Wibbly-wobbly timey-wimey... stuff

Disclaimer: this post quickly becomes quite technical, but it could give some extra STR intuition to those brave enough to stick with it.

I'll try to explain relativity geometrically. This is actually a very common approach, because space and time themselves are best understood geometrically. As discussed in the 'introduction', these notions are intricately related to our experience: our perceptions tell us that things can be in different places and can happen at different moments, the collection of which we call space-time. Very often, the need arises to quantify the places and moments in order for us to, you know, function as humans and as a society.  In our day-to-day lives, we go about this in an informal way, choosing just the right amount of vagueness that is needed for  particular situation. 

The formal (i.e. boring?), scientific way to describe positions is to use a system of coordinates, and the most commonly used one is Cartesian coordinates, introduced, just like the evil demon, by René Descartes. In two dimensions, these look in this probably familiar way:

What I mean by two dimensions is that the space described by the example above (e.g. the plane of your monitor) is such that any location can be precisely pinpointed by two numbers. These could be the x- and y-positions in the Cartesian frame above, but the concept of dimensionality is much more general. In fact, we are intuitively used to a very complex two-dimensional space, since we are tied by gravity to the surface of the Earth [1]. This surface has an extremely complicated shape, which the x-y plane above cannot capture well, but it is still two-dimensional, in the sense that any location can be uniquely defined by two numbers, as anybody who's used GPS coordinates knows. So 3D beings as we are, we do anyway spend our lives tied down to what is, to a good approximation, a 2D world. 

The Cartesian coordinate system seems very natural. Right angles make sense. Well, I don’t know if it is a priori aesthetic - in fact I think that a priori aesthetic is an oxymoron - but we humans definitely seem to like it a lot. We even like to organize our cities in this way. There's not much difference between saying '42nd and 5th' and saying 'at x = 3.2, y = 2', or something.

Now, to illustrate relativity, we're going to decrease the spatial dimensions to one. That means that we’re only going to keep the x-axis. It also means that anything living in this toy 1D world can only be in front or behind anything else. The reason for this simplification is because we want to use the other axis to represent time:

This is now a graphical representation of space-time, and already hints at the fact that those are kinda like different aspects of the same thing. Note that we always had some intuition that time is very similar to space, or in any case certainly before Einstein or Descartes. This can be seen in our language. Interestingly, there are different ways in which different cultures view time.  What is practically ubiquitous among cultures, however, is the fact that the words we use to speak about time are very similar to the ones we use for space: there is a conceptual metaphor of time as a path through space. The examples are numerous. Plenty of time ahead of us. At 8am. In French one even says, 'le moment où quelque chose s'est passé', which literally means 'the moment where something happened.' 

An important remark is due. The space-time reference frame shown above is neither unique nor absolute. It is in fact observer-dependent, so you should think of everyone as having one of those attached to them.

The Universe itself consists of events which are absolute in the sense I discussed here: all observers agree on them happening, but, to do that, every observer relies on their own reference frame. This is what this looks like.

This way of talking about events by naming both their spatial and temporal position is perhaps a bit awkward, but you’d better get used to it if you want to understand relativity. Space-time should always be thought of as indivisible.

The example above, where two observers at a different place compare their observations of an event is fairly straightforward to understand. It's also possible to compare reference frames at different times. This is slightly less intuitive, but still nothing too complicated.

In both cases, to see whether the observers agree, they simply have to properly add or subtract the distance between them (both in space and in time) from their corresponding observations. This transformation of the reference frames is mathematically known as a translation (technically, not related to language, but somehow this connotation is also appropriate here). All this was well understood already in pre-relativity physics. The main innovation of STR is what happens to the reference frames of observers that are moving with respect to one another. This is what we used to think the transformation looked like:

The two reference frames above correspond to two observers moving with respect to one another, which at t = 0, x = 0 (in both frames) find themselves at the same time and place. This is a graphical representation of what is known as a Galilean transformation. The fact that the time axis tilts while the space axis doesn't is deeply connected to our (wrong) understanding of time as something that is absolute for everybody. Consider this. If we have two events which happen at the same time with respect to observer one, then they also happen at the same time for observer two. 

This shared simultaneity of events means that we can come up with a common time-counting scheme for the two observers, such that t = t' for all events. This would represent the absolute-ness of time. However, if two events happen at the same place for observer one, they don’t happen at the same place for the second guy: space is relative. 

