April 2008. 153pp. PhD Thesis (Supervisor: Prof. Dr. Otto Nachtmann). Available online at: http://www.ub.uni-heidelberg.de/archiv/8385/ Replaced with PDFLaTeX version and improved figures
The averaging problem in general relativity concerns the difficulty of defining meaningful averages of tensor quantities and we consider various aspects of the problem. We first address cosmological backreaction which arises because the averaged Einstein tensor is not the same as the Einstein tensor of the averaged metric. It has been suggested that backreaction might account for the dark energy. We show numerically in the Buchert formalism that the corrections from (quasi)linear perturbations are only of the order of 10^-5 and act as a dark matter. We then focus on constructing averaged metrics and present a generally covariant averaging process which decomposes the metric into Vielbeins and parallel transports them with a relativistic Wegner-Wilson operator to a single point where they can then be averaged. The Vielbeins are chosen in a Lagrangian formalism. The functionality of the process is demonstrated in specific examples in two and three space dimensions. This involves the numerical solution of partial differential equations by the aid of the simulation toolkit Gascoigne.
We show that Renormalization Group extensions of the Einstein-Hilbert action for large scale physics are not, in general, a particular case of standard Scalar-Tensor (ST) gravity. We present a new class of ST actions, in which the potential is not necessarily fixed at the action level, and show that this extended ST theory formally contains the Renormalization Group case. We also propose here a Renormalization Group scale setting identification that is explicitly covariant and valid for arbitrary relativistic fluids.
General Relativity extensions based on Renormalization Group effects are motivated by a known physical principle and constitute a class of extended gravity theories that have some unexplored unique aspects. In this work we develop in detail the Newtonian and post Newtonian limits of a realisation called Renormalization Group extended General Relativity (RGGR). Special attention is taken to the external potential effect, which constitutes a type of screening mechanism typical of RGGR. In the Solar System, RGGR depends on a single dimensionless parameter $\bar ν_\odot$, and this parameter is suc…
Here we apply the full Will-Nordtvedt version of the Parameterized Post-Newtonian (PPN) formalism to a class of General Relativity extensions that are based on nontrivial renormalization group (RG) effects at large scales. We focus on a class of models in which the gravitational coupling constant $G$ is correlated with the Newtonian potential. A previous PPN analysis considered a specific realization of the RG effects, and only within the Eddington-Robertson-Schiff version of the PPN formalism, which is a less complete and robust PPN formulation. Here we find stronger, more precise bounds, and…
We review our contribution to infrared Renormalization Group (RG) effects to General Relativity in the context of galaxies. Considering the effective action approach to Quantum Field Theory in curved background, we argued that the proper RG energy scale, in the weak field limit, should be related to the Newtonian potential. In the galaxy context, even without dark matter, this led to a remarkably small gravitational coupling G variation (about or less than 10^{-12} of its value per light-year), while also capable of generating galaxy rotation curves about as good as the best phenomenological d…
The renormalization group framework can be applied to Quantum Field Theory on curved space-time, but there is no proof whether the beta-function of the gravitational coupling indeed goes to zero in the far infrared or not. In a recent paper we have shown that the amount of dark matter inside spiral galaxies may be negligible if a small running of the General Relativity coupling G is present. Here we extend the proposed model to elliptical galaxies and present a detailed analysis on the modeling of NGC 4494 (an ordinary elliptical) and NGC 4374 (a giant elliptical). In order to compare our resu…
In this paper, by applying Newman-Janis algorithm in spherical symmetric metrics, a class of embedded rotating solutions of field equations is presented. These solutions admit non-perfect fluids
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 present a derivation of 2PN/RM metric field equations from the Einstein field equation in General Relativity. We use the exponential parametrization and the $c^{-3}$ isotropic spatial coordinates such as in IAU2000 recommendations.
We investigate the consequences of the pseudo-complex General Relativity within a pseudo-complexified Roberston-Walker metric. A contribution to the energy-momentum tensor arises, which corresponds to a dark energy and may change with the radius of the universe, i.e., time. Only when the Hubble function $H$ does not change in time, the solution is consistent with a constant $Λ$.
