Numerical relativity

Numerical relativity solves Einstein's equations as an initial-value problem, usually after decomposing spacetime into spatial hypersurfaces and evolution in time.

A 3+1 decomposition writes $d{s}^2=-{\alpha }^{2}d{t}^2+{\gamma }_{ij}(d{x}^i+{\beta }^{i}dt)(d{x}^j+{\beta }^{j}dt)$ in terms of lapse, shift, and spatial metric.

A spacetime foliation separates a spatial metric, lapse and shift. The ADM formalism uses this decomposition to express Einstein evolution and constraint equations in Hamiltonian form.

For a minimally coupled scalar field with spatial induced metric $h_{ij}$, the field's derivative along the future unit normal is $\Pi$. If the shift vector enters as $d{x}^i-N^{i}dt$, then $\Pi =(\dot{\varphi }+N^i{\partial }_i\varphi )/N$. The canonical momentum of the density is ${\pi }_{\varphi }=\sqrt{h}\Pi$. Confusing this density with the normal derivative loses a volume factor.

For a minimally coupled scalar field, put $q^2=h^{ij}{D}_i\varphi D_j\varphi$ using the spatial covariant derivative. Spatial and normal projections of its stress-energy tensor give $\rho ={\Pi }^2/2+q^2/2+V$, ${\mathcal{J}}_i=-\Pi D_i\varphi$, and $S_{ij}=D_i\varphi D_j\varphi +h_{ij}({\Pi }^2/2-q^2/2-V)$. The trace-free stress is $D_i\varphi D_j\varphi -h_{ij}{q}^2/3$, with coefficient one. This makes the scalar anisotropic stress second order in a spatial gradient expansion.

In fixed spatial coordinates set $a_{\mathrm{loc}}=(\det h{)}^{1/6}$ and ${\gamma }_{ij}=a_{\mathrm{loc}}^{-2}{h}_{ij}$, so $\det \gamma =1$. This volume-shape decomposition is useful for separating isotropic expansion and anisotropic shape. The determinant-defined scale depends on the coordinate volume convention.

For a positive lapse function $\alpha$, shift vector ${\beta }^i$ and spatial induced metric ${\gamma }_{ij}$, the block metric has $g_{00}=-{\alpha }^2+{\gamma }_{ij}{\beta }^i{\beta }^j$ and $g_{0i}={\gamma }_{ij}{\beta }^j$. Its block determinant is $g=-{\alpha }^2\det \gamma$. Therefore the metric volume tensor contracted with the future unit normal restricts to the spatial volume tensor: $n^{\mu }{ϵ}_{\mu ijk}=\sqrt{\det \gamma }[ijk]$.

Using the lapse function $N$, shift vector $N^i$ and spatial metric $h_{ij}$, the Legendre transform in mechanics of the Einstein-Hilbert action gives, in units $16\pi G=1$,Here $\pi =h_{ij}{\pi }^{ij}$ and ${\pi }^{ij}$ is the canonical momentum of the spatial metric. Varying lapse and shift imposes the Hamiltonian constraint and momentum constraint. An asymptotic time translation requires an Arnowitt-Deser-Misner energy boundary term. The bulk constraints generate gauge evolution; the nonzero asymptotic energy comes from the boundary.

On a compact spatial Cauchy hypersurface without boundary, the gravitational Hamiltonian consists entirely of constraints and vanishes on solutions. There is no asymptotic time-translation boundary charge to add. This is the canonical statement that a closed universe has zero total energy; it does not assert that local matter energy density or local curvature vanishes.

The momentum constraint is the equation obtained by varying the shift vector in the Hamiltonian formulation of general relativity. It expresses the tangential-normal projection of the Einstein field equations and generates spatial coordinate transformations. Writing $p^{ij}={\pi }^{ij}/\sqrt{h}$ avoids ambiguities about covariant derivatives of tensor densities.

