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\begin{document}

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\section*{elasticity}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{cohesive_toposes}{}\paragraph*{{Cohesive $\infty$-Toposes}}\label{cohesive_toposes}

[[!include cohesive infinity-toposes - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{InSolidStatePhysics}{In solid state physics}\dotfill \pageref*{InSolidStatePhysics} \linebreak
\noindent\hyperlink{AnalogyWithAQualityOfSpace}{Analogy with a quality of space}\dotfill \pageref*{AnalogyWithAQualityOfSpace} \linebreak
\noindent\hyperlink{in_natural_philosophy}{In natural philosophy}\dotfill \pageref*{in_natural_philosophy} \linebreak
\noindent\hyperlink{InCategoricalLogic}{In categorical logic / topos theory}\dotfill \pageref*{InCategoricalLogic} \linebreak
\noindent\hyperlink{RubberSheetGeometry}{Rubber-sheet geometry}\dotfill \pageref*{RubberSheetGeometry} \linebreak
\noindent\hyperlink{RubberSheetAnalogyOfGravity}{Elasticity analogy of gravity}\dotfill \pageref*{RubberSheetAnalogyOfGravity} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{in_solid_state_physics_2}{In solid state physics}\dotfill \pageref*{in_solid_state_physics_2} \linebreak
\noindent\hyperlink{as_an_analogy_for_topologydifferential_geometry}{As an analogy for topology/differential geometry}\dotfill \pageref*{as_an_analogy_for_topologydifferential_geometry} \linebreak
\noindent\hyperlink{as_an_analogy_for_gravity}{As an analogy for gravity}\dotfill \pageref*{as_an_analogy_for_gravity} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

\hypertarget{InSolidStatePhysics}{}\subsubsection*{{In solid state physics}}\label{InSolidStatePhysics}

In [[physics]], specifically in \emph{[[solid state physics]]}, \emph{elasticity} is the tendency of solid materials to return to their original shape after being deformed.

Mathematical elasticity theory (\hyperlink{LandauLifshitz59}{Landau-Lifshitz 59}) encodes the [[forces]] acting on a [[solid]] by a [[stress tensor]], and the amount of deformation by a \emph{[[strain tensor]]}.

One distinguishes different types of elasticity:

\begin{itemize}%
\item linear elasticity: here the [[strain tensor]] is related by a [[linear equation]] to the [[stress tensor]] (a generalization of [[Hooke's law]]). To first [[infinitesimal]] order every type of material is approximately linearly elastic, the topic of \emph{\href{http://en.wikipedia.org/wiki/Infinitesimal_strain_theory}{infinitesimal strain theory}}.


\item non-linear elasticity

\begin{itemize}%
\item hyperelasticity: exhibited by (soft) rubber

\end{itemize}


\end{itemize}
\ldots{}

\hypertarget{AnalogyWithAQualityOfSpace}{}\subsubsection*{{Analogy with a quality of space}}\label{AnalogyWithAQualityOfSpace}

Elasticity is commonly used as an illustration via [[analogy]] of the nature of [[space]] in [[physics]], specifically that of [[manifolds]] and [[spacetimes]].

\hypertarget{in_natural_philosophy}{}\paragraph*{{In natural philosophy}}\label{in_natural_philosophy}

In [[Georg Hegel]]`s \emph{[[Encyclopedia of the Philosophical Sciences]]} (1817) there is discussion (in the section \emph{\href{Science%20of%20Logic#Kohaesion}{Physik -- Die Koh\"a{}sion}}) of the \emph{[[cohesion]]} and \emph{elasticity} of some [[substance]] which Hegel says is is the [[unity of opposites|unity]] of [[space]] and [[time]] (\href{Science%20of%20Logic#PN261}{PN\S{}261}).

\hypertarget{InCategoricalLogic}{}\paragraph*{{In categorical logic / topos theory}}\label{InCategoricalLogic}

[[William Lawvere]] argued that the ``[[objective logic]]'' of this discussion is to be formalized via [[categorical logic]] by the axiomatics of [[cohesive toposes]] (see there for references), i.e. by [[modal type theory]] equipped with [[shape modality]] and [[flat modality]].

