Understanding Motion and Relativity with Spacetime Diagrams

Motion in one dimension

Spacetime diagrams are a way to describe motion: to tell the story of how an object's position changes over time. As we'll see, they're great for intuitively understanding a whole story at once without the need for equations, and they're designed to be particularly helpful for illustrating the surprising relationships between space and time that Einstein described in his theory of special relativity.

One simplifying limitation is that spacetime diagrams usually only describe motion in one dimension: motion back and forth along a straight line, as shown above. (Formally, we'll call this the x-axis, with +x to the right.) Things would only get a little more complicated if we considered motion in 2D or 3D, but it would be much harder to draw.

The essential thing about motion is that an object's position changes with time. In an animation like the ones shown here, we're watching that as it happens: the object may move left or right, faster or slower, and it makes intuitive sense. But that means that we need to wait through the whole animation to see everything that happens, and we're limited in the kinds of reasoning that we can do to understand relationships between space and time.

Spacetime diagram basics

To describe and analyze the whole story at once, we will instead represent time graphically: we'll draw it as an axis on our diagram, perpendicular to the left-right (x) position axis, with +t pointing up. (This takes a little getting used to, since time is the "independent variable"—we write x(t), not t(x)—and graphs in math classes usually put the independent variable on the horizontal axis. I sometimes think of these diagrams like rock layers in geology, with the oldest on the bottom.) The motion of the object is represented as a curving path up through time. [To see this, scroll down a little.]

In the example at right, as the animation runs below the diagram, a yellow horizontal line moves up the graph to label the current time. (I sometimes call that line a "time slice", since it represents a "slice" of history at a given time.) At any particular moment, the position of the object is exactly where the yellow line intersects the object's path on the diagram. (This path is formally called the object's worldline. In the diagrams here, each worldline is labeled occasionally with its corresponding object.) It takes some getting used to, but with practice you can look at a spacetime diagram and understand the whole story at once.

Remember, the actual motion of the object is always just back and forth along the x-axis, as in the animation! It's common when learning this for the first time to mistakenly imagine the diagram as a top-down view of motion, like a map. But the right way to think of it is one horizontal "time slice" at a time. "Up" along the diagram means "later", not "north".

Relative velocities

Now that we've got the basic idea of how spacetime diagrams work, we can to build intuition about what they mean. In the example at right, an airplane flying over a house while a rocket zips by. We can immediately spot some patterns:

• Objects moving with constant velocity have straight lines on the diagram.
• The slope of the line corresponds to the speed: faster is flatter, slower is steeper. (This is the opposite of what you see in traditional graphs where time is horizontal.)
• An object sitting still has its worldline go straight up. In this case, the house's worldline is right on top of the t-axis at x = 0. Maybe this diagram was drawn by someone in the house!

That last observation raises an important point: every diagram makes an implicit choice of whose perspective it is from: some observer's reference frame. Because this diagram is in the house's reference frame, the house is shown sitting still (and at the origin of coordinates: x = 0).

Isn't it obvious that the house is sitting still? Well, let's look at this same story in the reference frame of someone on the plane. It's very easy for that person to think of themselves as sitting still, with the rest of the world drifting by outside the window. (I've made that mistake myself more than once, thinking the plane at the next gate over was pulling forward when actually my plane was pulling back. And high up in the sky I definitely don't feel like I'm moving at hundreds of miles per hour unless I look out the window, and maybe not even then.)

So in the plane's reference frame, what do we see? The plane is at rest at x = 0, with its worldline up along the t-axis. The house is drifting past the plane toward the left, and the rocket only very slowly comes up from behind and passes it by. (Our usual experience is that we just subtract to find the rocket's apparent speed in the plane's frame: if the plane's ground speed is 300 m/s and the rocket's ground speed is 500 m/s, then from inside the plane it looks like the rocket is going 200 m/s.)

As we'll discuss later, every reference frame moving at constant velocity has an equal claim to be "sitting still". (Maybe you're thinking, "Okay, but we know it's really the house that's sitting still." My answer is just to remind you that the Earth itself is blazing through space extremely fast as it orbits the Sun, and the Sun itself is orbiting the center of the galaxy, and so on.) So since we can't declare any one reference frame to be the "real" frame, we'll content ourselves with just picking one of them as our own chosen "Home Frame" where we'll mainly tell the story, and anything else we'll call an "Other Frame".

Acceleration: changing velocity

What does it look like on the diagram when motion isn't at constant velocity? Well, the object's worldline will have to get flatter and steeper as it moves faster and slower, so it won't be a straight line anymore. A curving worldline is a clear sign that the object is changing velocity: that it is accelerating in one direction or another. (If it curves flatter, it's speeding up; if it curves steeper, it's slowing down.)

