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Notes on Relativity
D. A. Edwards
Physics 128, Cornell, 1958
Preface
Many years ago as a junior faculty member at Cornell University I was teaching a course in second- year undergraduate physics for engineering students. I was sure that particle accelerators repre- sented a valuable and developing technology. Particle accelerators run according to Special Relativ- ity, because particle speeds approach that of light. I wanted to express my thoughts on the subject in a way that I could understand and therefore convey with some confidence to class members. Some of msey notes from that period appear below.
1 Frames of Reference – Inertial Systems
In discussion of a physical process - for example, the motion of a particle under the influence of known forces - we conventionally express the results made with respect to some set of coordinate axes. Thus, the motion of a particle may be specified by three functions of time, x(t), y(t), z(t) with respect to a particular Cartesian coordinate system. A coordinate system for the description of motion is also called a frame of reference
While certain frames of reference may seem natural for the discussion of a given physical problem, in principle, any coordinate system will do. In recording the motion of a billiard ball on a table, it seems most reasonable to record the positions of the ball with respect to a set of axes fixed to the table. One could, however, record positions with respect to a set of axes moving uniformly with respect to the table, or with respect to a set of axes rotating with respect to the table.
There are frames of reference in which the laws of motion take on a particularly simple form. The statement of Newtonian mechanics - Newton’s First Law -that a body initially at rest will remain at rest unless acted upon by a force is not true in all frames of reference. Someone living on a rotating disk will notice that objects set down will tend to slide away; from the perspective of this observer, Newton’s First Law is not valid.
The frames of reference for which Newton’s First Law holds are called inertial systems or inertial frames of reference. For many purposes, such as the discussion of the motion of a billiard ball on
a table, the earth’s surface is a very good approximation to an inertial frame, although for others such as missile ballistics the circumstance that the earth rotates must be taken into account. The notion of an inertial frame is in some sense an idealization or abstraction which must be examined in context.
Given one inertial frame, any coordinate system moving uniformly with respect to it will also be an inertial frame. For if an object is at rest in an inertial system, from the point of view of observers in a frame that is moving at constant velocity with respect to the first, the object will be moving at constant velocity with respect to their frame; that is Newton’s First Law will also be valid from their point of view. If one inertial frame, there will be many.
2 The Principle of Relativity
In applying Newton’s Laws of Motion it is necessary as indicated above that the frame of reference with respect to which coordinates are measured be an inertial one. No particular inertial frame is singled out from the many and our experience confirms that these laws work just as well in a uniformly moving train or airplane as they do for measurements made with respect to the surface of the earth. That this is so is consistent with the transformations which we normally apply to relate measurements made in one frame of reference to measurements to measurements made in a second frame of reference which is moving at constant velocity with respect to the first.
In Fig. 1 are two Cartesian coordinate systems. The system with axes x, y which we will refer to as S is at rest with respect to the paper. The system with axes x′, y′, which we will refer to as S′, is moving to the right with constant speed v. The x and x′^ axes coincide – for schematic purposes, they are shown with a small vertical separation. If the y and y′^ axes crossed at the zero of time, then we would expect the coordinates of a point in S to be related to its coordinates in S′^ to be
x′^ = x − vt (1) y′^ = y (2) t′^ = t (3)
The third relation, which would not have ordinarily write down in the days of Newton, asserts that the time as recorded by clocks everywhere in S is the same as the time of clocks everywhere in S′. These relations are often called the Galilean transformations.
Now in Fig. 1 suppose that m 1 and m 2 are the masses of two particles located at the points indicated. Using Newton’s Law of Gravitation, the equation of motion for the particle of mass m 1 would be
m 1 d^2 x 1 dt^2
= G
m 1 m 2 (x 1 − x 2 )^2
where we have written this equation with respect to S. Using the Galilean transformations, we can express Eq. 4 with respect to S′. Since
x′ 1 − x′ 2 = x 1 − x 2 (5) d^2 x′ 1 dt^2
d^2 x 1 dt^2
These considerations and others of a similar nature led Einstein, in a paper published in 1905, to investigate the consequences of asserting that the velocity of light is the same in all inertial frames, which had as an immediate consequence the abandonment of the Galilean transformations and modification to Newtonian mechanics. These consequences, while extensive, are not quite as destructive as they may sound, for we know very well that the Galilean transformations and Newton’s laws work very well for phenomena involving the moderate velocities with which we are familiar from everyday experience. As we shall see, Einstein’s assumptions lead to a modifications of these ideas for processes where one encounters velocities which are a significant fraction of the speed of light, and the older ideas appear as approximations – usually very good approximations – to the more complete notions of space, time, and mechanics to which these assumptions lead us.
