nLab Cauchy surface

Cauchy surfaces


Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

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



smooth space


The magic algebraic facts




infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular 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& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Riemannian geometry


physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics

Cauchy surfaces


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’.


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.



For X DX^D a smooth globally hyperbolic spacetime, it admits a smooth foliation by smooth spacelike Cauchy surfaces X dX^d, exhibited by a diffeomorphism

X D 0,1×X d, X^D \,\simeq\, \mathbb{R}^{0,1} \times X^d \,,

which is isometric with respect to a possibly non-product pseudo-Riemannian metric on the right

(Bernal & Sánchez 2005, Thm. 1.1, following Geroch 1970, reviewed in Sánchez 2022 (3)).


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).


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

in the topological category due to

and in the smooth category (needed in practice) due to

Last revised on December 21, 2023 at 15:15:03. See the history of this page for a list of all contributions to it.