Advances in High Energy Physics

Volume 2018, Article ID 2897419, 13 pages

https://doi.org/10.1155/2018/2897419

## Interesting Examples of Violation of the Classical Equivalence Principle but Not of the Weak One

Coordenação de Cosmologia, Astrofísica e Interações Fundamentais (COSMO), Centro Brasileiro de Pesquisas Físicas (CBPF), Rua Dr. Xavier Sigaud 150, Urca, 22290-180 Rio de Janeiro, RJ, Brazil

Correspondence should be addressed to Antonio Accioly; rb.fpbc@yloicca

Received 4 December 2017; Revised 20 April 2018; Accepted 3 June 2018; Published 8 July 2018

Academic Editor: Edward Sarkisyan-Grinbaum

Copyright © 2018 Antonio Accioly and Wallace Herdy. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

The equivalence principle (EP) and Schiff’s conjecture are discussed* en passant*, and the connection between the EP and quantum mechanics is then briefly analyzed. Two semiclassical violations of the classical equivalence principle (CEP) but not of the weak one (WEP), i.e., Greenberger gravitational Bohr atom and the tree-level scattering of different quantum particles by an external weak higher-order gravitational field, are thoroughly investigated afterwards. Next, two quantum examples of systems that agree with the WEP but not with the CEP, namely, COW experiment and free fall in a constant gravitational field of a massive object described by its wave-function , are discussed in detail. Keeping in mind that, among the four examples focused on in this work only COW experiment is based on an experimental test, some important details related to it are presented as well.

#### 1. Introduction

The equivalence principle (EP) is intrinsically connected to the history of gravitation theory and has played an important role in its development. Newton regarded this principle as such a cornerstone of mechanics that he devoted the opening paragraph of the* Principia* to it.

Before discussing some important aspects of the EP, we would like to explain the two different meanings usually attributed within the context of physics to the words* nonlocal* and* local*.

A* nonlocal* principle is valid everywhere in space and time (space-time) as far as nonrelativistic (relativistic) theories are concerned, while a* local* principle is restricted only to one point of space and time (space-time).

On the other hand, the so-called* principle of locality* states that an object is only directly influenced by its immediate surrounding. A theory which includes the principle of locality is said to be a* local theory*. Locality evolved out of field theories of classical physics. The concept is that, for an action at one point to have an influence at another point, something in the space between those points such as a field must mediate the action. To exert an influence, something, such as a wave or particle, must travel through the space between the two points, carrying the influence. It is worthy of note that the special theory of relativity limits the speed at which all such influences can travel to the speed of light . As a consequence, the principle of locality allows us to conclude that an event at one point cannot cause a simultaneous result at another point. In other words, an event at point cannot cause a result at point in a time less than , where is the distance between the points.

In 1935, Einstein, Podolsky, and Rosen in their EPR paradox theorized that quantum mechanics might not be a local theory since a measurement made on a pair of separated but entangled particles causes a simultaneous effect, the collapse of the wave-function, in the remote particle (i.e., an effect exceeding the speed of light). But because of the probabilistic nature of wave-function collapse, this violation of locality cannot be used to transmit information faster than light. In 1964 Bell formulated the “Bell inequality”, which if violated in actual experiments, implies that quantum mechanics violates either locality or realism, another principle which relates to the value of unmeasured quantities. The two principles are commonly referred to as a single principle,* local realism*.

Experimental tests of the Bell inequality show that quantum mechanics seem to violate either locality or realism. Nevertheless, critics have noted that these experiments contained “loopholes”, which prevented definite answers to this question. This might now be resolved: in 2015 Hanson at Delft University performed what has been called the first loophole free experiment [1].

In summary, the concepts of local and nonlocal principles are totally different from those concerning principles of locality and nonlocality. They cannot be used as synonymous.

We are now ready to examine, in passing, some relevant feature related to the EP in the framework of both Newton and Einstein gravity.

To begin with we discuss the notion of the EP in the context of Newtonian theory.

In the framework of Newton’s theory we have two equivalence principles:* Galileo’s equivalence principle* and* Newton’s equivalence principle*. The Galileo’s equivalence principle states that, in a given gravitational field, all test particles, of the same initial velocity, fall with the same acceleration. The requirement that all particles must be put at the given point with the same initial velocity is necessary because some pseudo-forces, such as the Coriolis force, are velocity-dependent. On the other hand, Newton’s equivalence principle asserts that the gravitational mass of any system equals its inertial mass, or, in more succint terms, . Note that the Galileo and the Newton principles should be regarded as* complementary*. Indeed, the former is a statement on the behavior of a special class of system (test particles) placed in arbitrary weak gravitational fields, while the latter is a statement on the behavior of an arbitrary system put in a weak gravitational field. These two principles have a* nonlocal* character; i.e., they are valid everywhere in space and time [2–5]. From now on we shall designate Newton’s equivalence principle as the* classical equivalence principle* (CEP) of Newtonian theory. Once again we call attention to the fact that the CEP is a nonlocal principle; i.e., it is valid everywhere in space and time.

