\begin{enumerate}
\item We imagine the rigid body, to which the place specification is
referred, supplemented in such a manner that the object whose position
-we require is reached by. the completed rigid body.
+we require is reached by the completed rigid body.
\item In locating the position of the object, we make use of a number
(here the length of the pole measured with the measuring-rod) instead
-\chapter{The Apparent Incompatability of the Law of Propagation of Light
+\chapter{The Apparent Incompatibility of the Law of Propagation of Light
with the Principle of Relativity}
phenomena connected with moving bodies show that experience in this
domain leads conclusively to a theory of electromagnetic phenomena, of
which the law of the constancy of the velocity of light in vacuo is a
-necessary consequence. Prominent theoretical physicists were theref
-ore more inclined to reject the principle of relativity, in spite of
+necessary consequence. Prominent theoretical physicists were therefore
+more inclined to reject the principle of relativity, in spite of
the fact that no empirical data had been found which were
contradictory to this principle.
At this juncture the theory of relativity entered the arena. As a
result of an analysis of the physical conceptions of time and space,
-it became evident that \emph{in realily there is not the least
-incompatibilitiy between the principle of relativity and the law of
+it became evident that \emph{in reality there is not the least
+incompatibility between the principle of relativity and the law of
propagation of light}, and that by systematically holding fast to both
these laws a logically rigid theory could be arrived at. This theory
has been called the \emph{special theory of relativity} to distinguish it
is simultaneous with C (simultaneous in the sense of the above
definition), then the criterion for the simultaneity of the pair of
events A, C is also satisfied. This assumption is a physical
-hypothesis about the the of propagation of light: it must certainly be
+hypothesis about the of propagation of light: it must certainly be
fulfilled if we are to maintain the law of the constancy of the
velocity of light in vacuo.}
The rigid rod is thus shorter when in motion than when at rest, and
the more quickly it is moving, the shorter is the rod. For the
velocity $v=c$ we should have $\sqrt{I-v^2/c^2} = 0$,
-and for stiII greater velocities the square-root becomes imaginary.
+and for still greater velocities the square-root becomes imaginary.
From this we conclude that in the theory of relativity the velocity $c$
plays the part of a limiting velocity, which can neither be reached
nor exceeded by any real body.
conclusiveness of the experiment as a crucial test in favour of the
theory of relativity, for the electrodynamics of Maxwell-Lorentz, on
which the original theory was based, in no way opposes the theory of
-relativity. Rather has the latter been developed trom electrodynamics
+relativity. Rather has the latter been developed from electrodynamics
as an astoundingly simple combination and generalisation of the
hypotheses, formerly independent of each other, on which
electrodynamics was built.
fields 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
+has considerably reduced the number of independent hypotheses 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
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
+laws of fundamental importance, namely, the law of the conservation 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
+The principle of relativity requires that the law of the conservation
of energy should hold not only with reference to a co-ordinate system
$K$, 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,
to another. Now assuming that the Maxwell-Lorentz equations hold for a
reference-body $K$, we then find that they do not hold for a
reference-body $K'$ moving uniformly with respect to $K$, if we assume
-that the relations of the Galileian transformstion exist between the
+that the relations of the Galileian transformation exist between the
co-ordinates of $K$ and $K'$. It thus appears that, of all Galileian
co-ordinate systems, one ($K$) corresponding to a particular state of
motion is physically unique. This result was interpreted physically by
theory from this difficulty by assuming that the motion of the body
relative to the æther produces a contraction of the body in the
direction of motion, the amount of contraction being just sufficient
-to compensate for the differeace in time mentioned above. Comparison
+to compensate for the difference in time mentioned above. Comparison
with the discussion in Section 11 shows that also from the
standpoint of the theory of relativity this solution of the difficulty
was the right one. But on the basis of the theory of relativity the
Space is a three-dimensional continuum. By this we mean that it is
possible to describe the position of a point (at rest) by means of
-three numbers (co-ordinales) $x, y, z$, and that there is an indefinite
+three numbers (co-ordinates) $x, y, z$, and that there is an indefinite
number of points in the neighbourhood of this one, the position of
which can be described by co-ordinates such as $x_1, y_1, z_1$, which
may be as near as we choose to the respective values of the
Up to the present, however, we have by no means maintained the
equivalence of all bodies of reference $K$ in connection with the
formulation of natural laws. Our course was more on the following
-Iines. In the first place, we started out from the assumption that
+lines. In the first place, we started out from the assumption that
there exists a reference-body $K$, whose condition of motion is such
that the Galileian law holds with respect to it: A particle left to
itself and sufficiently far removed from all other particles moves
consideration seems to suggest that, for the present at any rate,
there is little hope of success in such an attempt; Let us imagine
ourselves transferred to our old friend the railway carriage, which is
-travelling at a uniform rate. As long as it is moving unifromly, the
+travelling at a uniform rate. As long as it is moving uniformly, the
occupant of the carriage is not sensible of its motion, and it is for
this reason that he can without reluctance interpret the facts of the
case as indicating that the carriage is at rest, but the embankment in
not hold with respect to the non-uniformly moving carriage. Because of
this, we feel compelled at the present juncture to grant a kind of
absolute physical reality to non-uniform motion, in opposition to the
-general principle of relatvity. But in what follows we shall soon see
+general principle of relativity. But in what follows we shall soon see
that this conclusion cannot be maintained.
