Static and spherical spacetime (part 1) February 15, 2026
Symmetries of spacetime ¶ Solutions of Einstein equations are very difficult to find because they are non-linear PDEs.
How to define symmetries in a theory where coordinates have no physical meaning?
In classical physics, a quantity F F F F ( t , x , y , z ) F(t, x, y, z) F ( t , x , y , z )
F F F ∂ t F = 0 \partial_t F = 0 ∂ t F = 0
F F F ∂ θ F = ∂ ϕ F = 0 \partial_\theta F = \partial_\phi F = 0 ∂ θ F = ∂ ϕ F = 0
How to make these properties coordinate-independent?
Lie derivative ¶ Variation of a vector ¶ Let v = v μ ∂ μ v = v^\mu \partial_\mu v = v μ ∂ μ C ( λ ) C(\lambda) C ( λ ) λ ∈ R \lambda \in \mathbb{R} λ ∈ R u μ = d x μ d λ u^\mu = \dfrac{dx^\mu}{d\lambda} u μ = d λ d x μ C ( λ ) C(\lambda) C ( λ )
Variation of v v v ¶ δ v = v ( Q ) − v ( P ) = ( v μ ∂ μ ) ( Q ) − ( v μ ∂ μ ) ( P ) = v μ ( Q ) ∂ μ ( Q ) − v μ ( P ) ∂ μ ( P ) , \begin{aligned}
\delta v &= v(Q) - v(P) \\
&= \bigl(v^\mu \partial_\mu\bigr)(Q) - \bigl(v^\mu \partial_\mu\bigr)(P) \\
&= v^\mu(Q)\,\partial_\mu(Q) - v^\mu(P)\,\partial_\mu(P),
\end{aligned} δ v = v ( Q ) − v ( P ) = ( v μ ∂ μ ) ( Q ) − ( v μ ∂ μ ) ( P ) = v μ ( Q ) ∂ μ ( Q ) − v μ ( P ) ∂ μ ( P ) , with
∂ μ ( P ) \partial_\mu(P) ∂ μ ( P ) ∂ μ ( Q ) \partial_\mu(Q) ∂ μ ( Q ) P P P Q Q Q
P ( x μ ) P(x^\mu) P ( x μ )
Q ( x ′ μ = x μ + δ λ u μ ) Q(x'^\mu = x^\mu + \delta\lambda\, u^\mu) Q ( x ′ μ = x μ + δ λ u μ )
∂ μ ( Q ) = ∂ ∂ x ′ μ = ∂ x ν ∂ x ′ μ ∂ ∂ x ν = ( δ μ ν − δ λ ∂ μ u ν ) ∂ ∂ x ν = ∂ ∂ x μ − δ λ ( ∂ μ u ν ) ∂ ∂ x ν . \partial_\mu(Q) = \frac{\partial}{\partial x'^\mu}
= \frac{\partial x^\nu}{\partial x'^\mu}\frac{\partial}{\partial x^\nu}
= \bigl(\delta^\nu_\mu - \delta\lambda\,\partial_\mu u^\nu\bigr)\frac{\partial}{\partial x^\nu}
= \frac{\partial}{\partial x^\mu} - \delta\lambda\,(\partial_\mu u^\nu)\frac{\partial}{\partial x^\nu}. ∂ μ ( Q ) = ∂ x ′ μ ∂ = ∂ x ′ μ ∂ x ν ∂ x ν ∂ = ( δ μ ν − δ λ ∂ μ u ν ) ∂ x ν ∂ = ∂ x μ ∂ − δ λ ( ∂ μ u ν ) ∂ x ν ∂ . Variation of the components ¶ v μ ( Q ) = v μ ( x ν + δ λ u ν ) = v μ ( x ν ) + δ λ u ν ∂ ν v μ = v μ ( P ) + δ λ u ν ∂ ν v μ . \begin{aligned}
v^\mu(Q) &= v^\mu(x^\nu + \delta\lambda\, u^\nu) \\
&= v^\mu(x^\nu) + \delta\lambda\, u^\nu \partial_\nu v^\mu \\
&= v^\mu(P) + \delta\lambda\, u^\nu \partial_\nu v^\mu.