For scalar metric potentials $d{s}^2=a^2[-(1+2A)d{\tau }^2+2B_{,i}d\tau d{x}^i+((1-2\psi ){\delta }_{ij}+2E_{,ij})d{x}^{i}d{x}^j]$ and positive extrinsic curvature, $\delta K^i{}_j=a^{-1}[-({\psi }^{\prime }+\mathcal{H}A){\delta }^i{}_j+{\partial }^i{\partial }_j(E^{\prime }-B)]$. The scalar shear terms cancel between its divergence and trace gradient. With $j_i=-P^{\mu }{}_i{T}_{\mu \nu }{n}^{\nu }$, the momentum constraint becomes $2{\partial }_i({\psi }^{\prime }+\mathcal{H}A)/a=-8\pi G{j}_i$. Reversing the convention for $K$ reverses the geometric left side.

With the convention $K_{ij}=({\dot{h}}_{ij}-D_i{N}_j-D_j{N}_i)/(2N)$ for extrinsic curvature, differentiation of the gravitational Lagrangian gives ${\pi }^{ij}=\sqrt{h}(K^{ij}-K{h}^{ij})$ in units $16\pi G=1$. It is a weight-one tensor density. In three spatial dimensions, $\pi =-2\sqrt{h}K$, soThe lapse function and shift vector have no time derivatives in this action and consequently have vanishing conjugate momenta. Changing the sign convention for extrinsic curvature changes the corresponding momentum relation; it must not be silently combined with this formula.

For a unit timelike normal $n^{\mu }{n}_{\mu }=-1$, the spatial projection tensor annihilates $n^{\mu }$ and projects every spacetime tensor onto a spatial hypersurface.

Differentiating the spatial projection tensor and projecting its derivative and covariant indices leaves $-n^{\lambda }{K}_{\sigma \rho }$, where $K$ is the extrinsic curvature of a spatial hypersurface. Contracting $\lambda$ with $\rho$ gives zero because $K$ is transverse. This distinction separates the informative tensor identity, with a free normal index, from its vanishing trace. Hypersurface orthogonality implies symmetric extrinsic curvature.

With signature $+---$ and $n^{\mu }{n}_{\mu }=1$, the spatial projection tensor is $P^{\mu }{}_{\nu }={\delta }^{\mu }{}_{\nu }-n^{\mu }{n}_{\nu }$. Projecting the spacetime metric tensor gives $q_{\mu \nu }=g_{\mu \nu }-n_{\mu }{n}_{\nu }$, which is negative definite on spatial vectors. The positive spatial induced metric is $h_{\mu \nu }=-q_{\mu \nu }$. A plus sign in $g_{\mu \nu }+n_{\mu }{n}_{\nu }$ belongs to a different normal/signature convention and is not transverse in this one. Fully projected covariant derivatives of either spatial metric vanish by metric compatibility.

The spatial covariant derivative projects the spacetime covariant derivative and every free tensor index onto a spatial hypersurface. It is the Levi-Civita connection of the induced spatial metric.

For a unit timelike normal $n$, the induced metric is $h_{\mu \nu }=g_{\mu \nu }+n_{\mu }{n}_{\nu }$. The spatial covariant derivative projects the derivative and all free indices with the spatial projection tensor $P^{\mu }{}_{\nu }={\delta }^{\mu }{}_{\nu }+n^{\mu }{n}_{\nu }$. Metric compatibility gives ${\nabla }_{\lambda }{h}_{\mu \nu }=({\nabla }_{\lambda }{n}_{\mu })n_{\nu }+n_{\mu }{\nabla }_{\lambda }{n}_{\nu }$; projecting kills the normal factor in each term. Thus the spatial covariant derivative is compatible with the induced metric.

The normal acceleration measures failure of the hypersurface unit normals to be geodesics. It is spatial, $n^{\mu }{a}_{\mu }=0$, and in a 3+1 decomposition satisfies $a_i=D_i\log \alpha$.

The lapse function measures proper-time separation between neighboring spatial hypersurfaces along their unit normal.

The shift vector describes how spatial coordinates move tangentially between neighboring hypersurfaces.

With future unit normal $n^{\mu }$, the convention $K_{\mu \nu }=-{\perp }^{\rho }{}_{\mu }{\nabla }_{\rho }{n}_{\nu }$ measures how the hypersurface is embedded in spacetime.

