2.1  The Spacetime Interval

 

         …and then it was

There interposed a fly,

With blue, uncertain, stumbling buzz,

Between the light and me,

And then the windows failed, and then

I could not see to see.

                             Emily Dickinson, 1879

 

Although the theory of relativity is usually regarded as a classical theory, separate from and independent of quantum mechanics, many of its features are closely related to key aspects of quantum mechanics, and these features turned out to be essential for the development of quantum field theory. For example, as mentioned in Section 1.6, in special relativity the speed of light (in terms of any inertial coordinate system) is independent of the speed of the source, which is characteristic of a classical wave conception of light, but light also propagates isotropically in the rest frame of the source, which is characteristic of a particle conception of light. The wave-like and particle-like behavior of light in quantum mechanics could never have been reconciled in any context other than special relativity. Likewise the fact that the energy and frequency of a pulse of light transform in exact proportion to each other in special relativity is necessary to provide a coherent framework for the basic quantum relation E = hν. Fundamentally, in quantum field theory, we find that the phase of the quantum wave function of any physical entity or system passing uniformly from the event (t,x,y,z) to the event (t+dt, x+dx, y+dy, z+dz) is proportional to the value of dτ given by

 

 

where t,x,y,z are any system of inertial coordinates and c is a constant (the speed of light, equal to 300 meters per microsecond). Of course, special relativity teaches us that the quantity dτ is just the proper time for the interval between any two given events, which is invariant with respect to any system of inertial coordinates, consistent with the fact that such systems are related by Lorentz transformations. Since this invariance applies to the processes underlying the operation of ideal clocks (i.e., clocks corrected for any internally sensible effects, such as temperature, acceleration, etc.), it follows that dτ is the elapsed proper time for a characteristic process of any physical system, i.e., it represents the time that would be measured by an ideal clock co-moving with that system.

 

To illustrate, consider a muon, which has a radioactive half-life of roughly 2 μsec with respect to its inertial rest frame coordinates. In other words, between the appearance of a typical muon (arising from, say, the decay of a pion) and its decay, there is an interval of about 2 msec in terms of the time coordinate of the muon's rest frame, so the time and space components of this interval are (2,0,0,0), and the quantum phase of the particle advances by an amount proportional to Δτ, where

 

 

Now suppose we assess this same physical phenomenon with respect to a relatively moving system of inertial coordinates, say, a system with respect to which the muon moved from the spatial origin [0,0,0] all the way to the spatial position [980m, –750m, 1270m] before it decayed. With respect to these coordinates, the muon traveled a spatial distance of 1771 meters. Since the advance of the quantum wave function (i.e., the proper time) of a system or particle over any interval of its worldline is invariant, the corresponding time component of this physical interval with respect to these relatively moving inertial coordinates must be much greater than 2 μsec. Letting (ΔT, ΔX, ΔY, ΔZ) denote the components of this interval with respect to the relatively moving system of inertial coordinates, we must have

 

 

Solving for ΔT and substituting for the spatial components noted above, we have

 

 

This represents the time component of the muon decay interval with respect to the moving system of inertial coordinates. Since the muon has moved a spatial distance of 1771 meters in 6.23 μsec, we see that its velocity with respect to these coordinates is 284 m/μsec, which is 0.947c.

 

Likewise, the half-life of a muon moving at 0.947c in terms of our original inertial coordinate system is 6.23 μsec. In this case people sometimes ask how we know that the decay time for the moving particle is actually 2 μsec in terms of the inertial coordinate system co-moving with the particle. The answer, as noted above, is that inertial coordinate systems are defined in such a way that they conform to the manifest Lorentz invariance of all entities and processes. For example, a cesium clock is slowed by the same factor as the muon – and by the same factor as every other physical process. If this were not the case, it would be possible to detect differences in the characteristic times of physical phenomena at rest in different frames, which would imply that relativity is violated and we could detect absolute motion. But careful measurements of various physical phenomena in different states of motion (e.g., at different times of the year, when the Earth’s orbital motion is in different directions) have always failed to reveal any violations of relativity. Thus a clock co-moving with the muon does indeed measure a decay time of 2 μsec.

 

The identification of the spacetime interval with quantum phase applies to null intervals as well, consistent with the fact that the quantum phase of a photon does not advance at all between its emission and absorption. (See Section 9.9.) Hence the physical significance of a null spacetime interval is that the quantum state of any system is constant along that interval. In a sense, the interval represents a single quantum state of the system, so (for example) the emission and absorption of a photon can be regarded as, in some sense, a single quantum act.

