# Galilean transformations

For GTs we have *a* = *S* = 1, *в* = 0, which gives the line element

The value of the line element is that it gives a condition for light (null) geodesics, that is, we set *ds* = 0. Then we recover the Galilean relativity rule for the speed of light, that is, *c _{R}* =

*c*- v,

*c*=

_{L}*c*+ v, as expected.

# Einstein-Lorentz transformations

For ELTs we have *a* = *в* = *S* = *у*, which gives the standard line element
so setting *ds* = 0 leads to *c _{R}* =

*c*=

_{L}*c*as expected.

# Tangherlini-Chang-Rembielinski transformations

For TCRTs we have *a* = *у*^{-1}, *в* = 0, *S* = *у*, which gives the line element

Setting *ds* = 0 gives
as stated in Table 18.1.

# The splitting of causality

Special relativity in the form developed by Einstein makes a specific assumption about the one-way speed of light that impacts on the transformation rules between inertial frames. In this section we discuss a simple thought experiment that could in principle provide a test of Einstein-Lorentz synchronization (ELS). If ELS synchronization is upheld, then classical causality should appear to be split in an observable way.

The experiment involves signalling between two inertial frames in relative motion. Whenever signalling apparatus and detecting apparatus are not at rest in a common inertial frame of reference (or related by a simple translation or spatial rotation), we shall refer to such an experiment as an *interframe experiment.* Examples of interframe experiments are those involving Doppler shifts, such as in observational cosmology.

Consider an interframe experiment conducted over a finite interval of time, such that signals sent from apparatus *QP* at rest in frame *F* at time *t* = 0 are observed by apparatus *QP'* at rest in frame *F'* at time *T',* as measured in that frame. Figure 18.3 shows the essential details.

In Figure 18.3, event *O* is the common origin of spacetime coordinates, events *P* and *Q* are simultaneous in *F* at time *t* = 0 whilst events P'and *Q* are simultaneous *in F'* at time *t!* = *T' >* 0. We shall use the convention that an

**Fig. 18.3 ****Event B is a quantum (or information) horizon.**

event *P* has coordinates *(t*_{P}, *x*_{p}) in F, coordinates * *'_{P}, *x*'_{P}] in *F',* and we write *P* ~ *(t _{P}*,x

_{P}) ~ [tP,xp]. From Figure 18.3, it will be seen that a critical feature is event

*B,*where the lines of simultaneity

*t*= 0 and

*t*=

*T'*intersect. Assuming the velocity component

*v*is positive, then for

*B*we have

Event *B* is the focus of attention in this experiment. A novel and interesting interpretation of the significance of event *B* can be found from basic quantum mechanics. First, we recall that in de Broglie wave mechanics, the speed *w* of a pilot wave associated with any material physical particle moving with subluminal speed *v* satisfies the relation *vw* = c^{2} [de Broglie, 1924]. This suggests that such a pilot wave cannot be used to convey physical signals, because it travels at superluminal speed, given *v < c.* According to these ideas, event *B* may be regarded by observers in *F'* as the crest at time *T'* of such a pilot wave associated with a particle at rest in frame F, if it were sent out from *O* in the same direction as *F *appears to move. Note that this interpretation of event *B* should be regarded as no more than a mathematical curiosity, because any genuine de Broglie wave would not be localized at a single point. Moreover, the significance of event *O* as the common origin of coordinates is an artefact to do with the choice of coordinates. Nevertheless, we shall argue below that *B* does have something critical to do with quantum processes.

Several circumstances conspire to mask events such as *B* in conventional physics: (i) the speed of light *c* is large on ordinary laboratory scales; (ii) the relative speed | v | is usually small compared to c, (iii) signal detection is limited to a localized region, such as around event P'; and the time *T'* of observation of a signal at *P'* is usually relatively large. In consequence, event *B* will normally be beyond the limits of observation in typical experiments. In the case of the experiment under consideration, however, the architecture is different: non-localized detection is arranged in such a way that *B* is within the space-time region involved in the experiment.

