Articles in chronological order on two topics in general relativity: mathematical
cosmology and spacetime splittings.
Not all "[pdf]" links to the full text articles are available on-line
without institutional journal subscription intervention.
[List with No Abstracts]
An undergraduate introduction to the dynamics of class A spatially homogeneous cosmological models influenced by the work of Istvan Ozvath.
A study of Moncrief's gauge invariant perturbation theory applied to the diagonal class A spatially homogeneous cosmological models using the tools of Lie group harmonic analysis.
A review of Lie group theory and a study of the symmetry actions on matrix groups and inner product spaces and on tangent and tangent bundles, followed by an analysis of the Bianchi groups and their associated adjoint and automorphism groups, and isometric actions. The dynamics of spatially homogeneous cosmological models is then analyzed with this machinery.
The scalar, vector, and tensor harmonics on the 3-sphere are developed by its identification with SU(2), enabling familiar angular momentum techniques to be employed. The application to spatially homogeneous cosmology is discussed. In this context the classic work of Lifshitz and the recent approach of Hu are bridged. Finally spinor harmonics are introduced.
The true analogues of superspace and conformal superspace for spatially homogeneous cosmology are introduced and discussed in relation to the kinematics of the evolution of Cauchy data from a spatially homogeneous initial value surface using a spatially homogeneous lapse function. Having fixed the slicing to be the natural one, an obvious restriction on the freedom of choice of the shift vector field occurs, and its relation to the three-dimensional diffeomorphism gauge group of the problem is explained. In this context the minimal distortion shift equation of Smarr and York naturally arises. Finally these ideas are used to simplify the dynamics.
Lie group harmonic analysis is applied to the solution of tensor equations on LRS class A spatially homogeneous spacetimes. The techniques developed are used to discuss the Hamiltonian dynamics of the linearized vacuum Einstein equations and Moncrief's orthogonal decomposition of the linearized phase space adapted to the linearized gauge transformations. This leads to what Moncrief has called ``gauge invariant perturbation theory" for the class of spacetimes under consideration.
York's splitting of the tangent spaces to the space of Riemannian metrics over a 3-manifold is examined in the context of spatially homogeneous cosmology where it is generally found to have two distinct analogues in the noncompact case. This requires a modification of the theory of minimal distortion shifts as it applies to noncompact cosmology and clarifies the roles played by the ``kinematical" and ``dynamical" degrees of freedom in the evolution of spatially homogeneous Cauchy data.
Parameters which appear in the solutions of the dynamical equations of spatially homogeneous cosmology or in the dynamical equations themselves are subject to algebraic relations imposed by the constraint equations, i.e., are confined to a constraint hypersurface in parameter space. Values of these parameters off the constraint hypersurface often correspond to solutions which have an additional stiff perfect fluid source that may or may not be flowing orthogonally to the spatially homogeneous foliation or to a related inhomogeneous but spatially self-similar solution or to a combination of the two. These possibilities are studied and explicitly illustrated, leading to a uniform derivation of most of the known exact anisotropic spatially homogeneous or spatially self-similar solutions as well as some new ones.
The difficulties with the application of Lagrangian or Hamiltonian formulations to spatially homogeneous cosmology are examined from a new point of view and a simple explanation is given for the necessary modifications of those formulations. A rather natural restriction on the shift vector field freedom minimizes the extent of the required modifications, leaving the dynamical Einstein equations in the form of a Lagrangian/Hamiltonian system driven by a nonpotential force. The symmetry group of this classical mechanical system is a representation of the diffeomorphism group corresponding to the restricted class of shift vector fields.
A classification of inequivalent Belinsky-Zakharov coordinate systems is given for all spatially homogeneous and spatially self-similar vacuum and stiff perfect-fluid spacetimes which admit a 2-dimensional Abelian group of isometries acting orthogonally transitively on 2-dimensional spacelike surfaces, the symmetry which is necessary for the application of the Belinsky-Zakharov solution-generating technique. The Belinsky-Zakharov complex matrix function $\psi$ is then found for all known coordinate systems in which the component matrix of the spacetime is diagonal. Using only algebraic manipulations involving the $\psi$-matrix, one can in principle generate new exact (inhomogeneous) cosmological solutions containing an arbitrary number of ``solitons."
A brief summary of the article of the same title published in Nuovo Cimento after the meeting held in 1979.
Expressions are given for the energy and momentum of a free particle in a Friedmann universe and in an Einstein-Strauss universe. Special attention is given to the determination of the conserved quantities for the particle motion. The necessary conditions are given for a particle injected into a Schwarzschild geometry from an expanding open, closed or flat dust-filled Friedmann universe, matched to that geometry, to find itself in a bound orbit.
The growth of massive neutrino halos around a mass condensation in an expanding universe is considered using the simplified model of an Einstein-Strauss universe. This model illustrates the possibility of capturing freely moving particles in a homogeneous and isotropic gas by the local graviational field of a mass condensation. This capture process is particularly relevant to the neutrino background in a neutrino dominated university and explicit formulas are given relating the maximum capture rate by a neutrino condensation to the cosmological redshift and the mass of the condensation. Analogies between self-gravitating systems of neutrinos in an asymptotically flat spacetime and the condensations which maximize the capture rate are noted and observational consequences are considered.
An alternative Hamiltonian formulation is presented for the spatially homogeneous Einstein-Dirac system which in the nondegenerate case enables the number of gravitational degrees of freedom to be explicitly reduced to two.
Uniformly accelerated analogs of the stationary axially symmetric 2$n$-soliton solutions of the vacuum einstein equations obtained from flat spacetime are discussed and the ``static" case is studied in detail. The $C$-metric occurs as the $n=1$ solution while the Bonnor-Swaminarayan solution occurs as a certain $n=\infty$ limit.
The construction of a class of associative algebras $q_n$ on $R^4$ generalizing the well-known quaternions $Q$ provides an explicit representation of the universal enveloping algebra of thereal three-dimensional Lie algebras having tracefree adjoint representations (class A bianchi type Lie algebras). The identity components of the four-dimensional Lie groups GL$(q_n,1)\subset q_n$ (general linear group in one generalized quaternion dimension) which are generated by the Lie algebra of this class of quaternion algebras are diffeomorphic to the manifolds of spacetime homogeneous and spatially homogeneous spacetimes having simply transitive homogeneity isometry groups with tracefree Lie algebra adjoint representations. In almost all cases the complete group of isomoetries of such a spacetime is isomorphic to a subgroup of the group of left and right translations and automorphisms of the appropriate generalized quaternion algebra. Similar results hold for the single class B Lie algebra of Bianchi type V, characterized by its ``pure trace" adjoint representation.
Spatially homogeneous perfect fluid spacetimes are studied from a point of view which emphasizes the spatial geometry and the action of that subgroup of the spatial gauge group of the three-plus-one formulation of general relativity which is compatible with the spatial homogeneity. The specializations of the dynamics which correspond to the existence of additional spacetime symmetries are classified. An unconstrained set of gravitational and fluid variables is obtained by elimination of the gravitational constraints using an approach which obtains the gravitational evolution equations from a suitably modified Lagrangian/Hamiltonian formalism. A slightly different choice of variables is then described which allows one to take full advantage of the spatial gauge group and of the 1-parameter group of scale transformations of the unit of length.
The class of reference systems compatible with the symmetry of a spatially homogeneous perfect fluid spacetime is discussed together with the associated class of symmetry adapted comoving ADM frames (or computational frames). The fluid equations of motion are related to the four functions on the space of fluid flow lines discovered by Taub and which characterize an isentropic flow.
All spatially homogeneous spacetimes are described as a single parametrized family. Coupled with the explicit parametrization of the natural decomposition of the dynamical variables suggested by the spatial gauge freedom, one arrives at a parametrized unconstrained reduced Hamiltonian system of a point particle in 2-dimensions (in general) subject to various time-dependent potentials whose explicit form is very useful in deducing gross features of the evolution of the system.
