TIME IS OF THE ESSENCE IN SPECIAL RELATIVITY
by Dr. S. Peter Rosen
Office of Science
U.S. Department of Energy
When it comes to Einstein’s theory of special relativity, everyone thinks about the famous equation E = mc2, but the truly revolutionary aspect of the theory is the treatment of time. Prior to Einstein, time was regarded as a universal measure for all observers, no matter whether they were at rest, or traveling with high speed relative to each other. This point of view led to serious inconsistencies, and they were not resolved until Einstein allowed time to vary from one observer to another. In particular, he argued that simultaneity could not hold in all frames of reference: whereas one observer may see two events at different locations as occurring at the same time, a second observer moving relative to the first would see the events as occurring at different times. This was a profound change in thinking and it led to the concept that we live in the four-dimensional world of space-time, rather than a three dimensional world of space alone.
A hundred years ago, when Einstein wrote his original paper, there were two great theories of physics, Newtonian mechanics developed by Sir Isaac Newton in the 17th Century, and James Clerk Maxwell’s theory of electromagnetism developed in the latter part of the 19th Century. Newtonian mechanics treated time as universal, and so the speed of light had to depend on the motion of the observer. By contrast, Maxwell’s theory implied that light, and all other forms of electromagnetic waves travel at the same speed in all frames of reference, namely three hundred thousand kilometers per second. Einstein chose to explore the consequences of this aspect of Maxwell’s theory.
He used a series of “thought experiments” based upon sending and receiving light signals to develop the theory of special relativity. We can follow this path to show how the theory modified the concept of time. Our first step is to show that distance, or length, perpendicular to the relative motion of two observers Bob and Alice is unchanged by the relative motion.
Imagine that Alice is moving horizontally with speed v relative to Bob and that each of them holds a meter stick in the vertical direction. At the instant that Alice passes Bob, both can see that their meter sticks are exactly equal in length. Suppose now that Alice has attached a laser to each end of her stick with a switch to turn on the lasers at the same instant. She turns them on when she is well past Bob, and the signals from the lasers will always reach detectors located at the ends of Bob’s meter stick at exactly the same time. Bob will then conclude that both meter sticks are still equal in length.
Following Richard Feynman, our next step is to use this result to construct a rudimentary clock and analyse the relationship between the times measured by Bob and Alice. Suppose now that Alice removes one of the lasers and places mirrors at the ends of her meter stick and allows the laser beam to bounce back and forth between the mirrors. Every time the beam hits the mirror beside the remaining laser, a loud tick goes off. This defines the ticking of Alice ’s clock.
According to Alice , the time, tA it takes for the light signal to travel from one mirror to the other and back again is just twice the length of the meter stick divided by c, the velocity of light. According to Bob, however, the meter stick is moving with speed v in the horizontal x-direction and so he sees the light signal between the mirrors take a longer path. It has the back and forth motion in the vertical direction, exactly as in Alice’s frame, plus a component in the horizontal direction. Since, by the principles of special relativity, the velocity of the light is the same in both frames, the time tB that Bob measures must be longer than the time tA measured by Alice. Working out the relationship between the two times using simple geometry, we find that tB = g tA where g2 equals 1/ [1 – (v/c)2 ] and is always greater than 1 because v is always less than c.
This relationship between the two times has been described in various ways. One is to say that because tA is shorter than tB, clocks run slowly in moving frames. Alternatively, because tB is greater than tA, we can say that time is dilated in Bob’s frame. Whether we talk of clocks running slowly or time dilation, the physical consequences are the same. For example, the muon is a particle like the electron, but much heavier, and it decays into an electron in about 2 microseconds when it is at rest, or moving very slowly compared with the speed of light. When it is travelling with a high speed, however, it appears to live much longer than 2 microseconds. This fact enables muons created by cosmic rays high in the Earth’s atmosphere to travel down to the Earth’s surface and then penetrate a few thousand feet below the surface before decaying. It also enables us to create beams of muons at our large accelerators, and use them to probe the structure of neutrons and protons.
