The Riemann curvature tensor

The Christoffel symbols are not covariant tensors and cannot be easily used in variational principles. In this section we shall find a covariant tensor associated with the curvature of the space which can be used in constructing actions.

If we make a parallel transform (DA^a=0) of a vector A along a path dx, the components of the vector will generally change, dA^a=−Γ^a_bcdx^bA^c. The change ∆A^a=(∫)dA^a along an infinitesimal closed contour is apparently a covariant quantity. It must be proportional to the vector itself and the area of the surface enclosed by the contour. Let us calculate this change using a simple rectangular contour with sides dx and dx′. The vector A^a after a parallel transform along dx and then dx′ turns into

A^a(x+dx+dx′) = A^a−Γ^a_bcdx^bA^c

−(Γ^a_bc+Γ^a_bc,ddx^d)dx′^b(A^c−Γ^c_efdx^eA^f) .

(1)

Similarly, after a parallel transform first along dx′ and then along dx

A^a(x+dx′+dx) = A^a−Γ^a_bcdx′^bA^c

−(Γ^a_bc+Γ^a_bc,ddx′^d)dx^b(A^c−Γ^c_efdx′^eA^f) .

(2)

Subtracting these two and leaving only the lowest order term gives (after some index renaming)

∆A^a=A^a(x+dx+dx′)−A^a(x+dx′+dx)

= −(Γ^a_bd,c−Γ^a_bc,d+Γ^a_ceΓ^e_bd+Γ^a_deΓ^e_bc) A^bdx^cdx′^d

(3)

≡ −R^a_bcdA^bdx^cdx′^d .

The expression in brackets is called the Riemann tensor R^a_bcd,

R^a_bcd = Γ^a_bd,c − Γ^a_bc,d +Γ^a_ecΓ^e_bd − Γ^a_edΓ^e_bc .

(4)

It follows from (3) that if the Riemann tensor vanishes, the vector A^a does not change if parallely transported along an arbitrary infinitesimal closed path. Also after a parallel transform from x to x+δx the vector A^a(x+δx) is independent on which path is taken. The space where Riemann tensor vanishes everywhere is called flat.

The Riemann tensor defines also the commutator of covariant derivatives

(D_aD_b−D_bD_a)A_c = R^d_abcA_d,

(5)

where D_a ≡ [(D)/(dx^a)].

Properties of the curvature tensor

(Read about them in your textbook.)

R_abcd

−R_bacd,

(6)

R_abcd

−R_abdc,

(7)

R_abcd

R_cdab,

(8)

R_a[bcd]

(9)

R_ab[cd;e]

(10)

where the square brackets denote symmetrization over the indexes and the semi-colon is a covariant derivative. The fourth and fifth identities are sometimes called the älgebraic Bianchi identity" and the "differential Bianchi identity", respectively.

Ricci tensor

R_ab = R^d_adb

(11)

Ricci scalar

R = g^abR_ab

(12)

Exercises

Compute all the non-vanishing components of the Riemann tensor R_abcd (where a,b,c,d=θ,φ) for the metric

ds² = r²(dθ²+sin²θdφ²)
(13)
on a 2-dimensional sphere of radius r. Calculate also the Ricci tensor R_ab and the scalar curvature (Ricci scalar) R.
Answer:

R_θφθφ=r²sin²θ = R_φθφθ = −R_θφφθ = −R_φθθφ

File translated from T_EX by T_TH, version 3.77.
On 2 Oct 2007, 17:51.