TITLE: Dynamics of dielectric dipole during polarization?
QUESTION [0 upvotes]: When a dielectric material is placed in an exterior electric field, a process of polarization would occur.

During the stretching phase, does that happen uniformly(stretch evenly)or only in a specific direction(e.g. negative charge moving away from positive charge)?

Also, what limits the streching length per dipole? What limits it from going from $x$-$nm$ to $2x$-$nm$?-

REPLY [1 votes]: According to An Introduction to Electrodynamics The Fourth Edition by David Griffiths page 167, the positive nucleus and the electrons both move during polarization. So to your first question, the stretching is even: both move. On the same page, Griffiths states that the electron cloud and the nucleus reach a balance after they have become displaced by some distance. At this point, the force from the electron cloud to the nucleus balances the force of the external electric field on the nucleus and the electron cloud. If this external field is large enough, the electrons can be ripped from their nucleus.


TITLE: Absolute value of real numbers and inequality
QUESTION [0 upvotes]: Let $a$ and $b$ be real numbers. Show that
$\vert a-b\vert < \epsilon \Rightarrow \vert a\vert<\vert b\vert+\epsilon$
for $\epsilon>0$.
Looks quite easy but I'm not getting it. I tried to use triangular inequality in many forms but it doensn't come.

REPLY [1 votes]: You had the idea -- this is indeed the Triangle Inequality: $$|a| = |(a-b) + b| \leq |a-b| + |b| < \epsilon + |b|.$$

REPLY [1 votes]: $$|a - b| < \epsilon $$
$$|a| = |(a-b) + b| < |a-b| + |b| $$
Therefore by using (1) in (2)
$$|a| < |b| + \epsilon$$


TITLE: Rank of a 2 x 2 matrix
QUESTION [0 upvotes]: Prove that the rotation matrix is invertible.
Let
$$
\begin{pmatrix}
\cos 2\pi t &  \cos \frac{\pi}{6}t\\
\sin 2\pi t  &  \sin \frac{\pi}{6}t\\
\end{pmatrix}
$$

What is the rank of the last matrix for $0 \leq t < 12$ in particular for what values of t the matrix has rank 1.
I have prove $1$. and determine for which values the rotation matrix is not invertible but how can i prove $2$. Any hint will be appreciate it .Tahnxs.

REPLY [2 votes]: Hint:
$$\sin(a-b)=\sin(a)\cos(b)-\cos(a)\sin(b)$$ So $\det(A)=\sin\left(\frac{\pi}6-2\pi\right)t$. Now think about the values which make it vanished. In fact, find out which values of $t$ make the rank $2$ and which ones make it $1$.

REPLY [0 votes]: How about taking the determinant, setting it to zero, and solving for $t$ to make the equation true?


TITLE: In quantum dynamical simulations, what is the symmetric (Riemannian) analog of a Poisson bracket?
QUESTION [3 upvotes]: The question narrowly posed is:

What is the accepted name of the bracket operation that is obtained by replacing the (antisymmetric) symplectic structure of the Poisson bracket with a (symmetric) Riemann metric?

In the (likely?) event that this bracket operation is known by various names in various disciplines, preferred answer(s) will relate to algebras that are associated to Hamiltonian flows on (low-dimension) Kählerian varieties that are naturally immersed in a (large-dimension) Hilbert space, whose dynamical potentials are the operator symbol functions pulled back from that Hilbert space.
Specifically, in practical applications, the Kählerian varieties generically are (low-dimension) rank-$r$ secant varieties of $n$-factor (equivalently, $n$-particle) Segre varieties.
As explained below, this question is stimulated by Michael Nielsen's recent weblog post
Survey Notes on Fermi algebras and the Jordan-Wigner Transform, which points to Michaels's GitHub release titled The-Fermionic-canonical-commutation-relations-and-the-Jordan-Wigner-transform
Michael's write-up poses this broader question:

For what algebraic/geometric reason(s) are fermionic quantum dynamical flows (seemingly) harder to simulate on low-dimension varieties than bosonic quantum dynamical flows?

Background
The key ideas of Charles Slichter's classic 1963 textbook Principles of Magnetic Resonance (still in print, and presently in its 3rd edition) are developed in Chapter 3: Magnetic Dipolar Broadening In Rigid Lattices.
Mathematically speaking, a lot has happened since 1963, and it proves to be very instructive to supplement Slichter's Chapter 3 with material describing quantum spin dynamics in the language of Hamiltonian flow, and quantum simulation in the language of pullback (draft of "Slichter redux" here)
The key to reconciling the old and new ways of thinking about spin dynamics is summarized in the following theorem, which physically speaking, asserts that pullback onto low-dimension varieties preserves much of the algebraic and thermodynamic "quantum goodness" of Hilbert space:

(the above graphic is hosted on GitHub).
This theorem describes how to pullback operator commutator algebras onto low-dimension simulation varieties, and so it is natural to ask (inspired by Michael's GitHub notes):

Onto what algebraic varieties do canonical (fermionic) anticommutators pullback naturally?

A partial answer is supplied by the following lemma, namely, the above theorem goes through if in the Poisson bracket $\langle ds_{\mathcal{H}},dh_{\mathcal{H}}\rangle_{\phi^{-1}_\omega}$ the simple replacement $\omega\to g$ is made, that is, if we simply replace the canonical Kählerian symplectic structure with the canonical Kählerian metric structure.  Algebraically speaking, this means that anticommutation relations pullback just as naturally as commutation relations.
So in essence, we would like to extend our quantum pullback theorem, and its umbral discussion of practical applications, to encompass the fermionic dynamics of Michael Nielsen's notes, as well the spin dynamics of Charlie Slichter's textbook.
The practical problem is, it's not so easy (for me) to construct low-dimension algebras that realize (even approximately) the canonical anticommutation relations … whereas low-dimension constructions are easy for (say) angular momentum commutators … this is where expert mathematical advice (even starting name(s) for the symmetric bracket operation) would be welcome.
Applications
The above may seem pretty dry, but these algebraic/geometric considerations are of central importance in the practical pursuit of a goal set in the 1940s and 1950s by von Neumann, Wiener and Feynman (among many mathematicians and scientists of that generation), namely (in von Neumann's words) "to look at an $H$ atom."    Very broadly speaking, the relevance to the question asked is that bosonic (commutator-respecting) pullbacks characterize the quantum communication channels by which we see the atoms, while fermionic (anticommutator-respecting) pullbacks characterize their chemical dynamics.
To appreciate the attraction this challenge had for von Neumann, Wiener, and Feynman, it is instructive to start with Feynman's celebrated question: What good would it be to see individual atoms distinctly? and restrict it to: Which is numerically the greater challenge, to catalogue every individual star in the universe, or to catalogue every individual atom in the human body?
Then it is easy to compute, that if a 1.7 meter human body were scaled to the size of the observable universe (at present $\sim45.7\times10^9$ light-years), then the individual atoms would be separated by 4 light-years ... and so the answer is, the two great challenges are numerically comparable.
As Stephane Guisard's and Jose Salgado's awe-inspiring VLT timelapse video shows, the astronomers have made a very good start at their challenge … and as our practical mathematical understanding of spin-and-atom dynamics approaches the astronomer's practical understanding of photon dynamics (hopefully with help from MathOverflow) … well, it will be mighty interesting to participate in both of these great challenges.

REPLY [2 votes]: I will answer only your first question --- I hope that you get a more thorough answer from someone with more expertise than I have.  Your other questions, I think, are deeper, and I haven't had a chance yet to process them.
The short answer to your first question, I think, is that although there are many texts with words like "antibracket", whenever you come across a seemingly "symmetric" "Poisson bracket", it is probably still truly antisymmetric, for the right notion of "antisymmetric".  The reason is that it's hard to imagine a condition on a metric tensor $g^{ij}$ so that the bracket (biderivation)
$$ g^{ij}(x) \frac{\partial}{\partial x^i} \otimes \frac{\partial}{\partial x^j} $$
(always Einstein's summation convention) satisfies any sort of "Jacobi identity" --- at least, doing so is hard for me.  (Maybe this is some "flatness" condition?  An expert might be able to answer quickly.)
Much more common, I think, is the situation when $x^i$ are Fermionic variables, and the geometric structure is a symplectic one.  Note that from the "external" point of view of doing computations with homogeneous coordinates, a symplectic form on a Fermionic manifold looks "symmetric".  But the correct perspective is an internal one, where "symplectic" means, among other things, antisymmetric in the way that something acting on fermions should be.  If from an internal perspective the geometric structure is symplectic, then you should not invent extra language for it.  For details on the "internal" (aka "functor of points") approach to sign rules and Fermions and all that jazz, see the article "Notes on Supersymmetry" by Deligne and Morgan in QFT and Strings.

As an aside, let me also mention one more place where "symmetric" "antibrackets" show up, and try to explain why again there really should not be a new word for them.  Namely, in the Batalin-Vilkovisky approach to perturbative integrals, one thing that happens is that you have a ($\mathbb Z$-graded) supermanifold (choose local coordinates $x^i$) with a biderivation $P = P^{ij}(x) \frac{\partial}{\partial x^i} \otimes \frac{\partial}{\partial x^j}$.  This biderivation is required to satisfy that $P$ is symmetric in the internal sense, and that it satisfy a Jacobi-like requirement, and most importantly that $P$ itself is an odd operator for the Fermionic grading.  (In $\mathbb Z$-graded world you can do more interesting things with choice of which odd number.)  A typical example, where I denote by $\Pi$ the "parity reversal operator", are "odd cotangent bundles" $\Pi T^\ast X$ for $X$ any supermanifold (i.e. this is the supermanifold you get as the total space of a vector bundle over $X$, where the fiber over $x$ is the parity-reversal of the cotangent vector space $T^\ast_x X$).  Then $P$ is the "canonical Poisson bracket".
I claim that in an appropriate sense $P$ is still "antisymmetric".  Let $A$ denote the algebra of functions on your supermanifold --- it is a commutative algebra in the category of supervector spaces ("commutative" in the "internal" sense).  Note that the biderivation $P$ is not a morphism in the category of supervector spaces $A \otimes A \to A$, because the morphisms are only the "even" ones, and $P$ is odd.  It does define a morphism $\Pi A \otimes \Pi A \to \Pi A$; the conditions on $P$ above are that this morphism is a Lie bracket (in the internal sense).
In any case, the most basic statements about the geometry of an operator $P$ as above are very similar to the classical Poisson geometry.  So it's reasonable to say that "odd Poisson bracket" means something symmetric in the internal sense.


text:
\begin{document}

\maketitle

\begin{abstract}
We summarize the analytic theory of linear wave equations on globally
hyperbolic Lorentzian manifolds.
\end{abstract}

\tableofcontents

\section{Introduction}

In General Relativity \index{general relativity} spacetime \index{spacetime} is modelled by a Lorentzian manifold, \index{Lorentzian manifold}
see e.~g.\ \cite{EH,Wa}.
Many physical phenomena, such as electro-magnetic
radiation,\index{electro-magnetic radiation} are described by solutions to certain linear wave
equations defined on this spacetime manifold.
Thus a good understanding of the theory of wave equations is crucial.
This includes initial value problems (the Cauchy problem), fundamental
solutions, and inverse operators (Green's operators).
The classical textbooks on partial differential equations contain the relevant
results for small domains in Lorentzian manifolds or for very special
manifolds such as Minkowski space.

