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