From 6682fa5a49d16f64c0ceccaf60cf5dc93752997e Mon Sep 17 00:00:00 2001 From: Francis Russell Date: Tue, 29 Nov 2011 18:23:32 +0000 Subject: [PATCH] Numerous typographical corrections. --- relat10.tex | 108 ++++++++++++++++++++++++++-------------------------- 1 file changed, 54 insertions(+), 54 deletions(-) diff --git a/relat10.tex b/relat10.tex index d0dc70f..1c9655e 100644 --- a/relat10.tex +++ b/relat10.tex @@ -337,7 +337,7 @@ the conception of position has been developed. \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 @@ -612,7 +612,7 @@ being, however, we shall assume its correctness. -\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} @@ -684,15 +684,15 @@ investigations of H. A. Lorentz on the electrodynamical and optical 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 @@ -805,7 +805,7 @@ different places in such a manner that A is simultaneous with B and B 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.} @@ -1183,7 +1183,7 @@ direction of its length with a velocity $v$ is $\sqrt{I-v^2/c^2}$ of a metre. 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. @@ -1359,7 +1359,7 @@ circumstance, however, does not in the least diminish the 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. @@ -1423,7 +1423,7 @@ of relativity has grown out of electrodynamics and optics. In these 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 @@ -1469,14 +1469,14 @@ 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 +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, @@ -1650,7 +1650,7 @@ Galileian transformation for changing over from one body of reference 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 @@ -1684,7 +1684,7 @@ fact very perplexing to physicists. Lorentz and FitzGerald rescued the 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 @@ -1722,7 +1722,7 @@ four-dimensional space-time continuum. 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 @@ -1856,7 +1856,7 @@ experience can decide as to its correctness or incorrectness. 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 @@ -1888,7 +1888,7 @@ of relativity. But a simple and apparently quite reliable 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 @@ -1909,7 +1909,7 @@ uniform motion. At all events it is clear that the Galileian law does 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. @@ -2004,7 +2004,7 @@ postulate of relativity. \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 @@ -2032,7 +2032,7 @@ of the floor of the chest. He must therefore take up this pressure by 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 @@ -2045,7 +2045,7 @@ discussed in the preceding section), the man in the chest will thus 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. @@ -2073,7 +2073,7 @@ reference-body to be ``at rest." 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 @@ -2109,7 +2109,7 @@ such that, as judged from it, the gravitational field of the earth (in 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 @@ -2170,7 +2170,7 @@ Analogously, I seek in vain for a real something in classical 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 @@ -2248,7 +2248,7 @@ propagation of light varies with position. Now we might think that as 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). @@ -2274,7 +2274,7 @@ comprehensive theory, in which it lives on as a limiting case. 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 @@ -2289,7 +2289,7 @@ variable with respect to space and time.\footnotemark\ The character of this 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 @@ -2328,7 +2328,7 @@ used before. Let us consider a space time domain in which no 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 @@ -2385,7 +2385,7 @@ radius of the disc) tangentially to the edge of the disc, then, as 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 @@ -2507,7 +2507,7 @@ arbitrary? The method of Cartesian coordinates must then be 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 @@ -2689,7 +2689,7 @@ continuum under our notice. -\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} @@ -2739,7 +2739,7 @@ four-dimensional points. 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. @@ -2747,7 +2747,7 @@ 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).} @@ -2768,10 +2768,10 @@ according to this latter theory the velocity of light must always 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 @@ -2781,12 +2781,12 @@ impossible to construct a Cartesian co-ordinate system from equal 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 @@ -2827,7 +2827,7 @@ determine the corresponding values of the time by the observation of 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 @@ -2846,7 +2846,7 @@ continuum which has to be represented. \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 @@ -2950,10 +2950,10 @@ with reference to $K^1$ simply by mathematical transformation. We 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 @@ -2974,7 +2974,7 @@ following demands: 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} @@ -3031,7 +3031,7 @@ orbital ellipse was 43 seconds of arc per century, an amount ensured 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 @@ -3079,7 +3079,7 @@ the universe is infinite. There are stars everywhere, so that 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 @@ -3223,7 +3223,7 @@ world-sphere'' is a ``surface of constant curvature." 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 @@ -3263,7 +3263,7 @@ question arises for astronomers and physicists, and that is whether 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. @@ -3336,7 +3336,7 @@ with the Newtonian constant of gravitation.} 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 $x$ @@ -3535,7 +3535,7 @@ to $K$ should be in the direction of the $x$-axis. A simple consideration 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. @@ -3547,7 +3547,7 @@ of $x, y, x, t$, of such a kind that the relation $$x'^2 + y'^2 + z'^2 - c^2t'^2 = x^2 + y^2 + z^2 - c^2t^2 \quad . \quad . \quad . \quad \mbox{(11a)} $$ -\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. @@ -3647,7 +3647,7 @@ hypothesis of the hereditary transmission of acquired characters. 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 @@ -3794,7 +3794,7 @@ In practice, the question is tested in the following way. The stars in 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. @@ -3878,11 +3878,11 @@ the disc, then we have 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 @@ -3947,7 +3947,7 @@ bodies. \footnotemark 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} -- 2.47.3