In this text we summarize the global analytic results obtained in \cite{BGP},
see also Leray's unpublished lecture notes \cite{Leray} and Choquet-Bruhat's
exposition \cite{C-B}.
In order to obtain a good solution theory one has to impose certain geometric
conditions on the underlying manifold.
The situation is similar to the study of elliptic operators \index{elliptic operator}on Riemannian
manifolds.\index{Riemannian manifold}
In order to ensure that the Laplace-Beltrami operator \index{Laplace-Beltrami operator}on a Riemannian manifold
$M$ is essentially self-adjoint \index{essentially self-adjoint}one may make the natural assumption that $M$
be complete.
Unfortunately, there is no good notion of completeness \index{completeness}for Lorentzian
manifolds.
It will turn out that the analysis of wave operators works out nicely if one
assumes that the underlying Lorentzian manifold be globally hyperbolic.
Completeness of Riemannian manifolds and global hyperbolicity of Lorentzian
manifolds are indeed related.
If $(S,g_0)$ is a Riemannian manifold, then the Lorentzian cylinder \index{Lorentzian cylinder}$M=\R\times
S$ with product metric $g= -dt^2 + g_0$ is globally hyperbolic if and only if
$(S,g_0)$ is complete.

We will start by collecting some material on distributional sections in vector
bundles.
Then we will summarize the theory of globally hyperbolic Lorentzian manifolds.
Then we will define wave operators, also called normally hyperbolic operators,
and give some examples.
After that we consider the basic initial value problem, the Cauchy problem.
It turns out that on a globally hyperbolic manifold solutions exist and are
unique.
They depend continuously on the initial data.
The support of the solutions can be controlled which is physically nothing
than the statement that a wave can never propagate faster than with the speed
of light.
In the subsequent section we use the results on the Cauchy problem to show
existence and uniqueness of fundamental solutions.
This is closely related to the existence and uniqueness of Green's operators.

The author is very grateful for many helpful discussions with colleagues
including
Helga Baum,
Olaf M\"uller,
Nicolas Ginoux,
Frank Pf\"affle,
and Miguel S\'anchez.
The author also thanks the Deutsche Forschungsgemeinschaft for financial
support.

\section{Distributional sections in vector bundles}

Let us start by giving some definitions and by fixing the
terminology for distributions on manifolds.
We will confine ourselves to those facts that we will actually need later on.
A systematic and much more complete introduction may be found e.~g.\ in
\cite{FL1}.

\subsection{Preliminaries on distributional sections}

Let $M$ be a manifold equipped with a smooth volume density $\dV$.
Later on we will use the volume density induced by a Lorentzian metric
but this is irrelevant for now.
We consider a real or complex vector bundle $E\to M$.
We will always write $\K=\R$ or $\K=\Co$ depending on whether $E$ is real or
complex.
The space of compactly supported smooth sections in $E$ will be denoted by
$\mathcal{D}(M,E)$. \index{compactly supported smooth sections}
We equip $E$ and the cotangent bundle $T^*M$ with connections, both denoted by
$\nabla$. \index{connection}
They induce connections on the tensor bundles $T^*M\otimes\cdots\otimes T^*M
\otimes E$, again denoted by $\nabla$.
For a continuously differentiable section $\phi\in C^1(M,E)$ the covariant
derivative is a continuous section in $T^*M\otimes E$, $\nabla\phi\in
C^0(M,T^*M\otimes E)$.
More generally, for $\varphi\in C^k(M,E)$ we get $\nabla^k\phi\in
C^0(M,\underbrace{T^*M\otimes\cdots\otimes T^*M}_{k\,\, \mathrm{factors}}\otimes E)$.

We choose an auxiliary Riemannian metric on $T^*M$ and an auxiliary Riemannian
or Hermitian metric on $E$ depending on whether $E$ is real or complex.
This induces metrics on all bundles $T^*M\otimes\cdots\otimes
T^*M\otimes E$.
Hence the norm of $\nabla^k\phi$ is defined at all points of $M$.

For a subset $A \subset M$ and $\varphi\in C^k(M,E)$ we define the $C^k$-norm
by
\begin{equation}
\|\varphi\|_{C^k(A)} := \max_{j=0,\ldots,k} \;
\sup_{x\in A}\,|\nabla^j\varphi(x)| .
\label{defCkNorm}
\end{equation}
If $A$ is compact, then different choices of the metrics and the
connections yield equivalent norms $\|\cdot\|_{C^k(A)}$.
For this reason there will be no need to explicitly specify the
metrics and the connections.

The elements of $\DD(M,E)$ are referred to as test sections \index{test sections}in $E$.
We define a notion of convergence of test sections.\index{test sections!convergence of}

\Definition{
Let $\varphi,\varphi_n\in \mathcal{D}(M,E)$.
We say that the sequence $(\varphi_n)_n$ {\it converges to} $\varphi$ {\it in}
$\mathcal{D}(M,E)$ if the following two conditions hold:
\begin{enumerate}
 \item There is a compact set $K\subset M$ such that the supports of $\varphi$ and of all
 $\varphi_n$ are contained in $K$, i.~e.\ $\supp(\varphi), \supp(\varphi_n)\subset K$ for
 all $n$.
 \item The sequence $(\varphi_n)_n$ converges to $\varphi$ in all
 $C^k$-norms over $K$, i.~e.\ for each $k\in\N$
$$
\|\varphi-\varphi_n\|_{C^k(K)} \buil{\lra}{n\to\infty} 0.
$$
\end{enumerate}
}

We fix a finite-dimensional $\K$-vector space $W$.
Recall that $\K=\R$ or $\K=\Co$ depending on whether $E$ is real or complex.
Denote by $E^*$ the vector bundle over $M$ dual to $E$.

\Definition{\label{def:distr}
A $\K$-linear map $F:\mathcal{D}(M,E^*)\to W$ is called a
{\it distribution in $E$ with values in $W$} or a {\it distributional section
  in $E$ with values in $W$} \index{distributional section}if it is continuous in the sense
that for all convergent sequences $\varphi_n\to\varphi$ in
$\mathcal{D}(M,E^*)$  one has $ F[\varphi_n]\to F[\varphi]$.
We write $\DD'(M,E,W)$ for the space of all $W$-valued distributions in $E$.
}

Note that since $W$ is finite-dimensional all norms $|\cdot|$
on $W$ yield the same topology on $W$.
Hence there is no need to specify a norm on $W$ for Definition~\ref{def:distr}
to make sense.
Note moreover, that distributional sections in $E$ act on test sections in $E^*$.

\Example{
Pick a  bundle $E\to M$ and a point $x\in M$.
The {\it delta-distribution}\index{delta-distribution} $\delta_x$ is a distributional section in $E$
with values in $E_x^*$.
For $\varphi\in\mathcal{D}(M,E^*)$ it is defined by
\[ \delta_x[\varphi]=\varphi(x). \]
}

\Example{\label{ex:locint}
Every locally integrable section $f\in L^1_{\mathrm{loc}}(M,E)$ can be
regarded as a $\K$-valued distribution in $E$ by setting for any
$\varphi\in\mathcal{D}(M,E^*)$
\[ f[\varphi]:=\int_M \varphi(f) \dV.
\]
Here $\phi(f)$ denotes the $\K$-valued $L^1$-function with compact support on $M$
obtained by pointwise application of $\phi(x)\in E_x^*$ to $f(x)\in E_x$.
}

\subsection{Differential operators acting on distributions}
\label{subseq:DiffOpDist}

Let $E$ and $F$ be two $\K$-vector bundles over the manifold $M$, $\K=\R$ or
$\K=\Co$.
Consider a linear differential operator $P:C^\infty(M,E)\to
C^\infty(M,F)$.\index{linear differential operator}
There is a unique linear differential operator $P^*:C^\infty(M,F^*)\to
C^\infty(M,E^*)$ called the {\em formal adjoint of} $P$\index{linear differential operator!formal adjoint of} such that
for any $\varphi\in \DD(M,E)$ and $\psi\in\DD(M,F^*)$
\begin{equation}
\int_M \psi(P\varphi)\, \dV =
\int_M (P^*\psi)(\varphi)\, \dV .
\label{formaladjoint}
\end{equation}
If $P$ is of order $k$, then so is $P^*$ and (\ref{formaladjoint}) holds for
all $\varphi\in C^k(M,E)$ and $\psi\in C^k(M,F^*)$ such that
$\supp(\varphi)\cap\supp(\psi)$ is compact.
With respect to the canonical identification $E=(E^*)^*$ we have $(P^*)^*=P$.

Any linear differential operator $P:C^\infty(M,E)\to C^\infty(M,F)$ extends
canonically to a linear operator $P:\DD'(M,E,W)\to\DD'(M,F,W)$\index{linear differential operator!acting on distributions} by
$$
(PT)[\varphi]:=T[P^*\varphi]
$$
where $\phi\in\DD(M,F^*)$.
If a sequence $(\varphi_n)_n$ converges in $\mathcal{D}(M,F^*)$ to $0$,
then the sequence $(P^*\varphi_n)_n$ converges to $0$ as well
because $P^*$ is a differential operator.
Hence $(PT)[\varphi_n] = T[P^*\varphi_n] \to 0$.
Therefore $PT$ is indeed again a distribution.

