Null-Infinity Fixing and Gauge Choices

Null-infinity fixing keeps the conformal boundary at a time-independent coordinate location. It is a coordinate and conformal-gauge construction, not a freely imposed physical boundary condition. For a smooth conformal completion, one coordinates the lapse and shift with a prescribed conformal factor, then uses the remaining conformal freedom to control the expansion of $\mathscr I^+$ and the limits of formally singular source terms.

Required background. The Riemann Curvature Tensor supplies the conformal curvature identities behind the boundary limits.

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

Fixing the zero set of the conformal factor

Tildes denote physical fields, and

$$g_{\mu\nu}=\Omega^2\widetilde g_{\mu\nu}$$

is the regular unphysical metric. Take $\Omega\gt 0$ in the physical interior and prescribe it as a time-independent coordinate function, with $\mathscr I^+=\{\Omega=0,d\Omega\ne0\}$. Decompose the unphysical evolution vector as

$$t^\mu=\alpha n^\mu+\beta^\mu,$$

where $n^\mu$ is the future unit normal and $\beta^\mu$ is tangent to the hyperboloidal slice. Scri fixing requires

$$t(\Omega)\big|_{\mathscr I^+} =\alpha n(\Omega)+\beta^iD_i\Omega=0.$$

Let $s^i=D^i\Omega/|D\Omega|$ point toward increasing $\Omega$, hence into the physical interior. At future null infinity, $(\nabla\Omega)^2=0$ and future orientation gives

$$n(\Omega)=-|D\Omega|.$$

Writing $\beta^s=\beta^is_i$, the fixed-boundary condition becomes

$$\beta^s=\alpha \qquad\text{at }\mathscr I^+.$$

In general $t^\mu=\alpha(n^\mu+s^\mu)+\beta^\mu_{\parallel}$ there, where $\beta_{\parallel}$ is tangent to the two-dimensional cut. Scri fixing makes $t$ tangent to the boundary; it is tangent to the null generators themselves only after the tangential shift is set to zero. In that boundary-normal gauge, Andersson uses $N=1$ and $X=\eta$ in Andersson 2002, § 5.1, equations (13c)–(14), pp. 190–191. Reversing the spatial normal without reversing the shift component changes the fixed-boundary condition.

Characteristic directions at scri

The two radial null directions are $n+s$ and $n-s$. Relative to the time vector, their coordinate speeds along $s$ are

$$c_+=-\beta^s+\alpha, \qquad c_-=-\beta^s-\alpha.$$

At scri fixing, $c_+=0$: the generator $n+s$ remains on the boundary. The other speed is $c_-=-2\alpha$, directed toward decreasing $\Omega$, out of the physical domain $\Omega\ge0$. This causal observation explains why $\mathscr I^+$ is an outflow boundary for physical radiation.

It does not prescribe all boundary data. Gauge and constraint subsystems can have their own characteristic fields, and their treatment depends on the chosen hyperbolic reduction. A condition such as “no incoming radiation” must be formulated in physical characteristic variables; it is not synonymous with fixing $\Omega=0$.

Preferred conformal gauge

Scri fixing chooses the location of the zero set. A separate freedom remains:

$$g_{\mu\nu}\mapsto g'_{\mu\nu}=\omega^2g_{\mu\nu}, \qquad \Omega\mapsto\Omega'=\omega\Omega, \qquad \omega\gt 0.$$

Vacuum Einstein equations imply at a smooth $\mathscr I^+$ that the tracefree Hessian of $\Omega$ vanishes. The expansion, proportional to $\Box\Omega$, is gauge dependent. Along each generator one may solve

$$\nabla^\mu\Omega\nabla_\mu\ln\omega =-\frac14\Box\Omega \qquad\text{on }\mathscr I^+$$

to obtain the preferred conformal gauge

$$\Box'\Omega'\big|_{\mathscr I^+}=0, \qquad \nabla'_\mu\nabla'_\nu\Omega'\big|_{\mathscr I^+}=0.$$

The transformation and generator ODE are derived in Zenginoğlu 2008, § 3.1, equations (9)–(11), pp. 5–6. In a generalized harmonic reduction, use $\Omega$ itself as a coordinate near scri. Since its gauge source satisfies $F^\Omega=-\Box\Omega$, imposing $F^\Omega|_{\mathscr I^+}=0$ realizes the preferred gauge, as stated in Zenginoğlu 2008, § 3.2, Proposition 2, pp. 6–7.

These conclusions assume that the physical solution admits a smooth conformal compactification. They do not manufacture smooth scri from incompatible or polyhomogeneous data. Even in the preferred gauge, evaluating finite limits of separately singular terms can remain analytically and numerically delicate.

Minkowski characteristic check

For the compactified CMC hyperboloids of the preceding page,

$$ds^2=-\Omega^2d\tau^2+2a\,d\tau\,d\Omega +\frac{a^2}{1-\Omega^2}d\Omega^2 +a^2(1-\Omega^2)d\omega_2^2.$$

Here $\partial_\tau\Omega=0$. The $3+1$ fields are $\alpha=1$ and $\beta^\Omega=(1-\Omega^2)/a$. At $\Omega=0$, the shift has unit magnitude toward increasing $\Omega$, exactly matching the lapse. Radial null curves obey, at the boundary,

$$2a\frac{d\Omega}{d\tau} +a^2\left(\frac{d\Omega}{d\tau}\right)^2=0,$$

so their speeds are $0$ and $-2/a$. One direction is tangent and the other leaves $\Omega\ge0$; none enters from beyond scri.

Exercises

Starting from $t(\Omega)=0$ and $(\nabla\Omega)^2=0$, derive $\beta^s=\alpha$ for future null infinity with $s^i$ oriented toward increasing $\Omega$. What changes if $s^i$ is reversed?

Solution

The null condition decomposes as $-[n(\Omega)]^2+|D\Omega|^2=0$. Future orientation gives $n(\Omega)=-|D\Omega|$. Therefore

$$0=\alpha n(\Omega)+\beta^s|D\Omega| =(-\alpha+\beta^s)|D\Omega|,$$

and $d\Omega\ne0$ implies $\beta^s=\alpha$. With $s'=-s$, the component is $\beta^{s'}=-\alpha$; the geometric shift vector is unchanged.

References

• Andersson, Lars. “Construction of Hyperboloidal Initial Data.” In The Conformal Structure of Spacetimes: Geometry, Analysis, Numerics, edited by Jörg Frauendiener and Helmut Friedrich, Lecture Notes in Physics 604, 183–194. Springer, 2002. arXiv:gr-qc/0205083.
• Zenginoğlu, Anıl. “Hyperboloidal Evolution with the Einstein Equations.” Classical and Quantum Gravity 25 (2008): 195025. DOI. Open PDF.