Hypersurface In Spacetime Research Articles

Quantum Mechanical Probabilities on a General Spacetime-Surface

A theoretical aspect of quantum mechanical probabilities is studied. In nonrelativistic quantum mechanics (NRQM) for a particle, Schrodinger's wave function assigns probabilities to possible places where the particle is found on a surface (hypersurface in Newtonian spacetime) of constant time. We may then ask if quantum mechanical probabilities can bed defined for an exhaustive set of exclusive outcomes of observations made on a surface on which Newtonian time is not constant. Such a question may be said to belong to the study of the role of time in NRQM in a broad sense

Analysis of spatially inhomogeneous perturbations of the FRW cosmologies.

We use Bardeen's gauge-invariant formalism to analyze the behavior of, and relationship between, various geometric and physical quantities of cosmological interest at the linear level. This leads to a cosmologically oriented gauge-invariant characterization of the different perturbation modes that can arise. In particular a link is made between the existence of gravitational-wave modes and the conformal curvature of hypersurfaces in spacetime. We indicate how these results can be useful in the analysis of exact solutions of the Einstein field equations.

Junction conditions for null hypersurfaces

The authors derive an expression for the singular part of the stress-energy tensor on a hypersurface in spacetime in terms of the discontinuities in fundamental forms associated with the surface for both the non-null case, where the results are standard, and the null case. They then derive the minimum conditions which must be satisfied in order to glue two spacetimes together along such a hypersurface. In both cases, the essential requirement is only that the naturally induced (possibly degenerate) 3-metrics on the hypersurface must agree.

Null surface geometrodynamics

Investigates the dynamical structure of Einstein's theory of gravity using a time parameter whose level surfaces are null hypersurfaces in spacetime. A spacelike foliation of codimension two is used in order to give an invariant kinematic description. Dynamics are analysed via a constrained Hamiltonian formalism. It is found that the Hamiltonian formulation includes both first and second class constraints. The first class constraints are associated with the invariance of the theory with respect to diffeomorphisms of the null hypersurfaces. The second class constraints are direct consequences of the choice of null time parameter.

Colliding gravitational waves in expanding cosmologies

We discuss the singularity behavior and causal structure of expanding vacuum plane-symmetric cosmologies containing colliding gravitational waves. Penrose's theorem excluding global Cauchy hypersurfaces in plane-wave spacetimes does not hold for our expanding cosmologies. The singularity in our solutions occurs in the past and can be identified with the big-bang singularity. Analytical estimates based on a Green's-function analysis near the collision provide an explicit example which is verified by numerical computation.

Finite systems of observables in quantum field theory

The quantum description of field degrees of freedom on a space-like hypersurface in space-time is given. A method of construction of vacuum is discussed and tested in the free case.

Ondes de choc et hypothèses de compressibilité en magnétohydrodynamique relativiste

The principal purpose of this paper is to show that the normals to the hypersurfaces in space-time which represent the shock waves of relativistic magnetohydrodynamics are spacelike when the caloric equation of state of the medium satisfies the conditions of compressibility previously given in references [5] and [6]. Thus, the velocities of these waves are less than the local velocity of light and the theory is consistent with the postulates of a relativistic theory. A theorem previously given in reference [5] concerning the stability of Alfven shock waves, is corrected in this paper.

Hamiltonian formalism in einsteinian gravitation: Geometrical aspect

In the Hamiltonian formulation of relativistic theory of gravitation the gravitational field at a given time is defined on a spacelike hypersurface of Riemannian space-time by six gravitational potentials and as many conjugate momenta. The former characterize the metric of the hypersurface and it is shown that the momenta are constructed tensorially only with the two fundamental tensors of the hypersurface since Dirac's velocities are equated with the components of the second fundamental tensor. Invariants of the two fundamental tensors, geometrically interpretable, can then be introduced into the Hamiltonian. It is at once clear from the Gauss and Codazzi equations, relating the two fundamental tensors of a hypersurface that the four secondary constraints, which the conjugate dynamical variables satisfy, are equivalent to four Einstein gravitational equations.

Characteristic Hypersurfaces in General Relativity. I

Lightlike (null) hypersurfaces are treated by means of an intrinsic Ricci rotation coefficient technique. This provides an effective way of dealing with the various types of geometry on a null hypersurface. The formalism is used to examine inner affinities, differential invariants, local features such as asymptotic and shear directions and geodesic lines, and to give a short description of null hypersurfaces in flat space-time. Applications to gravitational radiation theory and cosmology are briefly mentioned.

Spacelike foliations by $(n - 1)$-umbilical hypersurfaces in spacetimes

Spacelike foliations by $(n - 1)$-umbilical hypersurfaces in spacetimes

The mass of spacelike hypersurfaces in asymptotically anti-de Sitter space-times

We give a Hamiltonian definition of mass for spacelike hypersurfaces in space-times with metrics which are asymptotic to the anti-de Sitter one, or to a class of generalizations thereof. We show that our definition provides a geometric invariant for a spacelike hypersurface embedded in a space-time.