Intrinsic symmetries in general relativity

A discussion is presented which indicates that a stagnation point is being reached in the standard applications of symmetries (especially isometries) in general relativity. In order to continue the advance of this area of research, an attractive alternative is suggested. This alternative involves the use of what are termed intrinsic symmetries. With this technique, emphasis is placed on underlying symmetries ofsubmanifolds. One particular set of symmetries gives rise to an invariant formulation of the Szekeres inhomogeneous cosmological models, and suggests quite natural generalizations.This essay received an honorable mention (1978) from the Gravity Research Foundation. (Ed.)     

Special conformal symmetries in general relativity

A geometrical discussion of special conformal vector fields in space-time is given. In particular, it is shown that if such a vector field is admitted, it is unique up to a constant scaling and the addition of a homothetic or a Killing vector field. In the case when the gradient of the conformal scalar associated with is non-null it is shown that other homothetic and affine symmetries are necessarily admitted by the space-time, that an intrinsic family of 2-dimensional flat submanifolds is determined in the space-time, that is, in general, hypersurface orthogonal and that the space-time, if non-flat, is necessarily (geodesically) incomplete. Other geometrical features of such space-times are also considered.     

Space-time symmetries and the gravitational-neutrino field equations in general relativity

We consider a neutrino field with geodesic rays in interaction with a gravitational field admitting a Killing vector field n^μ. It is found that for solutions of the Einstein-Weyl field equations the neutrino field ξ^A and the neutrino flux vector l^μ are restricted by the equations: $\text{L}{}_{\mathrm{n}}\text{ξ}{}^{\mathrm{<mtext>A\; =\; ?</mtext><mtext>1</mtext><mtext>2</mtext>}}\text{is}\text{ξ}{}^{\mathrm{A}}$ and $\text{L}{}_{\mathrm{n}}\text{l}{}^{\mathrm{\mu }}\text{= 0}$, whereas s is a real constant. In the case of pure radiation neutrino fields these equations become: $\text{L}\text{ξ}{}^{\mathrm{A}}\text{=}\text{case1}\text{2}\text{(p ? is)ξ}{}^{\mathrm{A}}$, $\text{L}{}_{\mathrm{n}}\text{l}{}^{\mathrm{\mu }}\text{= pl}{}^{\mathrm{\mu }}$, where p and s are in general real functions of the coordinates.     

Kinks in general relativity

The problem of classifying topologically distinct general relativistic metrics is discussed. For a wide class of parallelizable space-time manifolds it is shown that a certain integer-valued topological invariant n always exists, and that quantization when n is odd will lead to spinor wave functionals.     

Time in general relativity

In order to distinguish between physical and coordinate effects in an arbitrary gravitational field, the space coordinate system and the clock rates must be specified operationallya priori. Once this is done, it is no longer possible to set up an initial surface arbitrarily, since this operation must be consistent with certain physical experiments, whose results depend upon the particular physical situation. A method is given for setting up the initial surface, and the time evolution of the system is discussed.NASA Predoctoral Fellow.     

Tachyons in general relativity

The tachyonic version of the Schwarzschild (bradyonic) gravitational field within the framework of extended relativity is considered. The metric of a tachyonic black hole is obtained through superluminal transformations from a bradyonic metric. The extended space-time manifold of this geometry which includes both black and white tachyonic holes is analysed, and the differences between the tachyonic and bradyonic versions are noted. It is shown that the meanings of black holes, tachyons and bradyons depend on the character of the reference frame and are not absolute.     

Singularities in general relativity

The problem of singularities is examined from the stand-point of a local observer. A singularity is defined as a state with an infinite proper rest mass density. The approach consists of three steps: (i) The complete system of equations describing a non-symmetric motion of a perfect fluid under assumption of adiabatic thermodynamic processes and of no release of nuclear energy is reduced to six Einstein field equations and their four first integrals for six remaining unknown componentsgik. (ii) A differential relation for the behavior of the rest mass density is deduced. It shows that any inhomogeneity and anisotropy in the distribution and motion of a non-rotating ideal fluid accelerates collapse to a singularity which will be reached in a finite proper time. Collapse is also inevitable in a rotating fluid in the case of extremely high pressure when the relativistic limit of the equation of state must be applied. In the case of a lower or zero pressure the relation does not give an unambiguous answer if the matter is rotating. (iii) The influence of rotation on the motion of an incoherent matter is investigated. Some qualitative arguments are given for a possible existence of a narrow class of singularity-free solutions of Einstein equations. Assuming rotational symmetry the Einstein partial differential equations together with their first integrals are reduced to a system of simultaneous ordinary differential equations suitable for numerical integration. Without integrating this system the existence of the class of singularity-free solutions is confirmed and exactly delimited. These solutions, representing a new general relativistic effect, are, however, of no importance for the application in cosmology or astrophysics. It is proved that in all the other cases interesting from the point of view of application the occurrence of a point singularity in incoherent matter with a rotational symmetry is inevitable even if the rotation is present.Read on 15 May 1970 at the Gwatt Seminar on the Bearings of Topology upon General Relativity     

Discrete symmetries and the einstein principle of relativity

It is possible to limit the field of the data exchanged by observers so that discrete symmetry principles of a new kind could be valid. The decay of the K⁰_L into two pions exhibits such a symmetry. Some other consequences, unfortunately difficult to test, are presented. One of the main results is that the weak interactions should possess symmetries which correspond to TP and TC in addition to CP, but TCP is not necessarily a symmetry.     

