1. RELATIVISTIC KINEMATICS The one truth of which the human mind can be certain – indeed, this is the meaning of consciousness itself – is the recognition of its own existence. That we may be secure in this truth is assured us by Descartes’ famous axiom even if everything else, including Descartes, is a figment of our imagination. Nothing else can be proved. Our most fundamental belief, then, is that the universe exists around us, consisting of three spatial dimensions and time, and that while we can move about in the three spatial dimensions, time flows inexorably onward, everywhere the same. Newton himself said it: “…absolute, true, and mathematical time, of itself, and from its own nature, flows equably and without relation to anything external.” But he was wrong. The theory of relativity shows us that time and space do not have the meaning we thought they had. In the words of Weyl, “…we are to discard our belief in the objective meaning of simultaneity; it was the great achievement of Einstein in the field of the theory of knowledge that he banished this dogma from our minds, and this is what leads us to rank his name with that of Copernicus” (italics his). But the discovery of relativity by Einstein in 1905 was not a bolt from the blue. People had been concerned about the nature of space and time for at least hundreds of years before that, becoming more and more disturbed by the inconsistencies in our understanding of the physical world toward the end of the 19th century. Nevertheless, even after the true nature of space and time became clear, the theory of relativity so contradicted our most fundamental belief that it was rejected for years. Einstein himself never received the Nobel Prize for this work that was, in the words of Bertrand Russell, “probably the greatest synthetic achievement of the human intellect up to the present time.” Some sixteen years afterward he was reluctantly awarded the Nobel Prize for a lesser work because the greatest physicist of the century, known to more people than the President of the United States, could not be completely ignored. Yet, the truth Einstein taught us displayed once again nature’s tendency to assume the most beautiful, symmetric form, in spite of our objections. And while the truth is a merger of space and time that prevents us from ordering events absolutely in time, it does not result in chaos, but preserves those features that we cannot logically be denied. The principle of causality is never violated, and as each of us progresses through this four-dimensional space-time, our individual perception of time as moving always forward is not contradicted. 1.1. Einstein’s Postulates Einstein's solution to the dilemma of the velocity of light was as beautiful as it was radical. He chose the most symmetric form for nature, stating that all inertial reference frames are equivalent. He embodied this concept in his two postulates of special relativity, which state: 1. The laws of nature are identical in all inertial frames of reference. That is, if we transform the mathematical equations of physics from one inertial reference frame to another they remain in the same form. 2. The speed of light c is the same to all observers at rest in inertial frames of reference. A more general statement of this principle might be that the influence of one particle is 1 not felt instantaneously by another. Instead, the influence propagates at some (maximum) velocity c . These two postulates can be summed up in a single postulate that states that there is no experiment that can be done to distinguish the absolute velocity of any coordinate system. Einstein’s postulates appear beautifully simple and symmetric. In fact they are deceptively simple, since within them lie profound consequences and more than a few startling paradoxes. Fundamental to all the paradoxes is the fact that simultaneity is no longer an objective reality, as Weyl points out, but rather a subjective one that depends on the observer. A simple example, shown in Figure 1, illustrates this. Consider the following "gedanken experiment" (thought experiment; Einstein loved gedanken experiments): Figure 1 On the moon, Buffy and Bubba, representating the lunar colonies Alpha and Beta, are having a green-cheese eating contest. When the winner is declared, the news is radioed to the folks back home in Alpha and Beta, which are equidistant from the site of the contest. Each colony receives the news, which travels at the speed of light, at exactly the same time. Right? Of course. But Hilda and Wolfgang, who are passing the moon in their space ship on their way back to earth after a vacation, watch the events on the moon and come to a different conclusion. Since, as they view it, the moon is moving to the left, the news reaches Beta, which is moving toward the contest, before it reaches Alpha. Right? Of course. Who is right? Well, they both are. There is no absolute meaning to the concept of simultaneity. In fact, let’s check in with Edgar and Eloise, who are in a space ship going the opposite direction from Wolfie and Hildie, just starting their vacation. As viewed by E and E, the moon is going the opposite direction and the news reaches Alpha before it reaches Beta. We can’t even get agreement on which of two events (the news arriving at Alpha and at Beta) occurred first. In fact, there is really no absolute meaning to simultaneity. Time just doesn’t work that way, although the effects are usually so small that you never noticed it. You may (you should!) wonder what has happened to cause and effect. For example, if event A causes event B, what happens if someone else observes events A and B to occur in the reverse order. This is similar to the logical difficulty that occurs when people travel back in 2 time. Can Dr. No travel back in time and kill his mother so that he himself is never born? Well, of course not, regardless of whether you liked “Back to the Future” or not. In fact, the theory of relativity does not violate causality. Two events can be reordered in time by other observers only if the events happen so far apart in distance and so close together in time that neither light nor anything else (which must travel slower than light) can get from the first event to the second. Therefore, event A cannot have any influence on (or cause) event B, and the principle of causality is not violated. Clearly, the two events called “news reaching Alpha” and “news reaching Beta” are too far apart in distance to be connected by a single light pulse. It takes two light pulses to reach the two events, so it is OK that they can be reordered in time by different observers. One of these two events can never cause or influence the other. 1.2. Time dilation Let’s do another gedanken experiment. This time we put a simple (in concept, at least) clock on the space ship with Wolfie and Hildie. The clock sends a short laser pulse up to a mirror, and when it strikes the mirror and returns the clock ticks once and sends out the next pulse. If the distance to the mirror is L , the round-trip distance traveled by the laser pulse is dL= 2 . If c is the (universal) velocity of light, the clock ticks once in the time Δ=tdc/2/ = Lc. But what do Buffy and Bubba, standing on the moon, think of this? Figure 2 As they see it, the light makes a triangular trip up and down as the laser and the mirror move to the right at the velocity v . The total distance the light travels in one tick is found from Pythagoras' theorem: dLvt'2222= ()2 +Δ ' (1.1) But light travels at the velocity c , so the time for the clock to tick once is Δ=tdc''/. The moving clock goes too slow (that is, it is observed by Buffy and Bubba to take too long to tick) by the factor Δt ' = γ (1.2) Δt where 3 1 γ = (1.3) 1− β 2 β = vc/ (1.4) are the relativistic parameters. But this isn’t just a case of a clock going too slow. The clock is just fine. In fact, everything on the moving space ship is going too slow. Wolfie and Hildie’s hearts beat too slowly, and they are aging too slowly, at least according to Buffy and Bubba. Wolfie and Hildie don’t see anything wrong. There is no experiment they can do to detect their motion, after all. Should Bubba and Buffy be jealous that Wolfie and Hildie are getting old slower? Not at all; Wolfie and Hilda are not enjoying the extra time. Their thoughts, their days, everything is going slower for them. They don’t experience any extra time. In fact, when you think of it, Wolfie and Hildie see a clock belonging to Buffy and Bubba going too slow compared with their clock. After all, they see themselves as stationary and Buffy and Bubba moving past them on a (very large) space ship. So they see Buffy and Bubba getting old slower than they are. Each one sees the other’s clock as moving slower than their own! This is known as time dilation; time on a moving space ship is observed to be stretched out. Once again, this is not a problem caused by bad clocks.