Lorentz Invariance
In my last post, I showed off an algorithm that could create nicely irregular networks with integer dimension two, without using geometric information. In other words, I made lumpy spheres.
While I’m proud of this result, it doesn’t look much like the kind of spacetime networks that are used in almost all branches of discrete physics. That’s because the dimension of time is missing. And, as anyone who’s read a little Einstein will tell you, time and space are part of the same thing. They can’t really be uncoupled.
Except, of course, they can be uncoupled. It’s dead easy. It’s just that for most of the math that physicists do, it makes more sense to wedge them together.
I’ve briefly outlined in previous posts the way in which space and time can be unpacked from each other. However, after a fun conversation with a very math literate friend the other day revealed to me that I haven’t really done a good enough job of explaining. In this post, I’m going to try to set that straight.
Perhaps the easiest way of describing the relation between space and time that Einstein uncovered is using Minkowski space. In other words, space and time are connected by the following equation:
s^2 = t^2 – x^2 – y^2 – z^2
Where t is time, x, y, and z are the familiar dimensions of space, and s is the ‘spacetime interval’. What this relation says is that if someone is moving toward or away from you, how fast they’re doing it is going to affect how you both perceive time to be passing.
To illustrate this, we can invoke a classic example from science fiction. If you get on a spaceship and make a very fast trip to a distant star (let’s call it Distantia) and back at almost light speed, when you return almost no time will have passed for you. However, for us, it will seem as if decades have gone by. What matters here is who does the accelerating. If you go to Distantia, and a week later I join you there, then we will both perceive the same amount of elapsed time.
What’s chewy about this is that we know from experiment that we have to treat all reference frames as the same. Consider the following scenario: we discover a very fast moving planet in the heavens–called Speedia. Aliens from Speedia decide to visit the Earth. They show up, chat for a while, and then head home. From the perspective of the people of Speedia, the same relation should hold as did for our trips to Distantia. In other words, their travelers should stay young while the homebodies get creaky. (They should also see themselves as still, and everyone else as travelling fast.)
The way to fix this is to invoke some high-school algebra and move the terms in our equation about. In other words, we reframe the relation between space and time as follows:
t^2 = s^2 + x^2 + y^2 + z^2
At first, it’s not clear that we’ve done anything here. It’s the same formula as before, just the other way around. But let’s ask ourselves what the terms in this expression actually mean. What is a spacetime interval? What is the Minkowski relation actually saying?
It’s saying that for a pair of travelers approaching me from the same place, that the subjective experience of time needed to reach me will depend on how fast they’re going. In other words, what the spacetime interval defines is the experience of subjective time for those travelers, as taken from my perspective.
For relativity specialists out there, this may seem obvious. It may seem like I haven’t said anything yet. But here’s the kicker–once you’ve framed things this way round, you can pick a frame of reference for t and describe all the others in terms of it. In other words, so long as we have a way of encoding distance traveled in the s direction and if we maintain a fixed relation between the distance traveled in the s direction and the distance travelled in x, y, or z, we can describe everything in ordinary Euclidean space. (Note that the fixed relation here is key!)
Another way to think about this is that by turning the normal formula for spacetime around, we’ve created an external reference frame. Let’s call it Father Christmas’s reference frame. Nobody in the universe has access to FC’s frame. As far as they’re concerned, Minkowski space works are usual. All frames of reference are still equal and the math is exactly the same. Only FC can see this special view of the universe, which is handy as he needs to visit a very large number of chimneys very quickly and surprise everyone at Christmas.
The awesome thing for Father Christmas is that the universe has an unambiguous, objective geometry that encompasses everything that’s going on, and has this natty extra dimension s. For FC, creating a discrete model of spacetime is a breeze. He just divides everything up into a locally connected network. End of problem.
But hang on, we can’t see a direction s. And we haven’t detected it in any experiments! So how can I claim that this is a solution to the problem of relativity? It turns out that the physicists solved this for us years ago when they came up with something called Kaluza-Klein theory. The trick is that we roll the s direction up very tight into a little circle so that it’s invisible to us, but important at small scales. Sound familiar? It should do, this is exactly the trick that they use to make String Theory work. In fact, there’s exactly nothing new here. What I’m describing is old physics. If you can’t believe in a compact direction for s, you have to throw String Theory away too!