\chapter{The Equality of Inertial and Gravitational Mass
-as an Argument for the General Postule of Relativity}
+as an Argument for the General Postulate of Relativity}
We imagine a large portion of empty space, so far removed from stars
and other appreciable masses, that we have before us approximately the
means of his legs if he does not wish to be laid out full length on
the floor. He is then standing in the chest in exactly the same way as
anyone stands in a room of a home on our earth. If he releases a body
-which he previously had in his land, the accelertion of the chest will
+which he previously had in his land, the acceleration of the chest will
no longer be transmitted to this body, and for this reason the body
will approach the floor of the chest with an accelerated relative
motion. The observer will further convince himself that the
come to the conclusion that he and the chest are in a gravitational
field which is constant with regard to time. Of course he will be
puzzled for a moment as to why the chest does not fall in this
-gravitational field. just then, however, he discovers the hook in the
+gravitational field. Just then, however, he discovers the hook in the
middle of the lid of the chest and the rope which is attached to it,
and he consequently comes to the conclusion that the chest is
suspended at rest in the gravitational field.
Suppose that the man in the chest fixes a rope to the inner side of
the lid, and that he attaches a body to the free end of the rope. The
-result of this will be to strech the rope so that it will hang
+result of this will be to stretch the rope so that it will hang
``vertically'' downwards. If we ask for an opinion of the cause of
tension in the rope, the man in the chest will say: ``The suspended
body experiences a downward force in the gravitational field, and this
its entirety) vanishes.
We can now appreciate why that argument is not convincing, which we
-brought forward against the general principle of relativity at theend
+brought forward against the general principle of relativity at the end
of Section 18. It is certainly true that the observer in the
railway carriage experiences a jerk forwards as a result of the
application of the brake, and that he recognises, in this the
mechanics (or in the special theory of relativity) to which I can
attribute the different behaviour of bodies considered with respect to
the reference systems $K$ and $K$.\footnotemark\ Newton saw this objection and
-attempted to invalidate it, but without success. But E. Mach recognsed
+attempted to invalidate it, but without success. But E. Mach recognised
it most clearly of all, and because of this objection he claimed that
mechanics must be placed on a new basis. It can only be got rid of by
means of a physics which is conformable to the general principle of
a consequence of this, the special theory of relativity and with it
the whole theory of relativity would be laid in the dust. But in
reality this is not the case. We can only conclude that the special
-theory of relativity cannot claim an unlinlited domain of validity;
+theory of relativity cannot claim an unlimited domain of validity;
its results hold only so long as we are able to disregard the
influences of gravitational fields on the phenomena ({\it e.g.} of light).
In the example of the transmission of light just dealt with, we have
seen that the general theory of relativity enables us to derive
theoretically the influence of a gravitational field on the course of
-natural processes, the Iaws of which are already known when a
+natural processes, the laws of which are already known when a
gravitational field is absent. But the most attractive problem, to the
solution of which the general theory of relativity supplies the key,
concerns the investigation of the laws satisfied by the gravitational
field will of course depend on the motion chosen for $K'$. According to
the general theory of relativity, the general law of the gravitational
field must be satisfied for all gravitational fields obtainable in
-this way. Even though by no means all gravitationial fields can be
+this way. Even though by no means all gravitational fields can be
produced in this way, yet we may entertain the hope that the general
law of gravitation will be derivable from such gravitational fields of
a special kind. This hope has been realised in the most beautiful
gravitational field exists relative to a reference-body $K$ whose state
of motion has been suitably chosen. $K$ is then a Galileian
reference-body as regards the domain considered, and the results of
-the special theory of relativity hold relative to $K$. Let us supposse
+the special theory of relativity hold relative to $K$. Let us suppose
the same domain referred to a second body of reference $K'$, which is
rotating uniformly with respect to $K$. In order to fix our ideas, we
shall imagine $K'$ to be in the form of a plane circular disc, which
judged from the Galileian system, the length of this rod will be less
than I, since, according to Section 12, moving bodies suffer a
shortening in the direction of the motion. On the other hand, the
-measaring-rod will not experience a shortening in length, as judged
+measuring-rod will not experience a shortening in length, as judged
from $K$, if it is applied to the disc in the direction of the radius.
