Conformal Einstein Field Equations

The vacuum Einstein equation for the physical metric can be expressed using the regular unphysical metric $g_{\mu\nu}=\Omega^2\widetilde g_{\mu\nu}$. A direct substitution, however, leaves explicit divisions by $\Omega$. The conformal field equations remove those divisions by enlarging the set of unknowns: derivatives of $\Omega$, the Schouten tensor, and a rescaled Weyl tensor become independent fields tied together by regular differential and algebraic equations.

Required background. The Riemann Curvature Tensor supplies the Ricci, Weyl, and Bianchi decompositions used below.

Helpful background. We use The Einstein Equation, Hypersurface Normals and Tangential Projectors, Formulations of General Relativity, and Causality and Global Structure.

Conformal Einstein equations before regularization

Throughout, tildes mark physical fields and

$$g_{\mu\nu}=\Omega^2\widetilde g_{\mu\nu}, \qquad \Omega\gt 0$$

in the physical spacetime. We assume vacuum without cosmological constant on this page. Substitution of the conformal curvature formulas into $\widetilde G_{\mu\nu}=0$ gives

$$G_{\mu\nu}[g] =-\frac{2}{\Omega} \left(\nabla_\mu\nabla_\nu\Omega-g_{\mu\nu}\Box\Omega\right) -\frac{3}{\Omega^2}g_{\mu\nu}(\nabla\Omega)^2.$$

This equation is equivalent to physical vacuum Einstein wherever $\Omega\gt 0$, as derived in Zenginoğlu 2008, § 1, equations (1)–(2). It is not manifestly regular at $\Omega=0$. Smoothness of $g$ alone does not authorize setting $\Omega=0$ in the displayed fractions.

Two related strategies must therefore be distinguished. One may retain this equation and evaluate finite boundary limits using regularity plus a preferred conformal gauge. Alternatively, one may replace it by a larger system whose coefficients contain no negative powers of $\Omega$. The latter is the conformal field-equation strategy.

A regular vacuum conformal system

Define the unphysical Schouten tensor, Friedrich scalar, and rescaled Weyl tensor by

$$L_{\mu\nu} =\frac12\left(R_{\mu\nu}-\frac16Rg_{\mu\nu}\right), \qquad s=\frac14\Box\Omega+\frac1{24}R\Omega, \qquad d^\mu{}_{\nu\rho\sigma} =\Omega^{-1}C^\mu{}_{\nu\rho\sigma}[g].$$

For fields that extend smoothly, a metric conformal vacuum system is

$$\begin{aligned} \nabla_\mu\nabla_\nu\Omega &=-\Omega L_{\mu\nu}+s g_{\mu\nu},\\ \nabla_\mu s &=-L_{\mu\nu}\nabla^\nu\Omega,\\ \nabla_\mu L_{\nu\rho}-\nabla_\nu L_{\mu\rho} &=\nabla_\sigma\Omega\,d^\sigma{}_{\rho\mu\nu},\\ \nabla_\sigma d^\sigma{}_{\rho\mu\nu}&=0,\\ 6\Omega s-3(\nabla\Omega)^2&=0. \end{aligned}$$

The curvature decomposition closes the system,

$$R_{\mu\nu\rho\sigma} =\Omega d_{\mu\nu\rho\sigma} +2\left( g_{\mu[\rho}L_{\sigma]\nu} -g_{\nu[\rho}L_{\sigma]\mu} \right).$$

Here antisymmetrization has weight $1/2$, and the first index of $d^\mu{}_{\nu\rho\sigma}$ is lowered with the unphysical metric. These details make $2(g_{\mu[\rho}L_{\sigma]\nu}-g_{\nu[\rho}L_{\sigma]\mu})$ the full Ricci part of the Riemann tensor. The mixed-index Weyl tensor is conformally invariant, $C^\mu{}_{\nu\rho\sigma}[g]=C^\mu{}_{\nu\rho\sigma}[\widetilde g]$; the rescaled field $d$ is an additional regularity variable, not that invariant tensor itself.

