Talk:PlanetPhysics/General Results of the Theory

Original TeX Content from PlanetPhysics Archive

%%% This file is part of PlanetPhysics snapshot of 2011-09-01
%%% Primary Title: General Results of the Theory
%%% Primary Category Code: 03.30.+p
%%% Filename: GeneralResultsOfTheTheory.tex
%%% Version: 2
%%% Owner: bloftin
%%% Author(s): bloftin
%%% PlanetPhysics is released under the GNU Free Documentation License.
%%% You should have received a file called fdl.txt along with this file.        
%%% If not, please write to gnu@gnu.org.
\documentclass[12pt]{article}
\pagestyle{empty}
\setlength{\paperwidth}{8.5in}
\setlength{\paperheight}{11in}

\setlength{\topmargin}{0.00in}
\setlength{\headsep}{0.00in}
\setlength{\headheight}{0.00in}
\setlength{\evensidemargin}{0.00in}
\setlength{\oddsidemargin}{0.00in}
\setlength{\textwidth}{6.5in}
\setlength{\textheight}{9.00in}
\setlength{\voffset}{0.00in}
\setlength{\hoffset}{0.00in}
\setlength{\marginparwidth}{0.00in}
\setlength{\marginparsep}{0.00in}
\setlength{\parindent}{0.00in}
\setlength{\parskip}{0.15in}

\usepackage{html}

% this is the default PlanetMath preamble.  as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.

% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}

% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}

% there are many more packages, add them here as you need them

% define commands here

\begin{document}

 \subsection{General Results of the Theory}

From \htmladdnormallink{Relativity: The Special and General Theory}{http://planetphysics.us/encyclopedia/SpecialTheoryOfRelativity.html} by \htmladdnormallink{Albert Einstein}{http://planetphysics.us/encyclopedia/AlbertEinstein.html}
It is clear from our previous considerations that the (special) theory
of relativity has grown out of electrodynamics and optics. In these
\htmladdnormallink{fields}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} it has not appreciably altered the predictions of theory, but
it has considerably simplified the theoretical structure, {\it i.e.} the
derivation of laws, and---what is incomparably more important---it
has considerably reduced the number of independent hypothese forming
the basis of theory. The special theory of relativity has rendered the
Maxwell-Lorentz theory so plausible, that the latter would have been
generally accepted by physicists even if experiment had decided less
unequivocally in its favour.

\htmladdnormallink{Classical mechanics}{http://planetphysics.us/encyclopedia/MathematicalFoundationsOfQuantumTheories.html} required to be modified before it could come into
line with the demands of the special theory of relativity. For the
main part, however, this modification affects only the laws for rapid
\htmladdnormallink{motions}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html}, in which the \htmladdnormallink{velocities}{http://planetphysics.us/encyclopedia/Velocity.html} of matter $v$ are not very small as
compared with the velocity of light. We have experience of such rapid
motions only in the case of electrons and ions; for other motions the
variations from the laws of classical mechanics are too small to make
themselves evident in practice. We shall not consider the motion of
stars until we come to speak of the \htmladdnormallink{general theory}{http://planetphysics.us/encyclopedia/GeneralTheory.html} of relativity. In
accordance with the theory of relativity the \htmladdnormallink{kinetic energy}{http://planetphysics.us/encyclopedia/KineticEnergy.html} of a
material point of \htmladdnormallink{mass}{http://planetphysics.us/encyclopedia/Mass.html} m is no longer given by the well-known
expression

$$m\frac{v^2}{2}$$

\noindent but by the expression

$$\frac{mc^2}{\sqrt{I-\frac{v^2}{c^2}}}$$
~

This expression approaches infinity as the velocity $v$ approaches the
velocity of light $c$. The velocity must therefore always remain less
than $c$, however great may be the \htmladdnormallink{energies}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} used to produce the
\htmladdnormallink{acceleration}{http://planetphysics.us/encyclopedia/Acceleration.html}. If we develop the expression for the kinetic energy in
the form of a series, we obtain

$$mc^2 + m\frac{v^2}{2} + \frac{3}{8} m \frac{v^4}{c^2} + \cdots$$
~

When $v^2/c^2$ is small compared with unity, the third of these terms is
always small in comparison with the second,
which last is alone considered in classical mechanics. The first term
$mc^2$ does not contain the velocity, and requires no consideration if
we are only dealing with the question as to how the energy of a
point-mass; depends on the velocity. We shall speak of its essential
significance later.

