Tech Notes 4 and 5 Tech Notes 4 and 5 Let s prove that invariance of the spacetime interval the hard way. Suppose we begin with the equation (ds) 2 = (dt) 2 (dx) 2. We recall that the coordinate transformations between inertial frames in the special theory of relativity are the Lorentz transformations: y = y, z = z, x = x vt 1 (v 2 c 2 ), t = t vx c2 1 (v 2 c 2 ). For a fixed value of the relative velocity v, we see that the variable x is a function of both x and t, as is the variable t. We can then write this as x ' = x '(x,t), and t ' = t '(x,t),
economically using the same symbols for coordinates and for the names of functions of which they are the values (and of which x,t are the arguments). Next we remind ourselves of the formulas for total derivatives for these functions: dx ' = x ' x dt ' = t ' x dx + x ' t dt, dx + t ' t dt. We want to plug into these formulas, so first we have to find four partial derivatives. Fortunately, that is easy in this case. We have: x ' x = 1 1 v 2 x ' t = v 1 v 2, t ' x = v t ' 1- v 2 t = 1 1 v. 2 So we find that 2
(dt ') 2 (dx ') 2 = ( v 1 v dx + 1 2 1 v 2 dt)2 ( 1 1 v 2 dx + v 1 v 2 dt)2 = v2 1 v 2 dx2 2v 2 dt dxdt + 2 1 v 1 v ( 1 2 1 v 2 dx2 2v 1 v 2 dxdt + v2 1 v 2 dt 2 ) 1 v2 1 v2 = 1 v 2 dx2 + 1 v dt 2 = dt 2 dx 2, 2 which shows the invariance of the spacetime interval when coordinates in inertial frames are related by the Lorentz transformations. We can view the metric, as we wrote it above, (ds) 2 = (dt) 2 (dx) 2. as a kind of distance function. But it differs from our usual notion of distance in two ways. First, if we indicate the distance between events x and y as d(x,y), then it is usually the case if d(x,y)=0, then x=y. But with the metric above, there are many points or events in spacetime that have distance zero from a given or arbitrarily chosen point, O. If, as is usual in SR, we choose units such that light travels one spatial unit in one temporal unit (and so 3
has the invariant speed of 1 in vacuo), then every point on the path of a photon emitted from O or arriving at O has distance zero from O. It s easiest to understand this if O is the origin of a coordinate system, and so has coordinates (0,0,0,0). In this case, we can just look at the coordinates of various events in that coordinate system. For instance a photon leaving O along the x-axis will be at a distance of t spatial units at time t, in which case t 2 x 2 = t 2 - t 2 = 0. These points with distance 0 from O make up the light cone or the null cone of the O, as illustrated in Figure TN4.3 on page 440, where the point I call O (for origin, of a coordinate system) is called Here- Now. The light cone of Here-Now divides into two parts, which we may call future and past. We identify values of t > 0 with the future light cone and values of t < 0 with the past light cone. 4
The second way the interval or metric differs from the usual distance function is that it can be negative as well as positive (or zero). If the interval is positive, then t 2 > d 2. This divides into two cases: t > 0 and t < 0. In the first case, any point for which d < t is one in the future light cone (since t > 0) that can be reached from O by an object travelling slower than light, since d < t. Events within the (future) light cone called timelike separated (from Here-Now) and future to Here-Now. In particular we should note that a clock at Here- Now can reach any point in the future light cone without accelerating. Its world line will be just a straight line containing both that event and Here- Now, lying entirely within the future light cone. In other words, the world line of the clock can be thought of as the world line of an observer who experiences both that event and Here-Now or as the t-axis of an inertial frame. 5
Of course the same is true for points in the past light cone of Here-Now. For such points d < t, and a clock or inertial observer can experience both the event and Here-Now while being at rest or moving with velocity v < c. Notice that timelike separation is an invariant relation, one that holds for all inertial frames, given the invariance of the interval. It is often said that in the special theory no material body (and so presumably no causal influence) can propagate in spacetime at a speed greater than the speed of light in vacuo. Events in (or on) the past light cone of Here-Now can influence causally what happens there, and what happens there can influence causally what happens at other events in (or on) its future light cone. The light cone structure is sometimes called the causal structure of Minkowski spacetime. 6
