The dynamical approach to spin-2 gravity

Despite almost a century’s worth of study, it is still unclear how general relativity (GR) and quantum theory (QT) should be unified into a consistent theory. The conventional approach is to retain the foundational principles of QT, such as the superposition principle, and modify GR. This is referred to as ‘quantizing gravity’, resulting in a theory of ‘quantum gravity’. The opposite approach is ‘gravitizing QT’ where we attempt to keep the principles of GR, such as the equivalence principle, and consider how this leads to modifications of QT. What we are most lacking in understanding which route to take, if either, is experimental guidance. Here we consider using a Bose–Einstein condensate (BEC) to search for clues. In particular, we study how a single BEC in a superposition of two locations could test a gravitizing QT proposal where wavefunction collapse emerges from a unified theory as an objective process, resolving the measurement problem of QT. Such a modification to QT due to general relativistic principles is testable near the Planck mass scale, which is much closer to experiments than the Planck length scale where quantum, general relativistic effects are traditionally anticipated in quantum gravity theories. Furthermore, experimental tests of this proposal should be simpler to perform than recently suggested experiments that would test the quantizing gravity approach in the Newtonian gravity limit by searching for entanglement between two massive systems that are both in a superposition of two locations.

Einstein's equivalence principle was initially the equivalence of an accelerated frame and uniform gravity. In spite of being often challenged, Einstein insisted on the fundamental importance of his equivalence principle to general relativity. It is shown that existing criticisms, starting from Synge and Fock, are due to misunderstanding and misconceptions in physics, and/or inconsistent considerations. These include the misinterpretations of Pauli, Bergmann, Tolman, Landau & Lift­ shitz, Zel'dovich & Novikov, Dirac, Wheeler, Thome, Hawking, and others. It has been overlooked that Einstein's equivalence principle implies uniqueness ofthe gauge for given frame ofreference. The recent criticism by Hong has the distinction of starting from his intuitive, though inadequate, observation that a homogeneous field is characterized by the fact that any part of it is representa­ tive ofthe whole. It is pointed out that his notion of uniform gravity disagrees with experiment on the gravitational redshift. His arguments concerning acceleration also disagree with special relativ­ ity, while repeating the same mistake of Landau & Lifshitz. Moreover, it is pointed out that the cru­ cial role of Einstein's equivalence principle in general relativity is firmly established because the Maxwell-Newton Approximation, which is rigorously derived in the theoretical framework of gen­ eral relativity, is unambiguously supported by experiments. Thus, the Schwarzschild solution is ac­ tually invalid in physics.

Energy conservation and equivalence principle in General Relativity

Quantum Gravity (Cambridge Monographs on Mathematical Physics)

Within the lore of nearly every scientific theory, among the facts, laws, theorems, and such that constitute the “truths” it expresses, are one or more principles. Physics textbooks, I know, usually mention them at the beginning, or during some interlude given over to general remarks or speculation. As examples, the uncertainty principle and correspondance principle of quantum mechanics come to mind (see, e.g., Messiah 1960). And associated with the general theory of relativity, which is particularly rich in principles, are a principle of general covariance, a principle of equivalence (or two), Mach's principle, and a general principle of relativity (see, e.g., Misner, Thome, and Wheeler 1973).Where do principles come from? Generally, as I said, they are supposed to be “truths” about the working of the world: observations and experiments are appealed to as their sources. But this doesn't distinguish them from facts and laws.

We give a definition of rigid congruences in both General and Special Relativity, and we try to make the definition plausible. To this end we recall Fermat's principle in General Relativity and we show that this principle allows us to reinterpret the “quotient metric” as the quadratic form which defines the optical length in a gravitational field. We apply the definition to the Earth-Sun system in the post-Newtonian approximation. Furthermore we compute the Fermat tensor and the corresponding relative variation of the speed of light in a Michelson-Morley like experiment performed on the Earth's surface. According to all measurements to date, this quantity is extremely small (10 -13 ).

