Hermann Bondi

The contraction of gravitating spheres

Earlier ideas associating an invariant integral of the energy invariant with the number of nucleons in a gravitating body are shown to be fallacious, and thus do not provide a means of following through the contraction of such a body. It is shown how the full field equations of general relativity give a feasible and rigorous method of examining contracting models. Schwarzschild-type co-ordinates are introduced and are used to examine the slow adiabatic contraction of a sphere of constant density. The particle paths are found and the pressure-density relation permitting such slow adiabatic contraction is examined. It is shown that the simple 4/3 power law of Newtonian theory has to be replaced by a steeper dependence of pressure on density for high gravitational potentials. Radiation co-ordinates are introduced to examine radiating contracting systems, and equations fully specifying such a system are obtained. A simple example is given in outline to illustrate the method.

Conservation and Non-Conservation in General Relativity

The difficulties of conservation laws in general relativity are discussed, with special reference to the non-tangible nature of gravitational energy and its transformation into tangible forms of energy. Inductive transfer of energy is marked out as wholly distinct from wave transfer. Slow (adiabatic) changes are utilized to make clear, in the axi-symmetric case, that the mass of an isolated body is conserved irrespective of any local changes (e.g. of shape) and that in inductive transfer the movement of energy between two bodies can readily be traced by the changes in their masses.

Massive spheres in general relativity

The exact relativistic form of the equation of hydrostatic support by an isotropic pressure is found in an especially convenient form. A quantity u, the natural generalization of Schwarzschild’s m/r ratio at the surface, is used, and it is proved that the critical value u = ½ cannot be attained except perhaps under conditions of severe tension. It is shown that if u is everywhere less than different physical restrictions are significant in well-defined inner and outer zones as far as the attainment of high u values is concerned. A number of limiting configurations are derived for physically significant restrictions. In particular it is shown that if the density is nowhere negative then u < 0.485 ..., while if in addition the density is nowhere less than three times the pressure and nowhere increasing outwards, then u < 0.319.

The rigid body dynamics of unidirectional spin

A toy consists of a boat-shaped body showing great preference for spin in one direction only. Its sophisticated rigid body dynamics is examined in some detail, and fully accounts for this curious behaviour.

On the Physical Consequences of a General Excess of Charge

The possibility of a general excess of charge in the universe is proposed. If such exists, even to the extent of only 2 parts in 1018, sufficiently powerful electric forces result to produce the observed expansion of the universe on the basis of Newtonian mechanics. If the excess occurs as a slight difference in magnitude of the proton and electron charges, the hypothesis may be on the verge of what could be established by experiment. If creation of matter, and also necessarily charge, is assumed, the Maxwell equations must be modified to avoid strict conservation. The appropriate modification is shown to involve additional terms in the current and charge-density equations proportional to the vector and scalar potentials. When applied to a spherically symmetrical smoothed-out universe, the revised equations establish almost rigorously that electrical requirements imply a strict velocitydistance law for the mass motion of expansion. For agreement with observation, the requisite charge on the proton would be (1 + y) e where y = 2 x 10-18, or, if the charges are strictly equal in magnitude, it requires 1 + y protons for every electron, with the same value of y. The value of the Hubble constant and of the smoothed-out density of matter in the universe are shown to be simply related by the theory to the rate of creation. The same solution is shown to hold equally in de Sitter space-time, and the principle of complete equivalence of all observers at all times is thereby demonstrated to be a property of the solution. Construction of the corresponding stress-energy tensor enables the factor associated with the new terms in the Maxwell equations to be directly related to the observable radius of the universe. On the first form of the charge-excess hypothesis, galaxies and clusters of galaxies (with their haloes) arise as ionized condensation units within the general background distribution. Since the units are conducting, they remain electrically neutral, and therefore grow and are controlled by ordinary gravitational forces. Moreover, because they are conducting, the units will expel their excess charge in the form of free protons. It is shown that the electrostatic potential at the surface of a unit is maintained by creation at such a value that the protons are expelled with energies corresponding to the highest energy cosmic rays. These units will take part in the general expansion, not under the direct action of the electric repulsion, but because they form and grow from the expanding background material. Any small departure in velocity of a unit from the local value would be quickly removed through the gravitational braking action associated with accretion of further material. The gravitational potential at the surface of a unit is such that infalling hydrogen atoms will have energy of motion corresponding to temperature of the order of a million degrees, and the outer parts of the units at least will be at high temperature.

