Least Action and Classical Fields

2021 ◽  
pp. 286-325
Author(s):  
Moataz H. Emam
Keyword(s):  
General Relativity ◽  
Momentum Tensor ◽  
Point Particle ◽  
Kerr Metric ◽  
Field Equations ◽  
Tensor Fields ◽  
Particle Mechanics ◽  
Least Action ◽  
Principle Of Least Action ◽  
Maxwell Fields

We present the principle of least action and see how it is used in non-relativistic point particle mechanics, relativistic point particle mechanics, general relativity, derivation of field equations for scalar, vector and tensor fields as well as the energy momentum tensor. Towards the end we present examples of solutions of Einstein-Maxwell fields: The Reissner-Nordstrom metric, Kerr metric, and Kerr- Newman metric.

scholarly journals On Hilbert's world-function

1927 ◽  
Vol 113 (765) ◽  
pp. 496-511 ◽  
Keyword(s):  
Dynamical System ◽  
General Relativity ◽  
Minimum Principle ◽  
Relativity Theory ◽  
Field Equations ◽  
General Relativity Theory ◽  
Least Action ◽  
Principle Of Least Action ◽  
The Universe ◽  
World Function

The well-known theorem that the motion of any conservative dynamical system can be determined from the “Principle of Least Action” or “Hamilton’s Principle” was carried over into General Relativity-Theory in 1915 by Hilbert, who showed that the field-equations of gravitation can be deduced very simply from a minimum-principle. Hilbert generalised his ideas into the assertion that all physical happenings (gravitational electrical, etc.) in the universe are determined by a scalar “world-function” H, being, in fact, such as to annul the variation of the integral ∫∫∫∫H√(−g)dx 0 dx 1 dx 2 dx 3 where ( x 0 , x 1 , x 2 , x 3 ) are the generalised co-ordinates which specify place and time, and g is (in the usual notation of the relativity-theory) the determinant of the gravitational potentials g v q , which specify the metric by means of the equation dx 2 = ∑ p, q g vq dx v dx q . In Hilbert’s work, the variation of the above integral was supposed to be due to small changes in the g vq 's and in the electromagnetic potentials, regarded as functions of x 0 , x 1 , x 2 , x 3 .

General Relativity

2017 ◽  
Author(s):  
Nicholas Manton ◽  
Nicholas Mee
Keyword(s):  
Black Holes ◽  
General Relativity ◽  
Field Equation ◽  
Einstein Field Equation ◽  
Kerr Metric ◽  
Gravitational Lenses ◽  
Field Equations ◽  
Spacetime Geometry ◽  
Tests Of General Relativity ◽  
Rotating Black Holes

This chapter presents the physical motivation for general relativity, derives the Einstein field equation and gives concise derivations of the main results of the theory. It begins with the equivalence principle, tidal forces in Newtonian gravity and their connection to curved spacetime geometry. This leads to a derivation of the field equation. Tests of general relativity are considered: Mercury’s perihelion advance, gravitational redshift, the deflection of starlight and gravitational lenses. The exterior and interior Schwarzschild solutions are discussed. Eddington–Finkelstein coordinates are used to describe objects falling into non-rotating black holes. The Kerr metric is used to describe rotating black holes and their astrophysical consequences. Gravitational waves are described and used to explain the orbital decay of binary neutron stars. Their recent detection by LIGO and the beginning of a new era of gravitational wave astronomy is discussed. Finally, the gravitational field equations are derived from the Einstein–Hilbert action.

scholarly journals Extended harmonic mapping connects the equations in classical, statistical, fluid, quantum physics and general relativity

2020 ◽  
Vol 10 (1) ◽  
Author(s):  
Xiaobo Zhai ◽  
Changyu Huang ◽  
Gang Ren
Keyword(s):  
General Relativity ◽  
Quantum Physics ◽  
Harmonic Maps ◽  
Classical Mechanics ◽  
Harmonic Mapping ◽  
Geodesic Equation ◽  
Lagrange Equations ◽  
Least Action ◽  
Principle Of Least Action ◽  
Fundamental Equations

