Singularities in general relativity

1970 ◽  
Vol 48 (8) ◽  
pp. 970-980 ◽  
Author(s):  
J. Pachner
Keyword(s):  
High Pressure ◽  
General Relativity ◽  
Angular Velocity ◽  
Mass Density ◽  
Rest Mass ◽  
First Integrals ◽  
Einstein Equations ◽  
Rotating Fluid ◽  
Relativistic Limit ◽  
Local Observer

The problem of singularities is examined from the standpoint of a local observer. A singularity is defined as a state with an infinite proper rest mass density. It is proved that any inhomogeneity and anisotropy in the distribution and motion of a nonrotating ideal fluid accelerates collapse. Collapse is also inevitable in a rotating fluid in the case of extremely high pressure when the relativistic limit of the equation of state must be applied. In order to investigate the influence of rotation on the existence of singularities in incoherent matter the Einstein equations together with their first integrals are written out for the points on a vortex filament. They show that rotation decelerates the contraction of space not only in the direction perpendicular to the vector of the angular velocity, but indirectly also along this vector and can prevent the occurrence of a singularity. This conclusion is confirmed by the numerical integration of the Einstein equations. The paper concludes with a discussion of some cosmological implications.

Rotating fluid masses in general relativity. II

1966 ◽  
Vol 62 (3) ◽  
pp. 495-501 ◽  
Author(s):  
R. H. Boyer
Keyword(s):  
General Relativity ◽  
Variational Principle ◽  
First Integrals ◽  
Constant Angular Velocity ◽  
Hydrodynamic Equations ◽  
Rotating Fluid ◽  
Axially Symmetric ◽  
Field Equations ◽  
Full Field ◽  
Specific Entropy

AbstractWe describe some properties of a stationary, isolated, axially symmetric, rotating body of perfect fluid, according to general relativity. We first specialize to the case of constant specific entropy and constant angular velocity. The latter condition is equivalent to rigidity in the Born sense; both conditions are consequences of a simple variational principle. The hydrodynamic equations can then be integrated completely. Analogous first integrals are given also for the case of differential rotation. No use is made of the full field equations.

scholarly journals Variation in Magnetic Field Strength During a Specific Anisotropic Gravitational Collapse

1976 ◽  
Vol 29 (5) ◽  
pp. 461 ◽  
Author(s):  
DP Mason
Keyword(s):  
Magnetic Field ◽  
General Relativity ◽  
Energy Density ◽  
Field Strength ◽  
Gravitational Collapse ◽  
Mass Density ◽  
Rest Mass ◽  
Physical Significance ◽  
The Magnetic Field ◽  
Made In

The MHD approximation has been made in general relativity to derive expressions in terms of the fluid's total proper energy density and rest-mass density for the variation in the strength of the magnetic field during the anisotropic gravitational collapse in which the condition ?ab HaHb = 0 holds throughout the collapse, where ?ab is the expansion tensor. The physical significance of this condition is also examined.

scholarly journals Geometry and general relativity in the groupoid model with a finite structure group

2015 ◽  
Vol 93 (1) ◽  
pp. 43-54 ◽  
Author(s):  
Michael Heller ◽  
Tomasz Miller ◽  
Leszek Pysiak ◽  
Wiesław Sasin
Keyword(s):  
General Relativity ◽  
Quantum Mechanics ◽  
Lorentz Group ◽  
Space Time ◽  
Einstein Equations ◽  
Diagonal Form ◽  
Frame Bundle ◽  
Relativistic Limit ◽  
Theor Phys ◽  
A Finite Group

In a series of papers (M. Heller et al. J. Math. Phys. 38, 5840 (1997). doi:10.1063/1.532186 ; M. Heller and W. Sasin. Int. J. Theor. Phys. 38, 1619 (1999). doi:10.1023/A:1026617913754 ; M. Heller et al. Int. J. Theor. Phys. 44, 619 (2005). doi:10.1007/s10773-005-3992-7 ) we proposed a model unifying general relativity and quantum mechanics. The idea was to deduce both general relativity and quantum mechanics from a noncommutative algebra, [Formula: see text], defined on a transformation groupoid Γ determined by the action of the Lorentz group on the frame bundle (E, πM, M) over space–time M. In the present work, we construct a simplified version of the gravitational sector of this model in which the Lorentz group is replaced by a finite group, G, and the frame bundle is trivial E = M × G. The model is fully computable. We define the Einstein–Hilbert action, with the help of which we derive the generalized vacuum Einstein equations. When the equations are projected to space–time (giving the “general relativistic limit”), the extra terms that appear due to our generalization can be interpreted as “matter terms”, as in Kaluza–Klein-type models. To illustrate this effect we further simplify the metric matrix to a block diagonal form, compute for it the generalized Einstein equations and find two of their “Friedman-like” solutions for the special case when G = [Formula: see text]. One of them gives the flat Minkowski space–time (which, however, is not static), another, a hyperbolic, linearly expanding universe.

scholarly journals CAN f(R) GRAVITY MIMIC GENERAL RELATIVITY?

