Formulations of General Relativity

Einstein’s equation is compact in covariant form, but a covariant equation is not yet an initial-value problem, a characteristic hierarchy, a Hamiltonian system, or a prescription for data at a timelike boundary. Those descriptions reorganize the same gravitational dynamics so that different structures become calculable: gauge freedom, constraints, propagating fields, boundary flux, or asymptotic charges. This volume develops the translations and the conditions under which they are valid.

Helpful background. The volume assumes familiarity with the Einstein equation and uses the induced geometry developed in Hypersurface Normals and Tangential Projectors. The opening chapters provide local bridges when the corresponding PDE or frame language is unfamiliar.

One theory, many formulations

The covariant field equation

$$G_{\mu\nu}[g]+\Lambda g_{\mu\nu}=8\pi G\,T_{\mu\nu}$$

contains ten component equations, related by the four contracted Bianchi identities. Coordinate freedom prevents these from being ten independent evolution equations for ten physical fields. A formulation chooses fields, gauge conditions, constraint variables, and a domain on which some subset of the equations determines evolution while the remainder restricts or monitors data. Its value is that it exposes a mathematical task that the covariant equation leaves implicit.

For example, vacuum Cauchy data consist of six components of a spatial metric and six of its extrinsic curvature. Four constraints restrict those twelve functions. Four independent gauge directions then leave four local phase-space functions: two gravitational degrees of freedom, each with a configuration and a conjugate momentum. This count assumes a regular part of the constraint surface and refers to local bulk degrees of freedom; boundaries and global symmetries require separate treatment. The canonical constraint chapter develops the count instead of trying to subtract identities directly from Einstein’s ten equations.

Calling two formulations equivalent requires more than obtaining similar-looking equations. One must specify a field map and the region where it is invertible, impose the relevant gauge and constraints, translate initial and boundary data, and show that the subsidiary constraints propagate. A tetrad must remain nondegenerate; a null foliation must avoid caustics; a conformal factor must have the required regular zero; a symmetry ansatz must satisfy every discarded field equation. Failure of any one condition may lose Einstein solutions, add spurious solutions, or merely signal that a useful coordinate chart has ended.

This distinction separates three questions that are often conflated:

1. Geometric equivalence: does reconstruction give the same spacetime metric, up to diffeomorphism and any frame gauge?
2. Analytic fitness: does the reduced system have an appropriate principal symbol, constraint propagation law, and well-posed data problem?
3. Computational fitness: does a chosen discretization preserve those continuum properties accurately and stably?

This volume treats the first two. Discretization, convergence studies, and production algorithms belong to Computational Relativity. The continuum role of the principal part and a compatible energy estimate is summarized in Lehner 2001, §§ 4.4.1–4.4.2, pp. 31–33, PDF; later chapters define and distinguish weak, strong, and symmetric hyperbolicity for the systems in which those labels are used.

A first example: a moving spatial grid

Begin with flat spacetime in inertial coordinates $(T,X,Y,Z)$ and introduce

$$T=t,\qquad X=x+vt,\qquad Y=y,\qquad Z=z,$$

where $v$ is a constant and $c=1$. Substituting the coordinate differentials gives

$$ds^2=-dt^2+(dx+v\,dt)^2+dy^2+dz^2.$$

Compare this with the ADM metric decomposition: the lapse is $\alpha=1$, the shift is $\beta^i=(v,0,0)$, and the spatial metric is $\gamma_{ij}=\delta_{ij}$. The slices $t=\mathrm{const}$ are the original flat $T=\mathrm{const}$ slices, so their extrinsic curvature vanishes. The future unit normal is

$$n=\partial_t-v\partial_x=\partial_T, \qquad \partial_t=n+v\partial_x.$$

The shift measures how the coordinate grid slides within a slice. It does not create curvature or a gravitational wave. Even for $\lvert v\rvert\gt1$, the slices remain spacelike and $n$ remains timelike; only the coordinate lines $x,y,z=\mathrm{const}$ become spacelike. Such lines are labels, not possible observer worldlines. This elementary example illustrates why $g_{tt}$ alone does not decide whether a foliation is admissible. The lapse, shift, and normal interpretation follows Gourgoulhon 2007, §4.2, pp. 54–59, PDF.

Keep this example in mind when comparing formulations: first identify what changed in the geometry, what changed only in its labels, and which data are actually constrained. The gauge transformations page makes that distinction precise.

