Static models with spherical symmetry¶

Orthogonal Metrics¶

An orthogonal metric takes the form

\begin{array}{r} \begin{matrix} \tensor g_{α β} = 0 \forall α \neq β \\ i m p l y i n g t h a t t h e r e a r e n o c r o s s - t e r m s i n t h e i n v a r i a n t i n t e r v a l . T h u s \end{matrix} \end{array}

\begin{array}{r} \begin{matrix} \dd s^{2} = \tensor g_{α α} (\dd x^{α})^{2} \\ T h e c o m p o n e n t s o f a m e t r i c w i l l n o t b e o r t h o g o n a l i n e v e r y c o o r d i n a t e \end{matrix} \end{array}

system; suppose that $\tensor g_{α β}$ are the metric components in a particular coordinate system where the metric is orthogonal. Let $\tensor g^{'}_{μ ν}$ be the components of the metric in another coordinate system, then

Misplaced \cr

The orthogonal metric components are closely related to the question of whether the coordinate system has orthogonal basis vectors. If we have a coordinate system with basis vectors ${{\vec{e}}_{i}}$ , and two vectors $\vec{A} = A^{i} {\vec{e}}_{i}$ , $\vec{B} = B^{i} {\vec{e}}_{i}$ , then

\begin{array}{r} \begin{matrix} \vec{A} \vdot \vec{B} = (A^{i} {\vec{e}}_{i}) \vdot (B^{j} {\vec{e}}_{j}) = A^{i} B^{j} ({\vec{e}}_{i} \vdot {\vec{e}}_{j}) \\ s o i t f o l l o w s \end{matrix} \end{array}

\begin{array}{r} \begin{matrix} \tensor g_{i j} = {\vec{e}}_{i} \vdot {\vec{e}}_{j} \\ a n d : m a t h : ‘ \tensor g_{i j} = 0 ‘ i f f : m a t h : ‘ {\vec{e}}_{i} ‘ a n d \end{matrix} \end{array}

${\vec{e}}_{j}$ are orthogonal.

We normally want to choose a coordinate system in which the metric coefficients are orthogonal, to simplify the expressions for geometrical objects; if the contravariant metric components are orthogonal then the diagonal terms are simply the reciprocal of the covariant diagonal terms. To see this, we know

\begin{array}{r} \begin{matrix} g^{γ β} g_{γ γ} = δ_{γ}^{β} ∴ g^{γ γ} = \frac{1}{g_{γ γ}} \\ T h e C h r i s t o f f e l s y m b o l s a l s o t a k e a s i m p l e f o r m i n a n o r t h o g o n a l \end{matrix} \end{array}

metric,

\begin{array}{r} \begin{aligned} Γ_{μ ν}^{λ} & = 0 for λ, μ, ν all different. \\ Γ_{λ μ}^{λ} = Γ_{μ λ}^{λ} & = \frac{g_{λ λ, λ}}{2 g_{λ λ}} \\ Γ_{μ μ}^{λ} & = - \frac{g_{μ μ, λ}}{2 g_{λ λ}} \\ Γ_{λ λ}^{λ} & = \frac{g_{λ λ, λ}}{2 g_{λ λ}} \end{aligned} \end{array}

For an affine parameter $s$ the geodesic equation takes the form

\begin{array}{r} \begin{matrix} \dv s \qty (g_{λ ν} \dv x^{ν} s) - \half \pdv g_{μ ν} x^{λ} \dv x^{μ} s \dv x^{ν} s = 0 \\ f o r a n o r t h o g o n a l m e t r i c t h i s r e d u c e s t o \end{matrix} \end{array}

\dv s \qty (g_{λ λ} \dv x^{λ} s) - \half \pdv g_{μ μ} x^{λ} \qty (\dv x^{μ} s)^{2} = 0

Spherically symmetric metrics¶

Spherically symmetric solutions to the field equations are suitable for describing the spacetime inside and around stars. In flat Minkowski spacetime a polar coordinate system can be used to give an invariant interval

\begin{array}{r} \begin{matrix} \dd s^{2} = - \dd t^{2} + \dd r^{2} + r^{2} \qty (\dd θ^{2} + \sin^{2} θ \dd ϕ^{2}) \\ S u r f a c e s o f c o n s t a n t : m a t h : ‘ r ‘ a n d : m a t h : ‘ t ‘ h a v e t h e g e o m e t r y o f a \end{matrix} \end{array}

2-sphere with an interval

\begin{array}{r} \begin{matrix} \dd ℓ^{2} = r^{2} \qty (\dd θ^{2} + \sin^{2} θ \dd ϕ^{2}) \\ T h u s a s p a c e t i m e i s s p h e r i c a l l y s y m m e t r i c i f e v e r y p o i n t i n t h e \end{matrix} \end{array}

spacetime lies on a 2D surface which is a 2-sphere. Labelling the coordinates $(r^{'}, t, θ, ϕ)$ then every point in a spherically symmetric spacetime lies on a 2D surface, given by

\begin{array}{r} \begin{matrix} \dd ℓ^{2} = f (r^{'}, t) \qty [\dd θ^{2} + \sin^{2} θ \dd ϕ^{2}] \\ w i t h : m a t h : ‘ \sqrt{f} ‘ t h e r a d i u s o f c u r v a t u r e o f t h e 2 - s p h e r e . \end{matrix} \end{array}

