general-relativity.bigb

= General relativity
{wiki}

= Plane-fronted gravitational wave
{parent=General relativity}

For
$$
ds^2=2\,du\,dv+dx^2+dy^2+H(u,x,y)\,du^2,
$$
the only potentially nonzero Ricci component is
$$
R_{uu}=-\frac12(H_{xx}+H_{yy}).
$$
The vacuum equation is therefore the transverse Laplace equation.

= Covariant derivative
{parent=General relativity}
{wiki}

A connection defines covariant differentiation by adding one connection term for each contravariant index and subtracting one for each covariant index.

= Tensor type
{title2=$(r,s)$}
{parent=Covariant derivative}
{wiki=Tensor#As_multidimensional_arrays}

A tensor of type $(r,s)$ has $r$ contravariant and $s$ covariant indices. Taking a covariant derivative adds one covariant index, while an outer tensor product adds the corresponding contravariant and covariant index counts.

= Parallel transport
{parent=Covariant derivative}
{wiki}

A vector is parallel transported along a curve when its covariant derivative along the curve vanishes.

= Parallel transport around a latitude of the unit sphere
{parent=Parallel transport}

For $ds^2=d\theta^2+\sin^2\theta\,d\phi^2$, a vector initially having coordinate components $(V^\theta,V^\phi)=(1,0)$ returns after one constant-$\theta$ circuit with
$$
(V^\theta,V^\phi)
=\left(
\cos(2\pi\cos\theta),
-\frac{\sin(2\pi\cos\theta)}{\sin\theta}
\right).
$$

= Covariant derivative of a covariant tensor
{parent=Covariant derivative}

For $W_{\mu\nu}$,
$$
\nabla_\alpha W_{\mu\nu}
=\partial_\alpha W_{\mu\nu}
-\Gamma^\rho_{\alpha\mu}W_{\rho\nu}
-\Gamma^\rho_{\alpha\nu}W_{\mu\rho}.
$$

= Levi-Civita connection
{parent=Covariant derivative}
{c}
{wiki}

The Levi-Civita connection is the unique torsion-free connection compatible with the metric; its coefficients are the Christoffel symbols formed from first metric derivatives.

= Metric compatibility
{parent=Levi-Civita connection}
{wiki}

Metric compatibility is $\nabla_\alpha g_{\mu\nu}=0$, allowing the metric to pass through covariant derivatives and index operations.

= Riemann curvature tensor
{parent=General relativity}
{c}
{wiki}

The Riemann tensor measures the failure of covariant derivatives to commute.

= Curvature commutator on a covariant tensor
{parent=Riemann curvature tensor}

For a rank-two covariant tensor, the derivative commutator contributes one curvature action on each index:
$$
[\nabla_\alpha,\nabla_\beta]W_{\mu\nu}
=-R^\rho{}_{\mu\alpha\beta}W_{\rho\nu}
-R^\rho{}_{\nu\alpha\beta}W_{\mu\rho}.
$$

= Symmetric Hessian of a scalar field
{parent=Riemann curvature tensor}

For a torsion-free connection, $U_\mu=\partial_\mu\phi$ satisfies $\nabla_\alpha U_\beta=\nabla_\beta U_\alpha$.

= First Bianchi identity
{parent=Riemann curvature tensor}
{wiki=Bianchi_identity}

For a torsion-free connection, $R^\mu{}_{\alpha\beta\gamma}+R^\mu{}_{\beta\gamma\alpha}+R^\mu{}_{\gamma\alpha\beta}=0$.

= Weyl tensor
{title2=$C_{\alpha\beta\gamma\delta}$}
{c}
{parent=Riemann curvature tensor}
{wiki}

The Weyl tensor is the completely trace-free part of the <Riemann curvature tensor>. In four dimensions,
$$
\begin{aligned}
C_{\alpha\beta\gamma\delta}
={}&R_{\alpha\beta\gamma\delta}
-\frac12(
g_{\alpha\gamma}R_{\beta\delta}
+g_{\beta\delta}R_{\alpha\gamma}\\
&\qquad
-g_{\alpha\delta}R_{\beta\gamma}
-g_{\beta\gamma}R_{\alpha\delta})
+\frac R6(
g_{\alpha\gamma}g_{\beta\delta}
-g_{\alpha\delta}g_{\beta\gamma}).
\end{aligned}
$$

= Trace-free property of the Weyl tensor
{parent=Weyl tensor}
{c}

Every contraction of the Weyl tensor vanishes. Contracting its first and third indices directly cancels the Ricci and scalar-curvature terms; antisymmetry in each pair and symmetry under exchanging the two pairs reduce all other contractions to this one or make them vanish immediately.

