3.3 Energy flux due to
gravitational waves
Let us interpret the various terms appearing in the area balance
law (25).
The left side of this equation provides us with
the change in the horizon radius caused by the dynamical process
under consideration. Since the expansion
vanishes, this is
also the change in the Hawking mass as one moves from the cross section
to
. The first integral on the right side
of this equation is the flux
of matter
energy associated with the vector field
. The second term
is purely geometrical and accompanies the term representing the
matter energy flux. Hence it is interpreted as the flux
of
-energy
carried by the gravitational radiation:
A priori, it is surprising that there should
exist a meaningful expression for the gravitational energy flux in
the strong field regime where gravitational waves can no longer be
envisaged as ripples on a flat space-time. Therefore, it is
important to subject this interpretation to viability criteria
analogous to the ‘standard’ tests one uses to demonstrate the
viability of the Bondi flux formula at null infinity. It is known
that it passes most of these tests. However, to our knowledge, the
status is still partially open on one of these criteria. The
situation can be summarized as follows:
- Gauge
invariance
-
Since one did not have to introduce any structure, such as
coordinates or tetrads, which is auxiliary to the problem, the
expression is obviously gauge invariant. This is to be contrasted
with definitions involving pseudo-tensors or background
fields.
- Positivity
-
The energy flux (26) is manifestly
non-negative. In the case of the Bondi flux, positivity played a
key role in the early development of the gravitational radiation
theory. It was perhaps the most convincing evidence that
gravitational waves are not coordinate artifacts but carry physical energy. It is quite surprising that
a simple, manifestly non-negative expression can exist in the
strong field regime of DHs. One can of course apply our general
strategy to any space-like 3-surface
, foliated by
2-spheres. However, if
is not a DH, the sign of the
geometric terms in the integral over
can not be
controlled, not even when
lies in the black hole
region and is foliated by trapped (rather than marginally trapped)
surfaces
. Thus, the positivity of
is a rather
subtle property, not shared by 3-surfaces which are foliated by
non-trapped surfaces, nor those which are foliated by trapped
surfaces; one needs a foliation precisely by
marginally trapped surfaces. The property is delicately
matched to the definition of DHs [31
].
- Locality
-
All fields used in Equation (26) are defined by the
local geometrical structures on
cross-sections of
. This is a non-trivial property, shared
also by the Bondi-flux formula. However, it is not shared in other
contexts. For example, the proof of the positive energy theorem by
Witten [188] provides a positive definite energy
density on Cauchy surfaces. But since it is obtained by solving an
elliptic equation with appropriate boundary conditions at infinity,
this energy density is a highly non-local function of geometry.
Locality of
enables one to associate it with the
energy of gravitational waves instantaneously falling across any
cross section
.
- Vanishing in
spherical symmetry
-
The fourth criterion is that the flux should vanish in presence of
spherical symmetry. Suppose
is spherically symmetric.
Then one can show that each cross-section of
must be spherically symmetric. Now, since the only
spherically symmetric vector field and trace-free, second rank
tensor field on a 2-sphere are the zero fields,
and
.
- Balance
law
-
The Bondi-Sachs energy flux also has the important property that
there is a locally defined notion of
the Bondi energy
associated with any 2-sphere
cross-section
of future null infinity, and the
difference
equals the Bondi-Sachs flux
through the portion of null infinity bounded by
and
. Does the expression (26) share this property?
The answer is in the affirmative: As noted in the beginning of this
section, the integrated flux is precisely the difference between
the locally defined Hawking mass
associated with the cross-section. In Section 5 we
will extend these considerations to include angular momentum.
Taken together, the properties discussed above
provide a strong support in favor of the interpretation of
Equation (26) as the
-energy flux carried by gravitational waves into the
portion
of the DH. Nonetheless, it is important to continue
to think of new criteria and make sure that Equation (26) passes these tests.
For instance, in physically reasonable, stationary, vacuum
solutions to Einstein’s equations, one would expect that the flux
should vanish. However, on DHs the area must increase. Thus, one is
led to conjecture that these space-times do not admit DHs. While
special cases of this conjecture have been proved, a general proof
is still lacking. Situation is similar for non-spherical DHs in
spherically symmetric space-times.
We will conclude this section with two
remarks:
- The presence of the shear term
in the integrand of the flux formula (26) seems natural from
one’s expectations based on perturbation theory at the event
horizon of the Kerr family [108, 66]. But the term
is new and can arise only because
is space-like rather than null: On a null surface,
the analogous term vanishes identically. To bring out this point,
one can consider a more general case and allow the cross-sections
to lie on a horizon which is partially null and
partially space-like. Then, using a 2 + 2 formulation [117
] one can show that
flux on the null portion is given entirely by the term
[28]. However, on the space-like portion,
the term
does not vanish in general. Indeed, on
a DH, it cannot vanish in presence of
rotation: The angular momentum is given by the integral of
, where
is the rotational
symmetry.
- The flux refers to a specific vector field
and measures the change in the Hawking mass
associated with the cross-sections. However, this is not a good
measure of the mass in presence of angular momentum (see, e.g.,
[34
] for numerical
simulations). Generalization of the balance law to include angular
momentum is discussed in Section 4.2.