In curved spacetime there is no trivial relation between the angular coordinates of the two-sphere and the remaining coordinates at each point in spacetime, but if we define

\begin{array}{r} \begin{matrix} r^{2} = f (r^{'}, t) \\ a n d w e c a n l i n e - u p t h e o r i g i n s o f t h e 2 - s p h e r e c o o r d i n a t e s y s t e m s \end{matrix} \end{array}

$(θ, ϕ)$ for points in spacetime with different values of $r$ . Spherical symmetry requires that any radial path in the space is orthogonal to the 2D spheres on which the points along the radial path lie, implying

\begin{array}{r} \begin{matrix} g_{r θ} = g_{r ϕ} = 0 \\ S o t h e s p a c e t i m e m e t r i c i s r e d u c e d t o t h e f o r m \end{matrix} \end{array}

\begin{array}{r} \begin{matrix} \begin{matrix} \begin{matrix} \dd s^{2} = g_{t t} \dd t^{2} + 2 g_{t r} \dd r \dd t + 2 g_{t θ} \dd θ \dd t \\ + g_{r r} \dd r^{2} + r^{2} \qty (\dd θ^{2} + \sin [2] (θ) \dd ϕ^{2}) \end{matrix} \end{matrix} \\ C o n s i d e r i n g t h e c u r v e w i t h : m a t h : ‘ r ‘, : m a t h : ‘ θ ‘, a n d : m a t h : ‘ ϕ ‘ \end{matrix} \end{array}

are constant, which is a worldline of a particle in spacetime with constant spatial coordinates; this curve must also be orthogonal to 2-spheres on which each point lies, otherwise there would be a preferred direction in spacetime. Thus

\begin{array}{r} \begin{matrix} g_{t θ} = g_{t ϕ} = 0 \\ s o t h e g e n e r a l f o r m f o r a m e t r i c i n a s p h e r i c a l l y s y m m e t r i c s p a c e t i m e \end{matrix} \end{array}

\begin{array}{r} \begin{matrix} \begin{matrix} \begin{matrix} \dd s^{2} = g_{t t} \dd t^{2} + 2 g_{t r} \dd r \dd t + g_{r r} \dd r^{2} \\ + r^{2} \qty (\dd θ^{2} + \sin [2] (θ) \dd ϕ^{2}) \end{matrix} \end{matrix} \\ F o r : m a t h : ‘ g_{t t} ‘, : m a t h : ‘ g_{t r} ‘, a n d : m a t h : ‘ g_{r r} ‘ a r b i t r a r y \end{matrix} \end{array}

functions of $r$ and $t$ .

Static Spacetime¶

In s static spherically symmetric spacetime we can find a time coordinate $t$ where

all metric components are independent of $t$
the metric is unchanged under a time-reversal operation, :math:`t to
-t`.

The second property implies $g_{t r} = 0$ , meaning that the interval is

\begin{array}{r} \begin{matrix} \dd s^{2} = - e^{ν} \dd t^{2} + e^{λ} \dd r^{2} + r^{2} \qty (\dd θ^{2} + \sin [2] (θ) \dd ϕ^{2}) \\ w h i c h i s o r t h o g o n a l . T h e f u n c t i o n s : m a t h : ‘ ν (r) ‘ a n d \end{matrix} \end{array}

$λ (r)$ replace $g_{t t}$ and $g_{r r}$ ,since the exponential function is strictly positive for all $r$ , this is legitimate, provided $g_{t t} < 0$ and $g_{r r} > 0$ .

The Christoffel symbols for this spacetime are

\begin{array}{r} \begin{aligned} Γ_{r t}^{t} = Γ_{t r}^{t} & = \half ν^{'} & Γ_{r θ}^{θ} = Γ_{θ r}^{θ} & = \frac{1}{r} \\ Γ_{t t}^{r} & = \half ν^{'} e^{ν - λ} & Γ_{ϕ ϕ}^{θ} & = - \sin (θ) \cos (θ) \\ Γ_{r r}^{r} & = \half λ^{'} & Γ_{r ϕ}^{ϕ} = Γ_{ϕ r}^{ϕ} & = \frac{1}{r} \\ Γ_{θ θ}^{r} & = - r e^{- λ} & Γ_{θ ϕ}^{ϕ} = Γ_{ϕ θ}^{ϕ} & = \cot (θ) \end{aligned} \end{array}

Γ_{ϕ ϕ}^{r} = - r e^{- λ} \sin [2] (θ)

