This note provides a short guide to dimensional analysis in Lorentzian and general relativity and in differential geometry. It tries to revive Dorgelo and Schouten's notion of 'intrinsic' or 'absolute' dimension of a tensorial quantity. The intrinsic dimension is independent of the dimensions of the coordinates and expresses the physical and operational meaning of a tensor. The dimensional analysis of several important tensors and tensor operations is summarized. In particular it is shown that the components of a tensor need not have all the same dimension, and that the Riemann (once contravariant and thrice covariant), Ricci (twice covariant), and Einstein (twice covariant) curvature tensors are dimensionless. The relation between dimension and operational meaning for the metric and stress-energy-momentum tensors is discussed; and the possible conventions for the dimensions of these two tensors and of Einstein's constant $κ$, including the curious possibility $κ= 8πG$ without $c$ factors, are reviewed.
An attempt is made of giving a self-contained introduction to holomorphic ideas in general relativity, following work over the last thirty years by several authors. The main topics are complex manifolds, spinor and twistor methods, heaven spaces.
We discuss the recent proposal of implementing Doubly Special Relativity in configuration space by means of Finsler geometry. Although this formalism leads to a consistent description of the dynamics of a particle, it does not seem to give a complete description of the physics. In particular, the Finsler line element is not invariant under the deformed Lorentz transformations of Doubly Special Relativity. We study in detail some simple applications of the formalism.
Metric Relativity and the Dynamical Bridge: highlights of Riemannian geometry in physics
Instituto de Cosmologia Relatividade e Astrofisica (ICRA/CBPF)（宇宙学相对论与天体物理研究所）；CAPES Foundation, Ministry of Education of Brazil（巴西教育部CAPES基金会）；Physics Department, “La Sapienza” University of Rome（罗马大学物理系）
We review the language of differential forms and their applications to Riemannian Geometry with an orientation to General Relativity. Working with the principal algebraic and differential operations on forms, we obtain the structure equations and their symmetries in terms of a new product (the co-multiplication). It is showen how the Cartan - Grassmann algebra can be endowed with the structure of a Hopf algebra.
We explore the possibility of replacing point set topology by higher category theory and topos theory as the foundation for quantum general relativity. We discuss the BC model and problems of its interpretation, and connect with the construction of causal sites.
Relations between kinematics, geometry and law of reference frame motion are considered. We show, that kinematical tensors define geometry up to a space functional arbitrariness when integrability condition for spin tensor is satisfied. Some aspects of geometrization principle and geometrical conventionalism of Poincare are discussed in a light of the obtained results.
We show that the Schwarzschild solution can be embedded in a class of nonstandard solutions of the vacuum Einstein's equations with arbitrary rotation curves. These nonstandard solutions have to be taken as physical if dark matter as needed in the standard theory cannot be found. As a consequence general relativity is considered as a classical field theory in Minkowski space and not as a geometric theory in the sense of Einstein.
A formulation for stationary axisymmetric electromagnetic fields in general relativity is derived by casting them into the form of an anisotropic fluid. Several simplifications of the formalism are carried out in order to analyze different features of the fields, such as the derivation of electromagnetic sources for the Maxwell field in the form of thin layers, construction of new solutions, and generation techniques.
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.
In this paper we derive 4-dimensional General Relativity from three dimensions, using the intrinsic spatial geometry inherent in Yang--Mills theory which has been exposed by previous authors as well as as some properties of the Ashtekar variables. We provide various interesting relations, including the fact that General Relativity can be written as a Yang--Mills theory where the antiself-dual Weyl curvature replaces the Yang--Mills coupling constant. We have generalized the results of some previous authors, covering Einsteins spaces, to include more general spacetime geometries.
We discuss the physical features of two recent classes of analytical solutions of the Einstein equations sourced by an exotic perfect fluid with equation of state $ P=-ρ/5$. These geometries depend on up to four parameters and are static and spherically symmetric. They describe compact spaces with naked central singularities.
