (2) , i.e., in the normal plane to the gravitating body [28].

A. Time-independent black hole space-times
For the Schwarzschild space-time: using dimensionless coordinatesr = r/r H ,t = H 0 t and the quantity β 0 (21), we get, where, Since the line-element is independent oft and φ, we have the following invariant quantities The equation of motion (12) for the two electric field components E (1) , E (2) in the line-element (26) are: Note that in this case, the second and third terms in the LHS of Eq. (12) vanish because the Ricci tensor is zero. In order for the above equation to satisfy for arbitrary values of [E (1) , E (2) ], the LHS reduces to [28]:

Hr
To do that, we construct a local inertial frame corresponding to the Sultana-Dyer metric (20). Appendix (C) contains the details of the tetrads and the components of the Riemann tensor in the local frame. Unlike Schwarzschild and Kerr black holes, the Ricci tensor is non-zero for Sultana-Dyer black hole. Hence, in principle, the last two terms in the LHS of Eq. (12) will contribute to the dispersion relation. However, as we show in Appendix (D) the terms containing the Ricci tensor are negligible compared to the term containing Riemann tensor. Hence, in the rest of the analysis, we will not include the Ricci tensor terms.
Substituting the components of the Riemann tensor in the local frame (C4) in Eq. (12), the equations of motion for the two electric field components [E (1) , E (2) ] are: where, To satisfy Eq. (30) for arbitrary values of [E (1) , E (2) ] the L.H.S leads to the following conditions for the dispersion relation: Using the tetrad relations in Eq. (C2), we obtain the following two dispersion relations in the non-inertial frame: where, and and f i 's are defined above in Eq. In the next section, we calculate the deflection angle of the photon near the black hole.
We show that the deflection angle of the photon near Schwarzschild is linearly proportional to β 0 b; however, in the case of Sultana-Dyer, the deflection angle has a different dependence. We then compare the results between Schwarzschild and Sultana-Dyer black holes for different mass ranges.

V. DEFLECTION ANGLE
Potentially, the modification to the path taken by the photon leads to the following two observable quantities:

A. Schwarzschild space-time
To obtain photon-deflection angle, we need to rewrite the dispersion relation (29) in the non-inertial frame. To do this, substituting the tetrads for the line-element (26) in the dispersion relation (29) leads to: where, Solving p 1 in Eq. (37), we have: Dividing the above expression over dφ/dτ , the deflection angle dφ/dr is: Rewriting the LHS of the above expression in terms of r, we have: where, b =L Sch /Ẽ Sch . Note that −(+) sign corresponds to the deflection angle for the +(−) mode of polarization.
The modification to the dispersion relation is the largest close to the horizon and is negligible, very far from the horizon. We evaluate the difference between the deflection angles for two polarization modes at the radius of photon sphere (r = 3/2), i. e.: dφ dr This is the second important result regarding which we would like to stress the following points: First, the deflection angle depends on four parameters;λ, β 0 , r H and b. Interestingly deflection angle is linearly proportional to β 0 b. However, as we will show in Sec. (V) in the case of Sultana-Dyer space-time, the deflection angle has different β 0 b dependence. Hence, leading to potential observational signatures. Second, the deflection angle will be real if and only if, This implies thatλ, β 0 and b can not be arbitrary. However, we can not break the degeneracy of the choice of these values only using Schwarzschild space-time. We will be able to do this using constraint from Sultana-Dyer. We discuss the implications of this in detail in Sec.
(VI). We show that β 0 b < 1.8 for the range of acceptable values ofλ.

B. Sultana-Dyer black hole
In Sec. (IV B), we obtained quadratic and cubic dispersion relations (34,35) for the non-minimally coupled electromagnetic field in Sultana-Dyer black hole. In the rest of this section, we obtain the deflection angle for the two polarization modes.

Quadratic dispersion relation (34)
Like in the case of Schwarzschild, the derivation of deflection angle dφ dr requires prior knowledge of p 3 and p 1 -ratio of these two components of momentum leads to the deflection angle. To solve p 1 from the quadratic dispersion relation, we rewrite (35) as: where, very far from the horizon. As shown in Appendix (A), near the horizon,Ẽ/L >> 1. In this limit and near the horizon C 1 = β 0 r >> 1, the coefficients a 1 , a 2 are simplified and the dispersion relation reduces to: By solving p 1 from the above quadratic equation and taking the ratio p 3 /p 1 we get: Note that the +(−) sign corresponds to the deflection angle for the +(−) polarization mode.
where the explicit form of the coefficients D + n and D − n can be found in Appendix. (E). The difference between the deflection angle for two polarizations is: At the radius of photon sphere r = 3r H /2 andη =η 0 , the above expression reduces to: This is the third important result regarding which we would like to stress the following points: First, the deflection angle is inversely proportional to β. As mentioned earlier, in the case of Schwarzschild (42), the deflection angle is proportional to β 0 . Thus, our analysis implies that the deflection angle will be different for the two cases for the same black hole mass. Second, the deflection angle will be real if and only if λ < 729 16 (51) Comparing the above expression with (43), we see that in the case of Sultana-Dyer, the constraints on the coupling constant does not depend on β 0 and b. We want to stress that the above constraint is obtained by settingη =η 0 (current time of the Universe). At an arbitrary time, we haveλ <η 6 /16.
In Sec. (V C), we discuss the implications of this constraint for different black hole masses.

