The paper has heuristic character. The conceptual frame, based on the assumption of quantum uncertainty only, has been formerly introduced in two papers [S. Tosto, Il Nuovo Cimento B, vol. 111, n.2, 1996 and S. Tosto, Il Nuovo Cimento D, vol. 18, n.12, 1996]. Instead of attempting to increase the accuracy of some existing computational model through a new kind of approximation, these papers acknowledge since the beginning the lack of deterministic information about the local properties of the constituent particles, considered random, unknown and unpredictable and thus ignored in principle. The leading idea is that the physical properties of quantum systems could be inferred merely considering the delocalization ranges of dynamical variables, rather than their local values. In effect, despite the agnostic character of the approach proposed, both papers show that the kind of physical information reachable reproduces exactly in all cases examined that obtained solving the pertinent wave equations. The concept of quantum uncertainty is further extended in the present paper to both space and time coordinates, considering thus a unique spacetime delocalization range and still discarding since the beginning the local values of the conjugate dynamical variables. The paper shows an unexpected wealth of information obtainable simply extending the concept of space uncertainty to that of spacetime uncertainty: the results are inherently consistent with that of the operator formalism of wave mechanics and with the basic postulates of special relativity, both inferred as corollaries. Moreover, even the gravity appears to be essentially a quantum phenomenon. The most relevant outcomes of special and general relativity are achieved as straightforward consequence of the space-time delocalization of particles using the simple quantum formalism first introduced in the early papers.
A one-to-one correspondence is established between linearized space-time metrics of general relativity and the wave equations of quantum mechanics. Also, the key role of boundary conditions in distinguishing quantum mechanics from classical mechanics will emerge naturally from the procedure. Finally, we will find that the methodology will enable us to introduce not only test charges but also test masses by means of gauges.
In previous papers I expounded non-linear Schrodingerist quantum mechanics as a solution of the Measurement Problem. Here I show that NLQM is compatible with Einstein's theory of General Relativity. The extension to curved space-times presumes adoption of "space-time wavefunctions" (sometimes called "multi-time wavefunctions") and some additional algebraic structure: a "bitensor" supplementing Einstein's metric tensor. This kind of matter may violate the Strong Energy Condition even without a mass term, possibly with implications for the formation of singularities within Black Holes.
In this paper, starting from vortices we are finally lead to a treatment of Fermions as Kerr-Newman type Black Holes wherein we identify the horizon at the particle's Compton wavelength periphery. A naked singularity is avoided and the singular processes inside the horizon of the Black Hole are identified with Quantum Mechanical effects within the Compton wavelength. Inertial mass, gravitation, electromagnetism and even QCD type interactions emerge from such a description including relative strengths and also other features like the anomalous gyromagnetic ratio, the discreteness of the charge,…
The purpose of this contribution is to provide an introduction for a general physics audience to the recent results of Emile Grgin that unifies quantum mechanics and relativity into the same mathematical structure. This structure is the algebra of quantions, a non-division algebra that is the natural framework for electroweak theory on curved space-time. Similar with quaternions, quantions preserve the core features of associativity and complex conjugation while giving up the unnecessarily historically biased property of division. Lack of division makes possible structural unification with rel…
Using exact solutions, we show that it is in principle possible to regard waves and particles as representations of the same underlying geometry, thereby resolving the problem of wave-particle duality.
A quantum-mechanical Hamiltonian with a gravitational potential is derived in the framework of local times. This Hamiltonian is the one used by E. H. Lieb (Bull. Amer. Math. Soc. 22(1990), 1-49) in his explanation of stability and instability of cold stars. Our procedure deriving the Hamiltonian is based on an analysis of the process of the observation with the background of the notion of quantum-mechanical local times compatible with general theory of relativity.
The groupoid approach to noncommutative unification of general relativity with quantum mechanics is compared with the canonical gravity quantization. It is shown that by restricting the corresponding noncommutative algebra to its (commutative) subalgebra, which determines the space-time slicing, an algebraic counterpart of superspace (space of 3-metrics) can be obtained. It turns out that when this space-time slicing emerges the universe is already in its commutative regime. We explore the consequences of this result.
