General Relativity is today the best theory of gravity addressing a wide range of phenomena. Our understanding of physical laws, from cosmology to local scales, cannot be properly formulated without taking into account it. It is based on one of the most fundamental principles of Nature, the Equivalence Principle, which represents the core of the Einstein theory of gravity. The confirmation of its validity at different scales and in different contexts represents one of the main challenges of modern physics both from the theoretical and the experimental points of view. A major issue related to this principle is the fact that we actually do not know if it is valid at quantum level. Furthermore, recent progress on relativistic theories of gravity have to take into account new issues like Dark Matter and Dark Energy, as well as the validity of fundamental principles like local Lorentz and position invariance. Experiments allow to set stringent constraints on well established symmetry laws, on the physics beyond the Standard Model of particles and interactions, and on General Relativity and its possible extensions. In this review, we discuss precision tests of gravity in General Relativity and alternative theories and their relation with the Equivalence Principle. In the first part, we discuss the Einstein Equivalence Principle according to its weak and strong formulation. We recall some basic topics of General Relativity and the necessity of its extension. Some models of modified gravity are presented in some details. The second part of the paper is devoted to the experimental tests of the Equivalence Principle in its weak formulation. We present the results and methods used in high-precision experiments, and discuss the potential and prospects for future experimental tests.
In this essay it will be shown that Decoherence Model and Einstein Equivalence Principle are conceptually incompatible. In other words, assuming only the validity of the Weak Equivalence Principle the present work concludes that we face two possibilities: (i) if Decoherence Model provides a correct description of nature at quantum level, then there are systems which violate Local Position Invariance, or, (ii) if all the postulates behind Einstein Equivalence Principle are valid, even on quantum realm, then Decoherence Model breaks down in curved spacetimes. Finally, the present results are con…
Einstein's equivalence principle in classical physics is a rule stating that the effect of gravitation is locally equivalent to the acceleration of an observer. The principle determines the motion of test particles uniquely (modulo very broad general assumptions). We show that the same principle applied to a quantum particle described by a wave function on a Newtonian gravitational background determines its motion with a similar degree of uniqueness.
The computation of the phase shift in a symmetric atom interferometer in the presence of a gravitational field is reviewed. The difference of action-phase integrals between the two paths of the interferometer is zero for any Lagrangian which is at most quadratic in position and velocity. We emphasize that in a large class of theories of gravity the atom interferometer permits a test of the weak version of the equivalence principle (or universality of free fall) by comparing the acceleration of atoms with that of ordinary bodies, but is insensitive to that aspect of the equivalence principle kn…
The gedanken-experiment of Einstein's lift is analyzed in order of determining whether the free-falling observer inside the lift can detect the eventual topological non-triviality of space-time
The Einstein's equivalence principle is formulated in terms of the accuracy of measurements and its dependence of the size of the area of measurement. It is shown that different refinements of the statement 'the spacetime is locally flat' lead to different conculsions about the spacetime geometry.
We investigate leading order deviations from general relativity that violate the Einstein equivalence principle in the gravitational standard model extension. We show that redshift experiments based on matter waves and clock comparisons are equivalent to one another. Consideration of torsion balance tests, along with matter wave, microwave, optical, and Mössbauer clock tests, yields comprehensive limits on spin-independent Einstein equivalence principle-violating standard model extension terms at the $10^{-6}$ level.
The Lorentz invariance and Einstein equivalence principle violating effects in the narrow 7.8 eV transition in 229Th nucleus may be 100 000 times larger than in atoms. This transition may be investigated using high precision laser spectroscopy methods, has a very small width, and suppressed systematic effects. Similar values of the effects are expected in 73 eV 235U nuclear transition which is coming within the reach of the laser spectroscopy. Mossbauer transitions give another possibility.
