Constraints and the Contracted Bianchi Identity

Einstein initial data cannot be chosen freely. The normal projections of the field equation impose Hamiltonian and momentum constraints on every spacelike Cauchy hypersurface. The contracted Bianchi identity then supplies a homogeneous subsidiary evolution: if the constraints and gauge conditions hold initially and matter stress-energy is conserved, they continue to hold throughout the reduced solution’s domain of dependence.

Required background. Analysis for Mathematical Relativity supplies uniqueness for subsidiary equations, The Einstein Equation supplies the field equation, PDEs and Green Functions for Gravitation supplies domain-of-dependence arguments, and Tensor Fields and Natural Operations supplies tensor projections.

Helpful background. Causality and Global Structure explains global hyperbolicity and Cauchy development.

Normal projections of Einstein’s equation

Let $\Sigma$ be spacelike with future unit normal $n^\mu$, induced metric $h_{\mu\nu}=g_{\mu\nu}+n_\mu n_\nu$, and

$$K_{\mu\nu}=-h_\mu{}^\rho h_\nu{}^\sigma\nabla_\rho n_\sigma.$$

This sign is consequential. Write $D_i$ and $R(h)$ for the Levi-Civita derivative and scalar curvature of $h_{ij}$, and define

$$\rho=T_{\mu\nu}n^\mu n^\nu, \qquad j_i=-h_i{}^\mu n^\nu T_{\mu\nu}.$$

The Gauss relation contracted twice and the Codazzi relation contracted once give, in four spacetime dimensions,

$$R(h)+K^2-K_{ij}K^{ij}=16\pi G\rho+2\Lambda,$$ $$D^j(K_{ij}-Kh_{ij})=8\pi G j_i.$$

These are respectively the Hamiltonian and momentum constraints. They contain no second derivative in the normal direction and so restrict the Cauchy data $(h_{ij},K_{ij})$ rather than evolve them. The sign of the momentum equation agrees with the stated definitions of $K_{ij}$ and $j_i$; sources using $K_{ij}=+h_i{}^\mu h_j{}^\nu\nabla_\mu n_\nu$ must be translated as a whole.

The remaining tangential projection determines evolution only after lapse, shift, and gauge have been chosen. Bartnik and Isenberg derive the constraint system and its relation to developments in Bartnik and Isenberg 2004, §2, pp. 3–7 and §§3.1–3.4, pp. 7–13.

The Bianchi mechanism

Define the field-equation residual

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

The contracted Bianchi identity and matter conservation imply

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

In the generalized wave reduction used here, the reduced equations are

$$E_{\mu\nu}-\nabla_{(\mu}C_{\nu)} +\frac12g_{\mu\nu}\nabla_\alpha C^\alpha=0,$$

where $C^\mu$ is the gauge constraint. Taking a divergence and commuting derivatives yields

$$\Box_g C_\nu+R_\nu{}^\mu C_\mu=0.$$

Gauge initialization supplies $C|_\Sigma=0$; the Einstein constraints then supply $\nabla_nC|_\Sigma=0$ through the reduced equations. The second condition is therefore a consequence of compatible geometric and gauge data, not four additional freely imposed conditions on $(h,K)$. Uniqueness for this homogeneous wave equation gives $C=0$. Substitution into the reduced equation gives $E_{\mu\nu}=0$ everywhere in the domain of dependence. The derivation is presented in Friedrich and Rendall 2000, §2.4, equation (2.39) and the following subsidiary argument.

This proof has four distinct inputs:

1. the initial geometric constraints vanish;
2. the initial gauge constraint and its normal derivative vanish;
3. the reduced evolution has enough regularity for the divergence and uniqueness argument;
4. the matter equations imply $\nabla^\mu T_{\mu\nu}=0$.

Omitting any one changes the conclusion. Nonzero subsidiary initial data evolve by the same homogeneous equation and need not vanish. A numerical discretization adds truncation sources as well. Constraint damping may control such errors, but it is a formulation technique rather than the continuum proof given here.