The nexus between the gravitational field and the space-time metric was an essential element in Einstein's development of General Relativity and led him to his discovery of the field equations for the gravitational field/metric. I will argue here that the metric is in fact an inessential element of this theory and can be dispensed with entirely. Its sole function in the theory was to describe the space-time measurements made by ideal clocks and rods. However, the behavior of model clocks and measuring rods can be derived directly from the field equations of General Relativity using the Einstei…
This paper is devoted to the investigation of the energy-momentum problem in two theories, i.e., General Relativity and teleparallel gravity. We use Einstein, Landau-Lifshitz, Bergmann-Thomson and Möller's prescriptions to evaluate energy-momentum distribution of Bell-Szekeres metric in both the theories. It is shown that these prescriptions give the same energy-momentum density components in both General Relativity and teleparallel theory. Möller's prescription yields constant energy in both the theories.
A classical foundation for an idea of reality condition in the context of spin foams (Barrett-Crane models) is developed. I extract classical real general relativity (all signatures) from complex general relativity by imposing the area metric reality constraint; the area metric is real iff a non-degenerate metric is real or imaginary. First I review the Plebanski theory of complex general relativity starting from a complex vectorial action. Then I modify the theory by adding a Lagrange multiplier to impose the area metric reality condition and derive classical real general relativity. I invest…
The notion that the metric field in general relativity can be understood as a property of space-time rests on a feature of the theory sometimes called universal coupling -- the claim that rods and clocks "measure" the metric in a way that is independent of their constitution. It is pointed out that this feature is not strictly a consequence of the central dynamical tenets of the theory, and argued that the metric field would better be regarded as a (possibly emergent) field in space-time, rather than as the very fabric of space-time itself.
We perform a short comparison between the local and linear constitutive tensor $χ^{λνσκ}$ in four-dimensional electrodynamics (Sec.2), the elasticity tensor $c^{ijkl}$ in three-dimensional elasticity theory (Sec.3), and the DeWitt metric $G^{abcd}$ in general relativity, with ${a,b,\dots=1,2,3}$ (Sec.4). We find that the DeWitt metric has only six independent components.
Despite its remarkable agreement with gravity tests in the solar system, general relativity is challenged at larger scales. Large scale tests performed in the solar system by the Pioneer 10/11 probes have failed to confirm the expected gravity laws. We discuss here extensions of general relativity which preserve its geometrical nature while showing an ability to conciliate the Pioneer anomalies with other gravity tests. Further testable phenomenological consequences are also briefly discussed.
This work refers to the new formula for the superpotential Uikl in conservation laws in general relativity satisfying the integral and differential conservation laws within the Schwarzschild metric. The new superpotential is composed of two terms. The first term is based on Mollers concept and its a function of the metric gik and its first derivative only. The second term is the antisymmetric tensor density of weight plus one and it consists of higher derivatives of the metric gik. Although the new superpotential consists of higher derivatives of the metric gik it might bring a new evaluation …
The purpose of this paper is to evaluate the `Lorentzian pedagogy' defended by J.S. Bell in his essay ``How to teach special relativity'', and to explore its consistency with Einstein's thinking from 1905 to 1952. Some remarks are also made in this context on Weyl's philosophy of relativity and his 1918 gauge theory. Finally, it is argued that the Lorentzian pedagogy - which stresses the important connection between kinematics and dynamics - clarifies the role of rods and clocks in general relativity.
A reference frame consists of: a reference space, a time scale and a spatial metric. The geometric structure induced by these objects in spacetime is developed. The existence of a class of spatial metrics that are rigid, have free mobility and can be derived as a slight deformation of the radar metric, is shown.
In this paper we study some features of the Reissner-Nordström metric both from an analytic and a visual point of view. We perform an accurate ray tracing and study of null geodesics in various situations. Among the issues we focus on are (i) the comparison with the Schwarzschild case, (ii) the naked singularity case, where, if the electric charge is not too large, some dark shell appears on images despite there is no horizon in the metric, and (iii) the wormhole crossing case, i.e., a visual exploration of the maximal analytic extension of the metric.
Usually, General Relativity (GR) is known to be unrenormalizable perturbatively from the viewpoint of quantum field theory. But in the modern sense of renormalizability, there still remains the possibility to investigate whether GR is "nonpertubatively" renormalizable or not. Here I review the basics and results in this topic based on the "Effective Average Action" approach which was proposed by M.Reuter, and discuss its application to the balck hole geometry.
In this paper we study some features of the Kerr metric both from an analytic and a visual point of view by performing accurate raytracing in various situations. We focus on features that are unique to the maximal analytic extension of the Kerr metric as compared to that of the Schwarzschild or even the Reissner-Nordström one. A large number of new, yet underexplored phenomena appear, especially regarding the structure of bounded null geodesics and the aspect of the negative gravity regions whose visual characteristics are shown both from outside and inside it.