The relation between Einstein equivalence principle and a continuous quantum measurement is analyzed in the context of the recently proposed flavor-oscillation clocks, an idea pioneered by Ahluwalia and Burgard (Gen. Rel Grav. Errata 29, 681 (1997)). We will calculate the measurement outputs if a flavor-oscillation clock, which is immersed in a gravitational field, is subject to a continuous quantum measurement. Afterwards, resorting to the weak equivalence principle, we obtain the corresponding quantities in a freely falling reference frame. Finally, comparing this last result with the measur…
Validity of just a few physical conditions comprising the Einstein Equivalence Principle (EEP) suffices to ensure that gravity can be understood as space-time geometry. EEP is therefore subject to an ongoing experimental verification, with present day tests reaching the regime where quantum mechanics becomes relevant. Here we show that the classical formulation of the EEP does not apply in such a regime. The EEP requires equivalence between the total rest mass-energy of a system, the mass-energy that constitutes its inertia, and the mass-energy that constitutes its weight. In quantum mechanics…
We show, considering a specific f(R)-gravity model, that the Jordan frame and the Einstein frame are physically non-equivalent, although they are connected by a conformal transformation which yields a mathematical equivalence. Since all the calculations are performed analytically, this non-equivalence is shown in an unambiguous way.
We derive, in 3+1 spacetime dimensions, two alternative systems of quasi-linear wave equations, based on Friedrich's conformal field equations. We analyse their equivalence to Einstein's vacuum field equations when appropriate constraint equations are satisfied by the initial data. As an application, the characteristic initial value problem for the Einstein equations with data on past null infinity is reduced to a characteristic initial value problem for wave equations with data on an ordinary light-cone.
Long before the general theory of relativity was finally formulated in 1916, arguments based entirely on Einstein's equivalence principle predicted the well known phenomenon of the gravitational red shift. Precisely the same arguments are widely being used today to derive the same phenomenon. Accordingly, it is often claimed that the observed gravitational red shift is a verification of the equivalence principle rather than a verification of the full theory of general relativity. Here we show that, contrary to these claims, the arguments based on the equivalence principle are false and that \i…
The present work shows that the mathematical equivalence of Jordan frame and its conformally transformed version, the Einstein frame, so far as Brans-Dicke theory is concerned, survives a quantization of cosmological models in the theory. We work with the Wheeler-deWitt quantization scheme and take up quite a few anisotropic cosmological models as examples. We effectively show that the transformation from Jordan to Einstein frame is a canonical one and hence two frames are equivalent description of same physical scenario.
After a historical discussion of Einstein's 1907 principle of equivalence, a homogeneous gravitational field in Minkowski spacetime is constructed. It is pointed out that the reference frames in gravitational theory can be understood as spaces with a flat connection and torsion defined through teleparallelism. This kind of torsion was introduced by Einstein in 1928. The concept of torsion is discussed through simple examples and some historical observations.
We challenge the view that there is a basic conflict between the fundamental principles of Quantum Theory and General Relativity, and in particular the fact that a superposition of massive bodies would lead to a violation of the Equivalence Principle. It has been argued that this violation implies that such a superposition must inevitably spontaneously collapse (like in the Diósi-Penrose model). We identify the origin of such an assertion in the impossibility of finding a local, classical reference frame in which Einstein's Equivalence Principle would hold. In contrast, we argue that the formu…
A `no-hair' theorem for the Brans-Dicke field, in the domain of outer communications of a static black hole, is established, when the trace of the energy-momentum tensor, T, is a definite-sign scalar. This implies that Brans-Dicke theory of gravitation, for matter with T of definite sign, is equivalent to Einstein gravity with the constraint T=0.
The presence of gravity implies corrections to the Einstein-Planck formula $E=h ν$. This gives hope that the divergent blueshift in frequency, associated to the presence of a black hole horizon, could be smoothed out for the energy. Using simple arguments based on Einstein's equivalence principle we show that this is only possible if a black hole emits, in first approximation, not just a single particle, but thermal radiation.
