No more spacetime singularities?

We discuss the possibility that the issue of spacetime singularities in general relativity is solved by their stringy extensions.This essay received the joint fifth award from the Gravity Research Foundation, 1993—Ed.     

Stochastic Gravity

We give a summary of the status of currentresearch in stochastic semiclassical gravity and suggestdirections for further investigations. This theorygeneralizes the semiclassical Einstein equation to an Einstein-Langevin equation with a stochasticsource term arising from the fluctuations of theenergy-momentum tensor of quantum fields. We mentionrecent efforts in applying this theory to the study of black hole fluctuation and backreactionproblems, linear response of hot flat space, andstructure formation in inflationary cosmology. Toexplore the physical meaning and implications of thisstochastic regime in relation to both classical andquantum gravity, we find it useful to take the view thatsemiclassical gravity is mesoscopic physics and thatgeneral relativity is the hydrodynamic limit of certain spacetime quantum substructures. We view theclassical spacetime depicted by general relativity as acollective state and the metric or connection functionsas collective variables. Three basic issues —stochasticity, collectivity, correlations — andthree processes — dissipation, fluctuations,decoherence — underscore the transformation fromquantum microstructure and interaction to the emergenceof classical macrostructure and dynamics. We discuss ways toprobe into the high-energy activity from below and maketwo suggestions: via effective field theory and thecorrelation hierarchy. We discuss how stochastic behavior at low energy in an effective theoryand how correlation noise associated with coarse-grainedhigher correlation functions in an interacting quantumfield could carry nontrivial information about the high-energy sector. Finally, we describeprocesses deemed important at the Planck scale,including tunneling and pair creation, wave scatteringin random geometry, growth of fluctuations and forms, Planck-scale resonance states, and spacetimefoams.     

Conventionalism in special relativity

Reichenbach, Grünbaum, and others have argued that special relativity is based on arbitrary conventions concerning clock synchronizations. Here we present a mathematical framework which shows that this conventionality is almost equivalent to the arbitrariness in the choice of coordinates in an inertial system. Since preferred systems of coordinates can uniquely be defined by means of the Lorentz invariance of physical laws irrespective of the properties of light signals, a special clock synchronization—Einstein's standard synchrony—is selected by this principle. No further restrictions conerning light signal synchronization, as proposed, e.g., by Ellis and Bowman, are required in order to refute conventionalism in special relativity.     

The precessional frequency of a gyroscope in the quaternionic formulation of general relativity

The precessional frequency of a gyroscope in a reference frame that orbits about a gravitational body is compared between Einstein's tensor formulation of general relativity and the author's quaternion generalization—obtained from a factorization of the tensor form. The difference in predictions then suggests an experiment that could choose which of these formulations of general relativity is more valid in the analysis of gyroscopic motion.     

Global and local principles of relativity

The principles of relativity are assertions about the structure of physical laws, whose validity or nonvalidity can only be empirically confirmed or falsified. The weakest forms of those principles are the so-calledglobal propositions. They furnish statements as to which operations—assumed to be performed simultaneously throughout the whole universe—have no influence upon the physical events. Much stronger principles are those of alocal nature. These assert that the physical properties of a system do not change, when the relation of the system is altered vis-à-vis the universe at large. On formulating these local principles, we presuppose either that it is possible to eliminate any influence of the environment or that the influence can be compensated as in the case of universal forces (e.g., gravitational) which can principally not be removed. Still weaker, however, are those formulations of the relativity principles which postulate relativity only for infinitesimally small space-time domains or regions. This distinction yields clarification of all discussions about existence and meaning of a general relativity principle. Such an analysis was already performed by Einstein and Abraham in 1912.     