Nongeodesic motion in general relativity

The equations of motion of a spinning body in the gravitational field of a much larger mass are found using both the Corinaldesi-Papapetrou spin supplementary condition (SSC) and the Pirani SSC. These equations of motion are compared with our previous result derived from Gupta's quantum theory of Gravitation. It is found that the spin-dependent terms differ in each of the above three results due to a different location of the center of mass of the spinning body. As expected, these terms are not affected by the choice of either Schwarzschild or isotropic coordinates. Finally, for the presently planned Stanford gyroscope experiment, we find the maximum secular displacement of the orbit of the gyro with respect to the orbit of its non-rotating housing to be of the order of (10⁻⁷ cm/year)t, a result much smaller than Schiff's result which is proportional to time squared.     

Junction conditions in general relativity

We examine the three sets of junction conditions commonly used in general relativity: those due to Darmois, to O'Brien and Synge, and to Lichnerowicz. We show that those due to Darmois and Lichnerowicz are equivalent. The O'Brien and Synge set is stronger than the other two and is unsatisfactory in that it may rule out physically plausible junctions. We conclude that the Darmois set is the most convenient and reliable.     

Algebraic computing in general relativity

The purpose of this paper is to bring to the attention of potential users the existence of algebraic computing systems, and to illustrate their use by reviewing a number of problems for which such a system has been successfully used in General Relativity. In addition, some remarks are included which may be of help in the future design of these systems.     

Acceleration-dependent lagrangians in general relativity

In contrast to electromagnetic theory, in general relativity the elimination of acceleration terms in a lagrangian by substituting into the lagrangian the equations of motion which follow from the lagrangian is a correct procedure; it corresponds to a gauge transformation.     

The gauge in general relativity

The view is taken that the field equations of General Relativity, without a definition of congruence of length and time intervals at different events, are without physical content. The possibility is explored that the customary Einstein field equations are to be used but with a different congruence definition than is customary. When these resulting equations are, in turn, expressed with the customary congruence, they comprise a new set of field equations physically not equivalent to either Einstein's or Brans-Dicke's formulations of general relativity. Similarities with Einstein's and Brans-Dicke's formulations are discussed, and the possibility of experimental confirmation of these new equations is also briefly considered.     

Spin dynamics in general relativity

The energy-momentum tensor of a fluid composed of spinning particles is presented in a form that is structurally similar to that of a fluid with the energy flux q^a with respect to the fluid velocity u^a, isotropic pressure p and the trace-free anisotropic pressure π^ab. The effect of spin in the evolution of a shear free cosmological model with irrotational geodesic flow is considered.     

Global monopole in general relativity

We consider the gravitational properties of a global monopole on the basis of the simplest Higgs scalar triplet model in general relativity. We begin with establishing some common features of hedgehog-type solutions with a regular center, independent of the choice of the symmetry-breaking potential. There are six types of qualitative behaviors of the solutions; we show, in particular, that the metric can contain at most one simple horizon. For the standard Mexican hat potential, the previously known properties of the solutions are confirmed and some new results are obtained. Thus, we show analytically that solutions with the monotonically growing Higgs field and finite energy in the static region exist only in the interval 1 < λ < 3, where λ is the squared energy of spontaneous symmetry breaking in Planck units. The cosmological properties of these globally regular solutions apparently favor the idea that the standard Big Bang might be replaced with a nonsingular static core and a horizon appearing as a result of some symmetry-breaking phase transition at the Planck energy scale. In addition to the monotonic solutions, we present and analyze a sequence of families of new solutions with the oscillating Higgs field. These families are parametrized by n, the number of knots of the Higgs field, and exist for λ < γ_n=6/[(2n + 1)(n + 2)]; all such solutions possess a horizon and a singularity beyond it.     

Soliton concept in general relativity

Soliton physics has made considerable progress in solving nonlinear problems. This paper is meant to relate the soliten concept to the stationary axisymmetric vacuum fields in general relativity. We present a functional transformation which, working as a nonlinear creation operator, generates gravitational fields of isolated sources. When applied to flat space-time (gravitational vacuum) this operation leads to a nonlinear superposition of an arbitrary number of Kerr particles. This superposition also includes the Tomimatsu-Sato fields. The functional transformations form an infinite-parameter group which contains the Kinnersley-Geroch group as a subgroup.This essay received the second award from the Gravity Research Foundation for the year 1980.-Ed.