In general relativity, both non-gravitational matter energy as well as gravitational field energy are sources of the field. In this note we propose that a collapsing body turns into a black hole when contributions of matter energy and field energy become equal. For measure of field energy we use the Brown–York quasilocal energy while for the gravitational charge (measure of gravitational pull matter energy produces), the Gauss–Komar integral. This is an energetics definition, similar to the classical escape velocity argument, which is intuitively and physically appealing. One of the remarkable results that immediately follows is that an extremal hole can never be formed by collapse of a dispersed state as there is no net energy to drive it. This is however well-known from other considerations, but here it follows from simple energetics of collapse.

The measure of energy in general relativity (GR) is inherently ambiguous owing to contribution of gravitational field energy. The remarkable feature of GR is that its contribution is automatically taken care of by the curvature of space and hence remains generally hidden in the usual considerations. For instance, while deriving the Schwarzschild solution we ultimately solve the Laplace equation instead of the Poisson equation with the field energy density on the right. It turns out that the effect of field energy is taken care of by the curvature of the space part of the metric leaving the Laplace equation unaffected².

2-boundary embedded in the curved 3-hypersurface of a black hole spacetime and when embedded in (reference) asymptotically flat space. The quasilocal energy is the integral of this mean curvature difference over the 2-boundary. It brings out explicitly contribution of field energy and yields a very simple relation for energy enclosed by a black hole horizon, E(r₊) = r₊.

Let an energy distribution have total energy E(¥ ) = M in absolute dispersed state and as it collapses on its own under gravity it picks up gravitational field energy. The Brown–York quasilocal energy represents the sum of the field energy gained during collapse and the energy at infinity. This is the energy, E(r), contained inside the radius r.

We can hence define field energy as given by

E_F(r) = E(r) – M. (1)

where E(r) is given by³

(2)

Here s is the determinant of the 2-metric of the boundary B and K is the mean extrinsic curvature of B isometrically embedded in the curved 3-hypersurface C and 0 refers to B’s curvature when embedded in the asymptotic flat space as the reference. E is minus variation in action in a unit change in proper time separation between B and its neighbouring 2-surface. It is therefore the value of the Hamiltonian that generates unit time translations orthogonal to C at the boundary B. This is obviously the natural prescription for measure of energy. It is a covariant expression modulo reference spacetime and it also possesses additivity property.

Gravitational charge is defined by the Gauss integral for the red-shifted proper acceleration over the 2-surface^1,4–6 and it is given by

(3)

where g = –NÑ (lnN), and N is the lapse function. g is the red-shifted proper acceleration experienced by a free particle relative to infinity.

For a charged hole, eq. (2) yields

E(r) = r – (r² – 2Mr + Q²)^1/2, (4)

while eq. (3) yields the gravitational charge as^1,5

M_c(r) = M – Q²/r, (5)

which gives M_c(r₊) =(G = c = 1).

We propose that a black hole is defined by equality of gravitational charge and field energy which means

E_F = M_c. (6)

In view of eqs (1), (4), (5) and (6) we obtain the characterizing expression for a charged black hole as

(r² – 2Mr + Q²)(2Mr – Q²) = 0. (7)

This clearly defines the horizon, r₊ = M + . Thus the general characterizing relation for black hole is,

E(r) – E(¥ ) = M_c(r). (8)

This holds good for all coordinates, and in particular we have also verified it for isotropic coordinates. Though we cannot evaluate for a rotating black hole
E and M_c at any arbitrary r, they could be done at the horizon. There they do obey the above relation as E(r₊) – M = M_c(r₊) = The above characterizing relation is true in general.

In this case matter energy (coming through energy–momentum tensor) contained inside the radius r would be M – Q²/2r because the electrostatic energy Q²/2r is exterior to r. The equipartition of E into matter and nonmatter would mean

E(r) – (M – Q²/2r) = M – Q²/2r, (9)

which would also lead to the same relation eq. (7). That is alternatively a black hole is characterized by equipartition of the Brown–York quasilocal energy into matter and non-matter energy.

For the extremal hole (M² = Q²), it can be easily seen from eq. (4) that E = M everywhere and hence E_F vanishes indicating the absence of a driving force for collapse. That means, extremal black hole cannot be formed by collapse from a dispersed state. One has to do work to effect such an assembly because the net force (attractive due to M and repulsive due to Q²) vanishes. This is yet another way of getting at the third law of black hole dynamics that extremal hole cannot be created from a non-extremal one by a finite sequence of physical processes⁷. The remarkable thing here is that it is the basic energetics that prohibits it.

In the spirit of Christodoulou’s irreducible mass⁸, we can define irreducible energy as

(10)

which will lead to the equivalent relation

(11)

where E_ir = 2M_ir. Let us try to interpret the above relation in analogy with the special relativistic conservation law. Clearly M represents the total energy, J/E_ir the momentum and (E_ir + Q²/E_ir)/2 the rest energy. The electric field contributes to increasing rest energy while rotation, as expected, also contributes to kinetic energy of the hole. E_ir will be bounded as or The fraction of energy available for extraction would be (E_ir(Sch) – E_ir(extremal))/E_ir(Sch). It is 29% for the rotating and 50% for the charged black hole.

It is thus an interesting and insightful application of the Brown–York quasilocal energy in understanding the role of field energy in characterizing black hole. The fraction that can be put in field energy, E(r) – M £  (M² – a² – Q²)^1/2, has the value M for the Schwarzschild and zero for the extremal hole. The equality defines the horizon. Thus the field energy is bounded above by the total energy M. Further gravitational charge is conserved for the Schwarzschild hole while the Brown–York energy for extremal hole. The two indicate the two limits of collapse. Since collapse is driven by the field energy, which in contrast to the Schwarzschild case, vanishes for an extremal hole and hence it cannot be formed of gravitational collapse. Like the photon, extremal hole has to be born as such. Our definition (eq. (1)) for a black hole can in fact be established rigorously for stationary spacetimes by using the Gauss–Codacci relations⁹.

Finally, this is a physically and intuitively appealing characterization of black hole based on balance between gravitational charge and field energy. An excellent example of this kind of consideration is the escape velocity argument, which though gives the right result follows from an invalid relation. It only refers to escape of particles from the gravitational field and not to one-way character of a closed 2-surface, which is the characteristic feature of the horizon. The horizon marks irresistability of gravitational pull through gravitational charge, as well as one-way character, no escape out of closed surface, through space curvature caused by the field energy. These two features must define the same surface, as the photon is the limiting case of an ordinary particle.

Received 19 February 1998; revised accepted 21 December 1998