On the dynamical theory of the rotation of the earth

In the dynamical theory of the motion of the Earth relative to its centre of mass, the planet is usually regarded as a rigid or at most only slightly deformable body, and moments of inertia are adopted that are taken to refer to the Earth as a whole, while the motion itself at any instant is assumed capable of representation by a single angular velocity vector. This procedure, however, appears to involve unwarranted assumptions the recognition and removal of which may lead to conclusions of considerable importance. For it is well known from the theory of earthquake waves that the material of the central core of the Earth behaves like a liquid in that it transmits only longitudinal wave vibrations, while there is also other evidence suggesting that the material of the core is a true liquid (1). There is accordingly no a priori reason for supposing that the core will behave like a rigid body firmly attached to the surrounding shell if more or less permanent shearing forces are applied to it. In particular, in respect of any couple known to act on the outer shell, it is not permissible to assume, without examination of the assumption, that its effect will be transferred immediately to the inner core in a way preserving rigid-body rotation of the whole. If the material of the core behaves like a liquid where wave-motion is concerned, this suggests that it will probably also behave like a liquid whatever shearing forces act on it, and the extent to which changes in the rotatory motion of the outer shell can be communicated to the core, and what effects direct gravitational forces acting on the core may have, must in the first instance be questions of hydrodynamics and not rigid dynamics.

On Datta's spherically symmetric systems in general relativity

The problem tackled by B. K. Datta, [1] in a recent paper concerning non-static spherically symmetric systems in which the particle motion is, in a certain sense, purely transverse, is further developed and compared with the Newtonian case. A full classification of the possible motions is given.

Gravitational Waves in General Relativity. XIII. Caustic Property of Plane Waves

We study the effect of a plane gravitational wave of limited duration (‘sandwich wave’) on an infinite set of test particles at relative rest. We prove, at least for waves of fixed polarization, that all the particles strung out in a certain direction will collide after a finite time that is independent of how far apart they were originally. This we call the caustic property. The effect of the wave on null geodesics is such that this phenomenon does not require any particle to move faster than light. Indeed, an observer who has passed through such a wave will within a finite time have seen an infinite spatial volume lying in a space-like half hyperplane on the other side of the wave. For a certain (‘bicaustic’) type of wave the whole of that half-hyperplane will have become visible.

The Mass of Cylindrical Systems in General Relativity

It is shown through the study of slowly changing cylindrical systems that there is no conserved mass per unit length for a relativistic infinite cylinder. This non-conservation is found to be a result of gravitational induction.

TRANSFER OF ENERGY IN GENERAL RELATIVITY.

This paper studies time sequences of axially symmetric static configurations which can be continuously deformed into each other. We find the restriction which characterizes those time sequences which are physically allowed. This restriction can be interpreted as the law governing the non-radiative, or near field, transfer of gravitational energy, -m = ∯ P. dS and helps to clarify the concept of mass-energy in general relativity. We consider only the regions of space surrounding a source where, for quasi-static systems, the exact, static, empty space solutions of Weyl and Levi-Civita are good approximations at all times, and exact whenever the motion stops. Our results are then valid for arbitrarily strong fields. The restriction on the time sequences can be expressed as a restriction on the time dependence of the multipole moments Al, Bl (which are defined by the Weyl and Levi Civita solutions), becoming then -d/dt[ A0 + ½G ∑l = 0∞ (2l + 1) AlBl] = ½ G ∑l = 0∞ (AlBl - AlBl). The right-hand side of this expression corresponds exactly to the flux of energy across a surface, as given by Bondi’s Newtonian Poynting vector P = (1/8 πG) (ϕ∇ϕ-ϕ∇ϕ). It is natural to identify A0 + ½G ∑l = 0∞ (2l + 1)AlBl with the total mass-energy m enclosed by the surface.m can also be expressed as a surface intergal. Our restriction was derived from the vanishing of a surface integral in much the same manner as the equations of motion are derived in post-Newtonain approximations. However, it holds in arbitrarily strong fields, whereas the usual post-Newtonian methods use in lowest order a trivial solution (flat space) and can only be used in weak fields.