With an explicit example, we show that Jordan frame and the conformally transformed Einstein frames clearly lead to different physics for a non-minimally coupled theory of gravity, namely Brans-Dicke theory, at least at the quantum level. The example taken up is the spatially flat Friedmann cosmology in Brans-Dicke theory.
We find that noncritical gravity, a special class of higher derivative gravity, is classically equivalent to Einstein gravity at the full nonlinear level. We obtain the viscosity-to-entropy ratio and the second order transport coefficients of the dual fluid of noncritical gravity to all orders in the coupling of higher derivative terms. We also compute the holographic entanglement entropy in the dual CFT of noncritical gravity. All these results confirm the nonlinear equivalence between noncritical gravity and Einstein gravity at the classical level.
We review our recent theoretical results about inequivalence between passive gravitational mass and energy for a composite quantum body at a macroscopic level. In particular, we consider macroscopic ensembles of the simplest composite quantum bodies - hydrogen atoms. Our results are as follows. For the most ensembles, the Einstein's Equivalence Principle is valid. On the other hand, we discuss that for some special quantum ensembles - ensembles of the coherent superpositions of the stationary quantum states in the hydrogen atoms (which we call Gravitational demons) - the Equivalence Principle …
It is shown that the Jordan frame and its conformally transformed version, the Einstein frame of nonminimally coupled theories of gravity, are actually equivalent at the quantum level. The example of the theory taken up is the Brans-Dicke theory, and the wave packet calculations are done for a homogeneous and isotropic cosmological model in the purest form of the theory, i.e., in the absence of any additional matter sector. The calculations are clean and exact, and the result obtained are unambiguous.
We consider a class of condensed matter theories in a Newtonian framework with a Lagrange formalism related in a natural way with the classical conservation laws \partial_t ρ+ \partial_i (ρv^i) = 0 \partial_t (ρv^j) + \partial_i (ρv^i v^j + p^{ij}) = 0 We show that for an algebraically defined ``effective Lorentz metric'' g_{μν} and ``effective matter fields'' ϕthese theories are equivalent to material models of a metric theory of gravity with Lagrangian L = L_{GR} + L_{matter} - (8πG)^{-1}(Υg^{00}-Ξ(g^{11}+g^{22}+g^{33}))\sqrt{-g} which fulfils the Einstein equivalence principle and leads to …
Rastall gravity is a modified gravity proposal that incorporates a non-conserved energy momentum tensor (EMT). We study the equivalence between Rastall gravity and general relativity, analyzing its consequences for an EMT of dark matter and dark energy. We find that the translation between the Rastall and Einstein interpretations modifies the equation of state for each component. For instance, cold dark matter can translate into warm dark matter. If the EMT components are allowed to interact, the translation also changes the type of interaction between the components.
We report a joint test of local Lorentz invariance and the Einstein equivalence principle for electrons, using long-term measurements of the transition frequency between two nearly degenerate states of atomic dysprosium. We present many-body calculations which demonstrate that the energy splitting of these states is particularly sensitive to violations of both special and general relativity. We limit Lorentz violation for electrons at the level of $10^{-17}$, matching or improving the best laboratory and astrophysical limits by up to a factor of 10, and improve bounds on gravitational redshift…
In this note we consider the issue of the classical equivalence of scale-invariant gravity in the Einstein and in the Jordan frames. We first consider the simplest example $f(R)=R^{2}$ and show explicitly that the equivalence breaks down when dealing with Ricci-flat solutions. We discuss the link with the fact that flat solutions in quadratic gravity have zero energy. We also consider the case of scale-invariant tensor-scalar gravity and general $f(R)$ theories. We argue that all scale-invariant gravity models have Ricci flat solutions in the Jordan frame that cannot be mapped into the Einstei…
