Quasilocal energy exchange and the null cone

Energy is at best defined quasilocally in general relativity. Quasilocal energy definitions depend on the conditions one imposes on the boundary Hamiltonian, i.e., how a finite region of spacetime is “isolated.” Here, we propose a method to define and investigate systems in terms of their matter plus gravitational energy content. We adopt a generic construction, that involves embedding of an arbitrary dimensional world sheet into an arbitrary dimensional spacetime, to a $2+2$ picture. In our case, the closed 2-dimensional spacelike surface $\mathbb{S}$, that is orthogonal to the 2-dimensional timelike world sheet $\mathbb{T}$ at every point, encloses the system in question. The integrability conditions of $\mathbb{T}$ and $\mathbb{S}$ correspond to three null tetrad gauge conditions once we transform our notation to the one of the null cone observables. We interpret the Raychaudhuri equation of $\mathbb{T}$ as a work-energy relation for systems that are not in equilibrium with their surroundings. We achieve this by identifying the quasilocal charge densities corresponding to rotational and nonrotational degrees of freedom, in addition to a relative work density associated with tidal fields. We define the corresponding quasilocal charges that appear in our work-energy relation and which can potentially be exchanged with the surroundings. These charges and our tetrad conditions are invariant under type-III Lorentz transformations, i.e., the boosting of the observers in the directions orthogonal to $\mathbb{S}$. We apply our construction to a radiating Vaidya spacetime, a $C$-metric and the interior of a Lanczos-van Stockum dust metric. The delicate issues related to the axially symmetric stationary spacetimes and possible extensions to the Kerr geometry are also discussed.

Figure 1

Our quasilocal charges give $E_{\mathrm{K}1}=E_{\mathrm{Dil}}=W_{\mathrm{Tid}}$ and $E_{\mathrm{Rot}}=0$ for each $u$ value. Charges are given in units of $M_0$ and the time parameter is in units of $M_0/(ac)$ where the speed of light, $c$, is 1 throughout the paper.

Figure 2

Quasilocal charges of the $C$-metric which is parametrized with $A=\frac{1}{\sqrt{28}M}$. Those quasilocal charges are meaningful only in the region $2M<r<\sqrt{28}M\approx 5.29M$ due to the coordinate singularities.

Figure 3

Radial behavior of the gravitational potential of the $C$-metric, which is parametrized with $A=\frac{1}{\sqrt{28}M}$, plotted for observers located at different polar angles. Those potentials are meaningful only in the region $2M<r<\sqrt{28}M\approx 5.29M$ due to the coordinate singularities.

Figure 4

Acceleration parameter dependence of $\mathrm{\Phi }(r,\mathrm{const})$. For each acceleration parameter $A^{*}$, we consider only the region $2M<r<1/A^{*}$.

Figure 5

Radial and tangential dependence of the components of the acceleration vector.

Figure 6

Radial behavior of $a_r$ for observers at different polar angles. We consider the acceleration vector only in the region $2M<r<\sqrt{28}M\approx 5.29M$.

Figure 7

Quasilocal charges of the van Stockum dust. Charges are in length units which can be written as a function of individual mass of the dust particles, $m$, and the total number density, $n$.