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 and classical reference frame in which Einstein's Equivalence Principle would hold. In contrast, we argue that the formulation of the Equivalence Principle can be generalized so that it holds for reference frames that are associated with quantum systems in a superposition of spacetimes. The core of this new formulation is the introduction of a quantum diffeomorphism to such Quantum Reference Frames. This procedure reconciles the principle of linear superposition in Quantum Theory with the principle of general covariance and the Equivalence Principle of General Relativity. Hence, it is not necessary to invoke a gravity-induced spontaneous state reduction when a massive body is prepared in a spatial superposition.

In this paper, I discuss the impact of the positive cosmological constant on the interplay between the equivalence principle in general relativity, and the rules of quantum mechanics. At the nonrelativistic level, there is an ambiguity in the definition of a phase of a wave function measured by inertial and accelerating observes. This is the cosmological analogue of the Penrose effect, which can also be seen as a nonrelativistic limit of the Unruh effect. The symmetries of the associated Schrödinger equation are generated by the Newton–Hooke algebra, which arises from a nonrelativistic limit of a cosmological twistor space.

The seasonal polar caps of Mars can be used to test the equivalence principle in general relativity. The north and south caps, which are composed of carbon dioxide, wax and wane with the seasons. If the ratio of the inertial (passive) to gravitational (active) masses of the caps differs from the same ratio for the rest of Mars, then the equivalence principle fails, Newton's third law fails, and the caps will pull Mars one way and then the other with a force aligned with the planet's spin axis. This leads to a secular change in Mars's along-track position in its orbit about the Sun, and to a secular change in the orbit's semimajor axis. The caps are a poor E\"otv\"os test of the equivalence principle, being 4 orders-of-magnitude weaker than laboratory tests and 7 orders-of-magnitude weaker than that found by lunar laser ranging; the reason is the small mass of the caps compared to Mars as a whole. The principal virtue of using Mars is that the caps contain carbon, an element not normally considered in such experiments. The Earth with its seasonal snow cover can also be used for a similar test.

In this chapter I argue that the concept of proper time must be regarded as one of Minkowski’s enduring contributions to physics. I examine some confusions that still interfere with an appreciation of this, including a conflation of proper time with the co-ordinate time of the inertial frame of a system at rest, and the related mistaken notion that Special Relativity cannot be applied to accelerating systems. This sets the stage for a treatment of the so-called clock hypothesis, according to which the instantaneous rate of a clock depends only on its instantaneous speed. I argue that this does not have the status of an independent hypothesis, but is simply a description of the behaviour of an ideal clock as predicted by (classical, special and general) relativity theory. The question whether this hypothesis holds, moreover, must be distinguished from the question of whether the restorative acceleration of the mechanism within any real system acting as a clock is sufficiently great (relative to the acceleration undergone by the system) that the system will be able to approximate such an ideal clock. The failure of the clock hypothesis would entail the falsity of relativity theory in the form proposed by Einstein, as Weyl had sought to demonstrate with his unified theory of gravity and electromagnetism in 1918. I argue that it is the Strong Equivalence Principle in General Relativity that preserves the chronometric significance that the metric had in Special Relativity, and thereby preserves the relation of inertia to time assumed classically.

