[ home ] Riemann tensor. Ricci tensor. Ricci scalar. General Relativity. Note: « 5 »
Unnamed Web page The curl theorem, ∫∂SH⋅dl= ∫SrotH⋅dS, is a variant of the Stoke's theorem. The parallel transform of a vector is such, that the covariant derivative, DAa=dAa−ΓcabAcdxb, vanishes.

The Riemann curvature tensor

If we make a parallel transform (ie, DAa=0 along the way) of a vector Aa along an infinitesimal closed contour the components of the vector will generally change (ie, dAa!= 0) if the space is curved. The change ΔAa of the a−th component will be proportional to the vector itself and the area ΔSbc of the surface enclosed by the contour
ΔAa = 1/2 RdabcAdΔSbc.

The factor Rdabc here is called the Riemann tensor. As we have easily calculated

Rdabc = ∂bΓdac − ∂cΓdab +ΓdebΓeac − ΓdecΓeab .

The Riemann tensor defines also the commutator of covariant derivatives

(DaDb−DbDa)Ac = RdabcAd.

Properties of the curvature tensor

(read about them in your textbook).
Rabcd=−Rbacd,  Rabcd=−Rabdc,  Rabcd=Rcdab,  Ra[bcd]=0,  Rab[cd;e]=0,
where the square brackets denote symmetrisation over the indices and the semi−colon is a covariant derivative. The fourth and fifth identities are sometimes called the "algebraic Bianchi identity" and the "differential Bianchi identity", respectively.

Ricci tensor

Rab = Rdadb

Ricci scalar

R = gabRab

Exercises

  1. Compute all the non−vanishing components of the Riemann tensor Rabcd (where a,b,c,d=θ,φ) for the metric
    ds2 = r2(dθ2+sin2θdφ2)
    on a 2−dimensional sphere of radius r. Calculate also the Ricci tensor Rab and the scalar curvature R.

    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

Copyleft © 2005 D.V.Fedorov (fedorov @ phys au dk)