But Hegel goes on to speak of cohesion being refined to \emph{elasticity}:

\begin{quote}%
\href{Science+of+Logic#PN297Zusatz}{PN\S{}297Zusatz} Elasticity is the whole of cohesion.


\end{quote}
According to \href{Science+of+Logic#PN298}{PN\S{}298} this elasticity is related to the [[unity of opposites]] that consistute [[Zeno's paradox of motion]], hence to the modern concept of [[differentiation]] via a [[limit of a sequence]]. In terms of [[categorical logic]] this is precisely what is encoded in the [[infinitesimal shape modality]] and [[infinitesimal flat modality]] of ``[[differential cohesion]]'' (see there for detail).

Sticking to imagery from [[solid state physics]], these modalities are reminiscent of concepts in \href{http://en.wikipedia.org/wiki/Infinitesimal_strain_theory}{infinitesimal strain theory}. Notice that this applies to

\begin{quote}%
 structures built from relatively stiff elastic materials


\end{quote}
\hypertarget{RubberSheetGeometry}{}\paragraph*{{Rubber-sheet geometry}}\label{RubberSheetGeometry}

These modalities \hyperlink{InCategoricalLogic}{above} induce an axiomatization of \emph{\href{differential+cohesive+%28infinity%2C1%29-topos#EtaleObjects}{manifolds and \'e{}tale groupoids}} (``[[derived schemes]]'') for which a common imagery in mathematics is given by elastic rubber sheets, reflecting the fact that these spaces on top of their [[cohesion]] have a more rigid [[shape modality|shape]], namely [[infinitesimal shape modality|infinitesimal shape]].

In this context the [[topology]] of [[manifolds]] is often referred to as \emph{rubber-sheet geometry} (e.g.\href{http://simple.wikipedia.org/wiki/Topology}{Wikipedia}, \hyperlink{Britton}{Britton}).

\hypertarget{RubberSheetAnalogyOfGravity}{}\paragraph*{{Elasticity analogy of gravity}}\label{RubberSheetAnalogyOfGravity}

Moreover, these modalities \hyperlink{InCategoricalLogic}{above} induce an axiomatization of [[Cartan geometry]], hence in particular of [[pseudo-Riemannian geometry]] and hence of [[gravity]] (``[[general relativity]]'') on these manifolds.

A popular imagery for that illustrates the gravitational effect of massive bodies on light by a deformation of spacetime visualizes as an elastic wire frame. This is known as the \emph{rubber-sheet analogy} for gravity (e.g. \hyperlink{Das}{Das 11, p. 69}, \hyperlink{Gabor}{Gabor}, \hyperlink{Volk}{Volk, fig. 4 on p. 537}).

This seems to go back to the technical result of (\hyperlink{Sakharov67}{Sakharov 67}), which is informally summarized in the seminal textbook (\hyperlink{MisnerThorneWheeler73}{Misner-Thorne-Wheeler 73}) as saying that

\begin{quote}%
gravity is ``an elasticity of space that arises from particle physics''


\end{quote}
Further discussion of the analogy between the mathematical theory of elasticity and that of gravity include (\hyperlink{Tartaglia95}{Tartaglia 95},\hyperlink{MiddletonLangston13}{Middleton-Langston 13}, \hyperlink{WikipediaRubberSheetModelForGravity}{Wikipedia}).

In order to think of not just [[topology]] but [[Riemannian geometry]] in the \hyperlink{RubberSheetGeometry}{above} context of elasticity, the \emph{rigidity} mentioned \hyperlink{RelativelyStiffElasticity}{further above} seems advisable. A \emph{rigidly elastic} body is to be expected to produce sound when struck. This is a common imagery in Riemannian geometry, as in ``[[hearing the shape of a drum]]''.