= Weyl tensor of a conformally flat metric
{parent=Weyl tensor}
{c}

Every <conformally flat metric> in dimension at least four has identically zero Weyl tensor. The Weyl tensor is the obstruction to local conformal flatness in dimensions at least four.

= First-pair antisymmetry of the Riemann tensor
{parent=Riemann curvature tensor}

Applying the curvature commutator to the parallel metric gives $R_{\mu\alpha\beta\gamma}=-R_{\alpha\mu\beta\gamma}$.

= Ricci tensor
{parent=Riemann curvature tensor}
{c}
{wiki=Ricci_curvature}

The Ricci tensor is the contraction $R_{\alpha\beta}=R^\mu{}_{\alpha\mu\beta}$ and is symmetric for the Levi-Civita connection.

= Ricci tensor in two dimensions
{parent=Ricci tensor}

The algebraic symmetries of the <Riemann curvature tensor> leave only one independent curvature component in two dimensions and force
$$
R_{\alpha\beta}=\frac12g_{\alpha\beta}R.
$$
Consequently the <Einstein tensor> vanishes identically.

= Two-dimensional general relativity
{parent=Ricci tensor in two dimensions}

With no cosmological term, the two-dimensional <Einstein field equations> reduce identically on the geometric side to $G_{\alpha\beta}=0$. They impose $T_{\alpha\beta}=0$ and supply no local metric dynamics; the Einstein-Hilbert action is topological.

= Contracted Bianchi identity
{parent=Ricci tensor}
{wiki=Bianchi_identity\#Contracted_Bianchi_identity}

The Ricci tensor and scalar satisfy $\nabla^\alpha R_{\alpha\beta}=\tfrac12\nabla_\beta R$.

= Einstein tensor
{c}
{parent=Contracted Bianchi identity}
{wiki}

The Einstein tensor is
$$
G_{\mu\nu}=R_{\mu\nu}-\frac12Rg_{\mu\nu}.
$$
The contracted Bianchi identity and metric compatibility give
$\nabla^\nu G_{\mu\nu}=0$, ensuring compatibility of the Einstein equations with covariant stress-energy conservation.

= Flat FLRW Einstein-tensor divergence
{parent=Einstein tensor}

For $ds^2=-dt^2+a(t)^2d\mathbf x^2$, the nonzero connection coefficients are
$$
\Gamma^t_{ij}=a\dot a\,\delta_{ij},
\qquad
\Gamma^i_{tj}=\Gamma^i_{jt}=H\delta^i_j.
$$
With $G_t^t=-3H^2$ and
$G_i^j=-(2\ddot a/a+H^2)\delta_i^j$, direct substitution into the mixed-tensor covariant derivative gives zero.

= de Sitter scale factor in flat slicing
{c}
{parent=Flat FLRW Einstein-tensor divergence}

The vacuum equation with positive cosmological constant gives
$H^2=\Lambda/3$. Its expanding flat-slicing solution is
$$
a(t)=a_0e^{\sqrt{\Lambda/3}\,t}.
$$

= Schur theorem in pseudo-Riemannian geometry
{parent=Ricci tensor}
{c}
{wiki=Schur%27s_lemma_(Riemannian_geometry)}

In dimension greater than two, if sectional curvature is pointwise independent of the two-plane, the contracted Bianchi identity forces that curvature to be constant on each connected component.

= Normal coordinates
{parent=General relativity}
{wiki=Normal_coordinates}

At any spacetime point, normal coordinates make the metric equal to the Minkowski metric and all Christoffel symbols vanish at that point.

= Stress-energy tensor
{parent=General relativity}
{wiki}

The stress-energy tensor describes local energy density, momentum density, and stress. Compatibility with the <Einstein field equations> requires covariant conservation $\nabla^\mu T_{\mu\nu}=0$.

= Conserved Klein-Gordon scalar stress-energy ansatz
{parent=Stress-energy tensor}
{c}

For $\nabla_\alpha\nabla^\alpha\phi=m^2\phi$, consider
$$
T_{\mu\nu}=\frac12\left[
\nabla_\mu\phi\nabla_\nu\phi
+g_{\mu\nu}\left(A\nabla_\rho\phi\nabla^\rho\phi+B\phi^2\right)
\right].
$$
Covariant conservation for all solutions holds exactly when
$$
A=-\frac12,
\qquad
B=-\frac{m^2}{2}.
$$

= Weak energy condition
{parent=Stress-energy tensor}
{wiki}

The weak energy condition requires
$$
T_{\mu\nu}X^\mu X^\nu\geq0
$$
for every timelike vector $X$, so every observer measures nonnegative energy density.