It's worth pointing out that while sitting on an airplane that's cruising at constant velocity might feel like you're sitting still, if you're sitting in an accelerating airplane as it takes off, you definitely know it! You're pressed back into your seat. So unless we want to introduce a lot more complexity, we won't ever choose an accelerating observer as the definition of a reference frame. (The nice reference frames that aren't accelerating are formally called inertial frames.)

Why is that radical? Because our rule has always been to subtract velocities when you change to a different reference frame! We'd expect that if light was moving 3×10⁸ m/s as seen by the house, and we were on a rocket moving 2×10⁸ m/s, then on the rocket they'd see the light moving just 1×10⁸ m/s. But the principle of relativity says it doesn't work that way: the rocket sees the light moving at 3×10⁸ m/s exactly the same as the house does! (After all, they both measure the same values of those physical constants, and they both use the same law to put them together.) For that to be true, something truly bizarre must be going on with how different reference frames measure space and time.

For our diagrams, though, there's one immediate consequence: the slope of the worldline for anything moving at the speed of light must always be exactly the same. To make this especially convenient, we'll always choose to measure time and distance in "matching" units: if we measure time in years, we'll measure distance in light-years. If we measure time in seconds, we'll measure distance in light-seconds, and so on. That means that by definition, the speed of light will always be "1", whether that means "1 ly/y" or "1 ls/s". So on our diagrams, the worldline of light will always have a slope of 1 and tilt at a 45° angle (either left or right). (And since relativity tells us that nothing can move faster than light, every worldline is always less than 45° away from the vertical axis.)

Simultaneous events

Other scientists had noticed parts of this before, but Einstein's great achievement was to take it absolutely literally and work through the implications with extreme care, no matter how weird the results. We'll try to do the same thing here, though you'll need a formal textbook if you want to see the details. We'll hold tightly to two core principles: that the speed of light must be the same for every observer, and that every observer must agree about any physically measureable "event" (A bug hits the windshield, I scratch my nose, a detector goes "ping", etc.) Each observer will assign a specific time and a specific location to a given event, but we won't assume that they'll all assign the same time and location coordinates.

To start figuring out how time really works, a good first step is to ask, "How can I make sure two events happen at exactly the same time?" We'll want to set up a scenario based on observable physical events that everyone has to agree about (e.g. "has this firecracker exploded yet?"), and we'll want to base our reasoning on the motion of light (e.g. "the center of a room is the point where I can stand with a flash bulb so that a flash of light that bounces off mirrors on each side will get back to me from both directions at the same time.")

Here's the plan: in our coordinate frame, we stand at rest in the exact center of a room, and we place one firecracker on each side. The fuse on each one is attached to a detector that will detonate it when it detects light. At time t = 0, we activate a strobe light that we're holding. That flash of light (shown here as a blue dot going in each direction) will reach each firecracker at the same time, so they're guaranteed to go off simultaneously. (If you want specific numbers, we could say for simplicity that each firecracker is 3 light-seconds away, even though that's ridiculously big. That would mean that the grid lines on this graph are in units of 1 s (time) or 1 ls (distance). The light will arrive at t = 3 s.) For future reference, I've also labeled our central observer at this same instant in time (maybe we sneezed at that moment).

Our plan seems foolproof! But what I didn't tell you is that our room was actually in an interstellar shipping container loaded on a spaceship going half the speed of light. So an observer on the Earth watching our experiment will see us and both firecrackers moving at that same speed: parallel lines angling up the diagram. At some moment, they see us flash our strobe light, and they see the light travel in both directions: 45° lines on their diagram. But the flash going backward doesn't have to travel as far, because the firecracker is moving, too: they meet in the middle. On the other hand, the flash going forward has to travel much farther because it has to catch up to the front firecracker as it moves away. So even though we saw both firecrackers explode at the same instant, the observer on Earth sees the one in front explode substantially later than the one in back.

How could it be possible that two events are both simultaneous and not simultaneous depending on who's looking? Most people would reach that conclusion and say, "Either I've made a mistake, or one of my assumptions must be subtly wrong." But Einstein said, "I've checked my reasoning and I insist on my assumptions, so I guess this is just how the world works!" Observers in different reference frames can disagree on the order of events! (Remarkably, this doesn't break cause and effect, but explaining why is another story.)

Time and position in an Other Frame

To make sense of this, we can look at the graph and notice that the two explosions and the center observer label (all of which were all along the third (horizontal!) coordinate line in our Spaceship Frame diagram) still appear to lie along a straight line in this Earth Frame diagram, but now that line is tilted. If we'd flashed our light a little earlier or later, we would have gotten different tilted lines, all parallel to this one. Similarly, objects that were at fixed locations in the Spaceship Frame (like all three objects here) all follow parallel (tilted) worldlines up the Earth Frame diagram.