\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[formal smooth infinity-groupoid]]


\item [[super formal smooth infinity-groupoid]]


\item [[cohesion]], [[solidity]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{in_solid_state_physics_2}{}\subsubsection*{{In solid state physics}}\label{in_solid_state_physics_2}

\begin{itemize}%
\item [[Lev Landau]], [[Evgeny Lifshitz]], \emph{Theory of Elasticity}, part VII of \emph{[[Course of Theoretical Physics]]}, 1959, 1970

\end{itemize}
\hypertarget{as_an_analogy_for_topologydifferential_geometry}{}\subsubsection*{{As an analogy for topology/differential geometry}}\label{as_an_analogy_for_topologydifferential_geometry}

\begin{itemize}%
\item Jill Britton, \emph{\href{http://jwilson.coe.uga.edu/EMT668/EMAT6680.F99/Estes/unit/dayten/topology1.html}{Rubber sheet geometry}}

\end{itemize}
\hypertarget{as_an_analogy_for_gravity}{}\subsubsection*{{As an analogy for gravity}}\label{as_an_analogy_for_gravity}

The ``rubber-sheet analogy of gravity'' might go back to results in

\begin{itemize}%
\item [[Andrei Sakharov]], \emph{Vacvuum fluctuations in curved space and the theory of gravitation}, Doklady Akad. Nauk S.S.S.R. 177 70-71 (1967)

\end{itemize}
which were informally summarized in

\begin{itemize}%
\item [[Charles Misner]], [[Kip Thorne]], [[John Wheeler]], \emph{[[Gravitation]]}, 1973


\item [[John Wheeler]], \emph{Sakharov: A man of humility, understanding and leadership}, in \emph{Andrei Sakharov, Facets of a Life} (\href{https://books.google.co.uk/books?id=BTIziWmTX_0C&pg=PA647#v=onepage&q&f=false}{GoogleBooks})



\end{itemize}
as saying that gravity is ``an elasticity of space that arises from particle physics''.

The mathematical similarity between gravity and the physics of elasticity is discussed further in

\begin{itemize}%
\item A. Tartaglia, \emph{Four Dimensional Elasticity and General Relativity} (\href{http://arxiv.org/abs/gr-qc/9509043}{arXiv:gr-qc/9509043})


\item Chad A. Middleton, Michael Langston, \emph{Circular orbits on a warped spandex fabric} (\href{http://arxiv.org/abs/1312.3893}{arXiv:1312.3893}, \href{https://www.youtube.com/watch?v=F80HIolrsoc}{talk video})



\end{itemize}
The analogy is mentioned for expositional purpose for instance in

\begin{itemize}%
\item Wikipedia, \emph{\href{http://en.wikipedia.org/wiki/Gravity_well#The_rubber-sheet_model}{Gravity well -- The rubber-sheet model}}


\item Gabor, \emph{\href{http://theory.uwinnipeg.ca/users/gabor/black_holes/slide5.html}{Gravity: Space as a rubber sheet}}


\item Greg Volk, \emph{19th Natural Philosophy Alliance Proceedings} (\href{https://books.google.de/books?id=tZnZAwAAQBAJ&pg=PA537&lpg=PA537&dq=manifold+"rubber-sheet"&source=bl&ots=WAhzUMxkOR&sig=p2qNCPCR9-gnPi6_nb1S_Bz3-AI&hl=en&sa=X&ei=8Ob2VLmSIcvNygOTjoGIBA&ved=0CCIQ6AEwAA#v=onepage&q=manifold%20%22rubber-sheet%22&f=false}{online})


\item [[Ashok Das]], \emph{Lectures on Gravitation}, WorldScientific 2011 (\href{http://www.worldscientific.com/worldscibooks/10.1142/7990}{publisher}, \href{https://books.google.de/books?id=oupB7VUGzGcC&pg=PA69&lpg=PA69&dq=manifold+"rubber-sheet"&source=bl&ots=ajbiJu7T_m&sig=MPsA2KWds_Mmf_UCscieaGVJzJk&hl=en&sa=X&ei=8Ob2VLmSIcvNygOTjoGIBA&ved=0CCUQ6AEwAQ#v=onepage&q=manifold%20%22rubber-sheet%22&f=false}{GoogleBooks})



\end{itemize}
[[!redirects elasticity theory]]

[[!redirects rubber-sheet geometry]] [[!redirects rubber-sheet analogy of gravity]]

[[!redirects elastic]]



\end{document}