VariableG and the strong equivalence principle

A possible time variability ofG, implying a violation of the strong equivalence principle, was first proposed by P. A. M. Dirac in 1937. Since such a feature cannot be accommodated within either Newton’s or Einstein’s theories, a new theoretical framework is needed. In this paper we review one such possible scheme, the scale covariant theory, within which the consequences of a variableG on geophysics, astrophysics, and cosmology can be treated consistently. The global verdict is thatG may have varied by as much as a factor of 25 since the time of nucleosynthesis, without any disagreement emerging in any case. In spite of this result, we are not entitled to conclude from our analysis that a variableG has been shown to exist or that it is needed, but only that its variation iscompatible with known data. The proof thatG varies can in fact only come from direct observations. However, since the previous analyses had concluded that aG(t) would entail severe discrepancies with known data, the reversal of the verdict is believed to be significant, since it may hopefully spur new observational interest in this basic problem. Presented at the Dirac Symposium, Loyola University, New Orleans, May 1981.     

Energy conservation and the principle of equivalence

If the equivalence principle is violated, then observers performing local experiments can detect effects due to their position in an external gravitational environment (preferred-location effects) or can detect effects due to their velocity through some preferred frame (preferred-frame effects). We show that the principle of energy conservation implies a quantitative connection between such effects and structure-dependence of the gravitational acceleration of test bodies (violation of the Weak Equivalence Principle). We analyze this connection within a general theoretical framework that encompasses both non-gravitational local experiments and test bodies as well as gravitational experiments and test bodies, and we use it to discuss specific experimental tests of the equivalence principle, including non-gravitational tests such as gravitational redshift experiments, Eötvös experiments, the Hughes-Drever experiment, and the Turner-Hill experiment, and gravitational tests such as the lunar-laser-ranging “Eötvös” experiment, and measurements of anisotropies and variations in the gravitational constant. This framework is illustrated by analyses within two theoretical formalisms for studying gravitational theories: the PPN formalism, which deals with the motion of gravitating bodies within metric theories of gravity, and the TH?μ formalism that deals with the motion of charged particles within all metric theories and a broad class of non-metric theories of gravity.     

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.     

ADM electrons and the equivalence principle

Noting that the general relativistic ADM equation for the mass of a sphere of charged dust (with no angular momentum) reveals that the masses of point-like particles are determined solely by their electrical charge, electron models based on extended spheres of such purely electrical dust are examined. It is shown that for all realistic electron models of this type (where the observed electron mass is positive and many orders of magnitude smaller than either the Planck or ADM mass) the electron's bare active gravitational mass must be taken to be negative. Because of the negativity of the bare active gravitational mass, one of the two realistic models leads to a violation of the weak equivalence principle, but the other does not. A means of testing whether negative mass obeys the equivalence principle is mentioned.     

Physics with and without the equivalence principle

A differential manifold (d-manifold, for short) can be defined as a pair (M, C), where M is any set and C is a family of real functions on M which is (i) closed with respect to localization and (ii) closed with respect to superposition with smooth Euclidean functions; one also assumes that (iii) M is locally diffeomorphic to Rⁿ. These axioms have a straightforward physical interpretation. Axioms (i) and (ii) formalize certain compatibility conditions which usually are supposed to be assumed tacitly by physicists. Axiom (iii) may be though of as a (nonmetric) version of Einstein's equivalence principle. By dropping axiom (iii), one obtains a more general structure called a differential space (d-space). Every subset of Rⁿ turns out to be a d-space. Nevertheless it is mathematically a workable structure. It might be expected that somewhere in the neighborhood of the Big Bang there is a domain in which space-time is not a d-manifold but still continues to be a d-space. In such a domain we would have a physics without the (usual form of the) equivalence principle. Simple examples of d-spaces which are not d-manifolds elucidate the principal characteristics the resulting physics would manifest.on leave of absence from the Institute of Nuclear Physics, Department of Theoretical Physics, ul. Radzikowskiego 152, 31–342 Cracow, Poland.     

The maximum entropy principle as a consequence of the principle of Laplace

The maximum entropy principle states that the probability distribution which best represents our information is the one which maximizes the entropy with the given evidence as constraints. We prove that this principle is implied from the Laplace principle of equiprobabilities applied to the setS of allN-term sequences of results which are compatible with the given evidence. We generalize to the information gain method of Kullback.     

