Show that the coefficients of the connection Γαβγ on the coordinate system xα are related to the corresponding coefficients Γαβ ¯¯¯γ by Γαβ ¯¯¯γ =∂xα¯ ∂xα ∂xβ ∂xβ¯ ∂xγ ∂xγ¯Γαβγ+∂xα¯ ∂xα ∂2xα ∂xβ¯∂x¯γ

Physics 553 : Problem Set 8

Due Tuesday, Nov 7, 1998

Q1. Transformation property of connection coefficients:

a. Let M be a manifold, and let x^αand x^α¯ be two overlapping coordinate systems defined on M . Let ∇ be a connection on M . Show that the coefficients of the connection Γ^α_βγ on the coordinate system x^α are related to the corresponding coefficients Γ^α_{β ¯}^¯_¯γ by

Γ^αβ ¯^¯¯γ =∂x^α¯

∂x^α

∂x^β

∂x^β¯

∂x^γ

∂x^γ¯Γ^α_βγ+∂x^α¯

∂x^α

∂²x^α

∂x^β¯∂x^¯γ.

b. Consider three dimensional Euclidean space with Cartesian coordinates xⁱ= (x, y, z) and spherical polar coordinates x^¯i = (r, θ, ϕ). Let∇ be the connection whose connection coefficients Γⁱjk in the Cartesian coordinate system vanish. Compute the connection coefficients Γ^¯i_{¯j ¯k}. Derive a formula for the divergence

∇ivⁱ of the vector field

~v = v^r(r, θ, ϕ)∂r+ v^θ(r, θ, ϕ)∂θ+ v^ϕ(r, θ, ϕ)∂ϕ.

Q3. Local representations of metrics:

a. If g is a non-degenerate, covariant, symmetric two index tensor on a vector space V , show that one can always find a basis θ^αof V^*such that g = gαβθ^α⊗ θ^β, where gαβis diagonal with each diagonal element being +1 or−1.

b. LetP be a point in a manifold M, and let ~eα(P) be an arbitrary basis of the tangent space TP(M ).

Show that one can always find a coordinate system{x^α} such that

~e_α(P) =

 ∂

∂x^α



P

.

c. By combining a. and b., show that if g is a metric on a manifold M , given a pointP, one can always find coordinates x^αsuch that gαβ(P) is diagonal with diagonal elements ±1.

d. Show that one can further specialize the choice of coordinate system{x^α} such that

∂g_αβ

∂x^γ (P) = 0.

Such coordinates are called, in the context of general relativity, local Lorentz coordinates adapted to the pointP. [Hint: start with a general coordinate system with x^α(P) = 0, use a coordinate transformation of the form

y^α(x^β) = a^α+ b^α_βx^β+ c^α_βγx^βx^γ+ O(x³),

compute the metric in the new coordinate system, and choose the constants a^α, b^α_β and c^α_βγ suitably].

Q3. The two sphere: Consider the unit two-sphere{(x, y, z) | x²+ y²+ z²= 1} with coordinates (θ, ϕ).

a. Suppose that∇ is a connection on the two-sphere for which the geodesics are all great circles [i.e., circles in a plane which intersects the origin (x, y, z) = (0, 0, 0)]. Determine the connection coefficients of∇ in the coordinate system (θ, ϕ). Is this connection a flat connection?

b. Give an example of a flat connection in a local region on the two sphere.

c. Optional: Show that there is no globally defined, smooth, flat connection on the two sphere.