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\begin{document}

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\section*{energy}

\begin{quote}%
this entry needs attention


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

\hypertarget{physics}{}\paragraph*{{Physics}}\label{physics}

[[!include physicscontents]]

\hypertarget{energy}{}\section*{{Energy}}\label{energy}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{conservation}{Conservation}\dotfill \pageref*{conservation} \linebreak
\noindent\hyperlink{relationship_to_mass}{Relationship to mass}\dotfill \pageref*{relationship_to_mass} \linebreak
\noindent\hyperlink{in_special_relativity}{In special relativity}\dotfill \pageref*{in_special_relativity} \linebreak
\noindent\hyperlink{in_general_relativity}{In general relativity}\dotfill \pageref*{in_general_relativity} \linebreak
\noindent\hyperlink{enrgy_scales}{Enrgy scales}\dotfill \pageref*{enrgy_scales} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

In [[physics]], the \textbf{energy} of a [[physical system]] is one of its most basic [[observable]] properties, an [[extensive quantity]] roughly measuring how much stuff is in the system.

In [[experiment]] it is often measured in [[units]] of [[electronvolt]].

The term `energy' is also used in [[Riemannian geometry]] and [[statistics]] in vague analogy with the concept from physics.

\hypertarget{conservation}{}\subsection*{{Conservation}}\label{conservation}

As a system [[time evolution|evolves]] without interacting with its environment, its energy remains constant. If different systems [[interaction|interact]], then they may exchange energy, but the total energy remains constant; equivalently, we may combine the interacting systems into a supersystem whose energy is the sum of the energies of its parts, and this supersystem's energy is conserved.

Through [[Noether's theorem]], conservation of energy corresponds to [[time]] invariance. If a system's equations of motion are \emph{not} invariant with time, then the system will not exhibit energy conservation, and physically we conclude that the system must be interacting with something else.

In [[continuum physics]], conservation of energy may be treated locally: the change in energy in any region over a period of time must equal the total flux of energy through the region's boundary over that time. In [[general relativity]], this makes sense (in general) only as a [[differential equation]] involving both the [[stress-energy tensor]] and the [[Levi-Civita connection]].

\hypertarget{relationship_to_mass}{}\subsection*{{Relationship to mass}}\label{relationship_to_mass}

Energy is distinguished from [[mass]] in various ways:

\begin{itemize}%
\item \textbf{Relativistic mass} is the same as energy, although conventionally measured in different [[unit of measurement|units]]. We have $E = m c^2$ in conventional units, or simply $E = m$ in units where the [[speed of light]] is $c = 1$.


\item \textbf{[[rest mass|Rest mass]]} (the usual meaning of `mass' in modern physics) is the minimum energy inherent in a system; whereas energy is relative, rest mass is absolute. Here we have

\begin{equation}
E = m c^2 \sqrt{1 - \frac {v^2} {c^2}}
\label{emcg}\end{equation}
in conventional units, or

\begin{equation}
E = m \sqrt{1 - v^2}
\label{emg}\end{equation}
if $c = 1$.


\item In the non-relativistic limit, energy splits into two independently conserved quantities: mass (relativistic or rest, it makes no difference) and energy proper. That is, expanding \eqref{emcg} as a [[power series]] in $v$, we have

\begin{displaymath}
E = m c^2 + \frac 1 2 m v^2 + \frac 3 8 m \frac {v^4} {c^2} + \cdots ;
\end{displaymath}
as $c \to \infty$, this splits into the quantities $m$, $(1/2) m v^2$, $(3/8) m v^4$, etc. The first of these is the non-relativistic mass, the second is the [[kinetic energy]] $K$, the next is simply $3 K^2 / 2 m$, and all subsequent terms are also determined by $m$ and $K$.

However, the kinetic energy $K$ is not the \emph{total} energy in non-relativistic physics. Any entities with a rest mass of $0$ (or approaching zero in the limit) will still have energy and so must be treated separately from kinetic energy (which is proportional to mass). Non-relativistically, these may be viewed as [[field theory|fields]] with energy of their own, or (if the field strengths are derived entirely from interacting massive bodies) this energy may be attributed to the massive bodies in the system as [[potential energy]]. Thus the total energy of the system is $E = K + U$, where $K$ is total kinetic energy and $U$ is total potential or field energy.



\end{itemize}
\hypertarget{in_special_relativity}{}\subsection*{{In special relativity}}\label{in_special_relativity}

In relativistic physics, energy is relative, the timelike part of a [[4-vector]] quantity whose spacelike parts comprise [[linear momentum]]. If we also take into account the distribution of energy and momentum across space, this give a symmetric second-rank [[tensor]], the [[stress-energy tensor]].

Even in non-relativistic physics, there is an analogy between energy and linear momentum. In [[Hamiltonian mechanics]], both energy and linear momentum are examples of [[generalized momentum|generalised momenta]]; energy corresponds to [[time]] while linear momentum corresponds to position in [[physical space|space]]. However, the special treatment accorded to time and energy in the usual formulation of Hamiltonian mechanics obscures this; we also have (oddly) that the precise generalised momentum corresponding to the time variable $t$ is not the total energy $H$ but $-H$ instead.

This analogy from [[classical mechanics]] also applies to [[quantum mechanics]] and is particularly evident in Schr\"o{}dinger's original formulation of his quantum theory.

[[!include plane waves -- table]]

\hypertarget{in_general_relativity}{}\subsection*{{In general relativity}}\label{in_general_relativity}

In [[general relativity]], the concept of total energy of a system does not make sense, except in certain special cases. General relativity is a [[field theory]] in which energy appears only as one of $10$ (or $({n \atop 2})$ in $n$ [[dimensions]] of [[spacetime]]) independent components of the [[stress-energy tensor]]. Mathematically, it makes no sense to integrate just this component over a region of space to determine a total energy in that region. This makes it difficult to interpret general relativity in frameworks such as [[Hamiltonian mechanics]] and [[thermodynamics]] where energy is a primary concept. This is called the [[problem of time]] (since energy and [[time]] are conjugate observables and the problem occurs with both).

\hypertarget{enrgy_scales}{}\subsection*{{Enrgy scales}}\label{enrgy_scales}

[[energy]] $\,$ [[scales]]:

$\backslash$begin\{center\}  $\backslash$end\{center\}

\begin{quote}%
graphics grabbed from \href{http://sten.astronomycafe.net/the-particle-desert/}{here}


\end{quote}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[Dirichlet energy]], [[harmonic map]]

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

Serious references discussing energy in general relativity include

\begin{itemize}%
\item [[Edward Witten]], \emph{A new proof of the positive energy theorem}, Comm. Math. Phys. \textbf{80} (1981), no. 3, 381--402, \href{http://www.ams.org/mathscinet-getitem?mr=626707}{MR83e:83035}, \href{http://projecteuclid.org/getRecord?id=euclid.cmp/1103919981}{euclid}
\item A. Trautman, \emph{Conservation laws in general relativity. In: Gravitation: an introduction to current research}, Witten, L. (ed.). New York: Wiley 1962;
\item Misner, C., Thome, K.S., Wheeler, J.A.: Gravitation. San Francisco: W. H. Freeman 1973;
\item S. Weinberg, \emph{Gravitation and cosmology}, New York: Wiley 1972
\item R. Schoen, S. T. Yau, Comm. Math. Phys. 65 (1979), no. 1, 45--76; MR80j:83024

\end{itemize}
[[!redirects energy]] [[!redirects energies]]

[[!redirects relativistic mass]] [[!redirects relativistic masses]]



\end{document}