The Ricci tensor is given by

\begin{array}{r} \begin{matrix} R_{λ ν} = Γ_{λ ν}^{τ} Γ_{τ σ}^{σ} - Γ_{λ σ}^{τ} Γ_{τ ν}^{σ} + Γ_{λ ν, σ}^{σ} - Γ_{λ σ, ν}^{σ} \\ s o \end{matrix} \end{array}

\begin{array}{r} \begin{aligned} R_{t t} & = \half e^{ν - λ} \qty (ν^{″} + \half ν^{' 2} - \half ν^{'} λ^{'} + \frac{2}{r} ν^{'}) \\ R_{r r} & = - \half \qty (ν^{″} + \half ν^{' 2} - \half ν^{'} λ^{'} - \frac{2}{r} λ^{'}) \\ R_{θ θ} & = 1 - e^{- λ} \qty [1 + \frac{r}{2} (ν^{'} - λ^{'})] \\ R_{ϕ ϕ} & = R_{θ θ} \sin [2] (θ) \end{aligned} \end{array}

The Schwarzschild metric¶

We can derive the metric for the spacetime exterior to a star from the static spherically symmetric metric; the Schwarzchild metric; if the star is in an isolated region of space we can assume all components of the Ricci tensor to be zero, so

\begin{array}{r} \begin{matrix} e^{λ - ν} R_{t t} + R_{r r} = \frac{ν^{'} + λ^{'}}{r} = 0 \\ w h i c h i m p l i e s t h a t : m a t h : ‘ ν + λ ‘ i s c o n s t a n t . A t a l a r g e d i s t a n c e \end{matrix} \end{array}

from the star the metric should reduce to special relativity, so as

\begin{array}{r} \begin{matrix} r \to \infty, e^{ν} \to 1, e^{λ} \to 1 ⟹ ν \to 0, λ \to 0 \\ a n d s o : m a t h : ‘ ν + λ = 0 ‘, g i v i n g \end{matrix} \end{array}

\begin{array}{r} \begin{matrix} e^{ν} = e^{- λ} \\ T h i s l e t s u s e l i m i n a t e : m a t h : ‘ ν ‘ f r o m t h e : m a t h : ‘ R_{θ θ} ‘ \end{matrix} \end{array}

expression, equation , so

\begin{array}{r} \begin{matrix} e^{- λ} (1 - λ^{'} r) = 1 ⟹ \dv r \qty (r e^{- λ}) = 1 \\ I n t e g r a t i n g t h i s w e g e t \end{matrix} \end{array}

\begin{array}{r} \begin{matrix} e^{ν} = e^{- λ} = 1 + \frac{α}{r} \\ w h e r e : m a t h : ‘ α ‘ i s a c o n s t a n t . \end{matrix} \end{array}

Consider a material test particle, with so little rest mass that it does not disturb the metric, which is released from rest, then

\begin{array}{r} \begin{matrix} \dd x^{j} τ = 0 j = 1, 2, 3 \\ f o r : m a t h : ‘ τ ‘ t h e p r o p e r t i m e, a n d \end{matrix} \end{array}

\begin{array}{r} \begin{matrix} \dv x^{0} τ \equiv \dv t τ \neq 0 \\ R e a l l i n g t h a t \end{matrix} \end{array}

\begin{array}{r} \begin{matrix} g_{α β} \dv x^{α} τ \dv x^{β} τ = - 1 \\ t h e n \end{matrix} \end{array}

\begin{array}{r} \begin{matrix} \dv t τ = e^{- \frac{ν}{2}} \\ A p p l y i n g t h e g e o d e s i c e q u a t i o n s, e q u a t i o n, a t t h e i n s t a n c e t h a t t h e \end{matrix} \end{array}

particle is released this reduces to

\begin{array}{r} \begin{matrix} \dv [2] r τ + Γ_{t t}^{r} \qty (\dv t τ)^{2} = 0 \\ S u b s t i t u t i n g t h e C h r i s t o f f e l s y m b o l a n d e q u a t i o n w e o b t a i n \end{matrix} \end{array}

\begin{array}{r} \begin{matrix} \dv [2] r t = \frac{α}{2 r^{2}} \\ I n t h e l i m i t o f a w e a k f i e l d t h i s r e d u c e s t o N e w t o n i a n g r a v i t y, \end{matrix} \end{array}

\begin{array}{r} \begin{matrix} \dv [2] r t = - \frac{G M}{r^{2}} \\ f o r : m a t h : ‘ M ‘ t h e m a s s o f t h e s t a r, m e a n i n g : m a t h : ‘ α = - 2 G M = - 2 M ‘ \end{matrix} \end{array}

for $G = 1$ . Thus the invariant interval is

\begin{array}{r} \begin{matrix} \dd s^{2} = - \qty (1 - \frac{2 M}{r}) \dd t^{2} + \frac{\dd r^{2}}{\qty (1 - \frac{2 M}{r})} \\ + r^{2} \dd θ^{2} + r^{2} \sin [2] (θ) \dd ϕ^{2} \end{matrix} \end{array}