Another physical consequence concerns the relationship between lengths along the direction of relative motion as measured by Bob and by Alice. Suppose that a mark is made in Bob’s frame at every point where the light beam of Alice’s clock completes one round trip between the mirrors: Bob will say that Alice has traveled a distance LB = vtB between two such successive points, whereas Alice will say that Bob has receded a distance LA = vtA . From the relationship between the two times, we see that LA is much shorter than LB by a factor (1/g): LA = (1/g)LB.
We describe this phenomenon as “length contraction”: LB is the length as measured by Bob who is at rest with respect to any pair of successive marks made by Alice’s clock, whereas Alice is moving with respect to the marks and sees a shorter distance. It is not merely a coincidence that this contraction, also known as Lorentz-FitzGerald contraction, is exactly what was needed to explain the Michelson-Morley experiment.
In 1887, when Einstein was only 8 years old, two American scientists, Albert Michelson, a Polish-born physicist, and Edward Morley, a chemist, had tried to explore another aspect of Maxwell’s theory, namely the aether. Since light is an electromagnetic wave, then surely it must propagate through a medium, just as ocean waves propagated through water and sound waves through air. This medium was given the name “luminiferous aether” and Michelson and Morley set out to measure the speed of the Earth through the aether. They compared the time it took for a beam of light to travel a distance L and back in a direction parallel to the Earth’s motion with the time it took for light to travel the same distance and back perpendicular to the Earth’s motion. If time were universal, as Michelson and Morley assumed, then the speed of the Earth through the aether can be calculated from the difference in the two times, the parallel time being longer than the perpendicular one. But no matter how hard they tried, and no matter how many times, nor at what times of year, they made the measurement, they always came up with a null result: the time in the parallel direction was always the same as the time in the perpendicular direction.
To explain this null result, George Fitzgerald, an Irish physicist, put forward in 1889 an ad hoc hypothesis that the distance L parallel to the Earth’s motion should be contracted by an amount sufficient to make the two times equal. Independently, Hendrik Lorentz, in Holland, put forward the same hypothesis in 1904, but he based it on the fact that electromagnetic forces between charged particles are sensitive to their motion and cause a minute contraction in the size of moving bodies, just enough to explain the null results of the Michelson-Morley experiment. This is exactly the contraction derived by Einstein from special relativity, although it is not clear whether Einstein was actually aware of the Michelson-Morley experiment when he developed his theory.
A final outcome of Einstein’s reinterpretation of time concerns Newton’s Laws of motion, which, together with his theory of gravity, had successfully explained the motion of planets around the sun. Since the basic law is that force equals the rate of change of momentum as a function of time, and momentum is defined as the product of mass times velocity, it is clear that a change in our understanding of time requires a change in our thinking about Newton’s Laws. Of course, any change should not be at the cost of abandoning their successes. Einstein found, quite remarkably that the Laws could be maintained provided that one simple change was made in the definition of mass.
Instead of mass m being treated as being independent of velocity, Einstein replaced it with the expression m = m0g, g being the same factor as appears in the analysis of time and length above. The constant m0 is the value of the mass when the velocity is zero relative to the observer, and hence it is known as the rest-mass of the object. With this simple change, all of Newton’s Laws apply as before, and since the velocities of the planets around the sun are much less than the speed of light, the successful description of their orbits still holds to a very good approximation. Moreover, this apparently simple change leads to the familiar equation E = m0 c2 embodying the profound result of the equivalence of mass and energy. In round numbers, one gram of matter is equivalent to a very large amount of energy, namely 20 million kilowatt hours, or to 20 kilotons of TNT.
Having woven time into the fabric of space-time, Einstein went on to publish the general theory of relativity in 1915. In this general theory, gravity is represented by distortions of space-time caused by mass, just as a heavy sphere resting on a flat rubber sheet distorts the shape of the sheet. Sir Arthur Eddington, a British physicist, mounted an expedition to view the solar eclipse of 1919 and measure the deflection of starlight caused by the mass of the sun. General relativity passed this test of one of its important predictions with flying colors, and has continued to do so ever since. Gravity probe B, recently launched by NASA, will tell us whether this record of successful predictions will continue, but our views of space and time have been forever changed by the theories of Albert Einstein.
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