Deriving the Lorentz transformation

My last few posts have been all about visualizing the Lorentz transformation, the coordinate transformation in special relativity. But where does this transformation come from? In this post, I’ll derive it from basic principles. I saw this derivation first probably a year ago, and have since tried unsuccessfully to re-find the source.  It isn’t the algebraically simplest derivation I’ve seen, but it is the conceptually simplest. The principles we’ll use to derive the transformation should all seem extremely obvious to you.

So let’s dive straight in!

The Lorentz transformation in full generality is a 4D matrix that tells you how to transform spacetime coordinates in one inertial reference frame to spacetime coordinates in another inertial reference frame. It turns out that once you’ve found the Lorentz transformation for one spatial dimension, it’s quite simple to generalize it to three spatial dimensions, so for simplicity we’ll just stick to the 1D case. The Lorentz transformation also allows you to transform to a coordinate system that is both translated some distance and rotated some angle. Both of these are pretty straightforward, and work the way we intuitively think rotation and translation should work. So I’ll not consider them either. The interesting part of the Lorentz transformation is what happens when we translate to reference frames that are co-moving (moving with respect to one another). Strictly speaking, this is called a Lorentz boost. That’s what I’ll be deriving for you: the 1D Lorentz boost.

So, we start by imagine some reference frame, in which an event is labeled by its temporal and spatial coordinates: t and x. Then we look at a new reference frame moving at velocity v with respect to the starting reference frame. We describe the temporal and spatial coordinates of the same event in the new coordinate system: t’ and x’. In general, these new coordinates can be any function whatsoever of the starting coordinates and the velocity v.

Screen Shot 2018-12-09 at 10.31.11 PM.png

To narrow down what these functions f and g might be, we need to postulate some general relationship between the primed and unprimed coordinate system.

So, our first postulate!

1. Straight lines stay straight.

Our first postulate is that all observers in inertial reference frames will agree about if an object is moving at a constant velocity. Since objects moving at constant velocities are straight lines on diagrams of position vs time, this is equivalent to saying that a straight path through spacetime in one reference frame is a straight path through spacetime in all reference frames.

More formally, if x is proportional to t, then x’ is proportional to t’ (though the constant of proportionality may differ).

Screen Shot 2018-12-09 at 10.41.03 PM.png

This postulate turns out to be immensely powerful. There is a special name for the types of transformations that keep straight lines straight: they are linear transformations. (Note, by the way, that the linearity is only in the coordinates t and x, since those are the things that retain straightness. There is no guarantee that the dependence on v will be linear, and in fact it will turn out not to be.)

 These transformations are extremely simple, and can be represented by a matrix. Let’s write out the matrix in full generality:

Screen Shot 2018-12-09 at 10.45.02 PM.png

We’ve gone from two functions (f and g) to four (A, B, C, and D). But in exchange, each of these four functions is now only a function of one variable: the velocity v. For ease of future reference, I’ve chosen to name the matrix T(v).

So, our first postulate gives us linearity. On to the second!

2. An object at rest in the starting reference frame is moving with velocity -v in the moving reference frame

This is more or less definitional. If somebody tells you that they had a function that transformed coordinates from one reference frame to a moving reference frame, then the most basic check you can do to see if they’re telling the truth is verify that objects at rest in the starting reference frame end up moving in the final reference frame. And again, it seems to follow from what it means for the reference frame to be moving right at 1 m/s that the initially stationary objects should end up moving left at 1 m/s.

Let’s consider an object sitting at rest at x = 0 in the starting frame of reference. Then we have:

Screen Shot 2018-12-09 at 10.52.06 PM.png

We can plug this into our matrix to get a constraint on the functions A and C:

Screen Shot 2018-12-09 at 10.54.59 PM.png

Great! We’ve gone from four functions to three!

Screen Shot 2018-12-09 at 10.56.02 PM.png

3. Moving to the left at velocity v and to the right at the same velocity is the same as not moving at all

More specifically: Start with any reference frame. Now consider a new reference frame that is moving at velocity v with respect to the starting reference frame. Now, from this new reference frame, consider a third reference frame that is moving at velocity -v. This third reference frame should be identical to the one we started with. Got it?