In 5D, I take the metric in canonical form and define causality by null-paths. Then spacetime is modulated by a factor equivalent to the wave function, and the 5D geodesic equation gives the 4D Klein-Gordon equation. These results effectively show how general relativity and quantum mechanics may be unified in 5D.
In this paper a one to one correspondence is established between space-time metrics of general relativity and the wave equations of quantum mechanics. This is done by first taking the square root of the metric associated with a space and from there, passing directly to a corresponding expression in the dual space. It is shown that in the case of a massless particle, Maxwell's equation for a photon follows while in the case of a particle with mass, Dirac's equation results as a first approximation. Moreover, this one to one correspondence suggests a natural explanation of wave-particle duality.…
The Generalized Uncertainty Principle (GUP) has been directly applied to the motion of (macroscopic) test bodies on a given space-time in order to compute corrections to the classical orbits predicted in Newtonian Mechanics or General Relativity. These corrections generically violate the Equivalence Principle. The GUP has also been indirectly applied to the gravitational source by relating the GUP modified Hawking temperature to a deformation of the background metric. Such a deformed background metric determines new geodesic motions without violating the Equivalence Principle. We point out her…
We present an elastic constitutive model of gravity where we identify physical space with the mid-hypersurface of an elastic hyperplate called the "cosmic fabric" and spacetime with the fabric's world volume. Using a Lagrangian formulation, we show that the fabric's behavior as derived from Hooke's Law is analogous to that of spacetime per the Field Equations of General Relativity. The study is conducted in the limit of small strains, or analogously, in the limit of weak and nearly static gravitational fields. The Fabric's Lagrangian outside of inclusions is shown to have the same form as the …
The Poincaré-Snyder relativity was introduced in an earlier paper of ours as an extended form of Einstein relativity obtained by appropriate limiting setting of the full Quantum Relativity. The latter, with fundamental constants $\hbar$ and $G$ built into the symmetry, is supposed to be the relativity of quantum space-time. Studying the mechanics of Poincaré-Snyder relativity is an important means to get to confront the great challenge of constructing the dynamics of Quantum Relativity. The mechanics will also be of interest on its own, plausibly yielding prediction accessible to experiments. …
In a previous article a relationship was established between the linearized metrics of General Relativity associated with geodesics and the Dirac Equation of quantum mechanics. In this paper the extension of that result to arbitrary curves is investigated. The Dirac equation is derived and shown to be related to the Lie derivative of the momentum along the curve. In addition,the equations of motion are derived from the Hamilton-Jacobi equation associated with the metric and the wave equation associated with the Hamiltonian is then shown not to commute with the Dirac operator. Finally, the Maxw…
The Nelson stochastic mechanics is derived as a consequence of the basic physical principles such as the principle of relativity of observations and the invariance of the action quantum. The unitary group of quantum mechanics is represented as the transformations of the systems of perturbing devices. It is argued that the physical spacetime has a stochastic nature, and that quantum mechanics in Nelson's formulation correctly describes this stochasticity.
Some of the strategies which have been put forward in order to deal with the inconsistency between quantum mechanics and special relativity are examined. The EPR correlations are discussed as a simple example of quantum mechanical macroscopic effects with spacelike separation from their causes. It is shown that they can be used to convey information, whose reliability can be estimated by means of Bayes' theorem. Some of the current reasons advanced to deny that quantum mechanics contradicts special relativity are refuted, and an historical perspective is provided on the issue.
It is shown that, with some reasonable assumptions, the theory of general relativity can be made compatible with quantum mechanics by using the field equations of general relativity to construct a Robertson-Walker metric for a quantum particle so that the line element of the particle can be transformed entirely to that of the Minkowski spacetime, which is assumed by a quantum observer, and the spacetime dynamics of the particle described by a Minkowski observer takes the form of quantum mechanics. Spacetime structure of a quantum particle may have either positive or negative curvature. However…
I illustrate a simple hamiltonian formulation of general relativity, derived from the work of Esposito, Gionti and Stornaiolo, which is manifestly 4d generally covariant and is defined over a finite dimensional space. The spacetime coordinates drop out of the formalism, reflecting the fact that they are not related to observability. The formulation can be interpreted in terms of Toller's reference system transformations, and provides a physical interpretation for the spinnetwork to spinnetwork transition amplitudes computable in principle in loop quantum gravity and in the spin foam models.
