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The current standard time delay formula (CSTD) in gravitational lensing andits claimed relation to the lens equation through Fermat's principle (leasttime principle) have been puzzling to the author for some time. We find thatthe so-called geometric path difference term of the CSTD is an error, and itcauses a double counting of the correct time delay. We examined the deflectionangle and the time delay of a photon trajectory in the Schwarzschild metricthat allows exact perturbative calculations in the gravitational parameter $GM$in two coordinate systems -- the standard Schwarzschild coordinate system andthe isotropic Schwarzschild coordinate system. We identify a coordinatedependent term in the time delay which becomes irrelevant for the arrival timedifference of two images. It deems necessary to sort out unambiguously what iswhat we measure. We calculate the second order corrections for the deflectionangle and time delay. The CSTD does generate correct lens equations includingmultiple scattering lens equations under the variations and may be bestunderstood as a generating function. It is presently unclear what thesignificance is. We call to reanalyze the existing strong lensing data withtime delays.

Puzzles in Time Delay and Fermat Principle in Gravitational Lensing