The Einstein equations for a perfect fluid spatially homogeneous spacetime are studied in a unified manner by retaining the generality of certain parameters whose discrete values correspond to the various Bianchi types of spatial homogeneity. A parameter dependent decomposition of the metric variables adapted to the symmetry breaking effects of the nonabelian Bianchi types on the ``free dynamics" leads to a reduction of the equations of motion for those variables to a 2-dimensional time dependent Hamiltonian system containing various time dependent potentials which are explicitly described and diagrammed. These potentials are extremely useful in deducing the gross features of the evolution of the metric variables.
The definition of partially self-similar spacetimes introduced by one of the present authors is clarified and it is shown that compatibility with the Einstein equations reduced this class of spacetimes to the spatially self-similar case.
The relevance of harmonic mappings to the SU($N$) self-dual Yang-Mills field equations is clarified.
The finite dimensional Einstein-Maxwell-scalar field system, characterized by a spatially homogeneous or spatially self-similar gravitational field, is studied from a group theoretical point of view.
An exact power law metric is discussed which arises when one considers the reduced Einstein equations for certain scale invariant variables associated with a spatially homogeneous or spatially self-sismilar vacuum or nonvacuum spacetime. The metric contains a number of new solutions as well as many known ones.
[See the later clarification in 2003 by Pantelis S. Apostolopoulos]
Spatially homogeneous and non-exceptional self-similar spacetime metrics which are `exact power law metrics' are defined, explicitly parametrized and shown to have fixed conformal 3-geometry in the natural slicing of the spacetime by the orbits of the symmetry group and to admit a homothetic Killing vector field not tangent to that slicing. In fact the exact power law metrics are exactly those spatially homogeneous and spatially self-similar metrics which admit a homothetic Killing vector field not tangent to the spacelike orbits of the homogeneity or sel-similarity group. Such metrics arise as `singular point solutions' of gravitational field equations when formulated as a certain system of first-order ordinary differential equations. These special exact solutions play an important role in the qualitative behavior of the general solution of a given set of field equations and sources with these symmetries.
Spacetimes admitting a four-dimensional transitive similarity group are studied. It is shown that when the homogeneous hypersurfaces (which necessarily exist) are spacelike (spatially homogeneous case) then the connection components in a Lorentz frame adapted to the spatially homogeneous slicing are proportional to $t^{-1$ where $t$ is the proper time of the homogeneous slices. An analogous statement holds when those slices are timelike except that $t$ is a spacelike coordinate in that case. The transitively self-similar universes correspond exactly to the exact power law solutions in Wainwright's terminology. They are also closely connected to the regularized form of the Einstein field equations used in the qualitative approach to spatially homogeneous cosmology. In fact the transitively self-similar models are in one-one correspondence with those critical points of the regularized system which lie in the physical region of that system.
The relationship between the hypersurface-homogeneous slicing of an exact power law metric spacetime and slicings adapted to spatial self-similarities is discussed in a group theoretical setting.
Spacetimes admitting a transitive group of self-similarities are discussed. The relation to the conformally related homogeneous spacetime is given. Solutions of the Einstein equations with this symmetry are exact power law metrics. The parameters appearing in the metric can be related to the structure constants of the similarity group. As an example, that relation is given explicitly for the Kasner solution.
The finite dimensional Einstein-Maxwell-scalar field system, characterized by a spatially homogeneous or spatially self-similar gravitational field, is approached from a group theoretical point of view. The field equations are expressed as a parametrized finite dimensional mechanical system whose singularity behavior is readily apparent, using a method which allows one to consider all possible symmetry types simultaneously.
The problem of imposing symmetry on variational principles for higher dimensional theories is illustrated by considering spatially homogeneous solutions of Kaluza Klein theories. Various implications of the group theoretical nature of this specific situation are addressed.
A brief summary of the connection between exact power law metrics and the use of scale invariant variables in formulating the Hamiltonian dynamics of spatially homogeneous cosmological models.
The aim of this note is to clarify confusing notions of the word ``rotation" as applied to cosmological solutions of metric theories of gravity, both in general and in the specific case addressed by the article in which these confusing notions have recently reappeared.
A broader perspective is suggested for the study of higher dimensional cosmological models.
[1986: Considerations involving the Einstein constraints and the Ricci form of the evolution equations for spatially homogeneous spacetimes in 4 or more dimensions in the Taub time gauge (constant densitized lapse) are used to show how many of the special exact solutions found at random fit together into a larger picture in which the Taub spacetime solution plays an instructive role. Concludes with some remarks on chaos.]
A global picture is drawn tying together most exact cosmological solutions of gravitational theories in four or more dimensions.
The additional conditions satisfied by the generalized Kasner solution of the vacuum Einstein equations in $N+1$ spacetime dimensions are explicitly shown to have a simple geometric interpretation in the Hamiltonian formulation.
The Einstein equations for a perfect fluid filled spatially homogeneous spacetime are expressed in a reduced form which is as close as possible to a compactified regularized first order system of differential equations, while still respecting the scale invariance and spatial gauge symmetry of those equations. the present work is a generalization to all Bianchi types of the regularization procedure introduced by Rosquist for Bianchi types III and VI. Jantzen's unified Lie algebra automorphism formalism is used to reduce the gravitational phase space to a subspace parametrized by a minimal number of variables. Although motivated by the qualitative theory of differential equations, the reduced homogeneous Einstein system has also proved useful in the search for new exact solutions.
The choice of time function for cosmological solutions of gravitational field equations is related to the action of the group of independent scale transformations of the unit of length along orthogonal spatial directions. This is accomplished by the introduction of lapse functions which depend explicitly on the spatial metric in an appropriately defined power law fashion. The resulting power law lapse time gauges are the key to producing nearly all exact solutions of the class of models for which the field equations reduce to ordinary differential equations.
Most exact solutions of gravitational field equations which can be reduced to ordinary differential equations are fit together in a unifying manner by fixing the gauge of the related independent variable. Adopting a power law gauge condition adapted to the structure of those equations leads to at least one decoupled variable when an exact solution exists, and the behavior of this variable can often be reduced to a simple one dimensional scattering problem by a suitable redefinition of that variable.
Gravitoelectric and gravitomagnetic fields are placed in the context of a parametrized nonlinear reference frame on spacetime, in both the slicing and threading points of view.
Power law lapse and Jacobi time gauges are explored with an eye towards taking advantage of symmetries of the dynamics of Bianchi cosmological models.
Symmetries and time reparametrizations are used in a series of articles to study the Hamiltonian equations for certain vacuum and orthogonal perfect fluid Bianchi cosmologies. Using the framework of an extended phase space containing the time variable itself, one may couple the scale invariance symmmetry and the non-unimodular automorphism admitted by all the non-semisimple Bianchi types to obtain a variational symmetry whose associated constant of the motion in general depends explicitly on the time. In the vacuum case this is time independent for certain preferred choices of time gauge corresponding to preferred choices of lapse functions which may be constructed from symmetry invariants. One such preferred choice of lapse function is the Jacobi lapse, for which the field equations are reduced to a geodesic flow on a conformally flat Lorentz geometry. This article studies such preferred choices of lapse functions and the associated constants of the motion for the vacuum case.
Symmetry compatible time reparametrizations are calculated for orthogonal perfect fluid Hamiltonian Bianchi Cosmologies by using an extended phase space containing the time variable itself. As in the vacuum case treated in an earlier article, the scale symmetry and the non-unimodular automorphism can be coupled to yield a constant of the motion which in the perfect fluid case is necessarily explicitly time dependent except for the stiff matter equation of state. To obtain the symmetry compatible lapse functions it is necessary to start from one of three `source adapted' time gauges, two of which are the Bogoyvlensky-Novikov and Jacobi time gauges. The latter reduces the field equations to geodesic flows on certain conformally flat Lorentzian geometries. In these Jacobi geometries, the symmetry giving rise to the time-dependent constant of the motion is just a homothetic motion.