= Weak energy condition for a quadratic scalar stress-energy ansatz
{parent=Weak energy condition}

For the general scalar ansatz in <conserved Klein-Gordon scalar stress-energy ansatz>, work in a unit timelike frame. Then
$$
2T_{00}=(1+A)(\partial_0\phi)^2
-A|\nabla_{\rm s}\phi|^2-B\phi^2.
$$
The weak energy condition for arbitrary local field data is therefore equivalent to
$$
-1\leq A\leq0,
\qquad B\leq0.
$$

= Einstein field equations
{parent=General relativity}
{c}
{wiki}

With signature $(-,+,+,+)$ and vanishing cosmological constant, the Einstein field equations are
$$
R_{\mu\nu}-\frac12 g_{\mu\nu}R=\kappa T_{\mu\nu}.
$$
In four spacetime dimensions, contraction with $g^{\mu\nu}$ gives the useful trace relation $R=-\kappa T$.

= Trace-reversed Einstein field equations in D dimensions
{parent=Einstein field equations}
{c}

For $D>2$, tracing and substituting back gives
$$
R_{\mu\nu}
=\kappa\left(T_{\mu\nu}-\frac1{D-2}g_{\mu\nu}T\right).
$$

= Ricci curvature sourced by a Klein-Gordon scalar field
{parent=Trace-reversed Einstein field equations in D dimensions}
{c}

For the conserved scalar stress tensor above,
$$
R_{\mu\nu}
=\frac\kappa2\left[
\nabla_\mu\phi\nabla_\nu\phi
+\frac{m^2}{D-2}g_{\mu\nu}\phi^2
\right].
$$

= Linearized gravity
{parent=General relativity}
{wiki}

Linearized gravity writes $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ with $|h_{\mu\nu}|\ll1$ and discards terms quadratic in $h$ and its derivatives.

= Trace-reversed metric perturbation
{title2=$\bar h_{\mu\nu}$}
{parent=Linearized gravity}

In four dimensions, the trace-reversed perturbation is
$$
\bar h_{\mu\nu}=h_{\mu\nu}-\frac12\eta_{\mu\nu}h,
\qquad h=\eta^{\rho\sigma}h_{\rho\sigma}.
$$
Trace reversal simplifies the linearized Einstein equations.

= Lorenz gauge in linearized gravity
{c}
{parent=Trace-reversed metric perturbation}
{wiki=Gauge_fixing#Lorenz_gauge}

The Lorenz condition $\partial^\mu\bar h_{\mu\nu}=0$ reduces the vacuum field equation to $\Box\bar h_{\mu\nu}=0$. A plane-wave amplitude therefore satisfies $k^\mu H_{\mu\nu}=0$ and $k^\mu k_\mu=0$.

= Residual gauge symmetry of linearized gravity
{parent=Lorenz gauge in linearized gravity}

An infinitesimal coordinate change generated by $\xi_\mu$ changes
$$
\bar h_{\mu\nu}\mapsto
\bar h_{\mu\nu}-\partial_\mu\xi_\nu-\partial_\nu\xi_\mu
+\eta_{\mu\nu}\partial_\rho\xi^\rho.
$$
It preserves Lorenz gauge exactly when $\Box\xi_\mu=0$.

= Plane gravitational wave in linearized gravity
{parent=Lorenz gauge in linearized gravity}

A vacuum mode $\bar h_{\mu\nu}=H_{\mu\nu}e^{ik\cdot x}$ has a null wavevector and a transverse symmetric amplitude. Residual gauge freedom can further reduce it to two physical transverse-traceless polarizations.

