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Spacetime perspective of Schwarzschild lensing
General Relativity and Quantum Cosmology · Submitted 13 Jan 2000
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim We show that the strong-field thin-lens equation is remarkably accurate, even for lightrays that take several turns around the lens before reaching the observer.
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The abstract, sentence by sentence
We propose a definition of an exact lens equation without reference to a background spacetime, and construct the exact lens equation explicitly in the case of Schwarzschild spacetime. For the Schwarzschild case, we give exact expressions for the angular-diameter distance to the sources as well as for the magnification factor and time of arrival of the images. We compare the exact lens equation with the standard lens equation, derived under the thin-lens-weak-field assumption (where the light rays are geodesics of the background with sharp bending in the lens plane, and the gravitational field is weak), and verify the fact that the standard weak-field thin-lens equation is inadequate at small impact parameter. We show that the second-order correction to the weak-field thin-lens equation is inaccurate as well. Finally, we compare the exact lens equation with the recently proposed strong-field thin-lens equation, obtained under the assumption of straight paths but without the small angle approximation, i.e., with allowed large bending angles. We show that the strong-field thin-lens equation is remarkably accurate, even for lightrays that take several turns around the lens before reaching the observer.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We propose a definition of an exact lens equation without reference to a background spacetime, and construct the exact lens equation explicitly in the case of Schwarzschild spacetime. | contribution 1.00 | 0.96 | 0.26 | asserted 0.99 |
| s1 | For the Schwarzschild case, we give exact expressions for the angular-diameter distance to the sources as well as for the magnification factor and time of arrival of the images. | contribution 0.53 | 0.96 | 0.12 | asserted 1.00 |
| s2 | We compare the exact lens equation with the standard lens equation, derived under the thin-lens-weak-field assumption (where the light rays are geodesics of the background with sharp bending in the lens plane, and the gravitational field is weak), and verify the fact that the standard weak-field thin-lens equation is inadequate at small impact parameter. | result 0.76 | 0.91 | 0.21 | asserted 1.00 |
| s3 | We show that the second-order correction to the weak-field thin-lens equation is inaccurate as well. | result 0.99 | 0.93 | 0.08 | asserted 1.00 |
| s4 | Finally, we compare the exact lens equation with the recently proposed strong-field thin-lens equation, obtained under the assumption of straight paths but without the small angle approximation, i.e., with allowed large bending angles. | contribution 0.66 | 0.84 | 0.08 | asserted 0.70 |
| s5 | We show that the strong-field thin-lens equation is remarkably accurate, even for lightrays that take several turns around the lens before reaching the observer. | result 0.99 | 0.94 | 0.49 | asserted 1.00 |
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The paper
Original abstract
We propose a definition of an exact lens equation without reference to a background spacetime, and construct the exact lens equation explicitly in the case of Schwarzschild spacetime. For the Schwarzschild case, we give exact expressions for the angular-diameter distance to the sources as well as for the magnification factor and time of arrival of the images. We compare the exact lens equation with the standard lens equation, derived under the thin-lens-weak-field assumption (where the light rays are geodesics of the background with sharp bending in the lens plane, and the gravitational field is weak), and verify the fact that the standard weak-field thin-lens equation is inadequate at small impact parameter. We show that the second-order correction to the weak-field thin-lens equation is inaccurate as well. Finally, we compare the exact lens equation with the recently proposed strong-field thin-lens equation, obtained under the assumption of straight paths but without the small angle approximation, i.e., with allowed large bending angles. We show that the strong-field thin-lens equation is remarkably accurate, even for lightrays that take several turns around the lens before reaching the observer.
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References
How the literature uses it
29 arXiv papers cite this in 31 sentences
Papers citing it · 16 sentences without a confident label
Uses its method or tool 3 papers
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The lens equation could be solved analytically with the help of Frittelli, Killing, and Newman [ref] , and Bozza's SDL framework [ref] further developed these techniques.
Strong gravitational lensing by black hole in F(R) Euler Heisenberg Gravity's Rainbow · Introduction · 0.87
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To determine this, let the photon's null 4-velocity be written as $ẋ^μ=(ṫ,v⃗)$ where $v⃗$ is the space-like component of the 4-velocity, which is further split as [equation] When the photon reaches an arbitrary observer at $(t,r,θ,ϕ)$, we may define a local Euclidean orthonormal frame at that location as follows: [equation] The celestial sphere [ref] of the observer is parametrised by angles $ψ$ and $η$, where [equation] In the above equation, the dot products are understood as the inner product $a⃗·b⃗=g_ija^ib^j$, and $|a⃗|=√(g_ija^ia^j)$, where the indices $i, j$ run along the space-like coordinates.
Exact gravitational lensing in conformal gravity and Schwarzschild-de Sitter spacetime · Derivation of the bending angle formula · 0.87
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Based on lens equation [ref] , Bozza et al. [ref] developed a semi-analytical method to deal with it.
Null geodesics and gravitational lensing in a nonsingular spacetime · Introduction · 0.91
Cites it as background 11 papers
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In the last two decades the study of gravitational lensing in the strong deflection limit received revival and extensive elaboration [ref] [Recently, Virbadhra modeled the supermassive black hole $M87^*$ as a Schwarzschild lens and strudied its distorted (tangential, radial, and total) magnifications of images with respect to the change in angular source position and lens-source ratio distance [ref] .].
Strong lensing and shadow of Ayon-Beato-Garcia (ABG) nonsingular black hole · Introduction · 0.90
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Several attempts were made to obtain adequate lens equations for so called relativistic lensing in Schwarzschild space-time [ref] .
Black Hole Information Problem and Wave Bursts · Introduction · 0.95
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In Schwarzschild black hole lensing, Virbhadra and Ellis found a sequence of relativistic images on both sides of the optic axis apart from the primary and the secondary images [ref] , while Fritelli, Kling and Newman obtained the exact expressions for magnifications and time delays of relativistic images [ref] .
Investigating strong gravitational lensing with black hole metrics modified with an additional term · Introduction · 0.92
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