Based on the canonical quantization of $d>2$ dimensional general relativity (GR) via the Dirac constraint formalism (also termed as `constraint quantization'), we propose the loss of covariance as a fundamental property of the theory. This breakdown occurs for the first-order Einstein Hilbert action, whereby loss of diffeomorphism invariance, besides first class constraints, second class constriants also exist leading to non-standard ghost fields which render the path integral non-covariant. We also attempt, for the first time, the canonical quantization via calculation of the path integral for the equivalent Hamiltonian formulation of GR for which only first class constraints exist. However, loss of covariance still occurs in this action due to loss of diffeomorphism invariance and structures arising from non-covariant constraints in the path integral. In contrast, we find that covariance as a symmetry is restored and quantization with perturbative calculations is possible in the weak limit of the gravitational field of these actions. Hence, we firstly establish, for the first time, that the breakdown in space-time is a property of GR itself as its limitation, indicating it to be an Effective Field Theory (EFT). We further propose that the breakdown of space-time occurs as a non-perturbative feature of GR in the strong field limit of the theory. Besides GR, we also note that covariance is preserved when constraint quantization is conducted for non-Abelian gauge theories, such as the Yang-Mills theory. These findings are novel from a canonical gravity formalism and EFT approach, and are consistent with GR singularity theorems, yet are general and extend them, as the singularity theorems indicate breakdown at a strong field limit of GR in black holes.
We give a geometric criterion for the breakdown of an Einstein vacuum space-time foliated by a constant mean curvature, or maximal, foliation. More precisely we show that the foliated space-time can be extended as long as the the second fundamental form and the first derivatives of the logarithm of the lapse of the foliation remain uniformly bounded. No restrictions on the size of the initial data are made.
It is proposed that the event horizon of a black hole is a quantum phase transition of the vacuum of space-time analogous to the liquid-vapor critical point of a bose fluid. The equations of classical general relativity remain valid arbitrarily close to the horizon yet fail there through the divergence of a characteristic coherence length. The integrity of global time, required for conventional quantum mechanics to be defined, is maintained. The metric inside the event horizon is different from that predicted by classical general relativity and may be de Sitter space. The deviations from class…
This is the second of three papers on Conformal General Relativity (CGR). The conformal group is introduced here as the invariance group of the partial order of causal events in $n$D spacetime. Its general structure, discrete symmetries and field representations are described in detail. The spontaneous breakdown of conformal symmetry is then discussed and the role played by a ghost scalar field and a physical scalar field in 4D spacetime are evidenced. Kinematic--, conformal-- and proper--time hyperbolic coordinates are introduced in a negatively curved Milne spacetime for the purpose of provi…
Deformed Special Relativity (DSR) is a generalization of Special Relativity based on a deformed Minkowski space, i.e. a four-dimensional space-time with metric coefficients depending on the energy. We show that, in the DSR framework, it is possible to derive the value of the electron mass from the space-time geometry via the experimental knowledge of the parameter of local Lorentz invariance breakdown, and of the Minkowskian threshold energy $E_{0,em}$ for the electromagnetic interaction.
