scholarly journals Transformation Groups for a Schwarzschild-Type Geometry in f(R) Gravity

2016 ◽  
Vol 2016 ◽  
pp. 1-8
Author(s):  
Emre Dil ◽  
Talha Zafer
Keyword(s):  
Reference Frames ◽  
Inertial Frame ◽  
Flat Space ◽  
Space Time ◽  
Transformation Groups ◽  
Coordinate Transformations ◽  
Cartesian Coordinates ◽  
Curved Space ◽  
Inertial Frames ◽  
Accelerated Frames

We know that the Lorentz transformations are special relativistic coordinate transformations between inertial frames. What happens if we would like to find the coordinate transformations between noninertial reference frames? Noninertial frames are known to be accelerated frames with respect to an inertial frame. Therefore these should be considered in the framework of general relativity or its modified versions. We assume that the inertial frames are flat space-times and noninertial frames are curved space-times; then we investigate the deformation and coordinate transformation groups between a flat space-time and a curved space-time which is curved by a Schwarzschild-type black hole, in the framework of f(R) gravity. We firstly study the deformation transformation groups by relating the metrics of the flat and curved space-times in spherical coordinates; after the deformation transformations we concentrate on the coordinate transformations. Later on, we investigate the same deformation and coordinate transformations in Cartesian coordinates. Finally we obtain two different sets of transformation groups for the spherical and Cartesian coordinates.

scholarly journals Relativistic Reference Frames in Astrometry

1986 ◽  
Vol 7 ◽  
pp. 101-102
Author(s):  
C A Murray
Keyword(s):  
Coordinate System ◽  
Reference Frames ◽  
Inertial Frame ◽  
Space Time ◽  
Clock Time ◽  
Local Inertial Frame ◽  
Incoming Photon ◽  
Inertial Frames ◽  
Spatial Coordinates

Astrometry can be defined as the measurement of space-time coordinates of photon events. For example, in principle, in classical optical astrometry, we measure the components of velocity, and hence the direction, of an incoming photon with respect to an instrumental coordinate system, and the clock time, at the instant of detection. The observer’s coordinate system at any instant can be identified with a local inertial frame. In the case of interferometric observations, the measurements are of clock times of arrival of a wavefront at two detectors whose spatial coordinates are specified with respect to instantaneous inertial frames.

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 DEVELOPMENT OF THE GRAVITATIONAL AND YANG–MILLS FIELDS, AND THE TREATMENT OF ACCELERATED FRAMES

2005 ◽  
Vol 20 (32) ◽  
pp. 7485-7504 ◽  
Author(s):  
JONG-PING HSU ◽  
DANA FINE
Keyword(s):  
Quantum Gravity ◽  
Flat Space ◽  
Physical World ◽  
Space Time ◽  
Yang Mills ◽  
Inertial Frames ◽  
Theory Of Gravity ◽  
Mills Field ◽  
Time Transformations ◽  
Accelerated Frames

We discuss ideas and problems regarding classical and quantum gravity, gauge theory of gravity, and space–time transformations between accelerated frames. Both Einstein's theory of gravity and Yang–Mills theory are gauge invariant. The invariance principles are at the very heart of our understanding of the physical world. This paper attempts to survey the development and to reveal problems and limitations of various formulations to gravitational and Yang–Mills fields, and to space–time transformations of accelerated frames. Gravitational force and accelerated frames are two ingredients in Einstein's thought in the period around 1907. Accelerated frames are difficult to define and are not well developed. However, one cannot claim to have a complete understanding of the physical world, if one understands flat space–time physics only from the viewpoint of the special class of inertial frames and ignores the vast class of noninertial frames. The paper highlights three aspects: (1) ideas of gravity as a Yang–Mills field, first discussed by Utiyama; (2) problems of quantum gravity, discussed by Feynman, Dyson and others; (3) space–time properties and the physics of fields and particles in accelerated frames of reference. These unfulfilled aspects of Einstein and Yang–Mills' profound thoughts present a challenge to physicists and mathematicians in the 21st century.

