in space-time? The answer is yes, but the invariant quantity is slightly different, because the time axis rotates in the opposite direction to the space axes. We define the space-time interval as
This is the quantity that is invariant under a Lorentz transformation. Note that the Lorentz transformation can be derived by assuming that this quantity is invariant, and that the transformations be linear in the coordinates. When this is done, time dilation and length contraction can be derived from the Lorentz transformation rather than the other way around. This reinforces the idea that length contraction and time dilation are not artifacts of some particular measurement, but are an integral part of the nature of space-time.
To get a feeling for the physical meaning of As, consider an observer moving from one place to another in time At, as measured in the observer's rest frame. This means that At is the proper time interval. In the observer's rest frame, there is no change in position, so Ax = Ay = Az = 0. This means that As = cAt. Therefore, As is just the proper time interval (in units of length). Moreover, since it is an invariant, for any two events and any inertial reference frames As will always equal the proper time interval between the two events.
We can define three types of space-time intervals (Fig. 7.10a), depending on whether (As)2 is zero, positive or negative. Suppose our two events are the emission and absorption of a photon. A photon will move on the sphere whose surface is given by, (Ax)2 + (Ay)2 + (Az)2 = (c At)2. This means that (As)2 is zero for a photon traveling any distance in any direction. We call such intervals lightlike. Intervals for which (As)2 is positive are called timelike. The positions are close enough in space that a photon would have had more than enough time to travel from the first event to the second. This means that the first event could have caused the second. In the opposite case, when (As)2 is negative, we call the interval spacelike. A photon cannot traverse the distance in the time given. Unless a signal can be sent faster than the speed of light, there is no way the one event could have caused the other.
If we extend our space-time diagram to more dimensions, we call the surface defined by (As)2 = 0
the light cone (Fig. 7.10b). Events that could have caused the event at the origin of the cone are inside the past light cone. Events that could be caused by events at the origin of the cone are inside the future light cone. Events that are outside the light cone can have no causal connection with the event at the origin.
It should be noted that in our discussion of a space-time interval we could have defined it to be the negative of what it was in equation (7.12), and not changed any of the interpretation (apart from carrying through the minus sign). It is just a matter of convention to do it one way or the other, and you will find some authors who do it one way and some who do it the other. As long as they are internally consistent there is no problem. People who use one convention or the other then also differ in how they 'count' the time coordinate. That is, we can write (x, y, z) as (x1, x2, x3). If we use the space-time interval as given in equation (7.12), then we write ct as x0 and think of time as the 'zeroth coordinate'; if we use the space-time interval as the negative of that given in equation (7.12), then we write ct as x4 and think of time as the 'fourth coordinate'.
The space-time coordinates of an event are not the only quantities that transform according to the Lorentz transformation. For example, another important four-vector involves energy and momentum. To see the analogy with (ct, x, y, z), remember that for a photon moving in the x-direction, x = ct. The energy and momentum of a photon are related by E = cp, so, for a photon moving in the x-direction, E = cpx. This suggests that the energy-momentum four vector should be (E, cpx, cpy, cpz). These should then obey the Lorentz transformations:
The reverse transformation is given by
E' = y (E - 3cpx) cpx = y (cpx - 3E) cPy = p cpZ = cpz (7.14)
If we let the (') reference frame be one in which the particle is at rest, so that px' = 0, then the first thing we note is that E = y E', so, if the energy of the particle at rest, E', were equal to zero, then the energy, E', would always be zero. This is obviously not the case, since we know that moving particles must have some energy. This means that the rest energy, E0, cannot be zero. So, this gives us an expression for the relativistic energy:
We can then find the relativistic momentum as cpx = yp Eo (7.16)
In the non-relativistic (y close to one) limit, the momentum must give the classical expression, px = mvx . From equation (7.16) this can only occur if
where m0 is the rest mass of the particle.
We can rewrite the expressions for relativistic energy and momentum:
We can also define a kinetic energy as the difference between the total energy and the rest energy:
In the limit v V c, we can write y = [1 - (v/c)2]1/2 = 1 + v2/2 c2
where we have used the fact that, for x V 1, (1 + x)n = 1 + nx. The kinetic energy for v V c then becomes
which is the classical expression.
Since the energy-momentum four-vector obeys the Lorentz transformations, it must have an invariant length associated with it. It is
To give this quantity a physical meaning, we evaluate it in the rest frame of some particle. In that case, the momentum is zero and the energy is the rest energy. So the invariant length is simply moc2. Since this quantity is invariant, its value must be moc2 for any observer. (We just choose to work it out in an easy frame.)
Example 7.3 Rest energy of a proton What is the rest energy of a proton?
SOLUTION By equation (7.20)
To form an idea of how large this is, we express the answer in eV, to get 939 MeV (as compared, for example, with the 13.6 eV needed to ionize a hydrogen atom).
Note that, as v/c approaches unity, y approaches infinity. This means that it takes an infinite amount of energy to accelerate an object with non-zero rest mass to the speed of light. This means that the speed of light is a limiting speed.
Some physicists have speculated on particles that can travel faster than light. These particles have been given the name tachyons. The trick is that these particles, if they exist, can never go slower than the speed of light. The speed of light would seem to be a barrier for them as well, only from above. If tachyons do exist they can interact with photons, and make their presence known. All experiments to look for tachyons have indicated that they do not exist.
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