If, then, the observer first measures the circumference of the disc
with his measuring-rod and then the diameter of the disc, on dividing
discarded, and replaced by another which does not assume the validity
of Euclidean geometry for rigid bodies.\footnotemark\ The reader will notice
that the situation depicted here corresponds to the one brought about
-by the general postitlate of relativity (Section 23).
+by the general postulate of relativity (Section 23).
% Notes
-\chapter{The Space-Time Continuum of the Speical Theory of Relativity Considered as a
+\chapter{The Space-Time Continuum of the Special Theory of Relativity Considered as a
Euclidean Continuum}
Thus, if we choose as time-variable the imaginary variable $\sqrt{-I} \cdot ct$
instead of the real quantity $t$, we can regard the space-time
-contintium---accordance with the special theory of relativity---as a
+continuum---accordance with the special theory of relativity---as a
``Euclidean'' four-dimensional continuum, a result which follows from
the considerations of the preceding section.
% Notes
\footnotetext{Cf. Appendixes I and 2. The relations which are derived
-there for the co-ordlnates themselves are valid also for co-ordinate
+there for the co-ordinates themselves are valid also for co-ordinate
differences, and thus also for co-ordinate differentials (indefinitely
small differences).}
depend on the co-ordinates when a gravitational field is present. In
connection with a specific illustration in Section 23, we found
that the presence of a gravitational field invalidates the definition
-of the coordinates and the ifine, which led us to our objective in the
+of the coordinates and the time, which led us to our objective in the
special theory of relativity.
-In view of the resuIts of these considerations we are led to the
+In view of the results of these considerations we are led to the
conviction that, according to the general principle of relativity, the
space-time continuum cannot be regarded as a Euclidean one, but that
here we have the general case, corresponding to the marble slab with
rods, so here it is impossible to build up a system (reference-body)
from rigid bodies and clocks, which shall be of such a nature that
measuring-rods and clocks, arranged rigidly with respect to one
-another, shaIll indicate position and time directly. Such was the
+another, shall indicate position and time directly. Such was the
essence of the difficulty with which we were confronted in Section
23.
But the considerations of Sections 25 and 26 show us the way to
-surmount this difficulty. We refer the fourdimensional space-time
+surmount this difficulty. We refer the four-dimensional space-time
continuum in an arbitrary manner to Gauss co-ordinates. We assign to
every point of the continuum (event) four numbers, $x_1, x_2, x_3,
x_4$ (co-ordinates), which have not the least direct physical
encounters of the body with clocks, in conjunction with the
observation of the encounter of the hands of clocks with particular
points on the dials. It is just the same in the case of
-space-measurements by means of measuring-rods, as a litttle
+space-measurements by means of measuring-rods, as a little
consideration will show.
The following statements hold generally: Every physical description
\chapter{Exact Formulation of the General Principle of Relativity}
-We are now in a position to replace the pro. visional formulation of
+We are now in a position to replace the provisional formulation of
the general principle of relativity given in Section 18 by an exact
formulation. The form there used, ``All bodies of reference $K, K^1,$
etc., are equivalent for the description of natural phenomena
interpret this behaviour as the behaviour of measuring-rods, docks and
material points tinder the influence of the gravitational field $G$.
Hereupon we introduce a hypothesis: that the influence of the
-gravitational field on measuringrods, clocks and freely-moving
+gravitational field on measuring rods, clocks and freely-moving
material points continues to take place according to the same laws,
even in the case where the prevailing gravitational field is not
-derivable from the Galfleian special care, simply by means of a
+derivable from the Galileian special care, simply by means of a
transformation of co-ordinates.
The next step is to investigate the space-time behaviour of the
postulate of relativity.
\item If there is any matter in the domain under consideration, only its
inertial mass, and thus according to Section 15 only its energy is
-of importance for its etfect in exciting a field.
+of importance for its effect in exciting a field.