Affine variational principle for General Relativity, proposed in 1978 by one of us (J.K.), is a good remedy for the non-universal properties of the standard, metric formulation, arising when the matter Lagrangian depends upon the metric derivatives. Affine version of the theory cures the standard drawback of the metric version, where the leading (second order) term of the field equations depends upon matter fields and its causal structure violates the light cone structure of the metric. Choosing the affine connection (and not the metric one) as the gravitational configuration, simplifies consi…
It is shown that in general relativity some static metrics are able to simulate oscillatory motions. Their form depends on two arbitrary real parameters which determine the specific oscillation modes. The conclusion is that these metrics can be used for new geometric models of closed or even open bags. Pacs: 04.02.-q and 04.02.Jb
We use the Møller and Landau-Lifshitz energy-momentum definitions in General Relativity (GR) to evaluate the energy-momentum distribution of the phantom black hole space-time. The phantom black hole model was applied to the supermassive black hole at the Galactic Centre. We obtain that in both pseudotensorial prescriptions the energy distribution depends on the mass $M$ of the black hole, the phantom constant $p$ and the radial coordinate $r$. Further, all the calculated momenta are found to be zero. The limiting cases $r\rightarrow 0$, $r\rightarrow \infty $ and $r\rightarrow -\infty $ have a…
We explain the necessity of application of semi-metric in general relativity. A detailed discussion on the energy-momentum conservation in the general relativity is presented using the mathematical tool of semi-metric. By means of the general covariant spacetime translation transformation, the most general covariant conservation law of energy-momentum is obtained, which is valid for any coordinates and overcomes the flaws of the expressions of Einstain, Landau and Moller.
We consider conceptual issues of deep learning (DL) for metric detectors using test particle geodesics in curved spacetimes. Advantages of DL metric detectors are emphasized from a view point of general coordinate transformations. Two given metrics (two spacetimes) are defined to be conneted by a DL isometry if their geodesic image data cannot be discriminated by any DL metric detector at any time. The fundamental question of when the DL isometry appears is extensively explored. If the two spacetimes connected by the DL isometry are in superposition of quantum gravity theory, the post-measurem…
We consider the application of quantum corrections computed using renormalization group arguments in the astrophysical domain and show that, for the most natural interpretation of the renormalization group scale parameter, a gravitational coupling parameter $G$ varying $10^{-7}$ of its value across a galaxy (which is roughly a variation of $10^{-12}$ per light-year) is sufficient to generate galaxy rotation curves in agreement with the observations. The quality of the resulting fit is similar to the Isothermal profile quality once both the shape of the rotation curve and the mass-to-light rati…
We present an implementation of a ray tracing code in the Schwarzschild metric. We aim at building a numerical code with a correct implementation of both special (aberration, amplification, Doppler) and general (deflection of light, lensing, gravitational redshift) relativistic effects so as to simulate what an observer with arbitrary velocity would see near, or possibly within, the black hole. We also pay some specific attention to perform a satisfactory rendering of stars. Using this code, we then show several unexplored features of the maximal analytical extension of the metric. In particul…
We discuss a recently presented boosted Kerr black hole solution which had already been used by other authors. This boosted metric is based on wrong assumptions regarding asymptotic inertial observers and moreover the performed boost is not a proper Lorentz transformation. This note aims to clarify some of the issues when boosting black holes and the necessary care in order to interpret them. As it is wrongly claimed that the presented boosted Kerr metric is of Bondi-Sachs type, we recall out some of the necessary requirements and difficulties, when the casting the Kerr metric into a metric wi…
The present work has a double aim. On the one hand we call attention on the relationship existing between the Ashtekar formalism and other gauge-theoretical approaches to gravity, in particular the Poincaré Gauge Theory. On the other hand we study two kinds of solutions for the constraints of General Relativity, consisting of two mutually independent parts, namely a general three-metric-dependent contribution to the extrinsic curvature $K_{ab}$ in terms of the Cotton-York tensor, and besides it further metric independent contributions, which we analyze in particular in the presence of isotropi…
General relativity is reconsidered by starting from the unquestionable interpretation of special relativity, which (Klein 1910) is the theory of the invariants of the metric under the Poincaré group of collineations. This invariance property is physical and different from coordinate properties. Coordinates are physically empty (Kretschmann 1917) if not specified by physics, and one shall look for physics again through the invariance group of the metric. To find the invariance group for the metric, the Lie "Mitschleppen" is ideal for this task both in special and in general relativity. For a ge…
