scholarly journals Riemannian Space-Time, de Donder Conditions and Gravitational Field in Flat Space-Time

2013 ◽  
Vol 03 (01) ◽  
pp. 8-19 ◽  
Author(s):  
Gordon Liu
Keyword(s):  
Gravitational Field ◽  
Riemannian Space ◽  
Flat Space ◽  
Space Time

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.

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.

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.

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.

scholarly journals ON ELECTROGRAVITY DUALITY

1999 ◽  
Vol 14 (05) ◽  
pp. 337-342 ◽  
Author(s):  
NARESH DADHICH
Keyword(s):  
Black Hole ◽  
Gravitational Field ◽  
Topological Defect ◽  
Flat Space ◽  
Interesting Property ◽  
Space Time ◽  
Global Monopole ◽  
Duality Transformation ◽  
Original Space ◽  
Black Hole Solutions

By resolving the gravitational field into electric and magnetic parts, we define an electrogravity duality transformation and discover an interesting property of the field. Under the duality transformation, a vacuum/flat space–time maps into the original space–time with a topological defect of global monopole/texture. The electrogravity-duality is thus a topological defect generating process. It turns out that all black hole solutions possess dual solutions that imbibe a global monopole.

scholarly journals The Geodetic Interval in a Riemannian Space-Time in the Second Post-Minkowskian Approximation

1974 ◽  
Vol 64 ◽  
pp. 100-100
Author(s):  
Reiner Wilhelm John
Keyword(s):  
Differential Equation ◽  
Wave Propagation ◽  
Gravitational Field ◽  
Time Delay ◽  
Nonlinear Differential Equation ◽  
Riemannian Space ◽  
Perturbation Expansion ◽  
Space Time ◽  
Clear Cut ◽  
Explicit Criteria

The knowledge of the geodetic interval between two points in a Riemannian space-time with the metric gab is essential for statements on the time delay in a gravitational field represented by gab and makes possible to derive explicit criteria for clear-cut wave propagation. The nonlinear differential equation for the geodetic interval is integrated via perturbation expansion in the second post-Minkowskian approximation.

A Class of Conserved Tensors in an Arbitrary Gravitational Field

1962 ◽  
Vol 58 (2) ◽  
pp. 346-362 ◽  
Author(s):  
C. D. Collinson
Keyword(s):  
Gravitational Field ◽  
Riemannian Space ◽  
Empty Space ◽  
Gravitational Theory ◽  
Space Time ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Linear Combinations ◽  
Second Derivatives ◽  
Elementary Methods

ABSTRACTFour-index tensors with all possible terms either (a) quadratic in the Riemann curvature tensor, or (b) linear in its second derivatives, with coefficients constructed from products of metric tensors, are enumerated for the case of Riemannian space-time. All linear combinations of these which have zero divergence, and which might therefore be associated with the super-energy in gravitational theory, are found by elementary methods. The only independent such combination reduces in empty space-time to the tensor previously found by Bel and Robinson.

scholarly journals The principle of least action in Milne's kinematical relativty

1934 ◽  
Vol 147 (862) ◽  
pp. 478-490 ◽  
Keyword(s):  
Gravitational Field ◽  
Particle Density ◽  
Free Particle ◽  
Formal Theory ◽  
Flat Space ◽  
Space Time ◽  
Uniform Motion ◽  
Least Action ◽  
Theory Of Gravitation ◽  
Laws Of Motion

In a recent paper, Professor Milne has obtained a gravitational field with non-zero density of matter in flat space-time, the field in question being appropriate to the whole cosmos. It was obtained, without recourse to a formal theory of gravitation, by constructing a system of particles in motion satisfying Einstein’s cosmological principle applied to a set of fundamental observers in uniform relative motion: the result was a set of motions and a particle-density distribution which would be described in the same way by each one of the fundamental observers. This method of obtaining a gravitational field is fundamentally different from Einstein’s, the applicability being dependent upon explicit recognition of the priniciple that “an observer can either ( a ) select any one of the spaces of pure geometry presented to him by the mathematician, use it in order to describe the phenomena in the space; or alternatively ( b ) posit beforehand the laws of nature he wishes to see obeyed and then determine the space in which, in consequence, he must embed the phenomena he describes.” Einstein’s theory of gravitation consists essentially in obtaining a metric ds 2 such that a free particle obeys the law of nature δ ∫ ds = 0, and is an example of alternative ( b ). The gravitational field for the system described above begins by selecting Euclidean space and Newtonian time for any one observer, the different observer’s space and times being connected by Lorentz transformations, and then determines the laws of motion in this space; it is an example of alternative ( a ). The laws of motion were obtained as formulæ for the components of acceleration of a free particle as functions of the seven variables, t, x, y, z, u, v, w , reckoned from defined zeros. The alternative procedures have been recently stated very clearly by Milner. He wrote: ”Two courses are open to us. (1) We can modify the geometry assumed in the relation ds 2 = ∑ i=1 4 ( dx i ) 2 so that a mathematically straight track ( i. e ., its length obeys a stationery principle still continues to represent the non-uniform motion of a particle; this is the method of general relativity. (2) We can retain the four-fold with unaltered geometry and specify a curved track which represents the observed motion by weighting each element of its length so that the integral weighted length between two points is stationery; this is the method of ‘least action’. Both these methods of describing the motions of bodies must be considered equally logical when one remembers that a manifold (even when it is called ‘space-time’) is not the actual world, but a mental concept, in which real phenomena are represented symbolically.”

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.

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