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Symmetry 2018, 10(7), 246;

Analysis of the Angular Dependence of Time Delay in Gravitational Lensing

Dipartimento di Fisica, Università di Genova, Via Dodecaneso 33, I-16146 Genova, Italy
I.N.F.N.—Sezione di Genova, Via Dodecaneso 33, I-16146 Genova, Italy
DiSAT, Università Degli Studi dell’Insubria, Via Valleggio, 11, 22100 Como, Italy
INAF–Osservatorio Astronomico di Brera, Via E. Bianchi 46, I-23807 Merate, Italy
Author to whom correspondence should be addressed.
Received: 30 May 2018 / Revised: 22 June 2018 / Accepted: 27 June 2018 / Published: 1 July 2018
(This article belongs to the Special Issue Symmetry in Cosmology)
Full-Text   |   PDF [427 KB, uploaded 1 July 2018]   |  


We consider an alternative formula for time delay in gravitational lensing. Imposing a smoothness condition on the gravitationally deformed paths followed by the photons from the source to the observer, we show that our formula displays the same degrees of freedom as the standard one. In addition to this, it is shown that the standard expression for time delay is recovered when small angles are involved. These two features strongly support the claim that the formula for time delay studied in this paper is the generalization to the arbitrary angles of the standard one, which is valid at small angles. This could therefore result in a useful tool in Astrophysics and Cosmology which may be applied to investigate the discrepancy between the various estimates of the Hubble constant. As an aside, two interesting consequences of our proposal for time delay are discussed: the existence of a constraint on the gravitational potential generated by the lens and a formula for the mass of the lens in the case of central potential. View Full-Text
Keywords: classical general relativity; gravitational lenses; time delay classical general relativity; gravitational lenses; time delay

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Alchera, N.; Bonici, M.; Cardinale, R.; Domi, A.; Maggiore, N.; Righi, C.; Tosi, S. Analysis of the Angular Dependence of Time Delay in Gravitational Lensing. Symmetry 2018, 10, 246.

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