A new approach to the theory of relativity. II. The general theory of relativity

The considerations of Part I are extended and the experimental data and hypotheses that led to the establishment of the general theory of relativity are analyzed. It is found that one of the fundamental assumptions is that light is propagated homogeneously; i.e., by using arbitrary systems of coordinates, propagation of light can be represented by a homogeneous quadratic form. This is shown to be an assumption that can be verified by experiment, at least in principle. As a result of adding a number of further assumptions to this, the usual formalism of the general theory of relativity can be established. In the above point of view, the general theory of relativity—like any other theory—cannot be built upad hoc, but is built on distinct physical hypotheses, each of which can be subjected to test by experiment.     

On Gravitational Repulsion

The concepts of negative gravitational mass and gravitational repulsion are alien to general relativity. Still, we show here that small negative fluctuations—small dimples in the primordial density field—that act as if they have an effective negative gravitational mass, play a dominant role in shaping our Universe. These initially tiny perturbations repel matter surrounding them, expand and grow to become voids in the galaxy distribution. These voids—regions with a diameter of 40 h^-1 Mpc which are almost devoid of galaxies—are the largest objects in the Universe.     

A laboratory test of Mach's principle and strong-field relativistic gravity

A laboratory experiment that tests the validity of Mach's principle — the relativity and gravitational induction of inertia — and relativistic gravity in strong-field circumstances is described. It consists of looking for a stationary shift in the apparent weight of an object when a transient mass fluctuation is induced in one of its parts, that part then being subjected to a pulsed thrust. The transient mass fluctuation induced is of the order of a few tens of milligrams, and the stationary weight shift observed is several milligrams. Details of the apparatus used (capable of detecting an effect at the level of about a tenth of a milligram) are presented. Procedural protocols are laid out. The results obtained — signals some 10 to 15 times the standard error in magnitude — confirm to better than order of magnitude that the predicted effect is indeed present. The consequences of this confirmation of Mach's principle and relativistic gravity are briefly addressed. In particular, it is pointed out that in light of these results radical timelessness seems to be the correct way to understand reality and, from the practical point-of-view, it may prove possible to make traversable wormholes whenever we choose to devote sufficient resources to that end.     

A Kinetic Theory Approach to Quantum Gravity

We describe a kinetic theory approach to quantum gravity by which we mean a theory of the microscopic structure of space-time, not a theory obtained by quantizing general relativity. A figurative conception of this program is like building a ladder with two knotty poles: quantum matter field on the right and space-time on the left. Each rung connecting the corresponding knots represents a distinct level of structure. The lowest rung is hydrodynamics and general relativity; the next rung is semiclassical gravity, with the expectation value of quantum fields acting as source in the semiclassical Einstein equation. We recall how ideas from the statistical mechanics of interacting quantum fields helped us identify the existence of noise in the matter field and its effect on metric fluctuations, leading to the establishment of the third rung: stochastic gravity, described by the Einstein–Langevin equation. Our pathway from stochastic to quantum gravity is via the correlation hierarchy of noise and induced metric fluctuations. Three essential tasks beckon: (1) deduce the correlations of metric fluctuations from correlation noise in the matter field; (2) reconstituting quantum coherence—this is the reverse of decoherence—from these correlation functions; and (3) use the Boltzmann–Langevin equations to identify distinct collective variables depicting recognizable metastable structures in the kinetic and hydrodynamic regimes of quantum matter fields and how they demand of their corresponding space-time counterparts. This will give us a hierarchy of generalized stochastic equations—call them the Boltzmann–Einstein hierarchy of quantum gravity—for each level of space-time structure, from the the macroscopic (general relativity) through the mesoscopic (stochastic gravity) to the microscopic (quantum gravity).     