Abstract One potential pathway to find an ultimate rule governing our universe is to hunt for a connection among the fundamental equations in physics. Recently, Ren et al. reported that the harmonic maps with potential introduced by Duan, named extended harmonic mapping (EHM), connect the equations of general relativity, chaos and quantum mechanics via a universal geodesic equation. The equation, expressed as Euler–Lagrange equations on the Riemannian manifold, was obtained from the principle of least action. Here, we further demonstrate that more than ten fundamental equations, including that  of classical mechanics, fluid physics, statistical physics, astrophysics, quantum physics and general relativity, can be connected by the same universal geodesic equation. The connection sketches a family tree of the physics equations, and their intrinsic connections reflect an alternative ultimate rule of our universe, i.e., the principle of least action on a Finsler manifold.

scholarly journals Field Equations in C4 Space-Time

2018 ◽  
Author(s):  
Dimitris Mastoridis ◽  
K. Kalogirou
Keyword(s):  
General Relativity ◽  
Dark Energy ◽  
Complex Space ◽  
Energy Momentum Tensor ◽  
Real Space ◽  
Momentum Tensor ◽  
Space Time ◽  
Field Equations ◽  
Geometric Information

We explore the field equations in a 4-d complex space-time, in the same way, that general relativity does for our usual 4-d real space-time, forming this way, a new "general  relativity" in C4 space-time, free of sources. Afterwards, by embedding our usual 4-d real space-time in C4 space-time, we describe  geometrically the energy-momentum tensor Tμν as the lost geometric information of this embedding. We further give possible explanation of dark eld and dark energy.

The whole of physics

2017 ◽  
Author(s):  
Jennifer Coopersmith
Keyword(s):  
Physical Chemistry ◽  
General Relativity ◽  
Electromagnetic Waves ◽  
Materials Science ◽  
Classical Mechanics ◽  
Least Action ◽  
Principle Of Least Action ◽  
Theory Of Gravitation ◽  
Mathematical Equations ◽  
Deep Significance

How the Principle of Least Action underlies all physics (all physics that can be reduced to mathematical equations) is explained at a qualitative, semi-popular level. It even applies to smartphones. The domains of classical mechanics, continuum mechanics, materials science, light and electromagnetic waves, special and general relativity (Einstein’s Theory of Gravitation), electrodynamics, quantumelectrodynamics (QED), hydrodynamics, physical chemistry, statistical mechanics, and the quantum world, are examined. It is shown that the Principles of Least Time, Least Resistance, and Maximal Ageing, and Lenz’s Law are, in fact, examples of the Principle of Least Action. It is also shown how Planck’s constant is a measure of “absolute smallness,” and its units are the units of action. Never again, post quantum mechanics, can there be any doubt about the deep significance of action in physics.

Weyl static axially symmetric field equations in modified f(T) gravity

2017 ◽  
Vol 14 (04) ◽  
pp. 1750053 ◽  
Author(s):  
Saeed Nayeh ◽  
Mehrdad Ghominejad
Keyword(s):  
General Relativity ◽  
Symmetric Space ◽  
Energy Momentum Tensor ◽  
Radial Component ◽  
Momentum Tensor ◽  
Space Time ◽  
Axially Symmetric ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Torsion Scalar

In this paper, we obtain the field equations of Weyl static axially symmetric space-time in the framework of [Formula: see text] gravity, where [Formula: see text] is torsion scalar. We will see that, for [Formula: see text] related to teleparallel equivalent general relativity, these equations reduce to Einstein field equations. We show that if the components of energy–momentum tensor are symmetric, the scalar torsion must be either constant or only a function of radial component [Formula: see text]. The solutions of some functions [Formula: see text] in which [Formula: see text] is a function of [Formula: see text] are obtained.

FIELD EQUATIONS FOR SPACE–TIME–MATTER THEORY

2013 ◽  
Vol 10 (09) ◽  
pp. 1350038
Author(s):  
AUREL BEJANCU
Keyword(s):  
Tensor Field ◽  
Momentum Tensor ◽  
Brane World ◽  
Field Equations ◽  
Tensor Fields ◽  
Full Generality ◽  
Matter Theory ◽  
Warp Function ◽  
Brane Theory ◽  
Kaluza Klein

In the present paper we obtain, in a covariant form, and in their full generality, the field equations in a relativistic general Kaluza–Klein space. This is done by using the Riemannian horizontal connection defined in [3], and some 4D horizontal tensor fields, as for instance: horizontal Ricci tensor, horizontal Einstein gravitational tensor field, horizontal electromagnetic energy–momentum tensor field, etc. Also, we present some inter-relations between STM theory and brane-world theory. This enables us to introduce in brane theory some electromagnetic potentials constructed by means of the warp function.

