synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
physics, mathematical physics, philosophy of physics
theory (physics), model (physics)
experiment, measurement, computable physics
Axiomatizations
Tools
Structural phenomena
Types of quantum field thories
A Cauchy surface is a hypersurface in spacetime (so actually a $3$-dimensional region in our $4$-dimensional spacetime) that can profitably be seen as constituting ‘all of space at a given time’.
For $(X,g)$ a Lorentzian manifold, a Cauchy surface is an embedded submanifold $\Sigma \hookrightarrow X$ such that every timelike curve in $X$ may be extended to a timelike curve that intersects $\Sigma$ precisely in one point.
A Lorentzian manifold that does admit a Cauchy surface is called globally hyperbolic.
For $X^D$ a smooth globally hyperbolic spacetime, it admits a smooth foliation by smooth spacelike Cauchy surfaces $X^d$, exhibited by a diffeomorphism
which is isometric with respect to a possibly non-product pseudo-Riemannian metric on the right
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 $t$ to each surface $\Sigma$, so that we think of $\Sigma$ as ‘space at time $t$’. 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
Antonio N. Bernal, Miguel Sánchez, On smooth Cauchy hypersurfaces and Geroch’s splitting theorem, Commun. Math. Phys. 243 (2003) 461-470 [arXiv:gr-qc/0306108, doi:10.1007/s00220-003-0982-6]
Antonio N. Bernal, Miguel Sánchez, Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes, Commun. Math. Phys. 257 (2005) 43-50 [arXiv:gr-qc/0401112, doi:10.1007/s00220-005-1346-1]
Miguel Sánchez, Globally hyperbolic spacetimes: slicings, boundaries and counterexamples, Gen Relativ Gravit 54 124 (2022) [arXiv:2110.13672, doi:10.1007/s10714-022-03002-6]
Last revised on December 21, 2023 at 15:15:03. See the history of this page for a list of all contributions to it.