Hypotheses and a discriminating failure

Assume the reduced metric has the regularity needed to interpret the Ricci tensor and to apply uniqueness for the subsidiary wave equation; assume also that $\Sigma$ is spacelike and that the matter equations imply stress-energy conservation. Under these hypotheses, the proof is an energy-uniqueness argument: derive the homogeneous equation, identify zero Cauchy data, and conclude that the only subsidiary solution is zero.

A useful counterexample is an externally prescribed tensor with $\nabla^\mu T_{\mu\nu}=J_\nu\ne0$. Then the same divergence calculation gives $\Box_g C_\nu+R_\nu{}^\mu C_\mu=-16\pi GJ_\nu$. Even perfect initial constraints generally develop a nonzero violation sourced by $J_\nu$. The limitation is structural: the Bianchi identity enforces compatibility between geometry and matter rather than repairing an inconsistent source.

There is also a regularity limitation. If the metric is too rough for the commuted divergence identity or for uniqueness of the weak subsidiary equation, the formal calculation does not establish propagation. One must formulate both the Einstein residual and the subsidiary system in the same weak solution class.

Constraint propagation without wave gauge

The mechanism is geometric even though its explicit subsidiary system depends on formulation. In a $3+1$ split, let $\mathcal H$ and $\mathcal M_i$ denote the Hamiltonian and momentum residuals. Combining the evolution equations with $\nabla^\mu E_{\mu\nu}=0$ produces a first-order homogeneous system of the schematic form

$$\partial_t\mathcal H =\beta^iD_i\mathcal H-2\alpha D^i\mathcal M_i+\text{lower-order terms},$$ $$\partial_t\mathcal M_i =\beta^jD_j\mathcal M_i-\frac{\alpha}{2}D_i\mathcal H+\text{lower-order terms}.$$

If this subsidiary system is well posed, zero initial residuals remain zero. The precise coefficients depend on lapse, shift, and which multiples of the constraints were added to the evolution equations. What is invariant is the Bianchi identity that closes the system.

A common misconception is that the constraints need only be solved once because they are algebraic. They are differential equations on the initial slice, and their propagation is a theorem conditional on solving the chosen evolution and matter systems. The next chapter constructs data satisfying them; The Einstein Constraint Map makes their elliptic structure explicit.

Check your understanding

Verify the vacuum constraints for the standard $t=0$ slice of Minkowski spacetime.

Solution

The induced metric is Euclidean, so $R(h)=0$. The slice is totally geodesic, hence $K_{ij}=0$. With $\rho=j_i=\Lambda=0$, both $R(h)+K^2-K_{ij}K^{ij}=0$ and $D^j(K_{ij}-Kh_{ij})=0$ hold. Because every term vanishes, this is a baseline check but cannot discriminate the signs of the quadratic terms.

A stronger check uses the future Minkowski hyperboloid $t=\sqrt{r^2+a^2}$, $a\gt0$. Its intrinsic sectional curvature is $-a^{-2}$, so $R(h)=-6/a^2$, while the stated normal convention gives $K_{ij}=-h_{ij}/a$. Hence $K^2=9/a^2$ and $K_{ij}K^{ij}=3/a^2$: $-6+9-3=0$ verifies the nontrivial cancellation. The momentum equation vanishes because $D h=0$ and $a$ is constant. Neither example proves propagation; that still uses the subsidiary system.

References

• Bartnik, Robert, and Jim Isenberg. “The Constraint Equations.” In The Einstein Equations and the Large Scale Behavior of Gravitational Fields, 1–38. Birkhäuser, 2004. arXiv:gr-qc/0405092.
• Friedrich, Helmut, and Alan D. Rendall. “The Cauchy Problem for the Einstein Equations.” In Einstein’s Field Equations and Their Physical Implications, Lecture Notes in Physics 540, 127–224. Springer, 2000. arXiv:gr-qc/0002074.

Further reading

• Ringström, Hans. The Cauchy Problem in General Relativity. ESI Lectures in Mathematics and Physics. European Mathematical Society, 2009. Publisher record.