From: Francis Russell Date: Thu, 1 Dec 2011 02:52:12 +0000 (+0000) Subject: Fixes to formulae in chapters 25 and 26. X-Git-Url: https://git.unchartedbackwaters.co.uk/w/?a=commitdiff_plain;h=0de6e0778641e1c73ff21a39b9c2688d092381b4;p=francis%2Frelativity.git Fixes to formulae in chapters 25 and 26. Some of the formulae were broken in the Project Gutenberg LaTeX, so comparing these against a physical copy of the book is now required. --- diff --git a/relat10.tex b/relat10.tex index 1462a03..57c5bc7 100644 --- a/relat10.tex +++ b/relat10.tex @@ -2615,7 +2615,7 @@ indicate the distance (line-interval) between $P$ and $P_1$, as measured with a little rod, by means of the very small number $ds$. Then according to Gauss we have - $$ds_2 = g_{11}du^2 + 2g_{12}dudv = g_{22}dv^2$$ + $${ds}^2 = g_{11}{du}^2 + 2g_{12}dudv = g_{22}{dv}^2$$ \noindent where $g_{11}, g_{12}, g_{22}$, are magnitudes which depend in a perfectly definite way on $u$ and $v$. The magnitudes $g_{11}$, $g_{12}$ and $g_{22}$, @@ -2626,7 +2626,7 @@ continuum with reference to the measuring-rods, but only in this case, it is possible to draw the $u$-curves and $v$-curves and to attach numbers to them, in such a manner, that we simply have: - $$ds^2 = du^2 + dv^2$$ + $${ds}^2 = {du}^2 + {dv}^2$$ Under these conditions, the $u$-curves and $v$-curves are straight lines @@ -2648,19 +2648,19 @@ distance $ds$ is associated with the adjacent points $P$ and $P_1$, this distance being measurable and well defined from a physical point of view, then the following formula holds: -$$ds^2 = g_{11}dx_1^2 + 2g_{12}dx_1dx_2 . . . . g_{44}dx_4^2$$ +$${ds}^2 = g_{11}{dx_1}^2 + 2g_{12}dx_1dx_2 . . . . g_{44}{dx_4}^2$$ \noindent where the magnitudes $g_{11}$, etc., have values which vary with the position in the continuum. Only when the continuum is a Euclidean one is it possible to associate the co-ordinates $x_1 \ldots x_4$. with the points of the continuum so that we have simply -$$ds2 = dx_1^2 + dx_2^2 + dx_3^2 + dx_4^2$$ +$$ds^2 = {dx_1}^2 + {dx_2}^2 + {dx_3}^2 + {dx_4}^2$$ In this case relations hold in the four-dimensional continuum which are analogous to those holding in our three-dimensional measurements. -However, the Gauss treatment for $ds^2$ which we have given above is not +However, the Gauss treatment for ${ds}^2$ which we have given above is not always possible. It is only possible when sufficiently small regions of the continuum under consideration may be regarded as Euclidean continua. For example, this obviously holds in the case of the marble @@ -2698,8 +2698,8 @@ Minkowski, which was only vaguely indicated in Section 17. In accordance with the special theory of relativity, certain co-ordinate systems are given preference for the description of the four-dimensional, space-time continuum. We called these ``Galileian -co-ordinate systems." For these systems, the four co-ordinates $x, y, -z, t$, which determine an event or---in other words, a point of the +co-ordinate systems." For these systems, the four co-ordinates $x$, $y$, +$z$, $t$, which determine an event or---in other words, a point of the four-dimensional continuum---are defined physically in a simple manner, as set forth in detail in the first part of this book. For the transition from one Galileian system to another, which is moving @@ -2714,24 +2714,24 @@ Minkowski found that the Lorentz transformations satisfy the following simple conditions. Let us consider two neighbouring events, the relative position of which in the four-dimensional continuum is given with respect to a Galileian reference-body $K$ by the space co-ordinate -differences $dx, dy, dz$ and the time-difference $dt$. With reference to a +differences $dx$, $dy$, $dz$ and the time-difference $dt$. With reference to a second Galileian system we shall suppose that the corresponding -differences for these two events are $dx', dy', dz', dt'$. Then these +differences for these two events are $dx'$, $dy'$, $dz'$, $dt'$. Then these magnitudes always fulfil the condition\footnotemark. - $$dx^2 + dy^2 + dz^2 - c^2dt^2 = dx' 2 + dy' 2 + dz' 2 - c^2dt'^2$$ + $${dx}^2 + {dy}^2 + {dz}^2 - c^2{dt}^2 = {dx'}^2 + {dy'}^2 + {dz'}^2 - c^2{dt'}^2$$ The validity of the Lorentz transformation follows from this condition. We can express this as follows: The magnitude - $$ds^2 = dx^2 + dy^2 + dz^2 - c^2dt^2$$ + $${ds}^2 = {dx}^2 + {dy}^2 + {dz}^2 - c^2{dt}^2$$ \noindent which belongs to two adjacent points of the four-dimensional space-time continuum, has the same value for all selected (Galileian) -reference-bodies. If we replace $x, y, z$, $\sqrt{-I} \cdot ct$ , by $x_1, -x_2, x_3, x_4$, we also obtaill the result that +reference-bodies. If we replace $x$, $y$, $z$, $\sqrt{-I} \cdot ct$ , by $x_1$, +$x_2$, $x_3$, $x_4$, we also obtaill the result that - $$ds^2 = dx_1^2 + dx_2^2 + dx_3^2 + dx_4^2$$ + $${ds}^2 = {dx_1}^2 + {dx_2}^2 + {dx_3}^2 + {dx_4}^2$$ \noindent is independent of the choice of the body of reference. We call the magnitude ds the ``distance'' apart of the two events or @@ -2788,23 +2788,23 @@ essence of the difficulty with which we were confronted in Section But the considerations of Sections 25 and 26 show us the way to 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 +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 significance, but only serve the purpose of numbering the points of the continuum in a definite but arbitrary manner. This arrangement -does not even need to be of such a kind that we must regard $x_1, -x_2, x_3$, as ``space" co-ordinates and $x_4$, as a ``time'' +does not even need to be of such a kind that we must regard $x_1$, +$x_2$, $x_3$, as ``space" co-ordinates and $x_4$, as a ``time'' co-ordinate. The reader may think that such a description of the world would be quite inadequate. What does it mean to assign to an event the -particular co-ordinates $x_1, x_2, x_3, x_4$, if in themselves these +particular co-ordinates $x_1$, $x_2$, $x_3$, $x_4$, if in themselves these co-ordinates have no significance? More careful consideration shows, however, that this anxiety is unfounded. Let us consider, for instance, a material point with any kind of motion. If this point had only a momentary existence without duration, then it would to -described in space-time by a single system of values $x_1, x_2, x_3, -x_4$. Thus its permanent existence must be characterised by an +described in space-time by a single system of values $x_1$, $x_2$, $x_3$, +$x_4$. Thus its permanent existence must be characterised by an infinitely large number of such systems of values, the co-ordinate values of which are so close together as to give continuity; corresponding to the material point, we thus have a (uni-dimensional) @@ -2814,8 +2814,8 @@ statements having regard to these points which can claim a physical existence are in reality the statements about their encounters. In our mathematical treatment, such an encounter is expressed in the fact that the two lines which represent the motions of the points in -question have a particular system of co-ordinate values, $x_1, x_2, -x_3, x_4$, in common. After mature consideration the reader will +question have a particular system of co-ordinate values, $x_1$, $x_2$, +$x_3$, $x_4$, in common. After mature consideration the reader will doubtless admit that in reality such encounters constitute the only actual evidence of a time-space nature with which we meet in physical statements. @@ -2834,7 +2834,7 @@ The following statements hold generally: Every physical description resolves itself into a number of statements, each of which refers to the space-time coincidence of two events A and B. In terms of Gaussian co-ordinates, every such statement is expressed by the agreement of -their four co-ordinates $x_1, x_2, x_3, x_4$. Thus in reality, the +their four co-ordinates $x_1$, $x_2$, $x_3$, $x_4$. Thus in reality, the description of the time-space continuum by means of Gauss co-ordinates completely replaces the description with the aid of a body of reference, without suffering from the defects of the latter mode of @@ -2848,7 +2848,7 @@ continuum which has to be represented. 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 $K, K^1,$ +formulation. The form there used, ``All bodies of reference $K$, $K^1$, etc., are equivalent for the description of natural phenomena (formulation of the general laws of nature), whatever may be their state of motion," cannot be maintained, because the use of rigid @@ -2866,11 +2866,11 @@ in the form of the natural extension of the special principle of relativity. According to the special theory of relativity, the equations which express the general laws of nature pass over into equations of the same form when, by making use of the Lorentz -transformation, we replace the space-time variables $x, y, z, t$, of a -(Galileian) reference-body $K$ by the space-time variables $x^1, y^1, z^1, -t^1$, of a new reference-body $K^1$. According to the general theory of +transformation, we replace the space-time variables $x$, $y$, $z$, $t$, of a +(Galileian) reference-body $K$ by the space-time variables $x^1$, $y^1$, $z^1$, +$t^1$, of a new reference-body $K^1$. According to the general theory of relativity, on the other hand, by application of arbitrary -substitutions of the Gauss variables $x_1, x_2, x_3, x_4$, the +substitutions of the Gauss variables $x_1$, $x_2$, $x_3$, $x_4$, the equations must pass over into equations of the same form; for every transformation (not only the Lorentz transformation) corresponds to the transition of one Gauss co-ordinate system into another.