The map $P:\mathcal{D}'(M,E,W)\rightarrow \mathcal{D}'(M,F,W)$
is $\K$-linear.
If $P$ is of order $k$ and $\varphi$ is a $C^k$-section in $E$, seen as a
$\K$-valued distribution in $E$, then the distribution $P\varphi$ coincides
with the continuous section obtained by applying $P$ to $\varphi$ classically.

An important special case occurs when $P$ is of order $0$, i.~e.\
$P\in C^\infty(M,\Hom(E,F))$.
Then $P^*\in C^\infty(M,\Hom(F^*,E^*))$ is the pointwise adjoint.
In particular, for a function $f\in C^\infty(M,\K)$ we have
$$
(fT)[\varphi] = T[f\varphi].
$$

\subsection{Supports}

\Definition{
The {\em support} \index{distributional section!support of}of a distribution $T\in\DD'(M,E,W)$ is defined as the set
\begin{eqnarray*}
\lefteqn{\supp(T)} \\
&:=&
\{x\in M\,|\, \forall\mbox{ neighborhood $U$ of $x$ }\exists \,
\varphi\in \DD(M,E)\mbox{ with }\supp(\varphi)\subset U
\mbox{ and } T[\varphi]\not= 0\}.
\end{eqnarray*}
}
It follows from the definition that the support of $T$ is a closed subset of
$M$.
In case $T$ is a $L^1_{\mathrm{loc}}$-section this notion of support coincides
with the usual one for sections.

If for $\phi\in\DD(M,E^*)$ the supports of $\phi$ and $T$ are disjoint, then
$T[\phi]=0$.
Namely, for each $x\in\supp(\phi)$ there is a neighborhood $U$ of $x$ such
that $T[\psi]=0$ whenever $\supp(\psi)\subset U$.
Cover the compact set $\supp(\phi)$ by finitely many such open sets
$U_1,\ldots,U_k$.
Using a partition of unity one can write $\phi=\psi_1 + \cdots + \psi_k$ with
$\psi_j\in\DD(M,E^*)$ and $\supp(\psi_j)\subset U_j$.
Hence
$$
T[\phi]=T[\psi_1 + \cdots + \psi_k] = T[\psi_1] + \cdots + T[\psi_k]=0.
$$
Be aware that it is not sufficient to assume that $\phi$ vanishes on
$\supp(T)$ in order to ensure $T[\phi]=0$.
For example, if $M=\R$ and $E$ is the trivial $\K$-line bundle let
$T\in\DD'(\R,\K)$ be given by $T[\phi]=\phi'(0)$.
Then $\supp(T)=\{0\}$ but $T[\phi]=\phi'(0)$ may well be nonzero while
$\phi(0)=0$.

If $T\in\DD'(M,E,W)$ and $\varphi\in C^\infty(M,E^*)$, then the evaluation
$T[\varphi]$ can be defined if $\supp(T)\cap\supp(\varphi)$ is compact
even if the support of $\phi$ itself is noncompact.
To do this pick a function $\sigma\in \DD(M,\R)$ that is constant $1$ on
a neighborhood of $\supp(T)\cap\supp(\varphi)$ and put
$$
T[\varphi] := T[\sigma\varphi].
$$
This definition is independent of the choice of $\sigma$ since for another
choice $\sigma'$ we have
$$
T[\sigma\varphi] - T[\sigma'\varphi] =
T[(\sigma-\sigma')\varphi] = 0
$$
because $\supp((\sigma-\sigma')\varphi)$ and $\supp(T)$ are disjoint.

Let $T\in\DD'(M,E,W)$ and let $\Omega\subset M$ be an open subset.
Each test section $\varphi\in\DD(\Omega,E^*)$ can be extended by $0$ and yields
a test section $\varphi \in \DD(M,E^*)$.
This defines an embedding $\DD(\Omega,E^*)\subset\DD(M,E^*)$.
By the restriction of $T$ to $\Omega$ we mean its restriction from
$\DD(M,E^*)$ to $\DD(\Omega,E^*)$.

\Definition{
The {\em singular support}\index{distributional section!singular support of} $\ssupp(T)$ of a distribution $T\in\DD'(M,E,W)$
is the set of points which do not have a neighborhood restricted to which
$T$ coincides with a smooth section.
}

The singular support is also closed and we always have $\ssupp(T)
\subset\supp(T)$.

\Example{
For the delta-distribution $\delta_x$ we have
$\supp(\delta_x)=\ssupp(\delta_x)=\{x\}$.
}

\subsection{Convergence of distributions}

The space $\DD'(M,E)$ of distributions in $E$ will always be given the
{\em weak topology}.\index{weak topology}
This means that $T_n \to T$ in $\DD'(M,E,W)$ if and only if $T_n[\varphi]
\to T[\varphi]$ for all $\varphi\in\DD(M,E^*)$.
Linear differential operators $P$ are always continuous with respect to the
weak topology.
Namely, if $T_n \to T$, then we have for every $\varphi\in\DD(M,E^*)$
$$
PT_n[\varphi] = T_n[P^*\varphi] \to T[P^*\varphi] = PT[\varphi].
$$
Hence
$$
PT_n \to PT.
$$

\Remark{
Let $T_n,T\in C^0(M,E)$ and suppose $\|T_n - T\|_{C^0(M)} \to 0$.
Consider $T_n$ and $T$ as distributions.
Then $T_n \to T$ in $\DD'(M,E)$.
In particular, for every linear differential operator $P$
we have $PT_n \to PT$.
}

\section{Globally hyperbolic Lorentzian manifolds}

Next we summarize some notions and facts from Lorentzian geometry.
More comprehensive introductions can be found in \cite{BEE} and in
\cite{ONeill}.

By a {\em Lorentzian manifold}\index{Lorentzian manifold}  we mean a semi-Riemannian manifold whose
metric has signature $(-,+,\cdots,+)$.
We denote the Lorentzian metric \index{Lorentzian metric}by $g$ or by $\langle\cdot,\cdot\rangle$.
A tangent vector $X\in TM$ is called {\em timelike} \index{tangent vector!timelike} if $\langle X,X\rangle <
0$, {\em lightlike}\index{tangent vector!lightlike} if $\langle X,X\rangle = 0$ and $X\not=0$, {\em causal} \index{tangent vector!causal}if
it is timelike or lightlike, and {\em spacelike} \index{tangent vector!spacelike}otherwise.
At each point $p\in M$ the set of timelike vectors in $T_pM$ decomposes into
two connected components.
A {\em timeorientation} \index{timeorientation}on $M$ is a choice of one of the two
connected components of timelike vectors in $T_pM$ which depends continuously
on $p$.
This means that we can find a continuous timelike vector field on $M$ taking
values in the chosen connected components.
Tangent vectors in the chosen connected component are called {\em future
  directed},\index{tangent vector!future directed} those in the other component are called {\em past directed}.\index{tangent vector!past directed}

Let $M$ be a timeoriented Lorentzian manifold.
A piecewise $C^1$-curve in $M$ is called {\em timelike, lightlike, causal,
spacelike, future directed}, \index{curve!timelike}\index{curve!lightlike}\index{curve!causal}\index{curve!spacelike}\index{curve!future directed}\index{curve!past directed}or {\em
past directed} if its tangent vectors are timelike, lightlike, causal,
spacelike, future directed, or past directed respectively.

The {\em chronological future}\index{chronological future} $I_+^M(x)$ of a point $x\in M$ is the set of
points that can be reached from $x$ by future directed timelike curves.
Similarly, the {\em causal future}\index{causal future} $J_+^M(x)$ of a point $x\in M$ consists
of those points that can be reached from $x$ by causal curves and of $x$
itself.
The {\em chronological future} of a subset
$A\subset M$ is defined to be $I_+^M(A):=\buil{\cup}{x\in A}I_+^M(x)$.
Similarly, the {\em causal future} of $A$ is $J_+^M(A):=\buil{\cup}{x\in
  A}J_+^M(x)$.
The {\em chronological past}\index{chronological past} $I_-^M(A)$ and the {\em causal past}\index{causal past} $J_-^M(A)$
are defined by replacing future directed curves by past directed curves.
One has in general that $I_\pm^M(A)$ is the interior of $J^M_\pm(A)$ and
that $J_\pm^M(A)$ is contained in the closure of $I_\pm^M(A)$.
The chronological future and past are open subsets but the causal
future and past are not always closed even if $A$ is closed.

\begin{center}
\input{fig-futurepast}
\end{center}

We will also use the notation $J^M(A) := J_-^M(A) \cup J_+^M(A)$.
A subset $A\subset M$ is called {\em past compact}\index{past compact subset}  if $A\cap J_-^M(p)$
is compact for all $p\in M$.
Similarly, one defines {\em future compact} subsets.\index{future compact subset}

\begin{center}
\input{fig-pastcompact1}
\end{center}

\Definition{
A subset $S$ of a connected timeoriented Lorentzian manifold is called
{\em achronal}\index{achronal subset} if  each timelike
curve meets $S$ in at most one point.
A subset $S$ of a connected timeoriented Lorentzian manifold is called
\emph{acausal}\index{acausal subset} if  each causal
curve meets $S$ in at most one point.
A subset $S$ of a connected timeoriented Lorentzian manifold is a {\em
Cauchy hypersurface} \index{Cauchy hypersurface} if each inextendible timelike curve in $M$
meets $S$ at exactly one point.
}

\begin{center}
\input{fig-cauchy}
\end{center}

Obviously every acausal subset is achronal, but the reverse is wrong.
Any Cauchy hypersurface is achronal.
Moreover, it is a closed topological hypersurface and it is hit
by each inextendible causal curve in at least one point.
Any two Cauchy hypersurfaces in $M$ are homeomorphic.
Furthermore, the causal future and past of a Cauchy hypersurface is past
and future compact respectively.

\Definition{
A Lorentzian manifold is said to satisfy the \emph{causality condition} \index{causality condition}if
it does not contain any closed causal curve.

A Lorentzian manifold is said to satisfy the {\em strong causality condition}
\index{strong causality condition}if there are no almost closed causal curves.
More precisely, for each point $p\in M$ and for each open neighborhood
$U$ of $p$ there exists an open neighborhood $V\subset U$ of $p$ such
that each causal curve in $M$ starting and ending in $V$ is entirely
contained in $U$.
}

\begin{center}
\input{fig-strongcausal}
\end{center}

Obviously, the strong causality condition implies the causality condition.