We study stationary axially symmetric solutions of the Einstein vacuum field equations that can be used to describe the gravitational field of astrophysical compact objects in the limiting case of slow rotation and slight deformation. We derive explicitly the exterior Sedrakyan-Chubaryan approximate solution, and express it in analytical form, which makes it practical in the context of astrophysical applications. In the limiting case of vanishing angular momentum, the solution reduces to the well-known Schwarzschild solution in vacuum. We demonstrate that the new solution is equivalent to the …
Recent research has established that nonsymmetric gravitation theories like Moffat's NGT predict that a gravitational field singles out an orthogonal pair of polarization states of light that propagate with different phase velocities. We show that a much wider class of nonmetric theories encompassed by the $χg$ formalism predict such violations of the Einstein equivalence principle. This gravity-induced birefringence of space implies that propagation through a gravitational field can alter the polarization of light. We use data from polarization measurements of extragalactic sources to constra…
We report on a conceptually new test of the equivalence principle performed by measuring the acceleration in Earth's gravity field of two isotopes of strontium atoms, namely, the bosonic $^{88}$Sr isotope which has no spin vs the fermionic $^{87}$Sr isotope which has a half-integer spin. The effect of gravity upon the two atomic species has been probed by means of a precision differential measurement of the Bloch frequency for the two atomic matter waves in a vertical optical lattice. We obtain the values $η= (0.2\pm 1.6)\times10^{-7}$ for the Eötvös parameter and $k=(0.5\pm1.1)\times10^{-7}$ …
The Einstein equivalence principle is certainly a key element in the development of new enhanced theories of gravity. Although being an important building block in Einstein's general relativity, theoretically predicted violations of its validity are a main feature in alternative, nonmetric gravitation theories if they are able to incorporate quantum mechanical principles. We have investigated the metric-affine gauge theory of gravity which predict that a gravitational field singles out an orthogonal pair of polarization states of light that propagate with different phase velocities. This gravi…
The conformal equivalence between Jordan frame and Einstein frame can be used in order to search for exact solutions in general theories of gravity in which scalar fields are minimally or nonminimally coupled with geometry. In the cosmological arena a relevant role is played by the time parameter in which dynamics is described. In this paper we discuss such issues considering also if cosmological Noether symmetries in the ``point--like'' Lagrangian are conformally preserved. Through this analysis and through also a careful analysis of the cosmological parameters Ωand Λ, it is possible to contr…
In the framework of the Einstein-Palatini formalism, even though the projective transformation connecting the arbitrary connection with the Levi Civita connection has been floating in the literature for a long time and perhaps the result was implicitly known in the affine gravity community, yet as far as we know Julia and Silva were the first to realise its gauge character. We rederive this result by using the Rosenfeld-Dirac-Bergmann approach to constrained Hamiltonian systems and do a comprehensive self contained analysis establishing the equivalence of the Einstein-Palatini and the metric f…
We study the interplay of general relativity, the equivalence principle, and high-precision experiments involving atomic transitions and g factor measurements. In particular, we derive a generalized Dirac Hamiltonian, which describes both the gravitational coupling for weak fields, as well as the electromagnetic coupling, e.g., to a central Coulomb field. An approximate form of this Hamiltonian is used to derive the leading gravitational corrections to transition frequencies and g factors. The position-dependence of atomic transitions is shown to be compatible with the equivalence principle, u…
The $f(R)$ gravity and scalar-tensor theory are known to be equivalent at the classical level. We study if this equivalence is valid at the quantum level. There are two descriptions of the scalar-tensor theory in the Jordan and Einstein frames. It is shown that these three formulations of the theories give the same determinant or effective action on shell, and thus they are equivalent at the quantum one-loop level on shell in arbitrary dimensions. We also compute the one-loop divergence in $f(R)$ gravity.