The relativity principle and the nature of time

Old and recent ideas concerning the nature of time are reviewed, starting from Mach's refusal of Newton's absolute time. Many experiments show that the slowing down of moving clocks is a real phenomenon. Such must then also be the so-called “twin paradox,” which owes its name to its evident incompatibility with the philosophy of relativism (not to be confused with the theory of relativity). The Lorentz reformulation of special relativity started by postulating physical effects of the ether, but accepted Einstein's clock synchronization. Only because of this Lorentz could not understand the advantages of an easily deducible different theory. As stressed by Popper, one of the main problems of the usual approach is the introduction of a superdeterministic universe. Recent results obtained by the author show that a theory is possible, based on relative time but on absolute simultaneity, in which the conceptual difficulties of relativity are avoided.     

Entropy and superposition principle

Given a faithful normal state ? of a von Neumann algebra M, entropy and relative entropy for normal states of M are defined by Radon-Nikodyn derivatives of normal states with respect to ?. Most properties of entropy and relative entropy in finite quantum systems are shown to hold. It is also shown that the finiteness of relative entropy is related to the facial superposition principle in quantum theory [5].     

Mach's principle and the deceleration parameter

We relate the deceleration parameterq to the mass and “radius” of the observable universe, and set a theoretical estimate forq by means of Mach's principle as represented by Whitrow and Randall relationq∼1.     

Quantum aspects of the equivalence principle

Two thought experiments are discussed which suggest, first, a geometric interpretation of the concept of a (say, vector) potential (i.e., as a kinematic quantity associated with a transformation between moving frames of reference suitably related to the problem) and, second, that, in a quantum treatment one should extend the notion of the equivalence principle to include not only the equivalence of inertial forces with suitable real forces, but also the equivalence of potentials of such inertial forces and the potentials of suitable real forces. The two types of cancellation are physically independent of each other, because of the Aharonov-Bohm effect. Finally, we show that the latter effect itself can be understood geometrically as a kinematic effect arising upon the transformation between the two reference frames.On leave of absence from the Department of Physics, Tel-Aviv University, Israel, and the Department of Physics, Yeshiva University, New York.Supported by the NSF under Contract GP-14911.     

Mach’s principle and the notion of time

The role of time coordinate in the realization of March’s principle is highlighted. It is shown that Mach’s principle is linked to the definition of a ‘particle’. These results suggest a deep connection between quantum gravity and Mach’s principle.     

Gravitational redshift and the equivalence principle

Two problems have long been confused with each other: the gravitational redshift as discussed by the equivalence principle; and the Doppler shift observed by a detector which moves with constant proper acceleration away from a stationary source. We here distinguish these two problems and give for the first time a solution of the former which is exact within the context of the equivalence principle in a sense discussed in the paper. The equivalence principle leads to transformations between flat spacetimes. These are analyzed, and a generalized Lorentz transformation is proposed which covers transformations from inertial to uniformly accelerated frames of reference.     

Quantum black hole and the modified uncertainty principle

Recently Ali et al. (2009) [13] proposed a Generalized Uncertainty Principle (or GUP) with a linear term in momentum (accompanied by Planck length). Inspired by this idea we examine the Wheeler-DeWitt equation for a Schwarzschild black hole with a modified Heisenberg algebra which has a linear term in momentum. We found that the leading contribution to mass comes from the square root of the quantum number n which coincides with Bekenstein?s proposal. We also found that the mass of the black hole is directly proportional to the quantum number n when quantum gravity effects are taken into consideration via the modified uncertainty relation but it reduces the value of mass for a particular value of the quantum number.     

Mach's principle and the reciprocal symmetry of nature

This article develops the postulate that spacetime-charge inversion invariance reflects a fundamental reciprocal symmetry in nature between the two long range forces, from which the derivation of Mach's principle (i.e., the principle that the fundamental parameters of the electromagnetically elementary charged particle are related to those describing the electromagnetically observable universe) follows quite easily. Interpreting this result, it is argued that relativity and quantum mechanics can be made conceptually compatible and mathematically consistent by this reciprocal symmetry if one realizes that relativity isboth a macroscopic, semiclassical theory (i.e., the global half of relativity, described by Eq. (1.1), including special and general relativity) and a microscopic theory (i.e., the local half of relativity, described by Eq. (2.1), including relativistic quantum mechanics and field theory). The reciprocal symmetry of nature, then, promises unique (differential and/or integral) relationships between the coordinate variables of the observers of these tworeciprocally related theories, which implies unique, consistent numerical values for the scalar curvatureR, the massM, and the critical density for closure, _c, of the observable universe [derived from the elementary particle parameters (i.e., the electron mass and Coulomb radius)]. With this symmetry we also postulate a plausible mechanism for spontaneous generation of matter from the ubiquitous (zero-mass ether) nothingness of the Dirac sea of filled negative energy states, and can consistently interpret both the positive and negative-energy state solutions of Dirac's equation for massive, spin-1/2 (i.e., fermion) particles and both the advanced and retarded potential solutions of electromagnetic field equations. It is pointed out that, with this interpretation of the advanced potential solutions from electromagnetic field theory, one can actuallyderive causality from electromagnetic theory.     