Cubic dispersion relation (35)
To solve p 1 from the cubic dispersion relation, we rewrite Eq. (34) as: where,ā As mentioned earlier, the modification to the dispersion relation is largest close to the horizon and is negligible far from the horizon. Like in the case of quadratic dispersion relation, near the horizon,r ∼ 1,Ẽ L >> 1 and C 1 = β 0 r 1. Thus, the coefficientsā 0 ,ā 1 ,ā 2 simplify leading to the following dispersion relation: However, the dispersion relation is independent of the non-minimal coupling parameterλ.
In other words, in the limit C 1 >> 1 andẼ L >> 1, the deflection angle do not reflect any modification due to the non coupling term mentioned in Eq. (3). Hence, we only consider the effect due to the quadratic dispersion relation (50).
As our analysis is restricted to a non-rotating dynamical black hole, considering a rotating dynamical black hole might violate the limitẼ L >> 1. In that case, the deflection angle with the cubic dispersion relation plausibly leads to a better bound onλ.

C. Constraint on λ
In the case of Sultana-Dyer, the constraint onλ is given by Eq. (51). At the current time, this leads to 0 <λ < 45.56. Using the relation (31c) and (21), we have: where N is the factor corresponding to the difference from the solar mass M . Using the relation between the conformal time (η) and cosmic time (t), dt = a(η) dη, we have t = η 3 /(3η 2 0 ). Rewriting the above relation in terms of t, we have: Setting, t = t 0 and H 0 = t −1 0 , we have: This is the fourth important result regarding which we like to stress the following points: First, the coupling constant λ is an intrinsic property of the field and should be independent of the parameters of the model. However, from the above expression, we see that the constraint on λ depends on the mass of the black hole. On the face of it, this might look unphysical. However, we pointed that the energy density of the dust does not remain positive for all values of η (cf. Eq. (16)). This condition implies that after time η, the particles closest to the horizon become superluminal [30]. The condition (16) at the photon radius translates to η < 15r H /4. Since, we have fixed η = η 0 , this leads to the condition on λ.
Second, in Appendix F, we have evaluated the stress-tensor of the non-minimally coupled electromagnetic field action (3). We have shown that the energy-density to be positive in Sultana-Dyer black hole, λ < r 2 H (cf. Eq. F7). Thus, the above constraint on the coupling parameter λ ensures that the energy density of the electromagnetic field is non-negative.  The table below gives the upper limit on λ for different black hole mass ranges. Finally, it is interesting to compare the bounds on λ compared to obtained in the literature. In Ref. [28], the authors obtained constraints on λ considering photons in Schwarzschild space-time.
Considering signals from radar ranging past the Sun, the authors found λ ∼ 1.1 × 10 20 cm 2 , which is about three orders of magnitude more stringent than the one obtained in Ref. [29].
Our analysis shows that the value of λ for the Sultana-Dyer solar-mass black hole improves the bound on λ. Note that, in the case of signals coming from binary pulsar PSR B1534+12, the bound on λ ∼ 0.6 × 10 11 cm 2 [33,34]. This is the fifth key result of this work, and we would like to discuss the following points: First, for both the masses M and 10 −5 M and β 0 b ∼ 1, the difference between deflection angles for Schwarzchild black hole is larger than that of Sultana Dyer. This is because the deflection angle (42) for Schwarzschild is linearly related to β 0 b. Hence, the larger the value of b, the larger the correction for the Schwarzschild black-hole. Second, for both the masses M and 10 −5 M and β 0 b ∼ 10 −3 , the difference between deflection angles for Sultana-Dyer black hole is larger than that of Schwarzschild. This is because the deflection angle (50) for Sultana-Dyer is inversely related to β 0 . Hence, the smaller the value of b, the larger the correction for the Sultana-Dyer black-hole. Lastly, the difference in the deflection angle for the Schwarzschild black hole is always negative, while for Sultana-Dyer is always positive.
Thus, our analysis points out that the two black holes provide a distinct signature with the deflection angle irrespective of the black-hole mass.