On the Newtonian limit of general relativity

We establish rigorous results about the Newtonian limit of general relativity by applying to it the theory of different time scales for non-linear partial differential equations as developed in [4, 1, 8]. Roughly speaking, we obtain a priori estimates for solutions to the Einstein's equations, an intermediate, but fundamental, step to show that given a Newtonian solution there exist continuous one-parameter families of solutions to the full Einstein's equations — the parameter being the inverse of the speed of light — which for a finite amount of time are close to the Newtonian solution. These one-parameter families are chosen via aninitialization procedure applied to the initial data for the general relativistic solutions. This procedure allows one to choose the initial data in such a way as to obtain a relativistic solution close to the Newtonian solution in any a priori given Sobolev norm. In some intuitive sense these relativistic solutions, by being close to the Newtonian one, have little extra radiation content (although, actually, this should be so only in the case of the characteristic initial data formulation along future directed light cones).Our results are local, in the sense that they do not include the treatment of asymptotic regions; global results are admittedly very important — in particular they would say how differentiable the solutions are with respect to the parameter — but their treatment would involve the handling of tools even more technical than the ones used here. On the other hand, this local theory is all that is needed for most problems of practical numerical computation.     

Gravity as a Source of Phase Transitions

After going through several distinguished examples, we argue that gravity is definitely a source of phase transitions of quite different nature: usual scalar effective potential ones, chiral symmetry transitions and even transitions involving the chromomagnetic vacuum. In such context, we emphasize the fact that curvature-induced phase transitions of those kinds—where an interplay between general relativity and elementary particle physics occurs—can be used in the construction of new models of the inflationary universe.     

Motion of matter in metric-affine theory of gravitation

Based on the Lie derivative technique in a general space with affine connection (L₄, g), we show that in the metric-affine theory of gravitation, the law of conservation of the energy-momentum tensor for matter and consequently also the equations of motion for matter stemming from this law are (as in the general theory of relativity) a consequence of the gravitational field equations. We derive the hydrodynamic equation of motion for an ideal Weyssenhoff—Raabe spin fluid in Weyl space. We discuss the possibilities for observation of space—time nonmetricity.Moscow State Pedagogical University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 1, pp. 76–82, January, 1994.     

Erwin Schrödinger and the rise of wave mechanics. I. Schrödinger's scientific work before the creation of wave mechanics

This article is in three parts. Part I gives an account of Erwin Schrödinger's growing up and studies in Vienna, his scientific work—first in Vienna from 1911 to 1920, then in Zurich from 1920 to 1925—on the dielectric properties of matter, atmospheric electricity and radioactivity, general relativity, color theory and physiological optics, and on kinetic theory and statistical mechanics. Part II deals with the creation of the theory of wave mechanics by Schrödinger in Zurich during the early months of 1926; he laid the foundations of this theory in his first two communications toAnnalen der Physik. Part III deals with the early applications of wave mechanics to atomic problems—including the demonstration of equivalence of wave mechanics with the quantum mechanics of Born, Heisenberg, and Jordan, and that of Dirac—by Schrödinger himself and others. The new theory was immediately accepted by the scientific community.This article (in three parts) is an expanded version of the Schrödinger Centenary Lecture delivered by me at CERN (Organisation Européenne pour la Recherche Nucléaire), 1211 Geneva 23, Switzerland, on July 30, 1987.     

The Inertial Transformations and the Relativity Principle

Our recently proposed inertial transformations of the space and time variables based on absolute simultaneity imply the existence of a single isotropic inertial reference system (“privileged system”). We show, however, that aresynchronization of clocks in all inertial systems is possible leading to a different, arbitrarily chosen,isotropic “privileged” system. Such a resynchronization does not modify any one of the empirical consequences of the theory,which is thus compatible with a formulation of the relativity principle weaker than adopted in Einstein’s theory of special relativity.     

