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Christoffel symbols of the Schwarzschild exterior metric in product charts

Statement

Let $(M, g, N)$ be the Schwarzschild exterior datum of mass parameter $m_{*}$ with Schwarzschild radius $s = 2 G_{N} m_{*} / c^{2}$ and $f (r) = 1 - s / r$ (The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation), and let $\nabla$ be the metric-selected connection of $g$ (Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields). In every spherical-polar product chart $(x^{0}, r, ϑ, φ)$ with $0 < ϑ < π$ and $f^{'} = \frac{d f}{d r} = s / r^{2}$ , the Christoffel symbols $Γ^{a}_{b c}$ of $\nabla$ (Christoffel symbols of an affine connection) are:

(a) $Γ^{0}_{01} = Γ^{0}_{10} = \frac{f^{'}}{2 f}$ , and $Γ^{0}_{00} = Γ^{0}_{11} = Γ^{0}_{i j} = 0$ for all remaining index pairs;

(b) $Γ^{1}_{00} = \frac{f f^{'}}{2}$ , $Γ^{1}_{11} = - \frac{f^{'}}{2 f}$ , $Γ^{1}_{22} = - r f$ , $Γ^{1}_{33} = - r f \sin^{2} ϑ$ , and $Γ^{1}_{01} = Γ^{1}_{02} = Γ^{1}_{03} = Γ^{1}_{12} = Γ^{1}_{13} = Γ^{1}_{23} = 0$ ;

(c) $Γ^{2}_{12} = Γ^{2}_{21} = \frac{1}{r}$ , $Γ^{2}_{33} = - \sin ϑ \cos ϑ$ , and every other symbol with upper index $2$ vanishes;

(d) $Γ^{3}_{13} = Γ^{3}_{31} = \frac{1}{r}$ , $Γ^{3}_{23} = Γ^{3}_{32} = \cot ϑ$ , and every other symbol with upper index $3$ vanishes.

All components not listed vanish; equivalently, the only nonzero symbols are those displayed. For the model value $f (r) = 1 - s / r$ , so that $f^{'} = s / r^{2}$ , the listed symbols read

$Γ^{0}_{01} = Γ^{0}_{10} = \frac{s}{2 r^{2} f}, Γ^{1}_{00} = \frac{s f}{2 r^{2}}, Γ^{1}_{11} = - \frac{s}{2 r^{2} f},$

$Γ^{1}_{22} = - r f, Γ^{1}_{33} = - r f \sin^{2} ϑ, Γ^{2}_{12} = Γ^{3}_{13} = \frac{1}{r}, Γ^{2}_{33} = - \sin ϑ \cos ϑ, Γ^{3}_{23} = \cot ϑ .$

Facts & Assumptions

Given: The Schwarzschild exterior datum with Schwarzschild radius $s$ , the metric $g$ and its metric-selected connection $\nabla$ , and a product chart $(x^{0}, r, ϑ, φ)$ with $0 < ϑ < π$ .

[L1]

In the product chart the metric components are $g_{00} = - f$ , $g_{11} = f^{- 1}$ , $g_{22} = r^{2}$ , $g_{33} = r^{2} \sin^{2} ϑ$ and $g_{a b} = 0$ for $a \neq b$ , with $f = f (r) > 0$ smooth; the metric-selected connection exists and is the unique torsion-free $g$ -compatible affine connection (The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation, The Schwarzschild exterior datum is a smooth time-oriented Lorentzian four-manifold, Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields).

[L2]

The inverse-metric components are characterised by $\sum_{b} g^{a b} g_{b c} = δ^{a}_{c}$ ; for the diagonal matrix of [L1] they are $g^{00} = - f^{- 1}$ , $g^{11} = f$ , $g^{22} = r^{- 2}$ , $g^{33} = r^{- 2} \sin^{- 2} ϑ$ , all off-diagonal components vanishing (The inverse metric, musical isomorphisms and index raising, Tensor fields of type (r,s) on a Lorentzian four-manifold and their components).

[L3]

In every smooth chart the Christoffel symbols of the metric-selected connection are $Γ^{a}_{b c} = \frac{1}{2} \sum_{d} g^{a d} (\partial_{b} g_{d c} + \partial_{c} g_{b d} - \partial_{d} g_{b c})$ , and they are symmetric in the lower pair, $Γ^{a}_{b c} = Γ^{a}_{c b}$ (Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields clause (a), Christoffel symbols of an affine connection, Torsion freedom and metric compatibility in chart components).