VII. CONCLUSIONS AND DISCUSSIONS
In this work, we carried out a detailed analysis of the propagation of photons that are non-minimally coupled to gravity. To obtain a constraint on the coupling parameter, we calculated the deflection angle of a photon in the vicinity of a dynamical, spherically sym- We showed that the deflection angle for the Sultana-Dyer black hole is inversely proportional to β. However, in the case of Schwarzschild, the deflection angle is proportional to β 0 .
Further, we showed that for both the masses M and 10 −5 M and β 0 b ∼ 1, the difference between deflection angles for Schwarzchild black hole is larger than that of Sultana Dyer.
For both the masses M and 10 −5 M and β 0 b ∼ 10 −3 , the difference between deflection angles for Sultana-Dyer black hole is larger than that of Schwarzschild. We showed that the difference in the deflection angle for the Schwarzschild black hole is always negative, while for Sultana-Dyer is always positive. Thus, our analysis points out that the two black holes provide a distinct signature with the deflection angle irrespective of the black-hole mass.
We showed that Sultana-Dyer solar-mass black hole improves the bound on λ. In Ref. [28], the authors used photon dispersion relation and computed the photon's arrival time in Schwarzschild space-time. Considering signals from radar ranging past the Sun, the authors found λ ∼ 1.1 × 10 20 cm 2 , which is about three orders of magnitude more stringent than the one obtained in Ref. [29]. The bound on λ from the binary pulsar PSR B1534+12 is λ ∼ 0.6 × 10 11 cm 2 [33,34]. Using the fact that the energy density of the electromagnetic field must be positive in all regions in space-time provides a stringent bound for the solar mass black holes.
In the current analysis, we have not considered the frequency dependence of the polarization modes. However, the frequency dependence of the deflection angle can provide a useful way to test the non-minimal coupling strength and can potentially be measured in radio frequencies [35]. LIGO-VIRGO detectors have confirmed two NS-NS and one NS-BH merger events. These cataclysmic events will emit both gravitational waves and electromagnetic waves in all wave-band, including in the radio frequencies. With future radio telescopes, like SKA, aiming for higher sensitivity, the deflection angle produced by the electromagnetic field can confirm/infirm the non-minimal coupling. We hope to report this soon.
The entire analysis is focused on the spherically symmetric space-times. However, in the NS-NS and NS-BH mergers, the final black-hole is expected to be axially symmetric (like Kerr). Therefore, we need to extend the analysis for the rotating black-hole space-times. This is currently under investigation. India whose untiring work allowed the authors to continue this work during the COVID-19 outbreak.
Appendix A: Relation between conserved quantities in dimensionless and dimensionfull coordinates The Sultana-Dyer metric in non-geometric units is The conserved quantities in these coordinates are where α 0 and β 0 are defined in Eq. (21). For the Sultana-Dyer metric in dimensionless coordinates (19), the conserved quantities in Eqs. (22) and (23), can be rewritten as: Comparing Eqs. (A2a) and (A3a), and Eqs. (A2b) and (A3b), we get Using these two relations we have,L From Eqs. (A3a) and (A3b), we get, In Sec. (V), we show that the dispersion relation strongly depends on the value ofL/Ẽ.
In the non-inertial frame, the dispersion relation (29) leads to: Since this is different from that of Schwarzschild (37), this leads to a different time-of-arrival and deflection angle of photons compared to the Schwarzschild metric. As we show in Sec.
(IV B), in the case of Sultana-Dyer black hole, the dispersion relation is no more quadratic. The local inertial frame for the Sultana-Dyer black hole (20) is related to the non-inertial frame via the tetrads: where, η (µ)(ν) is the metric for local Minkowski space-time and g αβ is metric for the noninertial frame. For the line-element (20), we have: where Components of inverse tetrad, e µ (ν) can also be calculated. The four-momentum of photon in non-inertial (p µ ) and inertial (p (ν) ) frames are related by p (µ) = e ν (µ) p ν .
Close to the horizon we can expandr = (1 + ), where, | | < 1. Subsituting this in the above inequality, we haveλ <η 4 (F6) Rewriting the above expression in terms of the cosmic time and setting t = t 0 we get, It is interesting to note the above constraint is consistent with the constraint (57) obtained in a completely different way. In deriving the above constraint, we have assumed that the term with F 2 φθ is negligible compared to other components of F 2 µν because F φθ ∼ B r and the radial component of magnetic field falls off faster than its θ or φ components