4 The Relative Character of Simultaneity
It is clear from the arguments of the preceding section that Einstein’s assumption, the consequences of which are usually called the Special Theory of Relativity, lead to a modification of the law of addition of velocities. A simple example will indicate, however, that much more than this must be revised. Consider again, in Fig. ??, the two frames S and S′. At the instant the origins are superimposed, let a light flash be emitted from the common origin. Since the velocity of light is independent of motion of the source, it does not matter which frame we assume the source to be attached to. We have drawn the sketch at a time somewhat after the emission of the light flash, and the position of the wave front is drawn as a dotted line. But the word “time” presents a problem.
If x is the distance that the light flash has moved along the x-axis in the time interval t after the crossing of the origins, then x/t = c. From the point of view of an observer in S′, the light flash will have moved some distance x′^ along the x′^ axis, where x′^ is less than x. The time interval recorded by an observer moving with S′^ one might suppose would be t, the same as the time interval recorded by observers in S. But is x/t = c then x′/t cannot be c, in contradiction to the assumption that the velocity of light must be the same in all such frames. Therefore, the time interval recorded by observers in S′^ cannot be the same as t, but something less. We must abandon the notion of a uniform “public” time which is the same for all observers, regardless of their state of motion.
A further conclusion can be drawn from Fig. ??. We have drawn the wave front as a circle with center at the origin of S, as it must be from the point of view of observers in S. But since observers in S′^ have an equal right to the assertion that the speed of light is c in all directions, they must also see the wave front as a sphere with its center at the origin of S′, which it certainly is not in the sketch. This tells us that although the various points on the dotted line representing the wave front are all reached at the same time form the point of view of observers in S, these positions must be reached at different times from the point of view of observers in S′^ – that is, events which are simultaneous in one frame need not be simultaneous in another frame moving relatively to the first.
An example due to Einstein indicates rather clearly the relative character of simultaneity and certain of its consequences. Suppose that we wish to measure the length of a moving train. We can do this by stationing a number of observers by the side of the track with the instructions that at some specified time, say exactly 2:00 PM, the two observers who find an end of the train opposite
their positions are to record their position; we can assume that we have previously marked off a length scale along the track for this purpose. From the preceding example, we are aware that there may be difficulties in the specification of times for various observers; we should therefore specify exactly how the clocks of the observers situated along the track are to be synchronized. Fortunately, we have a standard velocity which can be used for this purpose. The observers are instructed that when the clock located at the zero position of the coordinate axis reads exactly 1:00 PM, a light signal will be sent out from this position. Since the speed of light is exactly c, the time at which the light signal reaches a distance x from the origin will be x/c later. So the observer at x sets his clock accordingly when he detects the light flash and is confident that his or her clock is synchronized with the one at the origin.
Now suppose that, in addition to recording the time at their positions, the two observers who find themselves at opposite ends of the train at 2:00 PM also send out light flashes (or radio signals) at that time. The observer who at 2:00 PM had been opposite the midpoint of the train in S would receive these signals simultaneously. An observer standing on the train at its midpoint would not detect the two light flashes simultaneously as we may see from Fig. ??. The leftmost of the sketches is drawn at 2:00 PM – 2:00 PM so far as the observers standing by the tracks are concerned. The observers C and D are confronting each other; C is standing on the train at its midpoint, and D is the observer standing by the track who is opposite the midpoint at 2:00 PM. The middle sketch shows the situation a short time later, with the light flashes moving away from their sources. In the rightmost sketch, still later, the two wave fronts are reaching D simultaneously. This implies that the flash from the front end of the train has already passed the observer C, while the flash from the back end of the train has not yet reached him or her. The observer on the train will of necessity conclude that the light flashes were not emitted at the same time. Thus, events that are simultaneous from observers in one inertial frame of reference need not be simultaneous for measurements conducted in another inertial frame.
5 Comparison of Lengths and Clock Rates
We must now give quantitative expression to the ideas raised above, and lay the groundwork for the coordinate transformations that will replace the Galilean transformations.
5.1 Distances at right angles to the direction of motion
Referring back to Fig. 1, if the distance from the point P from the x′^ axis is y′^ as measured in S′, then the Galilean transformation assures us that the corresponding distance as measured in S will be the same. That this relation will remain true we can see from the following argument.
Suppose that we have marked off distances on the y and y′^ axes with rulers which were identical when at rest with respect to each other. Now we place observers in the S frame at the origin and at the positions ±h on the y axis, These observers are given instructions to note the position on the y′^ axis which corresponds to theirs when the y′^ axis passes their position, and also, when this happens, to send out a light signal. If the observer at +h sees +h′^ on the y′^ axis, then the observer
signal returns to the x-axis as recorded by the clock at A will be
∆t = 2 s c
c
[
h^2 + v^2 ∆t^2 4
] 1 / 2
and so
∆t′^ = ∆t γ
, γ ≡
(1 − v 2 c^2 )
The time interval recorded by the single clock in S′^ is therefore shorter than this time difference between the two clocks in S′; this result is a direct consequence of the assertion that the velocity of light is the same in the two frames. The quantity γ is called the Lorentz factor.
