October 27, 2018 20:1  WSPC/Book Trim Size for 9in x 6in  18sol24

A^{Chapter 1}

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Special and General Relativity (PHZ 4601/5606) Fall 2018 Solutions Set 10

24. Christoffel Symbols.

(a)

ds^{2} = dx^{2} + dy^{2} = dr^{2} + r^{2} dθ^{2} .

(b)

(g_{ij}) = ( \begin{matrix}1 & 0 \\ 0 & r^{2} \end{matrix} ) .

(c)

(g^{ij}) = ( \begin{matrix}1 & 0 \\ 0 & 1/r^{2} \end{matrix} ) .

(d) We use

Γ_{jk}^{i} = \frac{1}{2} g^{im} (g_{mj,k} + g_{mk,j} − g_{jk,m})  (1.1)

to calculate the Christoffel symbols. Note that g_{ij} is constant except for g_{θθ}. Therefore, g_{ij,k} = 0 except for

g_{θθ,r} = 2 r .

In the following we use the summation convention for i, j, k and m, but not for θ and r.

Inspecting Eq. (1.1) for g_{θθ,r} contributions, we find

\frac{1}{2} g^{θ m} g_{mθ,r} = \frac{1}{2} g^{θθ} g_{θθ,r} = \frac{1}{r}
\frac{1}{2} g^{r m} g_{θθ,m} = \frac{1}{2} g^{r r} g_{θθ,r} = r .

Therefore,

Γ_{θ r}^{θ} = Γ_{r θ}^{θ} = \frac{1}{r} and Γ_{θθ}^{r} = r .

All other Γ_{jk}^{i} are zero.

(e) So the equations of motion

ẍ^{i} + Γ_{jk}^{i} ẋ^{j} ẋ^{k} = 0

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become

r̈ + Γ_{θθ}^{r} θ̇ θ̇ = r̈ + r θ̇ θ̇ = 0

and

θ̈ + Γ_{θ r}^{θ} θ̇ ṙ + Γ_{r θ}^{θ} ṙ θ̇ = θ̈ + \frac{2}{r} θ̇ ṙ = 0 .