Weak deflection gravitational lensing for photons coupled to Weyl tensor in a Schwarzschild black hole

Both null and timelike rays experience trajectory bending in a gravitational field. In this work, we systematically develop a perturbative method to compute the deflection angle of rays with general velocity v in arbitrary static and spherically symmetric spacetimes and in equatorial plane of arbitrary static and axisymmetric spacetimes. We show that the expansion in the large closest approach x0 limit depends on the asymptotic behavior of the metric functions only, and the generated integrand is always integrable, resulting in a deflection angle in a series form of either x0 or b, the impact parameter. Using this method, the deflection angles as series of both x0 and b are found in Schwarzschild, Reissner-Nordström and Kerr-Newman spacetimes to 17-th, 15-th and 6-th orders respectively, for both lightrays and particles with general velocity. The effects of the impact parameter, velocity and other parameters of the spacatimes are briefly analyzed. Moreover, we show that for spacetimes whose metric functions are only asymptotically known, the deflection angle in the weak field limit can also be calculated. Furthermore, it is shown that the deflection angle in general static and spherically symmetric spacetime and equatorial plane of static and axisymmetric spacetime to the lowest non-trivial order, depends only on the impact parameter, velocity of the particle, and the effective ADM mass of the spacetime but not on other parameters such as charge or angular momentum. These deflection angles are used in an exact gravitational lensing equation and the corresponding apparent angles of the images of the source are also solved perturbatively.

“…where we have substituted (67). Here E and L are still energy and angular momentum per unit mass of the particle at infinite ρ.…”

Section: Weak Deflection Angle In Equatorial Plane Of Sas Spactimesmentioning

confidence: 99%

The perturbative approach for the weak deflection angle

Jia

2020

“…It is worth mentioning that our convention on C(r ), see Eq. 15, is slightly different from the one of the work [73] about the photons coupled to the Weyl tensor in the Schwarzschild black hole. Before we proceed, it is necessary to determine the domain of {b,α • } which has not been intensively investigated in the previous work [88].…”

Section: Introductionmentioning

confidence: 80%

“…When the gravitational and electromagnetic fields are non-minimally coupled, a photon will never travel along the geodesics of the background spacetime while the geometrical-optics approximation might still be valid under some circumstances [44]. The gravitational lensing of the photons coupled to the Weyl tensor in the Schwarzschild black hole was investigated in both weak [72][73][74] and strong [75,76] deflection cases. The strong deflection gravitational lensing of the photons coupled to the Weyl tensor in the Kerr black hole was also studied [77].…”

Section: Introductionmentioning

confidence: 99%

“…The non-geodesic worldline of a photon in the spacetime (1) can be mapped into the geodesic one in the spacetime (13) which will be adopted in the following work. When b = 0, the effective metric (13) returns to the case of the photons coupled to the Weyl tensor in the Schwarzschild black hole and the behaviors of these photons have been widely studied [72][73][74][75]. It is worth mentioning that our convention on C(r ), see Eq.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Time delay of photons coupled to Weyl tensor in a regular phantom black hole

Xie

2020

Self Cite

Time delay of the photons coupled to the Weyl tensor in a regular phantom black hole is investigated in both weak and strong deflection gravitational lensing. We find that the time delay in the weak deflection lensing strongly depends on the phantom hair while the delay in the strong deflection lensing is significantly affected by the hair and the strength of the coupling. We suggest that it is necessary to measure these two kind of time signals for fully understanding and distinguishing such an interaction beyond the standard Einstein–Maxwell theory.

“…A large amount of insights about black holes can be provided by gravitational lensing [41]. With bending angle much smaller than 1, weak deflection lensing has already been a workhorse in astronomy [42][43][44][45] and gravitational physics [46][47][48][49][50][51]. With a key ingredient that a photon might be able to go around a black hole by at least one loop, strong deflection lensing can form an escape cone of light [52] (now popularly called "shadow") and relativistic images [53] (see Refs.…”

Section: Introductionmentioning

confidence: 99%

Weak and strong deflection gravitational lensing by a renormalization group improved Schwarzschild black hole

Xie

2019

Self Cite

Weak and strong deflection gravitational lensing by a renormalization group improved Schwarzschild black hole is investigated and its observables are found. By taking the supermassive black holes Sgr A* and M87* respectively in the Galactic Center and at the center of M87 as lenses, we estimate these observables and analyse possibility of detecting this quantum improvement. It is not feasible to distinguish such a black hole by most observables in the near future except for the apparent size of the shadow. We also note that directly using measured shadow of M87* to constrain this quantum effect requires great care.