Formally, this is simply saying the following:

Screen Shot 2018-12-09 at 11.01.36 PM.png

(I is the identity matrix.)

To make this equation useful, we need to say more about T(-v). In particular, it would be best if we could express T(-v) in terms of our three functions A(v), B(v), and D(v). We do this with our next postulate:

4. Moving at velocity -v is the same as turning 180°, then moving at velocity v, then turning 180° again.

Again, this is quite self-explanatory. As a geometric fact, the reference frame you end up with by turning around, moving at velocity v, and then turning back has got to be the same as the reference frame you’d end up with by moving at velocity -v. All we need to formalize this postulate is the matrix corresponding to rotating 180°.

Screen Shot 2018-12-09 at 11.07.28 PM.png

There we go! Rotating by 180° is the same as taking every position in the starting reference frame and flipping its sign. Now we can write our postulate more precisely:

Screen Shot 2018-12-09 at 11.09.47 PM

Screen Shot 2018-12-09 at 11.10.44 PM.png

Now we can finally use Postulate 3!

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Doing a little algebra, we get…

Screen Shot 2018-12-09 at 11.12.42 PM.png

(You might notice that we can only conclude that A = D if we reject the possibility that A = B = 0. We are allowed to do this because allowing A = B = 0 gives us a trivial result in which a moving reference frame experiences no time. Prove this for yourself!)

Now we have managed to express all four of our starting functions in terms of just one!

Screen Shot 2018-12-09 at 11.18.23 PM.png

So far our assumptions have been grounded by almost entirely a priori considerations about what we mean by velocity. It’s pretty amazing how far we got with so little! But to progress, we need to include one final a posteriori postulate, that which motivated Einstein to develop special relativity in the first place: the invariance of the speed of light.

5. Light’s velocity is c in all reference frames.

The motivation for this postulate comes from mountains of empirical evidence, as well as good theoretical arguments from the nature of light as an electromagnetic phenomenon. We can write it quite simply as:

Screen Shot 2018-12-09 at 11.43.23 PM

Plugging in our transformation, we get:

Screen Shot 2018-12-09 at 11.43.28 PM

Multiplying the time coordinate by c must give us the space coordinate:

Screen Shot 2018-12-10 at 3.27.16 AM

And we’re done with the derivation!

Summarizing our five postulates:

Screen Shot 2018-12-10 at 12.37.23 AM.png

And our final result:

Screen Shot 2018-12-10 at 3.29.09 AM.png

Swapping the past and future

There are a few more cool things you can visualize with the special relativity program from my last post.

First of all, a big theme of the last post was the ambiguity of temporal orderings. It’s easy to see the temporal ordering of events when there are only three, but gets harder when you have many many events. Let’s actually display the temporal order on the visualization, so that we can see how it changes for different frames of reference.

Display Order Of Three Events

Order of Many Events.gif

Looking at this second GIF, you can see the immense ambiguity that there is in the temporal order of events.

Now, where things get even more interesting is when we consider the spacetime coordinates of events that are not in your future light cone. Check this out:

Outside the Light Cone.gif

Here’s a more detailed image of the paths traced out by events as you change your velocity:

Screen Shot 2018-12-06 at 10.22.20 PM.png

Instead of just looking at events in your future light cone, we’re now also looking at events outside of your light cone!

We chose to look at a bunch of events that are initially all in your future (in the frame of reference where v = 0). Notice now that as we vary the velocity, some of these events end up at earlier times than you! In other words, by changing your frame of reference, events that were in your future can end up in your past. And vice versa; events in the past of one frame of reference can be in the future in the other.

We can see this very clearly by considering just two events.

Future Past Swap.gif

In the v = 0 frame, Red and Green are simultaneous with you. But for v > 0, Green is before Red is before you, and for v < 0, Green is after Red is after you. The lesson is the following: when considering events outside of your light cone there is no fact of the matter about what events are in your future and which ones are in your past.

Now, notice that in the above GIFs we never see events that are in causal contact leave causal contact, or vice versa. This holds true in general. While things certainly do get weirder when considering events outside your light cone, it is still the case that all observers will agree on what events are in causal contact with one another. And just like before, the temporal ordering of events in causal contact does not depend on your frame of reference. In other words, basketballs are always tossed before they go through the net, even outside your light cone.