We show that the local and deterministic mode of description is not only in conflict with the quantum theory, but also with relativity. We argue that elementary relativistic properties of spacetime lead to the emergence of a non-deterministic quantum-mechanical picture involving quantum superpositions and complex probability amplitudes.
We derive fundamental limits on measurements of position, arising from quantum mechanics and classical general relativity. First, we show that any primitive probe or target used in an experiment must be larger than the Planck length, $l_P$. This suggests a Planck-size {\it minimum ball} of uncertainty in any measurement. Next, we study interferometers (such as LIGO) whose precision is much finer than the size of any individual components and hence are not obviously limited by the minimum ball. Nevertheless, we deduce a fundamental limit on their accuracy of order $l_P$. Our results imply a {\i…
Using ideas from continuum mechanics we construct a theory of gravity. We show that this theory is equivalent to Einstein's theory of general relativity; it is also a much faster way of reaching general relativity than the conventional route. Our approach is simple and natural: we form a very general model and then apply two physical assumptions supported by experimental evidence. This easily reduces our construction to a model equivalent to general relativity. Finally, we suggest a simple way of modifying our theory to investigate non-standard space-time symmetries.
We construct a model unifying general relativity and quantum mechanics in a broader structure of noncommutative geometry. The geometry in question is that of a transformation groupoid given by the action of a finite group G on a space E. We define the algebra of smooth complex valued functions on the groupoid, with convolution as multiplication, in terms of which the groupoid geometry is developed. Owing to the fact that the group G is finite the model can be computed in full details. We show that by suitable averaging of noncommutative geometric quantities one recovers the standard space-time…
The main fundamental principles characterizing the vacuum field structure are formulated, the modeling of the related vacuum medium and charged point particle dynamics by means of devised field theoretic tools are analyzed. The Maxwell electrodynamic theory is revisited and newly derived from the suggested vacuum field structure principles, the classical special relativity theory relationship between the energy and the corresponding point particle mass is revisited and newly obtained. The Lorentz force expression with respect to arbitrary non-inertial reference frames is revisited and discusse…
It is shown that the Lax pair equation dL/dt = [L,A] can be given a neat tensorial interpretation for finite-dimensional quadratic Hamiltonians. The Lax matrices L and A are shown to arise from third rank tensors on the configuration space. The second Lax matrix A is related to a connection which characterizes the Hamiltonian system. The Toda lattice system is used to motivate the definition of the Lax pair tensors. The possible existence of solutions to the Einstein equations having the Lax pair property is discussed.
We discuss a way to obtain the doubly special relativity kinematical rules (the deformed energy-momentum relation and the nonlinear Lorentz transformations of momenta) starting from a singular Lagrangian action of a particle with linearly realized SO(1,4) symmetry group. The deformed energy-momentum relation appears in a special gauge of the model. The nonlinear transformations of momenta arise from the requirement of covariance of the chosen gauge.
The main fundamental principles characterizing the vacuum field structure are formulated and the modeling of the related vacuum medium and charged point particle dynamics by means of devised field theoretic tools are analyzed. The work is devoted to studying the vacuum structure, special relativity, electrodynamics of interacting charged point particles and quantum mechanics, and is a continuation of \cite{BPT,BRT1}. Based on the vacuum field theory no-geometry approach, the Lagrangian and Hamiltonian reformulation of some alternative classical electrodynamics models is devised. The Dirac type…
In relativity, two simultaneous events at two different places are not simultaneous for observers in different Lorentz frames. In the Einstein-Podolsky-Rosen experiment, two simultaneous measurements are taken at two different places. Would they still be simultaneous to observers in moving frames? It is a difficult question, but it is still possible to study this problem in the microscopic world. In the hydrogen atom, the uncertainty can be considered to be entirely associated with the ground-state. However, is there an uncertainty associated with the time-separation variable between the proto…
An experiment in Low Earth Orbit (LEO) is proposed to measure components of the Riemann curvature tensor using atom interferometry. We show that the difference in the quantum phase $Δϕ$ of an atom that can travel along two intersecting geodesics is given by $mR_{0i0j}/\hbar$ times the spacetime volume contained within the geodesics. Our expression for $Δϕ$ also holds for gravitational waves in the long wavelength limit.
