inconsistency in the equations for the momentum and the energy of a beam of radiation. The momentum of the beam of mass m we have given in equation (5) as
.
From our assumption that the energy of the beam is simply the kinetic energy of the moving mass m, we might expect from our knowledge of elementary mechanics to find for the energy the equation
;
whereas in fact we find from equations (4) and (5) that
| . | (9) |
We shall see, however, in the next section that this comparison of equations (5) and (9) instead of demolishing our theory actually furnishes a remarkably satisfactory argument in its favour.
Non-Newtonian Mechanics.
One of the interesting branches of modern mathematics has grown out of the study of those geometries which would result from the change of one or more of the axioms of Euclid. These non-Euclidian geometries present the properties of purely imaginary kinds of space and are therefore so far mere exercises in logic, without any physical significance. But their investigation was doubtless prompted in some cases by the belief that experiment itself may sometime show that there are deviations from the ordinary laws of space when these laws are subjected to tests of a different order from those of common mensuration. Indeed it is not unlikely that Euclidian geometry may prove inadequate when we are able to subject to an accurate metric investigation the vast stretches of interstellar space or the minute regions which we believe to be encompassed within an atom or an electron.
The science of mechanics, like geometry, has been built up from a set of simple axioms, which were laid down by Newton. But the conclusions of the previous section lead us to modify one of these axioms and thus lay the foundation of a system of non-Newtonian mechanics.
The axiom which we must surrender is the one which states that the mass of a body is independent of its velocity. We have concluded that mass is proportional to content of energy. When a body is set in motion it gains kinetic energy and therefore its mass must change with its velocity. In place of
