CAUSALITY AND SPECIAL RELATIVITY
An important step in the journey towards the Plexus is to understand why the conventional model of wave-function collapse is invalid. This will lead to the Many-Worlds Interpretation (which, in turn, we shall find is flawed, but it is a milestone that we have to visit on our journey). The concept of wave-function collapse itself is discussed in another page. Meanwhile, to see where the concept fails, you need to know about causality in the context of special relativity. That is what this page sets out to explain. The important result of the page is in the last paragraph in italics.
Figure 1 is essentially Figure 12 of the Block Universe page, except that now I have added a line connecting U and Q. Remember that a radio signal sent by Bob from P reached Q at the same moment (from Bob’s perspective) that a radio signal sent by Alice from S reached him at U. (This will be clearer if you go back to the Block Universe page.) Just as these events at Q and U happen at the same time for Bob, a line drawn through the two points connects all moments that happen at the same time for Bob. We can see from the geometry of Figure 1 that the angle that QU makes with the horizontal is the same as the angle to the vertical made by the the line PU charting Bob’s movement towards Alice. Just as the horizontal and vertical green lines are Alice’s space and time axes, so the slanted purple lines are Bob’s space and time axes.
You can use the same argument from the block-universe page to show that this slanting, or scissoring, always happens to the axes of a person moving with respect to a stationary observer (represented by Alice here). The scissoring is always symmetrical about the 45° light line. In Figure 2 you can see that Bob’s axes face the other way when he moves in the opposite direction. In Figure 2, I have drawn the origin of Alice’s axes coincident with that of Bob’s axes. This means that they are together at the same moment at O in the diagram, and they can both set their clocks to zero, before Bob moves off to the right.
I have drawn Figure 3 just to emphasise the point that everything above Alice’s space axis is in her future. You can see that the sunburst event at A happens at a distance x from Alice and at a time t. Since t is above the space axis, its value is positive, not negative, which means that the sunburst happens at a time t after Alice sets her watch to zero. In other words, event A is in Alice’s future.
In Figure 4, I have removed Alice’s space and time axes, leaving only Bob’s. The sunburst event at A is at a distance x’ from Bob’s space axis and at a time t’ below his time axis. I have labelled the space and time coordinates of the sunburst event for Bob with a superscript because they are not the same as for Alice. You can work out the exact values of Bob’s coordinates from Alice’s using the ideas of special relativity in the same way that we derived the 500 ns discrepancy in the Block Universe page. However, all I want to show here is that, for Bob, the time coordinate of the sunburst is negative, because it is below his space axis. In other words, event A happens before Bob’s watch is set to zero – it is already in his past when he sets his watch.
It is important to remember that we are talking about an event that is happening now for Bob, and not later, when he actually sees the event. It’s the same as saying that the Sun has just exploded – we won’t actually see that event until eight minutes later, but it may well be happening now!
In Figure 5 I have reunited Alice’s and Bob’s coordinates, still keeping the sunburst event at A. The line OA makes an angle theta with Alice’s space axis. As long as this angle theta is less than the angle that Bob’s space axis makes with the horizontal, you will see that A must always be in the past for Bob. Putting it another way, for Bob, event A occurs before event O.
Bob can travel faster, so that the scissors of his axis will close symmetrically on either side of the 45° light-line, but, while the angle theta remains less than the angle that Bob’s space axis makes with the horizontal, event A occurs before event O for Bob.
So this means that we can arrange for the angle theta to be anything up to 45° and still we shall be able to find a reference frame in which Bob will see event A occur before event O simply by ensuring that Bob travels fast enough that his space axis always makes a slightly greater angle than ϕ with Alice’s space axis.
Now imagine that, in Alice’s reference frame, the event at O has somehow caused the event at A to happen. For example, the event at O might be the emission of a neutron which is then transported very fast to A where it is inserted into the nucleus of the uranium-235 isotope which then splits (fissions), leading to a chain reaction observable as a nuclear explosion.
For this to happen, the speed at which the neutron is transported to A has to be faster than the speed of light – we can see this because of the way Alice’s axes are drawn. The faster something travels, the more its time axis is tilted away from the vertical. At the speed of light, it is tilted by 45°. To get to A from O, which is an extreme tilt, it must be travelling much faster than the speed of light. In general, as long as the angle theta is less than 45°, communication from O to A must exceed the speed of light.
So, putting all this together, whenever the angle theta is less than 45°, that is, whenever the communication between O and A is faster than the speed of light, we can always find a reference frame in which Bob will see event A occur before event O. In other words, if we can communicate faster than light between O and A, then there are always reference frames in which the explosion happens before it has been triggered by the neutron!
Patently, the universe cannot be constructed in such a way that causality can be so flagrantly broken. The resolution, of course, is that there can be no communication in our universe that is faster than light.
Just to be clear: there is no paradox in Alice recording event O happening before event A while Bob records event O as happening after event A – provided that A is not caused by O. In principle, Bob can always travel fast enough that, in his reference frame, the order of the events changes. What we can say, though, is that in such cases, the two events will always be so far apart that one cannot be caused by the other, because even light would not have time to travel between the events.
So, to put it slightly differently, if two events are separated in space so far apart that one cannot cause the other even using information travelling at the speed of light, then you can always find two frames of reference in which the events happen in a different order. I have put this in italics because, as we shall see in the Wave Function Collapse page, this will invalidate Bohr’s model of wave function collapse.
Click here to go to [5] Quantum Spin Filter page