Perturbative deflection angle, gravitational lensing in the strong field limit and the black hole shadow

Jia, J.; Huang, Ke

doi:10.1140/epjc/s10052-021-09026-7

Cited by 12 publications

(24 citation statements)

References 46 publications

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“…This total time (22) resembles the same form as the deflection angle in the SFL in Ref. [38], although their coefficients C k and D k will be different. When the source and detector radius r s,d are not infinite, then in the a → 0 limit, there is only one divergent term in t contributed by the O(a) 0 order.…”

Section: Perturbative Expansion Of Total Travel Timementioning

confidence: 83%

“…The perturbative method we used here to calculate the total travel time is adapted from Ref. [38] which was to calculate the deflection angle in the SFL, while here this method is used to a different integral. Therefore in this section, we will first recap the essential steps to help understanding the method, and then apply it to the suitable integral defining the total travel time of the signal.…”

Section: The Perturbation Methodsmentioning

confidence: 99%

“…In Ref. [38] we have developed a perturbative way to expand a similar integral for the computation of signal's deflection angle. Here we adapt that method to the computation of the total travel time in the SFL.…”

Section: The Perturbation Methodsmentioning

confidence: 99%

“…We choose this spacetime because its gravitational lensing (not including the time delay) in the SFL was studied previously in Ref. [38] and we can directly quote the expansion coefficients near r c of its metrics from there. Moreover, this spacetime also has a Schwarzschild limit l → 0 whose time delay in the SFL was studied for light rays [26].…”

Section: The Hayward Bh Spacetime Casementioning

confidence: 99%

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Time delay in the strong field limit for null and timelike signals and its simple interpretation

Liu

Jia

2021

Eur. Phys. J. C

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Gravitational lensing can happen not only for null signals but also timelike signals such as neutrinos and massive gravitational waves in some theories beyond GR. In this work we study the time delay between different relativistic images formed by signals with arbitrary asymptotic velocity v in general static and spherically symmetric spacetimes. A perturbative method is used to calculate the total travel time in the strong field limit, which is found to be a quasi-power series of the small parameter $$a=1-b_c/b$$ a = 1 - b c / b where b is the impact parameter and $$b_c$$ b c is its critical value. The coefficients of the series are completely fixed by the behaviour of the metric functions near the particle sphere $$r_c$$ r c and only the first term of the series contains a weak logarithmic divergence. The time delay $$\Delta t_{n,m}$$ Δ t n , m to the leading non-trivial order was shown to equal the particle sphere circumference divided by the local signal velocity and multiplied by the winding number and the redshift factor. By assuming the Sgr A* supermassive black hole is a Hayward one, we were able to validate the quasi-series form of the total time, and reveal the effects of the spacetime parameter l, the signal velocity v and the source/detector coordinate difference $$\Delta \phi _{sd}$$ Δ ϕ sd on the time delay. It is found that as l increases from 0 to its critical value $$l_c$$ l c , both $$r_c$$ r c and $$\Delta t_{n,m}$$ Δ t n , m decrease. The variation of $$\Delta t_{n+1,n}$$ Δ t n + 1 , n for l from 0 to $$l_c$$ l c can be as large as $$7.2\times 10^1$$ 7.2 × 10 1 [s], whose measurement then can be used to constrain the value of l. While for ultra-relativistic neutrino or gravitational wave, the variation of $$\Delta t_{n,m}$$ Δ t n , m is too small to be resolved. The dependence of $$\Delta t_{n,-n}$$ Δ t n , - n on $$\Delta \phi _{sd}$$ Δ ϕ sd shows that to temporally resolve the two sequences of images from opposite sides of the lens, $$|\Delta \phi _{sd}-\pi |$$ | Δ ϕ sd - π | has to be larger than a certain value, or equivalently if $$|\Delta \phi _{sd}-\pi |$$ | Δ ϕ sd - π | is small, the time resolution of the observatories has to be good.

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Section: Perturbative Expansion Of Total Travel Timementioning

confidence: 83%

Section: The Perturbation Methodsmentioning

confidence: 99%

Section: The Perturbation Methodsmentioning

confidence: 99%

Section: The Hayward Bh Spacetime Casementioning

confidence: 99%

See 2 more Smart Citations

Time delay in the strong field limit for null and timelike signals and its simple interpretation

Liu

Jia

2021

Eur. Phys. J. C

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“…[56]. There are also other works devoted to the study of some of the observable properties of the lensed images of a massive-particle source (e.g., [57][58][59]).…”

Section: Introductionmentioning

confidence: 99%

Kerr-Newman black hole lensing of relativistic massive particles in the weak field limit

He,

Lin

2021

Preprint

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The gravitational lensing of relativistic neutral massive particles caused by a Kerr-Newman black hole is investigated systematically in the weak field limit. Based on the Kerr-Newman metric in Boyer-Lindquist coordinates, we first derive the analytical form of the equatorial gravitational deflection angle of a massive particle in the third post-Minkowskian approximation. The resulting bending angle, which is found to be consistent with the result in the previous work, is adopted to solve the popular Virbhadra-Ellis lens equation. The analytical expressions for the main observable properties of the primary and secondary images of the particle source are thus obtained beyond the weak-deflection limit, within the framework of standard perturbation theory. The observables include the positions, magnifications, and gravitational time delays of the individual images, the differential time delay, and the total magnification and centroid position. The explicit forms of the velocitycorrectional effects induced by the deviation of the initial velocity of the massive particle from the speed of light on the observables of the lensed images are then achieved. Finally, serving as an application of the formalism, the supermassive black hole at the Galactic center, Sagittarius A * , is modeled to be a Kerr-Newman lens. The magnitudes of the velocity-correctional effects on the practical lensing observables as well as the possibilities to detect them in this scenario are also analyzed.

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Angular Location of the $$n^{th}$$ Einstein Ring at Large n

Minwalla

2024

Int J Theor Phys

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Perturbative deflection angle, gravitational lensing in the strong field limit and the black hole shadow

Abstract: A perturbative method to compute the deflection angle of both timelike and null rays in arbitrary static and spherically symmetric spacetimes in the strong field limit is proposed. The result takes a quasi-series form of $$(1-b_c/b)$$ ( 1 - b c / b ) whe… Show more

Cited by 12 publications

References 46 publications

Time delay in the strong field limit for null and timelike signals and its simple interpretation

Time delay in the strong field limit for null and timelike signals and its simple interpretation

Kerr-Newman black hole lensing of relativistic massive particles in the weak field limit

Angular Location of the $$n^{th}$$ Einstein Ring at Large n

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