Einstein's Special Theory of Relativity
Einstein's theory of relativity is now separated into two parts: Special Theory, and General Theory of Relativity. We will first examine the, more simple, Special Theory in the present section. General Theory of Relativity is discussed in the following section.
Ether and Michelson-Morley's Experiment
Sound waves are the result of oscillations of matter. When you speak, your vocal cords oscillate very much like a violin string. They cause the air in your throat to oscillate, which cause the air in front of your mouth to oscillate ... If it were not for the air in the room, your sound would not be heard! But light waves do travel in vacuum. Just look up in the night sky to see light from distant stars traveling though vacuum - space empty of matter: atoms, molecules, etc. But in 1887 people still believed that even vacuum was imbedded in a "background" called ether (not the same stuff as the chemical component used to put people into sleep!). So, even light from the distant stars traveled though ether. And so did the earth: it was thought to spin in ether. Michelson and Morley devised an experiment to measure the change in the value of the speed of light as it traveled with the earth in ether versus when it traveled opposite to earth in ether.
Speed of light is 300,000,000 meters per second. This is about 670,000,000 miles per hour. Earth's rotational speed is about as fast as it is, is only 1,000 miles per hour. This means that any experiment that wants to detect a measurable difference between the speed of light in the direction of motion (rotation) of earth versus one perpendicular to it (i.e. no motion with respect to ether) has to be accurate to about one part in million. Michelson and Morley's experiment was even more precise. But they could not find a difference! This "null" result became a new thorn in the back of classical physics. It was also this result that became the backbone of Einstein's Theory of Relativity and the beginning of a new physics.
This experiment essentially found that speed of light is measured to have the same value no matter how the light source or the measuring instrument is set to move. Please note how anti-intuitive this result is! In every day life speed of any object (or even waves) has a lot to do with relative motion of the object (source of wave) with the observer - person who is to make the measurement. If you run as fast as your friend, and in the same direction, you don't need to shout to speak to each other. Things will be the same as when you are both sitting down. In fact your friend's speed, with respect to you, will be zero; even though s/he is running. So, your friend's speed, with respect to you, has a lot to do with your speed. Michelson-Morley's result says that for light, it matters not whether you move with it (say with the same speed of 300,000,000 m/s) or move in the opposite direction from it, you'll always measure the same value for its speed! It is important to mention also that this experiment was performed with such high degree of precision that no one doubted its results. It was the implication of the experiment's result that made little sense.
Einstein's Postulates of Special Theory of Relativity
Einstein's first postulate is in fact the statement of Michelson-Morley's experiment:
speed of light in vacuum is measured to have the same value by all (inertial) observers independent of their relative motions.
Einstein's second postulate is a more philosophical one:
laws of physics must have the same form for all (inertial) observers.
The consequence of this second postulate is that if there are "laws" that don't have the same form for all observers, then they must not be "correct". Said differently, a "correct" theory then formulates the laws so that they will have the same mathematical form for all observers. Or, it at least it allows different observers of a given measurement to agree with each other's results, even though they are in relative motion with each other.
Let me define two commonly terms used in discussions on relativity. First of all, we refer to "observer" as any person or persons performing a measurement. The term is a bit misleading in that by "observing" we don't mean just "viewing"; we mean "measuring". In fact, one of the major contributions of Einstein's theory (after Earnst Mach) is that it is not good enough to assume you could measure something, but that you need to specify exactly how you would measure it! The second term is the notion of inertial observers. We'll get to this more later, but for now all we need to worry about is that these are observers that may move with respect to each other, but their relative speeds do not change (i.e. they don't go faster or slower; they keep their relative speeds a constant). Those observers whose relative speeds may change are referred to as non-inertial observers. It is the General Theory of Relativity that deals with non-inertial observers.
Please note that a major consequence of the first postulate is that indeed nothing can travel faster than light does in vacuum. (It was well known that light traveling in other media, say glass, slows down; so it travels its fastest in vacuum.) Because of this, two events can communicate with each other no faster than the speed of light. This clearly sets a limit on causality. If the star is 10 million light years away from us, then the light from it takes 10 million years to reach us. So, the only way that an event, A, on that star could have a PHYSICAL effect on an event, B, on earth today would be that event A took place 10 million years ago!