This is the first of three papers on Conformal General Relativity (CGR), which differs from Einstein's General Relativity (GR) in that it requires action--integral invariance under local scale transformations in addition to general coordinate transformations. The theory is here introduced in the semiclassical approximation as a preliminary approach to a quantum theoretical implementation. The idea of a conformal--invariant extension of GR was introduced by Weyl in 1919. For several decades it had little impact, as CGR implies that all fields are massless. Today this does not appear to be an un…
We investigate the importance of local gravity during preheating, the non-linear dynamics that may be responsible for starting the process of reheating the universe after inflation. We introduce three numerical methods that study a simple preheating scenario while relaxing gravitational assumptions, culminating in studying the process in full numerical relativity. We confirm that perturbation theory is no longer valid when one considers modes whose wavelengths are comparable to the size of the horizon at the end of inflation; however, this breakdown does not necessarily lead to a breakdown of …
It is shown that passive gravitational mass operator of a hydrogen atom in the post-Newtonian approximation of the general relativity does not commute with its energy operator, taken in the absence of gravitational field. Nevertheless, the equivalence between the expectation values of passive gravitational mass and energy is shown to survive at a macroscopic level for stationary quantum states. Breakdown of the equivalence between passive gravitational mass and energy at a microscopic level for stationary quantum states can be experimentally detected by studying unusual electromagnetic radiati…
Spacetime singularities in general relativity are commonly thought to be problematic, in that they signal a breakdown in the theory. We address the question of how to interpret this breakdown, restricting our attention to classical considerations (though our work has ramifications for more general classical metric theories of gravity, as well). In particular, we argue for a new claim: spacetime singularities in general relativity signal indeterminism. The usual manner in which indeterminism is formulated for physical theories can be traced back to Laplace. This formulation is based on the non-…
The simplest quantum composite body, a hydrogen atom, is considered in the presence of a weak external gravitational field. We define an operator for the passive gravitational mass of the atom in the post-Newtonian approximation of the general relativity and show that it does not commute with its energy operator. Nevertheless, the equivalence between the expectation values of the mass and energy is shown to survive at a macroscopic level for stationary quantum states. Breakdown of the equivalence between passive gravitational mass and energy at a microscopic level for stationary quantum states…
It is shown that weight operator of a composite quantum body in a weak external gravitational field in the post-Newtonian approximation of the General Relativity does not commute with its energy operator, taken in the absence of the field. Nevertheless, the weak equivalence between the expectations values of weight and energy is shown to survive at a macroscopic level for stationary quantum states for the simplest composite quantum body - a hydrogen atom. Breakdown of the weak equivalence between weight and energy at a microscopic level for stationary quantum states can be experimentally detec…
Asymptotic Simplicity and Scattering in General Relativity from Quantum Field Theory
Institut de Physique Théorique, CEA, CNRS, Université Paris-Saclay（巴黎萨克雷大学理论物理研究所）；Institute for Advanced Study（普林斯顿高等研究院）；ETH Zurich（苏黎世联邦理工学院）；Pennsylvania State University（宾夕法尼亚州立大学）；Fudan University（复旦大学）
Cosmic censorship is known to fail in some well-controlled phenomena, calling into question the predictive power of General Relativity and opening up the possibility of observing Planck-scale physics. We propose that the cosmic censorship conjecture can be amended so that its spirit prevails. Naked singularities that, classically, have zero mass are allowed. Physically, these are Planck-sized `black holes', which evaporate in a few Planck times. General Relativity fails only for a tiny interval in time, to then quickly regain control in a Miltonian evolution that returns us to the predictive p…
One of the most difficult questions in present-day physics concerns a fundamental theory of space, time, and matter that incorporates a consistent quantum description of gravity. There are various theoretical approaches to such a quantum-gravity theory. Nevertheless, experimental progress is hampered in this research field because many models predict deviations from established physics that are suppressed by some power of the Planck scale, which currently appears to be immeasurably small. However, tests of relativity theory provide one promising avenue to overcome this phenomenological obstacl…
We discuss how the perturbative particle paradigm fails in certain background with space-like singularity but asymptotically flat which should admit a S-matrix. The Feynman approach relies on the interaction picture. This approach means that we can interpret interactions as exchanges of particles. Particles are the modes of the quadratic part of the Lagrangian. In certain backgrounds with space-like singularity the interaction Hamiltonian is well defined but the perturbative expansion of the evolution operator through the singularity and the perturbative $S$ matrix do not exist. On the other h…
Massless interacting scalar fields in de Sitter space have long been known to experience large fluctuations over length scales larger than Hubble distances. A similar situation arises in condensed matter physics in the vicinity of a critical point, and in this better-understood situation these large fluctuations indicate the failure in this regime of mean-field methods. We argue that for non-Goldstone scalars in de Sitter space, these fluctuations can also be interpreted as signaling the complete breakdown of the semi-classical methods widely used throughout cosmology. By power-counting the in…
