Simple Nature - Light and Matter

At very low velocities, γ is close to 1, and the result is very nearlymv, as demanded by the correspondence principle. But at very highvelocities, γ gets very big — the small ball in figure b has a γ of5.0, and therefore has five times more inertia than we would expectnonrelativistically.This also explains the answer to another paradox often posedby beginners at relativity. Suppose you keep on applying a steadyforce to an object that’s already moving at 0.9999c. Why doesn’tit just keep on speeding up past c? The answer is that force is therate of change of momentum. At 0.9999c, an object already has a γof 71, and therefore has already sucked up 71 times the momentumyou’d expect at that speed. As its velocity gets closer and closer toc, its γ approaches infinity. To move at c, it would need an infinitemomentum, which could only be caused by an infinite force.7.3.2 Equivalence of mass and energyNow we’re ready to see why mass and energy must be equivalentas claimed in the famous E = mc 2 . So far we’ve only consideredcollisions in which none of the kinetic energy is converted into anyother form of energy, such as heat or sound. Let’s consider whathappens if a blob of putty moving at velocity v hits another blobthat is initially at rest, sticking to it. The nonrelativistic result isthat to obey conservation of momentum the two blobs must fly offtogether at v/2. Half of the initial kinetic energy has been convertedto heat. 4Relativistically, however, an interesting thing happens. A hotobject has more momentum than a cold object! This is becausethe relativistically correct expression for momentum is mγv, andthe more rapidly moving atoms in the hot object have higher valuesof γ. In our collision, the final combined blob must therefore bemoving a little more slowly than the expected v/2, since otherwisethe final momentum would have been a little greater than the initialmomentum. To an observer who believes in conservation of momentumand knows only about the overall motion of the objects and notabout their heat content, the low velocity after the collision wouldseem to be the result of a magical change in the mass, as if the massof two combined, hot blobs of putty was more than the sum of theirindividual masses.Now we know that the masses of all the atoms in the blobs mustbe the same as they always were. The change is due to the change inγ with heating, not to a change in mass. The heat energy, however,seems to be acting as if it was equivalent to some extra mass.4 A double-mass object moving at half the speed does not have the samekinetic energy. Kinetic energy depends on the square of the velocity, so cuttingthe velocity in half reduces the energy by a factor of 1/4, which, multiplied bythe doubled mass, makes 1/2 the original energy.Section 7.3 Dynamics 413