Aside from the limitations set on causality, discussed above, the result of Michelson and Morley also set a new requirement for the notion of simultaneity. Two events that may appear simultaneous to one observer, may happen at different times for another observer moving relative to the first observer. This is of no surprise in everyday life. It is the limitation that "nothing moves faster than c " that makes this notion of simultaneity even stranger. Because of this we cannot view time as independent of space. In fact Einstein showed us that time is the same as space. So, we now know that we really live in a (timeless) four-dimensional space.
Einstein used his two postulates, along with a mathematical relationship called Lorenz Transformation, to make sure that physics could be measured and formulated the same by all inertial observers. A direct consequence of his theory that required agreement among all inertial observers, turns out, is that different inertial observers may obtain different results for lengths and time intervals. For example, if you set a yardstick in relative motion to earth and then measure its length while it is moving, you'll find that it has a length shorter than a yard, it "contracts". The faster it moves, relative to you, the shorter it will appear to get! Similarly, time interval gets "dilated". So, any time measurement that one observer makes of the other, moving, observer will reveal that the "moving time" runs slower. It should be mentioned that these effects become measurably significant only when we are dealing with fast enough motions that are comparable to the speed of light. At every day speeds none of us need to be concerned with the seemingly odd effects of Relativity Theory. Still, both of these, length contraction and time dilation, have been clearly verified in laboratory experiments and today all physicists whose work deals with fast moving objects, such as nuclear and particle physicists, have to abide by the consequences of Special Relativity both in their calculations and in their measurements.
We've already seen that when matter is set to motion it gains kinetic energy. The faster it moves, the larger its kinetic energy. Also, of two objects of different mass, but equal speed the more massive one has the larger kinetic energy. But it took Einstein's Special Theory of Relativity to show that, as its consequence, mass itself is a form of energy. This surprising result came about when Einstein required that his second postulate must hold true even for Newton's laws of mechanics. (Both length contraction and time-interval dilation came about by requiring that simple kinematics should be the same for all inertial observers. That is to say, these results were not consequence of applications of physical laws.) More specifically, if two inertial observer make a measure of the same force and the same acceleration and connect these values using Newton's celebrated F = ma expression, then the only way that they can reach agreement with each other's measurements is to find that an object's mass, m, is not a constant value, as previously "assumed".
In fact, the faster objects move the more massive they get (i.e. the object's mass is measured by an inertial observer who is in relative motion to be more than what it is when the same observer makes the same measurement while, this time, at rest with respect to the object). Furthermore, when energy relations are balanced, same as in the classical physics, it is found that the energy associated with motion (kinetic energy) is not zero when the object comes to rest, but reaches a minimum value of moc2 instead. This is called the rest mass energy and is the most direct "statement" that inertial mass, which in Newtonian physics is a measure of object's resistance to motion, is really a measure of a new form of energy. Up to this point in time physics believed that there were two forms of energy: kinetic and potential. Kinetic energy is energy of motion. An object with an inertial mass m and velocity v has a kinetic energy equal to the product of its mass with the square of its velocity divided by two, i.e. 1/2 mv2. So, the faster the object moves, i.e. the larger its value of velocity v, the more will be its kinetic energy. Potential energy, energy of interaction, on the other hand, is a measure of the stored energy in a system (two or more things) that attracts and holds together its parts. The potential energy is simply the energy that is needed to separate these parts. It makes no sense to talk about the potential energy of a single thing, say even an electron. And if the electron is motionless, then it has no kinetic energy. It is in this context that the relativistic result, moc2 , is a new form of energy.
Is this form of energy real, or is it just a mathematical relation to satisfy Einstein's second postulate? What makes kinetic and potential energies "real" is that either one can change into the another. A falling apple loses as much potential energy as it gains kinetic energy by moving faster and faster as it falls. Equally, rest-mass energy, moc2 , can be converted to kinetic or potential forms of energy, as we will see in our studies of nuclear physics. It is indeed very real!
Questions on Einstein's Special Theory of Relativity
Einstein's Transformation Relations:
An observer in the (unprimed) frame F making measurements of position values x, y, z, and of time, t are related to the measurements of x', y', z', and t' of the observer in the (primed) frame F', that has a relative speed of v along the x-direction with respect to F, according to
x' = g (x - bct)
y' = y
z' = z
ct' = g(ct - b x)
where b = (v/c) and g = ( 1 - b2)-1/2 and c is the speed of light in vacuum.