\end{aligned} v μ ( Q ) = v μ ( x ν + δ λ u ν ) = v μ ( x ν ) + δ λ u ν ∂ ν v μ = v μ ( P ) + δ λ u ν ∂ ν v μ . Variation of v v v ¶ δ v = [ v μ ( P ) + δ λ u ν ∂ ν v μ ] [ ∂ μ ( P ) − δ λ ( ∂ μ u ν ) ∂ ν ( P ) ] = δ λ [ − v μ ( P ) ( ∂ μ u ν ) ∂ ν ( P ) + u ν ∂ ν v μ ∂ μ ( P ) ] + O ( δ λ 2 ) , \begin{aligned}
\delta v &= \bigl[v^\mu(P) + \delta\lambda\, u^\nu \partial_\nu v^\mu\bigr]
\bigl[\partial_\mu(P) - \delta\lambda\,(\partial_\mu u^\nu)\partial_\nu(P)\bigr] \\
&= \delta\lambda\Bigl[-v^\mu(P)(\partial_\mu u^\nu)\partial_\nu(P) + u^\nu \partial_\nu v^\mu \partial_\mu(P)\Bigr] + \mathcal{O}(\delta\lambda^2),
\end{aligned} δ v = [ v μ ( P ) + δ λ u ν ∂ ν v μ ] [ ∂ μ ( P ) − δ λ ( ∂ μ u ν ) ∂ ν ( P ) ] = δ λ [ − v μ ( P ) ( ∂ μ u ν ) ∂ ν ( P ) + u ν ∂ ν v μ ∂ μ ( P ) ] + O ( δ λ 2 ) , so
δ v = δ λ [ u ν ∂ ν v μ − v ν ∂ ν u μ ] ∂ μ ( P ) . \delta v = \delta\lambda\,\bigl[u^\nu \partial_\nu v^\mu - v^\nu \partial_\nu u^\mu\bigr] \partial_\mu(P). δ v = δ λ [ u ν ∂ ν v μ − v ν ∂ ν u μ ] ∂ μ ( P ) . Covariant version ¶ L u v μ = u ν ∇ ν v μ − v ν ∇ ν u μ . \mathcal{L}_u v^\mu = u^\nu \nabla_\nu v^\mu - v^\nu \nabla_\nu u^\mu. L u v μ = u ν ∇ ν v μ − v ν ∇ ν u μ . We have ∇ ν v μ = ∂ ν v μ + Γ ν ρ μ v ρ \nabla_\nu v^\mu = \partial_\nu v^\mu + \Gamma^\mu_{\nu\rho} v^\rho ∇ ν v μ = ∂ ν v μ + Γ ν ρ μ v ρ L u v μ = u ν ∂ ν v μ − v ν ∂ ν u μ \mathcal{L}_u v^\mu = u^\nu \partial_\nu v^\mu - v^\nu \partial_\nu u^\mu L u v μ = u ν ∂ ν v μ − v ν ∂ ν u μ
L u v μ = u ν ∇ ν v μ − v ν ∇ ν u μ + C μ , \mathcal{L}_u v^\mu = u^\nu \nabla_\nu v^\mu - v^\nu \nabla_\nu u^\mu + C^\mu, L u v μ = u ν ∇ ν v μ − v ν ∇ ν u μ + C μ , where C μ = Γ ν ρ μ ⏟ symmetric ( v ν u ρ − v ρ u ν ) ⏟ antisymmetric = 0 C^\mu = \underbrace{\Gamma^\mu_{\nu\rho}}_{\text{symmetric}}\;
\underbrace{(v^\nu u^\rho - v^\rho u^\nu)}_{\text{antisymmetric}} = 0 C μ = symmetric Γ ν ρ μ antisymmetric ( v ν u ρ − v ρ u ν ) = 0
Generalization to tensors ¶ We proceed the same way:
For a 1‑form T = T μ d x μ T = T_\mu dx^\mu T = T μ d x μ
