Quantum equivalence principle without mass superselection

The standard argument for the validity of Einstein?s equivalence principle in a non-relativistic quantum context involves the application of a mass superselection rule. The objective of this work is to show that, contrary to widespread opinion, the compatibility between the equivalence principle and quantum mechanics does not depend on the introduction of such a restriction. For this purpose, we develop a formalism based on the extended Galileo group, which allows for a consistent handling of superpositions of different masses, and show that, within such scheme, mass superpositions behave as they should in order to obey the equivalence principle.     

Questioning the equivalence principle

The Equivalence Principle (EP) is not one of the ‘universal’ principles of physics (like the action principle). It is a heuristic hypothesis which was introduced by Einstein in 1907, and used by him to construct his theory of general relativity. In modern language, the (Einsteinian) EP consists in assuming that the only long-range field with gravitational-strength couplings to matter is a massless spin-2 field. Modern unification theories, and notably string theory, suggest the existence of new fields (in particular, scalar fields: ‘dilaton’ and ‘moduli’) with gravitational-strength couplings. In most cases the couplings of these new fields ‘violate’ the EP. If the field is long-ranged, these EP violations lead to many observable consequences (variation of ‘constants’, non-universality of free fall, relative drift of atomic clocks, etc.). The best experimental probe of a possible violation of the EP is to compare the free-fall acceleration of different materials.     

The equivalence principle

The experimental basis of the equivalence principle is reviewed, and the implications for the gravitational interactions of elementary particles are studied within a special relativistic framework. The gravitational red shift is treated in detail and is used to show that antiparticles also obey the equivalence principle. The profound consequences of a violation of the equivalence principle are discussed.     

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.     

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.     

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.     

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.     

Scalar-tensor theories and the principle of equivalence

We investigate the restrictions on scalar-tensor theories of gravitation implied by the assumptions: (i) the field equations are derivable from an action principle, (ii) units of mass length and time are defined by atomic standards, and (iii) the principle of equivalence holds whenever gravitational self-energy can be neglected. We show that in all these theories the presence of gravitational energy in a system leads to violations of the principle of equivalence.     

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.     

Runaway dilaton and equivalence principle violations

In a recently proposed scenario, where the dilaton decouples while cosmologically attracted towards infinite bare string coupling, its residual interactions can be related to the amplitude of density fluctuations generated during inflation, and are large enough to be detectable through a modest improvement on present tests of free-fall universality. Provided it has significant couplings to either dark matter or dark energy, a runaway dilaton can also induce time variations of the natural "constants" within the reach of near-future experiments.     

The equivalence principle and Kaluza-Klein monopoles

It is reported that there is a unique static and spherically symmetric classical solution to the five-dimensional Kaluza-Klein theory which carries magnetic charge as well as electric charge, and satisfies the equivalence principle.     

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     

Limitations to testing the equivalence principle with satellite laser ranging

We consider the possibility of testing the equivalence principle (EP) in the gravitational field of the Earth from the orbits of LAGEOS and LAGEOS II satellites, which are very accurately tracked from ground by laser ranging. The orbital elements that are affected by an EP violation and can be used to measure the corresponding dimensionless parameter η are semimajor axis and argument of pericenter. We show that the best result is obtained from the semimajor axis, and it is limited—with all available ranging data to LAGEOS and LAGEOS II—to η ≃ 2 × 10⁻⁹, more than 3 orders of magnitude worse than experimental results provided by torsion balances. The experiment is limited because of the non uniformity of the gravitational field of the Earth and the error in the measurement of semimajor axis, precisely in the same way as they limit the measurement of the product GM of the Earth. A better use of the pericenter of LAGEOS II can be made if the data are analyzed searching for a new Yukawa-like interaction with a distance scale of one Earth radius. It is found that the pericenter of LAGEOS II is 3 orders of magnitude more sensitive to a composition dependent new interaction with this particular scale than it is to a composition dependent effect expressed by the η parameter only. Nevertheless, the result is still a factor 500 worse than EP tests with torsion balances in the gravitational field of the Earth (i.e. at comparable distance), though a detailed data analysis has yet to be performed. While EP tests with satellite laser ranging are not competitive, laser ranging to the Moon has been able to provide a test of the EP almost 1 order of magnitude better than torsion balances. We show that this is due to the much greater distance of the test masses (the Earth and the Moon) from the primary body (the Sun) and the correspondingly smaller gradients of its gravity field. We therefore consider a similar new experiment involving the orbit of LAGEOS: testing LAGEOS and the Earth for an EP violation in the gravitational field of the Sun. We show that this test may be of interest, though it is a factor 300 less sensitive than in the case of the Moon due to the fact that LAGEOS is closer to the Earth than the Moon and consequently its orbit is less affected by the Sun. The limitations we have pointed out for laser ranging can be overcome by flying in low Earth orbit a spacecraft carrying concentric test masses of different composition with the capability, already demonstrated in ground laboratories, to accurately sense in situ any differential effects between them. An erratum to this article can be found at