Curvature(s) of a light wavefront in a weak gravitational field

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. Comment: 3 pages, 1 fig, Talk given by JFPS at the 12th Marcel Grossmann conference (Paris, July, 2009), submitted to the Proceedings

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CURVATURE(S) OF A LIGHT WAVEFRONT IN A WEAK
GRAVITATIONAL FIELD
J.-F. PASCUAL-S´ANCHEZ, A. SAN MIGUEL and F. VICENTE
Dept. de Matem´ atica Aplicada, Universidad de Valladolid,
47005 Valladolid, Spain
E-mails: asmiguel@maf.uva.es, fvicente@maf.uva.es, jfpascua@maf.uva.es
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.
Keywords: Ray tracing; Light wavefront; Post-Newtonian formalism; Numerical Relativ-
ity.
1. Introduction
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. 1. In this work and in Ref. 2
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. 3) of this surface.
2. Light propagation in a weak gravitational field
Let us consider a spacetime (M,g) corresponding to a weak gravitational field
determined by a metric tensor given in a global coordinate system {(z,ct)} by
gαβ= ηαβ+ hαβ, with ηαβ= diag(1,1,1,−1). Where the coordinate components
of the metric perturbation hαβare:
hab= 2c−2κ?z?−1δab,
here κ := GM represents the gravitational constant of the Sun, located at Za(t),
and c represents the vacuum light speed. The null geodesics, z(t) =?z(t),t?satisfy
¨ za=1
−(h44,kδa
−(c−1h4k,j−1
0 = gαβ˙ zα˙ zβ.
ha4= −4c−3κ?z?−1˙Za,h44= 2c−2κ?z?−1,(1)
the following equations, see Ref. 4:
2c2h44,a− [1
2h44,tδa
l+ hak,l−1
2c−2hjk,t)˙ zj˙ zk˙ za,
k+ hak,t+ c(h4a,k− h4k,a)]˙ zk
2hkl,a)˙ zk˙ zl
The second equation is the isotropy constraint satisfied by the null geodesics.
arXiv:1001.2550v1 [gr-qc] 14 Jan 2010

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3. Local approximation of the wavefront
Let be S0a flat initial surface far from the Sun, formed by points (z1,z2,−ζ) (with
ζ > 0) in an asymptotically Cartesian coordinate system {z}. For the discretization
of S0a triangulation is constructed in such a form that each vertex is represented
by a complex number of the set:
¯V := {z = a1+ a2ω + a3ω2| a1,a2,a3∈ A, ω := exp(2πi/3)},
The initial triangulation by V induces a triangulation on the final wavefront St. The
evolution of a photon z0:= z(0) ∈ V with velocity ˙ z0:= (0,0,c) in phase space
u = (z, ˙ z) may be written as a first order differential system ˙ u = F(u,t). This
determines a flow, z(t) = ϕt(z0, ˙ z0), in the 3-dimensional curved quotient space of
M by the global timelike vector field ∂tassociated to the global coordinate system
used in the post-Newtonian formalism. For each time t, the flow ϕtdetermines a
2-dimensional curved wavefront St.
To compute the curvatures of the wavefront surface corresponding to the mesh 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 V the image under the flow ϕt. In a neighbourhood
of the image point the wavefront can be approximated by a least-squares fitting of
the data obtained as the quadric:
(2)
y3= f(y1,y2) :=1
2a1(y1)2+ a2y1y2+1
2a3(y2)2.(3)
using adapted normal coordinates {yi}.
4. Numerical integrator
We apply the ray tracing method, see Refs. 2,5, 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. 2,3, are computed
at each inner vertex of mesh on the light wavefront surface Stin the vicinity of the
Sun, by the implementation of the following pseudocode:
Data: u∗
for n = 1...N do
un:= Taylor(t,u∗
yn:= NormalCoordinates(zn)
for i = 0...6 do
yni:= Ring(yn)
end
(a1,a2,a3) = LeastSquares(yni)
γAB(xn) := Metric(a1,a2,a3,xn)
B = SecondFundamentalForm(xn)
(λ1,λ2) = Diagonalize(B)
(Krel,H) = Curvature(λ1,λ2)
end
n:= (z∗
n, ˙ z∗
n),n = 1,...N
n)

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Fig. 1.
field (a different scale is used for the vertical axis).
Wavefront surface and relative curvature (gray scale) deformed by a spherical gravitational
In Figure 1 the surface ST 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 Oz3–axis). One sees in this figure that the absolute value of
the relative curvature defined on ST increases as the distance between the photon
and the Oz3–axis, where the Sun is located, decreases.
Acknowledgements
This research was partially supported by the Spanish Ministry de Educaci´ on y
Ciencia, MEC-FEDER grant ESP2006-01263.
References
1. J. Samuel, Class. Quantum Grav., 21, L83 (2004).
2. A. San Miguel, F. Vicente and J.-F. Pascual-S´ anchez, Class. Quantum Grav. 26, 235004
(2009).
3. M. P. do Carmo, Riemannian Geometry (Boston: Birkh¨ auser, 1992).
4. V. A. Brumberg, Essential Relativistic Celestial Mechanics, (Bristol: Adam Hilger,
1991).
5. `A. Jorba and M. Zou, Experimental Mathematics 14, 99 (2005).
6. S. A. Klioner and M. Peip, Astron. Astrophys. 410, 1063 (2003).