In this addendum we clarify a point which strengthens one of the results from [the original paper]. Namely, we show that the algebra of the observables $F(r,θ)$ is yet simpler then it was described in [the original paper]. This is an important point, because with this simplification an important subalgebra becomes canonical, allowing for a natural reduction of the phase space.
The Hamiltonian formulation of general relativity on a null surface is established in the teleparallel geometry. No particular gauge conditons on the tetrads are imposed, such as the time gauge condition. By means of a 3+1 decomposition the resulting Hamiltonian arises as a completely constrained system. However, it is structurally different from the the standard Arnowitt-Deser-Misner (ADM) type formulation. In this geometrical framework the basic field quantities are tetrads that transform under the global SO(3,1) and the torsion tensor.
Recently, Choquet-Bruhat and York and Abrahams, Anderson, Choquet-Bruhat, and York (AACY) have cast the 3+1 evolution equations of general relativity in gauge-covariant and causal ``first-order symmetric hyperbolic form,'' thereby cleanly separating physical from gauge degrees of freedom in the Cauchy problem for general relativity. A key ingredient in their construction is a certain wave equation which governs the light-speed propagation of the extrinsic curvature tensor. Along a similar line, we construct a related wave equation which, as the key equation in a system, describes vacuum genera…
In this article we describe applications of Discrete Differential Forms in computational GR. In particular we consider the initial value problem in vacuum space-times that are spherically symmetric. The motivation to investigate this method is mainly its manifest coordinate independence. Three numerical schemes are introduced, the results of which are compared with the corresponding analytic solutions. The error of two schemes converges quadratically to zero. For one scheme the errors depend strongly on the initial data.
W.Lawvere suggested a approach to differential geometry and to others mathematical disciplines closed to physics, which allows to give definitions of derivatives, tangent vectors and tangent bundles without passages to the limits. This approach is based on a idea of consideration of all settings not in sets but in some cartesian closed category E, particular in some elementary topos. The synthetic differential geometry (SDG) is the theory developed by A.Kock in a context of Lawvere's ideas. In a base of the theory is an assumption of that a geometric line is not a filed of real numbers, but a …
We establish the Hamiltonian formulation of the teleparallel equivalent of general relativity, without fixing the time gauge condition, by rigorously performing the Legendre transform. The time gauge condition, previously considered, restricts the teleparallel geometry to the three-dimensional spacelike hypersurface. Geometrically, the teleparallel geometry is now extended to the four-dimensional space-time. The resulting Hamiltonian formulation is different from the standard ADM formulation in many aspects, the main one being that the dynamics is now governed by the Hamiltonian constraint H_0…
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 these lectures my aim is to review enough of conformal differential geometry in four dimensions to give an account of Penrose's conformal cyclic geometry.
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…
The new approach to quantize the gravity based on the notion of differential algebra is suggested. It is shown that the differential geometry of this object can not be described in terms of points. The spatialization procedure giving rise to points by loosing a part of the entire structure is discussed. The counterpart of the traditional objects of differential geometry are studied.
We examine some common features of minimal surfaces, nonzero constant mean curvature surfaces and marginally outer trapped surfaces, concerning their stability and rigidity, and consider some applications to Riemannian geometry and general relativity.
This article reviews some aspects in the current relationship between mathematical and numerical General Relativity. Focus is placed on the description of isolated systems, with a particular emphasis on recent developments in the study of black holes. Ideas concerning asymptotic flatness, the initial value problem, the constraint equations, evolution formalisms, geometric inequalities and quasi-local black hole horizons are discussed on the light of the interaction between numerical and mathematical relativists.