We develop a general formalism to determine the statistical equilibrium states of self-gravitating systems in general relativity and complete previous works on the subject. Our results are valid for an arbitrary form of entropy but, for illustration, we explicitly consider the Fermi-Dirac entropy for fermions. The maximization of entropy at fixed mass-energy and particle number determines the distribution function of the system and its equation of state. It also implies the Tolman-Oppenheimer-Volkoff equations of hydrostatic equilibrium and the Tolman-Klein relations. Our paper provides all th…
One of the conceptual tensions between quantum mechanics (QM) and general relativity (GR) arises from the clash between the spatial nonseparability} of entangled states in QM, and the complete spatial separability of all physical systems in GR, i.e., between the nonlocality implied by the superposition principle, and the locality implied by the equivalence principle. Possible experimental consequences of this conceptual tension will be discussed for macroscopically entangled, coherent quantum fluids, such as superconductors, superfluids, atomic Bose-Einstein condensates, and quantum Hall fluid…
In Gen. Rel. Grav. (36, 111-126 (2004); in press, gr-qc/0410010) we have proposed a model unifying general relativity and quantum mechanics based on a noncommutative geometry. This geometry was developed in terms of a noncommutative algebra A defined on a transformation groupoid given by the action of a group G on a space E. Owing to the fact that G was assumed to be finite it was possible to compute the model in full details. In the present paper we develop the model in the case when G is a noncompact group. It turns out that also in this case the model works well. The case is important since…
The existence of singularities in a closed FRW universe depends on the assumption that general relativity is valid for distances less than the Planck length. However, stationary state wave functions of the Schrodinger equation for a closed radiation-dominated FRW universe derived by Elbaz et al (General Relativity and Gravitation 29, 481, 1997) are zero at zero radius of curvature. Thus, even if general relativity is assumed valid at distances less than the Planck length, quantum mechanics seems to forbid singularities in a closed FRW universe.
One of the conceptual tensions between quantum mechanics (QM) and general relativity (GR) arises from the clash between the spatial nonseparability of entangled states in QM, and the complete spatial separability of all physical systems in GR, i.e., between the nonlocality implied by the superposition principle, and the locality implied by the equivalence principle. Possible experimental consequences of this conceptual tension will be discussed for macroscopically entangled, coherent quantum fluids, such as superconductors, superfluids, atomic Bose-Einstein condensates, and quantum Hall fluids…
First laws of black hole mechanics, or thermodynamics, come in a variety of different forms. In this paper, from a purely post-Newtonian (PN) analysis, we obtain a first law for binary systems of point masses moving along an exactly circular orbit. Our calculation is valid through 3PN order and includes, in addition, the contributions of logarithmic terms at 4PN and 5PN orders. This first law of binary point-particle mechanics is then derived from first principles in general relativity, and analogies are drawn with the single and binary black hole cases. Some consequences of the first law are …
A dynamical, non-Euclidean spacetime geometry in general relativity theory implies the possibility of gravitational radiation. Here we explore novel methods of detecting such radiation from astrophysical sources by means of matter-wave interferometers (MIGOs), using atomic beams emanating from supersonic atomic sources that are further cooled and collimated by means of optical molasses. While the sensitivities of such MIGOs compare favorably with LIGO and LISA, the sizes of MIGOs can be orders of magnitude smaller, and their bandwidths wider. Using a pedagogical approach, we place this problem…
The generation, as well as the detection, of gravitational radiation by means of charged superfluids is considered. One example of such a "charged superfluid" consists of a pair of Planck-mass-scale, ultracold "Millikan oil drops," each with a single electron on its surface, in which the oil of the drop is replaced by superfluid helium. When levitated in a magnetic trap, and subjected to microwave-frequency electromagnetic radiation, a pair of such "Millikan oil drops" separated by a microwave wavelength can become an efficient quantum transducer between quadrupolar electromagnetic and gravita…
This note is to bring to the reader's attention the fact that general relativity and quantum mechanics differ from each other in one main aspect. General relativity is based on the diffeomorphism covariant formulation of the laws of physics while quantum mechanics is constructed such that its fundamental laws remain invariant to a change of topology. It is the goal of this paper to show that in order to obtain a complete description of quantum gravity one has to extend the principle of diffeomorphism invariance from general relativity in the sense of quantum mechanics i.e. the laws of physics …
