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constants in (5), we obtain the first and the fourth of the equations given in Section 11.
Thus we have obtained the Lorentz transformation for events on the x-axis. It satisfies the
condition
x'2 - c2t'2 = x2 - c2t2 . . . (8a).
The extension of this result, to include events which take place outside the x-axis, is obtained by
retaining equations (8) and supplementing them by the relations
In this way we satisfy the postulate of the constancy of the velocity of light in vacuo for rays of light
of arbitrary direction, both for the system K and for the system K'. This may be shown in the
following manner.
We suppose a light-signal sent out from the origin of K at the time t = 0. It will be propagated
according to the equation
or, if we square this equation, according to the equation
x2 + y2 + z2 = c2t2 = 0 . . . (10).
It is required by the law of propagation of light, in conjunction with the postulate of relativity, that the
transmission of the signal in question should take place  as judged from K1  in accordance with
the corresponding formula
r' = ct'
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Relativity: The Special and General Theory
or,
x'2 + y'2 + z'2 - c2t'2 = 0 . . . (10a).
In order that equation (10a) may be a consequence of equation (10), we must have
x'2 + y'2 + z'2 - c2t'2 = � (x2 + y2 + z2 - c2t2) (11).
Since equation (8a) must hold for points on the x-axis, we thus have � = I. It is easily seen that the
Lorentz transformation really satisfies equation (11) for � = I; for (11) is a consequence of (8a) and
(9), and hence also of (8) and (9). We have thus derived the Lorentz transformation.
The Lorentz transformation represented by (8) and (9) still requires to be generalised. Obviously it
is immaterial whether the axes of K1 be chosen so that they are spatially parallel to those of K. It is
also not essential that the velocity of translation of K1 with respect 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, viz. from Lorentz transformations in the
special sense and from purely spatial transformations. which corresponds to the replacement of the
rectangular co-ordinate system by a new system with its axes pointing in other directions.
Mathematically, we can characterise the generalised Lorentz transformation thus :
It expresses x', y', x', t', in terms of linear homogeneous functions of x, y, x, t, of such a kind that the
relation
x'2 + y'2 + z'2 - c2t'2 = x2 + y2 + z2 - c2t2 (11a).
is satisficd 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.
Next: Appendix II: Minkowski's Four Dimensional Space
Relativity: The Special and General Theory
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Relativity: The Special and General Theory
Albert Einstein: Relativity
Appendix
Appendix II
Minkowski's Four-Dimensional Space ("World")
(supplementary to section 17)
We can characterise the Lorentz transformation still more simply if we introduce the imaginary
in place of t, as time-variable. If, in accordance with this, we insert
x1 = x
x2 = y
x3 = z
x4 =
and similarly for the accented system K1, then the condition which is identically satisfied by the
transformation can be expressed thus :
x1'2 + x2'2 + x3'2 + x4'2 = x12 + x22 + x32 + x42 (12).
That is, by the afore-mentioned choice of " coordinates," (11a) [see the end of Appendix II] is
transformed into this equation.
We see from (12) that the imaginary time co-ordinate x4, enters into the condition of transformation
in exactly the same way as the space co-ordinates x1, x2, x3. It is due to this fact that, according to
the theory of relativity, the " time "x4, enters into natural laws in the same form as the space co
ordinates x1, x2, x3.
A four-dimensional continuum described by the "co-ordinates" x1, x2, x3, x4, was called "world" by
Minkowski, who also termed a point-event a " world-point." From a "happening" in
three-dimensional space, physics becomes, as it were, an " existence " in the four-dimensional "
world."
This four-dimensional " world " bears a close similarity to the three-dimensional " space " of
(Euclidean) analytical geometry. If we introduce into the latter a new Cartesian co-ordinate system
(x'1, x'2, x'3) with the same origin, then x'1, x'2, x'3, are linear homogeneous functions of x1, x2,
x3 which identically satisfy the equation
x'12 + x'22 + x'32 = x12 + x22 + x32
The analogy with (12) is a complete one. We can regard Minkowski's " world " in a formal manner
as a four-dimensional Euclidean space (with an imaginary time coordinate) ; the Lorentz
transformation corresponds to a " rotation " of the co-ordinate system in the fourdimensional "
world."
Next: The Experimental Confirmation of the General Theory of Relativity
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Relativity: The Special and General Theory
Relativity: The Special and General Theory
76
Relativity: The Special and General Theory
Albert Einstein: Relativity
Appendix
Appendix III
The Experimental Confirmation of the General Theory of
Relativity [ Pobierz całość w formacie PDF ]