[L4]

All metric components of [L1] are independent of $x^{0}$ and $φ$ , so $\partial_{0} g_{b c} = 0$ and $\partial_{3} g_{b c} = 0$ for all indices; the coordinate partial derivatives of the components are those of Directional derivatives and partial derivatives of a map $U \subseteq R^{m} \to R^{n}$ , and $\frac{d}{d r} f = f^{'} = \frac{s}{r^{2}}$ for the model function $f (r) = 1 - s / r$ by The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set and the power rule.

[L5]

On the chart domain $0 < ϑ < π$ , so $\sin ϑ > 0$ , $\sin^{2} ϑ > 0$ and $r > s > 0$ ; the displayed expressions for $Γ$ are smooth there (Rational powers $a^{r}$ of a positive base).

Proof

technique · substitute the metric components of [L1] and the inverse components of [L2] into the Koszul formula of [L3], organised by the upper index

1.1L1L2L4algebra

(Metric derivatives and inverse components.) The only nonconstant components of [L1] are $g_{00} = - f (r)$ , $g_{11} = f (r)^{- 1}$ , $g_{22} = r^{2}$ , $g_{33} = r^{2} \sin^{2} ϑ$ , and their nonzero coordinate derivatives are $\partial_{1} g_{00} = - f^{'}$ , $\partial_{1} g_{11} = - f^{'} / f^{2}$ , $\partial_{1} g_{22} = 2 r$ , $\partial_{1} g_{33} = 2 r \sin^{2} ϑ$ , $\partial_{2} g_{33} = 2 r^{2} \sin ϑ \cos ϑ$ ; by [L4] every $\partial_{0}$ or $\partial_{3}$ derivative vanishes. The inverse components are $g^{00} = - 1 / f$ , $g^{11} = f$ , $g^{22} = 1 / r^{2}$ , $g^{33} = 1 / (r^{2} \sin^{2} ϑ)$ , as the diagonal matrix inverse of [L2].

2.1L3step 1.1algebra

(The family $Γ^{0}_{\cdot \cdot}$ .) By [L3], $Γ^{0}_{b c} = \frac{1}{2} g^{00} (\partial_{b} g_{0 c} + \partial_{c} g_{0 b} - \partial_{0} g_{b c})$ . If ${b, c}$ contains an index in ${2, 3}$ then $g_{0 c}$ or $g_{0 b}$ vanishes identically and $\partial_{0} g_{b c} = 0$ ; the cases $(b, c) = (0, 0)$ and $(1, 1)$ give $\partial_{0} g_{00} = 0$ and $\partial_{0} g_{11} = 0$ ; hence only $(b, c) \in {(0, 1), (1, 0)}$ can contribute, with value $\frac{1}{2} g^{00} \partial_{1} g_{00} = \frac{1}{2} (- 1 / f) (- f^{'}) = f^{'} / (2 f)$ . This is claim (a).

2.2L3step 1.1algebra

(The family $Γ^{1}_{\cdot \cdot}$ .) By [L3], $Γ^{1}_{b c} = \frac{1}{2} g^{11} (\partial_{b} g_{1 c} + \partial_{c} g_{1 b} - \partial_{1} g_{b c})$ . If $g_{1 c}$ or $g_{1 b}$ is nonzero the corresponding term contributes; since $g_{1 d} = 0$ unless $d = 1$ , only $b = c = 1$ retains a contribution and $Γ^{1}_{11} = \frac{1}{2} g^{11} \partial_{1} g_{11} = \frac{1}{2} f (- f^{'} / f^{2}) = - f^{'} / (2 f)$ . The cases $(b, c) = (0, 0)$ , $(2, 2)$ and $(3, 3)$ have $Γ^{1}_{b c} = - \frac{1}{2} g^{11} \partial_{1} g_{b c}$ , giving $Γ^{1}_{00} = \frac{1}{2} f f^{'} = f f^{'} / 2$ , $Γ^{1}_{22} = - \frac{1}{2} f (2 r) = - r f$ and $Γ^{1}_{33} = - r f \sin^{2} ϑ$ . Every mixed case $(b \neq c)$ has $\partial_{b} g_{1 c} + \partial_{c} g_{1 b} = 0$ and $\partial_{1} g_{b c} = 0$ for the off-diagonal components, hence vanishes. This is claim (b).