The same holds when considering interactions between a pair of events that straddle either side of your light cone:

Straddling No Cause.gif

Straddling With Cause

If A is in B’s light cone from one frame of reference, then A is in B’s light cone from all frames of reference. And if A is out of B’s light cone in one frame of reference, then it is out of B’s light cone in all frames of reference. Once again, we see that special relativity preserves as absolute our bedrock intuitions about causality, even when many of our intuitions about time’s objectivity fall away.

Now, all of the implications of special relativity that I’ve discussed so far have been related to time and causality. But there’s also some strange stuff that happens with space. For instance, let’s consider a series of events corresponding to an object sitting at rest some distance away from you. On our diagram this looks like the following:

Screen Shot 2018-12-08 at 11.12.10 PM.png

What does this look like when we if we are moving towards the object? Obviously the object should now be getting closer to us, so we expect the red line to tilt inwards towards the x = 0 point. Here’s what we see at 80% of the speed of light:

Screen Shot 2018-12-08 at 11.14.01 PM.png

As we expected, the object now rushes towards us from our frame of reference, and quickly passes us by and moves off to the left. But notice the spatial distortion in the image! At the present moment (t = 0), the object looks significantly closer than it was previously. (You can see this by starting from the center point and looking to the right to see how much distance you cover before intersecting with the object. This is the distance to the object at t = 0.)

This is extremely unusual! Remember, the moving frame of reference is at the exact same spatial position at t = 0 as the still frame of reference. So whether I am moving towards an object or standing still appears to change how far away the object presently is!

This is the famous phenomenon of length contraction. If we imagine placing two objects at different distances from the origin, each at rest with respect to the v = 0 frame, then moving towards them would result in both of them getting closer to us as well as each other, and thus shrinking! Evidently when we move, the universe shrinks!

Contraction

One last effect we can see in the diagram appears to be a little at odds with what I’ve just said. This is that the observed distance between yourself and the object increases as you move towards it (and as the actual distance shrinks). Why? Well, what you observe is dictated by the beams of light that make it to your eye. So at the moment t = 0, what you are observing is everything along the two diagonals in the bottom half of the images. And in the second image, where you are moving towards the object, the place where the object and diagonal intersect is much further away than it is in the first image! Evidently, moving towards an object makes it appear further away, even though in reality it is getting closer to you!

This holds as a general principle. The reason? When you observe an object, you are really observing it as it was some time in the past (however much time it took for light to reach your eye). And when you move towards an object, that past moment you are observing falls further into the past. (This is sort of the flip-side of time dilation.) Since you are moving towards the object, looking further into the past means looking at the object when it was further away from you. And so therefore the object ends up appearing more distant from you than before!

There’s a bunch more weird and fascinating effects that you can spot in these types of visualizations, but I’ll stop there for now.

Visualizing Special Relativity

I’ve been thinking a lot about special relativity recently, and wrote up a fun program for visualizing some of its stranger implications. Before going on to these visualizations, I want to recommend the Youtube channel MinutePhysics, which made a fantastic primer on the subject. I’ll link the first few of these here, as they might help with understanding the rest of the post. I highly recommend the entire series, even if you’re already pretty familiar with the subject.

Now, on to the pretty images! I’m still trying to determine whether it’s possible to embed applets in my posts, so that you can play with the program for yourself. Until I figure that out, GIFs will have to suffice.

lots of particles

Let me explain what’s going on in the image.

First of all, the vertical direction is time (up is the future, down is the past), and the horizontal direction is space (which is 1D for simplicity). What we’re looking at is the universe as described by an observer at a particular point in space and time. The point that this observer is at is right smack-dab in the center of the diagram, where the two black diagonal lines meet. These lines represent the observer’s light cone: the paths through spacetime that would be taken by beams of light emitted in either direction. And finally, the multicolored dots scattered in the upper quadrant represent other spacetime events in the observer’s future.