L u T μ = u ν ∇ ν T μ + T ν ∇ ν u μ . \mathcal{L}_u T_\mu = u^\nu \nabla_\nu T_\mu + T_\nu \nabla^\nu u_\mu. L u T μ = u ν ∇ ν T μ + T ν ∇ ν u μ . For a rank‑2 tensor T = T α β d x α ⊗ d x β T = T_{\alpha\beta}\, dx^\alpha \otimes dx^\beta T = T α β d x α ⊗ d x β
L u T α β = u ν ∇ ν T α β + ( ∇ α u μ ) T μ β + ( ∇ β u μ ) T α μ . \mathcal{L}_u T_{\alpha\beta} = u^\nu \nabla_\nu T_{\alpha\beta}
+ (\nabla_\alpha u^\mu) T_{\mu\beta}
+ (\nabla_\beta u^\mu) T_{\alpha\mu}. L u T α β = u ν ∇ ν T α β + ( ∇ α u μ ) T μ β + ( ∇ β u μ ) T αμ . Killing vector fields ¶ We are interested in symmetries of spacetime – symmetries of the metric.
Equivalently,
L k g α β = 0 ⇔ k μ ∇ μ g α β ⏟ = 0 + ( ∇ α k μ ) g μ β + ( ∇ β k μ ) g α μ = 0. \begin{aligned}
\mathcal{L}_k g_{\alpha\beta} &= 0 \\
\Leftrightarrow\quad k^\mu \underbrace{\nabla_\mu g_{\alpha\beta}}_{=0}
+ (\nabla_\alpha k^\mu) g_{\mu\beta} + (\nabla_\beta k^\mu) g_{\alpha\mu} &= 0.
\end{aligned} L k g α β ⇔ k μ = 0 ∇ μ g α β + ( ∇ α k μ ) g μ β + ( ∇ β k μ ) g αμ = 0 = 0. Now ( ∇ α k μ ) g μ β = ∇ α ( k μ g μ β ) = ∇ α k β (\nabla_\alpha k^\mu) g_{\mu\beta} = \nabla_\alpha (k^\mu g_{\mu\beta}) = \nabla_\alpha k_\beta ( ∇ α k μ ) g μ β = ∇ α ( k μ g μ β ) = ∇ α k β
L k g α β = ∇ α k β + ∇ β k α = 0. \mathcal{L}_k g_{\alpha\beta} = \nabla_\alpha k_\beta + \nabla_\beta k_\alpha = 0. L k g α β = ∇ α k β + ∇ β k α = 0. Physically: “g μ ν g_{\mu\nu} g μν k k k
Examples ¶ Consider a 2‑dimensional flat metric
d s 2 = d x 2 + d y 2 = g μ ν d x μ d x ν ⇒ g μ ν = ( 1 0 0 1 ) . ds^2 = dx^2 + dy^2 = g_{\mu\nu} dx^\mu dx^\nu
\quad\Rightarrow\quad g_{\mu\nu} = \begin{pmatrix}1&0\\0&1\end{pmatrix}. d s 2 = d x 2 + d y 2 = g μν d x μ d x ν ⇒ g μν = ( 1 0 0 1 ) . This metric admits two Killing vectors:
It does not depend on x x x k ( 1 ) = ∂ x k^{(1)} = \partial_x k ( 1 ) = ∂ x ⇔ \Leftrightarrow ⇔ k ( 1 ) x = 1 , k ( 1 ) y = 0 k^{(1)x}=1,\; k^{(1)y}=0 k ( 1 ) x = 1 , k ( 1 ) y = 0 ∇ α k β ( 1 ) + ∇ β k α ( 1 ) = ∂ α k β ( 1 ) + ∂ β k α ( 1 ) = 0 \nabla_\alpha k_\beta^{(1)}+\nabla_\beta k_\alpha^{(1)} = \partial_\alpha k_\beta^{(1)}+\partial_\beta k_\alpha^{(1)} = 0 ∇ α k β ( 1 ) + ∇ β k α ( 1 ) = ∂ α k β ( 1 ) + ∂ β k α ( 1 ) = 0