The reparametrization freedom in the choice of time variable in the dynamics of spatially homogeneous cosmological models is used to reformulate the field equations as a geodesic flow for a ``Jacobi geometry" in a particular time gauge called the Jacobi time gauge. For the diagonalizable models this Jacobi geometry is a conformally flat Lorentzian geometry. By choosing variables which are adapted to the symmetries of the Jacobi geometry, considerable simplification of the field equations is achieved, and one can explain the existence of all known exact solutions in terms of this analysis, as well as simplify the study of the qualitative behavior of the dynamics. In addition, certain ``hidden symmetries" which arise in the Jacobi formulation lead to a class of new exact solutions.
A single mathematical framework is introduced to discuss both the hypersurface and congruence approaches to splitting spacetime and to clarify their relationship to each other and to the invariant 4-geometry.
Some simple observations are made to clear up misconceptions regarding the behavior of spatially homegeneous monotonically expanding general relativistic cosmological models at very late times, as well as to emphasize some simple features of the dynamics in the direction away from an initial singularity even in the case of recollapse. A few examples illustrating these ideas are discussed, including the derivation of some approximate solutions of the late stage field equations.
Using the technique of decomposition of spacetime into ``space plus time,'' the gravitoelectric and gravitomagnetic fields are defined. The equation of motion for the spin of a test gyroscope is found in its exact form and previous results are recovered in the limit of weak gravitational field and small velocity. The geometrical interpretation of the results is discussed in detail.
An investigation of those cases of the generalized Friedmann equation which are solvable in terms of elementary or elliptic functions is undertaken together with a study of the time gauges which allow this to occur. This is accomplished by examining the natural choices of independent and dependent variables in this problem, using manipulations like those of the Kepler problem, which is shown to be equivalent to a generalized Friedmann problem, thus clarifying the similarities between the simplest solutions of each.
The numerous ways of introducing spatial gravitational forces are fit together in a single framework enabling their interrelationships to be clarified. This framework is then used to treat the ``acceleration equals force" equation and gyroscope precession, both of which are then discussed in the post-Newtonian approximation, followed by a brief examination of the Einstein equations themselves in that approximation.
Some brief remarks are made regarding the many possible approaches to splitting spacetime within general relativity and the subsequent definitions of related gravitoelectric and gravitomagnetic spatial gravitational force fields.
Applications of the various spacetime splitting formalisms as well as the corresponding gravitoelectromagnetism language are considered for the problem of circular geodesics in the ``rotating Minkowski," G\"odel, and Kerr spacetimes.
An underlying mechanism is discussed which explains why the use of power law variables and power law gauges has been so successful in finding exact hypersurface-homogeneous and hypersurface-self-similar models.
A mechanism for the occurrence of exact solutions in finite-dimensional Hamiltonian systems is discussed with special emphasis on minisuperspace applications. Examples are given for spatially homogeneous universes, gravitationally coupled scalar field solutions and relativistic star models (static spherically symmetric fields).
Drawing upon our knowledge of both special relativity and noninertial forces in nonrelativistic mechanics, the ``gravitoelectomagnetic" approach to general relativity is seen to be a marriage of both these well known topics on the arena of spacetime. The approach involves splitting spacetime into space plus time, i.e., interpreting spacetime quantities by the introduction of a family of test observers used to measure them.
The various spacetime splitting formalisms and corresponding gravitoelectromagnetic variables are introduced to study the three best known rotating spacetimes, ``rotating Minkowski," G\"odel, and Kerr. A careful analysis of the timelike circular geodesics in these spacetimes leads to a better understanding of the effects of rotation in them.
This was an article which later evolved into ``Exact Hypersurface-Homogeneous Solutions in Cosmology and Astrophysics" after additional work.
A brief summary of the motivation for gravitoelectromagnetism and its application to instructive example spacetimes which have played a key role in understanding rotation in general relativity.
A brief summary of the idea that a hierarchical structure exists on the space of solutions of the spatially homogeneous Einstein equations in which successively simpler models describe limiting behaviors of more complicated models.
A mechanism is presented for obtaining exact solutions of the Einstein equations for hypersurface-homogeneous scalar fields which unifies and generalizes recent results for inflaton fields in the spatially homogeneous case and for thick domain walls in the timelike-homogeneous case.
The well known Thomas precession effect is discussed in the context of the post-Newtonian approximation to general relativity using the language of gravitoelectromagnetism (3-plus-1 splitting of gravitational theory). Preliminary discussion anchors the post-Newtonian coordinate system and choice of gravitational variables in the mathematical structure of fully nonlinear general relativity, linking the post-Newtonian gravitoelectric and gravitomagnetic fields to kinematical properties of the associated observer congruence. The transformation laws for these fields under a change of post-Newtonian coordinate system are derived first within the post-Newtonian theory and then by taking the limit of their fully nonlinear form to reveal the interpretation of the various terms in the post-Newtonian case. These transformation laws are then used to make a case for the existence of a gravitational analog of the ordinary Thomas precession of the spin of a gyroscope.
A framework is introduced which explains the existence and similarities of most exact solutions of the Einstein equations with a wide range of sources for the class of hypersurface-homogeneous spacetimes which admit a Hamiltonian formulation. This class includes the spatially homogeneous cosmological models and the astrophysically interesting static spherically symmetric models as well as the stationary cylindrically symmetric models. The framework involves methods for finding and exploiting hidden symmetries and invariant submanifolds of the Hamiltonian formulation of the field equations. It unifies, simplifies and extends most known work on hypersurface-homogeneous exact solutions. It is shown that the same framework is also relevant to gravitational theories with a similar structure, like Brans-Dicke or higher-dimensional theories.
The straightforward reformulation of special relativistic concepts about relative observer kinematics in the context of the flat affine geometry of Minkowski spacetime so that they respect the manifold structure of that spacetime allows one to derive the general relativistic ``addition of acceleration law." This transformation law describes the relationship between the relative accelerations of a single test particle as seen by two different families of test observers.
Arguments are made in favor of broadening the scope of the various approaches to splitting spacetime into a single common framework in which measured quantities, derivative operations, and adapted coordinate systems are clearly understood in terms of associated test observer families. This ``relativity of splitting formalisms" for fully nonlinear gravitational theory has been tagged with the name ``gravitoelectromagnetism" because of the well known analogy between its linearization and electromagnetism, and it allows relationships between the various approaches to be better understood and makes it easier to extrapolate familiarity with one approach to the others. This is important since particular problems or particular features of those problems in gravitational theory are better suited to different approaches, and the present barriers between the proponents of each individual approach sometimes prevent the best match from occurring.
In General Relativity the gravitational field is often described in terms of gravitoelectric and gravitomagnetic fields stressing the analogy between Maxwell's theory of electromagnetism and Einstein's theory of gravitation. Recently a Newtonian analogy has been revived by Abramowicz et al which describes the gravitational field in terms of inertial forces like the centrifugal and Coriolis forces. Here we show how a definition of centripetal acceleration naturally arises within the more general framework of gravitoelectromagnetism, and how it is related to the centrifugal force defined by Abramowicz et al, and how the Newtonian and the Maxwellian analogies are related.
Spacetime splitting plays the important role of reintroducing into general relativity a sort of ``Newtonian terminology" or an ``electromagnetic terminology" which results in a better interface of the (spacetime) four geometry with our (space and time) intuition and experience. Here we show how the electromagnetic analogy, gravitoelectromagnetism (GEM), leads to rewriting the entire set of Einstein equations in a Maxwell-like form, once a Komar current for GEM fields is introduced. Motion (of particles, fluid, spinorial field) in the context of GEM is also discussed.
