We present a new method for embedding a causal set into an interval of Minkowski spacetime. The method uses spacetime volumes for causally related elements to define causal set analogs of Minkowski inner products. These are used to construct matrices of inner products which are then factored using the singular value decomposition to give coordinates in Minkowski spacetime. Results are presented showing good quality embeddings into Minkowski spacetime for dimensions d=2,3,4. The method applies in any dimension and does not require spacelike distances to be used as an input. It offers a new way to define spatial orientation and spacelike distances in a causal set.
A velocity of a point particle in the kappa-Minkowski spacetime is investigated. Characteristic points of the spacetime are that the Poincare group becomes a quantum group with kappa, which is a mass dimension parameter, and is a kind of non-commutative geometry. We consider a particle in a coordinate space instead of it in a momentum space which is discussed in many articles. We see that the particle's velocity has an uncertainty which depends on a length of particle's propagation.
Gravity is not only able to be mimicked in flat spacetimes, but also in curved spacetimes. We study analogue gravity models in curved spacetime by considering the relativistic Gross-Pitaevskii theory and Yang-Mills theory in the fixed background spacetime geometry. The results show that acoustic metrics can be emergent from curved spacetimes yielding a Hadamard product of a real metric-tensor and an analogue metric-tensor. Taking \emph{quantum vortices} as \emph{test particles}, we evaluate their released energy ratio during the "gravitational binding". The $2+1$-dimensional flat Minkowski met…
We develop a new description of the much-studied $κ$-Minkowski noncommutative spacetime, centered on representing on a single Hilbert space not only the $κ$-Minkowski coordinates, but also the associated differential calculus and the $κ$-Poincaré symmetry generators. In this "pregeometric" representation the relevant operators act on the kinematical Hilbert space of the covariant formulation of quantum mechanics, which we argue is the natural framework for studying the implications of the step from commuting spacetime coordinates to the $κ$-Minkowski case, where the spatial coordinates do not …
We compactify the ten-dimensional spacetime in heterotic supergravity leaving four-dimensional Minkowski spacetime. We search for nonsupersymmetric, non-Ricci-flat solutions of the equations of motion with the quadratic curvature term. By assuming that the extradimensional spaces are products of 2-manifolds, three types of solutions are found. They are $S^2\times T^2 \times H^2/Γ$, $S^2\times H^2/Γ\times H^2/Γ$, and $S^2\times S^2 \times H^2/Γ$, where $H^2/Γ$ denotes a compact hyperbolic manifold. The metrics can be written explicitly, and they can be applied to phenomenology.
We demonstrate that the static ground state atom, which interacts with a conformally coupled massless scalar field in the de Sitter invariant vacuum, can obtain a position-dependent energy-level shift and this shift could cause a Casimir-Polder-like force on it. Interestingly no such force arises on the inertial atom bathed in a thermal radiation in the Minkowski universe. Thus, although the energy-level shifts of the static atom for these two cases are structurally the same, whether the energy-level shift causes the Casimir-Polder-like force, in principle, could be as an indicator to distingu…
We discuss boosts in a deformed Minkowski space, i.e. a four-dimensional space-time with metric coefficients depending on non-metric coordinates (in particular on the energy). The general form of a boost in an arbitrary direction is derived in the case of space anisotropy. Two maximal 3-vector velocities are mathematically possible, an isotropic and an anisotropic one. However, only the anisotropic velocity has physical meaning, being invariant indeed under deformed boosts.
