Suppose I agree to meet my wife for dinner at 8 pm. She goes to El Pacífico (a local Mexican restaurant) whereas I go Sazón (another Mexican restaurant). The restaurants are a kilometer apart. I’ve made an error, of course.

The next week, we agree to meet at El Pacífico. She arrives at 7 pm, I get there at 8 pm. Oops, I’ve made another error, this time not in location, but in time.

Which error is worse?

Any student of special relativity will be familiar with the terms spacelike, timelike, and lightlike interval. Surprisingly, these terms are perfect for discussing my dinner date woes. But what do these terms *mean*, on an intuitive level? Are they even comprehensible in the realm of low, non-relativistic velocities?

Imagine two events, such as me clapping my hands, and you clapping yours. The events are separated *in space* by a distance Δ*R* and separated *in time* by a duration Δ*t*. It turns out that if you think of the cosmos as being 4-dimensional, there is then a relationship between Δ*R* and Δ*t*.

This relationship is the 4D distance formula, Δ*s ^{2} = *Δ

*R*Δ

^{2}– c^{2}*t*. Think about it as a sort of “Pythagorean theorem” for 4D spacetime. That is, its square root (

^{2}*Δs*) gives the “distance” between two events in spacetime, given that any event has coordinates

*(x,y,z,ct)*. (Unfortunately, sometimes Δ

*s*is negative, in which case you cannot take the square root. But that’s OK; we just talk about Δ

^{2 }*s*and don’t even worry about what Δ

^{2 }*s*“means”.)

OK, so why is *c* (the speed of light) in there? Well, for two reasons. One, there has to be *some* velocity as a conversion constant, so that the fourth coordinate *ct* has dimensions of distance (just like *x*, *y*, or *z*). Secondly, the 4D distance formula is *constructed* *explicitly* so that if you’re travelling at speed *c*, then your speed will always be *c* in *every other* reference frame. This is done to match experiment, but whether it’s justified or not depends upon whether the final result (the formula for *Δs ^{2}*) also has merit. In this case it

*does*, and it turns out that the quantity

*Δs*is an invariant: it stays the same regardless of the reference frame you’re in. Distance is relative; time is relative; but the unholy combination of distance and time,

^{2}*Δs*, is

^{2}*not*.

But we’re here to talk about intervals *Δs*, which represent spacetime distances between events. Let’s call our first event *P _{1}*; our second event is

*P*. Now,

_{2}*P*will be “I throw a ball.”

_{1}*P*will be “You catch the ball.” Let’s say we’re separated by Δ

_{2}*R*= 10 m and you catch the ball Δ

*t*= 0.4 s later. Then:

Δ*R ^{2} = 10^{2} = 100 m^{2},*

*c ^{2}*Δ

*t*

^{2}= (3 x 10^{8})^{2}(0.4)^{2}= 1.44 x 10^{16}m^{2}.Wow! The “time term” (in the Δ*s ^{2} *formula) dominates, so that

*Δs*, which is negative. In plain English,

^{2}=100 – 1.44 x 10^{16}= – 1.44 x 10^{16}m^{2}**the events are separated more by time than they are by space**. When this happens, the invariant interval Δ

*s*is negative. We call such an interval

^{2 }*timelike*. Another way to think of this is that

*P*and

_{1 }*P*can influence each other by cause and effect.

_{2 }Why is the time term so big? Basically, because light is so fast. Remember that we had to multiply time by the speed of light to make the fourth 4D spacetime coordinate have units of distance. So unless events are very, very far apart—or the time difference is very, very small—then Δ*s ^{2 }*will be negative and you will have a timelike separation.

Now, in your everyday life, distances are never *that* great. The biggest distance you will ever have between you and a friend is the diameter of the Earth; namely, Δ*R* = 12,700,000 m. Even at that distance, events will have a timelike separation unless Δ*t* is below a certain threshold:

*ΔR ^{2} > c^{2}*

*Δt*

^{2}*Δt < **ΔR/c = (12,700,000 m)/ (3 x 10 ^{8} m/s) = 0.042 s = 42 ms.*

I do something *P _{1 }*in North Carolina; something else

*P*happens to my friend Rick in Perth, Australia. In order for these events to be causally connected, at least 42 ms must past between

_{2 }*P*and

_{1 }*P*. If Δ

_{2}*t*is

*less*than 42 ms, then light doesn’t have enough time to get from me to Rick; the events

*cannot*be causally connected and we have a spacelike interval instead. In such a case Δ

*s*is always

^{2 }*greater*than zero. In plain English,

**the events are separated more by space than they are by time**.

(Note: 42 ms is a good value to keep in mind. On the Earth, for an interval between events to be spacelike, they have to be almost simultaneous: Δ*t *has to be less than 42 ms.)

Let’s apply our new terminology to the example that started this post. The events *P _{1 }*and

*P*are “I arrive at my destination” and “my wife arrives at her destination”. We arrive at the same time, 8 o’clock, but the restaurants are a kilometer apart. So:

_{2}*Δt = 0,*

*ΔR = 1000 m,*

Δ*s ^{2} = *Δ

*R*Δ

^{2}– c^{2}*t*

^{2}= 1000^{2}= 1,000,000 m^{2}.Seems like a big error! But the next week I get the time wrong (not the location) and we find:

*Δt = 1 hr. = 3600 s,*

*ΔR = 0,*

Δ*s ^{2} = *Δ

*R*Δ

^{2}– c^{2}*t*

^{2}= – c^{2}(3600)^{2}= –1,166,400,000,000,000,000,000,000 m^{2}.That’s worse. A lot worse. It represents a much bigger “interval” in spacetime. And, truth be told, our culture reflects this; it is much worse to be an hour late for dinner then to show up on time but 1 km away. And this makes sense: if you go to the wrong restaurant you can correct your error fairly easily; it wouldn’t take that long (maybe 1 minute?) to drive 1 km. If instead you’re an hour late, there’s not much you can do…

[Here’s the website for Sazón: http://www.sazoncullowhee.com/wordpressinstall/]

on May 1, 2013 at 3:52 PM |David MartinoExcellent post Matt. It doesn’t take a mathematician or physicist to conclude that both mistakes were your fault. Such things are known a prioi. Intuitively I thought that showing up an hour late would be a worse transgression because the other error is easily fixed. It is also interesting to see a representation of the value of an hour of time.

on May 1, 2013 at 4:30 PMMatthew RaveThe examples were hypothetical. I actually committed no such errors!

on May 1, 2013 at 4:47 PM |drakodocNow if you were early by 1 hour instead of late, then Δs2 would equally as large however since it would be your time wasted instead of your wife, the transgression would be minimal. An example of gender relativity????

on May 2, 2013 at 9:56 AMMatthew RaveNo; if Sarah were late or early she’d be in trouble (or not) in equal measure, so the trouble is invariant.

on May 5, 2013 at 11:49 PM |Rick SchokmanExcellent post. There are, however, occasions when I’m busting for a P2 and then it becomes very time like.

on November 19, 2013 at 10:15 AM |Sharing spacetime coordinates with Abraham Lincoln and Matt Damon | Many Worlds Theory[…] an earlier post I talked about events in spacetime, and about how an error in time is usually more grievous than an […]