With a future unit timelike normal in signature $(-,+,+,+)$, this convention makes the extrinsic curvature trace equal to the normal congruence's expansion, $K={\nabla }_{\mu }{n}^{\mu }$. An expanding flat Friedmann-Lemaître-Robertson-Walker metric has $K_{ij}=a^{\prime }{\delta }_{ij}$ and $K=3a^{\prime }/a^2$. This is the opposite sign to the negative-expansion convention used in many 3+1 decomposition of spacetime formulas. Projection identities and momentum constraints must use the same convention throughout.

In a 3+1 decomposition of spacetime with $K_{ij}=-({\partial }_t{\gamma }_{ij}-{\mathcal{L}}_{\beta }{\gamma }_{ij})/(2\alpha )$, contract the metric evolution with ${\gamma }^{ij}$ and use the Jacobi determinant derivative formula. Here $\gamma =\det {\gamma }_{ij}$, $K={\gamma }^{ij}{K}_{ij}$, $\alpha$ is the lapse function, and $\beta$ is the shift vector. The identity separates spatial transport and coordinate compression from the geometric volume change caused by extrinsic curvature.

For $d{s}^2=-N^{2}d{t}^2+h_{ij}(d{x}^i-N^{i}dt)(d{x}^j-N^{j}dt)$ and the future unit normal, the convention $K_{ij}=-n_{i;j}$ gives $K_{ij}=-({\dot{h}}_{ij}+D_i{N}_j+D_j{N}_i)/(2N)$. Lower the shift vector with $h$. The sign of the shift terms changes when the opposite threading convention is used.

For the threading convention $d{x}^i-N^{i}dt$ and $K_{ij}=-({\dot{\gamma }}_{ij}+D_i{N}_j+D_j{N}_i)/(2N)$, contract with ${\gamma }^{ij}$. Metric compatibility gives ${\gamma }^{ij}{D}_i{N}_j=D_i{N}^i$, and the Jacobi determinant derivative formula gives ${\gamma }^{ij}{\dot{\gamma }}_{ij}={\partial }_t\log \det \gamma$. With zero shift, $-K/3=(3N{)}^{-1}{\partial }_t\log \sqrt{\gamma }$ is the local volume Hubble parameter. Reversing the shift convention reverses both the normal's spatial component and the shift terms; the two conventions must not be mixed.

With signature $(-,+,+,+)$, future unit normal $n$ and $K_{ab}=-P^c{}_a{P}^d{}_b{\nabla }_c{n}_d$, the Gauss–Codazzi equations become $(\perp R{)}_{abcd}={\mathcal{R}}_{abcd}+K_{ac}{K}_{db}-K_{ad}{K}_{cb}$ and $(\perp R{)}_{abcd}{n}^d=-D_a{K}_{bc}+D_b{K}_{ac}$, where in the second expression only the first three indices are projected. The intrinsic curvature and spatial covariant derivative use the spatial induced metric. The normal is timelike, so the normal-sign convention differs from a spacelike normal in a positive-definite ambient metric.

For the positive-expansion convention for extrinsic curvature, write ${\nabla }_{\mu }{n}_{\nu }=K_{\mu \nu }-n_{\mu }{a}_{\nu }$ with normal acceleration $a_{\nu }=n^{\rho }{\nabla }_{\rho }{n}_{\nu }$. The Ricci commutator applied to $n$ gives the difference of its two second-derivative contractions. Projecting tangentially, the acceleration term in that commutator cancels exactly the acceleration term converting a spacetime divergence of $K$ into its fully projected spatial covariant derivative. Einstein's equations then identify the displayed equation as the momentum constraint.

For a Ricci-flat spacetime with the stated Gauss–Codazzi equations for a spatial hypersurface and induced orientation ${\widetilde{ϵ}}_{abc}=n^d{ϵ}_{dabc}$, the electric part of the Weyl tensor and magnetic part of the Weyl tensor satisfyThe electric relation comes from the vanishing spatial projection of the Ricci tensor. The magnetic relation combines the normal curvature projection with the sign ${ϵ}_{acde}{n}^c=-{\widetilde{ϵ}}_{ade}$.

