Does Gravitational Collapse Lead to Singularities? by Ernst Fischer

Dear Gary,

I do not think that a density in the classical sense can be attributed to individual elementary particles. Of course, by defining a characteristic dimension by the Compton length l=h/mc you can define a density by m/(4/3 pi l^3), but this can only be regarded as an order of magnitude estimation for the density, at which degeneracy pressure can play a role in the stabilisation of objects against gravitational collapse. Only when the mean distance between individual fermionic particles is of the order of their Compton length, this pressure occurs, as no two fermionic particles can be in the same quantum state. You are right that in the supposed final state of the collapsed objects, which we denote as neutron stars, the density should be in the order of the value mentioned above (>10^14 g/cm^3). But even if with increasing mass of the collapsed object this density increases, as more and more energy states of the fermions must be occupied, there will never be a limit, where a balance between gravitational attraction and degeneracy pressure is impossible.

Best regards,

Ernst

Dear Alan,

You say that to your opinion supernova explosions can recycle matter into space. But as far as we know from observations, all the matter which is expelled in this case is processed matter, consisting of heavier elements. The temperature in the interior of the exploding stars is determined by nuclear fusion, which is not sufficient to produce new hydrogen or helium. On contrary for jets emanating from neutron stars the energy of the particles in the interior of the star is so high that individual atoms can no longer exist and only during ejection new atomic nuclei are formed, preferably hydrogen and helium. Only under these conditions a process is possible, which recycles matter of primordial composition into space. From the existence of this process we must conclude that the present element composition of the universe must not necessarily be the consequence of element synthesis in a hot initial Big Bang.

Best regards,

Ernst

Dear Edwin Eugene Klingman,

I agree that it is the definition of potential energy, which is essential for the conclusions presented in my essay. But this definition is based on the fact that quantities like particle number should be conserved in curved space and that the particles, which constitute the sources of gravitation, have an invariant property, their rest mass. Under this proposition in any theoretical model, based on the covariant geometrical description of gravity, a correction term of the form given in my essay must be included in the energy tensor to account for the deviation of volume from Euclidean space. That this term can be identified with the potential energy of Newtonian theory makes it easier to understand its role in the purely geometrical context of GRT.

You remark that in a spinning black hole matter has one degree of freedom. I think that you will say that in this case rotational energy gives an additional contribution to the energy tensor, so that the geometry is not fixed by the total mass alone. This is true, of course, also if the matter or energy distribution is corrected for the influence of curvature. Modeling of such spinning black holes would be of high interest, as it is the basis to understand the cosmic jets emanating from collapsed systems. There remains much to do, but at least with the correct model of the spherically symmetric case we have a sound basis to attack this problem.

Best regards,

Ernst Fischer

Dear Tom,

Thank you for the hint to my misunderstanding of Edwins question on one-dimensional black holes. But if his remark is related to the Bekenstein-Mayo problem of entropy flux through the space-time discontinuity at the supposed surface of a black hole, it is of no practical relevance, as such discontinuities do not exist in nature according to the arguments given in my essay.

Ernst

Dear Vladimir

Things are easier than you think. To understand gravity you should look at a swinging pendulum on earth. In Newtonian theory gravitation is described by motion of an object in a field, which is caused by the mass of the earth. During each swinging period the object takes up kinetic energy by changing its position in the gravity field. The gravity field itself has no energy. If there is no pendulum, there is no energy. It is only the fact that the swinging object changes its position relative to the mass of the earth that changes its energy state. That is, what we call potential energy.

The difference in GR is that the dependence of the state of an object on the position relative to other objects is encoded in a deviation of geometry from Euclidean space so that motion under the influence of gravitation can be regarded as geodesic motion in the distorted geometry. But with this introduction of a distorted geometry we must keep in mind that the relation between coordinates and the size of areas or volume elements is changed. Thus our accustomed definitions of densities and their relation to the integral properties of objects change.

It is only this correction of the energy balance due to the change of geometry, which I have discussed in my essay. There is no "energy of the gravitational field immediately outside the star" which carries potential energy. The gravitational field itself or some equivalent in GR does not carry any energy. It is only the state of matter within this field, which contributes to the energy balance.

Ernst

Dear Jim,

I think that the matter jets emanating from black holes result from internal instabilities of rotating black holes and occur when additional matter is accreted from the surrounding galaxy. I suppose that the jet activity is coupled to the accretion rate and will come to an end, when all surrounding matter is consumed. The final dead black hole can be destroyed only in collisions with other celestial bodies. Maybe that such collisions are the cause of the mysterious gamma-ray bursts.

Ernst

Dear Colin,

Thank you for the valuable hint to the work of Vladimir Majernik. I think that the basic concept of his paper agrees with mine. Mach's idea that the energy of matter at some point contains a contribution from its relation to all the other matter in the universe is just the justification of the introduction of potential energy into the energy tensor, in this case expressed by the change of the volume element by the curvature of space. Together with the equivalence of mass and energy this constitutes the basis of GR. That means: all forms of energy, be it the energy equivalent of rest mass, field energy of massless fields like radiation or the potential energy of these fields, contribute to gravitation. I have not yet studied Majernik's paper in detail, but I think he is on the right track.

With respect to the stability of a static universe: Some years ago, based on the assumption that energy conservation is valid also in the case of the entire universe, I have tried to develop an equilibrium model of the universe, similar to Einstein's static solution. I have shown that it is possible to find alternative explanations for all the observations, which are usually considered as confirmations of the Big Bang model. The cosmological red shift, the microwave background, the chemical element composition and the formation of cosmic structures can all find an alternative interpretation without invoking such mysterious things like dark energy or an inflationary expansion. If you are interested in this stuff you can find it under arXiv:0708.3577.

Ernst Fischer