Black holes arising in the context of scalar-tensor gravity theories, where the scalar field is non-minimally coupled to the curvature term, have zero surface gravity. Hence, it is generally stated that their Hawking temperature is zero, irrespectivelly of their gravitational and scalar charges. The proper analysis of the Hawking temperature requires to study the propagation of quantum fields in the space-time determined by these objects. We study scalar fields in the vicinity of the horizon of these black holes. It is shown that the scalar modes do not form an orthonormal set. Hence, the Hilb…
Einstein's theory of general relativity models the physical universe using spacetimes which satisfy Einstein's gravitational field equations. To date, Einstein's theory has been enormously successful in modeling observed gravitational phenomena, both at the astrophysical and the cosmological levels. The collection of spacetime solutions of Einstein's equations which have been effectively used for modeling the physical universe is a very small subset of the full set of solutions. Among this larger set, there are many spacetimes in which strange phenomena related to time are present: There are s…
It is shown that the commonly accepted definition for the Casimir scalar operators of the Poincare group does not satisfy the properties of Casimir invariance when applied to the non-inertial motion of elementary particles while in the presence of external gravitational and electromagnetic fields, where general curvilinear co-ordinates are used to describe the momentum generators within a Fermi normal co-ordinate framework. Specific expressions of the Casimir scalar properties are presented for spin-1/2 to spin-2 particles inclusive. While the Casimir scalar for linear momentum remains a Loren…
While General Relativity ranks undoubtedly among the best physics theories ever developed, it is also among those with the most striking implications. In particular, General Relativity admits solutions which allow faster than light motion and consequently time travel. Here we shall consider a "pre-emptive" chronology protection mechanism that destabilises superluminal warp drives via quantum matter back-reaction and hence forbids even the conceptual possibility to use these solutions for building a time machine. This result will be considered both in standard quantum field theory in curved spa…
When it comes to performing thought experiments with black holes, Einstein-Bohr like discussions have to be re-opened. For instance one can ask what happens to the quantum state of a black hole when the wave function of a single ingoing particle is replaced by an other one that is orthogonal to the first, while keeping the total energy and momentum unaffected. Observers at $t\rightarrow\infty$ will not notice any difference, or so it seems in certain calculational schemes. If one argues that this cannot be correct for the complete theory because a black hole should behave in accordance with co…
This paper explores the properties of the Pauli-Lubanski spin vector for the general motion of spin-1/2 particles in curved space-time. Building upon previously determined results in flat space-time, it is shown that the associated Casimir scalar for spin possesses both gravitational contributions and frame-dependent contributions due to non-inertial motion, where the latter represents a possible quantum violation of Lorentz invariance that becomes significant at the Compton wavelength scale. When applied to muon decay near the event horizon of a microscopic Kerr black hole, it is shown that i…
It is argued that de Sitter space-times might be solutions of entangled relativity once the quantum trace anomaly from matter fields in curved space-times is taken into account. This hypothesis would be an elegant solution to the acceleration of the expansion of the universe within the rigid framework of entangled relativity.
We summarize some of the results we obtained in arXiv:1006.2126 (Physical Review Letters 106, 071301), arXiv:1102.4637 (Physics Letters B 700, 150-156) and in arXiv:1107.3334, giving complementary characterizations of the relativity of spacetime locality that affects certain Planck-scale-deformed phase-space descriptions of free particles.
In this article, we review and discuss different aspects of stability and genericity of some properties of space-times which occur in various contexts in the General Theory of Relativity. We also give argument supporting the conclusion that Linearization Stability is a generic property if we restrict space-times to the class of those which admit compact spacelike constant mean curvature hypersurfaces.
General relativity is incomplete because it cannot describe quantum effects of space-time. The complete theory of quantum gravity is not yet known and to date no observational evidence exists that space-time is quantized. However, in most approaches to quantum gravity the space-time manifold of general relativity is only an effective limit that, among other things like higher curvature terms, should receive corrections stemming from space-time defects. We here develop a modification of general relativity that describes local space-time defects and solve the Friedmann equations. From this, we o…
We study in this paper different topos-theoretical approaches to the problem of construction of General Theory of Relativity. In general case the resulting space-time theory will be non-classical, different from that of the usual Einstein theory of space-time. This is a new theory of space-time, created in a purely logical manner. Four possibitities are investigated: axiomatic approach to causal theory of space-time, the smooth toposes as a models of Theory of Relativity, Synthetic Theory of Relativity, and space-time as Grothendieck topos.