The four-dimensional Einstein-Gauss-Bonnet (4DEGB) gravity is proposed as a singular limit of the higher-dimensional EGB gravity with a rescaled coupling constant. Follow-up work indicates that the 4DEGB gravity may be equivalent to a special Horndeski gravity. However, the full equivalence of the two theories has not been proven. In this paper, we test the possible equivalence for Bianchi type I metric. We consider two symmetry choices of the extra dimensions in the 4DEGB gravity and obtain the explicit cosmological evolution equations of the theories. Our result shows that, for one symmetry …
The Einstein equivalence principle in the electromagnetic sector can be violated in modifications of gravity theory generated by a multiplicative coupling of a scalar field to the electromagnetic Lagrangian. In such theories, deviations of the standard result for the cosmic distance duality relation, and a variation of the fine structure constant are expected and are unequivocally intertwined. In this paper, we search for these possible cosmological signatures by using galaxy cluster gas mass fraction measurements and cosmic chronometers. No significant departure from general relativity is fou…
The issue of implementing the principle of general relativity in Einstein equations has been widely discussed, since Kretschmann's well-known criticism stated that general covariance of the Einstein equations is not suffice to express the principle of general relativity (the equivalence of all the coordinate systems). This failure is usually rooted in the fact that metric in Einstein equations is not univocally determined by the matter distribution. We show that the condition of univocal determination of the metric by the matter distribution is stronger than the requirement of equivalence of a…
The implementation of Einstein's principle of equivalence in Koopman-von Neumann (KvN) mechanics is discussed. The implementation is very similar to the implementation of this principle in quantum mechanics. This is not surprising, because KvN mechanics provides a Hilbert space formulation of classical mechanics that is very similar to the quantum mechanical formalism. Both in KvN mechanics and quantum mechanics, a propagator in a homogeneous gravitational field is simply related with a free propagator. As a result, the wave function in a homogeneous gravitational field in a freely falling ref…
The Einstein Equivalence Principle (EEP) carries a pivotal role in understanding theory of gravity and spacetime. It guarantees the gravity to be understood as geometric phenomenon. Considering gravitational coupling of matter in the standard model extension, we propose a novel scheme using frequency measurements to limit the equivalence principle violations in normal matter. The proposal consists of the comparison of high-precision clocks, comoving with the freely falling frame. The experimental comparison of identical kind of clocks on Earth surface can be used to carry out the proposed test…
We present in detail the scientific objectives in fundamental physics of the Space-Time Explorer and QUantum Equivalence Space Test (STE-QUEST) space mission. STE-QUEST was pre-selected by the European Space Agency together with four other missions for the cosmic vision M3 launch opportunity planned around 2024. It carries out tests of different aspects of the Einstein Equivalence Principle using atomic clocks, matter wave interferometry and long distance time/frequency links, providing fascinating science at the interface between quantum mechanics and gravitation that cannot be achieved, at t…
In the framework of a general scalar-tensor theory, we investigate the equivalence between two different parametrizations of fields that are commonly used in cosmology - the so-called Jordan frame and Einstein frame. While it is clear that both parametrizations are mathematically equivalent at the level of the classical action, the question about their mathematical equivalence at the quantum level as well as their physical equivalence is still a matter of debate in cosmology. We analyze whether the mathematical equivalence still holds when the first quantum corrections are taken into account. …
In this work, we present a complete analysis of the quantisation of the classical Brans-Dicke Theory using the method of affine quantisation in the Hamiltonian description of the theory. The affine quantisation method is based on the symmetry of the phase-space of the system, in this case the (positive) half-plane, that is identified with the affine group. We consider a FLRW-type spacetime, and since the scale factor is always positive, the affine method seems to be more suited than the canonincal quantisation for our Quantum Cosmology. We find the wave function of the Brans-Dicke universe, an…
