Narrow Results

In this paper, we establish the asymptotic behavior along outgoing and incoming radial geodesics, i.e. the peeling property for the tensorial Fackerell–Ipser and spin $\pm 1$ Teukolsky equations on Schwarzschild spacetime. Our method combines a conformal compactification with vector field techniques to prove the two-side estimates of the energies of tensorial fields through the future and past null infinity ${\mathcal{ℐ}}^{\pm }$ and the initial Cauchy hypersurface ${\mathrm{\Sigma }}_0=\{t=0\}$ in a neighborhood of spacelike infinity $i_0$ far away from the horizon and future timelike infinity. Our results obtain the optimal initial data which guarantees the peeling at all orders.

The theoretical basis for the energy carried away by gravitational waves that an isolated gravitating system emits was first formulated by Hermann Bondi during the ’60s. Recent findings from the observation of distant supernovae revealed that the rate of expansion of our universe is accelerating, which may be well explained by sticking a positive cosmological constant into the Einstein field equations for general relativity. By solving the Newman–Penrose equations (which are equivalent to the Einstein field equations), we generalize this notion of Bondi mass–energy and thereby provide a firm theoretical description of how an isolated gravitating system loses energy as it radiates gravitational waves, in a universe that expands at an accelerated rate. This is in line with the observational front of LIGO’s first announcement in February 2016 that gravitational waves from the merger of a binary black hole system have been detected.

The cosmological constant $\mathrm{\Lambda }$ used to be a freedom in Einstein’s theory of general relativity (GR), where one had a proclivity to set it to zero purely for convenience. The signs of $\mathrm{\Lambda }$ or $\mathrm{\Lambda }$ being zero would describe universes with different properties. For instance, the conformal structure of spacetime directly depends on $\mathrm{\Lambda }$: null infinity ${ℐ}$ is a spacelike, null, or timelike hypersurface, if $\mathrm{\Lambda }>0$, $\mathrm{\Lambda }=0$, or $\mathrm{\Lambda }<0$, respectively. Recent observations of distant supernovae have taught us that our universe expands at an accelerated rate, and this can be accounted for by choosing $\mathrm{\Lambda }>0$ in Einstein’s theory of GR. A quantity that depends on the conformal structure of spacetime, especially on the nature of ${ℐ}$, is the Bondi mass which in turn dictates the mass loss of an isolated gravitating system due to energy carried away by gravitational waves. This problem of extending the Bondi mass to a universe with $\mathrm{\Lambda }>0$ has spawned intense research activity over the past several years. Some aspects include a closer inspection on the conformal properties, working with linearization, attempts using a Hamiltonian formulation based on “linearized” asymptotic symmetries, as well as obtaining the general asymptotic solutions of de Sitter-like spacetimes. We consolidate on the progress thus far from the various approaches that have been undertaken, as well as discuss the current open problems and possible directions in this area.

We give a brief review of the definition of the Wang–Yau quasi-local mass and discuss the evaluation of this quantity on surfaces of unit size at null infinity of an axisymmetric spacetime in Bondi–van der Burg–Metzner coordinates.

We address the hyperboloidal initial value problem in the context of Numerical Relativity, motivated by its evolution on hyperboloidal slices: smooth spacelike slices that reach future null infinity, the “location” in spacetime where radiation is to be extracted. Our approach uses the BSSN and Z4 formulations and a time-independent conformal factor. The resulting system of PDEs includes formally diverging terms at null infinity. Here we discuss a regularized numerical scheme in spherical symmetry. A critical ingredient are the gauge conditions, which control the treatment of future null infinity. Stable numerical evolutions have been performed with regular and black hole initial data on a hyperboloidal slice. A sufficiently large scalar field perturbation will create a black hole, whose final stationary state is different from the trumpet initial data derived here.

The theoretical basis for the energy carried away by gravitational waves that an isolated gravitating system emits was first formulated by Hermann Bondi during the ’60s. Recent findings from the observation of distant supernovae revealed that the rate of expansion of our universe is accelerating, which may be well explained by sticking a positive cosmological constant into the Einstein field equations for general relativity. By solving the Newman–Penrose equations (which are equivalent to the Einstein field equations), we generalize this notion of Bondi mass–energy and thereby provide a firm theoretical description of how an isolated gravitating system loses energy as it radiates gravitational waves, in a universe that expands at an accelerated rate. This is in line with the observational front of LIGO’s first announcement in February 2016 that gravitational waves from the merger of a binary black hole system have been detected.