[Home] Introduction to General Relativity. Note « 5 »

N.B.
There will be NO teaching on week 42 (13-17/10).
There WILL be teaching on week 43 (20-24/10).
On Tuesday, 21/10 we have to change to another auditorium: 520-616.
The action for the matter in the presence of a gravitational field
It the action S for a physical system in special relativity is Sm = ∫ L dΩ, then in a gravitational field it must have basically the same form,
Sm = ∫ L[g]√(-g)dΩ,
where the notation L[g] indicates that all contractions in the Lagrangian L must be written explicitely through the metric tensor gab.
Variation of the action with respect to gab and the energy-momentum tensor of matter
The variation δSm of the matter action due to a variation δgab is given in terms of a symmetric tensor Tab,
δSm = 1/2Tabδgab√(-g)dΩ = -1/2Tabδgab√(-g)dΩ,
where
1/2√(-g)Tab = δ√(-g)L/δgab.
This tensor satisfies the equation Tab;b=0 which in a flat space turns into the energy-momentum conservation equation Tab,b=0 and we thus deduce that it must be the energy-momentum tensor.
Exercises
  1. Compute all the non-vanishing components of the Riemann tensor Rabcd (where a,b,c,d=θ,φ) for the metric on a 2-dimensional sphere of radius r. Calculate also the Ricci tensor Rab and the scalar curvature R.
    ds2 = r2(dθ2+sin2θ dφ2)
    Answer: Rθφθφ=r2Sin2θ= Rφθφθ= -Rθφφθ= -Rφθθφ
  2. In a suitable coordinate system the gravitational field of the earth is approximately (to the lowest order in M/r<<1)
    ds2 = (1-2M/r)dt2-(1+2M/r)(dx2+dy2+dz2)
    Suppose a satellite orbits the earth in a circular equatorial orbit (ur=uθ=0). What is the orbital period? What is the period in the Newtonian theory?

    Hints: Period=2π/ω, where ω=dφ/dt is the angular frequency which can be found from the geodesic equations Dur=0.

    Answer: ω2 = M/r3


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