Constraints, Evolution, and Gauge Content

Einstein’s equation has ten symmetric component equations for ten metric components, but this is not a count of ten physical waves. Four differential identities, four coordinate functions, and four constraint equations organize the system so that the linearized vacuum theory carries two propagating polarizations. This page gives that bounded structural count; constructing a gauge-fixed hyperbolic formulation and proving a nonlinear initial-value theorem belong to GR Formulations.

Required background. We use the Riemann curvature tensor for the Bianchi identity, Hilbert stress–energy for source projections, and the scope of the equivalence principles to distinguish coordinate freedom from physical local measurements.

Helpful background. Hypersurface normals and tangential projectors provide the geometric language of a spacelike slice; Minkowski geometry supports the polarization count; natural tensor operations clarify covariance; and presymplectic potentials prepare the constraint–gauge relation in phase space.

Ten components and four identities

Define the field-equation tensor

$$E_{\mu\nu} =G_{\mu\nu}+\Lambda g_{\mu\nu}-8\pi G T_{\mu\nu}.$$

It is symmetric, so $E_{\mu\nu}=0$ displays ten equations in four dimensions. On the matter equations, the contracted Bianchi identity gives

$$\nabla^\mu E_{\mu\nu}=0,$$

four differential relations among those equations. These are identities, not four coordinate conditions. Diffeomorphism covariance separately means that four freely chosen coordinate functions can describe the same geometry. Carroll’s component discussion makes both numbers visible in Carroll 1997, § 4, printed pp. 112–113, PDF, but subtracting them mechanically is only a heuristic until one identifies constraints and evolution.

Normal and tangential projections

Think of spacetime as a stack of spacelike slices $\Sigma_t$. The induced metric $h_{ij}$ measures geometry within one slice, its Levi-Civita derivative is $D_i$, and ${}^{(3)}R$ is its intrinsic scalar curvature. The future unit normal $n^\mu$ points from one slice toward the next, while the extrinsic curvature $K_{ij}$ records how the slice bends in spacetime. A coordinate time-flow vector decomposes as

$$(\partial_t)^\mu=Nn^\mu+\beta^\mu, \qquad n_\mu\beta^\mu=0.$$

The lapse $N$ controls proper-time advance in the normal direction; the shift $\beta^i$ moves spatial coordinate labels within the next slice. Neither is a new gravitational polarization. Hypersurface Normals and Tangential Projectors develops $n^\mu$, $h_{\mu\nu}$, and tangential projection; Extrinsic Curvature and the Weingarten Map develops $K_{ij}$. Lapse, shift, and their formulation-specific gauge choices are continued in GR Formulations.

Let $h_{\mu\nu}=g_{\mu\nu}+n_\mu n_\nu$. For this page fix

$$K_{\mu\nu} =h_\mu{}^\alpha h_\nu{}^\beta\nabla_\alpha n_\beta,$$

so $K_{ij}=\tfrac12\mathcal L_n h_{ij}$ in coordinates transported along the normal. The ten symmetric equations now split as $1+3+6$: one normal–normal projection, three normal–tangential projections, and six symmetric tangential–tangential projections. The first gives the Hamiltonian constraint

$${}^{(3)}R+K^2-K_{ij}K^{ij} =16\pi G\rho+2\Lambda, \qquad \rho=T_{\mu\nu}n^\mu n^\nu.$$

One normal and one tangential projection gives the three momentum constraints

$$D_jK^j{}_i-D_iK =8\pi G\,h_i{}^\mu T_{\mu\nu}n^\nu.$$

These four equations involve $h_{ij}$ and $K_{ij}$ but no second derivative normal to $\Sigma$. The six tangential–tangential projections contain the second normal derivatives and act as evolution equations after lapse and shift, hence a coordinate gauge, are chosen. The projection map is therefore $1$ Hamiltonian constraint, $3$ momentum constraints, and $6$ evolution equations. Reall derives the $\Lambda=0$ projections in this extrinsic-curvature convention and identifies their differential order in Reall 2013, § 11.4, printed p. 128, Eqs. (11.23)–(11.26), PDF. Projecting the additional $\Lambda g_{\mu\nu}$ term gives the displayed $+2\Lambda$ on the Hamiltonian constraint’s right side, while it contributes nothing to the momentum constraint because

$$h_i{}^\mu g_{\mu\nu}n^\nu=0.$$

The remaining signs follow the explicitly stated $K_{ij}$ convention; reversing $K_{ij}$ reverses the momentum equation’s left side but not the quadratic combination in the Hamiltonian constraint.

Different lapse and shift choices can produce very different component evolutions for the same spacetime geometry. A GR formulation supplements Einstein’s equations with gauge conditions and often introduces auxiliary variables so that the resulting partial differential system has controlled characteristic and constraint-propagation behavior.

The Bianchi identity makes constraint propagation compatible with the evolution equations: if the evolution and matter equations hold, the divergence relations govern how constraint violations propagate. It does not guarantee that arbitrary initial data satisfy the constraints.

Counting physical gravitational data

On a slice, $h_{ij}$ has six components and $K_{ij}$ supplies six velocity-like components, for twelve initial-data functions. The Hamiltonian and momentum constraints reduce these to eight. Quotienting by the four-function diffeomorphism freedom leaves four reduced phase-space functions: two canonical pairs, or two local configuration degrees of freedom.

The same result appears transparently around a Minkowski background. A symmetric perturbation $h_{\mu\nu}$ begins with ten components. Four infinitesimal gauge functions act as

$$h_{\mu\nu}\mapsto h_{\mu\nu} +\partial_\mu\xi_\nu+\partial_\nu\xi_\mu.$$

Four constraint or nondynamical combinations remove the remaining nonradiative components, leaving two transverse, trace-free polarizations. Gauge freedom and constraints are paired structures, so the mnemonic $10-4-4=2$ should not be read as three unrelated algebraic subtractions.

The initial-data count says the same thing in canonical language: six entries of $h_{ij}$ are configuration variables, and their six conjugate velocity-like entries reside in $K_{ij}$. Four first-class constraint directions both restrict the data and generate gauge transformations. Counting a constraint once as a restriction and once through its associated gauge orbit removes two phase-space functions per gauge direction, leaving a four-dimensional local reduced phase space.

Limits of the count

The canonical count gives two local gravitational configuration degrees of freedom in the regular four-dimensional Einstein system; around a Minkowski vacuum they appear as the two familiar wave polarizations. This does not make arbitrary nonlinear geometry a superposition of two waves. Matter adds its own modes; special backgrounds can have extra symmetries or degeneracies; boundaries can support additional data; and alternative gravitational actions can change the count. Most importantly, component counting does not choose a stable gauge, establish hyperbolicity, solve the constraints, or prove existence and uniqueness. Those questions require an explicit formulation and function-space assumptions.

Exercises

Explain why six freely specified components of $h_{ij}$ and six of $K_{ij}$ do not describe six propagating gravitational polarizations.

Solution

The twelve functions are phase-space data, not twelve configuration modes. Four constraint equations reduce them to eight. Four functions’ worth of diffeomorphism-related data describe the same physical geometry, leaving four reduced phase-space functions. A propagating configuration mode requires a canonical pair, so four phase-space functions correspond to two local gravitational modes. This count assumes a regular four-dimensional vacuum Einstein system and does not prove its evolution is well posed.

References

• Carroll, Sean M. “Lecture Notes on General Relativity.” NSF-ITP/97-147, arXiv:gr-qc/9712019v1 (1997). arXiv record.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes (2013), § 11.4, printed p. 128. PDF.