The proceedings of the Seventh Marcel Grossmann Meeting on General Relativity, more specifically, on recent developments in theoretical and experimental general relativity, gravitation, and relativistic field theories, which took place at Stanford University in July 1994. One of two trienniel international meetings regularly held in this field. Two volumes of nearly 1800 pages of almost 400 author contributions.
Everyday experience with centrifugal forces has always guided thinking on the close relationship between gravitational forces and accelerated systems of reference. Once spatial gravitational forces and accelerations are introduced into general relativity through a splitting of spacetime into space-plus-time associated with a family of test observers, one may further split the local rest space of those observers with respect to the direction of relative motion of a test particle world line in order to define longitudinal and transverse accelerations as well. The intrinsic covariant derivative (induced connection) along such a world line is the appropriate mathematical tool to analyze this problem, and by modifying this operator to correspond to the observer measurements, one understands more clearly the work of Abramowicz et al who define an ``optical centrifugal force'' in static axisymmetric spacetimes and attempt to generalize it and other inertial forces to arbitrary spacetimes. In a companion article the application of this framework to some familiar stationary axisymmetric spacetimes helps give a more intuitive picture of their rotational features including spin precession effects, and puts related work of de Felice and others on circular orbits in black hole spacetimes into a more general context.
The tools developed in a preceding article for interpreting spacetime geometry in terms of all possible space-plus-time splitting approaches are applied to circular orbits in some familiar stationary axisymmetric spacetimes. This helps give a more intuitive picture of their rotational features including spin precession effects, and puts related work of Abramowicz, de Felice, and others on circular orbits in black hole spacetimes into a more general context.
The automorphism group and frame commutator relations in the orthonormal frame approach to Bianchi cosmology are used to construct an explicit coordinate representation of the orthonormal frame itself (and hence of the spacetime metric) which depends algebraically on the connection coefficients. This is not possible in general inhomogeneous models where differential equations must instead be solved. The shift vector field required for this procedure is intimately related to the true Smarr–York minimal strain and minimal distortion shifts.
Given a nonlinear reference frame splitting spacetime into space plus time and an observer adapted frame associated with this splitting, one can induce from it an observer adapted frame for a new family of observers in a number of ways using the geometry of the relative observer plane, leading to a geometric interpretation of the Cattaneo-Ferrarese quasi-natural frames. Conversely, given an observer adapted frame for a new family of observers, the spacetime Frenet-Serret orthonormal frame being an example, one can induce from it an observer adapted frame for the original nonlinear reference frame observers in the same way. Similarly a relative Frenet-Serret observer adapted orthonormal frame may also be constructed using the properties of the relative motion of the two families of observers.
The somewhat fragmented body of current literature analyzing the properties of test particle motion in static and stationary spacetimes and in general spacetimes is pulled together and clarified using the framework of gravitoelectromagnetism.
Spacetime splitting approaches are applied to circular orbits in Kerr spacetime. This gives a more intuitive picture of the rotational features of the Kerr solution and clarifies the related works of Abramowicz, de Felice, and others on circular orbits in black hole spacetimes.
In nonrelativistic mechanics non-inertial observers studying accelerated test particle motion experience a centripetal acceleration which, once interpreted as a centrifugal force acting on the particle, allows writing the particle's equation of motion in a Newtonian form, simply by adding the inertial forces contribution to that of the external forces in the acceleration-equals-force equation. In general relativity centripetal and centrifugal acceleration generalizing the classical concepts must be properly (geometrically) defined. A useful tool, deeply used in this paper, is the relative Frenet-Serret frame approach.
A geometrical interpretation is given for the Frenet-Serret structure along constant speed circular orbits in orthogonally transitive stationary axisymmetric spacetimes. This gives a simple visualization of the acceleration of these orbits and of the Fermi-Walker angular velocity of the usual symmetry-adapted frame vectors along them and provides an elegant description of the various observers characterized by critical values of the variables parametrizing the acceleration.
Relative Frenet-Serret 3-frames are defined along a test particle world line with respect to a family of observers and a corresponding comoving relative Frenet-Serret 3-frame is introduced in the local rest space of the test particle itself. The latter frame provides the tools to geometrically define a generalized centrifugal force, tied to the rotation of the relative velocity of the test particle with respect to the observers relative to gyro-fixed axes in the test particle local rest space. For circular orbits in a stationary axisymmetric spacetime, all of these relative frames for any circular observers are closely related to the spacetime Frenet-Serret frame, but it is the comoving frame which reveals the geometric interpretation of the minimal intrinsic rotation observers, which are complementary to the extremely accelerated observers to which they reduce in the equatorial plane of rotating black hole spacetimes.
The Frenet-Serret approach is applied in several ways to a familiar but still not geometrically well understood example: circular orbits in black hole spacetimes. An invariant spacetime Frenet-Serret frame approach is useful in understanding the properties of these orbits and of Fermi-Walker transport along them, and provides a visual interpretation of the geometry of this family of orbits. Closely related to the spacetime frame for these special curves are the relative Frenet-Serret frames that may be defined with respect to a family of test observers on the spacetime. The latter connect more directly to our 3-dimensional intuition about the tangent, normal, and binormal to a curve in ordinary space. These absolute and relative frames together help interpret the effects of space curvature, and the gravitoelectric and gravitomagnetic effects on circular orbiting test particles and on their gyroscopic frames of reference.
An embedding diagram helps visualize the observer-dependent projected spatial geometry of the equatorial plane in the Kerr spacetime as seen by a family of circularly rotating observers. This is described for the various geometrically defined families of such observers, motivated by the attempt to better understand how the spatial geometry contributes to the properties of particle motion as seen by those observers. A new family of observers is introduced for which the equatorial plane appears to be flat.
The integral formulation of Maxwell's equations expressed in terms of an arbitrary observer family in a curved spacetime is developed and used to clarify the meaning of the lines of force associated with observer-dependent electric and magnetic fields.
Gravitoelectromagnetism has proven to be a powerful tool for observer-dependent analysis of phenomena in general relativity. Here a brief review is made of applications to test-particle dynamics, spinning test-particles, fluid dynamics, electromagnetism and scalar fields. Relative-observer embedding diagrams are briefly commented on as well.
The gravitomagnetic clock effect and the Sagnac effect for circularly rotating orbits in stationary axisymmetric spacetimes are studied from a relative observer point of view, clarifying their relationships and the roles played by special observer families. In particular Semer\'ak's recent characterization of extremely accelerated observers in terms of the two-clock clock effect is shown to be complemented by a similarly special property of the single-clock clock effect.
The first English translation from the Italian of the article classifying possible geometries for the spaces of spatially homogeneous but anisotropic cosmological models in general relativity, accompanied by a detailed summary and interpretation of its content, and a historical review of its application in cosmology.
On-line version of over 600 typewritten pages of 45 oral history transcripts recorded in 1984 by the remaining survivors of the Princeton Mathematics Community in the 1930s, a community comprised of the Department of Mathematics of Princeton University and the Department of Mathematical Sciences of the newly established Institute for Advanced Study, housed together in the same building, the original Fine Hall, during the period 1933-1939. The Oral History Project was carried out by Albert Tucker, William Aspray, and Rik Nebeker with the help of Charles Gillispie. Supporting on-line documents by Bob Jantzen place the story in context.
A short description of the Project and the way in which Bob Jantzen stumbled into it and produced its on-line version, making a plea to scientists to think about investing a bit more effort in recording the human side of science for posterity.
The definition of inertial forces in general relativity is briefly discussed for circular orbits in black hole spacetimes.
A new compact form of the Teukolsky Master Equation suggests a meaningful interpretation of the perturbation theory at every order in every vacuum spacetime for which the De Rham-Lichnerowicz Laplacian plays an essential role. Very recent work on this subject is recovered and extended. The nonvacuum case as well as the half-integer spin cases can be obtained following the same procedure.