From a discrete physics perspective, this trick is super-useful. This is because it means that so long as an object can only travel a fixed distance with each objective time step, special relativity will hold as long as we add a hidden Euclidean dimension. I’ve tested this and it works. For those of you who like videos, here’s a small demonstration. The flashing of each blob represents the time its experiencing. Note how slow blobs flash quickly, and fast blobs hardly flash at all. If you make the blobs send messages to each other at the speed of light, everything pans out just as Einstein would have predicted. (The video is a special superluminal Christmas treat, because you’re viewing it from FC’s reference frame.)
Note that this only works for simulations that are isotropic (the same in all directions). This means that, unless you’re being super clever, the same trick can’t work for cellular automata.
So where does this leave us? With a really nice tool for making thorny spacetime problems go away. However, we still need to build networks that have the extra magic direction s, and it still needs to magically relate to subjectively experienced time. The network we started off doesn’t have that direction, and we don’t have a way to encode the experience of particles, so lots more work is needed. But in the next post in this series, I’ll show you how to pull these tricks off too.
(By the way, if this post still doesn’t make a shred of sense, somebody please let me know and I’ll try again.)
How to Build A Universe 
I still don’t understand how time can be curled up into a little loop – wouldn’t this mean that the time dimension would keep looping round and round, returning back to the past? Why can’t it simply move off in a direction that we can’t see, but that we can sense through our experience of duration.
I still don’t fully grasp spacetime. It seems like a trick – “how are we going to relate space and time? Lets invent spacetime, and say that time squared equals spacetime squared plus space squared)”. You might as well say, “how are we going to relate zebras and tigers? Lets invent zebratigers, and say that zebras squared equal zebratigers squared plus tigers squared”. But that’s probably because I haven’t spent enough time (or spacetime?) on understanding the problem.
I’ve tried to visualise this space+time in a discrete network as follows: imagine space is a 2-D graph (for ease of visualisation) in the xy plane – lets say it’s all points (x,y) where x is in {0,1,2,3,4,5} and y is in {0,1,2,3,4,5} and x1y1 is connected to x2y2 if they’re only one unit apart, and then you have the same graph stacked upwards in the z direction. And z represents time. So then a given spacetime point is (x,y,z). Then you restrict the maximum distance you can move in the xy graph to something – for example you can only move from (3,4) to (3,5) if you also increase z by at least 1 – kind of a speed of light thing.
But I still don’t see what the discrete Minowksi relation would be for this network..
Also, is time going to be discrete in your model? And if so, what’s the shortest period of time possible going to be?
Hi Keir,
Theses are terrific questions. Thank you for being so straight-up about the things in my post I wasn’t clear about. I’m going to try to address as many of them as possible in the next post. Here, though, are some quick answers.
Re ‘a curled up time dimension’: What I’m proposing here is that the act of traveling in the secret direction s corresponds to the experience of subjective time, not that the time dimension itself is curled up. While this may seem like a pretty abstract distinction, hopefully the next post will make it clearer. (I don’t actually believe that there’s an extra dimension curled up this way. It’s just a handy mechanism that enables us to see what other kinds of physical model would give us relativistic effects.)
Re ‘grasping spacetime’: I think the problem should feel pretty clear. If it’s not, it means that those describing spacetime, myself included, haven’t done a good enough job yet. The whole notion of gluing space and time together into one medium is indeed a trick. However, it’s the only trick we know that adequately describes the weird time dilation effects we see in experiments. To my mind, it’s a trick that people have become a little overdependent on.
Re ‘visualizing space and time’: You’ve convinced me that I should put together some diagrams.
Re ‘is time going to be discrete in your model?’: Yes. Objective/Santa Claus time is discrete, and so, therefore is time for everyone else. Although the way in which time stretches for different reference frames means that it’s likely to be extremely hard to measure. Once again, diagrams will come in handy. And as for the shortest time interval, that’s likely to be the Planck-length divided by the speed of light. In other words, a very short time indeed.
I could always “crank the handle” when tackling GR questions during my maths degree days, and get the right answer, but I never saw what the bearing was of the formulas being applied on any natural phenomena. Mind you, I often had that problem (Laplace transforms were another – I just learned how to do them correctly, and when to use them, but they didn’t make any sense). So I could never formulate my own problems and apply the maths – I just learned when they were needed and manipulated them accordingly in order to pass exams.