 

Naturally the quantum phase is path dependent. Two particles at opposite ends of a lightlike (null) interval certainly do not share the same quantum state unless the second particle reached that event by passing along that null interval. This avoids conflict with the Pauli exclusion principle for fermions such as electrons, because even though two electrons can be null-separated, they cannot have separated along that null path, because they have non-zero rest mass. On the other hand, it is possible for photons at opposite ends of a null interval to represent "the same photon", but photons are bosons, and hence not excluded from occupying the same state. (In fact, the presence of one photon in a particular quantum state actually enhances the probability of another photon entering that state, which is responsible for stimulated emission and lasers.)

 

Neutrinos (like electrons) are fermions, meaning that they have anti-symmetric eigen functions, so they are subject to the Pauli exclusion principle. Nevertheless, for many years neutrinos were thought to be massless and hence to propagate along null intervals. One might argue that this violates the spirit of the exclusion principle, if multiple instances of a neutrino constitute a single null interval, all sharing the same quantum phase. However, we now know that neutrinos actually do have mass. (In theory, the Dirac equation has a massless solution, called a Weyl fermion, but no such fundamental particle – as distinct from quasiparticle – has ever been observed.)

 

Based on the identification of the invariant magnitude (i.e., proper time) of a timelike interval with the quantum phase along that interval, it follows that all physical processes and characteristic sequences of events will evolve in proportion to this quantity. The name "proper time" is appropriate because this quantity represents the most meaningful known measure of elapsed time along that interval, based on the fact that the quantum state is the most complete possible description of any physical system. Since not all spacetime intervals are timelike, the temporal relations between events induce only a partial ordering, as discussed in Section 1.2. A set of events can be totally ordered only if they are each inside the future or past null cone of each of the others. This doesn't hold if any of the pairwise intervals is spacelike. As a consequence of this partial ordering there exist timelike paths with different lapses of proper time between two fixed timelike-separated events.

 

Admittedly a partial ordering of events has been considered unacceptable by some people, who regard total temporal ordering in a classical Cartesian setting as an inviolable first principle. Rather than accept partial ordering, they prefer to imagine that one particular inertial reference system is the "true" one, as in Lorentz's original theory, and thereby asert an unambiguous total temporal ordering to events. They then account for the apparent differences in elapsed time (as in muon observations) by regarding them as effects of absolute velocity relative to the "true" frame of reference, again following Lorentz. However, unlike Lorentz, we now have a well-established theory of quantum mechanics, and the quantum state of a system gives (arguably) the most complete possible objective description of the system. Therefore, modern advocates of total temporal ordering face the daunting task of finding some mechanism underlying quantum mechanics (i.e., hidden variables) to provide a physical significance for their preferred total ordering. Unfortunately, the only prospects for a viable hidden-variable theory seem to be things like the explicitly non-local contrivance of David Bohm, which must surely be anathema to those who seek a physics based on classical Cartesian mechanisms. So, the success of quantum mechanics constitutes one of the strongest arguments for the relativistic interpretation of Lorentz invariance.

 

In order to identify the proper time interval with the reading of an ideal clock moving along that interval, it must be the case that two momentarily co-moving clocks keep time at the same instantaneous rate, even if one is accelerating and the other is not. This might be regarded as just a hypothesis, but another way of expressing this "clock hypothesis" is to say that an ideal clock is one that is corrected for acceleration and any other internally sensible effects (e.g., temperature), and to regard this as the definition of an "ideal clock". The physical significance of this definition arises from the empirical fact that acceleration is absolute, and therefore perfectly detectable (in principle). In contrast, velocity is empirically undetectable, which explains why we cannot define our "ideal clock" to compensate for velocity (or, for that matter, position). Thus relativity rests on both of the assumptions: (1) the zeroth and first derivatives of position are perfectly relative and undetectable, and (2) the second and higher derivatives of position are perfectly absolute and detectable. Most treatments of relativity emphasize the first assumption, but the second is no less important.

 

The notion of an ideal clock takes on even more physical significance from the fact that there exist physical entities (such a vibrating atoms, etc.) in which the intrinsic forces far exceed any accelerating forces we can apply, so that we have in fact (not just in principle) the ability to observe virtually ideal clocks. For example, in the Rebka and Pound experiments it was found that nuclear clocks were slowed by precisely the factor (1–(v/c)2)1/2, even though subject to accelerations up to 1016 g, which is huge in normal terms, but still small relative to nuclear forces.

