Fluid Ball Research Articles

Regge calculus and observations. II. Further applications

The method, developed in an earlier paper, for tracing geodesies of particles and light rays through Regge calculus space-times, is applied to a number of problems in the Schwarzschild geometry. It is possible to obtain accurate predictions of light bending by taking sufficiently small Regge blocks. Calculations of perihelion precession, Thomas precession, and the distortion of a ball of fluid moving on a geodesic can also show good agreement with the analytic solution. However difficulties arise in obtaining accurate predictions for general orbits in these space-times. Applications to other problems in general relativity are discussed briefly.

Cosmology: Matters of moderate gravity

Cosmology: Matters of moderate gravity

Axially symmetric gravitational two-body problem of Cooperstock, Lim, and Hobill

Cooperstock, Lim, and Hobill have constructed and analyzed a model spacetime describing the head-on free-fall from rest of two fluid balls under their mutual gravitational attraction. These authors find a gravitational radiation energy flux having a different dependence on the parameters of the problem than would be expected on the basis of the quadrupole formula. We show that this dependence is due to their having incorrectly solved Einstein's gravitational field equations. The correct solution is consistent with the quadrupole formula.

The unique spherically symmetric solution of the U 4-theory of gravity in the teleparallelism limit

Within the U 4-theory of gravity in the teleparallelism limit we determine the metric and torsion tensor of a spherically symmetric fluid ball. We uniquely derive a Schwarzschild solution thereby establishing the validity of the Birkhoff theorem in this framework.

A global analysis approach to the general relativistic fluid ball problem

That a self-gravitating perfect fluid in empty space has a spherical equilibrium configuration if it is static-i.e., nonrotating-is considered physically evident, but has not yet been rigorously derived from Einstein's field equations together with suitable asymptotic conditions. In this paper the global analysis techniques developed recently mainly by Fischer, Marsden, and Cantor are used to derive the result that if a family of static perfect fluid solutions with fixed total gravitational massm and fixed equation of stateϱ(p) satisfying 0 ⩽p ⩽ ϱ and 0 ⩽dϱ/dp < ∞ depends differentiably on a parameter and contains the spherically symmetric solution then it must consist of solutions diffeomorphic to the spherically symmetric one.

On the occurrence of naked singularities in general relativity. II

It is shown that naked “shell crossing” singularities can occur in the gravitational collapse of a spherically symmetric ball of perfect fluid for a large family of equations of state in which the pressure has an (arbitrarily large) upper bound, and, moreover, that this behaviour is stable with respect to spherically symmetric perturbations of the initial data, as well as with respect to perturbations of the equation of state.

A heuristic derivation of the Einstein equations

relate the curvature of the space-time manifold M* to the distribution of matter. In these equations R~j is the Ricci tensor of M 4, R=g~JRij=R~ is the scalar curvature, T~i is the stress-energy-momentum tensor, T=T~ ~ is the trace of Tij, x is the gravitational constant, and we have chosen units so that c, the speed of light, is 1. There have been a number of attempts at heuristic derivations of these equations, starting of course with EINSTEIN. An especially simple and geometric one is given in WHEELER'S paper [6]. WHEELER'S derivation is admittedly not completely satisfactory, except perhaps in the case of empty space (T~j= 0) in which case the equations reduce to R~=0. WHEELER has expressed the desire for a heuristic derivation that will lead to the full equations, including the important terms 89 R or 89 If such a derivation is given, then the divergence condition T~j = 0 is automatically satisfied, and these equations can then be interpreted as giving the equations of motion. In EINSTEIN'S original derivation he did not obtain the full equations; he later added the term 89 in order to achieve the divergence condition. In this note I propose to give a heuristic derivation of the full equations in the simplest physically realistic situation possible, namely a small ball of fluid sitting in an otherwise empty universe. In this derivation, no use will be made of the famous "geodesic hypothesis". There is much contact here with the 60 years of existing literature of general relativity; the starting point will be a paper of LEVI-CIVITA written in the year 1917. My object has been to follow the most direct route from Newtonian gravitation and special relativity to general relativity. Of course I will be guided by many things that came to light only after EINSTEIN discovered his remarkable equations.

General-relativistic interior metric for a static charged fluid ball with pressure

General-relativistic interior metric for a static charged fluid ball with pressure

Hydrostatic Equilibrium and Gravitational Collapse of Relativistic Charged Fluid Balls

We obtain the metric corresponding to an arbitrary spherically symmetric distribution of charged perfect fluid without restricting consideration to the static case, and we show that it joins onto the standard Reissner-Nordstr\om exterior solution of Einstein's equations. We also generalize the Oppenheimer-Volkoff equations of hydrostatic equilibrium to the charged case, and we discuss their applicability. Using our formalism, we then rederive a formula of Christodoulou which splits the mass of a charged black hole into an irreducible part and a reducible electrical part. Finally, we derive the equations for the gravitational collapse of a charged fluid ball, and we draw some general conclusions from them.

