The Feynman Lectures on Physics, Vol. I: The New Millennium Edition: Mainly Mechanics, Radiation, and Heat (Volume 1)

by Richard P. Feynman

One of the consequences of Maxwell’s equations is that if there is a disturbance in the field such that light is generated, these electromagnetic waves go out in all directions equally and at the same speed , or mi/sec. Another consequence of the equations is that if the source of the disturbance is moving, the light emitted goes through space at the same speed .

The Lorentz transformation

Einstein, following a suggestion originally made by Poincaré, then proposed that all the physical laws should be of such a kind that they remain unchanged under a Lorentz transformation.

The Michelson-Morley experiment

In carrying out the experiment, Michelson and Morley oriented the apparatus so that the line was nearly parallel to the earth’s motion in its orbit (at certain times of the day and night). This orbital speed is about miles per second, and any “ether drift” should be at least that much at some time of the day or night and at some time during the year. The apparatus was amply sensitive to observe such an effect, but no time difference was found—the velocity of the earth through the ether could not be detected. The result of the experiment was null.

Transformation of time

The Lorentz contraction

Simultaneity

Four-vectors

Relativistic dynamics

Equivalence of mass and energy

This theory of equivalence of mass and energy has been beautifully verified by experiments in which matter is annihilated—converted totally to energy: An electron and a positron come together at rest, each with a rest mass . When they come together they disintegrate and two gamma rays emerge, each with the measured energy of . This experiment furnishes a direct determination of the energy associated with the existence of the rest mass of a particle.

Relativistic Energy and Momentum

Relativity and the philosophers

Our inability to detect absolute motion is a result of experiment and not a result of plain thought, as we can easily illustrate.

Because it was not until Maxwell’s theory of electrodynamics was developed that there were physical laws that suggested that one could measure his velocity without looking outside; soon it was found experimentally that one could not.

The twin paradox

Transformation of velocities

The correct transformation law, that of Lorentz, is

These equations correspond to the relatively simple case in which the relative motion of the two observers is along their common -axes.

Relativistic mass

Now we shall try to demonstrate that the formula for must be , by arguing from the principle of relativity that the laws of physics must be the same in every coordinate system.

Now, let us accept that momentum is conserved and that the mass depends upon the velocity according to (16.10) and go on to find what else we can conclude. Let us consider what is commonly called an inelastic collision. For simplicity, we shall suppose that two objects of the same kind, moving oppositely with equal speeds , hit each other and stick together, to become some new, stationary object, as shown in Fig. 16–4(a).

Astonishing as that may seem, in order for the conservation of momentum to work when two objects come together, the mass that they form must be greater than the rest masses of the objects, even though the objects are at rest after the collision!

Relativistic energy

Then, although we might at first expect the mass to be , we have found that it is not , but . Since is what is put in, but are the rest masses of the things inside, the excess mass of the composite object is equal to the kinetic energy brought in.

This means, of course, that energy has inertia.

When the mass is different, we can tell that it is different. So, necessarily, the conservation of energy must go along with the conservation of momentum in the theory of relativity.

Space-Time

The geometry of space-time

A given point in space-time is called an event.

Space-time intervals

We would like, in other words, to put all our equations in a system of units in which . If time and space are measured in the same units, as suggested, then the equations are obviously much simplified.

Incidentally, we have just proved that if light travels with speed in one system, it travels with speed in another, for if the interval is the same in both systems, i.e., zero in one and zero in the other, then to state that the propagation speed of light is invariant is the same as saying that the interval is zero.

Past, present, and future

More about four-vectors

Four-vector algebra

To add four-vectors, we add the corresponding components.

The conservation of energy is the fourth equation which goes with the conservation of momentum to make a valid four-vector relationship in the geometry of space and time.

As Minkowski said, “Space of itself, and time of itself will sink into mere shadows, and only a kind of union between them shall survive.”

Rotation in Two Dimensions

The center of mass

Although there are all kinds of forces on the particles because of the strings, the wigglings, the pullings and pushings, and the atomic forces, and who knows what, and we have to add all these together, we are rescued by Newton’s Third Law. Between any two particles the action and reaction are equal, so that when we add all the equations together, if any two particles have forces between them it cancels out in the sum; therefore the net result is only those forces which arise from other particles which are not included in whatever object we decide to sum over.

Thus we find that the external force is the total mass times the acceleration of an imaginary point whose location is . This point is called the center of mass of the body.

First, if the external forces are zero, if the object were floating in empty space, it might whirl, and jiggle, and twist, and do all kinds of things. But the center of mass, this artificially invented, calculated position, somewhere in the middle, will move with a constant velocity.

Rotation of a rigid body

So rotation consists of a study of the variations of the angle with time.

So there are two conditions for equilibrium: that the sum of the forces is zero, and that the sum of the torques is zero.

Angular momentum

Conservation of angular momentum