Conventions and discriminating checks

The site uses signature $(-,+,+,+)$. For a spacelike foliation with future unit normal $n^\mu$, the conventions that connect the initial-data, evolution, and Hamiltonian chapters are

$$\gamma_{\mu\nu}=g_{\mu\nu}+n_\mu n_\nu, \qquad t^\mu=\alpha n^\mu+\beta^\mu, \qquad K_{ij}=-\frac12\mathcal L_n\gamma_{ij}.$$

Thus $(\partial_t-\mathcal L_\beta)\gamma_{ij}=-2\alpha K_{ij}$. Matter projections use $\rho=T_{\mu\nu}n^\mu n^\nu$ and $j_i=-\gamma_i{}^\mu n^\nu T_{\mu\nu}$. To keep that timelike unit normal distinct from a Newman–Penrose leg, null frames use $(\ell^\mu,n_{\mathrm{NP}}^\mu,m^\mu,\bar m^\mu)$ with $\ell\!\cdot n_{\mathrm{NP}}=-1$ and $m\!\cdot\bar m=1$. Conformal pages distinguish the physical metric $\widetilde g$ from the regular metric $g=\Omega^2\widetilde g$. These choices fix the sign chain from Gauss–Codazzi projections through canonical momentum, junction terms, and boundary charges; the complete 3+1 derivation in the same signature and extrinsic-curvature convention is given in Gourgoulhon 2007, Chapters 3–5, pp. 39–82, PDF.

Four examples recur because each can disprove a different kind of mistake. Minkowski spacetime distinguishes coordinate or tetrad effects from curvature. Schwarzschild spacetime tests alternative slicings, shell branches, characteristic variables, and asymptotic energy. Linearized gravity checks physical polarizations, principal symbols, residual gauge, and energy flux. Spherically symmetric Einstein–matter systems connect quotient geometry, constraints, null transport, and finite boundaries without hiding the distinction between an exact reduction and an approximation.

A successful flat-space check is necessary but not sufficient. A sign error can vanish when $K_{ij}=0$; a missing constraint can vanish under symmetry; and an incorrectly squared shell equation can reproduce a potential while losing the orientation branch. Each chapter therefore uses a check that retains the structure most likely to fail.

Chapter map

The chapters are ordered by the information a formulation must expose, from gauge and geometric splitting to boundary compatibility.

• Gauge Freedom and Coordinate Conditions distinguishes active and passive diffeomorphisms, develops harmonic, lapse, and shift conditions, and identifies residual freedom and gauge breakdown.
• Symmetry and Dimensional Reduction passes from group actions to quotient fields, spherical and stationary reductions, twist potentials, Kaluza–Klein variables, and the test for a consistent truncation.
• Hypersurface Geometry and Junction Conditions derives regular matching, nonnull and null surface layers, shell motion, joint terms, and matter matching across an interface.
• The 3+1 Decomposition introduces lapse, shift, spatial geometry, extrinsic curvature, the curvature projections, Einstein evolution equations, and matter sources.
• Constraint Equations and Initial Data treats the Hamiltonian and momentum constraints, York splitting, conformal methods, asymptotically flat and black-hole data, and gluing.
• Cauchy Evolution and Hyperbolicity compares ADM, BSSN, generalized harmonic, and Z4 systems through principal symbols, characteristics, constraint propagation, damping, and energy estimates.
• Characteristic and Null Formulations develops null hypersurface data, double-null and Bondi–Sachs hierarchies, the local characteristic problem, caustics, and Cauchy–characteristic interfaces.
• Hamiltonian General Relativity derives canonical momentum, the constraint Hamiltonian, hypersurface-deformation algebra, gauge generators, observables, and boundary Hamiltonians.
• Tetrad and Connection Formulations replaces the metric by coframes and Lorentz connections, then follows torsion, Palatini and Holst variation, and Ashtekar–Barbero variables.
• Spinor, Newman-Penrose, and GHP Methods projects curvature and differential identities onto a null tetrad while controlling boosts, spins, spin coefficients, and weighted derivatives.
• Conformal and Hyperboloidal Formulations regularizes null infinity with conformal field equations, hyperboloidal data, scri-fixing gauges, and explicit regularity tests.
• Covariant Phase Space, Boundaries, and Charges constructs the presymplectic current, Noether charge, degeneracy directions, flux, and the integrability conditions for a Hamiltonian generator.
• Initial-Boundary Value Problems in AdS and Finite Domains combines incoming physical modes, gauge data, constraint-preserving conditions, boundary energy, asymptotically AdS data, and corner compatibility.

Routes through the material

For a first encounter with the Cauchy problem, begin with lapse, shift, and the spatial metric, then extrinsic curvature and metric evolution, and then the Hamiltonian and momentum constraints. Work their solved examples before moving to gauge conditions, hyperbolicity, and finite-domain boundaries. This route answers a single continuous question: which data may be chosen, which are constrained, which variables evolve, and what may enter through a boundary?