Unit normalization gives $n^{\nu }{\nabla }_{\mu }{n}_{\nu }=0$. Hence $K_{\mu \nu }=-P^{\alpha }{}_{\mu }{P}^{\beta }{}_{\nu }{\nabla }_{\alpha }{n}_{\beta }=-P^{\alpha }{}_{\mu }{\nabla }_{\alpha }{n}_{\nu }$. Set $F_{\mu \nu }={\nabla }_{\mu }{n}_{\nu }-{\nabla }_{\nu }{n}_{\mu }$. Frobenius theorem for a hypersurface normal gives $n_{\alpha }{F}_{\beta \gamma }+n_{\beta }{F}_{\gamma \alpha }+n_{\gamma }{F}_{\alpha \beta }=0$. Contracting with $n^{\alpha }{P}^{\beta }{}_{\mu }{P}^{\gamma }{}_{\nu }$ leaves $-P^{\beta }{}_{\mu }{P}^{\gamma }{}_{\nu }{F}_{\beta \gamma }=0$, precisely the vanishing antisymmetric part of the extrinsic curvature of a spatial hypersurface.

For the convention $K_{\mu \nu }=-{\perp }^{\rho }{}_{\mu }{\nabla }_{\rho }{n}_{\nu }$, contraction of the Gauss equation gives

The Hamiltonian constraint is the normal-normal projection of the Einstein field equations onto a spatial hypersurface. With the extrinsic-curvature convention used here,

For a spatial initial slice with vanishing extrinsic curvature, the vacuum momentum constraint is automatic and the Hamiltonian constraint requires zero Ricci scalar. In three spatial dimensions, the indicated conformally flat metric has $\mathcal{R}=-8{\psi }^{-5}\Delta \psi$, with a nonzero conformal factor. Thus a harmonic function supplies the data on its domain. A positive example is $\psi =1+a/r$ on a punctured or exterior region. There is no nonconstant globally smooth example on all of Euclidean three-space tending to one at infinity, by the maximum principle for harmonic functions. Time symmetry of an initial slice does not imply that its entire evolved spacetime is static.

For a spatially flat Friedmann-Lemaître-Robertson-Walker metric, negative extrinsic curvature, and $K=-3H+\kappa$, the Hamiltonian constraint gives the displayed scalar perturbation equation. Indeed $K^2=9H^2-6H\kappa$ and $K^i{}_j{K}^j{}_i=3H^2-2H\kappa$ to first order. The spatial Ricci scalar is $4\Delta \Phi$; subtract the background equation $6H^2=16\pi G\bar{\rho }$ and divide by four.

The Z4 formulation introduces a spacetime vector $Z_{\mu }$ into Einstein's equations so that the Hamiltonian and momentum constraints become part of a dynamical system. Its normal projection is commonly written $\Theta =-n^{\mu }{Z}_{\mu }$.

For $J^{\mu }=Q{n}^{\mu }+Q^{\mu }$ with spatial $Q^{\mu }$, current conservation gives $n^{\mu }{\nabla }_{\mu }Q=KQ-D_{\mu }{Q}^{\mu }-Q^{\mu }{D}_{\mu }\log \alpha$ under the stated extrinsic-curvature convention.

Relative to a future unit normal $n^{\mu }$, the stress-energy tensor decomposes aswhere $j_{\mu }$ and both indices of $S_{\mu \nu }$ are spatial.

For signature $(+---)$ and future normal $n$, write $\Pi =n^{\mu }{\partial }_{\mu }\varphi$ and use the positive induced metric ${\gamma }_{ij}$ for $s^2={\gamma }^{ij}{D}_i\varphi D_j\varphi$. The Klein-Gordon scalar stress-energy tensor has normal-frame density $\rho =({\Pi }^2+s^2)/2+V$, momentum $J_i=-\Pi D_i\varphi$ and spatial stress $S_{ij}=D_i\varphi D_j\varphi +{\gamma }_{ij}[({\Pi }^2-s^2)/2-V]$. The spatial index convention and normal orientation must be specified.

