38 Propagation of light and horizons
In General Relativity, the same must be true in every local inertial coordinate frame.
Then, since the interval is invariant, the conditionds^2 =0 should be valid along
the light geodesic in any curved spacetime.
We consider mainly the radial propagation of light in an isotropic universe in a
coordinate system where the observer is located at the origin. The light trajectories
look especially simple if, instead of physical timet, we use the conformal time
η≡∫
dt
a(t)The metric (1.47) inη,χcoordinates is
ds^2 =a^2 (η)(dη^2 −dχ^2 − 2 (χ)(dθ^2 +sin^2 θdφ^2 )), (2.2)where
2 (χ)=⎧
⎨
⎩
sinh^2 χ, k=−1;
χ^2 , k=0;
sin^2 χ, k=+ 1.(2.3)
By symmetry, it is clear that the radial trajectoryθ,φ=const is a geodesic. The
functionχ(η) along the trajectory is then entirely determined by the condition
ds^2 =0, or
dη^2 −dχ^2 = 0. (2.4)Hence, radial light geodesics are described by
χ(η)=±η+const, (2.5)and correspond to straight lines at angles± 45 ̊in theη–χplane.
2.2 Horizons
Particle horizonIf the universe has a finite age, then light travels only a finite
distance in that time and the volume of space from which we can receive information
at a given moment of time is limited. The boundary of this volume is called the
particle horizon. Today, the universe is roughly 15 billion years old, so a naive
estimate for the particle horizon scale is 15 billion light years.
According to (2.5), the maximum comoving distance light can propagate is
χp(η)=η−ηi=∫ttidt
a