Flat space-times with gravitational fields

1959 ◽  
Vol 252 (1268) ◽  
pp. 45-50 ◽  
Keyword(s):  
Gravitational Field ◽  
Curvature Tensor ◽  
Flat Space ◽  
Space Time ◽  
Gravitational Fields ◽  
Extended Region

The geometry of an extended region of space-time is not fully determined by the vanishing of the Hiemann curvature tensor. This suggests the possible existence of a non-trivial gravitational field where space-time is flat. Two examples of such fields are considered with reference to their sources.

Self-conjugate gravitational fields. I

1963 ◽  
Vol 275 (1361) ◽  
pp. 245-256 ◽  
Keyword(s):  
Electromagnetic Field ◽  
Gravitational Field ◽  
Flat Space ◽  
Space Time ◽  
Second Order ◽  
Necessary Condition ◽  
Tensor Form ◽  
Gravitational Fields ◽  
Null Electromagnetic Field ◽  
Scalar Invariants

By splitting the curvature tensor R hijk into three 3-tensors of the second rank in a normal co-ordinate system, self-conjugate empty gravitational fields are defined in a manner analogous to that of the electromagnetic field. This formalism leads to three different types of self-conjugate gravitational fields, herein termed as types A, B and C . The condition that the gravitational field be self-conjugate of type A is expressed in a tensor form. It is shown that in such a field R hijk is propagated with the fundamental velocity and all the fourteen scalar invariants of the second order vanish. The structure of R hijk defines a null vector which can be identified as the vector defining the propagation of gravitational waves. It is found that a necessary condition for an empty gravitational field to be continued with a flat space-time across a null 3-space is that the field be self-conjugate of type A. The concept of the self-conjugate gravitational field is extended to the case when R ij # 0 but the scalar curvature R is zero. The condition in this case is also expressed in a tensor form. The necessary conditions that the space-time of an electromagnetic field be continued with an empty gravitational field or a flat space-time across a 3-space have been obtained. It is shown that for a null electromagnetic field whose gravitational field is self-conjugate of type A , all the fourteen scalar invariants of the second order vanish.

Time, Topology, and the Twin Paradox

2011 ◽  
Author(s):  
Jean‐Pierre Luminet
Keyword(s):  
High Speed ◽  
Reference Frames ◽  
Thought Experiment ◽  
Flat Space ◽  
Space Time ◽  
Twin Paradox ◽  
Theory Of Relativity ◽  
Apparent Paradox ◽  
Curved Space ◽  
Gravitational Fields

This chapter notes that the twin paradox is the best-known thought experiment associated with Einstein's theory of relativity. An astronaut who makes a journey into space in a high-speed rocket will return home to find he has aged less than his twin who stayed on Earth. This result appears puzzling, as the homebody twin can be considered to have done the travelling with respect to the traveller. Hence, it is called a “paradox”. In fact, there is no contradiction, and the apparent paradox has a simple resolution in special relativity with infinite flat space. In general relativity (dealing with gravitational fields and curved space-time), or in a compact space such as the hypersphere or a multiply connected finite space, the paradox is more complicated, but its resolution provides new insights about the structure of space–time and the limitations of the equivalence between inertial reference frames.

THE ASYMPTOTICALLY FREE AND ASYMPTOTICALLY CONFORMALLY INVARIANT GRAND UNIFICATION THEORIES IN CURVED SPACE-TIME

1990 ◽  
Vol 05 (20) ◽  
pp. 1599-1604 ◽  
Author(s):  
I.L. BUCHBINDER ◽  
I.L. SHAPIRO ◽  
E.G. YAGUNOV
Keyword(s):  
Renormalization Group ◽  
Gravitational Field ◽  
Gauge Group ◽  
Flat Space ◽  
Space Time ◽  
Grand Unification ◽  
Curved Space ◽  
Conformally Invariant ◽  
Renormalization Group Equations ◽  
The One

GUT’s in curved space-time is considered. The set of asymptotically free and asymptotically conformally invariant models based on the SU (N) gauge group is constructed. The general solutions of renormalization group equations are considered as the special ones. Several SU (2N) models, which are finite in flat space-time (on the one-loop level) and asymptotically conformally invariant in external gravitational field are also presented.