scholarly journals Blackhole in nonlocal gravity: comparing metric from Newmann–Janis algorithm with slowly rotating solution

2020 ◽  
Vol 80 (7) ◽  
Author(s):  
Utkarsh Kumar ◽  
Sukanta Panda ◽  
Avani Patel
Keyword(s):  
General Relativity ◽  
Gravitational Field ◽  
Modified Gravity ◽  
Kerr Metric ◽  
Slow Rotation ◽  
Field Equations ◽  
External Region ◽  
Modified Gravity Theories ◽  
Spacetime Metric ◽  
Gravity Theories

Abstract The strong gravitational field near massive blackhole is an interesting regime to test General Relativity (GR) and modified gravity theories. The knowledge of spacetime metric around a blackhole is a primary step for such tests. Solving field equations for rotating blackhole is extremely challenging task for the most modified gravity theories. Though the derivation of Kerr metric of GR is also demanding job, the magical Newmann–Janis algorithm does it without actually solving Einstein equation for rotating blackhole. Due to this notable success of Newmann–Janis algorithm in the case of Kerr metric, it has been being used to obtain rotating blackhole solution in modified gravity theories. In this work, we derive the spacetime metric for the external region of a rotating blackhole in a nonlocal gravity theory using Newmann–Janis algorithm. We also derive metric for a slowly rotating blackhole by perturbatively solving field equations of the theory. We discuss the applicability of Newmann–Janis algorithm to nonlocal gravity by comparing slow rotation limit of the metric obtained through Newmann–Janis algorithm with slowly rotating solution of the field equation.

scholarly journals Extension Rules of Newman–Janis Algorithm for Rotation Metrics in General Relativity

2020 ◽  
pp. 1-14
Author(s):  
Yu-Ching, Chou
Keyword(s):  
General Relativity ◽  
Exact Solutions ◽  
Kerr Metric ◽  
Tensor Structure ◽  
De Sitter ◽  
Field Equations ◽  
Null Tetrad ◽  
Axisymmetric Solutions ◽  
Metric Function ◽  
De Sitter Metric

Aims: The aim of this study is to extend the formula of Newman–Janis algorithm (NJA) and introduce the rules of the complexifying seed metric. The extension of NJA can help determine more generalized axisymmetric solutions in general relativity.Methodology: We perform the extended NJA in two parts: the tensor structure and the seed metric function. Regarding the tensor structure, there are two prescriptions, the Newman–Penrose null tetrad and the Giampieri prescription. Both are mathematically equivalent; however, the latter is more concise. Regarding the seed metric function, we propose the extended rules of a complex transformation by r2/Σ and combine the mass, charge, and cosmologic constant into a polynomial function of r. Results: We obtain a family of axisymmetric exact solutions to Einstein’s field equations, including the Kerr metric, Kerr–Newman metric, rotating–de Sitter, rotating Hayward metric, Kerr–de Sitter metric and Kerr–Newman–de Sitter metric. All the above solutions are embedded in ellipsoid- symmetric spacetime, and the energy-momentum tensors of all the above metrics satisfy the energy conservation equations. Conclusion: The extension rules of the NJA in this research avoid ambiguity during complexifying the transformation and successfully generate a family of axisymmetric exact solutions to Einsteins field equations in general relativity, which deserves further study.

scholarly journals Particle paths of general relativity as geodesics of an affine connection

1970 ◽  
Vol 3 (3) ◽  
pp. 325-335 ◽  
Author(s):  
R. Burman
Keyword(s):  
General Relativity ◽  
Energy Momentum Tensor ◽  
Equations Of Motion ◽  
Affine Connection ◽  
Momentum Tensor ◽  
Field Equations ◽  
Test Particles ◽  
Special Cases ◽  
Particle Paths ◽  
Arbitrary Energy

This paper deals with the motion of incoherent matter, and hence of test particles, in the presence of fields with an arbitrary energy-momentum tensor. The equations of motion are obtained from Einstein's field equations and are written in the form of geodesic equations of an affine connection. The special cases of the electromagnetic field, the Proca field and a scalar field are discussed.

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