It does not depend on y y y k ( 2 ) = ∂ y k^{(2)} = \partial_y k ( 2 ) = ∂ y
In polar coordinates ( r , θ ) (r,\theta) ( r , θ )
d s 2 = d r 2 + r 2 d θ 2 ⇒ g μ ν = ( 1 0 0 r 2 ) . ds^2 = dr^2 + r^2 d\theta^2
\quad\Rightarrow\quad g_{\mu\nu} = \begin{pmatrix}1&0\\0&r^2\end{pmatrix}. d s 2 = d r 2 + r 2 d θ 2 ⇒ g μν = ( 1 0 0 r 2 ) . Now there is one Killing vector because the metric does not depend on θ \theta θ
k ( 2 ) = ∂ θ = x ∂ y − y ∂ x . k^{(2)} = \partial_\theta = x\partial_y - y\partial_x. k ( 2 ) = ∂ θ = x ∂ y − y ∂ x . Check in Cartesian coordinates:
k x ( 3 ) = − y , k y ( 3 ) = x . k^{(3)}_x = -y,\qquad k^{(3)}_y = x. k x ( 3 ) = − y , k y ( 3 ) = x . Then
∇ α k β ( 3 ) − ∇ β k α ( 3 ) = ∂ α k β ( 3 ) − ∂ β k α ( 3 ) . \nabla_\alpha k_\beta^{(3)} - \nabla_\beta k_\alpha^{(3)} = \partial_\alpha k_\beta^{(3)} - \partial_\beta k_\alpha^{(3)}. ∇ α k β ( 3 ) − ∇ β k α ( 3 ) = ∂ α k β ( 3 ) − ∂ β k α ( 3 ) . Evaluating:
( α , β ) = ( x , x ) (\alpha,\beta)=(x,x) ( α , β ) = ( x , x ) ∂ x k x ( 3 ) + ∂ x k x ( 3 ) = 0 \partial_x k_x^{(3)} + \partial_x k_x^{(3)} = 0 ∂ x k x ( 3 ) + ∂ x k x ( 3 ) = 0
( α , β ) = ( y , y ) (\alpha,\beta)=(y,y) ( α , β ) = ( y , y ) ∂ y k y ( 3 ) + ∂ y k y ( 3 ) = 0 \partial_y k_y^{(3)} + \partial_y k_y^{(3)} = 0 ∂ y k y ( 3 ) + ∂ y k y ( 3 ) = 0
( α , β ) = ( x , y ) (\alpha,\beta)=(x,y) ( α , β ) = ( x , y ) ∂ x ( x ) + ∂ y ( − y ) = 1 − 1 = 0 \partial_x(x) + \partial_y(-y) = 1-1 = 0 ∂ x ( x ) + ∂ y ( − y ) = 1 − 1 = 0
Killing vectors and geodesics ¶ Killing vectors imply conserved quantities along geodesics (motion of a free particle).
Let k k k ∇ α k β + ∇ β k α = 0 \nabla_\alpha k_\beta + \nabla_\beta k_\alpha = 0 ∇ α k β + ∇ β k α = 0 u μ u^\mu u μ u ν ∇ ν u μ = 0 u^\nu \nabla_\nu u^\mu = 0 u ν ∇ ν u μ = 0
Then C = k μ u μ C = k^\mu u_\mu C = k μ u μ
Stationary spacetime ¶ Stationary spacetimes are expected to describe astrophysical objects at equilibrium (stars, black holes, ECOs).