We present a new observer-dependent integral formulation of Maxwell's equations in curved spacetime and give a classical interpretation of them.
The Cotton-York and Simon tensors are studied with the congruence approach. Their relation with the Papapetrou field defined by the stationary Killing congruence and with a recent characterization of the Kerr spacetime in terms of the alignment between the principal null directions of the Weyl tensor and those of the Papapetrou field is also investigated.
Three volumes of 2400 some pages of some 500(?) articles on the current state of physics touching upon classical and quantum gravity and relativistic astrophysics and space measurements. Complemented by an on-line version:
http://www.icra.it/MG/mg9/mg9.htm [World Scientific eProceedings]
While it is difficult to associate mental images with concepts on a general spacetime manifold, by making a local identification with Minkowski spacetime, we can transfer our space-plus-time experience in this simpler context, already upgraded to special relativity from Newtonian physics, to a general curved spacetime through a spacetime splitting scheme. This locally decomposes spacetime into ``space plus time" and permits the splitting of spacetime tensors and equations into the more familiar counterparts grounded in our Newtonian intuition.
Gravitoelectromagnetism has been used to analyse test particle motion (circular orbits) in black hole spacetimes, where it has been possible to study inertial forces according to the most appropriate general relativistic definition and compare with the corresponding Newtonian behaviour. Relative Frenet-Serret frames were introduced which generalize the spatial Frenet-Serret machinery needed for the analysis of test particle motion in classical mechanics. This approach and its application to general circular (nonequatorial orbits) in a Kerr spacetime are reviewed here.
An exhaustive list of references with titles on the general topic of spacetime splittings in general relativity and gravitoelectromagnetism.
Gravitomagnetic clock effects for circularly rotating orbits in black hole spacetimes are studied from a relative observer point of view, clarifying the roles played by special observer families.
The Cotton-York and Simon-Mars tensors in stationary spacetimes are studied in the language of the congruence approach pioneered by Hawking and Ellis. Their relationships with the Papapetrou field defined by the stationary Killing congruence and with a recent characterization of the Kerr spacetime in terms of the alignment between of the principal null directions of the Weyl tensor with those of the Papapetrou field are also investigated in this more transparent language.
Stationary axisymmetric spacetimes containing a pair of oppositely-rotating periodically-intersecting circular geodesics allow the study of various so-called `clock effects' by comparing either observer or geodesic proper time periods of orbital circuits defined by the observer or the geodesic crossing points. This can be extended from a comparison of clocks to a comparison of parallel transported vectors, leading to the study of special elements of the spacetime holonomy group. The band of holonomy invariance found for a dense subset of special geodesic orbits outside a certain radius in the static case does not exist in the nonstatic case. In the Kerr spacetime the dimensionless frequencies associated with parallel transport rotations can be expressed as ratios of the proper and average coordinate periods of the circular geodesics.
The historical origins of Fermi-Walker transport and Fermi coordinates and the construction of Fermi-Walker transported frames in black hole spacetimes are reviewed. For geodesics this reduces to parallel transport and these frames can be explicitly constructed using Killing-Yano tensors as shown by Marck. For accelerated or geodesic circular orbits in such spacetimes, both parallel and Fermi-Walker transported frames can be given, and allow one to study circular holonomy and related clock and spin transport effects. In particular the total angle of rotation that a spin vector undergoes around a closed loop can be expressed in a factored form, where each factor is due to a different relativistic effect, in contrast with the usual sum of terms decomposition. Finally the Thomas precession frequency is shown to be a special case of the simple relationship between the parallel transport and Fermi-Walker transport frequencies for stationary circular orbits.
For the still unpublished English translation of Fermi's original 1922 article "Sopra i fenomeni che avvengono in vicinanza di una linea oraria," see this link.
An exact wave equation satisfied by the Weyl tensor and the Maxwell tensor is studied in the Newman-Penrose formalism and in its Geroch-Held-Penrose variant in the case of vacuum spacetimes. Linearization of these wave equations gives all the well known results of perturbation theory, which are instead obtained by algebraic manipulation of those equations without a clear understanding of their true origin in the exact theory.
The transformation laws for the electric and magnetic parts of the electromagnetic 2-form and the Weyl tensor under a boost are studied using the complex vector approach, which shows the close analogy between the two cases. For a nonnull electromagnetic field, one can always find an observer which sees parallel electric and magnetic fields (vanishing Poynting vector) and also sees a minimum electromagnetic energy density (and minimum electric and magnetic field magnitudes) compared to other observers. For Weyl fields of all Petrov types except III, and boosts along certain directions of the Weyl principal tetrad, the more complicated Weyl transformation closely mimics the electromagnetic boost transformation, allowing one to extend the electromagnetic results directly to the families of boosts along those directions. In particular for black hole spacetimes, the alignment of the electric and magnetic parts of the Weyl tensor (vanishing super-Poynting vector) leads to minimal gravitational super-energy as seen by the Carter observer within the family of all circularly rotating observers at each spacetime point outside the horizon.
Parallel transport around closed circular orbits in the equatorial plane of the Taub-NUT spacetime is analyzed to reveal the effect of the gravitomagnetic monopole parameter on circular holonomy transformations. Investigating the boost/rotation decomposition of the connection 1-form matrix evaluated along these orbits, one finds a situation that reflects the behavior of the general orthogonally transitive stationary axisymmetric case and indeed along Killing trajectories in general.
A single master equation is given describing spin s<=2 test field gauge and tetrad invariant perturbations of the Taub-NUT spacetime. This solution of vacuum Einstein field equations describes a black hole with mass M and gravitomagnetic monopole moment l. This equation can be separated in its radial and angular parts. The behaviour of the radial functions at infinity and near the horizon is studied. The angular equation, solved in terms spherical harmonics of suitable weight resulting from the coupling of the spin-weight of the field and the gravitomagnetic monopole moment of the spacetime, is related with the total angular momentum operator associated with the spacetime's rotational symmetry. The results are compared with the Teukolsky master equation for the Kerr spacetime.
We study the motion of test particles and electromagnetic waves in the Kerr-Newman-Taub-NUT spacetime in order to elucidate some of the effects associated with the gravitomagnetic monopole moment of the source. In particular, we determine in the linear approximation the contribution of this monopole to the gravitational time delay and the rotation of the plane of the polarization of electromagentic waves. Moreover, we consider "spherical" orbits of uncharged test particles in the Kerr-Taub-NUT spacetime and discuss the modification of the Wilkins orbits due to the presence of the gravitomagnetic monopole.
A single master equation is given describing spin s <= 2 test fields that are gauge- and tetrad-invariant perturbations of the Kerr-Taub-NUT spacetime representing a source with mass M, gravitomagnetic monopole moment l and angular momentum per unit mass a. This equation can be separated into its radial and angular parts. The behavior of the radial functions at infinity and near the horizon is studied and used to examine the influence of l on the phenomenon of superradiance, while the angular equation leads to spin-weighted spheroidal harmonic solutions generalizing those of the Kerr spacetime. Finally the coupling between the spin of the perturbing field and the gravitomagnetic monopole moment is discussed.
The historical development of the Bianchi classification of homogeneous cosmological models is described with special emphasis on the contribution of Schűcking and Behr.
We review special timelike curves describing stationary axisymmetric circular orbits which arise in the discussion of some geometrical and physical features of the Kerr spacetime in its equatorial plane (i.e., transport laws for vectors, geodesics and accelerated orbits, clock effects, circular holonomy, etc.).
Princeton played an important role not only in the renaissance of general relativity that occurred in the 1960s and 1970s, but also in Remo's career. A brief overview is given of the background story through the lens of my own connections to Princeton and Remo and Rome and my subsequent work in Bianchi cosmology and the geometry of spacetime splittings.
