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Chapter 21 - Curvature and the Einstein Equation
from Part III - The Einstein Equation
James B. Hartle, University of California, Santa Barbara
Book:

Gravity

Published online:

03 September 2021

Print publication:

24 June 2021, pp 445-470
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Summary

The relation between local spacetime curvature and matter energy density is given by the Einstein equation – it is the field equation of general relativity in the way that Maxwell’s equations are the field equations of electromagnetism. Maxwell’s equations relate the electromagnetic field to its sources – charges and currents. Einstein’s equation relates spacetime curvature to its source – the mass-energy of matter. This chapter gives a very brief introduction to the Einstein equation; we consider the equation in the absence of matter sources (the vacuum Einstein equation) and will include matter sources in Chapter 22. Even the vacuum Einstein equation has important implications. Just as the field of a static point charge and electromagnetic waves are solutions of the source-free Maxwell’s equations, the Schwarzschild geometry and gravitational waves are solutions of the vacuum Einstein equation.

The initial-value problem for a fourth-order dispersive closed curve flow on the 2-sphere
Part of
- General higher-order equations and systems
- Equations of mathematical physics and other areas of application
Eiji Onodera
Journal:

Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 147 / Issue 6 / December 2017

Published online by Cambridge University Press:

14 August 2017, pp. 1243-1277

Print publication:

December 2017
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A closed curve flow on the 2-sphere evolved by a fourth-order nonlinear dispersive partial differential equation on the one-dimensional flat torus is studied. The governing equation arises in the field of physics in relation to the continuum limit of the Heisenberg spin chain systems or three-dimensional motion of the isolated vortex filament. The main result of the paper gives the local existence and uniqueness of a solution to the initial-value problem by overcoming loss of derivatives in the classical energy method and the absence of the local smoothing effect. The proof is based on the delicate analysis of the lower-order terms to find out the loss of derivatives and on the gauged energy method to eliminate the obstruction.

A Compactness Theorem for Yang-Mills Connections
Xi Zhang
Journal:

Canadian Mathematical Bulletin / Volume 47 / Issue 4 / 01 December 2004

Published online by Cambridge University Press:

20 November 2018, pp. 624-634

Print publication:

01 December 2004
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In this paper, we consider Yang-Mills connections on a vector bundle $E$ over a compact Riemannian manifold $M$ of dimension $m\,>\,4$, and we show that any set of Yang-Mills connections with the uniformly bounded ${{L}^{\frac{m}{2}}}$-norm of curvature is compact in ${{C}^{\infty }}$ topology.