Spherical Symmetry as a Two-Dimensional System

Exact spherical symmetry removes angular dependence but does not remove gravity. A four-dimensional metric becomes a Lorentzian metric on a two-dimensional orbit space together with one scalar—the areal radius—and invariant matter fields. The resulting system has constraints and evolution equations, yet no local vacuum gravitational-wave polarization; its nontrivial content lies in the radius, mass aspect, matter flux, and boundary data.

Required background. Use Killing fields and isometry groups to recognize the rotation action and the Einstein equation for the parent dynamics.

Helpful background. Hypersurface normals and tangential projectors provides the projection language used for orbit-space stress-energy and flux.

The warped-product geometry

Assume a four-dimensional spacetime admits an effective $SO(3)$ action whose generic orbits are spacelike two-spheres. Away from a symmetry center or other exceptional orbit, the metric has the invariant form

$$ds^2=g_{AB}(x)\,dx^A dx^B+r^2(x)\,d\Omega^2, \qquad d\Omega^2=d\theta^2+\sin^2\theta\,d\phi^2,$$

where $A,B\in\{0,1\}$, $g_{AB}$ has signature $(-,+)$, and $r$ is defined geometrically by the orbit area $4\pi r^2$. The scalar $r$ is therefore not a disposable coordinate. At a regular center, $r=0$ and elementary flatness supplies additional conditions; on a trapped surface, $r$ remains well defined even though its gradient changes causal character.

For this warped product the four-dimensional Ricci scalar is

$${}^{(4)}R =R[g]+\frac{2}{r^2} -\frac{4}{r}\Box r -\frac{2}{r^2}(\nabla r)^2,$$

where all derivatives on the right use $g_{AB}$. Substitution into the Einstein–Hilbert action and integration over the unit sphere give, up to a boundary term,

$$S_{\mathrm{grav}} =\frac{1}{4G}\int d^2x\,\sqrt{-g} \left[r^2R[g]+2(\nabla r)^2+2-2\Lambda r^2\right].$$

The two-dimensional metric by itself has no Einstein dynamics because its Einstein tensor vanishes identically. Its coupling to the dilaton-like field $r$ carries the original four-dimensional equations. Dropping $r$ would therefore destroy the reduction.

Reduced Einstein equations and matter data

Spherical invariance requires the stress tensor to split as

$$T_{\mu\nu}dx^\mu dx^\nu =T_{AB}(x)dx^A dx^B+p_T(x)r^2d\Omega^2.$$

The orbit-space components of the Einstein tensor are

$$G_{AB} =-\frac{2}{r}\nabla_A\nabla_Br +\frac{2}{r}g_{AB}\Box r +\frac{1}{r^2}g_{AB}\bigl((\nabla r)^2-1\bigr),$$

while the angular equation is equivalently

$$G^\theta{}_\theta =\frac{\Box r}{r}-\frac12R[g].$$

Together with $G_{\mu\nu}+\Lambda g_{\mu\nu}=8\pi GT_{\mu\nu}$, these equations determine $g_{AB}$ and $r$ subject to two-dimensional coordinate freedom. The angular pressure equation is not automatically redundant off shell; the contracted Bianchi identity relates it to the orbit equations only after matter conservation and appropriate nondegeneracy assumptions are used.

A particularly useful invariant is the spherical mass aspect. The original Misner–Sharp energy obeys $1-2Gm_{\mathrm{MS}}/r=(\nabla r)^2$; see Misner and Sharp 1964, pp. B572–B573. When $\Lambda$ is kept on the geometric side of Einstein’s equation, it is convenient to subtract its homogeneous vacuum energy and define $m=m_{\mathrm{MS}}-\Lambda r^3/(6G)$:

$$1-\frac{2Gm}{r}-\frac{\Lambda r^2}{3} =g^{AB}\nabla_A r\nabla_B r.$$

It packages the norm of the areal-radius gradient. In Minkowski spacetime, $(\nabla r)^2=1$ and $m=0$. In Schwarzschild–(anti-)de Sitter spacetime, $(\nabla r)^2=1-2GM/r-\Lambda r^2/3$, so $m=M$. This supplies an invariant check of the reduction without committing to Schwarzschild coordinates.

What symmetry removes—and what it retains

The two functions describing gravitational-wave polarizations are absent because a nonzero spin-2 tensor harmonic cannot be invariant under all rotations. Vacuum spherical gravity is correspondingly rigid: locally, a region with nonconstant $r$ is Schwarzschild–(anti-)de Sitter. This is a statement about vacuum dynamics, not a claim that every spherically symmetric spacetime is static. A scalar field, perfect fluid, null dust, or charged field supplies genuine time-dependent degrees of freedom through $T_{AB}$ and $p_T$.

The reduction fails if it is used as an approximation without controlling nonspherical modes. It also needs special treatment at $r=0$, where the orbit dimension changes; at shell crossings or distributional layers; and where a coordinate choice such as polar-areal gauge ceases to cover the geometry. Exact $SO(3)$ invariance is a statement about the fields, not merely the absence of angular coordinate symbols.

Exercise

Evaluate the Misner–Sharp mass for the spatially flat FLRW metric

$$ds^2=-dt^2+a^2(t)\bigl(d\chi^2+\chi^2d\Omega^2\bigr),$$

with $\Lambda=0$ and areal radius $r=a\chi$. Show that the Einstein equation gives the mass of a homogeneous ball.

Solution

On the orbit space, $\partial_t r=Hr$ and $\partial_\chi r=a$, so

$$(\nabla r)^2=-(Hr)^2+a^{-2}a^2=1-H^2r^2.$$

The defining relation therefore gives $2Gm/r=H^2r^2$, or

$$m=\frac{H^2r^3}{2G}.$$

For a spatially flat universe the Friedmann equation is $H^2=8\pi G\rho/3$. Hence

$$m=\frac{4\pi}{3}\rho r^3,$$

the energy contained in a ball of areal radius $r$. The calculation also shows why $m$ is dynamical when matter is present despite exact spherical symmetry.

References

• Misner, Charles W., and David H. Sharp. “Relativistic Equations for Adiabatic, Spherically Symmetric Gravitational Collapse.” Physical Review 136 (1964): B571–B576. DOI.