Areal Radius and Spherical Mass Functions

Spherical symmetry permits a geometric mass without stationarity: the Misner–Sharp mass is defined from the areal radius and its gradient. In four-dimensional Einstein gravity with $G$ explicit,

$$m=\frac{r}{2G}\left[1-(\nabla r)^2\right].$$

This scalar measures the departure of symmetry spheres from their flat-space embedding relation and evolves according to local energy density and flux once Einstein’s equation is imposed.

Required background. The Einstein Equation supplies the mass-gradient identity; Killing Fields and Isometry Groups supplies the spherical quotient and Kodama direction.

Helpful background. The Riemann Curvature Tensor supports the warped-product derivation; Formulations of General Relativity supplies complementary Hamiltonian and characteristic mass descriptions.

Areal radius and geometric mass

On a regular spherical region,

$$ds^2=h_{ab}dx^a dx^b+r^2d\Omega_2^2,$$

where $4\pi r^2$ is the invariant orbit area. Define $q=(\nabla r)^2_h$. The mass identity is equivalently

$$q=1-\frac{2Gm}{r}.$$

No Killing field appears, so $m$ is meaningful in collapsing, expanding, or radiating spherical geometries. The normalization is chosen so that Schwarzschild has $m=M$, a physical mass, when its metric function is $1-2GM/r$.

The definition has the right dimensions: $r/G$ is a mass in units with $c=1$. It is invariant under quotient coordinate changes and depends only on the geometry of symmetry spheres. It is not a general quasi-local mass for arbitrary nonspherical surfaces; spherical symmetry removes reference and embedding ambiguities that broader constructions must confront.

If a cosmological constant is treated geometrically rather than as stress–energy, it is often useful to subtract its homogeneous contribution:

$$M_\Lambda =\frac{r}{2G} \left(1-q-\frac{\Lambda r^2}{3}\right).$$

For Schwarzschild–(anti-)de Sitter, $M_\Lambda=M$ is constant, whereas the unsubtracted $m=M+\Lambda r^3/(6G)$. The convention must be stated before comparing “mass functions.”

Deriving the mass-gradient law

Let $T^{(2)}=h^{ab}T_{ab}$ be the trace over the orbit space. Combining the reduced $ab$ Einstein equations with the derivative of $m$ gives

$$\nabla_a m =4\pi r^2 \left(T_a{}^b-T^{(2)}\delta_a{}^b\right)\nabla_b r,$$

when $\Lambda$ is included in $T_{\mu\nu}$ if present. This identity is sometimes written as a work-plus-flux law. Define

$$w=-\frac12T^{(2)}, \qquad \psi_a=T_a{}^b\nabla_b r+w\nabla_a r,$$

and $A=4\pi r^2$, $V=4\pi r^3/3$. Then

$$\nabla_a m=A\psi_a+w\nabla_aV.$$

This is not a thermodynamic assumption; it is a rearrangement of the spherical Einstein equations. Hayward gives the invariant formulation and its trapping-horizon applications in Hayward 1996, §§II–III, pp. 1939–1943.

The identity also gives conditional monotonicity. Along an outward spacelike direction in an untrapped region, suitable energy conditions can make the projected right-hand side nonnegative. That conclusion requires a specified direction and stress tensor; it is not an algebraic property of $m$ alone. Along a null direction, the change records the corresponding energy flux and work. This is why a varying mass is compatible with exact spherical geometry but incompatible with vacuum.

For a static perfect fluid in areal coordinates, $T^a{}_b=\operatorname{diag}(-\rho,p)$ on $(t,r)$ and $\nabla_ar$ points radially. Hence

$$\frac{dm}{dr}=4\pi r^2\rho,$$

the familiar enclosed-mass relation. Pressure enters other reduced equations and the work term, but cancels from this radial derivative.

Sphere expansions and the mass threshold

For null normals $\ell\cdot n=-1$,

$$\theta_{(\ell)}\theta_{(n)} =-\frac{2}{r^2}\left(1-\frac{2Gm}{r}\right).$$

Thus $2Gm/r>1$ corresponds to spheres with expansions of the same sign; $2Gm/r=1$ makes the areal-radius gradient null. This is a local spherical statement. It does not by itself identify an event horizon, whose definition is global, nor choose between future trapped and past anti-trapped spheres.

Independent evaluations

For Schwarzschild,

$$q=g^{rr}=1-\frac{2GM}{r}$$

and the definition returns $m=M$. For spatially flat FLRW,

$$ds^2=-dt^2+a^2(t)(d\chi^2+\chi^2d\Omega_2^2), \qquad r=a\chi,$$

one finds $q=1-H^2r^2$. Therefore

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

Using the Friedmann equation $H^2=8\pi G\rho/3$ gives $m=4\pi r^3\rho/3$, exactly the matter energy inside the homogeneous sphere. This check shows both nonstationary applicability and dependence on Einstein’s equation for the matter interpretation.

At a constant-radius branch, $dr=0$ and $q=0$, so the definition gives $m=r/(2G)$ but cannot use $r$ as a coordinate or infer a Schwarzschild form. Such product branches must be handled separately.

Charged and homogeneous checks

For asymptotically flat Reissner–Nordström in Gaussian charge normalization, the definition yields

$$m=M-\frac{Q^2}{2r}.$$

The difference $Q^2/(2r)$ is the Maxwell energy outside the sphere. As the sphere expands toward infinity, the enclosed mass approaches the total parameter $M$. This independent interpretation fixes the sign of the charge term.

For FLRW with nonzero spatial curvature, the spatial-curvature contribution to $(\nabla r)^2$ combines with the full Friedmann constraint to give the same matter result: density times areal volume. Thus the mass does not depend on whether homogeneous slices were called closed, flat, or open.

If $\Lambda$ is moved between the geometric and matter sides of Einstein’s equation, both the definition named “matter mass” and its gradient law must be translated consistently. Subtracting $\Lambda$ in the definition while retaining it in the stress tensor would double-count vacuum energy.

Check your understanding

For Schwarzschild–de Sitter with $f(r)=1-2GM/r-\Lambda r^2/3$, compute $m$ and $M_\Lambda$.

Solution

Since $q=f(r)$,

$$m=\frac{r}{2G}(1-f) =M+\frac{\Lambda r^3}{6G}.$$

The unsubtracted mass includes the vacuum-energy contribution inside the sphere. Subtracting it gives

$$M_\Lambda=\frac{r}{2G} \left(1-f-\frac{\Lambda r^2}{3}\right)=M.$$

Neither convention is intrinsically wrong; a comparison that fails to state which one is used is ambiguous.

References

• Hayward, Sean A. “Gravitational Energy in Spherical Symmetry.” Physical Review D 53 (1996), 1938–1949. doi:10.1103/PhysRevD.53.1938.
• Misner, Charles W., and David H. Sharp. “Relativistic Equations for Adiabatic, Spherically Symmetric Gravitational Collapse.” Physical Review 136 (1964), B571–B576. doi:10.1103/PhysRev.136.B571.