We investigate the relation between the time-ordered vacuum correlation functions for interacting real scalar fields in Minkowski spacetime and in the Rindler wedge. The correlation functions are constructed perturbatively within the in-in formalism, often employed in calculations in more general spacetimes. We prove to all orders in perturbation theory that the time-ordered vacuum correlation functions can be calculated in the in-in formalism with internal vertices restricted to any Rindler wedge containing the external points. This implies that the Minkowski in-in (or in-out) perturbative ex…
We study the matching between the Hawking temperature of a large class of static D-dimensional black holes and the Unruh temperature of the corresponding higher dimensional Rindler spacetime. In order to accomplish this task we find the global embedding of the D-dimensional black holes into a higher dimensional Minkowskian spacetime, called the global embedding Minkowskian spacetime procedure (GEMS procedure). These global embedding transformations are important on their own, since they provide a powerful tool that simplifies the study of black hole physics by working instead, but equivalently…
We analyze the response of a detector with a uniform acceleration $α$ in $κ-$Minkowski spacetime using the first order perturbation theory. The monopole detector is coupled to a massless complex scalar field in such a way that it is sensitive to the Lorentz violation due to the noncommutativity of spacetime present in the $κ-$deformation. The response function deviates from the thermal distribution of Unruh temperature at the order of $1/κ$ and vanishes exponentially as the proper time of the detector exceeds a certain critical time, a logarithmic function of $κ$. This suggests that the Unruh …
Production of scalar particles by a relativistic, semi-transparent mirror in 1+3D Minkowski spacetime based on the Barton-Calogeracos (BC) action is investigated. The corresponding Bogoliubov coefficients are derived for a mirror with arbitrary trajectory. In particular, we apply our derived formula to the gravitational collapse trajectory. In addition, we identify the relation between the particle spectrum and the particle production probability, and we demonstrate the equivalence between our approach and the existing approach in the literature, which is restricted to 1+1D. In short, our trea…
We study the integrability of the equations of motion for the Nambu-Goto strings with a cohomogeneity-one symmetry in Minkowski spacetime. A cohomogeneity-one string has a world surface which is tangent to a Killing vector field. By virtue of the Killing vector, the equations of motion can be reduced to the geodesic equation in the orbit space. Cohomogeneity-one strings are classified into seven classes (Types I to VII). We investigate the integrability of the geodesic equations for all the classes and find that the geodesic equations are integrable. For Types I to VI, the integrability comes …
The doubly special relativity (DSR) theories are suggested in order to incorporate an observer-independent length scale in special theory of relativity. The Magueijo-Smolin proposal of DSR is realizable through a particular form of the noncommutative (NC) spacetime (known as $κ$-Minkowski spacetime) in which the Lorentz symmetry is preserved. In this framework, the NC parameter $κ$ provides the origin of natural cutoff energy scale. Using a nonlinear deformed relativistic dispersion relation along with the Lorentz transformations, we investigate some phenomenological facets of two-body collisi…
Within any anticipated unifying theory of quantum gravity, it should be meaningful to combine the fundamental notions of quantum superposition and spacetime to obtain so-called "spacetime superpositions": that is, quantum superpositions of different spacetimes not related by a global coordinate transformation. Here we consider the quantum-gravitational effects produced by superpositions of periodically identified Minkowski spacetime (i.e.\ Minkowski spacetime with a periodic boundary condition) with different characteristic lengths. By coupling relativistic quantum matter to fields on such a s…
In this letter we show that the ``preferred'' Klein-Gordon Quantum Field Theories (QFT's) on a d-dimensional de Sitter spacetime can be obtained from a Klein-Gordon QFT on a (d+1)-dimensional ``ambient'' Minkowski spacetime satisfying the spectral condition and, conversely, that a Klein-Gordon QFT on a (d+1)-dimensional ``ambient'' Minkowski spacetime satisfying the spectral condition can be obtained as superposition of d-dimensional de Sitter Klein-Gordon fields in the preferred vacuum. These results establish a correspondence between QFT's living on manifolds having different dimensions. The…
If Einstein's equations are to describe a field theory of gravity in Minkowski spacetime, then causality requires that the effective curved metric must respect the flat background metric's null cone. The kinematical problem is solved using a generalized eigenvector formalism based on the Segré classification of symmetric rank 2 tensors with respect to a Lorentzian metric. Securing the correct relationship between the two null cones dynamically plausibly is achieved using the naive gauge freedom. New variables tied to the generalized eigenvector formalism reduce the configuration space to the c…
We prove that the Minkowski spacetime is stable at nonlinear level and to all perturbative orders in the gravitational perturbation in a general class of nonlocal gravitational theories that are unitary and finite at quantum level.
The Fermionic Entanglement Entropy of Causal Diamonds in Two-Dimensional Minkowski Space
Fakultät für Mathematik Universität Regensburg（雷根斯堡大学数学系）；Institut für Theoretische Physik Universität Leipzig（莱比锡大学理论物理研究所）；Dipartimento di Matematica, Università di Genova（热那亚大学数学系）
We study a novel class of higher derivative theories for interacting massless gravitons in Minkowski spacetime. These theories were first discussed by Wald decades ago, and are characterized by scattering amplitudes essentially different from general relativity and many of its modifications. We discuss various aspects of these higher derivative theories, including the Lagrangian construction, violation of asymptotic causality, scattering amplitudes, non-renormalization, and possible implications on emergent gravitons from condensed matter systems.
A criticism sometimes made of the causal set quantum gravity program is that there is no practical scheme for identifying manifoldlike causal sets and finding embeddings of them into manifolds. A computational method for constructing an approximate embedding of a small manifoldlike causal set into Minkowski space (or any spacetime that is approximately flat at short scales) is given, and tested in the 2 dimensional case. This method can also be used to determine how manifoldlike a causal set is, and conversely to define scales of manifoldlikeness.
We examine embedding diagrams of hypersurfaces in the Reissner-Nordstrom black hole spacetime. These embedding diagrams serve as useful tools to visualize the geometry of the hypersurfaces and of the whole spacetime in general.