We present a new scheme of defining invariant observables for general relativistic systems. The scheme is based on the introduction of an observer which endowes the construction with a straightforward physical interpretation. The observables are invariant with respect to spatial diffeomorphisms which preserve the observer. The limited residual spatial gauge freedom is studied and fully understood. A full canonical analysis of the observables is presented: we analyze their variations, Poisson algebra and discuss their dynamics. Lastly, the observables are used to solve the vector constraint, wh…
We show that the general theory of relativity can be formulated in the language of Weyl geometry. We develop the concept of Weyl frames and point out that the new mathematical formalism may lead to different pictures of the same gravitational phenomena. We show that in an arbitrary Weyl frame general relativity, which takes the form of a scalar-tensor gravitational theory, is invariant with respect to Weyl tranformations. A kew point in the development of the formalism is to build an action that is manifestly invariant with respect to Weyl transformations. When this action is expressed in term…
Recently, there have been several applications of differential and algebraic topology to problems concerned with the global structure of spacetimes. In this paper, we derive obstructions to the existence of spin-Lorentz and pin-Lorentz cobordisms and we show that for compact spacetimes with non-empty boundary there is no relationship between the homotopy type of the Lorentz metric and the causal structure. We also point out that spin-Lorentz and tetrad cobordism are equivalent. Furthermore, because the original work [7] on metric homotopy and causality may not be known to a wide audience, we p…
The tetrad partial differential equations formulated by Buchman and Bardeen for vacuum gravity are shown to be well posed by calculation of the Cartan characters of an associated exterior differential system. Gauge specializations are discussed. A Cartan 4-form is found for this field theory, together with its intrinsic version the Lagrangian density.
It is well known that the Lorentzian length of a timelike curve in Minkowski spacetime is smaller than the Lorentzian length of the geodesic connecting its initial and final endpoints. The difference is known as the 'differential aging' and its calculation in terms of the proper acceleration history of the timelike curve would provide an important tool for the autonomous spacetime navigation of non-inertial observers. I give a solution in 3+1 dimensions which holds whenever the acceleration is decomposed with respect to a lightlike transported frame (lightlike transport will be defined), the a…
A differential bulk-surface relation of the lagrangian of General Relativity has been derived by Padmanabhan. This has relevance to gravitational information and degrees of freedom. An alternate derivation is given based on the differential form gauge theory formulation of gravity due to Gockeler and Schucker. Also an entropy functional of Padmanabhan and Paranjape can be rewritten as the Gockeler and Schucker lagrangian.
Modern formulation of Finsler geometry of a manifold M utilizes the equivalence between this geometry and the Riemannian geometry of VTM, the vertical bundle over the tangent bundle of M, treating TM as the base space. We argue that this approach is unsatisfactory when there is an indefinite metric on M because the corresponding Finsler fundamental function would not be differentiable over TM (even without its zero section) and therefore TM cannot serve as the base space. We then make the simple observation that for any differentiable Lorentzian metric on a smooth space-time, the corresponding…
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…
Ising models of emergent geometry are well known to possess ground states with many of the desired features of a low dimensional, Ricci flat vacuum. Further, excitations of these ground states can be shown to replicate the quantum dynamics of a free particle in the continuum limit. It would be a significant next step in the development of emergent Ising models to link them to an underlying physical theory that has General Relativity as its continuum limit. In this work we investigate how the canonical formulation of General Relativity can be used to construct such a discrete Hamiltonian using …
Deformed Special Relativity (DSR) is obtained by imposing a maximal energy to Special Relativity and deforming the Lorentz symmetry (more exactly the Poincaré symmetry) to accommodate this requirement. One can apply the same procedure deforming the Galilean symmetry in order to impose a maximal speed (the speed of light). This leads to a non-commutative space structure, to the expected deformations of composition of speed and conservation of energy-momentum. In doing so, one runs into most of the ambiguities that one stumbles onto in the DSR context. However, this time, Special Relativity is t…
The now-famous Majumdar-Papapetrou exact solution of the Einstein-Maxwell equations describes, in general, $N$ static, maximally charged black holes balanced under mutual gravitational and electrostatic interaction. When $N=2$, this solution defines the two-black-hole spacetime, and the relativistic two-center problem is the problem of geodesic motion on this static background. Contopoulos and a number of other workers have recently discovered through numerical experiments that in contrast with the Newtonian two-center problem, where the dynamics is completely integrable, relativistic null-geo…
We show that mass parameter and radial coordinate values can be indirectly measured in thought experiments performed in Schwarzschild spacetime, without using the Newtonian limit of general relativity or approximations based on Euclidean geometry. Our approach involves different proper time quantifcations as well as solutions to systems of algebraic equations, and aims to strengthen the conceptual independence of general relativity from Newtonian gravity.