In order to get a good analytical theory for wave operators we must impose
certain geometric conditions on the Lorentzian manifold.
Here are several equivalent formulations.

\begin{thm}\label{globhyp}
Let $M$ be a connected timeoriented Lorentzian manifold.
Then the following are equivalent:
\begin{itemize}
\item[(1)]
$M$ satisfies the strong causality condition and for all
$p,q\in M$ the intersection $J_+^M(p)\cap J_-^M(q)$ is compact.
\item[(2)]
There exists a Cauchy hypersurface in $M$.
\item[(3)]
There exists a smooth spacelike Cauchy hypersurface in $M$.
\item[(4)]
$M$ is foliated by smooth spacelike Cauchy hypersurfaces.
More precisely,
$M$ is isometric to $\R\times S$ with metric $-\beta dt^2 + g_t$ where
$\beta$ is a smooth positive function, $g_t$ is a Riemannian metric
on $S$ depending smoothly on $t\in\R$ and each $\{t\}\times S$ is a smooth
spacelike Cauchy hypersurface in $M$.
\end{itemize}
\end{thm}

That (1) implies (4) has been shown by Bernal and S\'anchez in
\cite[Thm.~1.1]{BS} using work of Geroch \cite[Thm.~11]{Geroch}.
See also \cite[Prop.~6.6.8]{EH} and \cite[p.~209]{Wa} for earlier mentionings
of this fact.
The implications $(4) \Rightarrow (3)$ and $(3) \Rightarrow (2)$ are trivial.
That (2) implies (1) is well-known, see e.~g.\ \cite[Cor.~39, p.~422]{ONeill}.

\Definition{
A connected timeoriented Lorentzian manifold satisfying one and hence all
conditions in Theorem~\ref{globhyp} is called {\em globally
hyperbolic}.\index{Lorentzian manifold!globally hyperbolic}
}

\Remark{\label{rem:globhypOmega}
If $M$ is a globally hyperbolic Lorentzian manifold, then a nonempty open
subset $\Omega\subset M$ is itself globally hyperbolic if and only if for any
$p,q\in \Omega$ the intersection $J_+^\Omega(p)\cap
J_-^\Omega(q)\subset\Omega$ is compact.
Indeed non-existence of almost closed causal curves in $M$ directly implies
non-existence of such curves in $\Omega$.
}

\Remark{\label{rem:globhypconf}
It should be noted that global hyperbolicity is a conformal notion.
The definition of a Cauchy hypersurface requires only causal concepts.
Hence if $(M,g)$ is globally hyperbolic and we replace the metric $g$ by a
conformally related metric $\hat g = f\cdot g$, $f$ a smooth positive function
on $M$, then $(M,\hat g)$ is again globally hyperbolic.
}

\Examples{\label{ex:globhyp}
Minkowski space \index{Minkowski space} is globally hyperbolic.
Every spacelike hyperplane is a Cauchy hypersurface.
One can write Minkowski space as $\R\times\R^{n-1}$ with the metric
$-\dt^2+g_t$ where $g_t$ is the Euclidean metric on $\R^{n-1}$ and does not
depend on $t$.

Let $(S,g_0)$ be a connected Riemannian manifold and $I\subset\R$ an interval.
The manifold $M=I\times S$ with the metric $g=-\dt^2 + g_0$ is
globally hyperbolic if and only if $(S,g_0)$ is complete.
This applies in particular if $S$ is compact.

More generally, if $f:I\to\R$ is a smooth positive function we may equip
$M=I\times S$ with the metric $g=-\dt^2 + f(t)^2\cdot g_0$.
Again, $(M,g)$ is globally hyperbolic if and only if $(S,g_0)$ is complete.
{\em Robertson-Walker spacetimes} \index{Robertson-Walker spacetime}and, in particular, {\em Friedmann
  cosmological models},\index{Friedmann cosmological model}  are of this type.
They are used to discuss big bang, \index{big bang}expansion of the universe, \index{expansion of the universe}and cosmological
redshift, \index{cosmological
redshift}compare \cite[Ch.~12]{ONeill}.
Another example of this type is {\em deSitter spacetime}, \index{deSitter spacetime}where $I=\R$,
$S=S^{n-1}$, $g_0$ is the canonical metric of $S^{n-1}$ of constant sectional
curvature $1$, and $f(t)=\cosh(t)$.
But {\em Anti-deSitter spacetime} \index{anti-deSitter spacetime}is not globally hyperbolic.

The interior and exterior {\em Schwarzschild spacetimes} \index{Schwarzschild spacetime}are globally
hyperbolic.
They model the universe in the neighborhood of a massive static rotionally
symmetric body such as a black hole.\index{black hole}
They are used to investigate perihelion advance of Mercury, the bending of
light near the sun and other astronomical phenomena, see
\cite[Ch.~13]{ONeill}.
}

\begin{lemma}\label{lJ+KJ-K'cpct}
Let $S$ be a Cauchy hypersurface in a globally hyperbolic Lorentzian manifold
$M$ and let $K,K'\subset M$ be compact.

Then $J_\pm^M(K)\cap S$, $J_\pm^M(K)\cap J_\mp^M(S)$, and $J_+^M(K)\cap
J_-^M(K')$ are compact.
\end{lemma}

\section{Wave operators}

Let ${M}$ be a Lorentzian manifold and let $E\to {M}$ be a real or complex
vector bundle.
A linear differential operator $P:\,C^{\infty}({M},E)\to
C^{\infty}({M},E)$ of second order will be called a {\it wave operator} \index{wave operator}or a
{\it normally hyperbolic operator} \index{normally hyperbolic operator} if its principal symbol is given by the
metric,
$$
\sigma_P(\xi) = -\la\xi,\xi\ra\cdot\id_{E_x}
$$
for all $x\in{M}$ and all $\xi \in T^*_x{M}$.
In other words, if we choose local coordinates $x^1,\ldots,x^n$ on $M$ and a
local trivialization of $E$, then
$$
P = -\sum_{i,j=1}^n g^{ij}(x)\frac{\partial^2}{\partial x^i \partial x^j}
+ \sum_{j=1}^n A_j(x) \frac{\partial}{\partial x^j} + B(x)
$$
where $A_j$ and $B$ are matrix-valued coefficients depending smoothly on
$x$ and $(g^{ij})_{ij}$ is the inverse matrix of  $(g_{ij})_{ij}$ with
$g_{ij}=\la\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j}\ra$.

\Example{
Let $E$ be the trivial line bundle so that sections in $E$ are just functions.
The d'Alembert operator \index{d'Alembert operator} $P=\Box=-\div\circ\grad$ is a wave operator.
}

\Example{
Let $E$ be a vector bundle and let $\nabla$ be a connection on $E$.
This connection together with the Levi-Civita connection on $T^*M$ induces a
connection on $T^*M\otimes E$, again denoted $\nabla$.
We define the {\em connection-d'Alembert operator} \index{connection-d'Alembert operator} $\Box^\nabla$ to be minus
the composition of the following three maps
$$
C^\infty(M,E) \stackrel{\nabla}{\longrightarrow}
C^\infty(M,T^*M\otimes E) \stackrel{\nabla}{\longrightarrow}
C^\infty(M,T^*M\otimes T^*M\otimes E)
\xrightarrow{\tr\otimes\id_E}
C^\infty(M,E)
$$
where $\tr:T^*M\otimes T^*M \to \R$ denotes the metric trace,
$\tr(\xi\otimes\eta)=\la\xi,\eta\ra$.
We compute the principal symbol,
$$
\sigma_{\Box^\nabla}(\xi)\phi
=
-(\tr\otimes\id_E)\circ\sigma_{\nabla}(\xi)\circ\sigma_{\nabla}(\xi)(\phi)
=
-(\tr\otimes\id_E)(\xi\otimes\xi\otimes\phi)
=
-\la\xi,\xi\ra\,\phi.
$$
Hence $\Box^\nabla$ is a wave operator.
}
\Example{
Let $E=\Lambda^kT^*M$ be the bundle of $k$-forms.
Exterior differentiation $d:C^\infty(M,\Lambda^kT^*M) \to
C^\infty(M,\Lambda^{k+1}T^*M)$ increases the degree by one while
the codifferential $\delta:C^\infty(M,\Lambda^{k}T^*M) \to
C^\infty(M,\Lambda^{k-1}T^*M)$ decreases the degree by one.
While $d$ is independent of the metric, the codifferential $\delta$ does
depend on the Lorentzian metric.
The operator $P=d\delta + \delta d$ is a wave operator.
}
\Example{
If $M$ carries a Lorentzian metric and a spin structure, then one can define
the spinor bundle $\Sigma M$ and the Dirac operator\index{Dirac operator}
$$
D : C^\infty(M,\Sigma M) \to C^\infty(M,\Sigma M) ,
$$
see \cite{Baum} or \cite{BGM} for the definitions.
The principal symbol of $D$ is given by Clifford multiplication,
$$
\sigma_D(\xi)\psi = \xi^\sharp \cdot \psi.
$$
Hence
$$
\sigma_{D^2}(\xi)\psi = \sigma_D(\xi)\sigma_D(\xi)\psi = \xi^\sharp \cdot
\xi^\sharp \cdot \psi = -\la\xi,\xi\ra\,\psi.
$$
Thus $P=D^2$ is a wave operator.
}

\section{The Cauchy problem}

We now come to the basic initial value problem \index{initial value problem}for wave operators, the
{\em Cauchy problem}.\index{Cauchy problem}
The local theory of linear hyperbolic operators can be found in basically any
textbook on partial differential equations.
In \cite{FL2} and \cite{Guenther} the local theory for wave operators on
Lorentzian manifolds is developed.
The results of this section are of global nature.
They make statements about solutions to the Cauchy problem which are defined
globally on a manifold.
Proofs of the results of this section can be found in \cite[Sec.~3.2]{BGP}.

\begin{thm}[Existence and uniqueness of solutions]\label{cauchyglobhyp}\index{Cauchy problem!existence and uniqueness of solutions}
Let $M$ be a globally hyperbolic Lorentzian manifold and let
 $S\subset M$ be a smooth spacelike Cauchy hypersurface.
Let $\nu$ be the future directed timelike unit normal field along $S$.
Let $E$ be a vector bundle over $M$ and let $P$ be a wave
operator acting on sections in $E$.

