Global energy conservation in nonlinear spherical characteristic evolutions

Associated to the unique 4-parametric subgroup of translations, normal to the Bondi–Metzner–Sachs group, there exists a generator of the temporal translation asymptotic symmetry. Such a descriptor of the motion along the conformal orbit near null infinity is propagated to finite regions. This allows us to observe the global energy conservation even in extreme situations near the critical behavior of the massless scalar field collapse in spherical symmetry.

Figure 1

Sectors of the initial–boundary value problem for the EKG system. Data for an ingoing pulse is specified on the initial outgoing null hypersurface ${\mathcal{J}}^{-}$. A black hole may form (right) or not (left), but always the pulse is scattered to ${\mathcal{I}}^{+}$, partially or totally. The final black hole mass depends on the initial amplitude. Evolution slows down near the critical (unstable) solution or in the vicinity of the future event horizon ${\mathcal{H}}^{+}$.

Figure 2

Three-dimensional drawing in which $\mathrm{\Sigma }$ appears as a closed loop. $k^{\mu }$ represent the outgoing null vector. Dashed lines depict trajectories linking $\mathrm{\Sigma }$ and ${\mathrm{\Sigma }}^{+}$.

Figure 3

Sequence of $\mathrm{\Phi }$ as a function of $x$ up to the black hole formation: $u_B\approx 0.0$ (1); 2.4 (2); 5.7 (3); 12.4 (4); 17,7 (5).

Figure 4

Energy conservation as a function of the Bondi time, up to the black hole formation, as displayed in Fig. 3. The descending curve corresponds to the Bondi mass given by Eq. (12); the ascending curve corresponds to the energy flow to infinity calculated as $M_{\text{out}}=-\int Pd{u}_B$, using Eq. (20). The horizontal curve corresponds to the algebraic conserved sum of both curves.

Figure 5

Energy conservation as a function of the Bondi time. The evolution is subcritical and mostly nonlinear. The descending curve corresponds to the Bondi mass given by Eq. (12); the ascending curve corresponds to the energy flow to infinity calculated as $M_{\text{out}}=-\int Pd{u}_B$, using Eq. (20). The horizontal curve corresponds to the algebraic conserved sum of both curves.

Figure 6

Evolution of $e^{2H}$ as a function of the Bondi time for the subcritical case. It is apparent that the evolution goes from the nonlinear regime to the linear regime.

Figure 7

Energy conservation as a function of the Bondi time, in a subcritical and nonlinear evolution. The upper curve corresponds to energy given by Eq. (12), but with limits between $r_{\text{in}}=1/3$ and $r_{\text{out}}=1/2$; the lower curve corresponds to the energy flow calculated as $M_{\text{in}}=\int P(u_B,r_{\text{in}})d{u}_B$, $M_{\text{out}}=-\int P(u_B,r_{\text{out}})d{u}_B$, using Eq. (19). The horizontal curve corresponds to the algebraic conserved sum $M+M_{\text{out}}-M_{\text{in}}=\text{constant}$.