A formulation of Einstein equations is presented that could yield advantages in the study of collisions of binary compact objects during regimes between linear-nonlinear transitions. The key idea behind this formulation is a separation of the dynamical variables into i) a fixed conformal 3-geometry, ii) a conformal factor possessing nonlinear dynamics and iii) transverse-traceless perturbations of the conformal 3-geometry.
In this we briefly cover the covariant approach to describe inertial forces in general relativity and in particular look at the behaviour of the cumulative drag index for stationary Kerr geometry.
General Very Special Relativity (GVSR) is the curved space-time of Very Special Relativity (VSR) proposed by Cohen and Glashow. The geometry of GVSR possesses a line element of Finsler Geometry proposed by Bogoslovsky. We calculate the Einstein field equations and derive a modified FRW cosmology, for an osculating Riemannian space. The Friedman equation of motion leads to an explanation of the cosmological acceleration in terms of an alternative non-Lorentz invariant theory. A first order approach for a primordial spurionic vector field introduced into the metric, gives back an estimation of t…
This work reports on the construction of a nonlinear distributional geometry (in the sense of Colombeau's special setting) and its applications to general relativity with a special focus on the distributional description of impulsive gravitational waves.
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…
A highly accurate, multi-domain spectral code is used in order to construct sequences of general relativistic, differentially rotating neutron stars in axisymmetry and stationarity. For bodies with a spheroidal topology and a homogeneous or an N=1 polytropic equation of state, we investigate the solution space corresponding to broad ranges of degree of differential rotation and stellar densities. In particular, starting from static and spherical configurations, we analyse the changes of the corresponding surface shapes as the rate of rotation is increased. For a sufficiently weak degree of dif…
Rigidly rotating stationary matter in general relativity has been investigated by Kramer by the Ernst coordinate method. A weakness of this approach is that the Ernst potential does not exist for differential rotation. We now generalize the techniques by the use of a nonholonomic and nonrigid frame. We apply these techniques for differentially rotating perfect fluids. We construct a complex analytic tensor, characterizing the class of matter states in which both the interior Schwarzschild and the Kerr solution are contained. We derive consistency relations for this class of perfect fluids. We …
In a recent work we derived the kinematic Hamiltonian and primary constraints of the new general relativity class of teleparallel gravity theories and showed that these theories can be grouped in 9 classes, based on the presence or absence of primary constraints in their Hamiltonian. Here we demonstrate an alternative approach towards this result, by using differential forms instead of tensor components throughout the calculation. We prove that also this alternative derivation yields the same results and show how they are related to each other.
Recently, an extension of teleparallelism to a Weyl geometry which allows us to easily establish conformal invariance and "geometrize" electromagnetism has been presented. In this paper, I extend a result which concerns the existence of the Schwarzschild solution to a particular class of this extension. In addition, I obtain the field equations of some models based on this extension, including the one which is equivalent to Einstein's field equations with a massless scalar field.