The paper considers the symmetries of a bit of information corresponding to one, two or three qubits of quantum information and identifiable as the three basic symmetries of the Standard model, U(1), SU(2), and SU(3) accordingly. They refer to “empty qubits” (or the free variable of quantum information), i.e. those in which no point is chosen (recorded). The choice of a certain point violates those symmetries. It can be represented furthermore as the choice of a privileged reference frame (e.g. that of the Big Bang), which can be described exhaustively by means of 16 numbers (4 for position, 4 for velocity, and 8 for acceleration) independently of time, but in space-time continuum, and still one, 17th number is necessary for the mass of rest of the observer in it. The same 17 numbers describing exhaustively a privileged reference frame thus granted to be “zero”, respectively a certain violation of all the three symmetries of the Standard model or the “record” in a qubit in general, can be represented as 17 elementary wave functions (or classes of wave functions) after the bijection of natural and transfinite natural (ordinal) numbers in Hilbert arithmetic and further identified as those corresponding to the 17 elementary of particles of the Standard model. Two generalizations of the relevant concepts of general relativity are introduced: (1) “discrete reference frame” to the class of all arbitrarily accelerated reference frame constituting a smooth manifold; (2) a still more general principle of relativity to the general principle of relativity, and meaning the conservation of quantum information as to all discrete reference frames as to the smooth manifold of all reference frames of general relativity. Then, the bijective transition from an accelerated reference frame to the 17 elementary wave functions of the Standard model can be interpreted by the still more general principle of relativity as the equivalent redescription of a privileged reference frame: smooth into a discrete one. The conservation of quantum information related to the generalization of the concept of reference frame can be interpreted as restoring the concept of the ether, an absolutely immovable medium and reference frame in Newtonian mechanics, to which the relative motion can be interpreted as an absolute one, or logically: the relations, as properties. The new ether is to consist of qubits (or quantum information). One can track the conceptual pathway of the “ether” from Newtonian mechanics via special relativity, via general relativity, via quantum mechanics to the theory of quantum information (or “quantum mechanics and information”). The identification of entanglement and gravity can be considered also as a ‘byproduct” implied by the transition from the smooth “ether of special and general relativity’ to the “flat” ether of quantum mechanics and information. The qubit ether is out of the “temporal screen” in general and is depicted on it as both matter and energy, both dark and visible.

It is well known that Newton’s work on mechanics depended in a crucial way on the previous observations of Galileo. The key insight of Galileo was that one can analyze the motion of bodies using experiments and mathematical equations. One experimental observation that roughly emerges from this work in modern terms is that two objects of different mass that are simultaneously released from rest and allowed to fall under the influence of gravity through a vacuum should hit the ground at the same time (this is essentially what is called the equivalence principle in general relativity). In popular legend, it is said that Galileo tested this by dropping two balls of different masses from the Leaning Tower of Pisa and showing that they hit the ground at the same time, but the historical evidence suggests that this is unlikely as he seems to have experimented mostly with balls rolling down inclined slopes.

We present calculations of the rate of deflection of light per unit central angle φ in a set of stationary frames along the light path in the gravitational field of the sun and in an equivalent (except for curvature) set of accelerated frames in flat spacetime in a study designed to further understanding of the equivalence principle in general relativity. The rate of deflection is emphasized in keeping with the local restriction of the equivalence principle in a metric theory of gravitation. In the sequence of stationary frames it is possible to distinguish the contribution from acceleration with respect to local inertial frames (the equivalence principle) from the total rate of deflection which includes the effect of spacetime curvature. Our results indicate that the deflection rate as a function of central angle can be expressed as dα/dφ=−(m/R)(1+2q)cos3 φ, where m is the geometric mass of the sun, R is the minimum radius at φ=0, and q is a curvature tagging parameter such that with q=0 we have only the effect of acceleration and with q=1 we have the full Schwarzschild curvature.

The article presents a critical analysis of the Special and General Relativity Theories. Both theories are found incomplete and need a revision. The general relativity principle is currently understood as meaning that there is no preferred frame of reference and equations of motion must be Lorentz invariant. These interpretations should be modified to the form that there is no global preferred frame of reference, but there is a local preferred frame of reference given by the local gravitation field, and the equations of motion should be invariant in conformal mappings. The reason why there is a local preferred frame of reference is that the Lorentz transform does not define a valid time for a moving frame and therefore cannot give a set of frames of reference where there is no preferred local frame of reference. As light travels along geodesics of the gravitational field, this local preferred frame is given by the local gravitation field. Much of relativity theory remains after this revision, including the geometric interpretation of gravity, the equivalence principle, time and length dilation in gravitational fields and accelerating frames, time and length dilation between frames in constant relative motion if the local gravitation field is strong enough.

It is argued that the boundary conditions that require the line element to degenerate to ds2=c2dt2 at infinity in the spatial co-ordinates enforce Mach’s principle in general relativity. Physically this leads to space closure at co-ordinate infinity.