= Linearized Ricci tensor and scalar
{parent=Linearized gravity}

To first order around flat spacetime,
$$
R_{\mu\nu}
=\frac12\left(
\partial_\rho\partial_\mu h^\rho{}_{\nu}
+\partial_\rho\partial_\nu h^\rho{}_{\mu}
-\mathop{\Box}h_{\mu\nu}
-\partial_\mu\partial_\nu h
\right),
$$
and
$$
R=\partial_\mu\partial_\nu h^{\mu\nu}-\mathop{\Box}h,
\qquad h=\eta^{\mu\nu}h_{\mu\nu}.
$$

= Newtonian limit of general relativity
{parent=Linearized gravity}
{c}
{wiki=Newtonian_limit}

For a weak, static field produced by nonrelativistic matter, $T_{00}=\rho$ is the dominant stress-energy component. The field equations and the slow-particle <geodesic equation> imply
$$
h_{00}=-2\Phi,
\qquad
\kappa=8\pi G,
$$
where the <Poisson equation> is $\nabla^2\Phi=4\pi G\rho$.

= Linearized point-mass metric in radial Cartesian form
{parent=Newtonian limit of general relativity}

For a point mass and $r>0$, write
$$
h_{00}=\frac{2GM}{r},
\qquad
h_{ij}=f(r)x_i x_j.
$$
The vacuum scalar-curvature equation gives $rf'+3f=0$, so $f=C/r^3$. The remaining vacuum equations fix $C=2GM$ in the corresponding Schwarzschild gauge.

= Schwarzschild spacetime
{parent=General relativity}
{c}
{wiki=Schwarzschild_metric}

In units $G=c=1$, the Schwarzschild exterior metric is
$$
ds^2=-\left(1-\frac{2M}{r}\right)dt^2
+\left(1-\frac{2M}{r}\right)^{-1}dr^2+r^2d\Omega^2.
$$

= Radial Weyl curvature of Schwarzschild spacetime
{title2=$C_{trtr}=-2M/r^3$}
{c}
{parent=Schwarzschild spacetime}

With the curvature convention
$$
R^\alpha{}_{\beta\gamma\delta}
=\partial_\gamma\Gamma^\alpha_{\beta\delta}
-\partial_\delta\Gamma^\alpha_{\beta\gamma}+\cdots,
$$
the vacuum Schwarzschild metric satisfies
$$
C_{trtr}=R_{trtr}=-\frac{2M}{r^3}.
$$
The opposite Riemann-sign convention reverses this sign.

= Tidal gravity and the equivalence principle
{parent=Radial Weyl curvature of Schwarzschild spacetime}

Weyl curvature produces relative acceleration between neighbouring freely falling trajectories and therefore tidal deformation. A freely falling frame removes gravity at one event, as required by the equivalence principle, but cannot remove curvature over an extended region.

= Equatorial null orbit in Schwarzschild spacetime
{parent=Schwarzschild spacetime}

For inverse radius $y=1/r$ and azimuth $\phi$, a Schwarzschild null geodesic obeys
$$
(y')^2+y^2-2My^3=\frac1{b^2},
\qquad
y''+y=3My^2,
$$
where $b=L/E$ is the <impact parameter>.

= Impact parameter
{title2=$b$}
{parent=Equatorial null orbit in Schwarzschild spacetime}
{wiki}

The impact parameter is the perpendicular miss distance of the corresponding asymptotic straight trajectory. For a null Schwarzschild orbit it is the ratio $b=L/E$ in geometrized units.

= Schwarzschild radius
{parent=Schwarzschild spacetime}
{c}
{wiki}

The Schwarzschild radius is $r_s=2GM/c^2$, or $2M$ in geometrized units.

= Schwarzschild tortoise coordinate
{parent=Schwarzschild spacetime}
{c}
{wiki=Eddington%E2%80%93Finkelstein_coordinates\#Tortoise_coordinate}

The tortoise coordinate satisfies
$$
\frac{dr_*}{dr}=\left(1-\frac{2M}{r}\right)^{-1},
\qquad
r_*=r+2M\log\left|\frac{r}{2M}-1\right|.
$$

= Ingoing Eddington-Finkelstein coordinates
{parent=Schwarzschild tortoise coordinate}
{c}
{wiki=Eddington%E2%80%93Finkelstein_coordinates}

The advanced coordinate $v=t+r_*$ puts the radial Schwarzschild metric into the form
$$
ds^2=-\left(1-\frac{2M}{r}\right)dv^2+2\,dv\,dr+r^2d\Omega^2,
$$
which is regular at the future horizon.

= Radial null trajectories in ingoing Eddington-Finkelstein coordinates
{parent=Ingoing Eddington-Finkelstein coordinates}

Radial null curves comprise the ingoing family $v=\text{constant}$ and the outgoing family
$$
\frac{dr}{dv}=\frac12\left(1-\frac{2M}{r}\right).
$$
Outside the horizon the outgoing family moves to larger $r$; inside it both future-directed families move to smaller $r$.