We give a critical analysis of the conceptual foundations of special relativity. We formulate a simple operational criterion for distinguishing between noninertial and inertial frames which is introduced prior to geometry. We associate the concept of maximal velocity with the existence of an upper bound for a set of rates of movers which travel in the same direction. We define the standard scale for reading the time flow. We refine the treatment of both Einstein's postulates, the principle of relativity and constancy of the velocity of light. The proposed ``reconstruction'' of the geometry of …
A lattice model for four dimensional Euclidean quantum general relativity is proposed for a simplicial spacetime. It is shown how this model can be expressed in terms of a sum over worldsheets of spin networks, and an interpretation of these worldsheets as spacetime geometries is given, based on the geometry defined by spin networks in canonical loop quantized GR. The spacetime geometry has a Planck scale discreteness which arises "naturally" from the discrete spectrum of spins of SU(2) representations (and not from the use of a spacetime lattice). The lattice model of the dynamics is a formal…
We discuss an alternative way of prescribing the spacetime geometry associated with a non-radiating distribution of charged matter. It is based on the possibility that the electrostatic self-energy does not reside on the Coulombian field but in a matter pressure term of electromagnetic origin localized at the sources. We work out completely the well controlled spherically symmetric case, questioning the realization of Reissner-Nordstrom geometry in nature. Finally, we sketch an experiment that could distinguish between the standard and the alternative scenario.
In Riemannian geometry, the Ricci flow is the analogue of heat diffusion; a deformation of the metric tensor driven by its Ricci curvature. As a step towards resolving the problem of time in quantum gravity, we attempt to merge the Ricci flow equation with the Hamilton-Jacobi equation for general relativity.
We briefly review a few aspects of the development of differential geometry which may be considered as being influenced by Einstein's general relativity. We focus on how Einstein's quest for a complete geometrization of matter and electromagnetism gave rise to an enormous amount of theoretical work both on physics and mathematics. In connection with this we also bring to light how recent investigation on theoretical physics has led to new results on some branches of modern differential geometry.
This paper reframes Riemannian geometry as a generalized Lie algebra allowing the equations of both RG and then General Relativity to be expressed as commutation relations among fundamental operators. We begin with an Abelian Lie algebra of n operators, X, whose simultaneous eigenvalues, y, define a real n-dimensional space. Then with n new operators defined as independent functions, we define contravariant and covariant tensors in terms of their eigenvalues, on a Hilbert space representation. We then define n additional operators, D, whose exponential map is to translate X as defined by a non…
The Problem of Time in Quantum Gravity is analyzed from a classical presymplectic perspective. In the first part of the paper the Three Space Approach to General Relativity is introduced via the Barbour-Foster-Ó Murchadha action and the dynamics corresponding to a theory of relativistic gravity where spacetime is not an a priori requirement. We also look into the nature and physical interpretation of the constraints in this theory and compare them with those of Standard ADM General Relativity. We then study the presymplectic phase space of three space general relativity and discuss briefly the…
We study the behavior of clocks in 1+1 spacetime assuming the relativity principle, the principle of constancy of the speed of light and the clock hypothesis. These requirements are satisfied by a class of Finslerian theories parametrized by a real coefficient $β$, special relativity being recovered for $β=0$. The effect of differential aging is studied for the different values of $β$. Below the critical values $|β| =1/c$ the differential aging has the usual direction - after a round trip the accelerated observer returns younger than the twin at rest in the inertial frame - while above the cri…
The notion that the geometry of our space-time is not only a static background but can be physically dynamic is well established in general relativity. Geometry can be described as shaped by the presence of matter, where such shaping manifests itself as gravitational force. We consider here probabilistic or atomistic models of such space-time, in which the active geometry emerges from a statistical distribution of 'atoms'. Such atoms are not to be confused with their chemical counterparts, however the shift of perspective obtained in analyzing a gas via its molecules rather than its bulk prope…
By employing an exact back-reaction geometry, Helliwell-Konkowski stability conjecture is shown to fail. This happens when a test null dust is inserted to the interaction region of cross-polarized Bell-Szekeres spacetime.
The following are expanded lecture notes for the course of eight one hour lectures given by the second author at the 2014 summer school Asymptotic Analysis in General Relativity held in Grenoble by the Institut Fourier. The first four lectures deal with conformal geometry and the conformal tractor calculus, taking as primary motivation the search for conformally invariant tensors and diffrerential operators. The final four lectures apply the conformal tractor calculus to the study of conformally compactified geometries, motivated by the conformal treatment of infinity in general relativity.