With $K_{\mu \nu }=-{\perp }^{\rho }{}_{\mu }{\nabla }_{\rho }{n}_{\nu }$ and $a_{\mu }=n^{\rho }{\nabla }_{\rho }{n}_{\mu }$, spatial projection of ${\nabla }_{\mu }{T}^{\mu \nu }=0$ gives

Geodesic slicing chooses unit lapse and zero shift, so the hypersurface normals are affinely parametrized timelike geodesics. It is simple but commonly develops coordinate singularities in finite time.

The Baumgarte–Shapiro–Shibata–Nakamura formulation rewrites the Einstein evolution equations using a conformal spatial metric, conformal traceless extrinsic curvature, the trace $K$, and conformal connection functions.

In vacuum geodesic slicing, discarding nonnegative shear leaves ${\partial }_{t}K=K^2/3$. Positive initial $K_0$ therefore gives $K=K_0/[1-K_0(t-t_0)/3]$ and blows up at $t=t_0+3/K_0$.

A harmonic coordinate satisfies ${\Delta }_g{x}^{\lambda }=0$ for a Riemannian metric, or ${\square }_g{x}^{\lambda }=0$ for a Lorentzian metric. Equivalently $g^{\mu \nu }{\Gamma }_{\mu \nu }^{\lambda }=0$ or ${\partial }_{\mu }(\sqrt{|\det g|}{g}^{\lambda \mu })=0$. This common coordinate condition applies in both Riemannian geometry and general relativity.

With the Riemann curvature tensor convention $R^{\mu }{}_{\nu \rho \sigma }={\partial }_{\rho }{\Gamma }^{\mu }{}_{\nu \sigma }-{\partial }_{\sigma }{\Gamma }^{\mu }{}_{\nu \rho }+\cdots$, harmonic coordinates remove the mixed second derivatives from the Ricci tensor. The remainder $Q_{ab}$ contains at most first derivatives of the metric tensor. The vacuum Einstein field equations become a system of quasilinear partial differential equations with a Lorentzian wave equation principal part. Harmonic coordinate constraints must be maintained when using the reduced system.

Harmonic slicing imposes the harmonic-coordinate condition only on time. In 3+1 form it obeys $({\partial }_t-{\beta }^i{\partial }_i)\alpha =-{\alpha }^{2}K$.

For a positive lapse function $\alpha$, shift vector ${\beta }^i$, and the convention $K_{ij}=-({\partial }_t{\gamma }_{ij}-{\mathcal{L}}_{\beta }{\gamma }_{ij})/(2\alpha )$ for extrinsic curvature of a spatial hypersurface, the time-coordinate harmonic slicing condition yields the displayed advective lapse equation. Combining ${\partial }_t(\sqrt{\gamma }/\alpha )={\partial }_i(\sqrt{\gamma }{\beta }^i/\alpha )$ with the spatial volume evolution identity proves it without setting the shift vector to zero.

The Bona--Masso family is $({\partial }_t-{\beta }^i{\partial }_i)\alpha =-{\alpha }^{2}f(\alpha )K$. Harmonic slicing is the choice $f=1$.

For the stationary conformally flat Schwarzschild slicing with areal radius $R=r+M$, lapse $\alpha =r/(r+M)$, and radial shift ${\beta }^r=Mr/(r+M{)}^2$, the Bona--Masso condition selects

In harmonic gauge, the vacuum Einstein equations have principal part $-g^{\mu \nu }{\partial }_{\mu }{\partial }_{\nu }{g}_{\alpha \beta }/2$, forming a quasilinear wave system supplemented by constraints.

The Arnowitt-Deser-Misner formalism expresses general relativity as constrained Hamiltonian evolution of a spatial metric and its conjugate momentum after a 3+1 decomposition of spacetime. Lapse and shift enforce the Hamiltonian and momentum constraints. For an asymptotically flat spacetime its boundary Hamiltonian gives the Arnowitt-Deser-Misner energy.