Let $\M_*=\cup_{t\in [t_0, t_*)} Σ_t$ be a part of vacuum globally hyperbolic space-time $(\bM, \bg)$, foliated by constant mean curvature hypersurfaces $Σ_t$ with $t_0<t_*<0$. We show that the foliation can be extended beyond $t_*$ if the second fundamental form $k$ and the lapse function $n$ satisfy $$ \int_{t_0}^{t_*}(\|k\|_{L^\infty(Σ_t)}+\|\nab \log n\|_{L^\infty(Σ_t)}) dt <\infty. $$ This improves the existing breakdown criteria for Einstein vacuum equations.
Commuting and noncommuting space-time coordinates in a class of deformed special relativity theories are investigated. Their momentum space representation, transformation behaviour, space-time algebra, invariants and the corresponding field theories are derived. Several coordinates require as a novel feature the introduction of deformed plane waves.
It has long been considered that conformal superspace is the natural configuration space for canonical general relativity. However, this was never definitively demonstrated. We have found that the standard conformal method of solving the Einstein constraints has an unexpected extra symmetry. This allows us to complete the project. We show that given a point and a velocity in conformal superspace, the Einstein equations generate a unique curve in conformal superspace.
General Relativity in the noncommutative spacetime of ${\cal M}^4 \times Z_2 \times Z_2$ is constructed based on a new type of fermions in addition to the known chiral quark-leptons. While this theory has a finite physical spectrum, all the known interactions of Nature and Higgs fields can be derived from it as components of gravity with only four free parameters.
Barbour, Foster and Ó Murchadha have recently developed a new framework, called here {\it{the 3-space approach}}, for the formulation of classical bosonic dynamics. Neither time nor a locally Minkowskian structure of spacetime are presupposed. Both arise as emergent features of the world from geodesic-type dynamics on a space of 3-dimensional metric--matter configurations. In fact gravity, the universal light cone and Abelian gauge theory minimally coupled to gravity all arise naturally through a single common mechanism. It yields relativity -- and more -- without presupposing relativity. This…
We argue that space-time geometry is not absolute with respect to the frame of reference being used. The space-time metric differential form $ds$ in noninertial frames of reference (NIFR) is caused by the properties of the used frames in accordance with the Berkley - Leibnitz - Mach - Poincaré ideas about relativity of space and time . It is shown that the Sagnac effect and the existence of inertial forces in NIFR can be considered from this point of view. An experimental test is proposed.
The black hole entropy formula applied to local Rindler horizon at each spacetime point has been used in the literature to derive the Einstein field equation as an equation of state of a thermodynamical system of spacetime. In the present paper we argue that due to the key role of causal structure and discrete spacetime in this approach the natural framework is unimodular relativity rather than general relativity. It is shown that the equation of state is trace free unimodular relativity field equation that uniquely determines only the traceless stress tensor. Recent generalization to nonequil…
We show how the Minkowskian space-time emerges from a topologically homogeneous causal network, presenting a simple analytical derivation of the Lorentz transformations, with metric as pure event-counting. The derivation holds generally for d=1 space dimension, however, it can be extended to d>1 for special causal networks.
The classifying space of inertial reference frames in special relativity is naturally hyperbolic. There is a remarkable interplay between central elements of hyperbolic geometry and those of special relativity -- which, to a certain extent, have already been observed in the past -- that we present and further discuss in the paper. We aim at a geometrization of special relativity at the level of kinematic space by giving to physical concepts/phenomena purely geometric definitions/descriptions. In this way, the differences between special relativity and classical mechanics can be seen as a manif…
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.
It is widely expected that generic black holes have a non-empty but weakly singular Cauchy horizon, due to mass inflation. Indeed this has been proven by the author in the spherical collapse of a charged scalar field, under decay assumptions of the field in the black exterior which are conjectured to be generic. A natural question then arises: can this weakly singular Cauchy horizon close off the space-time, or does the weak null singularity necessarily "break down", giving way to a different type of singularity? The main result of this paper is to prove that the Cauchy horizon cannot ever "cl…
In the present work, metrics which lead to projected closed orbits are found by comparing the relativistic differential equation of orbits with the corresponding classical differential equation. Physical and geometrical properties of these peculiar spacetimes are derived and discussed. It is also shown that some of these spacetimes belong to the broader class of the Bertrand spacetimes.