One perceives same response of a single-atom detector when placed at a point outside the horizon in Schwarzschild spacetime to that of a static single-atom detector in thermal Minkowski spacetime. So one cannot distinguish Schwarzschild spacetime from thermal Minkowski spacetime by using a single-atom detector. We show that, for Schwarzschild spacetime, beyond a characteristic length scale which is proportional to the inverse of the surface gravity $κ$, the \emph{Resonance Casimir-Polder interaction (RCPI)} between two entangled atoms is characterized by a $1/L^2$ power-law provided the atoms …
It is shown that cosmological spacetime manifold has the structure of a Lie group and a spinor space. This leads naturally to the Minkowski metric on tangent spaces and the Lorentzian metric on the manifold and makes it possible to dispense with double-valued representations.
We prove that every $(3+1)$-dimensional flat GHMC Minkowski spacetime which is not a translation spacetime or a Misner spacetime carries a unique foliation by spacelike hypersurfaces of constant scalar curvature. In otherwords, we prove that every such spacetime carries a unique time function with isochrones of constant scalar curvature. Furthermore, this time function is a smooth submersion.
We recently constructed the $R$-Poincaré algebra from an appropriate deformed Poisson brackets which reproduce the Fock coordinate transformation. We showed then that the spacetime of this transformation is the de Sitter one. In this paper, we derive in the $R$-Minkowski spacetime the Dirac equation and show that this is none other than the Dirac equation in the de Sitter spacetime given by its conformally flat metric. Furthermore, we propose a new approach for solving Dirac's equation in the de Sitter spacetime using the Schrödinger picture.
A formulation of Poincare symmetry as an inner symmetry of field theories defined on a fixed Minkowski spacetime is given. Local P gauge transformations and the corresponding covariant derivative with P gauge fields are introduced. The renormalization properties of scalar, spinor and vector fields in P gauge field backgrounds are determined. A minimal gauge field dynamics consistent with the renormalization constraints is given.
Quantum inequalities have been established for various quantum fields in both flat and curved spacetimes. In particular, for spin-3/2 fields, Yu and Wu have explicitly derived quantum inequalities for massive case. Employing the similar method developed by Fewster and colleagues, this paper provides an explicit formula of quantum inequalities for massless spin-3/2 field in four-dimensional Minkowski spacetime.
To extend previous results on the late time behavior of massive fields, for the Dirac field propagating in the D-dimensional Minkowski spacetime we calculate analytically its asymptotic tails. We find that the massive Dirac field has an oscillatory inverse power law tail. The frequency of the oscillations depends on the mass of the field and the power law decay rate depends on the dimension of the spacetime and the mode number of the angular eigenvalues. We also compare with previous results in curved spacetimes.
Solutions to the Yang-Mills field equations which describe exploding or imploding shells of gauge field and which may serve as approximations to the exploding sphaleron are discussed. The solutions are conformally related to $SO(2)\times SO(4)$ invariant Yang-Mills solutions. The behavior of fermions in the gauge field background and higher gauge groups are also considered.
We study exact tunneling solutions in scalar field theory for potential barriers composed of linear or quadratic patches. We analytically continue our solutions to imaginary Euclidean radius in order to study the profile of the scalar field inside the growing bubble. We find that generally there is a non-trivial profile of the scalar field, generating a stress-energy tensor, that depending on the form of the potential, can be a candidate for dark energy.
The conventional role of spacetime geometry in the description of gravity is pointed out. Global Poincar$\acute{\mbox{e}}$ symmetry as an inner symmetry of field theories defined on a fixed Minkowski spacetime is discussed. Its extension to local {\bf P\/} gauge symmetry and the corresponding {\bf P\/} gauge fields are introduced. Their minimal coupling to matter is obtained. The scaling behaviour of the partition function of a spinor in {\bf P\/} gauge field backgrounds is computed. The corresponding renormalization constraint is used to determine a minimal gauge field dynamics.
This is a short review of a series of papers which, in collaboration with Yue Ma, establish several novel existence results for systems of coupled wave-Klein-Gordon equation. Our method, the Hyperbolic Hyperboloidal Method, has allowed us to address the global evolution problem for the Einstein equations of general relativity and investigate the global geometry of matter spacetimes that are initially close to Minkowski spacetime.