Gravitational instability and collapse of charged fluid shells

A solution of Einstein’s field equations is derived, representing a thin spherical shell of charged fluid, falling in a spherically symmetric field due to mass and charge at its centre. No restrictions are placed on the equation of state. By integrating the equations of motion, the law of conservation of total energy is obtained, and used to study the equilibrium states of the system, and their stability against collapse. It is found, under reasonable assumptions, that for a given equation of state and a given entropy, there is a maximum equilibrium mass, and that instability sets in at a critical radius which is always larger (very much larger, for a shell of highly relativistic fluid) than the upper Nordstrom gravitational radius. For uncharged bodies, these results completely parallel, and serve as a simple illustration of, the results of the much more complicated analyses needed for fluid balls.

The role of abdominal pressure in relieving the pressure on the lumbar intervertebral discs.

1. Since the publication by Bradford and Spurling in 1945 of The Intervertebral Disc, there has been argument about the figure of 1,600 pounds that they calculated as the load on each lower lumbar intervertebral disc when a person lifts a heavy load with the trunk flexed, especially since experiments have shown that intervertebral discs subjected to increasing pressures yield at values well below this figure. In the author's experiments the discs were destroyed by pressures ranging from 350 to 1,400 pounds, with a mean of 710 pounds. 2. It occurred to the writer that the spine is not necessarily the only structure in the body that can transmit pressure forces from the shoulder to the pelvis. A raised intra-abdominal pressure impacts a thrust under the diaphragm, which will be transmitted to the thoracic spine and the shoulders by means of the ribs. This thrust can take care of part of the lifted weight and thus decrease the load on the spine. 3. In experiments in which the intra-abdominal pressure was measured by means of a small balloon in the stomach it was found that the pressure rose proportionally with the amount of weight lifted. 4. It is suggested that the abdominal fluid ball can exert a longitudinal force only if there is no contraction of the longitudinal muscles (at least anteriorly). Electromyographic studies of the abdominal muscles during weight lifting showed that the transverse and possibly the oblique abdominal muscles contract, but not the recti. 5. It thus seems that the load on the intervertebral discs is not necessarily so great as Bradford and Spurling calculated, but can remain within safe limits. It is hard to give accurate figures for the amount of load that is taken off the spine in this way, but an estimate would put it at several hundred pounds. The importance of a reflex contraction of the abdominal wall during effort as a protective mechanism for the spine must therefore be appreciated. Voluntary contraction may also be called upon to increase the intra-abdominal pressure and so reduce the load on the discs. This is done by many weight lifters.

On tidal energy in Newtonian two-body motion

In this work, which is based on an essential linear analysis carried out by Christodoulou, we study the evolution of tidal energy for the motion of two gravitating incompressible fluid balls with free boundaries obeying the Euler-Poisson equations. The orbital energy is defined as the mechanical energy of the two bodies' center of mass. According to the classical analysis of Kepler and Newton, when the fluids are replaced by point masses, the conic curve describing the trajectories of the masses is a hyperbola when the orbital energy is positive and an ellipse when the orbital energy is negative. The orbital energy is conserved in the case of point masses. If the point masses are initially very far, then the orbital energy is positive, corresponding to hyperbolic motion. However, in the motion of fluid bodies the orbital energy is no longer conserved because part of the conserved energy is used in deforming the boundaries of the bodies. In this case the total energy $\tilde{\mathcal{E}}$ can be decomposed into a sum $\tilde{\mathcal{E}}:=\widetilde{\mathcal{E}_{\mathrm{orbital}}}+\widetilde{\mathcal{E}_{\mathrm{tidal}}}$, with $\widetilde{\mathcal{E}_{\mathrm{tidal}}}$ measuring the energy used in deforming the boundaries, such that if $\widetilde{\mathcal{E}_{\mathrm{orbital}}} 0$, then the orbit of the bodies must be bounded. In this work we prove that under appropriate conditions on the initial configuration of the system, the fluid boundaries and velocity remain regular up to the point of the first closest approach in the orbit, and that the tidal energy $\widetilde{\mathcal{E}_{\mathrm{tidal}}}$ can be made arbitrarily large relative to the total energy $\tilde{\mathcal{E}}$. In particular under these conditions $\widetilde{\mathcal{E}_{\mathrm{orbital}}}$, which is initially positive, becomes negative before the point of the first closest approach.