No displayed equation divides by $\Omega$. Friedrich’s conformal vacuum formulation and its hyperboloidal initial-value interpretation are developed in Friedrich 1983, §§ 2–4, pp. 449–463. The equations still require coordinate, frame, and conformal gauge conditions before becoming a determined hyperbolic evolution system; “regular” does not mean “gauge free.”

The assumption that $d^\mu{}_{\nu\rho\sigma}$ is smooth is substantive. Because it equals $\Omega^{-1}C^\mu{}_{\nu\rho\sigma}$, an arbitrary asymptotically flat vacuum solution need not admit such a smooth extension. Polyhomogeneous completions and weaker peeling lie outside the smooth system stated here.

Reconstruction and the boundary equations

Where $\Omega\gt 0$, the enlarged equations recover physical vacuum rather than a different theory. Trace the Hessian equation and use the algebraic equation:

$$\Box\Omega=-\frac16R\Omega+4s, \qquad (\nabla\Omega)^2=2\Omega s.$$

Inserting these and $\nabla_\mu\nabla_\nu\Omega=-\Omega L_{\mu\nu}+sg_{\mu\nu}$ into the conformal Ricci transformation gives

$$\widetilde R_{\mu\nu} =R_{\mu\nu}-2L_{\mu\nu}-\frac16Rg_{\mu\nu}=0.$$

This reconstruction is a useful independent normalization check: an incorrect coefficient in any of the three conformal equations leaves an uncancelled $s/\Omega$ term.

At a regular boundary $\mathscr I=\{\Omega=0,d\Omega\ne0\}$, the algebraic equation yields

$$(\nabla\Omega)^2\big|_{\mathscr I}=0,$$

so $\mathscr I$ is null. The Hessian equation gives $\nabla_\mu\nabla_\nu\Omega=s g_{\mu\nu}$ there. A conformal-gauge rescaling $g\mapsto\omega^2g$, $\Omega\mapsto\omega\Omega$ can impose the preferred condition $\Box\Omega|_{\mathscr I}=0$, equivalently $s|_{\mathscr I}=0$, after which the Hessian itself vanishes at the boundary. Zenginoğlu derives this gauge and the finite limits of the direct source equation in Zenginoğlu 2008, § 3.1, equations (9)–(11).

Common pitfalls

Equating finite limits with manifest regularity. The direct rescaled Einstein equation can have finite limits in a suitable gauge, but it still contains explicit negative powers. The enlarged conformal system avoids those divisions by adding constrained variables.

Treating $d=\Omega^{-1}C$ as a definition with automatic smoothness. It is a definition in the physical region and a regularity requirement at the boundary. Smoothness of $g$ alone does not guarantee smoothness of $d$.

Exercises

Use only the first, fifth, and defining equations above to prove that $\widetilde R_{\mu\nu}=0$ wherever $\Omega\gt 0$.

Solution

The definition of $s$ and the trace of the first equation both give $\Box\Omega=-R\Omega/6+4s$. The fifth equation gives $(\nabla\Omega)^2=2\Omega s$. Substitution into

$$\widetilde R_{\mu\nu} =R_{\mu\nu} +2\Omega^{-1}\nabla_\mu\nabla_\nu\Omega +g_{\mu\nu}\Omega^{-1}\Box\Omega -3g_{\mu\nu}\Omega^{-2}(\nabla\Omega)^2$$

cancels all terms proportional to $s/\Omega$. The remainder is $R_{\mu\nu}-2L_{\mu\nu}-Rg_{\mu\nu}/6$, which vanishes by the definition of $L_{\mu\nu}$.

References

• Friedrich, Helmut. “Cauchy Problems for the Conformal Vacuum Field Equations in General Relativity.” Communications in Mathematical Physics 91 (1983): 445–472. DOI.
• Zenginoğlu, Anıl. “Hyperboloidal Evolution with the Einstein Equations.” Classical and Quantum Gravity 25 (2008): 195025. DOI. Open PDF.