Puzzles in Time Delay and Fermat Principle in Gravitational Lensing

The current standard time delay formula (CSTD) in gravitational lensing and its claimed relation to the lens equation through Fermat's principle (least time principle) have been puzzling to the author for some time. We find that the so-called geometric path difference term of the CSTD is an error, and it causes a double counting of the correct time delay. We examined the deflection angle and the time delay of a photon trajectory in the Schwarzschild metric that allows exact perturbative calculations in the gravitational parameter $GM$ in two coordinate systems -- the standard Schwarzschild coordinate system and the isotropic Schwarzschild coordinate system. We identify a coordinate dependent term in the time delay which becomes irrelevant for the arrival time difference of two images. It deems necessary to sort out unambiguously what is what we measure. We calculate the second order corrections for the deflection angle and time delay. The CSTD does generate correct lens equations including multiple scattering lens equations under the variations and may be best understood as a generating function. It is presently unclear what the significance is. We call to reanalyze the existing strong lensing data with time delays.Comment: 6 p., 1 fi

Generalized Legendre polynomials and related congruences modulo p2p^2

For any positive integer $n$ and variables $a$ and $x$ we define the generalized Legendre polynomial P_n(a,x)=\sum_{k=0}^n\b ak\b{-1-a}k(\frac{1-x}2)^k. Let $p$ be an odd prime. In the paper we prove many congruences modulo $p^2$ related to $P_{p-1}(a,x)$. For example, we show that P_{p-1}(a,x)\e (-1)^{_p}P_{p-1}(a,-x)\mod {p^2}, where $_p$ is the least nonnegative residue of $a$ modulo $p$. We also generalize some congruences of Zhi-Wei Sun, and determine $\sum_{k=0}^{p-1}\binom{2k}k\binom{3k}k{54^{-k}}$ and $\sum_{k=0}^{p-1}\binom ak\binom{b-a}k\mod {p^2}$, where $[x]$ is the greatest integer function. Finally we pose some supercongruences modulo $p^2$ concerning binary quadratic forms.Comment: 37 page

Jacobsthal sums, Legendre polynomials and binary quadratic forms

Let $p>3$ be a prime and $m,n\in\Bbb Z$ with $p\nmid mn$. Built on the work of Morton, in the paper we prove the uniform congruence: &\sum_{x=0}^{p-1}\Big(\frac{x^3+mx+n}p\Big) \equiv {-(-3m)^{\frac{p-1}4} \sum_{k=0}^{p-1}\binom{-\frac 1{12}}k\binom{-\frac 5{12}}k (\frac{4m^3+27n^2}{4m^3})^k\pmod p&\t{if $4\mid p-1$,} \frac{2m}{9n}(\frac{-3m}p)(-3m)^{\frac{p+1}4} \sum_{k=0}^{p-1}\binom{-\frac 1{12}}k\binom{-\frac 5{12}}k (\frac{4m^3+27n^2}{4m^3})^k\pmod p&\text{if $4\mid p-3$,} where $(\frac ap)$ is the Legendre symbol. We also establish many congruences for $x\pmod p$, where $x$ is given by $p=x^2+dy^2$ or $4p=x^2+dy^2$, and pose some conjectures on supercongruences modulo $p^2$ concerning binary quadratic forms.Comment: 35 page

Congruences concerning Legendre polynomials

Let $p$ be an odd prime. In the paper, by using the properties of Legendre polynomials we prove some congruences for $\sum_{k=0}^{\frac{p-1}2}\binom{2k}k^2m^{-k}\mod {p^2}$. In particular, we confirm several conjectures of Z.W. Sun. We also pose 13 conjectures on supercongruences.Comment: 16 page