Narrow Results

We present new procedures in REDUCE language using the EXCALC package (adapted for IBM-PC machines) for algebraic programming in the hamiltonian formulation of Einstein–Maxwell equations. The dynamic and the constraint equations containing specific terms of the interaction of gravity with source-free electromagnetic field are translated into computer procedures. The results obtained by processing some examples of space-time models are presented.

In this paper, to understand space–time dynamics in the canonical tensor model of quantum gravity for the positive cosmological constant case, we analytically and numerically study the phase profile of its exact wave function in a coordinate representation, instead of the momentum representation analyzed so far. A saddle point analysis shows that Lie group symmetric space–times are strongly favored due to abundance of continuously existing saddle points, giving an emergent fluid picture. The phase profile suggests that spatial sizes grow in “time,” where sizes are measured by the tensor-geometry correspondence previously introduced using tensor rank decomposition. Monte Carlo simulations are also performed for a few small $N$ cases by applying a re-weighting procedure to an oscillatory integral which expresses the wave function. The results agree well with the saddle point analysis, but the phase profile is subject to disturbances in a large space–time region, suggesting existence of light modes there and motivating future computations of primordial fluctuations from the perspective of canonical tensor model.

Strong coupling expansion is computed for the Einstein equations in vacuum in the Arnowitt–Deser–Misner (ADM) formalism. The series is given by the duality principle in perturbation theory as presented in M. Frasca, Phys. Rev. A58, 3439 (1998). An example of application is also given for a two-dimensional model of gravity expressed through the Liouville equation showing that the expansion is not trivial and consistent with the exact solution, in agreement with the general analysis. Application to the Einstein equations in vacuum in the ADM formalism shows that the space–time near singularities is driven by space homogeneous equations.

It is well known that moving puncture method and the specific gauge condition are critically important for successful simulation of binary black hole merging. On the contrary, the importance of formalism for numerical relativity is not very clear yet. Both generalized harmonic formalism and BSSN formalism work very well. So the simplicity of Einstein's equations in ADM formalism stimulates us to investigate a naive but interesting problem—can ADM formalism work as stably as BSSN formalism does with the moving puncture method and the advanced gauge condition, which were proved critical important for BSSN formalism's success. We apply this idea to Schwarzschild black hole simulation as a test example. Unfortunately, our result implies that ADM formalism has its intrinsic instable character which may be introduced by the property of the corresponding equations themselves instead of the gauge condition. And more, we find that the so called advanced gauge condition even make the situation worse. So we conclude that one gauge condition works well in one numerical formalism does not mean it works well also in other formalism. Through concrete examples, we give readers a primary sense on the roles that formalism, gauge condition, numerical method and other issues play in the problem of stability of numerical relativity.