To see what the situation looks like in relativistic physics, let’s first take care of a small technicality. In measuring and plotting x and t, we have to choose some units. We actually have some freedom in that, and the units we choose affect the scale of the axes. Let’s say we take the standard choice of measuring time in seconds. A standard choice for measuring positions is the metre, unless you have the misfortune to come from one of the three countries in the world where it isn't. For the purposes of relativity, however, it is particularly convenient to choose a different unit: the light-second. This is analogous to the light-year (which we would use if we chose to measure time in years instead of seconds) in that it’s defined as the distance that light travels for one second. With this choice of units, the reference frames for observers moving with respect to one another turn out to look like this:

We chose to use the 'proper' measurement units so that everything is nicely symmetrical. Physicists like stuff being symmetrical. The 45-degree line bisecting the reference frames marks the propagation of a light beam, which should be thought of as a series of absolute events. Note that this line bisects both reference frames. As opposed to the Galilean transformation above, in this relativistic (and correct) transformation, the x- and the t-axes of the moving reference frame tilt towards the light-line, at the same angle. Faster speeds mean a larger tilt, and in the limit of the speed of Observer 2 going to the speed of light, the two axes merge:

With this in mind, we can see where all the 'weird' stuff in relativity comes from. For example, simultaneity can no longer be defined in an absolute (i.e. observer-independent) way. Events that happen at the same time for one observer can happen at different times for another one:

Effects like length contraction and time dilation are very strongly related to the fact that simultaneity (and thus time) is relative. Those can also be inferred by looking at the graphs above and thinking a bit about how lengths and time intervals translate from one observer to a second, moving one. This is left as an exercise to the reader. *EVIL LAUGH*

The title of this post comes from a famous quote by the Doctor, namely, 'People assume that time is a strict progression of cause to effect, but actually from a non-linear, non-subjective viewpoint - it's more like a big ball of wibbly-wobbly timey-wimey... stuff.' One thing that Einsten always insisted upon, however, was kinda the opposite: the absolute nature of causality. In other words, he did subscribe to the assumption that the Doctor claims is wrong: the idea that cause always precedes effect. This is what is known as causality, and is usually held as a fundamental principle of nature. An illustration from everyday language is when we say that you cannot eat your cake and have it whole; this is because the cause - eating the cake - results in the effect - you no longer having it whole. Einstein made it clear that although some funky things happen according to STR, nothing as funky as eating a cake and having it whole can ever happen, in any reference frame. Pictorially, this looks like this.

It’s clear that regardless of the exact velocity of the second observer, the world-line of the cake always precedes the event of your eating it. Or does it? Remember how the faster the observer moves, the closer their x and t axes come to the light-line, until they practically merge? Well, if we let an observer move faster than the speed of light, and neglect a small imaginary detail (we can, cause it’s imaginary, right?), what happens is what you should expect to happen - the angle of the tilt of the axes continues to increase, and they switch places! The space-axis comes above the light line, while the time-axis pops up below. The crazy thing is that if you now map the cake, in this new reference frame the event of eating it precedes the series of events of having it whole:

What is perceived as cause becomes an effect, and vice versa: according to this super-luminal observer, people eat their cakes in order to make them! And then I guess they enjoy them in a moment of un-cooking that succeeds both eating the cake and having it whole. This sounds like nonsense, and is one of the arguments against the possibility of anything moving faster than the speed of light. But this is a philosophical argument: should we dismiss something just because it sounds like nonsense? There is another strong argument about why we ourselves could never achieve such speeds: the acceleration would require infinite energy, and then some more. But there is no conclusive argument against the existence of matter that is already moving faster than the speed of light. In a similar fashion to us, such matter would not be allowed to decelerate to below that speed, as that would require more than infinite energy. Still, such matter could in principle exist, and, a few decades ago, the theoretical study of tachyons - i.e. particles moving faster than the speed of light - was very hot.

The enthusiasm has by now somewhat died out, though. Even though we cannot prove them impossible, most physicists don't consider the existence of tachyons very likely. More precisely, the possibility to observe these particles even if they do exist is considered unlikely, which is practically an equivalent statement. That's some food for thought for you: in terms of physics, is there any difference between something not existing, and something existing but not interacting with us?

Einstein would have agreed that tachyons cannot be a part of our world. Basically, interacting with such particles would immediately break causality and result in a ton of paradoxes of the kill-your-grandfather type (and the even more disturbing Futurama version). Again, this is technically not a proof of the impossibility of backwards-in-time travel. But, while strictly speaking nothing can be proven impossible, some things just look mighty improbable. In any case, Einsten considered the cause-and-effect realtionship a fundamental principle of nature, and its breaking - impossible. In fact he was much more concerned with that than with determinism, despite his famous 'God does not play dice' quote. And I must say that once again I find it easy to agree with the great man. I would say that it is, in fact, safe to assume that time is a strict progression of cause to effect, at least until we have seen even the tiniest reason to think otherwise. Which, right now, we haven't.

Bottom line, while Einstein's theory does illustrate that time is wibbly-wobbly (i.e. not absolute), I don't think he would've been much of a fan of the Doctor's dismissal of causality.