VII: Modern Physics
The Special Theory of Relativity
The theory of relativity is usually associated with the name Einstein. It might therefore come as a surprise that the concept of relativity did not originate with Einstein. The honor belongs to Galileo. In Two New Sciences, Galileo (1914) discussed the problem of the behavior of falling bodies on a moving earth, arriving at the conclusion that they fall exactly as they w'ould appear to do if the earth were not moving. Galileo argued that you cannot tell whether the earth is moving or at rest by watching an object fall.
In Two New Sciences, Galileo discussed the simpler problem of uniform linear motion. According to Galileo, if we are in a ship moving along a straight line with constant speed and drop a ball from the crow’s nest, the ball will fall straight down, hitting the deck at the foot of the mast, not the water. But an observer on shore will thus see the ball falling down following a curved path and not a straight line (Figure 19.1).
If we drop a ball inside a closed room, the ball will fall straight down whether the room is in our house, a cabin in a cruise ship, or the closed bathroom of an airplane. In fact, no experiment can be performed inside a closed room that will reveal to us whether or not the room is at rest or moving along a straight line at a constant speed. If we are in the cabin of a ship that is moving steadily and drop a ball from the middle of the ceiling, it will drop in the middle of the floor, just as if we were
FIGURE 19.1 (a) An observer in a moving ship will see a ball falling straight down, as if the ship were not
moving, (b) A person on shore will see the ball move along a curved path.
doing the experiment in our room at home. Motion in a straight line at a constant speed, according to Galileo, has no discoverable effects. The only way we can tell whether or not we are moving is by looking out of the window to determine whether there is relative motion between us and the earth. Sitting in an airplane we have just boarded, we are sometimes fooled into thinking it is moving when the engines are running and another airplane is taxiing nearby. Unless we catch a glimpse of a building or are able to see the ground from where we sit, the vibrations from the engine and the motion of the other airplane make it impossible to decide who is moving. A similar experience occurs when, tired during a long car trip, we may find ourselves sitting low in the back seat when our driver gets stuck in a traffic jam. Seeing only the tops of the other automobiles on the other lanes, we might think we are finally moving, only to discover that the cars traveling in the opposite direction were the only ones moving.
In a sense, then, all motion is relative. In everyday situations, we refer motion to the earth. Although we would say that when we are sitting in our room reading we are not moving, a hypothetical observer traveling through the solar system would affirm that our room is actually rotating along with the earth and revolving around the sun. The description of motion depends on the particular reference frame to which we refer it. If we are on an airplane that is traveling at a steady 800 km/h, we would consider the book we are reading to be at rest with respect to us, the other passengers, and the plane (Figure 19.2). Of course, the book is moving along with us, the other passengers, and the plane at 800 km/h relative to the ground. Which view is the correct one? Both are. The book is at rest in the reference frame of the plane and moving at 800 km/h in the reference frame of the ground. While the plane is moving steadily, your coffee will not spill and your pen will not roll off the foldout table. The law of inertia holds and, as we have said, nothing, other than looking out the window, will tell you that you are moving. A reference frame in which the law of inertia holds is called an inertial reference frame.
Suppose now that, while a plane is flying with a constant velocity at 800 km/h, a flight attendant walks from the back of the plane to the front at a steady pace. Assume that she walks at a speed of 2 km/h (Figure 19.3). This, of course, is her speed in the reference frame of the plane. If the velocity of the flight attendant were to be measured from the ground, we would find it to be 802 km/h. This is hardly surprising to us. It is not uncommon to see people in a hurry walking on escalators. Their velocities w'ith respect to the building where the escalator is located are greater than the velocity of someone who merely rides the escalator. We all have seen children walking down an “up” escalator so that they remain stationary in relation to the building. In this case, their velocities are the negative of the escalator velocity.
FIGURE 19.2 The plane and all its contents are traveling at 800 km/h relative to the ground. The book that the passenger is reading and the passenger herself are at rest relative to the plane.
FIGURE 19.3 While the plane flies at a constant speed of 800 krn/h relative to the ground, a light attendant walks down the aisle at a speed of 2 krn/h relative to the plane.
If a reference frame S’ is moving with a velocity vF relative to a second reference frame S, then the velocity v of an object relative to 5 is equal to its velocity v’ in frame S’ plus vF; that is,
Returning to our example of the flight attendant, the velocity of the reference frame S' of the airplane relative to the frame of the ground is vF= 800 km/h, the flight attendant’s velocity relative to the plane is v’ = 2 km/h, and her velocity relative to the reference frame S of the ground is v = 2 + 800 km/h = 802 km/h.
Since inertial frames of reference move at constant velocities, the acceleration of an inertial frame is zero. Therefore, the acceleration of an object in one reference frame is the same as in any other inertial frame. The object we drop from the ceiling of a cabin in a steadily moving ship not only falls straight down to the ground relative to the people on the ship, but also accelerates at 9.8 m/s2. This is the same acceleration that an observer on shore would measure for the falling object if this observer could see it. Thus, not only the law of inertia—Newton’s first law—holds for inertial frames of reference but the second law and the universal law of gravitation as well. In fact, all the laws of mechanics are the same for all observers moving at a constant velocity relative to each other, as Newton himself recognized. This statement is implicit in Galileo’s own statement that uniform motion has no discoverable effects and is known today as the Galilean principle of relativity. We can state this as follows:
The laws of mechanics are the same in all inertial frames of reference.
This principle means that there is no special or absolute reference frame; all reference frames are equivalent. Thus, there is no absolute standard of rest; uniform motion has to be referred to an inertial frame.
A word of caution before we leave this section is in order. We have implicitly said that the earth is an inertial reference frame. This is not exactly true, because the earth is rotating, so any point on its surface is always accelerating. However, this acceleration is very small, and the rotational effects of the earth can be neglected. The earth can thus be considered an inertial frame of reference for our purposes.