We consider the quantization of matter fields in a background described by the teleparallel equivalent to general relativity. The presence of local Lorentz and gauge symmetries gives rise to different coupling prescriptions, which we analyse separately. As expected, quantum matter fields produce divergences that cannot be absorbed by terms in the background action of teleparallel equivalent to general relativity. Nonetheless, the formulation of teleparallel gravity allows one to find out the source of the problem. By imposing local Lorentz invariance after quantization, we show that a modified…
We outline unimodular conformal and projective relativity (UCPR), an extension of unimodular relativity in which the conformal and projective structures play central roles. Under $SL(4,\mathbb{R})$ symmetry group, the pseudo-Riemannian metric naturally decomposes into a four-volume element and a conformal metric; and the affine connection decomposes into a one-form and a trace-free projective connection. In UCPR, these four space-time structures are treated as independent fields that have clear physical interpretations. A Palatini-type variational principle for the usual general relativity Lag…
The canonical description is based on the prior choice of a spacelike foliation, hence making a reference to a spacetime metric. However, the metric is expected to be a dynamical, fluctuating quantity in quantum gravity. After presenting the developments in the History Projection Operator histories theory in the last seven years--giving special emphasis on the novel temporal structure of the formalism--we show how this problem can be solved in the histories formulation of general relativity. We implement the 3+1 decomposition using metric-dependent foliations which remain spacelike with respec…
The possibility of spherically symmetric solutions in bi-metric theory of gravity is examined. It is shown that two possible black hole type solutions exists in the model. Spherically symmetric solution of general theory of relativity is recovered in the absence of the second metric. The result is compared with other bi-metric models as well as general theory of relativity.
The case of asymptotic Minkowskian space-times is considered. A special class of asymptotic rectilinear coordinates at the spatial infinity, related to a specific system of free falling observers, is chosen. This choice is applied in particular to the Schwarzschild metric, obtaining a vanishing energy for this space-time. This result is compared with the result of some known theorems on the uniqueness of the energy of any asymptotic Minkowskian space, showing that there is no contradiction between both results, the differences becoming from the use of coordinates with different operational mea…
In this contribution, we calculate the light deflection, perihelium shift, time delay and gravitational redshift using an approximate metric that contains the Kerr metric and an approximaction of the Erez-Rosen spacetime. The results were obtained directly using Mathematica. The results agree with the ones presented in the literature, but they are extended until second order terms of mass, angular momentum and mass quadrupole.
It seems to be not well known that the metrics of general relativity (GR) can be obtained without integrating Einstein equations. To that, we need only define a unit for GR-interval $Δs$, and observe 10 geodesics (out of which at least one must be nonnull). Even without using any unit, we can have $κg_{μν}(x^ρ)$, where $κ=$const. Our notes attempt to simplify the articles of E. Kretschmann (1917) and of H.A. Lorentz (1923) about this last subject. The text of this article in English will soon be available, in LaTeX. Please ask the author. ----- /Sajne estas malmulte konata ke la metrikoj de /g…
I show that all inertial systems are not equivalent, and the Lorentz transformation is not the space-time transformation over two inertial systems moving with relative constant velocity. To do this, I consider imaginary signals travelling over any inertial system K with arbitrarily large velocities. The travelling of an imaginary signal over K is just a time lapse over K. Then I present an example to show that all coordinate systems are not equivalent when the related theory is restricted over tensor-based coordinate transformations, i.e., the genereal relativity principle is not valid. Instea…
We show how Newtonian gravity with effective (actually observed) masses, obeying the mass-energy relation of special relativity, can explain all observations used to test General relativity. Dynamics of a gravitationally coupled binary system is considered in detail, and the effective masses of constituents are determined. Interpreting our results in terms of motion in a curved space-time background, we are led, using the Lagrangian formalism, to consider the exponential metric as a natural space-time metric of Newtonian gravity.
We present a renormalization group analysis to Einstein-Rosen waves or vacuum spacetimes with whole-cylinder symmetry. It is found that self-similar solutions appear as fixed points in the renormalization group transformation. These solutions correspond to the explosive gravitational waves and the collapsing gravitational waves at late times and early times, respectively. Based on the linear perturbation analysis of the self-similar solutions, we conclude that the self-similar evolution is stable as explosive gravitational waves under the condition of no incoming waves, while it is weakly unst…
We consider two most popular definitions of velocities of remote objects in General Relativity. Our work has two motivations. From a research point of view, we generalize the formula connecting these two velocities in FRW metrics found by Chodorowski to arbitrary synchronous spherically symmetric metrics. From a methodological point of view, our goal is to outline certain counter-intuitive properties of the definitions in question, which would allow to use them when it is reasonable and to avoid incorrect statements, based on inappropriate use of intuition.
