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)
Let $X$ be a smooth manifold and let $D$ be a differential operator on (smooth sections of) the trivial line bundle over $X$ (or more generally a properly supported pseudo-differential operator). Then the principal symbol $q(D)$ of $D$ is equivalently a smooth function on the cotangent bundle $T^\ast X$ (by this example). With the cotangent bundle canonically regarded as a symplectic manifold, let
be the corresponding Hamiltonian vector field.
The bicharacteristic flow of $D$ is the Hamiltonian flow of the Hamiltonian vector field $v_{q(D)}$ inside the submanifold defined by $q = 0$. Moreover:
A single flow line in $T^\ast X$ is called a bicharacteristic strip of $D$,
the projection of such to a curve in $X$ is called a bicharacteristic curve.
The relation $C$ on $T^\ast X$ given by
is called the bicharacteristic relation.
(bicharacteristic curves of wave operator/Klein-Gordon operators are the lightlike geodesics)
Let $(X,g)$ be a Lorentzian manifold and let $D \coloneqq \Box_g - m^2$ be its wave operator/Klein-Gordon operator.
Then the bicharacteristic curves of $D$ (def. ) are precisely the lightlike geodesics of $(X,e)$, and the bicharacteristic strips are precisely these geodesices with their cotangent vectors.
Accordingly two cotangent vectors are bicharacteristically related $(x_1,k_1) \sim (x_2,k_2)$ precisely if there is a lightlike geodesic connecting the points, with $k_1$ and $k_2$ the corresponding cotangents, hence one the result of parallel transport of the other along the geodesic.
(Radzikowski 96, prop. 4.2 and below (6))
Specifically on Minkowski spacetime:
(bicharacteristic flow of Klein-Gordon operator on Minkowski spacetime)
Let $\mathbb{R}^{p,1}$ be Minkowski spacetime of dimension $p+1$ consider the Klein-Gordon operator
Its principal symbol is the function
Hence $q(k) = 0$ is the condition that the wave vector $k$ be lightlike.
The Hamiltonian vector field corresponding to $q$ is
in that
It follows that the bicharacteristic curves are precisely the lightlike curves
and the corresponding bicharacteristic strips are these with their lightlike contangent vector constantly carried along
The propagation of singularities theorem says that the wave front set of a distributional solution to the differential equation of a sufficiently nice differential operator (or generally of a properly supported pseudo-differential operator) is preserved by the bicharacteristic flow.
Review in the context of the free scalar field on globally hyperbolic spacetimes (with $Q$ the wave operator/Klein-Gordon operator) is in
Last revised on August 27, 2018 at 18:58:09. See the history of this page for a list of all contributions to it.