We may obtain from Eq. 10 the time interval recorded by a clock attached to a body which is moving in any manner with respect to an inertial frame S. Even though the velocity of the body may not be uniform with respect to S, for a sufficiently short time interval dt, as measured in S, we may consider the body to be at rest in some inertial frame which is in motion with respect to the body at that instant. The time interval dt′^ for the clock attached to the body corresponding to dt will be
dt′^ = dt γ(v)
If we conceptually perform this operation throughout the motion of the body, the total time elapsed for the moving clock will be
∆t′^ =
∫ (^) t 2
t 1
dt γ(v)
there t 2 − t 1 = ∆t is the time interval recorded by clocks in S. The time recorded by a clock attached to a body is called the proper time. From Eq. 10 we see that proper time intervals are always shorter than time intervals recorded by clocks in frames with respect to which one is in motion.
A striking confirmation of the predictions above is provided by the rapidly moving radioactive particles such as pi mesons. The lifetime of such mesons which are moving with velocities at a significant fraction of the speed of light are found to be greater than the same variety of particle at rest by just by the factor predicted above.
5.3 Distances parallel to the direction of motion; the Lorentz contraction
Suppose we lay out a distance L on the x- axis of S. An observer located at the origin of S′ can measure this interval by noting the time ∆t′^ required for this piece of the x-axis to go by the positions; since the relative velocity of the frames is v, the observer would conclude that the interval is of length L′^ = v∆t′. In S, the time for the origin of S′^ to travel the distance L would be L/v. From the preceding discussion of clock rates, however, the time interval ∆t′^ recorded by the single clock at the origin of S′^ must be related to the time interval ∆t by the Lorentz factor. So L′^ = L/γ.
This astonishing result was put forth by Lorentz and Fitzgerald in order to account for the negative result of the Michelson-Morley experiment, and preceded Einstein’s conclusive 1905 paper. So the effect is often referred to as the Lorentz-Fitzgerald contraction.
6 The Lorentz Transformation
6.1 Transformation equations
We may make use of the results of the preceding section to find the new coordinate transformations. Suppose that the point P in Fig. ?? has the coordinates x′, y′^ at time t′, all as measured in S′. The figure is drawn at the time t for observers in S. As measured in S, x′^ will be shortened by the factor γ; the x coordinate off P will therefore be x = vt + x′/γ. After rearrangement, x′^ = γ(x − vt). This result differs from the Galilean transformation by the factor of γ and reduces to the Galilean transformation for speeds much less than c as one would expect.
Similarly, the inverse relation which for any x′^ and t′^ yields x will be x = γ(x′^ + vt′). Finally, the time t′^ recorded by a clock in S′^ at the position corresponding to the x, t in S may be found be elimination of x′^ from these last two equations, yielding t′^ = γ(t − xv/c^2 ). The replacements to the Galilean transformations are
x′^ = γ(x − vt) (13) y′^ = y (14) t′^ = γ
[
t −
xv c^2
]
These relations were obtained by H. A. Lorentz in 1904, although their full significance for the relativity of motion was not realized at that time. Einstein, unaware of the work of Lorentz, derived them independently in his 1905 treatment of relativity.
6.2 Transformation of velocity
If no velocity can exceed c, then the Galiean addition of velocities needs to be replaced. Suppose that a particle is traveling to the right in S′^ parallel to the x′^ axis with speed u′ x as measured in S′. Without loss of generality, we can assume that the particle started from x′^ = 0 at t′^ = 0. Then x′^ = u′ xt′^ gives the position of the particle in S′^ as a function of time. Applying the Lorentz transformations yields
x =
u′ x + v 1 + u′ xv/c^2 t. (16)
so for observers in S, the particle is moving with speed
ux = u′ x + v 1 + u ′ xv c^2
As u′ x approaches c then so does ux also, as consistent with the limit of the speed of light.
If there is a vertical velocity component, y′^ = u′ yt′, then application of the Lorentz transformation yields
uy =
u′ y 1 + u
′ xv c^2
Then in what was a wonderfully imaginative leap, Einstein endowed a particle with a “rest energy” E = mc^2 in recognition of the prospects of radioactivity only then underway and other processes even now awaiting exploration. The “total energy” becomes γmc^2 which with the replacement of γ with the momentum leads to the more general expression
E =
p^2 c^2 + m^2 c^4. (23)