Now, what is being varied is the velocity of the observer. Again, keep in mind that the observer is not actually moving through time in this visualization. What is being shown is the way that other events would be arranged spatially and temporally if the observer had different velocities.

Take a second to reflect on how you would expect this diagram to look classically. Obviously the temporal positions of events would not depend upon your velocity. What about the spatial positions of events? Well, if you move to the right, events in your future and to the right of you should be nearer to you than they would be had you not been in motion. And similarly, events in your future left should be further to the left. We can easily visualize this by plugging in the classical Galilean transformation:

Classical Transformation.gif

Just as we expected, time positions stay constant and spatial positions shift according to your velocity! Positive velocity (moving to the right) moves future events to the left, and negative velocity moves them to the right. Now, technically this image is wrong. I’ve kept the light paths constant, but even these would shift under the classical transformation. In reality we’d get something like this:

Classical Corrected.gif

Of course, the empirical falsity of this prediction that the speed of light should vary according to your own velocity is what drove Einstein to formulate special relativity. Here’s what happens with just a few particles when we vary the velocity:

RGB Transform

What I love about this is how you can see so many effects in one short gif. First of all, the speed of light stays constant. That’s a good sign! A constant speed of light is pretty much the whole point of special relativity. Secondly, and incredibly bizarrely, the temporal positions of objects depend on your velocity!! Objects to your future right don’t just get further away spatially when you move away from them, they also get further away temporally!

Another thing that you can see in this visualization is the relativity of simultaneity. When the velocity is zero, Red and Blue are at the same moment of time. But if our velocity is greater than zero, Red falls behind Blue in temporal order. And if we travel at a negative velocity (to the left), then we would observe Red as occurring after Blue in time. In fact, you can find a velocity that makes any two of these three points simultaneous!

This leads to the next observation we can make: The temporal order of events is relative! The orderings of events that you can observe include Red-Green-Blue, Green-Red-Blue, Green-Blue-Red, and Blue-Green-Red. See if you can spot them all!

This is probably the most bonkers consequence of special relativity. In general, we cannot say without ambiguity that Event A occurred before or after Event B. The notion of an objective temporal ordering of events simply must be discarded if we are to hold onto the observation of a constant speed of light.

Are there any constraints on the possible temporal orderings of events? Or does special relativity commit us to having to say that from some valid frames of reference, the basketball going through the net preceded the throwing of the ball? Well, notice that above we didn’t get all possible orders… in particular we didn’t have Red-Blue-Green or Blue-Red-Green. It turns out that in general, there are some constraints we can place on temporal orderings.

Just for fun, we can add in the future light cones of each of the three events:

RGB with Light Cones.gif

Two things to notice: First, all three events are outside each others’ light cones. And second, no event ever crosses over into another event’s light cone. This makes some intuitive sense, and gives us a constant that will hold true in all reference frames: Events that are outside each others’ light cones from one perspective, are outside each others’ light cones from all perspectives. Same thing for events that are inside each others’ light cones.

Conceptually, events being inside each others’ light cones corresponds to them being in causal contact. So another way we can say this is that all observers will agree on what the possible causal relationships in the universe are. (For the purposes of this post, I’m completely disregarding the craziness that comes up when we consider quantum entanglement and “spooky action at a distance.”) 

Now, is it ever possible for events in causal contact to switch temporal order upon a change in reference frame? Or, in other words, could effects precede their causes? Let’s look at a diagram in which one event is contained inside the light cone of another:

RGB Causal

Looking at this visualization, it becomes quite obvious that this is just not possible! Blue is fully contained inside the future light cone of Red, and no matter what frame of reference we choose, it cannot escape this. Even though we haven’t formally proved it, I think that the visualization gives the beginnings of an intuition about why this is so. Let’s postulate this as another absolute truth: If Event A is contained within the light cone of Event B, all observers will agree on the temporal order of the two events. Or, in plainer language, there can be no controversy over whether a cause precedes its effects.

I’ll leave you with some pretty visualizations of hundreds of colorful events transforming as you change reference frames:

Pretty Transforms LQ

And finally, let’s trace out the set of possible space-time locations of each event.

Hyperbolas

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Try to guess what geometric shape these paths are! (They’re not parabolas.) Hint.