Abstract:  Gravitational collapse of diffuse material has been investigated using a new solution of Einstein's equations of general relativity. This replaces the theory of black-hole formation developed for the standard vacuum solution of Schwarzschild. The bodies which now form have reasonable physical properties, such as nuclear hard core density in collapsed stars, or 104kg/l in galactic centres, and only 1kg/l in quasars. Accreting material converts to kinetic energy and radiation, so that a singularity cannot be produced. This record was migrated from the OpenDepot repository service in June, 2017 before shutting down

WAYTE, RICHARD

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   On gravitational collapse  R. Wayte 29 Audley Way, Ascot, Berkshire, SL5 8EE, England, UK. email: rwayte@googlemail.com   Research artical submitted to PMC Physics A.    29 Oct 2010   Abstract:  Gravitational collapse of diffuse material has been investigated using a new solution of Einstein’s equations of general relativity. This replaces the theory of black-hole formation developed for the standard vacuum solution of Schwarzschild. The bodies which now form have reasonable physical properties, such as nuclear hard core density in collapsed stars, or 104kg/l in galactic centres, and only 1kg/l in quasars. Accreting material converts to kinetic energy and radiation, so that a singularity cannot be produced.   PACS Codes: 98.62.Mw; 98.35.Mp; 97.10.Gz    1.  Introduction         A recent paper has revealed that the observed precession of planet Mercury’s orbit is no longer compatible with standard General Relativity theory, [1]. That is, the orthodox vacuum solution with its concomitant space-time curvature interpretation may not be physically meaningful. Consequently, black-holes cannot exist, so the end state of stellar evolution theory now needs to be reconsidered in the light of another solution of Einstein’s Equations.  2Some years ago [2], Paper 1, it was shown how Einstein’s equations could be interpreted in straight-forward physical terms by explicit introduction of gravitational field energy as gravitons, analogous to the electromagnetic field. These gravitons of the field produce the gravitational force by direct interactions, all conducted in flat space-time. Metric tensor components then describe the variation in dimensions and time-rate or energy of particles in a field, not so-called continuum curvature. Agreement with the ideas found in Special Relativity theory and accelerated frames is thereby guaranteed, and gravity is no longer detached from other forces. The old problem of justifying the equality of space-time manifold curvature Rµν and physical matter Tµν is rendered obsolete. Consequently, mass particles and their field gravitons are expressions of the same material, (energy), existing in empty flat space-time.  2.  Building a massive body from diffuse matter. In Paper 1, Einstein’s equations were solved for the exterior solution of the spherically-symmetric static field of energetic gravitons in polar coordinates. The line element was then found to be:   222222222222 dtrcGM1dsinrdr)rc/GM1(drds −+ϕθ−θ−−−=    . (1) Logically, it was argued that the gravitational field induces mass particles to fall by converting their own mass (potential energy) into kinetic energy. Upon impact, the particle KE is radiated away, leaving the particle with reduced rest mass:       )rc/GM1(mm 2or −=    .       (2) We can calculate how much free diffuse matter is required to build a body mass M up to its ultimate gravitational radius (R0 = GM0/c2). If the final density is to be nuclear hard-core density ρn throughout, then let M = (4/3)πρnr3. And for an originally diffuse mass element dm, only:       )rc/GM1(dmdM 2−=    ,       (3) is added to M. Upon eliminating r, this may be integrated to find the total amount of diffuse mass required to build a body of mass M:  3     ( )∞−== →∫∫→ 0MMM03/20T]M/M1[dMdmM  .    (4) Thus, it is impossible to build a body up to its gravitational radius because infalling matter is increasingly less effective at adding mass. Figure 1 illustrates the amount of diffuse matter necessary to build the body up to any particular mass M.   012340 0.2 0.4 0.6 0.8 1M / M0MT / M0 Figure 1  The total diffuse mass MT necessary to build a body from zero to     mass M, relative to the maximum possible body mass M0 .  If during the building process the core material should collapse to a denser state, the mass to kinetic energy to radiation conversion will continue further and still prevent a singular surface from forming. This means that it is impossible to build the black-holes peculiar to the Schwarzschild solution. More energy is actually liberated by this process of accretion than that predicted by the Schwarzschild solution. Furthermore, in this theory, energetic γ-rays may be produced by total conversion of infalling matter, and these will subsequently leave the gravitational field without loss of energy. The bulk density of collapsed bodies must vary greatly, from that of quasars, galactic centres, through white dwarfs, neutron stars, and quark stars. For bodies  4approaching their gravitational radius, their instantaneous masses may be related to their mean densities ρo by applying an approximate classical calculation, as follows. Let : 3o2 R (4/3) M and ,GM/cR πρ=≈ ,        (5) then:  ( )[ ] 2/1o33 3/4GcM −πρ≈ .            (6) Thus, body mass must decrease with increasing density because infalling matter converts to free radiation more and more as it compacts. It follows that a body of large mass with relatively low density must be supported by radiation pressure at high temperature. As it burns-away, its mass will decrease and density increase. Collapse will only cease when the core material gas pressure is able to resist self-gravity, see Figure 2. A final body of nuclear density has a mass around 1.1MΘ, while another of quark density (say 1022kg/l ) has four times Jupiter's mass. Consequently, collapsing bodies can act like photon factories, unless they collapse violently and blast their material into space. -3-2-101234567890 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23log10[Density kg/l]log 10[M / Msun]nuclear densityGalactic centrequasarsquark density Figure 2  Variation of body mass (in Solar mass units) with density, for bodies     near to their gravitational radius.  5 3.  Interior properties of a massive body  The interior properties of a collapsed body, near to its gravitational radius, may be calculated using the new solution of Einstein’s equations, (see Paper 1, Section 16). For example, a dense body of mass 3.6×106MΘ  but low luminosity has been predicted at the centre of our galaxy, see [3-5]. It could be named a ‘black-corps’, which may be near to its gravitational radius (R ≈ GM/c2 ≈ 5.3 x 106km), with average density around 1.1 x 104 kg/l. This density is very high and implies that the material would behave like a fluid in hydrostatic equilibrium. It is commonly believed that quasars contain black-holes of up to 108 MΘ , but these would be black-corps with lesser density around 10 kg/l . The line element for the interior of a spherically-symmetric static body consisting of a “perfect fluid” will be expressed in isotropic form as:     ( ) 22222222 dtedsinrdrdreds νµ +ϕθ+θ+−=    .    (7) For the energy-momentum tensor components we shall take the mechanical local hydrostatic pressure po and constant local mass density ρoo :     ooo440330220110 T,andpTTT ρ====    .     (8) Solution of Einstein’s equations then yields the spatial metric tensor component:     3/8kfor),4/kr1(e oo22/ πρ=+=µ−    ,     (9) and temporal metric tensor component:     −−−= µµµν2e3e1e3e 2/2/2/2/mm   ,         (10) where 2/meµ  pertains to the maximum radius rm at the surface. Both these equations automatically designate the centre of the fluid sphere as the coordinate reference frame of special relativity. We see that on going from the surface towards the centre of the body, a material element is compressed isotropically according to Eq.(9), and slows down internally according to Eq.(10), due to loss of potential energy (mass). The local (pressure/density) ratio increases, on going from the surface inwards:  6    −−−=ρ µµµµ2/2/2/2/oooe1e3eepmm  .         (11) Figure 3 illustrates the variation of central pressure of a body (where 1e 2/ =µ ), in terms of its actual size rm relative to its theoretical gravitational radius R0 .   0123456789100 0.2 0.4 0.6 0.8 1R0 / rmPressure / Density Figure 3  Pressure / density ratio at the centre of a massive body in hydrostatic    equilibrium. A body of mass M and maximum radius rm has a theoretical    gravitational radius R0 = GM / c2 .  These results are interesting because the body material behaves sensibly, even if an exterior observer would lose sight of the ultimate black-corps. In the most extreme case, the time-rate at the surface may approach infinity relative to the centre time-rate, if according to Eq.(10) we have:      02e3 2/m =−µ   .           (12) After introducing Eq.(9) and practical units, this means:      1r34cG 2moo2 =ρπ  ,          (13) so a test particle would lose all its mass on falling from the surface to the centre. This  7equation is compatible with Eq.(5) although ρoo is the locally measured density, which corresponds with a reduced coordinate (real) density. Confirmation of this comes from considering the particle energy which decreases with eν/2 towards zero relative to the surface, whereas particle size can only decrease with e−µ/2  by a factor of 3/2. According to Eq.(11) the local pressure experienced by particles increases inwards and could cause them to rupture then collapse suddenly to a denser state. The release of radiation energy would probably cause some outer material to be blown away, while inner material would be compressed by the explosion, as predicted in a super-nova event.  4.  Conclusion The new solution of Einstein’s equations has been employed to describe how gravitational collapse of diffuse material may produce very dense bodies of low luminosity. Mass converts to kinetic energy during the contraction, and is lost from the system as radiation upon impact with the stationary core. Black holes cannot exist so, at last after 75 years of wasted labour, the ''stellar buffoonery'' ridiculed by Eddington and Einstein has been hurled into the abyss for ''hideous fantasies''; freeing the lives of bright physicists to pursue inspired praiseworthy research, increasing true knowledge.  Acknowledgements I would like to thank Imperial College Libraries, and L. Gao for typing. References 1. Wayte R: submitted 2010, and www.vixra.org/abs/1002.0019 2. Wayte R: Astrophys Space Sci 1983, 91:345, (Paper I) 3. Genzel R et al: Nature 2003, 425:934. 4. Eckart A. et al: Astro Astrophys 2004, 427:1. 5. Ghez AM. et al: 2005 Astrophys J 2005, 620:744. 

ON GRAVITATIONAL COLLAPSE

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ON GRAVITATIONAL COLLAPSE

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