12.3 Expressions in static spacetimes
Update
12.3.1 Tolman’s energy for static spacetimes
Let
be a hypersurface-orthogonal timelike Killing vector field,
a spacelike hypersurface to which
is orthogonal, and
. Then
, a field equation for
,
follows from Einstein equations (see, e.g., pp. 71 – 74 of [240
] or [199
]). Here
and
, the energy density and the average spatial pressure of the matter fields, respectively, seen
by the observer at rest with respect to
(or
).
In the study of (‘quasi-static’) equilibrium configurations of self-gravitating systems Tolman [520, 521]
found the integral
to be the energy of the system. Here
is a compact domain with smooth boundary
,
is
the outward pointing unit normal of
in
; and the second expression follows from the field equation
for
above.
can in fact be interpreted as some form of a quasi-local energy [2
, 3
], called the
Tolman energy. Clearly, for matter fields with non-negative energy density and average pressure and
non-positive cosmological constant this is non-negative on the domain where
is timelike. Using the
defining equation of
in terms of the two-surface integral, one can show that in asymptotically flat
spacetimes it tends to the ADM energy as a non-decreasing set function. The second expression in
Eq. (12.1) implies that, similarly to Komar’s expression, in vacuum
depends only on the homology
class of the 2-surface
. Thus, in particular, it associates zero energy with vacuum domains. For
spherically symmetric configurations on round spheres with the area radius
(see Section 4.2.1)
it is
, which in vacuum reduces to the Misner–Sharp
energy.
The Tolman energy appeared to be a useful tool in practice: By means of
Abreu and Visser gave
remarkable entropy bounds for localized, but uncollapsed bodies [2
, 3
]. (We discuss this bound in
Section 13.4.3.)
12.3.2 The Katz–Lynden-Bell–Israel energy for static spacetimes
Let
, the set of those points of
where the length of the Killing field is the value
,
i.e.,
are the equipotential surfaces in
. Let
be the set of those points where the
magnitude of
is not greater than
. Suppose that
is compact and connected. Katz,
Lynden-Bell, and Israel [309
] associate a quasi-local energy to the two-surfaces
as follows. Suppose
that the matter fields can be removed from
and concentrated into a thin shell on
in such a
way that the space inside is flat but the geometry outside remains the same. Then, denoting
the (necessarily distributional) energy-momentum tensor of the shell by
and assuming
that it satisfies the weak energy condition, the total energy of the shell,
, is
positive. Here
is the future-directed unit normal to
. Then, using the Einstein equations,
the energy of the shell can be rewritten in terms of geometric objects on the two-surface as
where
is the jump across the two-surface of the trace of the extrinsic curvatures of the two-surface
itself in
. Remarkably enough, the Katz–Lynden-Bell–Israel quasi-local energy
in the
form (12.2), associated with the equipotential surface
, is independent of any distributional matter
field, and can also be interpreted as follows. Let
be the metric on
,
the extrinsic curvature of
in
and
. Then, suppose that there is a flat metric
on
such that the induced metric from
on
coincides with that induced from
, and
matches continuously to
on
. (Thus, in particular, the induced area element
determined on
by
, and
coincide.) Let the extrinsic curvature of
in
be
, and
. Then
is the integral on
of
times
the difference
. Apart from the overall factor
, this is essentially the Brown–York
energy.
In asymptotically flat spacetimes
tends to the ADM energy [309]. However, it does not
reduce to the round-sphere energy in spherically-symmetric spacetimes [374], and, in particular, gives zero
for the event horizon of a Schwarzschild black hole.
12.3.3 Static spacetimes and post-Newtonian approximation
The Newtonian limit of general relativity is defined in [240
], pp. 71 – 74, via static and (at spatial infinity)
asymptotically flat spacetimes. The Newtonian scalar potential
is identified with the logarithm of the
length of the Killing vector, i.e., (in traditional units) it is
. Decomposing the energy density
as the sum of the rest-mass energy and the internal energy,
, Einstein’s equations yield
the field equation
Identifying the last term as
, the sum of the energy density and three times of the average
spatial stress of the field
, (see the definitions (3.2) in the Newtonian case), the exact field equation
(12.3) can be compared with the naïve, relativistically corrected Newtonian field equation (3.3); and one
can read off the various relativistic corrections [199
]. Then (3.4) motivates the definition of an ‘effective’
quasi-local energy as the integral of all the effective source terms on the right hand side of the exact field
equation (12.3):
If the spacetime is asymptotically flat at spatial infinity (in which case
) such that the hypersurface
extends to spatial infinity, then, using e.g. the result of [60], one can show that
tends to the ADM
energy as
is enlarged to exhaust the whole
[199
]. Since in the vacuum region the integrand of the
3-dimensional integral is negative definite, near infinity
tends to the ADM energy as a
monotonically decreasing set function. However, because of the extra relativistic correction term
in the source, the rate of change of this set function deviates from the one in the
naïve relativistically corrected Newtonian theory of Section 3.1.1. In fact, for a two-sphere of
radius
in the Schwarzschild spacetime with mass parameter
the quasi-local energy,
for large
, is
, rather than
(see
Section 3.1.1).
Though
is negative in the vacuum regime, for spherically symmetric configurations, when the
material source of the gravitational ‘field’ is contained in
, it is positive if an energy condition is
satisfied; and it is zero if and only if the domain of dependence of
in the spacetime is flat. (For the
details see [199].)