Solution

Write

f (r) = 1 + r^{2} / a^{2}

. The geodesic Lagrangian is

L = \frac{1}{2} [- f \dot{t}^{2} + f^{- 1} \overset{r}{˙}^{2} + r^{2} \dot{θ}^{2} + r^{2} sin^{2} θ \dot{ϕ}^{2}] .

Its

θ

equation is

\frac{d}{d τ} (r^{2} \dot{θ}) - r^{2} sin θ cos θ \dot{ϕ}^{2} = 0.

Spherical symmetry lets us rotate the conserved angular-momentum plane into

θ = π /2

. Equivalently, initial data

θ = π /2

\dot{θ} = 0

solve this equation for all

τ

The cyclic coordinates give

E = f \dot{t}, h = r^{2} \dot{ϕ} .

Using the timelike normalization

g_{μν} \overset{x}{˙}^{μ} \overset{x}{˙}^{ν} = - 1

gives

\overset{r}{˙}^{2} + f (1 + \frac{h ^{2}}{r ^{2}}) = E^{2} .

After separating the constant terms,

\frac{1}{2} \overset{r}{˙}^{2} + V (r) = \frac{1}{2} (E^{2} - 1 - \frac{h ^{2}}{a ^{2}}), V (r) = \frac{1}{2} (\frac{r ^{2}}{a ^{2}} + \frac{h ^{2}}{r ^{2}}) .

Solved by gpt-5.6-sol high.

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