THE LOGIC OF RELATIVITY |
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where c is the velocity of light,:, and d is the distance between the twoclocks as measured by the observer on 5j.
By means of the foregoing theorems we may readily obtain the formulae for the celebrated Lorentz transformation of space and time coordinates. (The non-mathematical reader may omit the remainder of this paragraph.) Let two systems of reference 5 and
S' have the relative velocity v in the line /. Let systems of rectangular coordinates be attached to the systems of reference 5 and S' in such a way that the x-axis of each system is in the line / and that the two *-axes have the same positive direction, and let the y-axis and the z-axis of one system be parallel to the y-axis and z-axis respectively of the other system and have their positive senses in the same directions. Let these two systems of axes coincide at the time zero. Furthermore, for the sake of distinction, denote the space and time coordinates on S by x, y, z, t, and those on Let us suppose that S' moves with respect to 5 in thedirection of increasing values of
x. Then it turns out that the foregoing theorems imply the following relations between the two systems of coordinates:whereand
c is the velocity of light.The foregoing theorems, or (in more compact language) the foregoing equations of transformation, furnish the effective means for developing the whole of the restricted theory of relativity. Our purpose does not require us to follow that development further in detail. But we may mention a few of the remarkable conclusions which now emerge readily. If two velocities, each of which is less than c, are combined the resultant velocity is also less than
c. The mass of a body increases with an increase in its velocity relative to the system on which the mass is measured. The mass of a body at rest appears to be the measure of its internal energy. Mass and energy in general appear to be essentially convertible terms. The velocity of light is a maximum which the velocity of a material body may approach but can never equal or exceed.The development by Einstein in 1905 of the foregoing restricted theory of relativity led to a fresh analysis of the whole foundations of physics. This was made inevitable by its effective attack upon such fundamental notions as those of length and time and mass and velocity. Einstein himself succeeded in 1915 in greatly extending the range of his theory, developing what has since |
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