Gravitation, according to General Relativity, is an attribute of space-time's geometry and hence not a force in the Newtonian sense. This is a consequence of Einstein's equivalence principle, which so far passed all experimental tests with high precision. However, the search for possible violations continues, for they, as the case may be, are expected to point towards more fundamental theoretical extensions of General Relativity. In particular, it is expected that useful insights are gained by studying the interaction between gravitational fields and genuine quantum-mechanical systems, like at…
This work explores a classical mechanical theory under two further assumptions: (a) there is a universal dry friction force (Aristotelian mechanics), and (b) the variation of the mass of a body due to wear is proportional to the work done by the friction force on the body (Reye's hypothesis). It is shown that mass depends on velocity as in Special Relativity, and that the velocity is constant for a particular characteristic value. In the limit of vanishing friction the theory satisfies a relativity principle as bodies do not decelerate and, therefore, the absolute frame becomes unobservable. H…
We study the statistical mechanics of classical self-gravitating systems confined within a box of radius $R$ in general relativity. It has been found that the caloric curve $T_{\infty}(E)$ has the form of a double spiral whose shape depends on the compactness parameter $ν=GNm/Rc^2$. The double spiral shrinks as $ν$ increases and finally disappears when $ν_{\rm max}=0.1764$. Therefore, general relativistic effects render the system more unstable. On the other hand, the cold spiral and the hot spiral move away from each other as $ν$ decreases. Using a normalization $Λ=-ER/GN^2m^2$ and $η=GNm^2/R…
The classical and continuum limit of a quantum gravitational setting could lead, at mesoscopic regimes, to a very different notion of geometry w.r.t. the pseudo-Riemannian one of special and general relativity. A possible way to characterize this modified space-time notion is by a momentum dependent metric, in such a way that particles with different energies could probe different spacetimes. Indeed, doubly special relativity theories, deforming the special relativistic kinematics while maintaining a relativity principle, have been understood within a geometrical context, by considering a curv…
Based on the doubly special relativity we find a new type of generalized uncertainty principle (GUP) where the coordinate remain unaltered at the high energy while the momentum is deformed at the high energy so that it may be bounded from the above. For this GUP, we discuss some quantum mechanical problems in one dimension such as box problem, momentum wave function, and harmonic oscillator problem.
Inspired in metamaterials, we present a covariant mechanics for particles in Kleinian spacetime and show some of its effects, such as time contraction and length dilatation. We present the new expressions for relativistic momentum and energy for a point-like particle. To illustrate the new mechanics, we describe the particle motion under a uniform Newtonian gravitational field. We also revisit the free spin-half particle problem in Kleinian spacetime, discuss some quantum implications, like the constraint on the dispersion relation for Weyl fermions, and adapt a metamaterial analog system to K…
A system of two interacting protofields with generic parameters is unstable with respect to unceasing cycles of nonlinear squeeze (reduction) to randomly chosen centres and reverse extension which form the causally probabilistic process of quantum beat observed as elementary particle (quant-ph/9902015). Here we show that the emerging wave-particle duality, space, and time lead to the equations of special relativity and quantum mechanics thus providing their causal extension and unification. Relativistic inertial mass (energy) is universally defined as the temporal rate (frequency) of the chaot…
This paper studies the nature of fractional linear transformations in a general relativity context as well as in a quantum theoretical framework. Two features are found to deserve special attention: the first is the possibility of separating the limit-point condition at infinity into loxodromic, hyperbolic, parabolic and elliptic cases. This is useful in a context in which one wants to look for a correspondence between essentially self-adjoint spherically symmetric Hamiltonians of quantum physics and the theory of Bondi-Metzner-Sachs transformations in general relativity. The analogy therefore…