Then for each  $u_0, u_1 \in \DD(S,E)$ and for each
$f\in\DD(M,E)$ there exists a unique $u\in C^\infty(M,E)$ satisfying
$Pu=f$, $u|_S = u_0$, and $\nabla_\nu u|_S = u_1$.
\end{thm}

It is unclear how to even formulate the Cauchy problem on a Lorentzian
manifold which is not globally hyperbolic.
One would have to replace the concept of a Cauchy hypersurface by something
different to impose the initial conditions upon.
Here are two examples which illustrate what can typically go wrong.

\Example{
Let $M=S^1 \times \R^{n-1}$ with the metric $g=-d\theta^2 + g_0$ where
$d\theta^2$ is the standard metric on $S^1$ of length $1$ and $g_0$ is the
Euclidean metric on $\R^{n-1}$.
The universal covering of $M$ is Minkowski space.

Let us try to impose a Cauchy problem on $\{\theta_0\} \times \R^{n-1}$ which
is the image of a Cauchy hypersurface in Minkowski space.
Such a solution would lift to Minkowski space where it indeed exists uniquely
due to Theorem~\ref{cauchyglobhyp}.
But such a solution on Minkowski space is in general not time periodic, hence
does not descend to a solution on $M$.

Therefore existence of solutions fails.
The problem is here that $M$ violates the causality condition, i.~e.\ there
are closed causal curves.
}

\Remark{
Compact Lorentzian manifolds always possess closed timelike curves and are
therefore never well suited for the analysis of wave operators.
}

\Example{
Let $M$ be a timelike strip in $2$-dimensional Minkowski space, i.~e.\
$M=\R\times (0,1)$ with metric $g=-dt^2 + dx^2$.
Let $S:= \{0\}\times(0,1)$.
Given any $u_0, u_1 \in \DD(S,E)$ and any $f\in\DD(M,E)$, there exists a
solution $u$ to the Cauchy problem.
One can simply take the solution in Minkowski space and restrict it to $M$.
But this solution is not unique in $M$.
Choose $x$ in Minkowski space, $x\not\in M$, such that
$J_+^{\mathrm{Mink}}(x)$ intersects $M$ in the future of $S$ and of $\supp(f)$.
The advanced fundamental solution $w=F_+(x)$ (see next section) has support
contained in $J_+^{\mathrm{Mink}}(x)$ and satisfies $Pw=0$ away from $x$.
Hence $u+w$ restricted to $M$ is again a solution to the Cauchy problem on $M$
with the same initial data.

\begin{center}
\input{fig-zeitstreifen}
\end{center}

The problem is here that $S$ is acausal but not a Cauchy hypersurface.
Physically, a wave ``from outside the manifold'' enters into $M$.
}

The physical statement that a wave can never propagate faster than with the
speed of light is contained in the following.

\begin{thm}[Finite propagation speed]\index{finite propagation speed}
The solution $u$ from Theorem~\ref{cauchyglobhyp} satisfies $\supp(u) \subset
J^M(K)$ where $K=\supp(u_0)\cup \supp(u_1) \cup \supp(f)$.
\end{thm}

The solution to the Cauchy problem depends continuously on the data.

\begin{thm}[Stability]\label{cauchystetig}\index{Cauchy problem!stability of solutions}
Let $M$ be a globally hyperbolic Lorentzian manifold and let
$S\subset M$ be a smooth spacelike Cauchy hypersurface.
Let $\nu$ be the future directed timelike unit normal field along $S$.
Let $E$ be a vector bundle over $M$ and let $P$ be a wave
operator acting on sections in $E$.

Then the map $\DD(M,E) \oplus \DD(S,E) \oplus \DD(S,E) \to C^\infty(M,E)$
sending $(f,u_0,u_1)$ to the unique solution $u$ of the Cauchy
problem $Pu=f$, $u|_S=u|_0$, $\nabla_\nu u = u_1$ is linear continuous.
\end{thm}

This is essentially an application of the open mapping theorem for Fr\'echet
spaces.

\section{Fundamental solutions}

\Definition{\label{deffundsol}
Let $M$ be a timeoriented Lorentzian manifold, let $E\to M$ be a
vector bundle and let $P:\,C^{\infty}(M,E)\to C^{\infty}(M,E)$ be
a wave operator.
Let $x\in M$.
A {\em fundamental solution} \index{fundamental solution}of $P$ at $x$ is a distribution
$F\in\DD'(M,E,E_x^*)$ such that
$$
PF=\delta_x.
$$
In other words, for all $\varphi\in\DD(M,E^*)$ we have
$$
F[P^*\varphi] = \varphi(x).
$$
If $\supp(F(x))\subset J_+^M(x)$, then we call $F$ an {\em advanced
fundamental solution}, \index{fundamental solution!advanced}if $\supp(F(x))\subset J_-^M(x)$, then we call $F$
a {\em retarded fundamental solution}.\index{fundamental solution!retarded}
}

Using the knowlegde about the Cauchy problem from the
previous section it is now not hard to find global fundamental solutions
on a globally hyperbolic manifold.

\begin{thm}\label{globhypexist}
Let $M$ be a globally hyperbolic Lorentzian manifold.
Let $P$ be a wave operator acting on sections in a
vector bundle $E$ over $M$.

Then for every $x\in M$ there is exactly one
fundamental solution $F_+(x)$ for $P$ at $x$ with past compact
support and exactly one fundamental solution $F_-(x)$ for $P$ at $x$ with
future compact support.
They satisfy
\begin{enumerate}
\item
$\supp(F_\pm(x))\subset J_\pm^M(x)$,
\item
for each $\varphi\in\DD(M,E^*)$ the maps $x\mapsto F_\pm(x)[\varphi]$
are smooth sections in $E^*$ satisfying the differential equation
$P^*(F_\pm(\cdot)[\varphi])=\varphi$.
\end{enumerate}
\end{thm}

\begin{proof}[Sketch of proof.]
We do not do the uniqueness part.
To show existence fix a foliation of $M$ by spacelike Cauchy hypersurfaces
$S_t$, $t\in\R$ as in Theorem~\ref{globhyp}.
Let $\nu$ be the future directed unit normal field along the leaves $S_t$.
Let $\varphi\in\DD(M,E^*)$.
Choose $t$ so large that $\supp(\varphi)\subset I_-^M(S_t)$.
By Theorem~\ref{cauchyglobhyp} there exists a unique $\chi_\varphi\in
C^\infty(M,E^*)$ such that $P^*\chi_\varphi=\varphi$ and $\chi_\varphi|_{S_t}
= (\nabla_\nu\chi_\varphi)|_{S_t} = 0$.
One can check that $\chi_\varphi$ does not depend on the choice of $t$.

Fix $x\in M$.
By Theorem~\ref{cauchystetig} $\chi_\varphi$ depends continuously on
$\varphi$.
Since the evaluation map $C^\infty(M,E) \to E_x$ is continuous, the
map $\DD(M,E^*) \to E_x^*$, $\varphi \mapsto \chi_\varphi(x)$, is also
continuous.
Thus $F_+(x)[\varphi] := \chi_\varphi(x)$ defines a distribution.
By definition $P^*(F_+(\cdot)[\varphi]) = P^*\chi_\varphi = \varphi$.

Now $P^*\chi_{P^*\varphi} = P^*\varphi$, hence
$P^*(\chi_{P^*\varphi}-\varphi)=0$.
Since both $\chi_{P^*\varphi}$ and $\varphi$ vanish along $S_t$ the uniqueness
part which we have omitted shows $\chi_{P^*\varphi}=\varphi$.
Thus
$$
(PF_+(x))[\varphi] = F_+(x)[P^*\varphi] = \chi_{P^*\varphi}(x) = \varphi(x)
= \delta_x[\varphi].
$$
Hence $F_+(x)$ is a fundamental solution of $P$ at $x$.

It remains to show $\supp(F_+(x)) \subset J_+^M(x)$.
Let $y\in M\setminus J_+^M(x)$.
We have to construct a neighborhood of $y$ such that for each test section
$\varphi\in\DD(M,E^*)$ whose support is contained in this neighborhood
we have $F_+(x)[\varphi] = \chi_\varphi(x) = 0$.
Since $M$ is globally hyperbolic $J_+^M(x)$ is closed and therefore $J_+^M(x)\cap
J_-^M(y') = \emptyset$ for all $y'$ sufficiently close to $y$.
We choose $y'\in I_+^M(y)$ and $y''\in I_-^M(y)$ so close that $J_+^M(x)\cap
J_-^M(y') = \emptyset$ and $\left(J_+^M(y'') \cap\bigcup_{t \leq t'}S_t\right)\cap
J_+^M(x) = \emptyset$ where $t'\in\R$ is such that $y'\in S_{t'}$.

\begin{center}
\input{fig-fundsolglobhyp}
\end{center}

Now $K:=J_-^M(y')\cap J_+^M(y'')$ is a compact neighborhood of $y$.
Let $\varphi\in\DD(M,E^*)$ be such that $\supp(\varphi)\subset K$.
By Theorem~\ref{cauchyglobhyp} $\supp(\chi_\varphi) \subset J_+^M(K)\cup J_-^M(K)
\subset J_+^M(y'') \cup J_-^M(y')$.
By the independence of $\chi_\varphi$ of the choice of $t>t'$ we have that
$\chi_\varphi$ vanishes on $\bigcup_{t> t'}S_t$.
Hence $\supp(\chi_\varphi) \subset \left(J_+^M(y'') \cap \bigcup_{t \leq
t'}S_t\right) \cup J_-^M(y')$ and is therefore disjoint from $J_+^M(x)$.
Thus $F_+(x)[\varphi] = \chi_\varphi(x) = 0$ as required.
\end{proof}

For a complete proof see \cite[Sec.~3.3]{BGP}.

\section{Green's operators}

Now we want to find ``solution operators'' for a given wave
operator $P$.
More precisely, we want to find operators which are inverses of $P$ when
restricted to suitable spaces of sections.
We will see that existence of such operators is basically equivalent to the
existence of fundamental solutions.

