Narrow Results

We derive the Einstein equation from the condition that every small causal diamond is a variation of a flat empty diamond with the same free conformal energy, as would be expected for a near-equilibrium state. The attractiveness of gravity hinges on the negativity of the absolute temperature of these diamonds, a property we infer from the generalized entropy.

Finite difference and pseudo-spectral methods have been widely used in the numerical relativity to solve the Einstein equations. As the third major category method to solve partial differential equations, finite element method is less frequently used in numerical relativity. In this paper, we design a finite element algorithm to solve the evolution part of the Einstein equations. This paper is the second one of a systematic investigation of applying adaptive finite element method to the Einstein equations, especially aiming for binary compact objects simulations. The first paper of this series has been contributed to the constrained part of the Einstein equations for initial data. Since applying finite element method to the Einstein equations is a big project, we mainly propose the theoretical framework of a finite element algorithm together with local discontinuous Galerkin method for the Einstein equations in the current work. In addition, we have tested our algorithm based on the spherical symmetric spacetime evolution. In order to simplify our numerical tests, we have reduced the problem to a one-dimensional space problem by taking the advantage of the spherical symmetry. Our reduced equation system is a new formalism for spherical symmetric spacetime simulation. Based on our test results, we find that our finite element method can capture the shock formation which is introduced by numerical error. In contrast, such shock is smoothed out by numerical dissipation within the finite difference method. We suspect this is partly the reason that the accuracy of finite element method is higher than the finite difference method. At the same time, different kinds of formulation parameters setting are also discussed.

On a Riemannian or a semi-Riemannian manifold, the metric determines invariants like the Levi-Civita connection and the Riemann curvature. If the metric becomes degenerate (as in singular semi-Riemannian geometry), these constructions no longer work, because they are based on the inverse of the metric, and on related operations like the contraction between covariant indices. In this paper, we develop the geometry of singular semi-Riemannian manifolds. First, we introduce an invariant and canonical contraction between covariant indices, applicable even for degenerate metrics. This contraction applies to a special type of tensor fields, which are radical-annihilator in the contracted indices. Then, we use this contraction and the Koszul form to define the covariant derivative for radical-annihilator indices of covariant tensor fields, on a class of singular semi-Riemannian manifolds named radical-stationary. We use this covariant derivative to construct the Riemann curvature, and show that on a class of singular semi-Riemannian manifolds, named semi-regular, the Riemann curvature is smooth. We apply these results to construct a version of Einstein's tensor whose density of weight 2 remains smooth even in the presence of semi-regular singularities. We can thus write a densitized version of Einstein's equation, which is smooth, and which is equivalent to the standard Einstein equation if the metric is non-degenerate.

We classify all warped product space-times in three categories as (i) generalized twisted product structures, (ii) base conformal warped product structures and (iii) generalized static space-times and then we obtain the Einstein equations with the corresponding cosmological constant by which we can determine uniquely the warp functions in these warped product space-times.

In this paper, we discuss about a set of geometric and physical properties of hyper-generalized quasi-Einstein spacetime. In the beginning, we discuss about pseudosymmetry over a hyper-generalized quasi-Einstein spacetime. Here, we discuss about $W_2$-Ricci pseudosymmetry, $Z$-Ricci pseudosymmetry, Ricci pseudosymmetry and projective Ricci pseudosymmetry over a hyper-generalized quasi-Einstein spacetime. Later on, we take over Ricci symmetric hyper-generalized quasi-Einstein spacetime and derive a set of important geometric and physical theorems over it. Moving further we consider some physical applications of the hyper-generalized quasi-Einstein spacetime. Last, we prove the existence of a hyper-generalized quasi-Einstein spacetime by constructing a non-trivial example.

This chapter presents an overview of cosmological framework that is necessary to perform cosmological simulations. First, we start with a brief history of cosmological studies of the Universe, such as the discovery of Hubble’s law and cosmic microwave background radiation which constitute the major observational evidence of expanding Big Bang cosmology. Second, we present the basics of General Relativity theory and Friedmann models that describe the expanding universe. Under this theoretical framework, we introduce various cosmological parameters and current best-fit Λ cold dark matter (CDM) model. Third, we discuss the history and development of computational cosmology which was achieved concurrently with the evolution of supercomputers.

Cosmological N-body simulations play an important role in modern cosmology by providing vital information regarding the evolution of the dark matter: its clustering and motion, and properties of dark matter halos. The simulations are instrumental for the transition of the theoretical cosmology from an inspiring but speculative part of astronomy to the modern precision cosmology. In spite of more than 50 yrs of development, N-body methods are still a thriving field with the invention of more powerful methods providing more accurate theoretical predictions. Here, we review different numerical methods (PM, Tree, AMR) and ideas used in this field.

The cosmology has made enormous progress after the construction of General Relativity (GR) in 1915. The theoretical predictions of GR — such as the existence of black holes and gravitational waves — have been directly/indirectly confirmed by observations. Now, we know that GR is sufficiently dependable to describe the gravitational law in the solar system.

The following sections are included:

• Basics
  • Friedmann–Lemaître–Robertson–Walker Universe
  • Energy forms
  • Future
  • Density perturbation
• Thermal production
  • Number density history
  • Decoupling temperature
  • Interaction and decay rates
• Cosmological nucleosynthesis
• Baryon number in the Universe
• Thermal WIMP production in the Universe
  • Heavy neutrino
  • Forces for weakly interacting massive particles
  • WIMP relic density
• Nonthermal production
  • Non-thermal WIMP production
  • E-WIMP production
  • ADM production
  • Axion production
  • Heavy lepton as WIMP
• References