General relativity has two fundamental problems that render it unsuitable for tackling the gravitational field's quantization. The first problem is the lack of a genuine gravitational variable representing gravitation only, inertial forces apart. The second problem is its incompatibility with quantum mechanics, a problem inherited from the more fundamental conflict of special relativity with quantum mechanics. A procedure to overcome these difficulties is outlined, which amounts to replacing general relativity with its teleparallel equivalent and the Poincaré-invariant special relativity with …
The arena normally used in black holes thermodynamics was recently generalized to incorporate a broad class of physically interesting situations. The key idea is to replace the notion of stationary event horizons by that of `isolated horizons.' Unlike event horizons, isolated horizons can be located in a space-time quasi-locally. Furthermore, they need not be Killing horizons. In particular, a space-time representing a black hole which is itself in equilibrium, but whose exterior contains radiation, admits an isolated horizon. In spite of this generality, the zeroth and first laws of black hol…
In classical Hamiltonian theories, entropy may be understood either as a statistical property of canonical systems, or as a mechanical property, that is, as a monotonic function of the phase space along trajectories. In classical mechanics, there are theorems which have been proposed for proving the non-existence of entropy in the latter sense. We explicate, clarify and extend the proofs of these theorems to some standard matter (scalar and electromagnetic) field theories in curved spacetime, and then we show why these proofs fail in general relativity; due to properties of the gravitational H…
Having started with the general formulation of the quantum theory of the real scalar field (QFT) in the general Riemannian space--time $ V_{1,3} $, the general--covariant quasinonrelativistic quantum mechanics of a point-like spinless particle in $ V_{1,3} $ is constructed. To this end, for any normal geodesic 1+3--foliation of $ V_{1,3} $, a space $Φ^-$ of asymptotic in $c^{-1}$ solutions of the field equation is specified, which can be mapped to a space $Ψ$ of solutions of a Schrödinger equation with an (asymptotically) Hermitean hamiltonian and the Born probabilistic interpretation of the v…
Let m(i) be the mass of i-th planet and M be the Solar mass. From astronomical data it is known that ratios r(i)=m(i)/(m(i)+M) are of order 10^(-3)-10^(-6) for all planets. The same is true for all satellites of heavy planets. These results suggest that Einstein's treatment of Mercury dynamics can be extended to almost any object in the Solar System. This fact does not explain the existing order in the Solar System. Indeed, all planets lie in the same (Sun's equatorial) plane and move in the same direction coinciding with that for the rotating Sun.The same is true for regular satellites of hea…
The rules of quantum mechanics require a time coordinate for their formulation. However, a notion of time is in general possible only when a classical spacetime geometry exists. Such a geometry is itself produced by classical matter sources. Thus it could be said that the currently known formulation of quantum mechanics pre-assumes the presence of classical matter fields. A more fundamental formulation of quantum mechanics should exist, which avoids having to use a notion of time. In this paper we discuss as to how such a fundamental formulation could be constructed for single particle, non-re…
One of the conceptual tensions between quantum mechanics (QM) and general relativity (GR) arises from the clash between the spatial nonseparability of entangled states in QM, and the complete spatial separability of all physical systems in GR, i.e., between the nonlocality implied by the superposition principle, and the locality implied by the equivalence principle. Experimental consequences of this conceptual tension will be explored for macroscopically coherent quantum fluids, such as superconductors, superfluids, and atomic Bose-Einstein condensates (BECs), subjected to tidal and Lense-Thir…
The "external" or "bulk" motion of extended bodies is studied in general relativity. Compact material objects of essentially arbitrary shape, spin, internal composition, and velocity are allowed as long as there is no direct (non-gravitational) contact with other sources of stress-energy. Physically reasonable linear and angular momenta are proposed for such bodies and exact equations describing their evolution are derived. Changes in the momenta depend on a certain "effective metric" that is closely related to a non-perturbative generalization of the Detweiler-Whiting R-field originally intro…
Some concepts of real and complex projective geometry are applied to the fundamental physical notions that relate to Minkowski space and the Lorentz group. In particular, it is shown that the transition from an infinite speed of propagation for light waves to a finite one entails the replacement of a hyperplane at infinity with a light cone and the replacement of an affine hyperplane - or rest space - with a proper time hyperboloid. The transition from the metric theory of electromagnetism to the pre-metric theory is discussed in the context of complex projective geometry, and ultimately it is…