Rotating observers and circular test particle orbits in Minkowski spacetime are used to illustrate the transport laws and derivative operators needed to define the various ``inertial forces" one can introduce using the natural relative observer approach to describing spacetime. Various centripetal accelerations (often called centrifugal forces when multiplied by the mass) are evaluated and compared with the familiar value v^{2}/r of nonrelativistic physics.
Accelerated circular orbits in the equatorial plane of the Taub-NUT spacetime are analyzed to investigate the effects of its gravitomagnetic monopole source. The effect of a small gravitomagnetic monopole on these orbits is compared to the corresponding orbits pushed slightly off the equatorial plane in the absense of the monopole.
The de Rham Laplacian Δ_{dR} for differential forms is a geometric generalization of the usual covariant Laplacian Δ, and it may be extended naturally to tensor-valued p-forms using the exterior covariant derivative associated with a metric connection. Using it the wave equation satisfied by the curvature tensors in general relativity takes its most compact form. This wave equation leads to the Teukolsky equations describing integral spin perturbations of black hole spacetimes.
For nonvacuum stationary spacetimes, the Simon tensor and the Simon-Mars tensor differ by source terms which enter the Bianchi identities through the imposition of the field equations on the Ricci curvature terms. For the Einstein-Maxwell case, these source terms can be absorbed into a redefinition of the Simon tensor using the Ernst potential, leading to a situation similar to the vacuum case. This is illustrated for the Kerr-Newmann spacetime, where the Simon-Mars tensor vanishes due to the simultaneous alignment of the principal null directions of the Weyl tensor, the Papapetrou field associated with the timelike Killing vector field and the electromagnetic field of the spacetime.
The relativistic definition of ``inertial forces" is briefly reviewed.
The master equation describing massless fields of different spin in the Kerr-Taub-NUT spacetime representing a source with a mass plus gravitomagnetic monopole and dipole moments is studied. This equation can be separated into its radial and angular parts. The behaviour of the radial functions at infinity and near the horizon can be used to examine the influence of the NUT parameter on the phenomenon of superradiance. Moreover, we discuss the coupling between the spin of the perturbing field and the gravitomagnetic monopole moment.
Parallel transport along circular orbits in orthogonally transitive stationary axisymmetric spacetimes is described explicitly relative to Lie transport in terms of the electric and magnetic parts of the induced connection. The influence of both the gravitoelectromagnetic fields associated with the zero angular momentum observers and of the Frenet-Serret parameters of these orbits as a function of their angular velocity is seen on the behavior of parallel transport through its representation as a parameter-dependent Lorentz transformation between these two inner-product preserving transports which is generated by the induced connection. This extends the analysis of parallel transport in the equatorial plane of the Kerr spacetime to the entire spacetime outside the black hole horizon, and helps give an intuitive picture of how competing ``central attraction forces" and centripetal accelerations contribute with gravitomagnetic effects to explain the behavior of the 4-acceleration of circular orbits in that spacetime.
Abstract on the same topic as the following paper.
The role of the Taub time gauge in cosmology is linked to the use of the densitized lapse function instead of the lapse function in the variational principle approach to the Einstein equations. The spatial metric variational equations then become the Ricci evolution equations, which are then supplemented by the Einstein constraints which result from the variation with respect to the densitized lapse and the usual shift vector field. In those spatially homogeneous cases where the least disconnect occurs between the general theory and the restricted symmetry scenario, the recent adjustment of the conformal approach to solving the initial value problem resulting from densitized lapse considerations is seen to be inherent in the use of symmetry-adapted metric variables. The minimal distortion shift vector field is a natural vector potential for the new York thin sandwich initial data approach to the constraints, which in this case corresponds to the diagonal spatial metric gauge. For generic spacetimes, the new approach suggests defining a new minimal distortion shift gauge which agrees with the old gauge in the Taub time gauge, but which also makes its defining differential equation agree with the vector potential equation for solving the supermomentum constraint.
For stationary vacuum spacetimes the Bianchi identities of the second kind equate the Simon tensor to the Simon-Mars tensor, the latter having a clear geometrical interpretation. The equivalence of these two tensors is broken in the nonvacuum case by additional source energy-momentum terms, but absorbing these source terms into a redefinition of the Simon tensor restores the equality. Explicit examples are discussed for electrovacuum and rigidly rotating matter fields.
A recent paper claiming to have found the true general vacuum solution of the locally rotationally symmetric Bianchi type IX spatially homogeneous Einstein equations already found by Taub in 1951 rediscovers that solution.
We study the behavior of nonzero rest mass spinning test particles moving along circular orbits in the Schwarzschild spacetime in the case in which the components of the spin tensor are allowed to vary along the orbit, generalizing some previous work. We find that only Pirani's supplementary conditions are compatible with this symmetry specialization.
The coordinate transformation which maps the Kerr metric written in standard Boyer-Lindquist coordinates to its corresponding form adapted to the natural local coordinates of an observer at rest at a fixed position in the equatorial plane, i.e., Fermi coordinates for the neighborhood of a static observer world line, is derived and discussed in a way which extends to any uniformly circularly orbiting observer there.
Based on the recent understanding of the role of the densitized lapse function in Einstein's equations and of the proper way to pose the thin sandwich problem, a slight readjustment of the minimal distortion shift gauge in the $3+1$ approach to the dynamics of general relativity allows this shift vector to serve as the vector potential for the longitudinal part of the extrinsic curvature tensor in the new approach to the initial value problem, thus extending the initial value decomposition of gravitational variables to play a role in the evolution as well. The new shift vector globally minimizes the changes in the conformal 3-metric with respect to the spacetime measure rather than the spatial measure on the time coordinate hypersurfaces, as the old shift vector did.
The Frenet-Serret curve analysis and the closely associated Fermi-Walker transport is extended from nonnull to null trajectories in a generic spacetime using the Newman-Penrose formalism and is then used to discuss the motion of massless particles along null circular orbits in stationary axisymmetric spacetimes using the Kerr spacetime as a concrete example.
The motion of massless spinning test particles is investigated using the Newman-Penrose formalism within the Mathisson-Papapetrou model extended to massless particles by Mashhoon and supplemented by the Pirani condition. When the "multipole reduction world line" lies along a principal null direction of an algebraically special vacuum spacetime, the equations of motion can be explicitly integrated. Examples are given for some familiar spacetimes of this type in the interest of shedding some light on the consequences of this model.
The circular motion of spinning massive test particles in the equatorial plane of a rotating black hole is investigated in the case where the components of the spin tensor are allowed to vary along the orbit.
Three natural classes of orthonormal frames, namely Frenet-Serret, Fermi-Walker and parallel transported frames, exist along any timelike world line in spacetime. Their relationships are investigated for timelike circular orbits in stationary axisymmetric spacetimes, and illustrated for black hole spacetimes.
The motion of test particles along circular orbits in the vacuum C metric is studied in the Frenet-Serret formalism. Special orbits and corresponding intrinsically defined geometrically relevant properties are selectively studied.
The speciality index function S for any Petrov type I, II or D spacetime
is shown to be a natural function of a single complex scalar quantity μ (natural modulo permutation symmetries). For the family of Kasner spacetimes, this quantity is a function of the Kasner indices alone which coincides with the real Lifshitz-Khalatnikov parameter u for those indices.
The scale invariant Petrov classification of the Weyl tensor is linked to the scale invariant combination of the Kasner index constraints, and the Lifshitz-Khalatnikov Kasner index parametrization scheme turns out to be a natural way of adapting to this symmetry, while hiding the permutation symmetry that is instead made manifest by the Misner parametrization scheme. While not so interesting for the Kasner spacetime by itself, it gives a geometrical meaning to the famous Kasner map transitioning between Kasner epochs and Kasner eras, equivalently bouncing between curvature walls, in the BLK-Mixmaster dynamics exhibited by spatially homogeneous cosmologies approaching the initial cosmological singularity and the inhomogeneous generalization of this dynamics.