According to the socalled "quasi-metric" framework developed elsewhere, the cosmic expansion applies directly to gravitationally bound systems. This prediction has a number of observable consequences, none of which are in conflict with observation. In this paper we compare test particle motion in the nonstatic gravitational field outside a spherically symmetric source (as predicted by a quasi-metric theory of gravity) to test particle motion in the Schwarzschild geometry. It is found that if one incorrectly uses the Schwarzschild geometry (to the relevant accuracy) to represent the nonstatic q…
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 this paper, we use the Killing vector method to formulate the de Sitter/Anti-de Sitter invariant special relativity (dS/AdS-SR). Through solving the Einstein equation with $Λ\neq 0$, the basic inertial metric for dS/AdS-SR is determined to be the Beltrami metric $B_{μν}(x)$. The corresponding Killing equations are system of ten simultaneous partial differential equations of first order. Their most general solutions were obtained, and all the ten independent Killing vectors were found out. These results confirm that the Beltrami metric has maximal spacetime symmetry. The ten Killing-Noether …
A very general class of axially-symmetric metrics in general relativity (GR) that includes rotations is used to discuss the dynamics of rotationally-supported galaxies. The exact vacuum solutions of the Einstein equations for this extended Weyl class of metrics allow us to deduce rigorously the following: (i) GR rotational velocity always exceeds the Newtonian velocity (thanks to Lenz's law in GR); (ii) A non-vanishing intrinsic angular momentum ($J$) for a galaxy demands the asymptotic constancy of the Weyl (vectorial) length parameter ($a$) -a behavior identical to that found for the Kerr me…
Einstein's general relativity with both metric and vielbein treated as independent fields is considered, demonstrating the existence of a consistent variational principle and deriving a Hamiltonian formalism that treats the spatial metric and spatial vielbein as canonical coordinates. This results in a Hamiltonian in standard form that consists of Hamiltonian and momentum constraints as well as constraints that generate spatial frame transformations---all appearing as primary, first class constraints on phase space. The formalism encompasses the standard coordinate frame and vielbein approache…
A new geometric interpretation for General Relativity (GR) is proposed. We show that in the presence of an arbitrary affine connection, the gravitational field is described as nonmetricity of the affine connection. An affine connection can be interpreted as induced by a frame of reference (FR). Although the gravitational field equations are identical to Einstein's equations of GR, this formulation leads to a covariant tensor (instead of the pseudotensor) of energy-momentum of the gravitational field and covariant conservation laws. We further develop a geometric representation of FR as a metri…
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…
The metrics of general relativity generally fall into two categories: Those which are solutions of the Einstein equations for a given source energy-momentum tensor, and the "reverse engineered" metrics -- metrics bespoke for a certain purpose. Their energy-momentum tensors are then calculated by inserting these into the Einstein equations. This latter approach has found frequent use when confronted with creative input from fiction, wormholes and warp drives being the most famous examples. In this paper, we shall again take inspiration from fiction, and see what general relativity can tell us a…
General relativity poses serious problems for counterfactual propositions peculiar to it as a physical theory. Because these problems arise solely from the dynamical nature of spacetime geometry, they are shared by all schools of thought on how counterfactuals should be interpreted and understood. Given the role of counterfactuals in the characterization of, inter alia, many accounts of scientific laws, theory confirmation and causation, general relativity once again presents us with idiosyncratic puzzles any attempt to analyze and understand the nature of scientific knowledge must face.
The initial-value problem is posed by giving a conformal three-metric on each of two nearby spacelike hypersurfaces, their proper-time separation up to a multiplier to be determined, and the mean (extrinsic) curvature of one slice. The resulting equations have the {\it same} elliptic form as does the one-hypersurface formulation. The metrical roots of this form are revealed by a conformal ``thin sandwich'' viewpoint coupled with the transformation properties of the lapse function.
It has been shown that the extension of the elasticity theory in more than three dimensions allows a description of space-time as a properly stressed medium, even recovering the Minkowski metric in the case of uniaxial stress. The fundamental equation for the metric in the theory is shown to be the equilibrium equation for the medium. Examples of spherical and cylindrical symmetries in four dimensions are considered, evidencing convergencies and divergencies with the classical general relativity theory. Finally the possible meaning of the dynamics of the four dimensional elastic medium is disc…
