A method of reduction of Einstein's equations of evolution and a natural symplectic structure on the space of gravitational degrees of freedom

1996 ◽  
Vol 28 (2) ◽  
pp. 207-219 ◽  
Author(s):  
Arthur E. Fischer ◽  
Vincent Moncrief
Keyword(s):  
Degrees Of Freedom ◽  
Symplectic Structure ◽  
Einstein’S Equations ◽  
Einstein's Equations

scholarly journals Solution-generating methods of Einstein’s equations by Hamiltonian reduction

2018 ◽  
Vol 33 (18) ◽  
pp. 1850101
Author(s):  
Seung Hun Oh ◽  
Kyoungtae Kimm ◽  
Yongmin Cho ◽  
Jong Hyuk Yoon
Keyword(s):  
Exact Solutions ◽  
Degrees Of Freedom ◽  
Hamiltonian Reduction ◽  
Einstein’S Equations ◽  
Schwarzschild Solution ◽  
Einstein's Equations ◽  
Constraint Equations ◽  
Dimensional Spacetime ◽  
Free Data ◽  
Privileged Coordinates

The purpose of this paper is to demonstrate a new method of generating exact solutions to Einstein’s equations obtained by the Hamiltonian reduction. The key element to the successful Hamiltonian reduction is finding the privileged spacetime coordinates in which physical degrees of freedom manifestly reside in the conformal two-metric, and all the other metric components are determined by the conformal two-metric. In the privileged coordinates, Einstein’s constraint equations become trivial; the Hamiltonian and momentum constraints are simply the defining equations of a nonvanishing gravitational Hamiltonian and momentum densities in terms of conformal two-metric and its conjugate momentum, respectively. Thus, given any conformal two-metric, which is a constraint-free data, one can construct the whole four-dimensional spacetime by integrating the first-order superpotential equations. As the first examples of using Hamiltonian reduction in solving Einstein’s equations, we found two exact solutions to Einstein’s equations in the privileged coordinates. Suitable coordinate transformations from the privileged to the standard coordinates show that they are just the Einstein–Rosen wave and the Schwarzschild solution. The local gravitational Hamiltonian and momentum densities of these spacetimes are also presented in the privileged coordinates.

scholarly journals GRAVITY AS ELASTICITY OF SPACETIME: A PARADIGM TO UNDERSTAND HORIZON THERMODYNAMICS AND COSMOLOGICAL CONSTANT

2004 ◽  
Vol 13 (10) ◽  
pp. 2293-2298 ◽  
Author(s):  
T. PADMANABHAN
Keyword(s):  
Cosmological Constant ◽  
Degrees Of Freedom ◽  
Vacuum Energy ◽  
Consistency Condition ◽  
Coordinate Transformations ◽  
Einstein’S Equations ◽  
Einstein's Equations ◽  
Long Wavelength ◽  
Horizon Thermodynamics ◽  
Pure Surface

It is very likely that the quantum description of spacetime is quite different from what we perceive at large scales, l≫(Gℏ/c3)1/2. The long wavelength description of spacetime, based on Einstein's equations, is similar to the description of a continuum solid made of a large number of microscopic degrees of freedom. This paradigm provides a novel interpretation of coordinate transformations as deformations of "spacetime solid" and allows one to obtain Einstein's equations as a consistency condition in the long wavelength limit. The entropy contributed by the microscopic degrees of freedom reduces to a pure surface contribution when Einstein's equations are satisfied. The horizons arises as "defects" in the "spacetime solid" (in the sense of well-defined singular points) and contributes an entropy which is one quarter of the horizon area. Finally, the response of the microstructure to vacuum energy leads to a near cancellation of the cosmological constant, leaving behind a tiny fluctuation which matches with the observed value.

scholarly journals What drives the time evolution of the spacetime geometry?

2014 ◽  
Vol 23 (12) ◽  
pp. 1441003 ◽  
Author(s):  
T. Padmanabhan
Keyword(s):  
Time Evolution ◽  
Degrees Of Freedom ◽  
Rate Of Change ◽  
Einstein’S Equations ◽  
Einstein's Equations ◽  
Spacetime Geometry ◽  
The Difference ◽  
Holographic Description ◽  
Gravitational Momentum ◽  
General Dynamic

I show that in a general, dynamic spacetime, the rate of change of gravitational momentum is related to the difference between the number of bulk and boundary degrees of freedom. All static spacetimes maintain holographic equipartition; i.e. in these spacetimes, the number of degrees of freedom in the boundary is equal to the number of degrees of freedom in the bulk. It is the departure from holographic equipartition that drives the time evolution of the spacetime. This result, which is equivalent to Einstein's equations, provides an elegant, holographic, description of spacetime dynamics.

COVARIANT CONFORMAL DECOMPOSITION OF EINSTEIN EQUATIONS

2002 ◽  
Vol 17 (20) ◽  
pp. 2762-2762
Author(s):  
E. GOURGOULHON ◽  
J. NOVAK
Keyword(s):  
Numerical Simulations ◽  
Extrinsic Curvature ◽  
Einstein Equations ◽  
Numerical Implementation ◽  
Einstein’S Equations ◽  
Einstein's Equations ◽  
Tensor Density ◽  
Dirac Gauge ◽  
Ill Posed ◽  
Isolated Systems

It has been shown1,2 that the usual 3+1 form of Einstein's equations may be ill-posed. This result has been previously observed in numerical simulations3,4. We present a 3+1 type formalism inspired by these works to decompose Einstein's equations. This decomposition is motivated by the aim of stable numerical implementation and resolution of the equations. We introduce the conformal 3-"metric" (scaled by the determinant of the usual 3-metric) which is a tensor density of weight -2/3. The Einstein equations are then derived in terms of this "metric", of the conformal extrinsic curvature and in terms of the associated derivative. We also introduce a flat 3-metric (the asymptotic metric for isolated systems) and the associated derivative. Finally, the generalized Dirac gauge (introduced by Smarr and York5) is used in this formalism and some examples of formulation of Einstein's equations are shown.

scholarly journals LOCAL COSMIC STRING AND C-FIELD

2006 ◽  
Vol 21 (18) ◽  
pp. 3727-3732 ◽  
Author(s):  
F. RAHAMAN ◽  
R. MONDAL ◽  
M. KALAM
Keyword(s):  
Cosmic String ◽  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
Einstein’S Equations ◽  
Einstein's Equations

We investigate a local cosmic string with a phenomenological energy–momentum tensor as prescribed by Vilenkin, in the presence of C-field. The solutions of full nonlinear Einstein's equations for exterior and interior regions of such a string are presented.

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