The most important result of a general character to which the special
theory of relativity has led is concerned with the conception of mass.
Before the advent of relativity, physics recognised two conservation
laws of fundamental importance, namely, the law of the canservation of
energy and the law of the conservation of mass these two fundamental
laws appeared to be quite independent of each other. By means of the
theory of relativity they have been united into one law. We shall now
briefly consider how this unification came about, and what meaning is
to be attached to it.

The principle of relativity requires that the law of the concervation
of energy should hold not only with reference to a co-ordinate \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/GenericityInOpenSystems.html} $K$, but also with respect to every co-ordinate system $K'$ which is in a
state of uniform motion of translation relative to $K$, or, briefly,
relative to every ``Galileian'' system of co-ordinates. In contrast to
classical mechanics; \htmladdnormallink{The Lorentz transformation}{http://planetphysics.us/encyclopedia/LorentzTransformation.html} is the deciding factor
in the transition from one such system to another.

By means of comparatively simple considerations we are led to draw the
following conclusion from these premises, in conjunction with the
fundamental equations of the electrodynamics of Maxwell: A body moving
with the velocity $v$, which absorbs \footnotemark\ an amount of energy $E_0$ in
the form of \htmladdnormallink{radiation}{http://planetphysics.us/encyclopedia/Cyclotron.html} without suffering an alteration in velocity in
the process, has, as a consequence, its energy increased by an amount

$$\frac{E_0}{\sqrt{I-\frac{v^2}{c^2}}}$$
~

In consideration of the expression given above for the kinetic energy
of the body, the required energy of the body comes out to be

$$\frac{\left(m+\frac{E_0}{c^2}\right)c^2}{\sqrt{I-\frac{v^2}{c^2}}}$$
~

\noindent Thus the body has the same energy as a body of mass

$$\left(m+\frac{E_0}{c^2}\right)$$
~

\noindent moving with the velocity $v$. Hence we can say: If a body takes up an
amount of energy $E_0$, then its \htmladdnormallink{inertial mass}{http://planetphysics.us/encyclopedia/Mass.html} increases by an amount

$$\frac{E_0}{c^2}$$
~

\noindent the inertial mass of a body is not a constant but varies according to
the change in the energy of the body. The inertial mass of a system of
bodies can even be regarded as a measure of its energy. The law of the
conservation of the mass of a system becomes identical with the law of
the conservation of energy, and is only valid provided that the system
neither takes up nor sends out energy. Writing the expression for the
energy in the form

$$\frac{mc^2+E_0}{\sqrt{I-\frac{v^2}{c^2}}}$$
~

\noindent we see that the term $mc^2$, which has hitherto attracted our attention,
is nothing else than the energy possessed by the body \footnotemark\ before it
absorbed the energy $E_0$.

A direct comparison of this \htmladdnormallink{relation}{http://planetphysics.us/encyclopedia/Bijective.html} with experiment is not possible
at the present time (1920; see \footnotemark\ Note, p. 48), owing to the fact that
the changes in energy E[0] to which we can Subject a system are not
large enough to make themselves perceptible as a change in the
inertial mass of the system.

$$\frac{E_0}{c^2}$$
~

\noindent is too small in comparison with the mass $m$, which was present before
the alteration of the energy. It is owing to this circumstance that
classical mechanics was able to establish successfully the
conservation of mass as a law of independent validity.

Let me add a final remark of a fundamental nature. The success of the
Faraday-Maxwell interpretation of electromagnetic action at a distance
resulted in physicists becoming convinced that there are no such
things as instantaneous actions at a distance (not involving an
intermediary medium) of the \htmladdnormallink{type}{http://planetphysics.us/encyclopedia/Bijective.html} of \htmladdnormallink{Newton's law of gravitation}{http://planetphysics.us/encyclopedia/PrincipleOfCorrespondingStates.html}.
According to the theory of relativity, action at a distance with the
velocity of light always takes the place of instantaneous action at a
distance or of action at a distance with an infinite velocity of
\htmladdnormallink{transmission}{http://planetphysics.us/encyclopedia/FluorescenceCrossCorrelationSpectroscopy.html}. This is connected with the fact that the velocity c
plays a fundamental role in this theory. In Part II we shall see in
what way this result becomes modified in the general theory of
relativity.


\footnotetext[1]{$E_0$ is the energy taken up, as judged from a co-ordinate system
moving with the body.}

\footnotetext[2]{As judged from a co-ordinate system moving with the body.}

\footnotetext[3]{The equation $E = mc^2$ has been thoroughly proved time and
again since this time.}

\subsection{References}
This article is derived from the Einstein Reference Archive (marxists.org) 1999, 2002. \htmladdnormallink{Einstein Reference Archive}{http://www.marxists.org/reference/archive/einstein/index.htm} which is under the FDL copyright.

\end{document}