2.3L3L4step 1.1algebra

(The families $Γ^{2}_{\cdot \cdot}$ and $Γ^{3}_{\cdot \cdot}$ .) For the upper index $2$ , [L3] gives $Γ^{2}_{b c} = \frac{1}{2} g^{22} (\partial_{b} g_{2 c} + \partial_{c} g_{2 b} - \partial_{2} g_{b c})$ with $g^{22} = 1 / r^{2}$ . The only nonzero $g_{2 c}$ is $c = 2$ , so only $b = c = 2$ and the pairs with $g_{b c}$ depending on $ϑ$ contribute: $Γ^{2}_{22} = \frac{1}{2} r^{- 2} \partial_{2} g_{22} = 0$ and $Γ^{2}_{33} = - \frac{1}{2} r^{- 2} \partial_{2} g_{33} = - \frac{1}{2} r^{- 2} (2 r^{2} \sin ϑ \cos ϑ) = - \sin ϑ \cos ϑ$ , while $Γ^{2}_{12} = Γ^{2}_{21} = \frac{1}{2} r^{- 2} \partial_{1} g_{22} = r^{- 2} \cdot r = 1 / r$ and the $(0, 0)$ case vanishes. For the upper index $3$ , $g^{33} = 1 / (r^{2} \sin^{2} ϑ)$ and only $g_{33}$ is nonzero among the $g_{3 c}$ : $Γ^{3}_{13} = Γ^{3}_{31} = \frac{1}{2} (r^{2} \sin^{2} ϑ)^{- 1} \partial_{1} g_{33} = \frac{1}{2} (r^{2} \sin^{2} ϑ)^{- 1} (2 r \sin^{2} ϑ) = 1 / r$ , $Γ^{3}_{23} = Γ^{3}_{32} = \frac{1}{2} (r^{2} \sin^{2} ϑ)^{- 1} \partial_{2} g_{33} = \frac{1}{2} (r^{2} \sin^{2} ϑ)^{- 1} (2 r^{2} \sin ϑ \cos ϑ) = \cot ϑ$ , and $Γ^{3}_{33} = \frac{1}{2} g^{33} \partial_{3} g_{33} = 0$ by [L4]; all cases with an index $0$ or $1$ in $b, c$ other than the displayed ones vanish because the corresponding $g_{3 c}$ or $g_{b c}$ is constant. Together with the lower-pair symmetry of [L3], these are claims (c) and (d).

3.1L3L4L5step 2.1step 2.2step 2.3∎

(Exhaustiveness and the model values.) The index triples $(a, b, c)$ with $a, b, c \in {0, 1, 2, 3}$ split into the families treated in step 1.1 and steps 2.1-2.3; each family was computed from the same unique connection $\nabla$ of [L1], so the displays (a)-(d) list exactly the nonzero symbols of $\nabla$ in the chart. Substituting $f = 1 - s / r$ and $f^{'} = s / r^{2}$ of [L4] into the general expressions (a)-(d) gives the second display of the Statement. All symbols are smooth on the chart domain by [L5], and the expressions are those of the metric-selected connection, not of an arbitrary connection.

Remarks

Sign check at the leading order. For large $r$ one has $f = 1 - s / r + O (r^{- 2})$ and $f^{'} = s / r^{2}$ , so $Γ^{1}_{00} = \frac{1}{2} f f^{'} = \frac{s}{2 r^{2}} + O (r^{- 3})$ ; this is the coefficient $+ c^{- 2} \partial_{r} Φ$ with the Newtonian potential $Φ = - G_{N} m_{*} / r$ of the flat and Newtonian bridge page, whose spatial acceleration reads $- \partial_{r} Φ = - G_{N} m_{*} / r^{2}$ and matches the geodesic acceleration $- c^{2} Γ^{1}_{00}$ computed in the weak-field bridge lemma of the flat-and-Newtonian page in the overlapping regime. The comparison is made with the round metric of the same chart and is not a statement of this lemma.
Distance to the Newtonian form. The symbols here are written with $x^{0} = c t$ as the time coordinate; in the coordinate $t = x^{0} / c$ the components acquire factors of $c$ by the chain rule and the invariant statements of the following lemmas are used rather than the raw symbols.

Christoffel symbols of the Schwarzschild exterior metric in product charts

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources