---
abstract: 'The geometry of a light wavefront evolving from a flat wavefront under the action of weak gravity field in the 3-space associated to a post-Newtonian relativistic spacetime, is studied numerically by means of the ray tracing method.'
address: |
    Dept. de Matemática Aplicada, Universidad de Valladolid,\
    47005 Valladolid, Spain\
    E-mails: asmiguel@maf.uva.es, fvicente@maf.uva.es, jfpascua@maf.uva.es
author:
- 'J.-F. PASCUAL-SÁNCHEZ, A. SAN MIGUEL and F. VICENTE'
title: 'CURVATURE(S) OF A LIGHT WAVEFRONT IN A WEAK GRAVITATIONAL FIELD'
---

Introduction {#aba:sec1}
============

The curvature of initially plane light wavefronts by a gravity field is a purely general relativistic effect that has no special relativistic analogue. In order to obtain an experimental measurement of the curvature of a light wavefront, Samuel recently proposed a method based on the relation between the differences of arrival time recorded at four points on the Earth and the volume of a parallelepiped determined by four points in the curved wavefront surface, see Ref. . In this work and in Ref.  with more detail, we study a discretized model of the wavefront surface by means of a regular triangulation for the study of the curvature(s) (mean and relative, see Ref. ) of this surface.

Light propagation in a weak gravitational field
===============================================

Let us consider a spacetime $(\mathcal{M},g)$ corresponding to a weak gravitational field determined by a metric tensor given in a global coordinate system $\{(\bm{z},ct)\}$ by $g_{\alpha\beta}=\eta_{\alpha\beta}+h_{\alpha\beta}$, with $\eta_{\alpha\beta}=\diag(1,1,1,-1)$. Where the coordinate components of the metric perturbation $h_{\alpha\beta}$ are: $$h_{ab} =  2c^{-2}\kappa \|\bm{z}\|^{-1}\delta_{ab},\quad
            h_{a4} = -4c^{-3}\kappa \|\bm{z}\|^{-1}\dot{Z}_a, \quad
            h_{44} =  2c^{-2}\kappa \|\bm{z}\|^{-1},$$ here $\kappa:=GM$ represents the gravitational constant of the Sun, located at $Z^a(t)$, and $c$ represents the vacuum light speed. The null geodesics, $z(t)=\big(\bm{z}(t),t\big)$ satisfy the following equations, see Ref. : $$\begin{aligned}
          \ddot{\bm{z}}^a & =& \mfrac{1}{2} c^2h_{44,a} -[\mfrac{1}{2}h_{44,t}\delta^a_k+h_{ak,t}+c(h_{4a,k}-h_{4k,a})]\dot{\bm{z}}^k\nonumber\\
        & & -(h_{44,k}\delta^a_l+h_{ak,l}-\mfrac{1}{2}h_{kl,a})\dot{\bm{z}}^k\dot{\bm{z}}^l \nonumber\\
        & & -(c^{-1}h_{4k,j}-\mfrac{1}{2}c^{-2}h_{jk,t})\dot{\bm{z}}^j\dot{\bm{z}}^k\dot{\bm{z}}^a,\label{PN1a}\\[.1in]
0 & = &  g_{\alpha\beta}\dot{z}^\alpha\dot{z}^\beta.\label{PN1b}\end{aligned}$$ The second equation is the isotropy constraint satisfied by the null geodesics.

Local approximation of the wavefront
====================================

Let be $\mathcal{S}_0$ a flat initial surface far from the Sun, formed by points $(z_1,z_2,-\zeta)$ (with $\zeta>0$) in an asymptotically Cartesian coordinate system $\{z\}$. For the discretization of $\mathcal{S}_0$ a triangulation is constructed in such a form that each vertex is represented by a complex number of the set: $$\label{vertices1}
\bar{\mathcal{V}}:=\{z=a_1+a_2\omega+a_3\omega^2\;|\;\; a_1,a_2,a_3\in\mathcal{A},\; \omega:=\exp(2\pi \uniC/3)\},$$ The initial triangulation by $\mathcal{V}$ induces a triangulation on the final wavefront $\mathcal{S}_t$. The evolution of a photon $\bm{z}_0:=\bm{z}(0)\in\mathcal{V}$ with velocity $\dot{\bm{z}}_0:=(0,0,c)$ in phase space $\bm{u}=(\bm{z},\dot{\bm{z}})$ may be written as a [first order]{} differential system [$\dot{\bm{u}}=\bm{F}(\bm{u},t)$]{}. This determines a flow, $\bm{z}(t) =\varphi_t(\bm{z}_0,\dot{\bm{z}}_0)$, in the 3-dimensional curved quotient space of $ \mathcal{M} $ by the global timelike vector field $ \partial_t$ associated to the global coordinate system used in the post-Newtonian formalism. For each time $t$, the flow $\varphi_t$ determines a 2-dimensional curved wavefront $\mathcal{S}_t$.

To compute the curvatures of the wavefront surface corresponding to the mesh $\mathcal{V}$ at each inner vertex, we consider a 1–ring formed by the six vertices closest. For each 1–ring, one obtains on the mesh $\mathcal{V}$ the image under the flow $\varphi_t$. In a neighbourhood of the image point the wavefront can be approximated by a least-squares fitting of the data obtained as the quadric: $$\label{2}
            y^3=f(y^1,y^2):=\mfrac{1}{2}a_1(y^1)^2 + a_2y^1 y^2 + \mfrac{1}{2}a_3(y^2)^2.$$ using adapted normal coordinates $\{y^i\}$.

Numerical integrator
====================

We apply the ray tracing method, see Refs. , to a tubular region of light wavefront region supposing a gravitational model generated by a static Sun, considered as a point. The mean and relative total curvatures, defined in Refs. , are computed at each inner vertex of mesh on the light wavefront surface $\mathcal {S}_t$ in the vicinity of the Sun, by the implementation of the following pseudocode:

  ------------------------------------------------------------------------- --
  [**Data:** ]{} $\bm{u}^*_n:=(\bm{z}^*_n,\dot{\bm{z}}^*_n), n=1,\dots N$   
  [**for** ]{} $n=1\dots N$ [**do**]{}                                      
  $\bm{u}_n :=\rm{\tt Taylor}(t,\bm{u}_n^*)$                                
  $\bm{y}_n := \rm{\tt NormalCoordinates}(\bm{z}_n)$                        
  $i=0\dots 6$ [**do**]{}                                                   
  $\bm{y}_{n_i} := \rm{\tt Ring}(\bm{y}_n)$                                 
                                                                            
  $(a_1,a_2,a_3) = {\tt LeastSquares}(\bm{y}_{n_i})$                        
  $\gamma_{AB}(\bm{x}_n) := \rm{\tt Metric}(a_1,a_2,a_3,\bm{x}_n)$          
  $B= {\tt SecondFundamentalForm}(\bm{x}_n)$                                
  $(\lambda_1,\lambda_2) = {\tt Diagonalize}(B)$                            
  $(K_{\rm{rel}},H) = {\tt Curvature}(\lambda_1,\lambda_2)$                 
  [**end**]{}                                                               
  ------------------------------------------------------------------------- --

In Figure \[figure4\] the surface $\mathcal{S}_T$ at the time when the wavefront arrives at the Earth is shown using a gray-scale to represent the relative curvature (note we have used a different scale on the $Oz^3$–axis). One sees in this figure that the absolute value of the relative curvature defined on $\mathcal{S}_T$ increases as the distance between the photon and the $Oz^3$–axis, where the Sun is located, decreases.

Acknowledgements {#acknowledgements .unnumbered}
================

This research was partially supported by the Spanish Ministry de Educación y Ciencia, MEC-FEDER grant ESP2006-01263.

[9]{} J. Samuel, [*Class. Quantum Grav.*]{}, [**21**]{}, L83 (2004). A. San Miguel, F. Vicente and J.-F. Pascual-Sánchez, [*Class. Quantum Grav.*]{} [**26**]{}, 235004 (2009). M. P. do Carmo, [*Riemannian Geometry*]{} (Boston: Birkhäuser, 1992). V. A. Brumberg, [*Essential Relativistic Celestial Mechanics*]{}, (Bristol: Adam Hilger, 1991). À. Jorba and M. Zou, [*Experimental Mathematics*]{} [**14**]{}, 99 (2005). S. A. Klioner and M. Peip, [*Astron. Astrophys.*]{} [**410**]{}, 1063 (2003).
