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1.8 Particularly important results |
In order to use the hyperbolae to derive the effects of time dilation and Lorentz contraction, as we do in the next section, we must point out a simple but important property of the tangent to the hyperbolae.
In Fig. 1.12(a) we have drawn a hyperbola and its tangent at x = 0, which is obviously a line of simultaneity t = const. In Fig. 1.12(b) we have drawn the same curves from the point of view of observer O¯ who moves to the left relative to O. The event P has been shifted to the right: it could be shifted anywhere on the hyperbola by choosing the Lorentz transformation properly. The lesson of Fig. 1.12(b) is that the tangent to a hyperbola at any event P is a line of simultaneity of the Lorentz frame whose time axis joins P to the origin. If this frame has velocity v, the tangent has slope v.
1.8 Pa r t i c u l a r l y i m p o r t a n t re s u l t s
Time dilation
From Fig. 1.11 and the calculation following it, we deduce that when a clock moving on the
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Notice that ¯t is the time actually measured by a single clock, which moves on a world line from the origin to B, while t is the difference in the readings of two clocks at rest in O; one on a world line through the origin and one on a world line through B. We shall return to this observation later. For now, we define the proper time between events B and the origin to be the time ticked off by a clock which actually passes through both events. It is a directly measurable quantity, and it is closely related to the interval. Let the clock be at rest in frame O¯ , so that the proper time τ is the same as the coordinate time ¯t. Then, since the clock is at rest in O¯ , we have x¯ = y¯ = z¯ = 0, so
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The proper time is just the square root of the negative of the interval. By expressing the interval in terms of O’s coordinates we get
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This is the time dilation all over again.
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Figure 1.13 The proper length of AC is the length of the rod in its rest frame, while that of AB is its length in O.
Lorentz contraction
In Fig. 1.13 we show the world path of a rod at rest in O¯ . Its length in O¯ is the square root of s2AC, while its length in O is the square root of s2AB. If event C has coordinates ¯t = 0, x¯ = l, then by the identical calculation from before it has x coordinate in O
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This is the Lorentz contraction.
Conventions
The interval s2 is one of the most important mathematical concepts of SR but there is no universal agreement on its definition: many authors define s2 = ( t)2 − ( x)2 −
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1.8 Particularly important results |
( y)2 − ( z)2. This overall sign is a matter of convention (like the use of Latin and Greek indices we referred to earlier), since invariance of s2 implies invariance of − s2. The physical result of importance is just this invariance, which arises from the difference in sign between the ( t)2 and [( x)2 + ( y)2 + ( z)2] parts. As with other conventions, students should ensure they know which sign is being used: it affects all sorts of formulae, for example Eq. (1.9).
Failure of relativity?
Newcomers to SR, and others who don’t understand it well enough, often worry at this point that the theory is inconsistent. We began by assuming the principle of relativity, which asserts that all observers are equivalent. Now we have shown that if O¯ moves relative to O, the clocks of O¯ will be measured by O to be running more slowly than those of O. So isn’t it therefore the case that O¯ will measure O’s clocks to be running faster than his own? If so, this violates the principle of relativity, since we could as easily have begun with O¯ and deduced that O’s clocks run more slowly than O¯ ’s.
This is what is known as a ‘paradox’, but like all ‘paradoxes’ in SR, this comes from not having reasoned correctly. We will now demonstrate, using spacetime diagrams, that O¯ measures O’s clocks to be running more slowly. Clearly, we could simply draw the spacetime diagram from O¯ ’s point of view, and the result would follow. But it is more instructive to stay in O’s spacetime diagram.
Different observers will agree on the outcome of certain kinds of experiments. For example, if A flips a coin, every observer will agree on the result. Similarly, if two clocks are right next to each other, all observers will agree which is reading an earlier time than the other. But the question of the rate at which clocks run can only be settled by comparing the same two clocks on two different occasions, and if the clocks are moving relative to one another, then they can be next to each other on only one of these occasions. On the other occasion they must be compared over some distance, and different observers may draw different conclusions. The reason for this is that they actually perform different and inequivalent experiments. In the following analysis, we will see that each observer uses two of his own clocks and one of the other’s. This asymmetry in the ‘design’ of the experiment gives the asymmetric result.
Let us analyze O’s measurement first, in Fig. 1.14. This consists of comparing the reading on a single clock of O¯ (which travels from A to B) with two clocks of his own: the first is the clock at the origin, which reads O¯ ’s clock at event A; and the second is the clock which is at F at t = 0 and coincides with O¯ ’s clock at B. This second clock of O moves parallel to the first one, on the vertical dashed line. What O says is that both clocks at A read t = 0, while at B the clock of O¯ reads ¯t = 1, while that of O reads a later time, t = (1 − v2)−1/2. Clearly, O¯ agrees with this, as he is just as capable of looking at clock dials as O is. But for O to claim that O¯ ’s clock is running slowly, he must be sure that his own two clocks are synchronized, for otherwise there is no particular significance in observing that at B the clock of O¯ lags behind that of O. Now, from O’s point of view, his
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Figure 1.14
Special relativity
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The proper length of AB is the time ticked by a clock at rest in O¯ , while that of AC is the time it takes to do so as measured by O.
clocks are synchronized, and the measurement and its conclusion are valid. Indeed, they are the only conclusions he can properly make.
But O¯ need not accept them, because to him O’s clocks are not synchronized. The dotted line through B is the locus of events that O¯ regards as simultaneous to B. Event E is on this line, and is the tick of O’s first clock, which O¯ measures to be simultaneous with event B. A simple calculation shows this to be at t = (1 − v2)1/2, earlier than O’s other clock at B, which is reading (1 − v2)−1/2. So O¯ can reject O’s measurement since the clocks involved aren’t synchronized. Moreover, if O¯ studies O’s first clock, he concludes that it ticks from t = 0 to t = (1 − v2)1/2 (i.e. from A to B) in the time it takes his own clock to tick from ¯t = 0 to ¯t = 1 (i.e. from A to B). So he regards O’s clocks as running more slowly than his own.
It follows that the principle of relativity is not contradicted: each observer measures the other’s clock to be running slowly. The reason they seem to disagree is that they measure different things. Observer O compares the interval from A to B with that from A to C. The other observer compares that from A to B with that from A to E. All observers agree on the values of the intervals involved. What they disagree on is which pair to use in order to decide on the rate at which a clock is running. This disagreement arises directly from the fact that the observers do not agree on which events are simultaneous. And, to reiterate a point that needs to be understood, simultaneity (clock synchronization) is at the heart of clock comparisons: O uses two of his clocks to ‘time’ the rate of O¯ ’s one clock, whereas O¯ uses two of his own clocks to time one clock of O.
Is this disagreement worrisome? It should not be, but it should make the student very cautious. The fact that different observers disagree on clock rates or simultaneity just means that such concepts are not invariant: they are coordinate dependent. It does not prevent any given observer from using such concepts consistently himself. For example, O can say that A and F are simultaneous, and he is correct in the sense that they have the same value of the coordinate t. For him this is a useful thing to know, as it helps locate the events in spacetime.