For radiation at null infinity, begin with gauge and characteristic formulations, continue to conformal and hyperboloidal methods, and finish with covariant phase space. The first chapter controls the labels, the second identifies radiative transport, the third regularizes the asymptotic location, and the fourth decides whether a proposed symmetry variation has a finite, integrable charge. Winicour’s worldtube–null-cone construction gives the characteristic side of this seam in Winicour 2012, §§ 2.1, 6.1, and 6.6, PDF.

For canonical or frame methods, take 3+1 decomposition before Hamiltonian GR, then read tetrad and connection variables before NP/GHP. Covariant phase space comes last because its surface charges can be compared with canonical boundary Hamiltonians only after orientations, allowed variations, and boundary conditions have been fixed.

Researchers can instead enter through a task. For admissible black-hole slices, use the constraint chapter. For a continuum evolution system, compare the characteristic matrices in the hyperbolicity chapter before selecting boundary data. For matching an interior to a vacuum exterior, use the junction chapter and then decide whether the interface is part of initial data, an evolved timelike boundary, or a material shell. For asymptotically AdS evolution, combine the conformal and initial-boundary chapters; neither a Fefferman–Graham expansion alone nor a hyperbolic reduction alone supplies a complete dynamical problem.

How neighboring formulations meet

A formulation is most vulnerable at its seams. Initial data produced by an elliptic conformal method must satisfy the sign convention used by the evolution system. Characteristic data on two null hypersurfaces must agree at their corner. A finite timelike boundary must control the incoming characteristic fields while preserving the harmonic or Z4 subsidiary constraints. A charge variation must be evaluated on the same phase space selected by those boundary conditions.

A Cauchy–characteristic matching problem has a Cauchy interior and an outgoing Bondi–Sachs characteristic exterior meeting at a timelike worldtube. In addition to the Cauchy data, one must specify the freely chosen characteristic data on an initial outgoing null hypersurface and impose compatibility at the initial two-surface where that hypersurface, the worldtube, and the initial Cauchy slice meet. The worldtube embedding and its coordinate Jacobian must reconstruct the same metric and derivatives on both sides. The Bondi–Sachs radial hierarchy also needs worldtube integration data that satisfy the worldtube constraints, and the null-coordinate domain must end before the outgoing congruence forms a caustic or its Jacobian degenerates; see Winicour 2012, §§ 2.1, 5.1–5.8, and 6.6, PDF.

Incoming fields are reduction-dependent: the Cauchy and characteristic systems must each identify them from its own principal symbol and then translate them through common interface variables. A proof of coupled well-posedness would additionally require an interface energy estimate controlling the shared flux without prescribing the same outgoing datum twice. Separate well-posedness results for the Cauchy reduction and characteristic hierarchy, even together with compatible reconstruction data, do not by themselves prove well-posedness of the full nonlinear Einstein Cauchy–characteristic matching problem.

Canonical and covariant boundary operations answer different questions. In Hamiltonian GR, varying a bulk generator generally leaves a surface variation, so a surface functional is added to make the generator differentiable on the chosen canonical phase space. In covariant phase space, $L\mapsto L+d\mu$ preserves the bulk Euler–Lagrange equations while inducing the corresponding shift of the symplectic potential, and the independent ambiguity $\boldsymbol\Theta\mapsto\boldsymbol\Theta+d\boldsymbol Y$ moves presymplectic and charge expressions by boundary or corner terms. Boundary action and corner forms must therefore be carried consistently. These variational choices are distinct from PDE boundary conditions, which control incoming characteristic fields and select the admissible solution space. Iyer and Wald derive the local Noether-current and surface-charge structure and its ambiguities in Iyer and Wald 1994, §§3–5, pp. 6–13, PDF; Harlow and Wu treat total derivatives, spatial-boundary actions, corner potentials, and Hamiltonians systematically in Harlow and Wu 2020, §§ 2.2–2.4, pp. 11–23, PDF. Canonical and covariant generators agree only after their common solution space, orientations, reference normalization, and allowed boundary variations are matched.

The strongest theorem in one formulation also does not automatically transfer to another. A smooth invertible change of variables preserves a local solution, but derivative-dependent maps, gauge fixing, boundaries, and loss of regularity can change the function spaces or compatibility conditions. Friedrich and Nagy’s vacuum initial-boundary theorem assumes a smooth, orientable, compact three-manifold with nonempty boundary, smooth constraint-satisfying vacuum initial data, smooth boundary data and associated gauge-source functions, and consistency conditions of every order at the initial-boundary corner. Within that setting it supplies the stated reduced system, maximally dissipative boundary data, and local existence. Its local geometric-uniqueness conclusion is further restricted to constant boundary mean curvature $\chi=\chi_0$ and vanishing gauge source $F=0$ on the timelike boundary $T$. See Friedrich and Nagy 1999, §§3–8, pp. 624–652, especially Definition 7.2 and Theorem 8.1, pp. 650–652.