We have left then the case that d > t. In that case no object at Here-Now can reach the point (t,d,0,0) unless it travels at a speed faster than light. Such events are usually called spacelike separated (from Here-Now), and in Figure TN4.3 they are in the region labeled Elsewhere. For any given event in the elsewhere, there is exactly one inertial frame in which it is simultaneous with the Here-now, an infinity of inertial frames in which it occurs before Here-Now and another infinity of frames in which it occurs after Here-Now. The time-order of spacelike separated events in not invariant. What Do Clocks Measure? We consider a rest frame in which a clock is moving uniformly at some speed v. Consider two events in the history of the clock (on the world line of the clock, that is), E 1 and E 2. 7
In the frame of the clock, these events happen at the same place, so we will say that the distance between them (in the clock s frame) is D 0. The time difference as read by the clock from E 1 to E 2 we will call T 0. In the rest frame we considered initially, we will let the distance between the two events be D and the time between the two events be T. By construction, the distance D 0 between E 1 and E 2 in the clock frame is 0, since both events happen at the clock. In this frame, then, the square of the time difference between E 1 and E 2 as indicated by the clock is equal to the interval between the two events. Given the invariance or frame-independence of the interval, we then know that (T 0 ) 2 = T 2 - D 2 /c 2. We can re-write this as 8
(T 0 ) 2 + D 2 /c 2 = T 2, and then we can divide both sides by T 2 to arrive at (T 0 ) 2 /T 2 + D 2 /c 2 T 2 = 1. The quantity D/T is the ratio of the distance the clock moves to the time it takes to get from E 1 to E 2 in the initial rest frame, the frame in which the clock moves. This is just the velocity of the clock, v, in the initial rest frame. If we remind ourselves that we are using units in which c = 1, we can re-write the last equation in the simple form (T 0 ) 2 /T 2 + v 2 = 1. (8.11) It is worthwhile to try to understand what (8.11) means. The ratio T 0 /T is the rate at which a moving clock ticks (in nanoseconds, say) in terms of the rate at 9
which clocks tick (= time lapses?) in the rest frame, the frame in which the clock moves with velocity v. Given our units, by the way, the absolute value of v, v, is 1. As N. David Mermin says (in his book It s About Time ), The relation (8.11) tells us that the sum of the square of the speed at which a uniformly moving clock runs (in nanoseconds of clock reading per nanosecond of time [time in a chosen rest frame, that is]) plus the square of the speed at which the clock moves through space (in feet of space per nanosecond of time) is 1. (86) The key fact is just that 1 is a constant. So a clock that is not moving in the initial rest frame must tick at the rate of 1 nanosecond (of its time) per nanosecond (of the rest frame time). But if the clock moves in that frame, then, as Mermin says, there is a trade-off. (87) As v increases, the ratio T 0 /T must decrease, if the sum of the two is to remain constant (as 8.11 requires). Therefore, as v approaches 1 as 10
a limit (or v 1, in symbols), the ratio T 0 /T 0. The faster the clock moves through space, the slower it moves through time. Conversely, the slower the clock moves through space, the faster it moves through time. At clock at rest moves only through time (and so at its maximum rate through time, 1). It is as if the clock is always moving through a union of space and time spacetime at the speed of light. From this point of view, qualitatively, a clock at rest will register the maximum time between two (timelike separated) events, and the twin or clock paradox simply reflects that fact that a moving clock will register less time. When forces and accelerations are involved, one does not contradict the principle of relativity by considering just one of the clocks involved to be moving. 11