What that means is that we can sketch lines of constant time and position as seen by the Spaceship Frame as a tilted coordinate grid when drawn on the Earth Frame diagram. (Traditionally, we call the coordinates of the diagram's own Home Frame t and x, and the coordinates of the tilted Other Frame t′ and x′, pronounced "t prime" and "x prime". For convenience, we typically define both coordinate systems so the origin is the same event in each of them.) If you look carefully, you'll notice that the Other Frame's time axis tilts down from vertical by exactly the same angle that its position axis tilts up from horizontal. Nifty!

Thinking through this tilted Spaceship Frame grid can lead us to another strange conclusion about time in relativity: the Earth Frame observer sees time passing slower on the spaceship than it's passing on Earth! The diagram illustrates this in a straightforward way. First, we'll add a worldline for our Earth observer (at rest here, with its worldline straight up the t-axis), and we'll mark a dot along that path for their heart beating exactly once every second: every time the Earth crosses one of the horizontal Home Frame coordinate lines. (We'll count the heartbeats in the animation, too.) (Most people call these "clock ticks", but I think "heartbeats" is more viscerally relatable.)

If we'd done the same thing on our original Spaceship Frame diagram, we would have drawn a dot every time our central worldline crossed one of those coordinate lines. But since heartbeats are observable, physical events, on the Earth Frame diagram each of our heartbeats must happen when our tilted worldline crosses one of the tilted coordinate lines. Comparing the two paths (we've coordinated so that both hearts beat at the origin event as we pass the Earth), we can see that the Earth observer sees our spaceship heartbeats lagging behind their own. They conclude that our clocks (and heartbeats) are running slow. (And the faster the spaceship travels, the more the coordinates tilt and the slower the moving clock seems to run.)

But now let's go back to our Spaceship Frame diagram, this time including the Earth observer's worldline and their coordinate grid (which is tilted the opposite way). Now our heartbeats happen at the expected coordinate intervals, and we see the Earth observer's heartbeats running slow! This seems like a paradox: how is it possible that we each believe that the other's clocks are running slow?

It would take us too far afield to give a satisfying answer to that puzzle here. But the essential solution to this apparent paradox (and many others) is that it's not reliable to compare events that are far apart in space, more or less because different observers disagree on which events are simultaneous. (If you look at the Earth observer's third heartbeat event here and follow its tilted coordinate line over to our spaceship observer, you'll see that line intersect before the spaceship observer's third heartbeat, just as we saw in the Earth frame diagram.)

The Twin "Paradox"

The classic illustration of these ideas that does work reliably by comparing events at the same location is often called the "Twin Paradox". Two twins grow up on Earth, but one stays home while the other becomes an astronaut. On their mutual 20th birthday, the astronaut twin departs on a rocket to visit a distant star four light-years away. The rocket is very fast: it travels there and back averaging 2/3 the speed of light, so (as measured by the twin on Earth) the rocket arrives at its destination six years after leaving. The travelling twin takes some quick data and immediately comes back home. But as illustrated here, because the rocket is moving so fast, the Earth twin sees the astronaut twin's clock and hearbeats running slow during travel: the astronaut twin is aging more slowly. By the time they get back to Earth, the Earth twin is 32 years old while the astronaut twin is only about 28. (And if the rocket had traveled even faster, the astronaut could have been just 21 when they got home, or 20.01.)

That's a weird and remarkable result, almost as if the astronaut twin has time-traveled into the future. (We've done lots of experiments to verify that this really works, often by measuring the lifespans of tiny particles moving close to light speed.) But surprising as it may be, this is not actually the "paradox" in the name: it's not an inconsistency that would disprove the theory, just an unexpected result. The supposed paradox instead goes like this: "But what if we told this story from the astronaut twin's point of view? They should see the Earth twin as the one who's moving and think that Earth clocks are the ones running slow. So the astronaut twin and the Earth twin should make opposite predictions about who will be younger when they get home. Since that's a physically observable outcome that they ought to agree on, this is a contradiction." (The implication would then be that relativity is not a self-consistent theory and would therefore be incorrect.)

Physically, the essential idea that resolves this apparent contradiction is that the astronaut twin's path is not an inertial reference frame: at the distant star, their rocket must accelerate back toward Earth to turn around and go home. (They presumably had to accelerate up to speed away from Earth, too, and to slow down when they got back.) And since acceleration is something we can feel, the astronaut's experience is measurably different than the Earth twin's. That means that the "paradox's" premise that the twins should both be able to use special relativity in their own reference frames isn't valid: the astronaut twin is not in an inertial frame.