Scope and limits

The volume develops continuum formulations of classical Einstein gravity. It does not rank numerical codes, prescribe mesh or spectral algorithms, prove the sharpest low-regularity theorems, catalog exact metrics, derive waveform models, or quantize Ashtekar variables. It also does not treat every formal field redefinition as useful: a formulation must state its reconstruction map and failure set.

Several negative statements are just as important as the equations. Coordinate blow-up is not necessarily curvature blow-up. Constraint damping does not remove the need for constraint-compatible data. A characteristic chart cannot pass through a caustic merely because the spacetime remains smooth. A finite charge variation need not be integrable. Reflective AdS boundary conditions are a choice, not a consequence of asymptotic AdS geometry alone. A stable discretization cannot repair an ill-posed continuum system.

For object-first lookup, use the Black Hole Atlas. The Research Atlas carries dated assessments of open formulation and evolution questions, while Reference provides compact conventions and formulas. Numerical implementation, convergence, and code-facing validation continue in Computational Relativity.

Review the volume

1. Coordinate time and proper time. In the moving-grid example, compute the squared norm of $\partial_t$ and the proper time of a fixed-grid observer when $\lvert v\rvert\lt1$. Why does this differ from the unit lapse?

Solution

Since $\partial_t=\partial_T+v\partial_X$, its squared norm is $-1+v^2$. Along a timelike fixed-grid line, $d\tau=\sqrt{1-v^2}\,dt$. The lapse instead measures proper time along the normal observers: they have $dx=-v\,dt$, so $d\tau=dt$. The two observer families coincide only at $v=0$. A coordinate transformation has changed neither flatness nor the geometry of the slices.

2. Combining formulations.

Consider vacuum general relativity in a finite region. Initial data $(\gamma_{ij},K_{ij})$ are supplied on a spacelike slice, generalized harmonic variables evolve a Cauchy interior, that interior is coupled across a timelike worldtube to an outgoing Bondi–Sachs characteristic exterior, and a time-translation charge is requested. List the compatibility questions that must be answered before these pieces define one problem.

Solution

First, $(\gamma_{ij},K_{ij})$ must satisfy the Hamiltonian and momentum constraints and must be converted consistently into the generalized-harmonic metric and first-derivative variables, including initial gauge-source data. Second, the reduced harmonic constraints and their normal derivatives must have compatible initial values. Third, the timelike portion of the boundary requires a count of incoming characteristic fields, with physical, gauge, and constraint-preserving conditions arranged in a boundary hierarchy; its values must agree with the initial data at the corner.

The characteristic exterior also needs freely specified data on its initial outgoing null hypersurface, compatible with the Cauchy data and worldtube data at their initial two-surface. At the timelike worldtube, the embedding map and a nonsingular coordinate Jacobian must carry the induced two-geometry, null-frame normalization, and the metric derivatives required by the Bondi–Sachs hierarchy between the two descriptions. Its radial integration data must satisfy the worldtube constraints, and the outgoing congruence must remain caustic-free throughout the intended characteristic domain.

Each chosen Cauchy or characteristic reduction must identify its own incoming fields. Matching must exchange those fields through common interface variables without prescribing outgoing data twice, and a coupled proof would need an interface energy estimate that controls the shared flux. Separate well-posedness of the two reductions is not a proof of the full nonlinear Einstein matching problem. Finally, a time-translation charge requires a specified phase space and boundary conditions, finite presymplectic flux, an integrable surface variation, an orientation, and a reference normalization. Passing each question separately is not enough: the reconstructed metric and boundary data must agree on every overlap.

References

• Friedrich, Helmut, and Gabriel Nagy. “The Initial Boundary Value Problem for Einstein’s Vacuum Field Equation.” Communications in Mathematical Physics 201 (1999): 619–655. DOI.
• Gourgoulhon, Éric. 3+1 Formalism and Bases of Numerical Relativity. Lecture notes, 2007. arXiv:gr-qc/0703035. Open PDF.
• Harlow, Daniel, and Jie-qiang Wu. “Covariant Phase Space with Boundaries.” Journal of High Energy Physics 2020, article 146 (2020). DOI. Open PDF.
• Iyer, Vivek, and Robert M. Wald. “Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy.” Physical Review D 50 (1994): 846–864. DOI. Open PDF.
• Lehner, Luis. “Numerical Relativity: A Review.” Classical and Quantum Gravity 18 (2001): R25–R86. DOI. Open PDF.
• Winicour, Jeffrey. “Characteristic Evolution and Matching.” Living Reviews in Relativity 15 (2012): article 2. DOI. Open PDF.