\Definition{ \label{defGfunc}
Let $M$ be a timeoriented connected Lorentzian manifold.
Let $P$ be a wave operator acting on sections in a
vector bundle $E$ over $M$.
A linear map $G_+ : \DD(M,E) \to C^\infty(M,E)$ satisfying
\begin{itemize}
\item[(i)]
$P\circ G_+ = \id_{\DD(M,E)}$,
\item[(ii)]
$G_+ \circ P|_{\DD(M,E)} = \id_{\DD(M,E)}$,
\item[(iii)]
$\supp(G_+\varphi) \subset J_+^M(\supp(\varphi))$ for all
$\varphi\in\DD(M,E)$,
\end{itemize}
is called an {\em advanced Green's operator for} \index{Green's operator!advanced}$P$.
Similarly, a linear map $G_- : \DD(M,E) \to C^\infty(M,E)$ satisfying
(i), (ii), and
\begin{itemize}
\item[(iii')]
$\supp(G_-\varphi) \subset J_-^M(\supp(\varphi))$ for all
$\varphi\in\DD(M,E)$
\end{itemize}
instead of (iii) is called a {\em retarded Green's operator for} $P$.\index{Green's operator!retarded}
}

Fundamental solutions and Green's operators are closely related.

\begin{thm}\label{functorsolve}
Let $M$ be a globally hyperbolic Lorentzian manifold.
Let $P$ be a wave operator acting on sections in a
vector bundle $E$ over $M$.

Then there exist unique advanced and retarded Green's operators $G_\pm:
\DD(M,E) \to C^\infty(M,E)$ for $P$.
\end{thm}

\begin{proof}
By Theorem~\ref{globhypexist} there exist families $F_\pm(x)$ of advanced and
retarded fundamental solutions for the adjoint operator $P^*$ respectively.
We know that $F_\pm(x)$ depend smoothly on $x$ and the
differential equation $P(F_\pm(\cdot)[\varphi])=\varphi$ holds.
By definition we have
$$
P(G_\pm\varphi) = P(F_\mp(\cdot)[\varphi]) = \varphi
$$
thus showing (i).
Assertion (ii) follows from the fact that the $F_\pm(x)$ are fundamental
solutions,
$$
G_\pm(P\varphi)(x) = F_\mp(x)[P\varphi] = P^*F_\mp(x)[\varphi]
= \delta_x[\varphi] = \varphi(x).
$$
To show (iii) let $x\in M$ such that $(G_+\varphi)(x) \not= 0$.
Since $\supp(F_-(x))\subset J_-^M(x)$ the support of $\varphi$ must hit
$J_-^M(x)$.
Hence $x\in J_+^M(\supp(\varphi))$ and therefore $\supp(G_+\varphi)\subset
J_+^M(\supp(\varphi))$.
The argument for $G_-$ is analogous.
\end{proof}

We have seen that existence of fundamental solutions for $P^*$ depending
nicely on $x$ implies existence of Green's operators for $P$.
This construction can be reversed.
Then uniqueness of fundamental solutions in Theorem~\ref{globhypexist}
implies uniqueness of Green's operators.

\begin{lemma}\label{Greenadjungiert}
Let $M$ be a globally hyperbolic Lorentzian manifold.
Let $P$ be a wave operator acting on sections in a
vector bundle $E$ over $M$.
Let $G_\pm$ be the Green's operators for $P$ and $G^*_\pm$ the Green's
operators for the adjoint operator $P^*$.
Then
\begin{equation}
\int_M (G^*_\pm\varphi)\cdot\psi \dV = \int_M \varphi\cdot(G_\mp\psi) \dV
\label{Gsa}
\end{equation}
holds for all $\varphi\in\DD(M,E^*)$ and $\psi\in\DD(M,E)$.
\end{lemma}

\begin{proof}
For the Green's operators we have $PG_\pm=\id_{\DD(M,E)}$ and
$P^*G_\pm^*=\id_{\DD(M,E^*)}$ and hence
\begin{eqnarray*}
\int_M (G_\pm^*\varphi)\cdot\psi \dV
&=& \int_M (G_\pm^*\varphi)\cdot(PG_\mp\psi) \dV  \\
&=& \int_M (P^*G_\pm^*\varphi)\cdot(G_\mp\psi) \dV \\
&=& \int_M \varphi\cdot(G_\mp\psi) \dV .
\end{eqnarray*}
Notice that $\supp(G_\pm\phi) \cap \supp(G_\mp\psi)\subset
J_\pm^M(\supp(\phi))\cap J_\mp^M(\supp(\psi))$ is compact in a globally hyperbolic
manifold so that the partial integration in the second equation is justified.
\end{proof}

\Notation{
We write $\Csc(M,E)$ for the set of all $\phi\in C^\infty(M,E)$ for which
there exists a compact subset $K \subset M$ such that $\supp(\phi) \subset
J^M(K)$.
Obviously, $\Csc(M,E)$ is a vector subspace of $C^\infty(M,E)$.

The subscript ``sc'' should remind the reader of ``space-like compact''.
Namely, if $M$ is globally hyperbolic and $\phi\in\Csc(M,E)$, then for every
Cauchy hypersurface $S\subset M$ the support of $\phi|_{S}$ is contained in
$S\cap J^M(K)$ hence compact by Lemma~\ref{lJ+KJ-K'cpct}.
In this sense sections in $\Csc(M,E)$ have space-like compact support.\index{space-like compact support}
}

\Definition{
We say that a sequence of elements $\phi_j\in \Csc(M,E)$ {\em converges in
 $\Csc(M,E)$ to} $\phi\in \Csc(M,E)$ if there exists a compact subset $K
\subset M$ such that
$$
\supp(\phi) \subset J^M(K)
\mbox{ and }
\supp(\phi_j) \subset J^M(K)
$$
for all $j$ and
$$
\|\phi_j - \phi\|_{C^k(K',E)}\to 0
$$
for all $k\in\N$ and all compact subsets $K'\subset M$.
}

If $G_+$ and $G_-$ are advanced and retarded Green's operators for $P$
respectively, then we get a linear map
$$
G := G_+ - G_- : \DD(M,E) \to \Csc(M,E).
$$

Much of the solution theory of wave operators on globally hyperbolic
Lorentzian manifolds is collected in the following theorem.

\begin{thm}\label{thmExSeq}
Let $M$ be a globally hyperbolic Lorentzian manifold.
Let $P$ be a wave operator acting on sections in a
vector bundle $E$ over $M$.
Let $G_+$ and $G_-$ be advanced and retarded Green's operators for $P$ respectively.

Then
\begin{equation}
0 \to \DD(M,E) \stackrel{P}{\longrightarrow} \DD(M,E)
\stackrel{G}{\longrightarrow} \Csc(M,E) \stackrel{P}{\longrightarrow}\Csc(M,E)
\label{eqExSeq}
\end{equation}
is an exact sequence of linear maps.
\end{thm}

\begin{proof}
Properties (i) and (ii) in Definition~\ref{defGfunc} of Green's operators
directly yield $G\circ P=0$ and $P\circ G=0$, both on $\DD(M,E)$.
Properties (iii) and (iii') ensure that $G$ maps $\DD(M,E)$ to $\Csc(M,E)$.
Hence the sequence of linear maps forms a complex.

Exactness at the first $\DD(M,E)$ means that
$$
P : \DD(M,E) \to \DD(M,E)
$$
is injective.
To see injectivity let $\phi\in\DD(M,E)$ with $P\phi=0$.
Then $\phi= G_+P\phi=G_+0=0$.

Next let $\phi\in\DD(M,E)$ with $G\phi=0$, i.~e.\ $G_+\varphi=G_-\varphi$.
We put $\psi:=G_+\varphi=G_-\varphi \in C^\infty(M,E)$ and we see
$\supp(\psi)=\supp(G_+\phi)\cap\supp(G_-\phi)\subset J_+^M(\supp(\varphi))\cap
J_-^M(\supp(\varphi))$.
Since $(M,g)$ is globally hyperbolic $J_+^M(\supp(\varphi))\cap
J_-^M(\supp(\varphi))$ is compact, hence $\psi\in \DD(M,E)$.
From $P(\psi) = P(G_+(\phi)) = \phi$ we see that $\phi\in P(\DD(M,E))$.
This shows exactness at the second $\DD(M,E)$.

Finally, let $\phi\in\Csc(M,E)$ such that $P\phi=0$.
Without loss of generality we may assume that $\supp(\phi)\subset I_+^M(K)\cup I_-^M(K)$ for
a compact subset $K$ of $M$.
Using a partition of unity subordinated to the open covering $\{I_+^M(K),I_-^M(K)\}$ write $\phi$ as $\phi=\phi_1 + \phi_2$
where $\supp(\phi_1)\subset I_-^M(K)\subset J_-^M(K)$ and $\supp(\phi_2)\subset I_+^M(K)\subset J_+^M(K)$.
For $\psi:=-P\phi_1=P\phi_2$ we see that $\supp(\psi) \subset J_-^M(K) \cap
J_+^M(K)$, hence $\psi\in\DD(M,E)$.

We check that $G_+\psi=\phi_2$.
For all $\chi\in\DD(M,E^*)$ we have
$$
\int_M \chi\cdot(G_+P\phi_2) \dV =
\int_M (G_-^{*}\chi)\cdot(P\phi_2) \dV =
\int_M (P^*G_-^{*}\chi)\cdot\phi_2 \dV =
\int_M \chi\cdot\phi_2 \dV
$$
where $G_-^*$ is the Green's operator for the adjoint operator $P^*$ according
to Lemma~\ref{Greenadjungiert}.
Notice that for the second equation we use the fact that $\supp(\phi_2) \cap
\supp(G^*_-\chi) \subset J^M_+(K)\cap J^M_-(\supp(\chi))$ is compact.
Similarly, one shows $G_-\psi=-\phi_1$.

Now $G\psi = G_+\psi - G_-\psi = \phi_2 + \phi_1 = \phi$, hence $\phi$ is in
the image of $G$.
\end{proof}

\begin{prop}\label{PGstetig}\index{Green's operator!continuity of}
Let $M$ be a globally hyperbolic Lorentzian manifold,
let $P$ be a wave operator acting on sections in a
vector bundle $E$ over $M$.
Let $G_+$ and $G_-$ be the advanced and retarded Green's operators for $P$
respectively.

Then $G_\pm:\DD(M,E) \to \Csc(M,E)$ and all maps in the complex
(\ref{eqExSeq}) are continuous.
\end{prop}

\begin{proof}
The maps $P:\DD(M,E) \to \DD(M,E)$ and $P:\Csc(M,E) \to \Csc(M,E)$ are
continuous simply because $P$ is a differential operator.
It remains to show that $G:\DD(M,E) \to \Csc(M,E)$ is continuous.

