black hole spacetimes | vanishing angular momentum | positive angular momentum |
---|---|---|
vanishing charge | Schwarzschild spacetime | Kerr spacetime |
positive charge | Reissner-Nordstrom spacetime | Kerr-Newman spacetime |
physics, mathematical physics, philosophy of physics
theory (physics), model (physics)
experiment, measurement, computable physics
Axiomatizations
Tools
Structural phenomena
Types of quantum field thories
equality (definitional, propositional, computational, judgemental, extensional, intensional, decidable)
isomorphism, weak equivalence, homotopy equivalence, weak homotopy equivalence, equivalence in an (∞,1)-category
Examples.
What are called Einstein’s equations are the equations of motion of gravity: the Euler-Lagrange equations induced by the Einstein-Hilbert action.
They say that the Einstein tensor $G$ of the metric/the field of gravity equals the energy-momentum tensor $T$ of the remaining force- and matter-fields:
Given a choice of Cauchy surface $\Sigma$, the initial value problem for Einstein’s differential equations of motion is determined by a choice of Riemannian metric on $\Sigma$ and a second fundamental form along $\Sigma$.
With this data a solution to the equation exists and is unique. (Klainerman-Nicolo 03).
A general discssion is for instance in section 11 of
A discussion of the vacuum Einstein equations (only gravity, no other fields) in terms of synthetic differential geometry is in
Genuine PDE theory for Einstein’s equations goes back to local existence results by Yvonne Choquet-Bruhat in the 1950s. Global existence in the presence of a Cauchy surface was then shown in
For further developments see
H. Friedrich, A. D. Rendall, The Cauchy Problem for the Einstein Equations (arXiv:gr-qc/0002074)
Alan D. Rendall, Partial differential equations in general relativity, Oxford University press 2008 (web)
Hans Ringström, The Cauchy Problem in General Relativity, ESI Lectures in Mathematics and Physics 2009 (web)
Yvonne Choquet-Bruhat, General relativity and the Einstein equations. Oxford University Press (2008) (publisher)
See also The Cauchy Problem in Classical Supergravity