For causal graphs we propose a definition of proper time which for small scales is based on the concept of volume, while for large scales the usual definition of length is applied. The scale where the change from "volume" to "length" occurs is related to the size of a dynamical clock and defines a natural cut-off for this type of clock. By changing the cut-off volume we may probe the geometry of the causal graph on different scales and therey define a continuum limit. This provides an alternative to the standard coarse graining procedures. For regular causal lattice (like e.g. the 2-dim. light…
Bubble universes and traversable wormholes in general relativity can be realized as two sides of the same concept. To exemplify, we find, display, and study in a unified manner a Minkowski-Minkowski closed universe and a Minkowski-Minkowski traversable wormhole. By joining two 3-dimensional flat balls along a thin shell two-sphere of matter, i.e., a spherical domain wall, into a single spacetime one gets a Minkowski-Minkowski static closed universe, i.e., a bubble universe. By joining two 3-dimensional complements of flat balls along a thin shell two-sphere of matter, i.e., a spherical throat,…
Cobalt nanoparticle-based ferrofluid in the presence of external magnetic field forms a self-assembled hyperbolic metamaterial. Wave equation describing propagation of extraordinary light inside the ferrofluid exhibits 2+1 dimensional Lorentz symmetry. The role of time in the corresponding effective 3D Minkowski spacetime is played by the spatial coordinate directed along the periodic nanoparticle chains aligned by the magnetic field. Here we present a microscopic study of point, linear, planar and volume defects of the nanoparticle chain structure and demonstrate that they may exhibit strong …
Confined Klein-Gordon oscillators in Minkowski spacetime and a pseudo-Minkowski spacetime with a space-like dislocation: PDM KG-oscillators, isospectrality and invariance
The rich structure of solid state physics provides us with Dirac materials the effective theory of which enjoys the Lorentz symmetry. In non-symmorphic lattices, the Lorentz symmetry will be deformed in a way that the null energy-momentum vectors will correspond to on-shell condition for tilted Dirac cone dispersion. In this sense, tilted Dirac/Weyl materials can be viewed as solid state systems where the effective spacetime is non-Minkowski. In this work, we show that the polarization tensor for tilted Dirac cone systems acquires a covariant from only when the spacetime is considered to be an…
A time-flat condition on spacelike 2-surfaces in spacetime is considered here. This condition is analogous to constant torsion condition for curves in three dimensional space and has been studied in [2, 4, 5, 12, 13]. In particular, any 2-surface in a static slice of a static spacetime is time-flat. In this article, we address the question in the title and prove several local and global rigidity theorems for such surfaces in the Minkowski spacetime.
In this note, we study a geometric property of asymptotically Minkowski spacetimes and an analytic property of the Klein-Gordon operator. Precisely, our first main results show that asymptotically Minkowski spacetimes are geodesically complete under a null non-trapping condition. Secondly, we prove that Sobolev index of a real principal type estimate used in the previous work is actually optimal.
This talk reports on recent progress toward the semiglobal study of asymptotically flat spacetimes within numerical relativity. The development of a 3D solver for asymptotically Minkowski-like hyperboloidal initial data has rendered possible the application of Friedrich's conformal field equations to astrophysically interesting spacetimes. As a first application, the whole future of a hyperboloidal set of weak initial data has been studied, including future null and timelike infinity. Using this example we sketch the numerical techniques employed and highlight some of the unique capabilities o…
The nonlinear instability of anti-de Sitter spacetime has recently been established with the striking result that generic initial data collapses to form black holes. This outcome suggests that confined matter generically collapses, and that collapse can only be halted -- at most -- by nonlinear bound states. Here we provide evidence that such mechanism can operate even in asymptotically flat spacetimes, by studying the evolution of the Einstein-Klein-Gordon system for a self-interacting scalar field. We show that (i) configurations which do not collapse promptly can do so after successive refl…
We study the existence of surfaces with constant or prescribed Gauss curvature in certain Lorentzian spacetimes. We prove in particular that every (non-elementary) 3-dimensional maximal globally hyperbolic spatially compact spacetime with constant non-negative curvature is foliated by compact spacelike surfaces with constant Gauss curvature. In the constant negative curvature case, such a foliation exists outside the convex core. The existence of these foliations, together with a theorem of C. Gerhardt, yield several corollaries. For example, they allow to solve the Minkowski problem in the 3-…
The causal set program as well as the Wolfram physics project leave open the problem of how a graph that is a (3+1)-dimensional Minkowski-spacetime according to its simple geodesic distances, could be generated solely from simple deterministic rules. This paper provides a solution by describing simple rules that characterize discrete Lorentz boosts between 4D lattice graphs, which combine further to form Wigner rotations that produce isotropy and lead to the emergence of the continuous Lorentz group and the (3+1)-dimensional Minkowski-spacetime. On such graphs, the speed of light, the proper t…
Using the methods of ordinary quantum mechanics we study $κ$-Minkowski space as a quantum space described by noncommuting self-adjoint operators, following and enlarging arXiv:1811.08409. We see how the role of Fourier transforms is played in this case by Mellin transforms. We briefly discuss the role of transformations and observers.
