The Geometry of Causality Script (Patreon)

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 spacetime.

Recently on Space Time we’ve been talking about the weirdness of space 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 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 spacetime.

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 spacetime relationships. There’s this thing called the spacetime interval that relates observer-dependent perspectives on the length and duration of any journey that the ALL observers agree on, even if they don’t agree on the Delta-x and Delta-t. 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, it defines the flow of causality.

In relativity, 3-D space and 1-D time become a single 4-D entity called spacetime. To preserve our sanity we represent this on a spacetime diagram, plotting time and only one dimension of space. We’ll see our causal geometry emerge plain as day even in in this simplified picture. There is no standing still on the spacetime 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 a group of spacetime travelers starts from the origin where x and t=0. They race away to the left and right for 5 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 spacetime is called their worldline. My worldline is only through time, and the tick-marks on the time axis correspond to my own clock ticks. The faster a traveler moves, the longer their worldline. That’s not just because of their speed; 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 spacetime 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 actually the contours defining the gradient of causality, down which time flows, etched into spacetime by the equations of special relativity. To understand why, we need to see how these appear to our other spacetime travelers. But instead of doing it with equations, we can see it with geometry.

First we need to draw the spacetime 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 worldline. 

And their x-axis? From my stationary point of view, I define MY x-axis as a long string of spacetime events at different distances, but that all occur simultaneously at time t=zero. To observe those points I just wait around until their light has had time to reach me. At every future tick of my clock, a signal arrives from the left and right that I use to build up a set of simultaneous events defining my t=0 x-axis. 

Our traveler does the same thing. But from MY point of view their clock is slow, so I see them register signals at a different rate. At the same time, they are moving away from the signals coming from the left and towards the ones originating on the right, affecting which signals are seen at a given instant. The traveler infers a set of simultaneous events that to me are NOT simultaneous. But there is no preferred reference frame. Their sloped x-axis is right for them. Even just doing this graphically we see that the traveler’s x-axis is rotated by the same angle as their time axis. That comes from insisting that we all see the same speed of light - 45 degrees on the spacetime diagram.

Moving between these reference frames is now a simple matter of squaring up our traveler’s axes. In fact we grid up the diagram with a set of lines parallel to these new axes and square up everything while maintaining the intersection points. My worldline is now speeding off to the left, while our traveler is motionless. We just performed a Lorentz transformation but using geometry rather than math. This transformation allows you to calculate how properties - distance, time, velocity, even mass and energy shift between reference frames. But check out what happens if I attach pens to all of the intersections when I transform between frames. They trace out our hyperbolae.

Those intersections represent specific spacetime events. They will ALWAYS land on the same hyperbola no matter the observer’s reference frame. I told you that these contours show where clocks moving from the origin reach the same proper-time count. But more generally, each represents a single value for the spacetime interval. The Delta-x and Delta-t showing the event at the end-point of a traveler’s worldline might change depending on who is watching, but the hyperbolic contour they landed on - the spacetime interval - will not. This is because the spacetime interval itself comes directly from the Lorentz transformation, as the only measurement of spacetime separation that is unchanging - invariant - under that transformation. 

Now we can finally get to why this thing is so important, and what it really represents. 

It may seem counterintuitive that an event very close to the origin in both space and time can be “separated” from that origin by the same spacetime interval as an event that is very distant in both space and time. The hyperbolic shape seems to demand that. But remember, it takes the same amount of proper time to travel from the origin to a nearby, near-future event compared to a distant, far-future event on the same contour. From the point of view of a particle communicating some causal influence, those points are equivalent. The spacetime interval tracks this causal proximity.

We can think of these lines as contours of a sort of causal geography. I defined the spacetime interval to become increasingly negative in the forward-time direction, so we can  represent this as a valley dropping away from me, here at the origin. I naturally roll through time by the steepest path, straight down; I can change that path by expending energy to change my velocity, although doing so realigns these contours so I ALWAYS roll down the steepest path. There’s no point anywhere downhill that I can’t reach, as long as I can get close enough to the speed of light. In fact the downhill contours define the forward light cone for anyone, anywhere on my spacetime diagram. But uphill is impossible, as long as the cosmic speed limit is maintained.

Breaking that speed limit and rolling up the causal hill are equivalent. To reverse the direction of your changing spacetime interval is to reverse the direction of causality: to travel backwards in time. The spacetime diagram we looked at today was for a flat, or Minkowski space, in which faster-than-light travel is the only way to flip your spacetime interval. But in the crazy curved space within a black hole, it gets flipped for you. We’ll soon see how this requirement of forward causal evolution leads to some incredible predictions when we try to calculate a sub-event-horizon interval of spacetime.