= Schwarzschild event horizon
{parent=Schwarzschild spacetime}
{c}
{wiki=Event_horizon}

The surface $r=2M$ is the future event horizon of Schwarzschild spacetime. Its singularity in Schwarzschild coordinates is removable, whereas $r=0$ is a curvature singularity.

= Gravitational redshift between static Schwarzschild observers
{parent=Schwarzschild spacetime}
{wiki=Gravitational_redshift}

For static observers at radii $r_A$ and $r_B$, equal Schwarzschild-coordinate time intervals correspond to proper times $d\tau=\sqrt{1-2M/r}\,dt$. Hence
$$
\frac{\Delta\tau_B}{\Delta\tau_A}
=\sqrt{\frac{1-2M/r_B}{1-2M/r_A}}.
$$

= Proper acceleration of a static Schwarzschild observer
{parent=Schwarzschild spacetime}

A static observer at Schwarzschild radius $r>2M$ must maintain the outward <proper acceleration>
$$
a=\frac{M}{r^2\sqrt{1-2M/r}}.
$$
It approaches the Newtonian value $M/r^2$ at large radius and diverges at the horizon.

= Birkhoff theorem
{c}
{parent=Schwarzschild spacetime}
{wiki=Birkhoff%27s_theorem_(relativity)}

Every spherically symmetric vacuum solution of the Einstein field equation is locally a portion of Schwarzschild spacetime. In particular, a spherically symmetric body's exterior field depends only on its total mass.

= Redshifted gravitational mass of a static thin shell
{parent=Birkhoff theorem}

A thin spherical shell of proper surface mass density $\rho$ held at Schwarzschild radius $r_{\rm st}$ around mass $M$ contributes, to first order in its mass,
$$
\Delta M_\infty
=4\pi r_{\rm st}^2\rho\sqrt{1-\frac{2M}{r_{\rm st}}}
$$
to the exterior Schwarzschild mass parameter.

= Schwarzschild light deflection
{c}
{parent=Schwarzschild spacetime}
{wiki=Gravitational_lens}

A null Schwarzschild geodesic with $b\gg r_s$ is deflected through $2r_s/b=4GM/(bc^2)$ to first order.

= Perturbative derivation of Schwarzschild light deflection
{parent=Schwarzschild light deflection}

Writing $y=b^{-1}\sin\phi+\Delta y$ in $y''+y=3My^2$ and retaining first order in $M$ gives
$$
\Delta y''+\Delta y=\frac{3M}{2b^2}(1-\cos2\phi).
$$
The displacement of the outgoing zero is $4M/b$.

= Nordstrom theory of gravitation
{c}
{parent=General relativity}
{wiki=Nordstr%C3%B6m%27s_theory_of_gravitation}

Nordstrom's scalar theory restricts the spacetime metric to the conformally flat form $g_{\mu\nu}=\varphi^2\eta_{\mu\nu}$. Its null geodesics have the same unparametrized paths as those of the <Minkowski metric>, so the theory predicts no gravitational bending of light.

= Conformally flat metric
{parent=Nordstrom theory of gravitation}
{wiki=Conformally_flat_manifold}

A metric is conformally flat when it is locally a positive scalar multiple of a flat metric. Multiplication by the conformal factor preserves null cones and unparametrized null geodesics.

= Geodesic equation for a conformally flat metric
{parent=Nordstrom theory of gravitation}

For $g_{\mu\nu}=\varphi^2\eta_{\mu\nu}$,
$$
\Gamma^\mu_{\alpha\beta}
=\delta^\mu_\alpha\partial_\beta\log\varphi
+\delta^\mu_\beta\partial_\alpha\log\varphi
-\eta_{\alpha\beta}\partial^\mu\log\varphi.
$$
For a null curve, the last term drops from the <geodesic equation>; the remaining acceleration is parallel to the tangent and can be removed by reparametrization.

= Anti-de Sitter spacetime
{parent=General relativity}
{c}
{wiki}

Anti-de Sitter spacetime is the maximally symmetric Lorentzian spacetime of constant negative curvature.

= Static coordinates on anti-de Sitter spacetime
{parent=Anti-de Sitter spacetime}

In four dimensions, global static coordinates have
$$ds^2=-(1+r^2/a^2)dt^2+(1+r^2/a^2)^{-1}dr^2+r^2d\Omega^2.$$

= Radial timelike geodesic in anti-de Sitter spacetime
{parent=Static coordinates on anti-de Sitter spacetime}

A radial timelike geodesic through the origin obeys a harmonic radial equation and returns to the origin after proper time $\pi a$, independently of its energy.