No Preferred Reference Frame at the Foundation of Quantum Mechanics

Quantum information theorists have created axiomatic reconstructions of quantum mechanics (QM) that are very successful at identifying precisely what distinguishes quantum probability theory from classical and more general probability theories in terms of information-theoretic principles. Herein, we show how one such principle, Information Invariance and Continuity, at the foundation of those axiomatic reconstructions, maps to “no preferred reference frame” (NPRF, aka “the relativity principle”) as it pertains to the invariant measurement of Planck’s constant h for Stern-Gerlach (SG) spin measurements. This is in exact analogy to the relativity principle as it pertains to the invariant measurement of the speed of light c at the foundation of special relativity (SR). Essentially, quantum information theorists have extended Einstein’s use of NPRF from the boost invariance of measurements of c to include the SO(3) invariance of measurements of h between different reference frames of mutually complementary spin measurements via the principle of Information Invariance and Continuity. Consequently, the “mystery” of the Bell states is understood to result from conservation per Information Invariance and Continuity between different reference frames of mutually complementary qubit measurements, and this maps to conservation per NPRF in spacetime. If one falsely conflates the relativity principle with the classical theory of SR, then it may seem impossible that the relativity principle resides at the foundation of non-relativisitic QM. In fact, there is nothing inherently classical or quantum about NPRF. Thus, the axiomatic reconstructions of QM have succeeded in producing a principle account of QM that reveals as much about Nature as the postulates of SR.     

Cosmological aspects of the theory of relativistic bi-Hamiltonian systems. 1. Formation of the space-time continuum of the general theory of relativity

Starting from the fundamental propositions of the nonunitary quantum theory of relativistic bi-Hamiltonian systems, an attempt is made to explain the nature of the big bang by identifying it with a total ff transition occurring in an ensemble of relativistic bi-Hamiltonian systems. The main principles controlling the formation of the space—time continuum of the general theory of relativity are formulated here.NNTs, Khar'kov Engineering Physics Institute. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 106–115, February, 1995.     

How unique are the Friedmann-Robertson-Walker models of the universe

By accepting the validity of certain conjectures in classical general relativity and kinetic theory, it is argued that, in a sense, the spatially homogeneous and isotropic Friedmann-Robertson-Walker (FRW) cosmological models are unique. This is accomplished in two steps. First, there is reason to believe that kinetic theory requires perfect fluids to be shear-free. Second, it seems that general relativity constrains expanding shear-free fluids to be irrotational. The uniqueness of the FRW models then follows, since it has already been established that they are the only space-times which represent an expanding shear-free irrotational perfect fluid that are physically reasonable on a global scale.This essay received an honorable mention (1986) from the Gravity Research Foundation—Ed.     

Topology Change and Context Dependence

The nonclassical features of quantum mechanicsare reproduced using models constructed with a classicaltheory — general relativity. The inability todefine complete initial data consistently andindependently of future measurements, nonlocality, and thenon-Boolean logical structure are reproduced by theseexamples. The key feature of the models is the role oftopology change. It is the breakdown of causal structure associated with topology change that leads tothe apparently nonclassical behavior. For geons,topology change is required to describe the interactionof particles. It is therefore natural to regard topology change as an essential part of the measurementprocess. This leads to models in which the measurementimposes additional nonredundant boundary conditions. Theinitial state cannot be described independently of the measurement and there is a causalconnection between the measurement and the initialstate.     

A pulsar model from an oscillating black hole

The first part of this paper examines conditions in accord with Einstein's criterion of regularity on the field solutions everywhere that would correspond to the existence of a black hole star, following from solutions of his (nonvacuum) field equations. Black hole is defined here as a star whose matter is so condensed as to correspond to a complete family of spatially closed geodesics. The condition imposed is that the angular momentum of a test body in each of the closed geodesics is a constant of the motion. The second part of the paper examines the implications in the problem of the condensed star of a generalized (factorized) version of the metrical field equations, discovered earlier by the author. It is found that in general relativity stars should naturally pulsate, and in its succeeding cycles the gravitational radius of the star is attenuated by a factor exp(–0.349T), where T is the pulsation period. Conditions are discussed for the possibility that the (relatively) regular emissions of radiation from a pulsar may be dynamically rooted in a (smaller) part of the pulsation cycle when the star is out of the black hole state (less dense open geodesics)—when radiation would be emitted to the outside world—and the (greater) part of the cycle when it is in the black hole state (more dense closed geodesics)—when radiation would not be emitted.