We can still analyze the astronaut twin's motion using the rules of special relativity as we've learned them, as long as we draw our diagrams in the Earth twin's inertial frame. But it would be a lot more satisfying if we had a sense of what the story does look like from the astronaut's point of view. In principle, you could turn around really quickly! How could a few minutes out of a years-long journey completely reverse the outcome?

Time slices for an accelerated frame

I won't try here to explain the elegant but complicated mathematics of general relativity that we'd need if we wanted to carefully analyze motion in an accelerated frame, but it turns out that we can still use spacetime diagram reasoning to get some intuition about how the astronaut observer would see things. Let's tentatively assume that at each moment along the astronaut's worldline, we look at their velocity and draw a "time slice" through that point with the angle it would have for an inertial observer. As illustrated here, that means that the time slices change their angles from moment to moment. (I've drawn time slices through every heartbeat along the astronaut's path, but when the angle is changing quickly I've also drawn dashed line slices for each quarter heartbeat to make the pattern easier to see.) It's intriguing that during constant acceleration, all these tilting time slices turn out to intersect at a single point! (Ignore the shaded wedges past those points: the time slices overlap and the coordinates get ambiguous there.) [For the experts out there, this approach turns out to be equivalent to "Rindler coordinates" for an accelerated observer in general relativity.]

How does this help us to understand what's going on? Well, when the twins are right next to each other, the main effect of the tilting time slices is exactly what we've already talked about: the accelerating twin sees the Earth twin's clock ticking a little slow. But when the astronaut turns around at the far end of the journey, the changing tilt makes a huge difference because the line swings up very fast. In this example, while the astronaut measures just two years or so while they're turn around, their tilted time slices sweep up through almost eight years along the Earth twin's worldline! (And you can see that no matter how fast they turned around, the angle would still sweep up just as far in the end.) So if you buy this argument, the astronaut twin's story would be "My Earth twin aged normally with me before I left, then they aged slowly while I traveled away, then they aged really fast while I turned around, and then slowly again as I came home. Their aging in the middle was so very fast that they ended up older than me when I got home."

[The Earth twin's story would instead have been "My astronaut twin aged normally with me before they left, then they aged slowly while they traveled away, then they aged pretty normally as they turned aroud, and then slowly again on the way back. So I was older than them when they got home." Entirely different details, but the final conclusion once they were back together to compare the end results was the same.]

A careful answer to why time seems to pass differently in different places during acceleration would take us much farther into the complexity of general relativity than is reasonable here. But I'll go ahead and give a hint of intuition. General relativity is really a theory about gravity, and a key result is that if you get close to the strong gravity of a massive object, time passes more slowly for you. (Spend a day hovering close to a black hole, and years might have passed on Earth when you get home.) Another key result is that acceleration is secretly equivalent to gravity. (Are you pressed into the back of your seat because the car is zooming away from a stop sign, or is your seat just tilted on its back? General relativity says that deep down, those are actually the same.) So in our example, when the astronaut twin turns around and accelerates back toward Earth, their feeling of being pushed into the back of their seat is just as if there were a giant massive object nearby to their right. (Right around the corner of that shaded wedge, in fact.) So from the astronaut's perspective, it's as if they're right next to a black hole and their twin is far away: of course their twin will age faster!

So what's next?

There are so many additional directions out there to explore: the correct rules for how velocity changes in an Other Frame, or the fact that a moving object appears to have a shorter length than it does at rest, or the rules of cause and effect... the list goes on and on. There are great courses and great books out there about relativity. (I've been teaching for years out of the Six Ideas That Shaped Physics textbooks by Thomas Moore: Unit R of that series covers special relativity, with a particular focus on spacetime diagrams. I also very much enjoyed learning the subject myself from Special Relativity by Thomas Helliwell, my undergrad advisor.) I know there are great online resources, too: videos and tutorials and interactive texts.

Whether you follow a formal textbook or just want to play around with these ideas on your own, I encourage you to try the main interactive spacetime diagram app on this site. There are instructions and loadable examples on that main page, and you can experiment with scenarios like the ones here. One option that this introduction hasn't illustrated is the ability to display two diagrams of the same scenario in different frames side by side. (There's a checkbox in the controls.) You can manipulate the events in a trajectory directly with a mouse or a touch interface, or you can have fine control over each event's coordinates (in either frame) and over the trajectory's velocity as it leads into the event or departs out of it (options which usually correspond to accelerated motion). Try it out, and have some fun! My hope is that by playing around with these diagrams, you'll wind up with a deeper understanding of the weird and wonderful way that our world works.