A Class of Energetically Rigid Gravitational Fields

1974 ◽  
Vol 29 (11) ◽  
pp. 1527-1530 ◽  
Author(s):  
H. Goenner
Keyword(s):  
Flat Space ◽  
Second Fundamental Form ◽  
Momentum Tensor ◽  
Space Time ◽  
Curved Space ◽  
Geometrical Properties ◽  
Gravitational Fields ◽  
The Second Fundamental Form ◽  
Perfect Fluids ◽  
Higher Dimensional

In Einstein's theory, the physics of gravitational fields is reflected by the geometry of the curved space-time manifold. One of the methods for a study of the geometrical properties of space-time consists in regarding it, locally, as embedded in a higher-dimensional flat space. In this paper, metrics admitting a 3-parameter group of motion are considered which form a generalization of spherically symmetric gravitational fields. A subclass of such metrics can be embedded into a five- dimensional flat space. It is shown that the second fundamental form governing the embedding can be expressed entirely by the energy-momentum tensor of matter and the cosmological constant. Such gravitational fields are called energetically rigid. As an application gravitating perfect fluids are discussed.

New conservation laws for zero rest-mass fields in asymptotically flat space-time

1968 ◽  
Vol 305 (1481) ◽  
pp. 175-204 ◽  
Keyword(s):  
Conservation Laws ◽  
Rest Mass ◽  
Power Series Expansion ◽  
Flat Space ◽  
Arbitrary Spin ◽  
Space Time ◽  
Maxwell Theory ◽  
Asymptotically Flat ◽  
Gravitational Fields ◽  
Higher Spin

Some recently discovered exact conservation laws for asymptotically flat gravitational fields are discussed in detail. The analogous conservation laws for zero rest-mass fields of arbitrary spin s = 0,½,1,...) in flat or asymptotically flat space-time are also considered and their connexion with a generalization of Kirchoff’s integral is pointed out. In flat space-time, an infinite hierarchy of such conservation laws exists for each spin value, but these have a somewhat trivial interpretation, describing the asymptotic incoming field (in fact giving the coefficients of a power series expansion of the incoming field). The Maxwell and linearized Einstein theories are analysed here particularly. In asymptotically flat space-time, only the first set of quantities of the hierarchy remain absolutely conserved. These are 4 s + 2 real quantities, for spin s , giving a D ( s , 0) representation of the Bondi-Metzner-Sachs group. But even for these quantities the simple interpretation in terms of incoming waves no longer holds good: it emerges from a study of the stationary gravitational fields that a contribution to the quantities involving the gravitational multipole structure of the field must also be present. Only the vacuum Einstein theory is analysed in this connexion here, the corresponding discussions of the Einstein-Maxwell theory (by Exton and the authors) and the Einstein-Maxwell-neutrino theory (by Exton) being given elsewhere. (A discussion of fields of higher spin in curved space-time along these lines would encounter the familiar difficulties first pointed out by Buchdahl.) One consequence of the discussion given here is that a stationary asymptotically flat gravitational field cannot become radiative and then stationary again after a finite time, except possibly if a certain (origin independent) quadratic combination of multipole moments returns to its original value. This indicates the existence of ‘tails’ to the outgoing waves (or back-scattered field),which destroys the stationary nature of the final field.

scholarly journals TACHYON MONOPOLE

2005 ◽  
Vol 20 (23) ◽  
pp. 5491-5499 ◽  
Author(s):  
XIN-ZHOU LI ◽  
DAO-JUN LIU
Keyword(s):  
Exact Solution ◽  
Approximate Solution ◽  
Gravitational Field ◽  
Flat Space ◽  
Space Time ◽  
Global Monopole ◽  
Tachyon Field ◽  
Field Equations ◽  
Monopole Solution ◽  
Static Space

