Energy flux due to gravitational waves

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 $Θ(ℓ)$ vanishes, this is also the change in the Hawking mass as one moves from the cross section $S1$ to $S2$ . The first integral on the right side of this equation is the flux $ℱ (mRa)tter$ of matter energy associated with the vector field $ξa(R)$ . The second term is purely geometrical and accompanies the term representing the matter energy flux. Hence it is interpreted as the flux $(R ) ℱgrav$ of $a ξ(R)$ -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 $¯ H$ , foliated by 2-spheres. However, if $¯ H$ is not a DH, the sign of the geometric terms in the integral over $Δ ¯H$ can not be controlled, not even when $¯H$ lies in the black hole region and is foliated by trapped (rather than marginally trapped) surfaces $¯S$ . Thus, the positivity of $(R) ℱ grav$ 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 [30 ].

Locality: All fields used in Equation (26 ) are defined by the local geometrical structures on cross-sections of $H$ . 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 $(R) ℱ grav$ enables one to associate it with the energy of gravitational waves instantaneously falling across any cross section $S$ .

Vanishing in spherical symmetry: The fourth criterion is that the flux should vanish in presence of spherical symmetry. Suppose $H$ is spherically symmetric. Then one can show that each cross-section of $S$ 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, $σab = 0$ and $ζa = 0$ .

Balance law: The Bondi–Sachs energy flux also has the important property that there is a locally defined notion of the Bondi energy $E (C)$ associated with any 2-sphere cross-section $C$ of future null infinity, and the difference $E (C1) − E (C2)$ equals the Bondi–Sachs flux through the portion of null infinity bounded by $C 2$ and $C 1$ . 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 $ξ (R)$ -energy flux carried by gravitational waves into the portion $ΔH$ 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 $|σ|2$ 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 , 64 ]. But the term $|ζ|2$ is new and can arise only because $H$ 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 $S$ to lie on a horizon which is partially null and partially space-like. Then, using a 2 + 2 formulation [116 ] one can show that flux on the null portion is given entirely by the term $2 |σ |$ [27 ]. However, on the space-like portion, the term $|ζ|2$ 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 $ζaϕa$ , where $ϕa$ is the rotational symmetry.
The flux refers to a specific vector field $a ξ(R)$ 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., [33 ] for numerical simulations). Generalization of the balance law to include angular momentum is discussed in Section 4.2.