As far as Einstein gravity is concerned, two EP are generally contemplated: the weak equivalence principle (WEP) and the Einstein one (EEP). The WEP asserts that* locally* we cannot distinguish between inertial and gravitational fields through ‘falling body experiments’. Since the WEP and the CEP are locally identical, the difficulty, at the first sight, of differentiating them in an easy way increases. Consequently, some researchers are led to the common misconception that they coincide even nonlocally (see for instance [6–10]).

Undoubtedly, the two great pillars of modern physics are quantum mechanics and general relativity. Nevertheless, the merging of these theories has not been accomplished despite the herculean efforts employed by so many distinguished physicists.* What is the rationale for such an incompatibility between these outstanding theories? Perhaps it is the conceptual difficulty of reconciling the local character of the WEP with the nonlocal character of the uncertainty principle*.

We call attention to the fact that the validity of both general relativity and WEP can be characterized by the PPN parameter , which reflects the level of space curvature per unit rest mass (for a recent review see [3]). As is well known, as far as general relativity is concerned. On the other hand, the measurement of the absolute value of has reached high accuracy in Astronomy. In fact, Krauss and Tremaine [11], as well as Longo [12], by observing a neutrino burst within 3 h of the associate optical burst from supernova 1987A in the Large Magellanic Cloud, were able to devise a new test of the WEP, by demonstrating that neutrinos and photons follow the same trajectory in the gravitational field of the galaxy. Recently, Wei et al. [13] probed WEP with fast ray bursts, while Yu, Xi, and Wang [14] found a robust method to test the WEP. More recently Yu and Wang [15] tested WEP with strongly lensed cosmic transients. It is worth noticing that every test of general relativity is a potentially deadly test or a possible probe for new physics. It is amazing that this theory, which came to light 100 years ago out of almost pure thought, has managed to survive every test. Of course, the possibility of finding a discrepancy will continue to drive experiments for years to come and these experiments will search for new physics beyond Einstein at many different scales: the large distance scales of the astrophysical, galactic, and cosmological realms; scales of very short distances or high energy; and scales related to strong or dynamical gravity [3].

EEP, on the other hand, embodies* WEP*,* local Lorentz invariance*—the outcome of any local non-gravitational experiment is independent of the velocity of the freely falling reference frame in which it is performed—and* local position invariance*—the outcome of any local non-gravitational experiment is independent of where and when it is performed [3]. EEP may be considered in the broadest sense of the term as the heart and soul of gravity theory. It would not be an exaggeration to say that if the EEP holds, then gravitation must necessarily be a ‘curved space-time’ phenomenon; in other words, the effects of gravity must be equivalent to the effects of living in a curved space-time [3]. Around 1960, Schiff conjectured that* any complete and self consistent theory of gravity that obeys the WEP must also, unavoidably, obey the EEP* [16]. This surmise is known as Schiff’s conjecture. According to it the validity of the WEP alone should guarantee the validity of the local Lorentz and position invariance, and thus of the EEP. However, a rigorous proof of Schiff’s conjecture is improbable. In fact, some special counterexamples are available in the literature [17–20]. Nevertheless, there are some powerful arguments of ‘plausibility’, such as the assumption of energy conservation [21] and the formalism [22], among others, that can be formulated.

In the seminal work by Will [3] entitled “The Confrontation between General Relativity and Experiment” it is asserted that EEP is well supported by experiments such as the Eötvos experiment, tests of local invariance, and clock experiments. In the aforementioned work, ongoing tests of EEP and the inverse square law with the aim of searching for new interactions arising from unification or quantum gravity are also discussed. Actually, EEP and related tests are currently viewed as ways to discover or place constraints on new physical interactions, or as a branch of “non-accelerator particle physics”, searching for the possible imprints of high-energy particle effects in the low-energy realm of gravity. On the phenomenological side, the idea of using EEP tests in this way has its origin in the middle 1980s, with the search for a “fifth” force. In 1986 Fischbach et al. suggested the existence of a fifth force in nature [23]. Nevertheless, the physical community came to the conclusion that there was no credible evidence for a fifth force of nature of a type and range proposed by Fischbach et al. (for a review see, for instance, [24–26]). As far as short-range modifications of Newtonian gravity are concerned, no deviations from the inverse square law have been found up to now at distances between tens of nanometers and 10 mm [27–29].