Aims: Recently, cosmological fast radio bursts (FRBs) have been used to provide the most stringent limit up to date on Einstein's Equivalence Principle (EEP). We study how to further test EEP with FRBs. Methods: Future systematic radio surveys will certainly find abundant FRBs at cosmological distances and some of them will inevitably be located behind clusters of galaxies. Here we suggest to use those FRBs to further test EEP. Results: We find that the robustness and accuracy of testing EEP can be improved further by orders of magnitude with these FRBs. The same methodology can also be applie…
Comparing the parameterized post-Newtonian parameter $γ$ values for different types of particles, or the same type of particles with different energies is an important method to test the Einstein Equivalence Principle (EEP). Assuming that the observed time delays are dominated by the gravitational potential of the Laniakea supercluster of galaxies, better results of EEP constraints can be obtained. In this paper, we apply photons from three kinds of cosmic transients, including TeV blazars, gamma-ray bursts as well as fast radio bursts to constrain EEP. With a gravitational field far more stro…
We analyze black hole thermodynamics in a generalized theory of gravity whose Lagrangian is an arbitrary function of the metric, the Ricci tensor and a scalar field. We can convert the theory into the Einstein frame via a "Legendre" transformation or a conformal transformation. We calculate thermodynamical variables both in the original frame and in the Einstein frame, following the Iyer--Wald definition which satisfies the first law of thermodynamics. We show that all thermodynamical variables defined in the original frame are the same as those in the Einstein frame, if the spacetimes in both…
The S-Stars in the Galactic-center region are found to be on near-perfect Keplerian orbits around presumably a supermassive black hole, with periods of 15-50 yr. Since these stars reach a few percent of light speed at pericenter, various relativistic effects are expected, and have been discussed in the literature. We argue that an elegant test of the Einstein equivalence principle should be possible with existing instruments, through spectroscopic monitoring of an S-star concentrated during the months around pericenter, supplemented with an already-adequate astrometric determination of the inc…
Einstein's weak equivalence principle (EEP) can be tested through the arrival time delay between photons with different frequencies. Assuming that the arrival time delay is solely caused by the gravitational potential of the Milky Way, we show that a "nano-shot" giant pulse with a time delay between energies corrected for all known effects, e.g. $Δt<0.4~\rm{ns}$, from the Crab pulsar poses a new upper limit on the deviation from EEP, i.e. $Δγ< (0.6-1.8)\times 10^{-15}$. This result provides the hitherto most stringent constraint on the EEP, improving by at least 2 to 3 orders of magnitude from…
The Einstein Equivalence Principle~(EEP) can be verified by the measurement of the spectral distortions of the Cosmic Microwave Background (CMB). The existence of energy-dependency in the cosmological redshift effect means the EEP violation. Introducing the energy-dependent Friedmann-Robertson-Walker metric motivated by rainbow gravity, we show that the energy-dependent redshift effect causes the CMB spectral distortions. Assuming the simple energy-dependent form of the metric, we evaluate the distortions. From the COBE/FIRAS bound, we find that the deviation degree from the EEP, which is comp…
Recent results have shown that a field non-minimally coupled to the electromagnetic Lagrangian can induce a violation of the Einstein equivalence principle. { This kind of coupling is present in a very wide class of gravitation theories.} In a cosmological context, this would break the validity of the cosmic distance duality relation as well as cause a time variation of the fine structure constant. Here, we improve constraints on this scenario by using four different observables: the luminosity distance of type Ia supernovae, the angular diameter distance of galaxy clusters, the gas mass fract…
We consider the problem of testing the Einstein Equivalence Principle (EEP) by measuring the gravitational redshift with two Earth-orbiting stable atomic clocks. For a reasonably restricted class of orbits we find an optimal experiment configuration that provides for the maximum accuracy of measuring the relevant EEP violation parameter. The perigee height of such orbits is $\sim$~1,000~km and the period is 3--5~hr, depending on the clock type. For the two of the current best space-qualified clocks, the VCH-1010 hydrogen maser and the PHARAO cesium fountain clock, the achievable experiment acc…
In this communication, we consider a wide class of extensions to General Relativity that break explicitly the Einstein Equivalence Principle by introducing a multiplicative coupling between a scalar field and the electromagnetic Lagrangian. In these theories, we show that 4 cosmological observables are intimately related to each other: a temporal variation of the fine structure constant, a violation of the distance-duality relation, the evolution of the cosmic microwave background (CMB) temperature and CMB spectral distortions. This enables one to put very stringent constraints on possible vio…
