36D (General Relativity)

L = \frac{1}{2} (- λ^{2} \dot{t}^{2} + μ^{2} \overset{r}{˙}^{2} + r^{2} \dot{θ}^{2} + r^{2} sin^{2} θ \dot{ϕ}^{2}),

(1)

where here the overdot on a coordinate means differentiation with respect to the affine parameter. Applying the Euler-Lagrange equation to each coordinate and comparing with the geodesic equation

\overset{x}{¨}^{α} + Γ_{β γ}^{α} \overset{x}{˙}^{β} \overset{x}{˙}^{γ} = 0

(2)

determines the Christoffel symbols. To display derivatives of the metric coefficients unambiguously, write

λ_{t} = \partial_{t} λ

λ_{r} = \partial_{r} λ

, and similarly for

μ

. The nonzero symbols, together with those obtained by symmetry in the two lower indices, are

Γ_{tt}^{t} = \frac{λ _{t}}{λ}, Γ_{tt}^{r} = \frac{λ λ _{r}}{μ ^{2}}, Γ_{θθ}^{r} = - \frac{r}{μ ^{2}}, Γ_{r θ}^{θ} = Γ_{θ r}^{θ} = \frac{1}{r}, Γ_{r ϕ}^{ϕ} = Γ_{ϕ r}^{ϕ} = \frac{1}{r}, Γ_{t r}^{t} = Γ_{r t}^{t} = \frac{λ _{r}}{λ}, Γ_{t r}^{r} = Γ_{r t}^{r} = \frac{μ _{t}}{μ}, Γ_{ϕϕ}^{r} = - \frac{r sin ^{2} θ}{μ ^{2}}, Γ_{ϕϕ}^{θ} = - sin θ cos θ, Γ_{θϕ}^{ϕ} = Γ_{ϕθ}^{ϕ} = cot θ . Γ_{rr}^{t} = \frac{μ μ _{t}}{λ ^{2}}, Γ_{rr}^{r} = \frac{μ _{r}}{μ},

(3)

This is the Christoffel symbols of a diagonal spherical spacetime metric.

b

 0  0

Solution

 0  0

For the constant density

ρ_{0}

m (r) = 4 π \int_{0}^{r} ρ_{0} s^{2} d s = \frac{4 π}{3} ρ_{0} r^{3} .

(1)

The definition of

m

then gives

\frac{1}{μ ^{2}} = 1 - \frac{2 m ( r )}{r} = 1 - \frac{8 π}{3} ρ_{0} r^{2} .

(2)

Substitution in the Tolman–Oppenheimer–Volkoff equation yields

\frac{d P}{d r} = - \frac{4 π r}{1 - \frac{8 π}{3} ρ _{0} r ^{2}} (P + \frac{ρ _{0}}{3}) (P + ρ_{0}) .

(3)

The minus sign follows both from the stated TOV equation and from the outward decrease of pressure; a version of the requested intermediate formula without that sign is a typographical error.

Set