\item Gravitational field and matter together must satisfy the law of
the conservation of energy (and of impulse).
\end{enumerate}
to be correct to within a few seconds of arc. This effect can be
explained by means of classical mechanics only on the assumption of
hypotheses which have little probability, and which were devised
-solely for this purponse.
+solely for this purpose.
On the basis of the general theory of relativity, it is found that the
ellipse of every planet round the sun must necessarily rotate in the
density of matter, although very variable in detail, is nevertheless
on the average everywhere the same. In other words: However far we
might travel through space, we should find everywhere an attenuated
-swarm of fixed stars of approrimately the same kind and density.
+swarm of fixed stars of approximately the same kind and density.
This view is not in harmony with the theory of Newton. The latter
theory rather requires that the universe should have a kind of centre
To this two-dimensional sphere-universe there is a three-dimensional
analogy, namely, the three-dimensional spherical space which was
-discovered by Riemann. its points are likewise all equivalent. It
+discovered by Riemann. Its points are likewise all equivalent. It
possesses a finite volume, which is determined by its ``radius"
($2\pi^2R^3$). Is it possible to imagine a spherical space? To imagine a
space means nothing else than that we imagine an epitome of our
the universe in which we live is infinite, or whether it is finite in
the manner of the spherical universe. Our experience is far from being
sufficient to enable us to answer this question. But the general
-theory of relativity permits of our answering it with a moduate degree
+theory of relativity permits of our answering it with a moderate degree
of certainty, and in this connection the difficulty mentioned in
Section 30 finds its solution.
For the relative orientation of the co-ordinate systems indicated in
-Fig. 2, the x-axes of both systems pernumently coincide. In the
+Fig. 2, the x-axes of both systems permanently coincide. In the
present case we can divide the problem into parts by considering first
only events which are localised on the $x$-axis. Any such event is
represented with respect to the co-ordinate system $K$ by the abscissa $x$
shows that we are able to construct the Lorentz transformation in this
general sense from two kinds of transformations, {\it viz.} from Lorentz
transformations in the special sense and from purely spatial
-transformations. which corresponds to the replacement of the
+transformations, which corresponds to the replacement of the
rectangular co-ordinate system by a new system with its axes pointing
in other directions.
$$x'^2 + y'^2 + z'^2 - c^2t'^2 = x^2 + y^2 + z^2 - c^2t^2 \quad . \quad . \quad . \quad \mbox{(11a)} $$
-\noindent is satisficd identically. That is to say: If we substitute their
+\noindent is satisfied identically. That is to say: If we substitute their
expressions in $x, y, x, t$, in place of $x', y', x', t'$, on the
left-hand side, then the left-hand side of (11a) agrees with the
right-hand side.
We have another instance of far-reaching agreement between the
deductions from two theories in Newtonian mechanics on the one hand,
and the general theory of relativity on the other. This agreement goes
-so far, that up to the preseat we have been able to find only a few
+so far, that up to the present we have been able to find only a few
deductions from the general theory of relativity which are capable of
investigation, and to which the physics of pre-relativity days does
not also lead, and this despite the profound difference in the
the neighborhood of the sun are photographed during a solar eclipse.
In addition, a second photograph of the same stars is taken when the
sun is situated at another position in the sky, {\it i.e.} a few months
-earlier or later. As compared whh the standard photograph, the
+earlier or later. As compared with the standard photograph, the
positions of the stars on the eclipse-photograph ought to appear
displaced radially outwards (away from the centre of the sun) by an
amount corresponding to the angle a.
In the first place, we see from this expression that two clocks of
identical construction will go at different rates when situated at
-different distances from the centre of the disc. This result is aiso
+different distances from the centre of the disc. This result is also
valid from the standpoint of an observer who is rotating with the
disc.
-Now, as judged from the disc, the latter is in a gravititional field
+Now, as judged from the disc, the latter is in a gravitational field
of potential $\phi$, hence the result we have obtained will hold quite
generally for gravitational fields. Furthermore, we can regard an atom
which is emitting spectral lines as a clock, so that the following
almost an exact circle, which makes it more difficult to locate the
perihelion with precision.}
-\footnotetext[2]{The displacentent of spectral lines towards the red end of the
+\footnotetext[2]{The displacement of spectral lines towards the red end of the
spectrum was definitely established by Adams in 1924, by observations
on the dense companion of Sirius, for which the effect is about thirty
times greater than for the Sun. R.W.L. -- translator}
projects / francis / relativity.git / commitdiff