For this experiment, we imagine that a quantum state has been prepared by apparatus Aq_{P}, at rest in frame F, and a contingent quantum outcome subsequently detected by apparatus A!q_{P}, , at rest in frame *F'.*

The critical word here is *‘subsequently*’. Quantum physics, as it is performed in real laboratories, can discuss only the possibility of information travelling forwards in time. Both signal emitter and signal detector in any quantum experiment must agree that the former acts before the latter. Otherwise, the physical significance of the Born probability rule would be completely undermined. In quantum theory and in the real world, we cannot know the outcome of an experiment before it is performed. We shall call the requirement that *P* is earlier than *P* in both frames of reference *quantum causality.*

From Figure 18.3, it is clear that there is no problem with quantum causality as far as events *P* and *P* are concerned. But consider events *Q* and *Q* on the other side of *B.* If quantum causality is valid, then signals prepared at *Q* cannot be received by Q'. In essence, event *B* acts a barrier to quantum causality, and on this account we shall refer to *B* as a *quantum horizon.*

Ordinarily, such horizons are ignored in conventional physics because, under most circumstances, *B* appears to be very far from events such as *P* and *P*. For example, in experiments looking at the transmission of quantum information, speeds in excess of 10^{5} *c* have been reported [Scarani *et al.,* 2000]. In practice, high-energy particle theory conventionally takes the scattering limit *T' ^* X, *v* = 0 in the calculation of scattering matrix elements. Finite-time processes and inter-frame experiments of the sort discussed by us here are generally avoided, presumably because it is assumed that there is no significant novel physics involved. A consequence of this assumption is that this scattering limit simplifies the calculations, because all quantum horizons are at spatial infinity.

We now consider the implications of the relativity principle and ask the following question: if according to the relativity principle frames *F* and *F'* are ‘just as good as each other’, why does the quantum horizon *B* appear to distinguish between the two?

A little thought soon resolves the question. If the relativity principle is valid, then there must be a symmetry between the two frames. There is no doubt that a quantum signal can be prepared at *P* and received at P', if *P* is in or on the forwards lightcone with vertex P. Quantum causality rules out the transmission of a quantum signal from *P* to P, and the transmission of a signal from *Q* to Q'. But nothing currently known in physics forbids the possibility of a physical signal being sent from *Q* to Q, if *Q* is in the forwards lightcone with vertex Q'. Indeed, symmetry demands such a possibility. This is the essence of the split causality experiment proposed here.

Based on the above considerations, we propose the following experimental test of the principle of special relativity. It will undoubtedly be difficult to perform, but would test the principle of relativity in a spectacular and convincing way.

We envisage the use of four spacecraft P, Q, P', and Q', sufficiently far from gravitating bodies to justify the use of the SR transformation rule. *P* and *Q* are in the same rest frame *F* and situated at some distance from each other. By prior signalling arrangement, clocks on *P* and *Q* craft have been synchronized. Likewise, *P* and *Q* are in their own rest frame *F'* and all their clocks have been synchronized.

With reference to Figure 18.3, spacetime homogeneity means that we may always transfer the origin of spacetime coordinates in both frames F, *F'* to the quantum horizon *B*. This means that the hyperplanes of simultaneity involved in the experiment are now at times *t* = 0 in *F* and *t* = 0 in *F',* as shown in Figure 18.4.

The experiment consists of *P* sending a brief light pulse signal towards *P* at time *t* = 0, whilst simultaneously in *F*, *Q* opens a detector in order to receive

**Fig. 18.4 ****O is a quantum horizon. Shaded regions are forwards lightcones. Arrowed signals shown with subluminal transmission speeds, which do not alter the overall conclusions.**

light from *Q* for a similar brief period. In addition, the same protocol is carried out in frame *F'* at time *t* = 0: *Q* sends a brief light pulse towards *Q* whilst simultaneously in *F', P* opens a detector to receive a signal from P. The whole experiment is illustrated in Figure 18.4.

After the signals have been sent and received, observers from all spacecraft can meet at leisure and compare results. If it turns out that *P* sent a signal at the same time *t* = 0 that *Q* received a signal, and that *P* received a signal at the same time *t'* = 0 that *Q* sent a signal, then ELS would be upheld. If not, then Tangherlini synchronization with all its implications would be preferred. If ELS holds, then quantum causality would have been respected but classical causality would appear to be split in a remarkable and counterintuitive fashion. On the other hand, if no such result was ever detected despite repeated attempts, this would rule out ELS, with implications for physics.