The property and gravitational field of global monopole of tachyon are investigated in a four-dimensional static space–time. We give an exact solution of the tachyon field in the flat space–time background. Using the linearized approximation of gravity, we get the approximate solution of the metric. We also solve analytically the coupled Einstein and tachyon field equations which is beyond the linearized approximation to determine the gravitational properties of the monopole solution. We find that the metric of tachyon monopole represents an asymptotically AdS space–time with a small effective mass at the origin. We show that this relatively tiny mass is actually negative, as it is in the case of ordinary scalar field.

scholarly journals KLEINIAN GEOMETRY AND THE N = 2 SUPERSTRING

1994 ◽  
Vol 09 (09) ◽  
pp. 1457-1493 ◽  
Author(s):  
J. BARRETT ◽  
G. W. GIBBONS ◽  
M. J. PERRY ◽  
C. N. POPE ◽  
P. RUBACK
Keyword(s):  
Gravitational Field ◽  
Dual Space ◽  
Flat Space ◽  
Space Time ◽  
Degree Of Freedom ◽  
Twistor Theory ◽  
Β Function

This paper is devoted to the exploration of some of the geometrical issues raised by the N = 2 superstring. We begin by reviewing the reasons that β functions for the N = 2 superstring require it to live in a four-dimensional self-dual space–time of signature (− − + +), together with some of the arguments as to why the only degree of freedom in the theory is that described by the gravitational field. We move on to describe at length the geometry of flat space, and how a real version of twistor theory is relevant to it. We then describe some of the more complicated space–times that satisfy the β function equations. Finally we speculate on the deeper significance of some of these space–times.

THE GRAVITATIONAL FIELD OF A PLANE SLAB

2009 ◽  
Vol 24 (28n29) ◽  
pp. 5381-5405 ◽  
Author(s):  
RICARDO E. GAMBOA SARAVÍ
Keyword(s):  
Gravitational Field ◽  
Mirror Symmetry ◽  
Energy Condition ◽  
External Solution ◽  
Finite Depth ◽  
Space Time ◽  
Dominant Energy Condition ◽  
Positive Density ◽  
Plane Symmetric ◽  
Gravitational Fields

We discuss the exact solution to Einstein's equation corresponding to a static and plane symmetric distribution of matter with constant positive density located below z = 0 matched to vacuum solutions. The internal solution depends essentially on two constants: the density ρ and a parameter κ. We show that these space–times finish down below at an inner singularity at finite depth [Formula: see text]. We show that for κ ≥ 0.3513 ⋯, the dominant energy condition is satisfied all over the space–time. We match these singular solutions to the vacuum one and compute the external gravitational field in terms of slab's parameters. Depending on the value of κ, these slabs are either attractive, repulsive or neutral. The external solution turns out to be a Rindler's space–time. Repulsive slabs explicitly show how negative, but finite pressure can dominate the attraction of the matter. In this case, the presence of horizons in the vacuum shows that there are null geodesics which never reach the surface of the slab. We also consider a static and plane symmetric nonsingular distribution of matter with constant positive density ρ and thickness [Formula: see text] surrounded by two external vacuums. We explicitly write down the pressure and the external gravitational fields in terms of ρ and d. The solution turns out to be attractive and remarkably asymmetric: the "upper" solution is Rindler's vacuum, whereas the "lower" one is the singular part of Taub's plane symmetric solution. Inside the slab, the pressure is positive and bounded, presenting a maximum at an asymmetrical position between the boundaries. We show that if [Formula: see text], the dominant energy condition is satisfied all over the space–time. We also show how the mirror symmetry is restored at the Newtonian limit. We also find thinner repulsive slabs by matching a singular slice of the inner solution to the vacuum. We also discuss solutions in which an attractive slab and a repulsive one, and two neutral ones are joined. We also discuss how to assemble a "gravitational capacitor" by inserting a slice of vacuum between two such slabs.

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