74 Tests of General Relativity
The ray-tracing process mapping a single source into its image can be expressed
by the Jacobian matrix between the source-plane coordinates and the observer-plane
coordinates:
J(훼)=(
1 −휅−훾 1 −훾 2
−훾 2 1 −휅+훾 1
)
, (4.13)
where휅is theconvergenceof the lens and훾=훾 1 +i훾 2 is the shear. The matrixJ(훼)
transforms a circular source into an ellipse with semi-axes stretched by the factor
( 1 −휅±|훾|)−^1. The convergence affects the isotropic magnification or the projected
mass density divided by the critical density, whereas the shear affects the shape of the
image. The magnification is given by
휇=(detJ)−^1 =[( 1 −휅)^2 −훾^2 ]−^1. (4.14)Clearly, there are locations where휇can become infinite. These points in the source
plane are calledcausticsand they lie on the intersections ofcritical curves.
Weak Lensing Surveys. Background galaxies would be ideal tracers of distortions
if they were intrinsically circular. Any measured ellipticity would then directly reflect
the action of the gravitational tidal field of the interposed lensing matter, and the
statistical properties of the cosmic-shear field would reflect the statistical properties
of the matter distribution. But many galaxies are actually intrinsically elliptical, and
the ellipses are randomly oriented. These intrinsic ellipticities introduce noise into
the inference of the tidal field from observed ellipticities.
The sky is covered with a ‘wall paper’ of faint and distant blue galaxies, about
20 000–40 000 on an area of the size of the full moon. This fine-grained pattern of
the sky makes statistical weak-lensing studies possible, because it allows the detec-
tion of the coherent distortions imprinted by gravitational lensing on the images of
the faint-blue-galaxy population. Large collaborations carrying out such surveys have
reported statistically significant observations of cosmic shear and thus of the distri-
bution of interposed lensing dark matter. We shall come back to this discussion in the
chapter on dark matter.
To test general relativity versus alternative theories of gravitation, the best way is to
probe the gravitational potential far away from visible matter, and weak galaxy–galaxy
lensing being a good approach to this end because it is accurate on scales where all
other methods fail, and it is simple if galaxies are treated as point masses. Alternative
theories may predict an isotropic signal where general relativity predicts an azimuthal
variation. The current knowledge favors anisotropy and thus general relativity.
4.4 Gravitational Waves
Einstein noted in 1916 that his general relativity predicted the existence of gravita-
tional radiation, but its possible observation is still in the future. As we explained in
Section 4.2, the slowdown of binary pulsars is indirect evidence that the system loses
its energy by radiating gravitational waves.