A natural question must now be posed: what is the connection between the EP and quantum mechanics? As is well known, quantum tests of the EP are radically different from the classical ones because classical and quantum descriptions of motion are fundamentally unlike. In particular, the universality of free fall (UFF) possesses a clear significance in the classical context. Now, how both UFF and WEP are to be understood in quantum mechanics is a much more subtle point. It is generally implicitly assumed that quantum mechanics is valid in the freely falling frame associated with classical test bodies. Nonetheless, an unavoidable problem regarding quantum objects is the existence of half integer spins, which have no classical counterpart. For integer spin particles, the EP can be accounted for by a minimal coupling principle (see Sections A.1 and A.3 of Appendix A), while the procedure to couple a spin field to gravity is much more complex and requires the use of a spinorial representation of the Lorentz group (see Section A.2 of Appendix A).

On the other hand, the most cited scientific experiment claimed to support the idea that, at least in some cases, quantum mechanics and the WEP can be reconciled, that is COW experiment [7]. Although this test, as we shall prove, is in accordance with the WEP, it is in disagreement with the CEP. Another example of a possible quantum mechanical violation of the CEP but not of the WEP is provided by analyzing free fall in a constant gravitational field of a massive object described by its wave-function .

At the semiclassical level an interesting event in which the CEP is also supposed to be violated but not the WEP is the tree-level deflection of different quantum particles by an external weak higher-order gravitational field. We recall beforehand that in Einstein theory the scattering of any particle by an external weak gravitational field is nondispersive which, of course, is in agreement with the WEP. In other words, the deflection angle of all massive particles will be exactly equal. The same is valid for the massless particles. Obviously, the deflection angle will be different whether the particle is massive or massless. A crucial question must then be posed: why to study at the tree level the bending of quantum particles in the framework of higher-derivative gravity? It is not difficult to answer this question. Higher-derivative gravity is the only model that is known to be renormalizable along its matter couplings up to now [30]. Nonetheless, since this system is renormalizable, it is compulsorily nonunitary [31, 32]. We call attention to the fact that the breaking down of unitarity is indeed a serious problem. Fortunately, we shall only deal with the linearized version of higher-derivative gravity, which is stable [33]. The reason why it does not explode is because the ghost cannot accelerate owing to energy conservation. Another way of seeing this is by analyzing the free-wave solutions. We remark that this model is not in disagreement with the result found by Sotiriou and Faraoni [34]. In fact, despite containing a massive spin-2 ghost, as asserted by these authors, the alluded ghost cannot cause trouble [35]. Another probable example at the tree level of violation of the CEP but not of the WEP is provided by Greenberger gravitational Bohr atom [6].

Our main goal here is to explicitly show that, in all situations described above, the WEP is not violated but the CEP is.

The article is organized as follows.

In Section 2 we study the following semiclassical examples:(i)Greenberger gravitational Bohr atom.(ii)Tree-level scattering of different quantum particles by an external weak higher-order gravitational field.

After a careful investigation of both models, we came to the conclusion that they do not violate at all the WEP but are not in accordance with the CEP. As far as the second example is concerned, it is worthy of note that the resulting deflection angles are dependent on both spin and energy. In addition the well-known deflection angles (related to both massive and massless particles) predicted by general relativity are recovered through a suitable limit process.

In Section 3 we analyze two quantum examples: COW experiment and free fall in a constant gravitational field of a massive object described in quantum mechanics by the wave-function . Again, these systems are in accordance with the WEP but not with the CEP.

Our comments are presented in Section 4.

The lengthy calculations concerning the computation of unpolarized cross sections for the scattering of different quantum particles by an external weak higher-order gravitational field are put in Appendix A.

We use natural units throughout and our Minkowski metric is diag (1, -1, -1,-1).

#### 2. Two Examples of Semiclassical Violation of the CEP but Not of the WEP

We analyze in the following two examples of semiclassical violation of the CEP but not of the WEP in a gravitational field.

##### 2.1. Greenberger Gravitational Bohr Atom

As far as we know, Greenberger [6] was the first to foresee the existence of mass-dependent interference effects related to a particle bound in an external gravitational field.

Here we are particularly interested in analyzing Greenberger gravitational Bohr atom, which from the classical point of view consists of a small mass bound to a very much larger mass by the potential , in the limit where all recoil effects may be neglected. If we restrict ourselves to circular orbits, we arrive at the conclusion that classically (see Figure 1).