nLab
Cauchy surface

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Riemannian geometry

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Cauchy surfaces

Idea

A Cauchy surface is a hypersurface? in spacetime (so actually a 33-dimensional region in our 44-dimensional spacetime) that can profitably be seen as constituting ‘all of space at a given time’.

Definition

For (X,g)(X,g) a Lorentzian manifold, a Cauchy surface is an embedded submanifold ΣX\Sigma \hookrightarrow X such that every timelike curve in XX may be extended to a timelike curve that intersects Σ\Sigma precisely in one point.

A Lorentzian manifold that does admit a Cauchy surface is called globally hyperbolic.

Applications

One way to formulate causality in physics is that the values of all observables at all points on a single Cauchy surface in spacetime is enough information (in the sense of a boundary condition to apply to a differential equation constituting a relevant physical theory) to determine the values of all observables at all points of spacetime. (This is not always an actual theorem of differential equations.) Stated more intuitively, the state of the universe at any given time is enough information to determine the state of the universe at all times.

If spacetime can be equipped with a foliation of Cauchy surfaces, then we may assign a real number tt to each surface Σ\Sigma, so that we think of Σ\Sigma as ‘space at time tt’. Of course, there are typically many ways to do this (if any), in accordance with the principle of relativity of simultaneity?. On the other hand, for some spacetimes, this may not be possible at all (because they are not globally hyperbolic).

References

The existence of a smooth splitting of globally hyperbolic spacetimes into Cauchy surfaces is in

  • Antonio N. Bernal, Miguel Sánchez, On smooth Cauchy hypersurfaces and Geroch’s splitting theorem (arXiv:gr-qc/0306108v2)

Revised on December 7, 2015 13:53:58 by Urs Schreiber (195.37.209.180)