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## Late for dinner: a timelike error

If you’re ever in Cullowhee, NC, give Sazón a try.

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 ΔR and separated in time by a duration Δt.  It turns out that if you think of the cosmos as being 4-dimensional, there is then a relationship between ΔR and Δt.

This relationship is the 4D distance formula, Δs2 = ΔR2 – c2Δt2.  Think about it as a sort of “Pythagorean theorem” for 4D spacetime.  That is, its square root (Δs) gives the “distance” between two events in spacetime, given that any event has coordinates (x,y,z,ct).  (Unfortunately, sometimes Δs2 is negative, in which case you cannot take the square root.  But that’s OK; we just talk about Δs2 and don’t even worry about what Δs “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 Δs2) also has merit.  In this case it does, and it turns out that the quantity Δs2 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, Δs2, is not.

But we’re here to talk about intervals Δs, which represent spacetime distances between events.  Let’s call our first event P1;  our second event is P2.  Now, P1 will be “I throw a ball.”  P2 will be “You catch the ball.”  Let’s say we’re separated by ΔR = 10 m and you catch the ball Δt  = 0.4 s later.  Then:

ΔR2 = 102 = 100 m2,

c2Δt2 = (3 x 108)2 (0.4)2 = 1.44 x 1016 m2.

Wow!  The “time term” (in the Δs2 formula) dominates, so that Δs2=100 – 1.44 x 1016 = – 1.44 x 1016m2, which is negative.  In plain English, the events are separated more by time than they are by space.  When this happens, the invariant interval Δs2 is negative.  We call such an interval timelike.  Another way to think of this is that P1 and P2 can influence each other by cause and effect.

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 Δs2 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, ΔR = 12,700,000 m.  Even at that distance, events will have a timelike separation unless Δt is below a certain threshold:

ΔR2 > c2Δt2

Δt < ΔR/c = (12,700,000 m)/ (3 x 108 m/s) = 0.042 s = 42 ms.

I do something Pin North Carolina; something else Phappens to my friend Rick in Perth, Australia.  In order for these events to be causally connected, at least 42 ms must past between P1 and P2.  If Δt 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 Δs2 is always 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: Δt has to be less than 42 ms.)

Let’s apply our new terminology to the example that started this post.  The events P1 and P2 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:

Δt = 0,

ΔR = 1000 m,

Δs2 = ΔR2 – c2Δt2 = 10002 = 1,000,000 m2.

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,

Δs2 = ΔR2 – c2Δt2 = – c2 (3600)2 = –1,166,400,000,000,000,000,000,000 m2.

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/]