Let $\phi_j,\phi\in\DD(M,E)$ and $\phi_j\to \phi$ in $\DD(M,E)$ for all $j$.
Then there exists a compact subset $K\subset M$ such that
$\supp(\phi_j) \subset K$ for all $j$ and $\supp(\phi) \subset K$.
Hence $\supp(G\phi_j)\subset J^M(K)$ for all $j$ and $\supp(G\phi)\subset J^M(K)$.
From the proof of Theorem~\ref{globhypexist} we know that $G_+\phi$ coincides
with  the solution $u$ to the Cauchy problem $Pu=\phi$ with initial conditions
$u|_{S_-} = (\nabla_\nu u)|_{S_-}=0$ where $S_- \subset M$ is a spacelike
Cauchy hypersurface such that $K \subset I_+^M(S_-)$.
Theorem~\ref{cauchystetig}  tells us that if $\phi_j\to\phi$ in $\DD(M,E)$,
then the solutions $G_+\phi_j\to G_+\phi$ in $C^\infty(M,E)$.
The proof for $G_-$ is analogous and the statement for $G$ follows.
\end{proof}

config: arxiv


TITLE: Represent $f(x)=\ln x$ as power series, in powers of $(x-4)$
QUESTION [1 upvotes]: I was requested to represent $f(x)=\ln x$ as a power series, in powers of $(x-4)$. I only started studying power series today, and though I was able to represent different functions as power series I can't seem to work this one out, specially with the condition that the series must be in powers of $(x-4)$.
Any ideas on how to solve this problem?

REPLY [0 votes]: $$f(x)=\ln (x)=\ln (4)-\sum_{n=1}^\infty \dfrac{(-1)^n}{n\cdot 4^n}\cdot (x-4)^n,$$
power series that converges if $|x-4|<4$.


TITLE: What is the result of $\lim_{n\to +\infty}\sum_{k=1}^{n}\frac{\sin\left(\frac{k}{n}\right)}{n+\frac{1}{k}}$?
QUESTION [0 upvotes]: What is the result of the following:
$$ \lim_{n \rightarrow  \infty }  \frac{\sin( \frac{1}{n}) }{n+1}+ \frac{\sin( \frac{2}{n} )}{n+ \frac{1}{2} }+\cdots+ \frac{\sin( \frac{n}{n}) }{n+ \frac{1}{n} }=\text{?}$$

REPLY [1 votes]: For first,
$$ \lim_{n\to +\infty}\frac{1}{n}\sum_{k=1}^{n}\sin\left(\frac{k}{n}\right) = \int_{0}^{1}\sin(x)\,dx = 1-\cos(1) = \color{red}{2\sin^2\left(\frac{1}{2}\right)}$$
by Riemann sums. However,
$$ 0\leq\sum_{k=1}^{n}\sin\left(\frac{k}{n}\right)\left(\frac{1}{n}-\frac{1}{n+\frac{1}{k}}\right)\leq\sum_{k=1}^{n}\frac{k}{n}\cdot\frac{\frac{1}{k}}{n\left(n+\frac{1}{k}\right)}\leq\sum_{k=1}^{n}\frac{1}{n^3}=\frac{1}{n^2}$$
hence we have already computed our limit.


TITLE: Simultaneous eigenkets of angular momentum operators in QM
QUESTION [0 upvotes]: Is it the case that if you have simultaneous eigenkets say $|l m \rangle$ then we have both $$\sum_{l} |l m \rangle \langle l m \rangle = I ~~\text{ and }~~ \sum_{m} |l m \rangle \langle l m \rangle = I ?$$ Also, is there always degeneracy when having shared eigenstates?  Consider the simultaneous eigenkets of the angular momentum operators $J^{2}$ and $J_{z}$, since we have that $|l, m \rangle$ and $|l', m \rangle$ are both eigenstates of the angular momentum operator $J_{z}$ with $J_{z}|l, m \rangle = m \hbar |l, m \rangle$  and $J_{z}|l', m \rangle = m \hbar |l', m \rangle$ clearly the operators $J_{z}$ and $J^2$ are degenerate.

REPLY [1 votes]: No.  The unit operator is
$$
\hat 1= \sum_{\ell m}\vert \ell m\rangle\langle \ell m\vert\, .
$$
The simplest intuition into this is to compare with spherical harmonics, for which the completeness relation involves a sum on both $\ell$ and $m$.  Of course you may have a priori knowledge of a state $\vert\psi\rangle$ and know it's an eigenstate of $\hat L_z$ with eigenvalue $M\hbar$.  Then of course
\begin{align}
\vert\psi\rangle&=\sum_{\ell m}\vert \ell m\rangle\langle \ell m\vert\psi\rangle=\sum_{\ell m}\vert \ell m\rangle\langle \ell m\vert\psi\rangle\,\delta_{Mm}\, ,\\
&=\sum_{\ell}\vert \ell M\rangle\langle \ell M\vert\psi\rangle
\end{align}
just like you may know that $F(\theta,\phi)=f(\theta)e^{iM\phi}$ and so restrict the sum to the appropriate $m$ but this is because you are implicitly using additional information on the eigenvalue to simplify your expansion.  The same general type of argument holds for restricting to a single $\ell$.  Thus type of simplification will not work in general.


TITLE: Write $\csc(x)$ in terms of $\sec(x)$
QUESTION [1 upvotes]: I was working on trig homework and came across a question that I didn't understand how to even begin to approach.  It asked us to use trigonometric identities to write $\csc(x)$ in terms of $\sec(x)$.
I'm not sure what I can do.  I had though the cofunction identity $\sec \left(\frac{\pi}{2} - x \right) = \csc x$ was the answer, but I was told I was wrong, with no explanation.  This identity was on a handout provided by the teacher.
How would I go about solving this problem and problems like it?  Also, why was my initial answer wrong?  Any help would be greatly appreciated.

REPLY [5 votes]: It’s wrong because you didn’t write $\csc x$ in terms of $\sec x$: $\frac{\pi}2-x$ isn’t $x$. I suspect that you were intended to do something like this:
$$\csc x=\frac1{\sin x}\;,$$ and $$\sec x=\frac1{\cos x}\;,$$ so
$$\begin{align*}&\frac1{\csc^2 x}+\frac1{\sec^2x}=1\;,\\
&\frac1{\csc^2x}=1-\frac1{\sec^2x}=\frac{\sec^2x-1}{\sec^2x}\;,\\
&\csc^2x=\frac{\sec^2x}{\sec^2x-1}\;,\\
&\csc x=\pm\frac{\sec x}{\sqrt{\sec^2x-1}}\;.
\end{align*}$$
For a complete answer you still have to sort out where $\csc x$ is positive and where it’s negative, which is just a bit messy.


TITLE: Show a $\sigma$-algebra contains the Borel sets : with $(a,\infty)$ or $(-\infty,b)$?
QUESTION [6 upvotes]: For a certain $\sigma$-algebra $A$ on the real line, I would like to show that it contains the Borel sets. I can show that $A$ contains the left and right half-line $(a,\infty)$ and $(-\infty,b)$ for any real numbers $a$ and $b$. My question is : can I infer that $A$ contains the Borel sets by only prooving that it contains the left half-line or is it mandatory to show that $A$ contains both half-line?
I'm not clear on how the Borel sets are generated from half-line and open intervals.

REPLY [6 votes]: What is the complement of one of these half-lines?  Then consider intersections, unions, etc. In other words, yes, if you can show your $\sigma$-algebra contains $(a, \infty)$ for any $a$, that is enough... but it sounds like it would be a good exercise for you to prove this.  Here are some hints:

What is the complement of $(a, \infty)$?
For $a< b$, what is the intersection of $(a, \infty)$ and $(-\infty, b]$?
Show that a $\sigma$-algebra containing all half-open intervals $(a, b]$ contains all Borel sets.  In fact, this is sometimes taken as the definition -- i.e. the Borel $\sigma$-algebra is the $\sigma$-algebra generated by the half-open intervals.  So, you are either done, or you need to show that this is equivalent to whatever definition you are using.
(Hint: at this point you have all the intervals $(a, b-1/n]$, for $n\in \mathbb N$ in your $\sigma$-algebra.)


TITLE: How can I find the initial state vector from state space model?
QUESTION [0 upvotes]: Assume that we have input $u(k)\in \Re $ and output $y(k) \in \Re$ and we estimate the black box model by using subspace identification method.
$$x (k+1) = Ax (k)  + Bu(k) $$
$$y (k) = Cx (k) $$
If we know $A \in \Re^{2x2} ,B \in \Re^{2x1},C  \in \Re^{1x2}$ and the input and output.
How can we find $x (k) \in \Re^{2}$ at $y (k) $ when $k > 0$ ?

REPLY [1 votes]: I assume you do not know the initial condition and/or have uncertainty and noise. Otherwise you can use the state equations to propagate the state vector. If the system is observable
https://en.wikipedia.org/wiki/Observability,
then you can use an observer:
https://en.wikipedia.org/wiki/State_observer.
If you have noise in the output and uncertainty in the model, you can use the Kalman filter:
https://en.wikipedia.org/wiki/Kalman_filter
Better resource on KF:
http://www.cs.unc.edu/~tracker/media/pdf/SIGGRAPH2001_CoursePack_08.pdf
If your system is not observable, then the state cannot be estimated from the information you have.


TITLE: Surface bound charge in the electric displacement
QUESTION [0 upvotes]: In Griffiths, the electric displacement is written with just volume charge density, and that is because, he says, 'we cannot apply gauss law precisely at the surface of a dielectric, for here volume charge density blows up, taking the divergence of $E$ with it'.
So I understand this with idea that volume charge density is charge/volume and at the surface $v$ goes to zero and the density blows up.
However, in problem 4.15(3ed), gauss law is used to find the electric field of a spherical shell with polarization. In this process, $Q$ total includes surface charge density!
So, my question is: how do I deal with the surface charge density when I study the dielectric?
One more. There is a comment like volume charge density is made by nonuniform polarization while uniform accumulates surface charge density. Then both charge density cannot exist simultaneously, but it's not. what is wrong in this?