Three volumes of nearly 3000 some pages of some 500(?) articles on the current state of physics touching upon classical and quantum gravity and relativistic astrophysics and space measurements. Complemented by an on-line version:
http://www.icra.it/MG/mg11/mg11.htm [World Scientific eProceedings]
Apollonius studied the normals to an ellipse which pass through a given point in the plane, introducing what is now known as the evolute of the ellipse, concepts which arise in answering the question of what is the minimum and maximum distance to the ellipse from the given point in the plane. The points on the ellipse connected to this point via the normals are the intersections with the hyperbola of Apollonius. The number of such such normals changes from 4 to 3 to 2 as the point crosses the evolute of the ellipse, which is the locus of centers of the osculating circles to the ellipse. Maple is used to bring this discussion to life with still and animated graphics, and brings the associated calculations within the grasp of an undergraduate student.
The general relativistic version is developed for Robertson's discussion of the Poynting-Robertson effect that he based on special relativity and Newtonian gravity for point radiation sources like stars. The general relativistic model uses a test radiation field of photons in outward radial motion with zero angular momentum in the equatorial plane of the exterior Schwarzschild or Kerr spacetime.
The Mixmaster dynamics is revisited in a new light as revealing a series of transitions in the complex scale invariant scalar invariant of the Weyl curvature tensor best represented by the speciality index $\mathcal{S}$, which gives a 4-dimensional measure of the evolution of the spacetime independent of all the 3-dimensional gauge-dependent variables except for the time used to parametrize it. Its graph versus time characterized by correlated isolated pulses in its real and imaginary parts corresponding to curvature wall collisions serves as a sort of electrocardiogram of the Mixmaster universe, with each such pulse pair arising from a single circuit or ``complex pulse'' around the origin in the complex plane. These pulses in the speciality index and their limiting points on the real axis seem to invariantly characterize some of the so called spike solutions in inhomogeneous cosmology and should play an important role as a gauge invariant lens through which to view current investigations of inhomogeneous Mixmaster dynamics.
We study the motion of a test particle in a stationary, axially and reflection symmetric spacetime of a central compact object, as affected by interaction with a test radiation field of the same symmetries. Considering the radiation flux with fixed but arbitrary (non-zero) angular momentum, we extend previous results limited to an equatorial motion within a zero-angular-momentum photon flux in the Kerr and Schwarzschild backgrounds. While a unique equilibrium circular orbit exists if the photon flux has zero angular momentum, multiple such orbits appear if the photon angular momentum is sufficiently high.
The deviation of the path of a spinning particle from a circular geodesic in the Schwarzschild spacetime is studied by an extension of the idea of geodesic deviation. Within the Mathisson-Papapetrou-Dixon model and assuming the spin parameter to be sufficiently small so that it makes sense to linearize the equations of motion in the spin variables as well as in the geodesic deviation, the spin-curvature force adds an additional driving term to the second order system of linear differential equations satisfied by nearby geodesics. Choosing initial conditions for geodesic motion leads to solutions for which the deviations are entirely due to the spin-curvature force, and one finds that the spinning particle position for a given fixed total spin is confined roughly to an ellipsoid about the corresponding geodesic position, modulo a secular drift in the azimuthal angle.
Fermi coordinates are constructed as exact functions of the Schwarzschild coordinates around the world line of a static observer in the equatorial plane of the Schwarzschild spacetime modulo a single impact parameter determined implicitly as a function of the latter coordinates. This illustrates the difficulty of constructing explicit exact Fermi coordinates even along simple world lines in highly symmetric spacetimes.
Motion of test particles in the nonvacuum spherically symmetric radiating Vaidya spacetime is investigated, allowing for physical interaction of the particles with the radiation field in terms of which the source energy-momentum tensor is interpreted. This ``Poynting-Robertson effect" is modeled by the usual effective term describing a Thomson-type radiation drag. The equations of motion are studied for simple types of motion including free motion (without interaction), purely radial and purely azimuthal (circular) motion, and for the particular case of ``static" equilibrium; appropriate solutions are given where possible. The results---mainly those on the possible existence of equilibrium positions---are compared with their counterparts obtained previously for a test spherically symmetric radiation field in a vacuum Schwarzschild background.
A simple observation about the action for geodesics in a stationary spacetime with separable geodesic equations leads to a natural class of slicings of that spacetime whose orthogonal geodesic trajectories represent freely falling observers. The time coordinate function can then be taken to be the observer proper time, leading to a unit lapse function. This explains some of the properties of the original Painlev\'e-Gullstrand coordinates on the Schwarzschild spacetime and their generalization to the Kerr-Newman family of spacetimes, reproducible also locally for the G\"odel spacetime. For the static spherically symmetric case the slicing can be chosen to be intrinsically flat with spherically symmetric geodesic observers, leaving all the gravitational field information in the shift vector field.
We discuss the solution proposed by Fermi to the so called ``4/3 problem'' in the classical theory of the electron, a problem which puzzled the physics community for many decades before and after his contribution to the discussion. Unfortunately his early resolution of the problem in 1922--1923 published in three articles in Italian and German journals went largely unnoticed, and even recent texts devoted to classical electron theory still do not present his argument or acknowledge the actual content of those articles.
Fermi's analysis of the contribution of the electromagnetic field to the inertial mass of the classical electron within special relativity is brought to its logical conclusion, leading to the conservation of the total 4-momentum of the field plus mechanical mass system as seen by the sequence of inertial observers in terms of which the accelerated electron is momentarily at rest.
Since these elementary considerations about the black hole mass formula were not given in the original articles, nor seem to be found in any reviews since those early days of the renaissance of general relativity, it seems useful to present them now. [See Misner, Thorne Wheeler, Gravitation, section 33.8, p.907, in particular the key differential equation 33.56]
Three volumes of about 2600 some pages of some 500(?) articles on the current state of physics touching upon classical and quantum gravity and relativistic astrophysics and space measurements. Complemented by an on-line version:
http://www.icra.it/MG/mg12/ [World Scientific eProceedings]
A two parameter model for the Lorenz curve describing income distribution interpolating between self-similar behavior at the low and high ends of the income spectrum naturally leads to two separate Gini indices describing the low and high ends individually. These new indices accurately capture realistic data on income distribution and give a better picture of how income data is shifting over time.
The features of the scattering of massive neutral particles propagating in the field of a gravitational plane wave are compared with those characterizing their interaction with an electromagnetic radiation field. The motion is geodesic in the former case, whereas in the case of an electromagnetic pulse it is accelerated by the radiation field filling the associated spacetime region. The interaction with the radiation field is modeled by a force term entering the equations of motion proportional to the 4-momentum density of radiation observed in the particle's rest frame. The corresponding classical scattering cross sections are evaluated too.
In considering the mathematical problem of describing the geodesics on a torus or any other surface of revolution, there is a tremendous advantage in conceptual understanding that derives from taking the point of view of a physicist by interpreting parametrized geodesics as the paths traced out in time by the motion of a point in the surface, identifying the parameter with the time. Considering energy levels in an effective potential for the reduced motion then proves to be an extremely useful tool in studying the behavior and properties of the geodesics. The same approach can be easily tweaked to extend to both the nonrelativistic and relativistic Kepler problems. The spectrum of closed geodesics on the torus is analogous to the quantization of energy levels in models of atoms.
A 3-parameter family of helical tubular surfaces obtained by screw revolving a circle provides a useful pedagogical example of how to study geodesics on a surface that admits a 1-parameter symmetry group, but is not as simple as a surface of revolution like the torus which it contains as a special case. It serves as a simple example of helically symmetric surfaces which are the generalizations of surfaces of revolution in which an initial plane curve is screw-revolved around an axis in its plane. The physics description of geodesic motion on these surfaces requires a slightly more involved effective potential approach than the torus case due to the nonorthogonal coordinate grid necessary to describe this problem. Amazingly this discussion allows one to very nicely describe the geodesics of the surface of the more complicated ridged cavatappi pasta.