 

It was emphasized in Section 1 that a pulse of light has no inertial rest frame, but this may seem puzzling at first. The pulse has a well-defined spatial position at each time with respect to some inertial coordinate system, and a fixed finite velocity c relative to that system, so one might think we could simply transform to another system of inertial coordinates in terms of which the pulse is at rest. It is certainly possible to define a system of coordinates in which a light pulse in vacuum is at rest, but it cannot be an inertial coordinate system in the full sense described in Section 1.3. The reason is that the transformations between inertial coordinate systems (i.e., Lorentz transformations) are singular for the velocity c, so there is no inertial coordinate system moving at speed c relative to any inertial frame. The singular behavior of the transformation corresponds to the fact that the absolute magnitude of the spacetime intervals along lightlike paths is null. The singularity in the inertial coordinates at the speed c suggests that the conception of light as an entity in itself may be somewhat misleading. It is often useful to regard light as simply an interaction between two massive bodies along a null spacetime interval.

 

Discussions of special relativity often refer to the use of clocks and reflected light signals for the evaluation of spacetime intervals. For example, suppose two identical clocks are moving uniformly with speeds +v and –v along the x axis of a given inertial coordinate system, and these clocks are set to zero at the intersection of their worldlines. When the leftward reaches event A it emits a pulse of light, which bounces off the rightward clock when that clock is at event B and arrives back at the leftward clock when that clock is at event C, as depicted in the drawing below.

 

 

Letting τOA, τOB, and τOC denote the magnitudes of the intervals OA, OB, and OC respectively, by similar triangles we immediately have τOBOA = τOCOB, and thus τOB2 = τOAτOC. This same relation holds good in Galilean spacetime as well (not to mention Euclidean plane geometry, using distances instead of time intervals), and the reflected signal need not be a light pulse. Any object moving at the same speed (angle) in both directions with respect to this coordinate system would serve just as well, and would lead to the same result that τOB is the geometric mean of τOA and τOC. Naturally if we apply any Minkowskian, Galilean, or Euclidean transformation (respectively), the pictorial angles of the lines will differ, but the three absolute intervals will remain unchanged.

 

It is, of course, possible to distinguish between the Galilean and Minkowskian cases based just on the values of the elapsed times, provided we know the relative speeds of the clocks and the signal. In Galilean spacetime each proper time interval Δτ equals the corresponding coordinate time interval Δt, regardless of the Δx for the interval, whereas in Minkowski spacetime (with c = 1) it equals (Δt2 – Δx2)1/2 where Δx = ±vΔt. Hence the proper time interval Δτ in Minkowski spacetime is Δt(1 – v2)1/2. This might seem to imply that the ratios of proper times are the same in the Galilean and Minkowskian cases, but in fact we have not made a valid comparison for equal relative speeds between the clocks. In this example each clock is moving with speed v away from the midpoint, which implies that the relative speed is 2v in the Galilean case, but only 2v/(1 + v2) in the Minkowskian case.

 

To give a valid comparison for equal relative speeds between the clocks, let's consider a case such that the left-hand clock is stationary and the right-hand clock is moving at the speed v. This v represents the magnitude of the actual relative speed between the two clocks (i.e., the speed of each clock in terms of the rest frame of the other clock). We also stipulate that the original signal is moving with speed u in terms of the inertial coordinates in which the left-hand clock is at rest, and the reflected signal is moving with speed –u in terms of the inertial coordinates in which the right-hand clock is at rest. The situation is illustrated in the figure below.

 

 

The speed, with respect to these coordinates, of the reflected signal is what distinguishes the Galilean from the Minkowskian case. Letting xB and tB denote the coordinates of the reflection event, and noting that τOA = tA and τOC = tC, we have v = xB/tB and u = xB/(tB –τOA). We also have

 

 

Dividing the numerator and denominator of the expression for u by tB, and replacing xB/tB with v, gives u = v/[1 – (τOA/tB)]. Likewise the above expressions can be written as

 

 

Solving these equations for the time ratios, we have

 

 

Consequently, depending on whether the metric is Galilean or Minkowskian, the ratio of τOC over τOA is given by

 

 

respectively. If u happens to be unity (meaning that the signals propagate at the speed of light), these expressions reduce to the squares of the Galilean and relativistic Doppler shift factors, i.e., 1/(1–v)2 and (1+v)/(1–v), discussed more fully in Section 2.4.

 

Another distinguishing factor between the two metrics is that with the Minkowski metric the speed of light is invariant with respect to any system of inertial coordinates, so (arguably) we can even say that it represents the same "u" relative to a spacelike interval as it does relative to a timelike interval, in order to adhere to our stipulation that the reflected signal has the speed u relative to "the rest frame of the right-hand clock". Of course, a spacelike interval cannot actually be the worldline of a clock (or any other material object), but the invariance of the speed of light under Minkowskian transformations enables us to rationally apply the same "geometric mean" formula to determine the magnitudes of spacelike intervals, provided we use light-like signals, as illustrated below.

 

 

In this case we have τOA = –τOC, so τOB2 = –τOC2, meaning that τOB is a spacelike interval, and that squared spacelike intervals are negative.

 

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