The Geometry of Causality

Matt O'Dowd (0:02)

This episode is sponsored by Audible. The special theory of relativity tells us that one person's past may be another's future when time is relative paradoxes threaten. Today, we peer deeper into Einstein's theory to find that the immutable ordering of cause and effect emerges when we discover the causal geography of space time.

Recently, we've been talking about the weirdness of space-time in the vicinity of a black hole's event horizon. Very soon, we'll be dropping below that horizon to peer at the interior of the black hole. There, space and time switch roles. But to truly understand that bizarre statement, we need to think a little bit more about how the flow of time is described in relativity. Today, we're going to look at the amazing geometric structure that time, or more accurately causality, imprints on the fabric of space-time. First, let's recap a little bit of Einstein's special theory of relativity. There are two previous episodes in particular that will be useful here if you find you need more background. Special relativity tells us that our experience of both distance and time are, well, relative. If I accelerate my rocket ship to half the speed of light, the distance I need to travel to a neighboring star shrinks dramatically from my point of view. An observer I leave behind, with an amazing telescope, observes me traveling the entire original distance but will perceive my clock as having slowed. The combination of this length contraction and time dilation allows both moving and stationary observers to agree on how much older everyone looks at the end of the journey. Everyone agrees on the number of ticks that occurred on everyone else's clock. They just don't agree on the duration of all of those ticks. Reminder, time measured by a moving observer on their own clock is called proper time. But counting those clock ticks isn't the best way for everyone to agree on space-time relationships. There's this thing called the space-time interval that relates observer-dependent perspectives on the length and duration of any journey that all observers will agree on, even if they don't agree on the delta x and delta t of that journey. We've talked about it before, but it's a tricky concept to understand intuitively. But we want that intuition, because more than proper time, the space-time interval defines the flow of causality. In relativity, 3D space and 1D time become a single 4D entity called space-time. To preserve our sanity, we represent this on a space-time diagram plotting time and only one dimension of space. We'll see our causal geometry emerge plain as day even in this simplified picture.

There is no standing still on a space-time diagram. If I don't move through space, I still travel forward in time at a speed of exactly one second per second, according to my proper time clock. Motion at a constant velocity appears as a sloped line, and the time axis is scaled so that the speed of light is a 45 degree line. Now, let's say we have a group of space-time travelers. They start at the origin where x and t equals They race away to the left and the right for five seconds, according to their own watches. They all travel at different speeds, some close to the speed of light, but never faster. The path they cut through space-time is called their world line. My world line is only through time, and the tick marks on the time axis correspond to my own proper time clock ticks.

The faster a traveler moves, the longer their world line. That's not just because of their speed, though. To me, their clocks tick slow.

They time their journey on these slow clocks, so I perceive them traveling for longer. Accounting for this, we find that our space-time travelers are arranged on a curve that looks like this. This shape is a hyperbola. Drawing a connecting line at the tick of every traveler's proper time clock gives us a set of nested hyperbolae. But these aren't just a pretty pattern. These curves are kind of the contours defining the gradient of causality. Down which time flows and etched into space-time by the equations of special relativity. But to understand why, we need to see how these proper time contours appear to other space-time travelers.

Instead of doing that with equations, we can see it with geometry. First, we need to draw the space-time diagram from the perspective of one of the other travelers. To transform the diagram, we need to figure out what they see as their space and time axes.

Time is easy. They see themselves as stationary, so their time axis is just their own constant velocity world line. And their x-axis? Well, from my stationary point of view, I define my x-axis as a long string of space-time events at different distances, but that all occur simultaneously at time t equals zero.