= Stable circular timelike geodesic in anti-de Sitter spacetime
{parent=Static coordinates on anti-de Sitter spacetime}

At every radius $r_0>0$, global anti-de Sitter spacetime has stable circular timelike geodesics with $|\dot\phi|=1/a$ and $\dot t=1$.

= Planarity of geodesics in spherical symmetry
{parent=General relativity}

Rotational symmetry conserves the angular-momentum direction, so every nonradial geodesic lies in a plane through the symmetry centre; coordinates may place that plane at $\theta=\pi/2$.

= Timelike geodesic effective potential
{parent=General relativity}

After using Killing constants and four-velocity normalization, radial timelike geodesic motion can often be written as kinetic energy plus an effective potential equal to a constant.

= Effective-potential stability of a circular orbit
{parent=Timelike geodesic effective potential}

A circular orbit at a strict local minimum of the radial effective potential is stable under small radial perturbations.

= Effective potential for timelike Schwarzschild geodesics
{parent=Timelike geodesic effective potential}

For equatorial timelike motion with specific angular momentum $h$,
$$
\frac12\dot r^2+V_{\rm eff}(r)=\frac{E^2-1}{2},
\qquad
V_{\rm eff}=-\frac mr+\frac{h^2}{2r^2}-\frac{mh^2}{r^3}.
$$
A circular orbit has $h^2=mr^2/(r-3m)$ and therefore requires $r>3m$.

= Schwarzschild perihelion precession
{parent=Effective potential for timelike Schwarzschild geodesics}
{c}
{wiki=Apsidal_precession#General_relativity}

The nearly circular orbit equation
$$
u''+u=\frac m{h^2}+3mu^2
$$
has radial angular frequency $1-3m^2/h^2$ to leading order. The perihelion advances by $6\pi m^2/h^2$ per orbit.

= Metric tensor
{title2=$g_{\mu\nu}$}
{parent=General relativity}
{wiki}

A metric tensor is a <nondegenerate bilinear form>[nondegenerate] symmetric <covariant tensor> that defines squared line elements by $ds^2=g_{\mu\nu}dx^\mu dx^\nu$.

= Inverse metric
{title2=$g^{\mu\nu}$}
{parent=Metric tensor}
{wiki=Metric_tensor#Inverse_metric}

The inverse metric satisfies $g^{\mu\rho}g_{\rho\nu}=\delta^\mu_\nu$ and raises covariant tensor indices.

= Two-dimensional hyperbolic metric in polar coordinates
{title2=$ds^2=a^2(dr^2+\sinh^2r\,d\phi^2)$}
{parent=General relativity}

This metric is the <hyperbolic plane> with curvature scale $a$. Its nonzero connection coefficients are
$$
\Gamma^r_{\phi\phi}=-\sinh r\cosh r,
\qquad
\Gamma^\phi_{r\phi}=\Gamma^\phi_{\phi r}=\coth r.
$$

= Coordinate geodesics of the hyperbolic polar metric
{parent=Two-dimensional hyperbolic metric in polar coordinates}

Every radial coordinate line $\phi=\text{constant}$ is a geodesic. No nonconstant coordinate circle $r=r_0>0$ is a geodesic.

= Curvature of the two-dimensional hyperbolic polar metric
{parent=Two-dimensional hyperbolic metric in polar coordinates}

For the curvature convention
$$
R^\alpha{}_{\beta\gamma\delta}
=\partial_\gamma\Gamma^\alpha_{\beta\delta}
-\partial_\delta\Gamma^\alpha_{\beta\gamma}
+\Gamma^\mu_{\beta\delta}\Gamma^\alpha_{\mu\gamma}
-\Gamma^\mu_{\beta\gamma}\Gamma^\alpha_{\mu\delta},
$$
the independent nonzero mixed components are
$$
R^r{}_{\phi r\phi}=-\sinh^2r,
\qquad
R^\phi{}_{r\phi r}=-1,
$$
and the Ricci scalar is $R=-2/a^2$.

= Ricci scalar
{title2=$R$}
{parent=General relativity}
{c}
{wiki=Scalar_curvature}

The Ricci scalar is the full metric contraction of the <Ricci tensor>:
$$
R=g^{\mu\nu}R_{\mu\nu}.
$$