REPLY [1 votes]: First question:
I do not have Griffiths book but I hope this is what you mean.
You can use Gauss law on a surface that encloses the surface charge, you just can't use it on the surface itself, because the electric field is infinite there. Mathematically you can state:
$$\iint \mathrm{d}S \, \, \vec{n} \cdot \vec{D}= \sum_i \iiint_{V_i} \mathrm{d}V \, \, \rho(\vec{r})+ \iint_{S_i} \mathrm{d}S \, \, \pi(\vec{r})$$
Here you take the sum over the different homogeneous materials and there surface.
Actually you can see this intuitively considering a 1D situation. In the case of a surface charge there will be a jump in the electric field, which can be denoted by a heaviside function $E(x)=\mathbb{H}(x)$. The derivative of  this electric field becomes a delta distribution, which represents a surface charge. Integrating this will again give you a heaviside function, but at $x=0$ this integration is not simply defined.
Second question:
Maxwell's laws states $\nabla \cdot \epsilon_0\vec{E}=\rho-\nabla\cdot \vec{P}$
The right hand side represents the total charge. If there is only polarization and the polarization is homogeneous the divergence becomes zero, and there is no net charge. However at the boundary of two materials there will be a jump in polarization and a polarization charge is created.
When a material is not homogeneous there will be a volume charge, but there can still be a jump in polarization at the boundary and therefore both can exist simultaneous.


text:
\section{Conclusion and Perspectives}\label{sec:perspective}
\subsection{Summary and Conclusion}
All parts presented in this thesis are fully implemented and integrated into the \Sage project and are officially available since version 9. With over $14,\!000$ lines of new code, this includes: general improvements of existing code, vector bundles, sections, bundle connections, mixed differential forms and finally characteristic classes. All these extensions yield new tools for both teaching and research. In education, they can be used to create new examples to illustrate the notion of vector bundles and characteristic classes. With respect to research, recall that on manifolds without boundary, two characteristic forms of the same characteristic class have \emph{topologically invariant integrals}. That is, integrating each top form over the whole manifold yields the same result; which follows immediately from Stokes's theorem. Since all computations are based on explicit expressions, the corresponding integrals can be at least evaluated numerically. Therefore, our current implementation provides a functional tool for applications in differential topology as well as index theory and allows for the comparison of differential geometric and topological invariants on manifolds with an empty or, at least, specifically chosen boundary. A possible extension to manifolds with arbitrary, non-empty boundary is discussed in the following section.

\begin{table}[t]
	\centering
	\begin{tabular}{lr}
		\toprule
		\textbf{Example} & \textbf{Wall Time} \\
		\midrule
		\hyperref[in:78]{\color{incolor}\texttt{In [78]}} & \texttt{299\,ms} \\
		\hyperref[in:86]{\color{incolor}\texttt{In [86]}} & \texttt{560\,ms} \\
		\hyperref[in:98]{\color{incolor}\texttt{In [98]}} & \texttt{2.2\,s} \\
		\hyperref[in:117]{\color{incolor}\texttt{In [117]}} & \texttt{1\,h\ 2\,min\ 55\,s} \\
		\bottomrule
	\end{tabular}
	\caption{Preformed on an \texttt{Intel(R) Core(TM) i3-3120M CPU @ 2.50GHz} machine using \texttt{Ubuntu 18.04.3 LTS} and \Sage version \texttt{9.0.beta1} without parallelization.}\label{tab:con:times}
\end{table}

\paragraph{Wall Times.}
In Table~\ref{tab:con:times}, we list the wall times of our computations of characteristic forms performed in Section~\ref{sec:char:ch_char},~\ref{sec:char:taut},~\ref{sec:char:euler} and~\ref{sec:char:berger}. The first three examples yield short computation times as expected. The dimension of the base space and the rank of the vector bundle do not exceed two, which is computationally simple for \Sage. In contrast, the last example yields a wall time of more than one hour. Here, we are dealing with four dimensions and more complicated symbolic expressions. On top of that, parallelization is deactivated and division free algorithms are used in the background. These algorithms are not optimized for speed. Notice that $\Sphere^3$ is a Lie group, and using structure coefficients would provide a shortcut in the calculation. However, since general Lie groups are currently not supported, stating the global frame in stereographic coordinates is the only viable option.

\subsection{Outlook and Future Prospects}
As an open-source piece of software, \Sage is always in a state of progress; and there are tickets still open at this point.\footnote{As of 02/2020, the tickets \ticket{28629}, \ticket{28640}, \ticket{28854} and \ticket{28963} are still open.} Nevertheless, our modifications provide a solid starting point and already provide a powerful tool. In this section, we give a glimpse of possible extensions to our work and collect some ideas for future versions of \Sage.

\paragraph{Vector Bundles.} Regarding vector bundles, there is still much work to do. Many features are yet unimplemented. This specifically involves bundle metrics and bundle automorphisms as well as tensor product bundles, duals and pullbacks.\footnote{The latter are not a severe restriction as they can still be considered as separate vector bundles and initialized manually.} To achieve this, the current tensor field code can be generalized to vector bundles again. Undertaking this task, it is equally beneficial to merge tensor fields entirely into the setup of vector bundles. This particularly includes structured inheritance trees in order to remove code~duplication.

Another issue concerns bundle connections. We are naturally interested in applying them to local sections. This could be accomplished as follows. Suppose we have a local section and a vector field defined on the same domain. By using the decomposition in~\eqref{eq:vec:sec_decomp} and making use of~\eqref{eq:vec:def_con_form} together with Leibniz's rule, we can derive a straightforward formula in terms of connection forms. This certainly involves actions of one forms on vector fields. Fortunately, they are integrated in \Sage already. Since each bundle connection is designed to store its connection forms, this formula could easily be implemented.

\paragraph{Characteristic Classes.} With respect to the Chern--Weil theory, our implementation already provides a comprehensive tool for applications. Nevertheless, there are still some things left to do. Currently, skew-symmetric curvature matrices must be inserted manually to obtain the characteristic form of a Pfaffian class. This, however, is not necessary once an orientation is chosen. In fact, one might find a general formula which could be implemented directly. Such a formula exists at least by applying the Gram--Schmidt procedure to local frames. Still, there might be a more concise formula that is easier to apply.

As Table~\ref{tab:con:times} indicates, computation times rise quickly with increasing complexity. For that purpose, it might be useful to support parallelization with respect to characteristic forms. One might also think about more sophisticated matrix algorithms for background tasks, especially in view of mixed differential forms. This could decrease computation times and make our implementation even more useful for applications in general relativity and mathematics.

As aforementioned, if our manifold has non-empty boundary, two characteristic forms in general no longer yield topologically invariant integrals. More precisely, both forms differ by an exact differential form which is the exterior derivative of the so-called \defstyle{transgression form}. Applying Stokes's theorem, its integral over the boundary yields the difference of these integrals. Transgression forms play an important role in gravitational physics as they occur, for example, in the index formula for the Dirac operator on Lorentzian manifolds~\cite{baer:math_chiral}. A concrete formula that could be embedded into \Sage is presented in~\cite[p.~299, Eq.~(6.4)]{eguchi:grav}. The way we designed characteristic classes within \Sage provides a solid foundation for this purpose.

Unfortunately, Stiefel--Whitney classes are still missing in \Sage. They are characteristic classes over the trivial field $\ZZ / 2 \ZZ$ and encode essential information about the underlying vector bundle. For example, the first Stiefel--Whitney class describes whether a vector bundle is orientable or not~\cite[Thm.~1.2]{lawson:spin}. The second Stiefel--Whitney class, on the other hand, provides us with information about the existence of a spin structure on an orientable manifold~\cite[Thm.~1.7]{lawson:spin}. However, as the de~Rham cohomology has coefficients in $\RR$ or $\CC$, there is little hope to obtain these classes by adopting our approach in Chern--Weil theory.
\paragraph{Lie Groups.}
Having the additional structure of a Lie group, most computations might get simpler. In particular, the Levi--Civita connection can be easily computed when using structure coefficients. Up to now, \Sage supports only \emph{nilpotent} Lie groups~\cite{sagemath:reference:nilpotent}. The general case would be a tremendous advantage to facilitate computations and enhance usability. Possible benefits can be inferred directly from Section~\ref{sec:char:berger} where the computation has seen to be expensive.


TITLE: Can heat cause flourescence or phosphorescence at STP?
QUESTION [0 upvotes]: I am aware that chemical reactions may cause fluorescence.
Most fluorescence results from adding light and then watching the captured light re-emit. Is there any that doesn't require adding light, but instead relies on heat aka thermal fluorescence at or near room temperature? Standard Temperature and Pressure=STP.

REPLY [1 votes]: Depending on how loosely one interprets 'fluorescence', a thermal excitation
can indeed produce light, at almost any given temperature.
The trick is, some fluorescence-like behavior involves a metastable electron
excited state, and this is usually called 'phosphorescence' to distinguish
its slow-release light emission (occurring after a delay, rather than
immediately on excitation) from 'normal' fluorescence.
Phosphorescence time delay depends on disturbing factors (such as temperature)
and can be very long (centuries) at low temperatures.   Some plastic
scintillator-type materials are used as radiation sensors, in radiation
badges, which are read out by simply warming them in an oven.
The pulse of light that comes out indicates the radiation dose that
was accumulated in preceeding weeks.   Thermoluminescence is the
name given to this warm-it-up-and-it-glows behavior.
Such a metastable scintillator might be excited by electrons (beta rays)
or light, or other energy quantum with higher energy than a  light
photon (so room-temperature thermal motion, circa 0.025 electron volt,
falls short by about 1 electron volt).   It needs an energy source
in addition to heat to make a visible glow, heat
merely triggers emission of stored energy from the other energy source.


TITLE: Closed form for the Stirling numbers of the second kind.
QUESTION [3 upvotes]: I have realized through inspection that ${n \brace 2}=2^{n-1}-1$ and I have figured out with the help of Pedro Tamaroff that ${n \brace 3}=\frac{1}{6}(3^{n}-3\cdot2^n+3)$. For what other values of $k$ can we do a similar characterization of ${n \brace k}$? can we do it for all positive integers?

REPLY [3 votes]: Hint: taken with an appropriate index, the Stirling numbers of the second kind form the exponential Riordan array $[1, e^x - 1]$. (See e.g. P. Barry, Generalized Stirling Numbers, Exponential Riordan Arrays, and Toda Chain Equations, Journal of Integer Sequences, 17 (2014)).