2012 ◽  
Vol 10 ◽  
pp. 63-68 ◽  
Author(s):  
JE-AN GU
Keyword(s):  
General Relativity ◽  
Differential Equations ◽  
Early Time ◽  
Einstein Equations ◽  
Modified Theories Of Gravity ◽  
Cosmological Perturbations ◽  
Long Run ◽  
Theories Of Gravity ◽  
The Stability ◽  
Higher Order Differential Equations

We discuss the stability of the general-relativity (GR) limit in modified theories of gravity, particularly the f(R) theory. The problem of approximating the higher-order differential equations in modified gravity with the Einstein equations (2nd-order differential equations) in GR is elaborated. We demonstrate this problem with a heuristic example involving a simple ordinary differential equation. With this example we further present the iteration method that may serve as a better approximation for solving the equation, meanwhile providing a criterion for assessing the validity of the approximation. We then discuss our previous numerical analyses of the early-time evolution of the cosmological perturbations in f(R) gravity, following the similar ideas demonstrated by the heuristic example. The results of the analyses indicated the possible instability of the GR limit that might make the GR approximation inaccurate in describing the evolution of the cosmological perturbations in the long run.

scholarly journals Lagrangian formulation of Newtonian cosmology

2014 ◽  
Vol 36 (3) ◽  
Author(s):  
H.S. Vieira ◽  
V.B. Bezerra
Keyword(s):  
General Relativity ◽  
Differential Equations ◽  
Classical Mechanics ◽  
Lagrangian Formulation ◽  
Einstein Equations ◽  
Lagrangian Formalism ◽  
Newtonian Cosmology

In this paper, we use the Lagrangian formalism of classical mechanics and some assumptions to obtain cosmological differential equations analogous to Friedmann and Einstein equations, obtained from the theory of general relativity. This method can be used to a universe constituted of incoherent matter, that is, the cosmologic substratum is comprised of dust.

An interior solution in general relativity

1968 ◽  
Vol 306 (1484) ◽  
pp. 1-3 ◽  
Keyword(s):  
General Relativity ◽  
Mass Density ◽  
Gravitational Mass ◽  
Interior Solution

Formulae are given for the field of a sphere of constant gravitational mass density.

scholarly journals On the Dependence of the Relativistic Angular Momentum of a Uniform Ball on the Radius and Angular Velocity of Rotation

2020 ◽  
Vol 15 ◽  
pp. 9-14
Author(s):  
Sergey G. Fedosin
Keyword(s):  
Angular Momentum ◽  
Neutron Star ◽  
Angular Velocity ◽  
Mass Density ◽  
Nonlinear Function ◽  
Kerr Black Hole ◽  
Special Theory ◽  
Spherical Body ◽  
Theory Of Relativity ◽  
Speed Of Light

In the framework of the special theory of relativity, elementary formulas are used to derive the formula for determining the relativistic angular momentum of a rotating ideal uniform ball. The moment of inertia of such a ball turns out to be a nonlinear function of the angular velocity of rotation. Application of this formula to the neutron star PSR J1614-2230 shows that due to relativistic corrections the angular momentum of the star increases tenfold as compared to the nonrelativistic formula. For the proton and neutron star PSR J1748-2446ad the velocities of their surface’s motion are calculated, which reach the values of the order of 30% and 19% of the speed of light, respectively. Using the formula for the relativistic angular momentum of a uniform ball, it is easy to obtain the formula for the angular momentum of a thin spherical shell depending on its thickness, radius, mass density, and angular velocity of rotation. As a result, considering a spherical body consisting of a set of such shells it becomes possible to accurately determine its angular momentum as the sum of the angular momenta of all the body’s shells. Two expressions are provided for the maximum possible angular momentum of the ball based on the rotation of the ball’s surface at the speed of light and based on the condition of integrity of the gravitationally bound body at the balance of the gravitational and centripetal forces. Comparison with the results of the general theory of relativity shows the difference in angular momentum of the order of 25% for an extremal Kerr black hole.

scholarly journals Fine Structure Constant derived from Principle Theory.

2021 ◽  
Author(s):  
Manfred Geilhaupt
Keyword(s):  
General Relativity ◽  
Fine Structure ◽  
General Theory ◽  
Rest Mass ◽  
Structure Constant ◽  
Fine Structure Constant ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
The Standard Model ◽  
Principle Theory

Abstract Derivation of mass (m), charge (e) and fine structure constant (FSC) from theory are unsolved problems in physics up to now. Neither the Standard Model (SM) nor the General theory of Relativity (GR) has provided a complete explanation for mass, charge and FSC. The question “of what is rest mass” is therefore still essentially unanswered. We will show that the combination of two Principle Theories, General Relativity and Thermodynamics (TD), is able to derive the restmass of an electron (m) which surprisingly depends on the (Sommerfeld) FSC (same for the charge (e)).

Sign in / Sign up

Export Citation Format

Share Document