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Compact Stars in the Nonsymmetric Gravitational Theory by Lyle Compact Stars in the Nonsymmetric Gravitational Theory by Lyle McLean Campbell A Thesis submitted in conformity with the requirements for the Degree of Doctor of Philosophy in the University of Toronto © Lyle McLean Campbell 1988 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Abstract Stable white dwarfs and neutron stars axe shown to exist in the Nonsym- metric Gravitational Theory (NGT). They are modelled as static spherically symmetric bodies of charge-neutral perfect fluid matter, with standard equa­ tions of state. The particle number model for the NGT conserved current, S 11, is used. The effects of NGT reduce the stability of these compact stars com­ pared to similar stars modelled using General Relativity or Newtonian grav­ ity. There is a decrease in the maximum mass of both white dwarfs and neutron stars. The central densities are greater and the radii are smalls- In all these ways, it can be seen that NGT produces a greater gravitational force in compact stars than General Relativity. NGT also decreases the surface gravitational redshift. From examination of the solutions for compact stars, constraints are placed on the £2 charges, (/" -f/") and f 2, of protons, neutrons and electrons. If matter composed only of these particles is considered, the constraints keep the £2 charge of the Sim so small that the NGT effects it produces in the solar system are unobservable at present. Similarly, the NGT terms in the periastron precession of eclipsing binary star systems, such as DI Herculis, would be so small that NGT could not explain the anomalies found there. Extended models for S11 are considered. One of these, based on cosmions (wimps), might allow £2 charges for the Sun and DI Herculis to be large enough to be interesting while keeping the £2 charges of compact stars small. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table of Contents A b s t r a c t ............................................................................................................ i Acknowledgements ............................................................................................... » Chapter 1 Introduction to the Nonsymmetric Gravitational Theory 1 Chapter 2 Modelling Stars in NGT ......................................................10 Section 1: Setting Up the P ro b lem ...............................................................10 Section 2: The Model for S ........................................................................... 15 Section 3: Derivation of the Numerical Equations ................................21 Section 4: Density and Pressure V ariables ................................................. 36 Section 5: Initial Data for the Integration ................................................. 43 Section 6: Derivation of the Stability C onditions .................................... 47 Chapter 3 White Dwarf Stars in N G T ................................................... 59 Section 1: The White Dwarf Equation of Sta te .......................................59 Section 2: GR White Dwarf Stars ........................................................64 Section 3: NGT White Dwarf S tars ........................................................70 C h a p te r 4 Neutron Stars in N G T ....................................................... 82 Section 1: The Mean Field Equation of State ............................................82 Section 2: GR N eutron Stars ............................................................94 Section 3: NGT Neutron Stars ..........................................................100 Chapter 5 Conclusions ................................................................ 112 Section 1: Approximations ............................................................112 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Section 2: Consequences of the B o u n d s .....................................................116 Section 3: The Perihelion Precession of M ercu ry ....................................113 Section 4: The Anomalous Periastron Shift of DI Herculis .... 120 Section 5: Extended Models for ..............................................................124 Section 6: Summary ....................................................................................... 129 Appendix 1 The Perfect Fluid in NGT ................................................ 131 A p p e n d ix 2 Conservation Laws and the Total E n e r g y .......................145 R e f e r e n c e s ............................................................................................................. 161 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C hapter 4 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Figure 4 . 5 ....................................................................................................................95 Figure 4 . 6 ....................................................................................................................96 Figure 4 . 7 ....................................................................................................................97 Figure 4 . 8 ....................................................................................................................98 Figure 4 . 9 ....................................................................................................................99 Figure 4.10 101 Figure 4.11 103 Figure 4.12 104 Figure 4.13 106 Figure 4.14 107 Figure 4.15 108 Figure 4.16 109 Figure 4.17 110 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. I To my wife Jane, who has infinite patience, and to my parents. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. C H A P T E R 1 Introduction to the Nonsymmetric Gravitational Theory The Nonsymmetric Gravitational Theory (NGT) t1-3! is a theory of gravity which generalizes the structure of General Relativity (GR). In GR, gravity arises from the geometry of spacetime which is described by the metric tensor, a symmetric tensor. In NGT, this g is extended to a nonsymmetric fundamental tensor by dropping the symmetry restriction, thus including additional fields into the theory. The connection, W£v, is also generalized to include an antisymmetric part called the torsion tensor. This nonsymmetric structure has its origins in Einstein’s unified field theory, M which tried to include electromagnetism into the framework of GR. In the unified field theory the antisymmetric part of g was interpreted as the electromagnetic field strength tensor. The hope was that this would produce a theory explaining both forces as part of a single coherent whole. This program failed and was abandoned because of its inability to reproduce the Lorentz force ^ in the equation of motion of a charged test body. Although the structure produced a consistent classical field theory, it failed to live up to the interpretation given the extra fields. NGT takes this same structure, but changes the interpretation of the antisymmetric fields. This avoids the problem encountered by Einstein, since there are now no preconceived ideas of how the extra fields should behave. There are other important differences between NGT and Einstein’s uni­ fied field theory besides interpretation. NGT has an additional matter field, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5^, as well as the usual This was introduced into the theory for the following reason. One of the field equations in the unified field theory shows th at * yj—g g^v'- is a conserved current of that theory, but one not generated by any matter source. It seems natural to provide such a source, S^. The antisymmetry of \J—g <7^ then ensures that S'1* itself is conserved. The charge associated with S^ is I2 = f y/=f S' d3x. (1.1) J body It has units of area, and is defined to be i2 so that £ has units of length. Note, however, that £2 cam be either positive or negative. In NGT, any bodvrs interaction with the gravitational field is governed by its I2 charge as well as its mass. Choosing a model for S*. and therefore £2, is extremely important. The predictions which the theory makes depend crucially on the £2 charges of the bodies involved. The standard model of S'* takes it to be a linear combination of conserved particle numbers. The NGT charges of the elementary particles are then the only degrees of freedom in the model. More will be said about this in Section 2.2. In order that the reader become familiar with the fields and equations of NGT a short mathematical review is in order. The theory can be derived from the variation of the following Lagrangian: ^ C = y f T g ^ R ^ W ) - 8 - g ^ T ^ + y (1.2) * Throughout this work the following convention is used for symmetric and antisymmetric parts of a tensor: X^u = -f , where X ^ ^ = oiX^u + X VI1) and = ^[X ^ — _Y„M]. Also, units with c = G = 1 are used throughout. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Here. g>iU is the inverse of g ^ defined by g*x g »A = gx>L g\u = 8$. (1.3) Note that this is only one of two possible definitions for g ^ . The lack of symmetry of g ^ and in NGT forces constant care in tbe ordering of indices.
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