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.

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.

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.

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.

scholarly journals Enabling unification of the general theory of relativity and quantum electro-dynamics by visualizing a particle energy configuration

2016 ◽  
Vol 2 (11) ◽  
pp. 288
Author(s):  
William S. Oakley
Keyword(s):  
General Theory ◽  
Radial Strain ◽  
Space Time ◽  
Particle Energy ◽  
Two Dimensions ◽  
Emergent Properties ◽  
Theory Of Relativity ◽  
General Theory Of Relativity ◽  
Curved Space ◽  
Strong Force

<p class="abstract">The long standing major issue in physics has been the inability to unify the two main theories of quantum electro-dynamics (QED) and the general theory of relativity (GTR), both of which are well proven and cannot accommodate significant change. The problem is resolved by combining the precepts of GTR and QED in a conceptual model describing the electron as electromagnetic (EM) energy localized in relativistic quantum loops near an event horizon. EM energy is localized by propagating in highly curved space-time of closed geometry, the local metric index increases, and the energy is thus relativistic to the observer at velocity v &lt; c, with the curved space-time thereby evidencing gravity. The presence of gravity leads to the observer notion of mass. Particle energy is in dynamic equilibrium with relativistic loop circumferential metric strain at the strong force scale opposed by radial metric strain. The resulting particle is a quantum black hole with the circumferential strong force in the curved metric orthogonal in two dimensions to all particle radials. The presence of energy E is thus evident in observer space reduced by c<sup>2</sup> to E/c<sup>2</sup> = mass. The circumferential strain diminishes as it extends into the surrounding metric as the particle’s gravitational field. The radial strain projects outward into observer space and is therein evident as electric field. Gravity, unit charge, and their associated fields are emergent properties and Strong and electric forces are equal within the particle, quantizing gravity and satisfying the Planck scale criteria of force equality. A derived scaling factor produces the gravity effect experienced by the observer and the GRT-QED unification issue is thereby largely resolved.</p>

scholarly journals A New Problem with the Twin Paradox

2017 ◽  
Vol 9 (2) ◽  
pp. 77
Author(s):  
Koshun Suto
Keyword(s):  
Coordinate System ◽  
Constant Velocity ◽  
Inertial Frame ◽  
Thought Experiment ◽  
Special Theory ◽  
Twin Paradox ◽  
Theory Of Relativity ◽  
Special Theory Of Relativity ◽  
Coordinate Systems ◽  
Accelerated Motion

This paper discusses the “triplet thought experiment” in which accelerated motion is eliminated from the famous twin paradox thought experiment of the special theory of relativity (STR). The author considers the coordinate systems of an inertial frame M and rocket A moving at constant speed relative to each other. First, an observer in inertial frame M performs the triplet thought experiment, and it is confirmed that the delay in time which elapses in the moving system agrees with the predictions of the STR. However, the delay in time predicted by the STR is observed even in the case when an observer A in rocket A carries out the triplet thought experiment. Before starting movement at constant velocity, rocket A experiences accelerated motion. The coordinate system of rocket A cannot be regarded physically as a stationary system. Even so, observer A observes the delay predicted by the STR. If the previous, traditional interpretation is assumed to be correct, observer A will never observe a delay in time agreeing with the predictions of the STR. To avoid paradox, the previously proposed traditional interpretation must be revised.

Sine-Gordon model in (1 + 1)-dimensional curved space: Schrödinger picture approach

1996 ◽  
Vol 74 (9-10) ◽  
pp. 626-633
Author(s):  
Anjana Sinha ◽  
Rajkumar Roychoudhury
Keyword(s):  
Effective Potential ◽  
Flat Space ◽  
Space Time ◽  
Curved Space ◽  
Curvature Term ◽  
Sine Gordon Model ◽  
Schrödinger Picture

The effective potential for the sine-Gordon model in a curved space-time, given by [Formula: see text], has been calculated using the Schrödinger picture formalism. It has been shown that when α(x) → 1 our method reproduces the flat-space results. To show the effect of the curvature term, the effective potential Veff has been calculated numerically for several values of the parameter M, where α(x) has been taken to be of the form [Formula: see text].

scholarly journals Four interactions in the sedenion curved spaces

2019 ◽  
Vol 16 (02) ◽  
pp. 1950019 ◽  
Author(s):  
Zi-Hua Weng
Keyword(s):  
General Relativity ◽  
Quantum Mechanics ◽  
Classical Mechanics ◽  
Flat Space ◽  
Field Equations ◽  
Curved Space ◽  
Gravitational Fields ◽  
Curved Spaces ◽  
Spatial Parameters ◽  
Physical Quantities

The paper aims to apply the complex-sedenions to explore the field equations of four fundamental interactions, which are relevant to the classical mechanics and quantum mechanics, in the curved spaces. Maxwell was the first to utilize the quaternions to describe the property of electromagnetic fields. Nowadays, the scholars introduce the complex-octonions to depict the electromagnetic and gravitational fields. And the complex-sedenions can be applied to study the field equations of the four interactions in the classical mechanics and quantum mechanics. Further, it is able to extend the field equations from the flat space into the curved space described with the complex-sedenions, by means of the tangent-frames and tensors. The research states that a few physical quantities will make a contribution to certain spatial parameters of the curved spaces. These spatial parameters may exert an influence on some operators (such as, divergence, gradient, and curl), impacting the field equations in the curved spaces, especially, the field equations of the four quantum-fields in the quantum mechanics. Apparently, the paper and General Relativity both confirm and succeed to the Cartesian academic thought of ‘the space is the extension of substance’.

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