Neutrino oscillations in non-inertial frames and the violation of the equivalence principle Neutrino mixing induced by the equivalence principle violation

Neutrino oscillations are analyzed in an accelerating and rotating reference frame, assuming that the gravitational coupling of neutrinos is flavor dependent, which implies a violation of the equivalence principle. Unlike the usual studies in which a constant gravitational field is considered, such frames could represent a more suitable framework for testing if a breakdown of the equivalence principle occurs, due to the possibility to modulate the (simulated) gravitational field. The violation of the equivalence principle implies, for the case of a maximal gravitational mixing angle, the presence of an off-diagonal term in the mass matrix. The consequences on the evolution of flavor (mass) eigenstates of such a term are analyzed for solar (oscillations in the vacuum) and atmospheric neutrinos. We calculate the flavor oscillation probability in the non-inertial frame, which does depend on its angular velocity and linear acceleration, as well as on the energy of neutrinos, the mass-squared difference between two mass eigenstates, and on the measure of the degree of violation of the equivalence principle (). In particular, we find that the energy dependence disappears for vanishing mass-squared difference, unlike the result obtained by Gasperini, Halprin, Leung, and other physical mechanisms proposed as a viable explanation of neutrino oscillations. Estimations on the upper values of are inferred for a rotating observer (with vanishing linear acceleration) comoving with the earth, hence rad/sec, and all other alternative mechanisms generating the oscillation phenomena have been neglected. In this case we find that the constraints on are given by for solar neutrinos and for atmospheric neutrinos. Received: 14 December 2000 / Published online: 15 March 2001     

The opacity of the universe and the strong equivalence principle

A possible explanation of why the advanced (i.e. acausal) solutions of Maxwell's equations are not observed in nature is by way of absorption by an opaque universe. As Davies has shown, the ever expanding, general relativistic cosmological models fail to provide the needed absorption. This is perhaps not surprising. In fact, the absorption mechanism calling for an interplay between local physics and cosmology, is usually developed adopting the strong equivalence principle, SEP, which precludes such interplay. We show that complete absorption of electromagnetic radiation by ionized intergalactic plasma is obtained provided a violation of the SEP, of the order of the Hubble's constant, is allowed to occur. The same degree of violation was previously found to be compatible with a large body of observational data.     

The principle of reciprocity

The circumstances surrounding the realisation that NMR signal reception could be quantified in a simple fundamental manner using Lorentz’s Principle of Reciprocity are described. The poor signal-to-noise ratio of the first European superconducting magnet is identified as a major motivating factor, together with the author’s need to understand phenomena at a basic level. A summary is then given of the thought processes leading to the very simple pseudo-static formula that has been the basis of signal-to-noise calculations for over a generation.     

Quantum theory of the generalised uncertainty principle

We extend significantly previous works on the Hilbert space representations of the generalized uncertainty principle (GUP) in 3 + 1 dimensions of the form \([X_i,P_j] = i F_{ij}\) where \(F_{ij} = f({\mathbf {P}}^2) \delta _{ij} + g({\mathbf {P}}^2) P_i P_j\) for any functions f. However, we restrict our study to the case of commuting X’s. We focus in particular on the symmetries of the theory, and the minimal length that emerge in some cases. We first show that, at the algebraic level, there exists an unambiguous mapping between the GUP with a deformed quantum algebra and a quadratic Hamiltonian into a standard, Heisenberg algebra of operators and an aquadratic Hamiltonian, provided the boost sector of the symmetries is modified accordingly. The theory can also be mapped to a completely standard Quantum Mechanics with standard symmetries, but with momentum dependent position operators. Next, we investigate the Hilbert space representations of these algebraically equivalent models, and focus specifically on whether they exhibit a minimal length. We carry the functional analysis of the various operators involved, and show that the appearance of a minimal length critically depends on the relationship between the generators of translations and the physical momenta. In particular, because this relationship is preserved by the algebraic mapping presented in this paper, when a minimal length is present in the standard GUP, it is also present in the corresponding Aquadratic Hamiltonian formulation, despite the perfectly standard algebra of this model. In general, a minimal length requires bounded generators of translations, i.e. a specific kind of quantization of space, and this depends on the precise shape of the function f defined previously. This result provides an elegant and unambiguous classification of which universal quantum gravity corrections lead to the emergence of a minimal length.     

Equivalence principle and light deflection

The deflection of light is discussed with regard to its significance for the structure of general relativity. It is argued that this effect is no consequence of the equivalence principle, but is a real test for general relativity.Einstein's elevator model andFock's criticism of the equivalence principle are investigated.