SIMPLE GENERALIZATIONS OF LORENTZ TRANSFORMATIONS AND THEIR IMPLICATIONS FOR HIGH ENERGY EXPERIMENTS

2005 ◽  
Vol 20 (26) ◽  
pp. 5989-6006 ◽  
Author(s):  
DANIEL T. SCHMITT ◽  
JONG-PING HSU
Keyword(s):  
Reference Frames ◽  
High Energy ◽  
Space Time ◽  
Decay Length ◽  
Lorentz Transformations ◽  
Lorentz And Poincaré Invariance ◽  
Inertial Frames ◽  
Accelerated Lifetime ◽  
Time Transformations ◽  
Accelerated Frames

Based on Lorentz and Poincaré invariance, we discuss reference frames with constant-linear-accelerations and their generalized space–time transformations with minimal departure from the Lorentz transformations. The requirement of limiting four-dimensional symmetry of the Lorentz and Poincaré groups assures that the generalized transformations reduce to the Lorentz transformations in the limit of zero acceleration. This suggests that the space–time coordinates xμ of accelerated frames are as meaningful as those of inertial frames. A flexibility and "gauge invariance" of the time for noninertial frames are discussed. These properties and the changes in the space–time of accelerated frames are shown graphically, including singularities and horizons. Physical implications of various accelerated transformations related to accelerated lifetime or decay-length dilations are predicted for experimental test.

Zum Übergang von der Wellenoptik zur geometrischen Optik in der allgemeinen Relativitätstheorie

1967 ◽  
Vol 22 (9) ◽  
pp. 1328-1332 ◽  
Author(s):  
Jürgen Ehlers
Keyword(s):  
General Relativity ◽  
Maxwell Equations ◽  
Expansion Method ◽  
Flat Space ◽  
Space Time ◽  
Moving Medium ◽  
Formal Similarity ◽  
Curved Space ◽  
Optical Metric ◽  
Series Expansion Method

The transition from the (covariantly generalized) MAXWELL equations to the geometrical optics limit is discussed in the context of general relativity, by adapting the classical series expansion method to the case of curved space time. An arbitrarily moving ideal medium is also taken into account, and a close formal similarity between wave propagation in a moving medium in flat space time and in an empty, gravitationally curved space-time is established by means of a normal hyperbolic optical metric.

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.

Comment on: Curling rock dynamics - The motion of a curling rock: inertial vs. noninertial reference frames

2000 ◽  
Vol 77 (11) ◽  
pp. 903-922 ◽  
Author(s):  
MRA Shegelski ◽  
M Reid
Keyword(s):  
Smooth Surface ◽  
Reference Frames ◽  
Inertial Frame ◽  
Center Of Mass ◽  
Rotating Cylinder ◽  
Lateral Motion ◽  
Noninertial Frame ◽  
Nonzero Component ◽  
Inertial Frames ◽  
Left Right Asymmetry

We examine the approach used and the results presented in a recent publication(Can. J. Phys. 76, 295 (1998))in which (i) a noninertial reference frame is used to examine the motion ofa curling rock, and (ii) the lateral motion of a curling rock isattributed to left-right asymmetry in the force acting on the rock.We point out the important differences between describing the motionin an inertial frame as opposed to a noninertial frame.We show that a force exhibiting left-right asymmetryin an inertial frame cannot explain the lateral motion of a curlingrock. We also examine, as was apparently done in the recent publication,an effective force that has left-right asymmetry in a noninertial, rotating frame. We show that such a force is not left-right asymmetric in an inertial frame, and that anylateral motion of a curling rock attributed to the effective forcein the noninertial frame is actually due to a real force, in aninertial frame, which has a net nonzero component transverse to the velocityof the center of mass. We inquire as to the physical basis for thetransverse component of this real force. We also examine the motion ofa rotating cylinder sliding over a smooth surface for which there isno melting: we show that the motion is easily analyzed in an inertialframe and that there is little to be gained by considering a rotating frame.We relate the results for this simple case to the more involved problemof the motion of a curling rock: we find that the motion of curling rocksis best studied in inertial frames. Perhaps most importantly, we showthat the approach taken and the results presented in the recent publicationlead to predicted motions of curling rocks that are indisagreement with observed motions of real curling rocks.PACS Nos.: 46.00, 01.80+b

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.

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