This concept will be analyzed further in the next chapters, where we shall find that if the ideas of the preceding paragraph are applied to momentum, the transformation gives three space parts that are like ordinary momentum com- ponents, and a fourth component, the time part, which is the energy. 15-8 Relativistic dynamics We are now ready to investigate, more generally, what form the laws of echanics take under the Lorentz transformation. [We have thus far explained how length and time change, but not how we get the modified formula for m (Eq. 15.1). We shall do this in the next chapter. To see the consequences of Einstein' s modification of m for Newtonian mechanics we start with the Newtonian law that force is the rate of change of momentum, or F Momentum is still given by mu, but when we use the new m this becomes This is Einsteins modification of Newton's laws. Under this modification, if action and reaction are still equal(which they may not be in detail, but are in the long run), there will be conservation of momentum in the same way as before, but the quantity that is being conserved is not the old my with its constant mass, but instead is the quantity shown in(15. 10), which has the modified mass. When this change is made in the formula for momentum, conservation of momentum still works Now let us see how momentum varies with speed. In Newtonian mechanics it is proportional to the speed and according to(15. 10), over a considerable range of speed, but small compared with c, it is nearly the same in relativistic mechanics, because the square root expression differs only slightly from 1. But when v is Imost equal to c, the square-root expression approaches zero, and the momentum therefore goes toward infinity t' What happens if a constant force acts on a body for a long time? In Newtonian mechanics the body keeps picking up speed until it goes faster than light. But this impossible in relativistic mechanics. In relativity, the body keeps picking up, not speed, but momentum, which can continually increase because the mass is increasing. After a while there is practically no acceleration in the sense of a change of velocity, but the momentum continues to increase Of course, whenever a force produces very little change in the velocity of a body, we say that the body has a great deal of inertia, and that is exactly what our formula for relativistic mass ays(see Eq. 15. 10)it says that the inertia is very great when v is nearly as great as c. As an example of this effect, to deflect the high-speed electrons in the syn chrotron that is used here at Caltech, we need a magnetic field that is 2000 times stronger than would be expected on the basis of Newtons laws. In other words, the mass of the electrons in the synchrotron is 2000 times as great as their normal mass, and is as great as that of a proton! That m should be 2000 times mo means that 1-u2/c2 must be 1/4,000,000, and that means that v2/c2 differs from I by one part in 4,000,000 or that v difers from c by one part in 8,000,000, so the electrons are getting pretty close to the speed of light. If the electrons and light were both to start from the synchrotron (estimated as 700 feet away) and rush ut to Bridge Lab, which would arrive first? The light, of course, because light always travels faster How much earlier? That is too hard to tell--instead, we tell by what distance the light is ahead it is about 1/1000 of an inch, or t the thick- ess of a piece of paper! When the electrons are going that fast their masses are enormous, but their speed cannot exceed the speed of light The electrons would actually win the race versus visible light because of the index of refraction of air a gamma ray would make out better 159