Innocent musing on geodesics on the surface of helical pasta shapes leads to a single continuous 4-parameter family of surfaces invariant under at least a 1-parameter symmetry group and which contains as various limits spheres, tori, helical tubes, and cylinders, all useful for illustrating various aspects of geometry in a visualizable setting that are important in special and general relativity. In this family the most aesthetically pleasing surfaces come from screw-rotating a plane cross-sectional curve perpendicular to itself, i.e., orthogonal to the tangent vector to a helix. If we impose instead this orthogonality in the Lorentzian geometry of 3-dimensional Minkowski spacetime with a timelike helical ``central" world line representing a circular orbit, we can model the Fermi Born rigid model of the classical electron in such an orbit around the nucleus, and visualize the Fermi coordinate grid and its intersection with the world tube of the equatorial circle of the spherical surface of the electron (suppressing one spatial dimension). This leads what we might playfully term ``relativistic pasta." This is a useful 2-dimensional stationary spacetime with closed spatial slices on which to illustrate the slicing and threading splittings of general relativity relative to a Killing congruence, like stationary axisymmetric spacetimes including the rotating black hole family.
Motivated by the picture of a thin accretion disc around a black hole, radiating mainly in the direction perpendicular to its plane, we study the motion of test particles interacting with a test geodesic radiation flux originating in the equatorial plane of a Schwarzschild space-time and propagating initially in the perpendicular direction. We assume that the interaction with the test particles is modelled by an effective term corresponding to the Thomson-type interaction which governs the Poynting-Robertson effect. After approximating the individual photon trajectories adequately, we solve the continuity equation approximately in order to find a consistent flux density with a certain plausible prescribed equatorial profile. The combined effects of gravity and radiation are illustrated in several typical figures which confirm that the particles are generically strongly influenced by the flux. In particular, they are both collimated and accelerated in the direction perpendicular to the disc, but this acceleration is not enough to explain highly relativistic outflows emanating from some black hole-disc sources. The model can however be improved in a number of ways before posing further questions which are summarized in concluding remarks.
A general framework is developed to investigate the properties of useful choices of stationary spacelike slicings of stationary spacetimes whose congruences of timelike orthogonal trajectories are interpreted as the world lines of an associated family of observers, the kinematical properties of which in turn may be used to geometrically characterize the original slicings. On the other hand properties of the slicings themselves can directly characterize their utility motivated instead by other considerations like the initial value and evolution problems in the 3-plus-1 approach to general relativity. An attempt is made to categorize the various slicing conditions or ``time gauges" used in the literature for the most familiar stationary spacetimes: black holes and their flat spacetime limit.
Three volumes of about 2700 some pages of some 500(?) articles on the current state of physics touching upon classical and quantum gravity and relativistic astrophysics and space measurements. Complemented by an on-line version:
http://www.icra.it/MG/mg13/ [World Scientific eProceedings]
The precession of a test gyroscope along stable bound equatorial plane orbits around a Kerr black hole is analyzed and the precession angular velocity of the gyro's parallel transported spin vector and the increment in precession angle after one orbital period is evaluated. The parallel transported Marck frame which enters this discussion is shown to have an elegant geometrical explanation in terms of the electric and magnetic parts of the Killing-Yano 2-form and a Wigner rotation effect.
The precession of a test gyroscope along unbound equatorial plane geodesic orbits around a Kerr black hole is analyzed with respect to a static reference frame whose axes point towards the ``fixed stars." The accumulated precession angle after a complete scattering process is evaluated and compared with the corresponding change in the orbital angle. Limiting results for the non-rotating Schwarzschild black hole case are also discussed.
The precession angular velocity of a gyroscope moving along a general geodesic in the Kerr spacetime is analyzed using the geometric properties of the spacetime. Natural frames along the gyroscope world line are explicitly constructed by boosting frames adapted to fundamental observers. A novel geometrical description is given to Marck's construction of a parallel propagated orthonormal frame along a general geodesic, identifying and clarifying the special role played by the Carter family of observers in this general context, thus extending previous discussion for the equatorial plane case.
Position determination of photon emitters and associated strong field parallax effects are investigated using relativistic optics when the photon orbits are confined to the equatorial plane of the Schwarzschild spacetime. We assume the emitter is at a fixed space position and the receiver moves along a circular geodesic orbit. This study requires solving the inverse problem of determining the (spatial) intersection point of two null geodesic initial data problems, serving as a simplified model for applications in relativistic astrometry as well as in radar and satellite communications.
Open access e-book of 4400 some pages of some nearly 600 articles on the current state of physics touching upon classical and quantum gravity and relativistic astrophysics and space measurements.
http://www.icra.it/MG/mg14/ [World Scientific eProceedings]
The Wigner rotations arising from the combination of boosts along two different directions are rederived from a relative boost point of view and applied to gyroscope spin precession along timelike geodesics in a Kerr spacetime, clarifying the geometrical properties of Marck's recipe for describing parallel transport along such world lines expressed in terms of the constants of the motion. His final angular velocity isolates the cumulative spin precession angular velocity independent of the spacetime tilting required to keep the spin 4-vector orthogonal to the gyro 4-velocity.
As an explicit example the cumulative precession effects are computed for a test gyroscope moving along both bound and unbound equatorial plane geodesic orbits.
Using standard cylindrical-like coordinates naturally adapted to the cylindrical symmetry of the Gödel spacetime, we study ellipticlike geodesic motion on hyperplanes orthogonal to the symmetry axis through an eccentricity-semi-latus rectum parametrization which is familiar from the Newtonian description of a two-body system. We compute several quantities which summarize the main features of the motion, namely the coordinate time and proper time periods of the radial motion, the frequency of the azimuthal motion, the full variation of the azimuthal angle over a period, and so on. Exact as well as approximate (i.e., Taylor-expanded in the limit of small eccentricity) analytic expressions of all these quantities are obtained. Finally, we consider their application to the gyroscope precession frequency along these orbits, generalizing the existing results for the circular case.
Timelike geodesics on a hyperplane orthogonal to the symmetry axis of the G\"odel spacetime appear to be elliptic-like if standard coordinates naturally adapted to the cylindrical symmetry are used. The orbit can then be suitably described through an eccentricity-semi-latus rectum parametrization, familiar from the Newtonian dynamics of a two-body system. However, changing coordinates such planar geodesics all become explicitly circular, as exhibited by Kundt's form of the G\"odel metric. We derive here a one-to-one correspondence between the constants of the motion along these geodesics as well as between the parameter spaces of elliptic-like versus circular geodesics. We also show how to connect the two equivalent descriptions of particle motion by introducing a pair of complex coordinates in the 2-planes orthogonal to the symmetry axis, which brings the metric into a form which is invariant under M\"obius transformations preserving the symmetries of the orbit, i.e., taking circles to circles.
Open access e-book of 2160 some pages of hundreds of articles on the current state of physics touching upon classical and quantum gravity and relativistic astrophysics and space measurements.
http://www.icra.it/MG/mg15/ [World Scientific eProceedings] [May 2022 to be updated]
The class of Petrov type I curvature tensors is further divided into those for which the span of the set of distinct principal null directions has dimension four (maximally spanning type I) or dimension three (nonmaximally spanning type I).
Research monograph. Perhaps to remain forever a web document.
Short historical overview of spacetime splitting methods in general relativity, with a long detailed development of a framework into which all such approaches may be fit and compared to each other, leading to a relativity of spacetime splittings which has been referred to as gravitoelectromagnetism. Long appendix summarizing differential geometry formulas. Long list of references on the topic of spacetime splittings and gravitoelectromagnetism.