We choose a coordinate system such that k = ∂ t k = \partial_t k = ∂ t
d s 2 = g 00 ( x a ) d t 2 + g 0 i ( x a ) d t d x i + g i j ( x a ) d x i d x j , ds^2 = g_{00}(x^a)\, dt^2 + g_{0i}(x^a)\, dt\, dx^i + g_{ij}(x^a)\, dx^i dx^j, d s 2 = g 00 ( x a ) d t 2 + g 0 i ( x a ) d t d x i + g ij ( x a ) d x i d x j , where g μ ν g_{\mu\nu} g μν t t t
Static spacetime ¶ This implies g 0 i = 0 g_{0i}=0 g 0 i = 0
d s 2 = g 00 ( x a ) d t 2 + g i j ( x a ) d x i d x j . ds^2 = g_{00}(x^a)\, dt^2 + g_{ij}(x^a)\, dx^i dx^j. d s 2 = g 00 ( x a ) d t 2 + g ij ( x a ) d x i d x j . Remark: Such a metric cannot describe rotating astrophysical objects.
Spherical symmetry ¶ A static and spherically symmetric spacetime is associated to a point O O O
d s 2 = − A ( r ) d t 2 + B ( r ) d r 2 + r 2 d Ω 2 , d Ω 2 = d θ 2 + sin 2 θ d φ 2 . \begin{aligned}
ds^2 &= -A(r)\, dt^2 + B(r)\, dr^2 + r^2 d\Omega^2, \\
d\Omega^2 &= d\theta^2 + \sin^2\theta\, d\varphi^2.
\end{aligned} d s 2 d Ω 2 = − A ( r ) d t 2 + B ( r ) d r 2 + r 2 d Ω 2 , = d θ 2 + sin 2 θ d φ 2 . It has two Killing vectors:
∂ t = k ( 1 ) \partial_t = k^{(1)} ∂ t = k ( 1 )
∂ φ = k ( 2 ) \partial_\varphi = k^{(2)} ∂ φ = k ( 2 )
Geodesic motion ¶ A particle follows a geodesic x μ ( λ ) = ( t ( λ ) , r ( λ ) , θ ( λ ) , φ ( λ ) ) x^\mu(\lambda) = \bigl(t(\lambda), r(\lambda), \theta(\lambda), \varphi(\lambda)\bigr) x μ ( λ ) = ( t ( λ ) , r ( λ ) , θ ( λ ) , φ ( λ ) )
u μ = ( t ˙ , r ˙ , θ ˙ , φ ˙ ) , ˙ = d d λ . u^\mu = (\dot{t}, \dot{r}, \dot{\theta}, \dot{\varphi}), \qquad \dot{} = \frac{d}{d\lambda}. u μ = ( t ˙ , r ˙ , θ ˙ , φ ˙ ) , ˙ = d λ d . Two conserved quantities exist:
Energy:
E = − k μ ( 1 ) u μ = − k ( 1 ) t u t = − 1 ⋅ g t t u t = A ( r ) t ˙ . E = -k_\mu^{(1)} u^\mu = -k^{(1)t} u_t = -1 \cdot g_{tt} u^t = A(r)\,\dot{t}. E = − k μ ( 1 ) u μ = − k ( 1 ) t u t = − 1 ⋅ g tt u t = A ( r ) t ˙ . Angular momentum:
L = k μ ( 2 ) u μ = k φ ( 2 ) u φ = k ( 2 ) φ g φ φ u φ = 1 ⋅ r 2 sin 2 θ φ ˙ . L = k_\mu^{(2)} u^\mu = k_\varphi^{(2)} u^\varphi = k^{(2)\varphi} g_{\varphi\varphi} u^\varphi = 1 \cdot r^2\sin^2\theta \,\dot{\varphi}. L = k μ ( 2 ) u μ = k φ ( 2 ) u φ = k ( 2 ) φ g φφ u φ = 1 ⋅ r 2 sin 2 θ φ ˙ .