The Einstein field equation

2021 ◽  
pp. 213-226
Author(s):  
Andrew M. Steane
Keyword(s):  
Field Theory ◽  
Field Equation ◽  
Einstein Field Equation ◽  
Equation Of Motion ◽  
Energy Conditions ◽  
Gravitational Fields ◽  
Electromagnetic Field Theory ◽  
Stationary Conditions ◽  
Energy And Momentum ◽  
Insight Into

Various aspects of the Einstein field equation are presented. First the field equation is obtained by arguing that it is the simplest equation that respects the fundamental geometric insight into gravity. Then we consider whether the equation is stable, and introduce the weak energy and dominant energy conditions. The connection between inertial motion and the distant universe (Mach’s principle) is discussed. The equation of motion of matter is obtained from the field equation, and a comparison made with electromagnetic field theory. The energy and momentum of gravitational fields in stationary conditions is discussed, and the Komar energy obtained.

The classical field theory of matter and electricity - The electromagnetic theory of particles

1960 ◽  
Vol 253 (1025) ◽  
pp. 205-226 ◽  
Keyword(s):  
Field Theory ◽  
Wave Equation ◽  
Classical Field Theory ◽  
Relativity Theory ◽  
Modern Theory ◽  
Field Equations ◽  
Electromagnetic Field Theory ◽  
Energy And Momentum ◽  
Static Form ◽  
Classical Particles

The electromagnetic field theory developed in the previous paper is here applied to the problem of devising systems which behave as classical particles. It is found that spherically symmetrical systems can exist which, when they are stationary: (1) satisfy the static form of the extended equations at every point of space; and (2) are characterized mechanically by being everywhere in equilibrium under the sole action of the Maxwellian stress of their own field—thus they are pure electromagnetic systems subsisting free of external constraint. (3) When they are transformed so as to be in motion, the energy and momentum they possess are exactly those required for material particles by relativity theory. A rather obvious restriction made on the generality of the conditions for particle existence brought to light the possibility of a solution denoting an ‘atomic’ system built up of successive shells, each of which must contain the same energy, and net charge, as the others. The reason for such a result is that, when their very great generality is restricted in the most straightforward way, the field equations reduce to the form of a wave equation. The relation of this to the wave equation of modern theory is briefly discussed. The transformation behaviour of the field equations when a Lorentz transformation is applied to the co-ordinates is dealt with in this paper; it is found that they remain invariant in form under wider transformations of the field variables than are permitted by the classical equations. The variables may be submitted to a certain transformation without the co-ordinates being transformed at all. The physical meaning of this is investigated and an explanation of it found.

The classical field theory of matter and electricity - An approach from first principles

1960 ◽  
Vol 253 (1025) ◽  
pp. 185-204 ◽  
Keyword(s):  
Field Theory ◽  
First Principles ◽  
Classical Field Theory ◽  
Classical Field ◽  
Field Equations ◽  
Additional Energy ◽  
Conservation Of Energy ◽  
Electromagnetic Field Theory ◽  
Energy And Momentum ◽  
Accepted Form

The most desirable classical field theory of the fundamental continuous substratum of matter, from which we can imagine particles are formed, would generally be considered to be the electromagnetic equations but for the fact that these are not consistent with the permanent existence of electrons. Instead of attempting (as has been usual) to modify the equations by special assumptions for the purpose, the problem is attacked here by deriving from first principles field equations which represent conserved matter; for the failure of the standard equations can be traced to the fact that they do not admit conservation of energy and momentum in general, but only in simple cases. The new equations are found to be identical with those of standard electromagnetic theory except that they contain two extra variables, which indicate the existence of additional energy, momentum and stress in the field. The two variables, however, come into the equations in a way which allows them to be included in the charge and current terms, so that they become there concealed and leave the form of the equations virtually unchanged. Consequently they do not affect the ordinary practical use which is made of the electromagnetic equations; they only come into open play in fundamental theory and in the presence of charge and current in the field, and there they remove the difficulties which the electromagnetic field theory in its accepted form presents.

ON A PERFECT FLUID K¨ AHLER SPACETIME

2016 ◽  
Vol 12 (3) ◽  
pp. 4350-4355
Author(s):  
VIBHA SRIVASTAVA ◽  
P. N. PANDEY
Keyword(s):  
Field Equation ◽  
Perfect Fluid ◽  
Sectional Curvature ◽  
Einstein Field Equation ◽  
Cosmological Term ◽  
Conformal Killing ◽  
Killing Vectors ◽  
Projectively Flat ◽  
Conformal Killing Vectors

The object of the present paper is to study a perfect fluid K¨ahlerspacetime. A perfect fluid K¨ahler spacetime satisfying the Einstein field equation with a cosmological term has been studied and the existence of killingand conformal killing vectors have been discussed. Certain results related to sectional curvature for pseudo projectively flat perfect fluid K¨ahler spacetime have been obtained. Dust model for perfect fluid K¨ahler spacetime has also been studied.

A Review of Conformal Field Theory in 2D

2014 ◽  
Vol 6 (2) ◽  
pp. 1079-1105
Author(s):  
Rahul Nigam
Keyword(s):  
Field Theory ◽  
Quantum Field Theory ◽  
Conformal Field Theory ◽  
Statistical Mechanics ◽  
Critical Behavior ◽  
Verma Module ◽  
Virasoro Algebra ◽  
Quantum Field ◽  
Conformal Field ◽  
Insight Into

In this review we study the elementary structure of Conformal Field Theory in which is a recipe for further studies of critical behavior of various systems in statistical mechanics and quantum field theory. We briefly review CFT in dimensions which plays a prominent role for example in the well-known duality AdS/CFT in string theory where the CFT lives on the AdS boundary. We also describe the mapping of the theory from the cylinder to a complex plane which will help us gain an insight into the process of radial quantization and radial ordering. Finally we will develop the representation of the Virasoro algebra which is the well-known "Verma module".  

Energy and Momentum

2018 ◽  
Author(s):  
David M. Wittman
Keyword(s):  
Time Dilation ◽  
Speed Of Light ◽  
Massless Particles ◽  
The Everyday ◽  
Low Speeds ◽  
Energy And Momentum ◽  
Deep Relationship ◽  
Insight Into ◽  
Momentum Relation

Tis chapter explains the famous equation E = mc2 as part of a wider relationship between energy, mass, and momentum. We start by defning energy and momentum in the everyday sense. We then build on the stretching‐triangle picture of spacetime vectors developed in Chapter 11 to see how energy, mass, and momentum have a deep relationship that is not obvious at everyday low speeds. When momentum is zero (a mass is at rest) this energy‐momentum relation simplifes to E = mc2, which implies that mass at rest quietly stores tremendous amounts of energy. Te energymomentum relation also implies that traveling near the speed of light (e.g., to take advantage of time dilation for interstellar journeys) will require tremendous amounts of energy. Finally, we look at the simplifed form of the energy‐momentum relation when the mass is zero. Tis gives us insight into the behavior of massless particles such as the photon.

scholarly journals Positive energy conditions in 4D conformal field theory

2016 ◽  
Vol 2016 (10) ◽  
Author(s):  
Kara Farnsworth ◽  
Markus A. Luty ◽  
Valentina Prilepina
Keyword(s):  
Field Theory ◽  
Conformal Field Theory ◽  
Energy Conditions ◽  
Positive Energy ◽  
Conformal Field
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