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shakespeareanAngelita
shakespeareanAngelita" <ewttp@ceres.dti.ne.jp
Tue, 05 Oct 2004 01:23:00 -0600
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<font color=3D"#fffffD">Moreover, according to this equation the time diff=
erence t' of two events with respect to K' does not in general vanish, eve=
n when the time difference t of the same events with reference to K vanish=
es. Pure =93space-distance=94 of two events with respect to K results in =93=
time-distance=94 of the same events with respect to K'. But the discovery,=
of Minkowski, which was of importance for the formal development of the t=
heory of relativity, does not lie here. It is to be found rather in the fa=
ct =B1=F2[=D3is recognition that the four-dimensional space-time continuum=
of the theory of relativity, in its most essential formal properties, sho=
ws a pronounced relationship to the three-dimensional continuum of Euclide=
an geometrical space. 1 In order to give due prominence to this relationsh=
ip, however, we must replace the usual time co-ordinate t by an imaginary =
magnitude=20 Ought we to smile at the man and say that he errs in his co=
nclusion? I do not believe we ought if we wish to remain consistent; we mu=
st rather admit that his mode of grasping the situation violates neither r=
eason nor known mechanical laws. Even though it is being accelerated with =
respect to the =93Galileian space=94 first considered, we can nevertheless=
regard the chest as being at rest. We have thus good grounds for extendin=
g the principle of relativity to include bodies of reference which are acc=
elerated with respect to each other, and as a result we have gained a powe=
rful argument for a generalised postulate of relativity. 5=20 In prac=
tice, the rigid surfaces which constitute the system of co-ordinates are g=
enerally not available; furthermore, the magnitudes of the co-ordinates ar=
e not actually determined by constructions with rigid rods, but by indirec=
t means. If the results of physics and astronomy are to maintain their cle=
arness, the physical meaning of specifications of position must always be =
sought in accordance with the above considerations. 4 9=20</font>
<font color=3D"#fffffE"> But this result comes into conflict with the pri=
nciple of relativity set forth in Section V. For, like every other general=
law of nature, the law of the transmission of light in vacuo must, accord=
ing to the principle of relativity, be the same for the railway carriage a=
s reference-body as when the rails are the body of reference. But, from ou=
r above consideration, this would appear to be impossible. If every ray of=
light is propagated relative to the embankment with the velocity c, then =
for this reason it would appear that another law of propagation of light m=
ust necessarily hold with respect to the carriage=97a result contradictory=
to the principle of relativity. 4=20 We can now appreciate why that =
argument is not convincing, which we brought forward against the general p=
rinciple of relativity at the end of Section XVIII. It is certainly true t=
hat 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=
non-uniformity of motion (retardation) of the carriage. But he is compell=
ed by nobody to refer this jerk to a =93real=94 acceleration (retardation)=
of the carriage. He might also interpret his experience thus: =93My body =
of reference (the carriage) remains permanently at rest. With reference to=
it, however, there exists (during the period of application of the brakes=
) a gravitational field which is directed forwards and which is variable w=
ith respect to time. Under the influence of this field, the embankment tog=
ether with the earth moves non-uniformly in such a manner that their origi=
nal velocity in the backwards direction is continuously reduced.=94=20 ct =
proportional to it. Under these conditions, the natural laws satisfying th=
e demands of the (special) theory of relativity assume mathematical forms,=
in which the time co-ordinate plays exactly the same r=F4le as the three =
space co-ordinates. Formally, these four co-ordinates correspond exactly t=
o the three space co-ordinates in Euclidean geometry. It must be clear eve=
n to the non-mathematician that, as a consequence of this purely formal ad=
dition to our knowledge, the theory perforce gained clearness in no mean m=
easure. 4=20</font>
<font color=3D"#fffff8"> It is not clear what is to be understood here by=
=93position=94 and =93space.=94 I stand at the window of a railway carria=
ge which is travelling uniformly, and drop a stone on the embankment, with=
out throwing it. Then, disregarding the influence of the air resistance, I=
see the stone descend in a straight line. A pedestrian who observes the m=
isdeed from the footpath notices that the stone falls to earth in a parabo=
lic curve. I now ask: Do the =93positions=94 traversed by the stone lie =93=
in reality=94 on a straight line or on a parabola? Moreover, what is meant=
here by motion =93in space=94? From the considerations of the previous se=
ction the answer is self-evident. In the first place, we entirely shun the=
vague word =93space,=94 of which, we must honestly acknowledge, we cannot=
form the slightest conception, and we replace it by =93motion relative to=
a practically rigid body of reference.=94 The positions relative to the b=
ody of reference (railway carriage or embankment) have already been define=
d in detail in the preceding section. If instead of =93body of reference=94=
we insert =93system of co-ordinates,=94 which is a useful idea for mathem=
atical description, we are in a position to say: The stone traverses a str=
aight line relative to a system of co-ordinates rigidly attached to the ca=
rriage, but relative to a system of co-ordinates rigidly attached to the g=
round (embankment) it describes a parabola. With the aid of this example i=
t is clearly seen that there is no such thing as an independently existing=
trajectory (lit. =93path-curve=94 1), but only a trajectory relative to a=
particular body of reference. 2=20 We thus obtain the following resu=
lt: Every description of events in space involves the use of a rigid body =
to which such events have to be referred. The resulting relationship takes=
for granted that the laws of Euclidean geometry hold for =93distances,=94=
the =93distance=94 being represented physically by means of the conventio=
n of two marks on a rigid body.=20 Space is a three-dimensional continuu=
m. By this we mean that it is possible to describe the position of a point=
(at rest) by means of three numbers (co-ordinates) x, y, z, and that ther=
e is an indefinite number of points in the neighbourhood of this one, the =
position of which can be described by co-ordinates such as x1, y1, z1, whi=
ch may be as near as we choose to the respective values of the co-ordinate=
s x, y, z of the first point. In virtue of the latter property we speak of=
a =93continuum,=94 and owing to the fact that there are three co-ordinate=
s we speak of it as being =93three-dimensional.=94 2=20</font>
<font color=3D"#fffffD"> But there are two classes of experimental facts =
hitherto obtained which can be represented in the Maxwell-Lorentz theory o=
nly by the introduction of an auxiliary hypothesis, which in itself=97i.e.=
without making use of the theory of relativity=97appears extraneous. 2=
=20 IN the first part of this book we were able to make use of space-time =
co-ordinates which allowed of a simple and direct physical interpretation,=
and which, according to Section XXVI, can be regarded as four-dimensional=
Cartesian co-ordinates. This was possible on the basis of the law of the =
constancy of the velocity of light. But according to Section XXI, the gene=
ral theory of relativity cannot retain this law. On the contrary, we arriv=
ed at the result that according to this latter theory the velocity of ligh=
t must always depend on the coordinates when a gravitational field is pres=
ent. In connection with a specific illustration in Section XXIII, we found=
that the presence of a gravitational field invalidates the definition of =
the co-ordinates and the time, which led us to our objective in the specia=
l theory of relativity. 1=20 But this result comes into conflict with=
the principle of relativity set forth in Section V. For, like every other=
general law of nature, the law of the transmission of light in vacuo must=
, according to the principle of relativity, be the same for the railway ca=
rriage as reference-body as when the rails are the body of reference. But,=
from our above consideration, this would appear to be impossible. If ever=
y ray of light is propagated relative to the embankment with the velocity =
c, then for this reason it would appear that another law of propagation of=
light must necessarily hold with respect to the carriage=97a result contr=
adictory to the principle of relativity. 4=20</font>
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