Introduction to General Relativity. Note 10

Introduction to General Relativity. Note « 10 »

Gravitational red shift Gravitational red shift is a change of the frequency of the electro-magnetic radiation as it passes through a gravitational field. It is a direct consequence of the equivalence principle.

Indeed the connection between the proper time interval τ and the world time interval Δt (here we only consider stationary gravitational fields where such world time can be introduced) is Δτ = √(g₀₀)Δt.

Since frequencies are inversely proportional to the time intervals the corresponding connection between world frequency ω₀ and the locally measured frequency ω is ω = ^ω₀/_√(g₀₀). In a weak gravitational field g₀₀=1+2φ and therefore ω = ω₀(1 - φ).

A photon emitted from a point with φ₁ and received at a point with φ₂ will be shifted by Δω = (φ₁-φ₂)ω.

Relativistic cosmology The subject of cosmology is the universe as a whole at scales so large that the local irregularities of the matter distribution should be disregarded.

We assume that the universe is and always was homogeneous and isotropic. Therefore our three-dimensional space must be a space with constant curvature where the line element dl² is equal

dl² = (1 - ^r²/_a²)^-1dr² + r²(dθ²+sin²θ dφ²) .

If a²>0 with a substitution r=a sinχ we obtain a line element

dl² = a²(dχ² + sin²χ dΩ²) ,

which corresponds to a closed universe. If a²<0, the line element turns to that of an open universe

dl² = a²(dχ² + sinh²χ dΩ²) ,

where now r=a sinhχ.

Exercises

What is the relative shift of the frequency of a photon which travelled 10m upwards in the Earth's field?
A spacetime contains no matter and is everywhere isotropic. Prove that it is flat Minkowski space.