But enough about me, and back to the post – in the equation what’s being said is that time is a space vector with an imaginary coefficient, so when squared it’s negative. And then this “s” component is introduced with the right units so in some kind of “space squared plus time squared” Pythagorus theorem, the square of the distance travelled plus the square of the time taken equals the square of this spacetime?
But what does it “mean”?
I like the question: “What does it mean?”, and I have a hand-wavey answer for you. In fact, this question is probably deep enough that it deserves its own blog post. (I’ve started writing it.) I think this question forces us to ask, ‘what is physical math for in the first place?’
Digging deeper into this spacetime stuff and doing some dimensional analysis on the equation. velocity = distance/time. But if time is i*c*d, then velocity is dimensionless. This also doesn’t make sense to me.
Okay, I made a mistake in the dimensional analysis, and I think I now get it. You’re dealing with distances the whole time. If a stationary “event” occurs in space and time relative to some observers frame, with the observer being at the origin, then the space and time position of the event is given as (x,y,z,ict) in the Minowski metric. Therefor the “distance” to the event is the square root of x^2+y^2+z^2+(ict)^2, and the last term becomes -c^2t^2. Because the speed of light is in there, the last term has the same dimensions as the previous three (distance squared), but the sign is reversed. Call this “distance” to the event s, for spacetime, and you can start looking at what it means.
If x^2+y^2+z^2 is bigger than (ict)^2, then s is real, as light has time to get from the event to the observer and the spacetime interval is described as spacelike.
If the two are equal, then they are exactly separated by the speed of light, i.e. the information of the event can only just be observed at the origin of the frame at some point (or perhaps only just not observed), and the spacetime interval is described as lightlike.
If x^2+y^2+z^2 is small than (ict)^2, then s is imaginary, the event lies outside the observable space, and the spacetime interval is described as timelike.
So the equation allows you to determine whether observers at two different points in space and time can interact.
That sounds about right to me, except that you may have your space and time likenesses mixed up. An interval is timelike if there’s more time involved than space. In other words, if it’s somewhere I could get by moving. Conversely, an interval is spacelike if you can’t get there no matter how fast you try to go.
Maybe this is what you meant, but here’s a nice link anyway.
Click to access Relativity_4.pdf
And yes, the whole thing is meant to tell you whether two things can be causally connected or not.
For those of a more digital persuasion, some of the introductory papers reachable via the Causal Sets Wikipedia page provide a lovely, and very approachable, way to think about this stuff.
http://en.wikipedia.org/wiki/Causal_sets
This might also give you a way to think about the whole business of spacetime that feels more intuitively reasonable.
The first document helps a lot. I wish I’d had access to it in 1990. Thanks for the pointer.
The second requires a lot more study on my part, which I unfortunately will not have time for in the near future.
An alien from Fatlia visits and hands you one of a pair of walkie talkies so you can keep in touch. If you try to use your handset when the mute button on the other handset is activated, it will immediately express a busy signal. The time between when you press the talk button and when the busy signal sounds is constant, no matter where the other handset is located in space.
Strangely, neither special relativity nor causality can be violated using this pair of walkie talkies.
Make sure your system is ultimately consistent with this (or it won’t be consistent with reality 😉
Are we talking about quantum-entangled walkie-talkies here, or is Fatlia generally pronounced with a space between the syllables?
It’s just a walkie talkie, you don’t know how it works. Its behavior is observable. It can’t be using Quantum Entanglement, because that shit don’t work.
The Causal Sets article seems to ignore the primary “consequence” of Relativity, that the poset is observationally a non-chain (“anti-chain”, but I hate that locution). The non-chain buys you easier acceptance of local finiteness (even though that “consequence” of Relativity is a fudge), in fact between (Einsteinian) comparable elements there is a finite chain, I think that might even be literally true. Of course the big bang and the big crunch are comparable, that’s how we know they are irreal.
A lot of it reads like people think partial and non-chain are the same thing.
Even though nearly everything about the theory is wrong… big step up from Classical Physics! Keep exploring 😉
If the Wikipedia article isn’t doing it for you, I’d take a look at some of the papers in the reference section. A lot of them are very approachable and quite interesting. While the Causal Set program still has plenty of ground to cover, I wouldn’t rule its validity out just yet. It’s still the most fully discretized model in the mainstream and there are some really first-rate people working on it.