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 $g_{22}$,
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
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
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
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
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)
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.
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
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
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.
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