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Posts Tagged ‘spacetime’

In an earlier post I talked about events in spacetime, and about how an error in time is usually more grievous than an error in space.

Let’s now talk about the coincidence of spacetime coordinates.  Specifically, how significant is it if you share one, two, three, or even four coordinates with a famous person?

Gettysburg Address

First, some preliminary discussion.  An event is a point (x,y,z,ct) in spacetime.  Technically, you are not an event; you are a series of (unfortunate?) events smoothly snaking its way forward in time.  As you sit there, reading this post, your x, y, and z are probably staying constant while ct is continually increasing.  (Of course if you are reading this on the bus, then x, y, and z may be changing as well.)  Note that I will use a relative coordinate system where x and y are measured with respect to the Earth (they are effectively longitude and latitude) and z is height above sea level.  This way, we don’t have to deal with the annoying detail that the Earth is spinning, and orbiting the Sun, and that the solar system is hurtling through space.

Now the act of you reading this is an event; let’s say it has the coordinates (x,y,z,ct) in spacetime.  But let’s also suppose that when you read that word, Matt Damon was eating a bagel with cream cheese.  That event had the coordinate (X,Y,Z,cT), say.  Unless you happened to have been with Matt Damon just then, your spatial coordinates did not coincide.  However, it should be obvious that t=T.  This means that it is no big deal to share a time coordinate with a celebrity.  You currently share a time coordinate with every living celebrity.  Right now, as you read this, Quentin Tarantino is doing something.  So is cricketer Michael Clarke.  So is chess grandmaster Magnus Carlsen.

michael-clarke

What are the spacetime coordinates of the Ashes?

But how significant is it if one spatial coordinate (x, y, or z) coincides with a celebrity?  Or two spatial coordinates?  Can we sort this out?

Here are some other possible cases:

x or y (and t) coincide: this is not likely to be true for you at this instant, but it happens with great frequency.  It means that either your longitude or latitude is the same as a celebrity, such as Christopher Walken.  Let’s say you’re currently in Jacksonville, FL whereas Walken is in Los Angeles.  Obviously, your x’s are very different and your y’s, although close, are also different.  But you decide to drive to Raleigh, NC for a friend’s wedding.  At some point along your drive on I-95 your y-coordinate will be the same as Walken’s, as the line of your latitude sweeps through 34 degrees North.  (If you’re curious, it will happen a little before you stop for lunch at Pedro’s South of the Border.)  On a flight from Seattle to Miami, your lines of x and y will coincide (at different times) with a majority of celebrities in the USA.

z (and t) coincide: this is also quite common.  It means that you and a celebrity (such as chess grandmaster Hikaru Nakamura) share an altitude.  I am currently at z = 645 m (2116 ft.) in elevation…well, scratch that, I am three floors up, so it’s closer to z = 657 m.  Anyway, if Nakamura drives from Saint Louis (Z = 142 m) to Denver (Z = 1600 m) on I-70 then our elevations will coincide at some point along his drive (presumably a little bit past Hays, KS).

x or y, with z and t: this is much rarer, but does happen.  For this to occur, your line of longitude or latitude would have to sweep through a celebrity (such as quarterback Cam Newton), but you would also have to coincidentally be at the same altitude.  Now, if you live in the same city as the celebrity (in this case, Charlotte, NC) then a simple trip across town to visit Trader Joe’s would probably be sufficient to achieve x=X (or y=Y) along with z=Z and t=T.  However, for someone like me who lives at an arbitrary (and uncommon) elevation such as 645 m, this does not happen often.

x, y, z….but not t: this means that you have visited the exact location that a famous person has visited, but not at the same time.  This probably happens hundreds of times in your life.  An obvious example is when you go to a famous location: maybe Dealey Plaza in Dallas, maybe the Blarney Stone, maybe the location of Lincoln’s Gettysburg address.  (By the way, today is the 150th anniversary of that speech!)  A not-so-obvious example (but much more common) is when you drive along a much-used road.  I have driven I-95 for huge stretches, for example, and I am sure many celebrities have driven that highway as well.  At some point along my drives, I will have “visited” the same location as another celebrity (Tina Fey, let’s say) when she decided to drive down to Savannah for the weekend.  I’m sure she stopped at Pedro’s South of the Border, and so have I.

pedro's

Proof that I went there.

x,y,z and t: this is the holy grail of celebrity coincidence.  It means you met the person.  Now, of course, humans are not bosons, so the spatial coordinates cannot be exactly the same, but if you meet the person I will say that the coordinates are close enough.  My (x,y,z,ct) were once the same as Al Gore.  My (x,y,z,ct) were once the same as Alan Dershowitz.  My (x,y,z,ct) were once (almost) the same as Hikaru Nakamura.  That’s about it.

I have left out several cases (such as x and/or z coinciding, without t) because they are trivial and uninteresting.  Imagine the entire world line of a celebrity such as Winston Churchill, who traveled all over the world.  If his spatial coordinates were projected onto the ground (painted bright yellow, say) then this looping curvy line would be a huge mess, spanning the globe, and covering huge swaths of England like spaghetti.  As I live my life, at any given instant I am pretty sure that one or two of my coordinates match some part of this snaky line.  No big deal.

It’s not like he was Matt Damon or anything.

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If you enjoyed this post, you may also enjoy my book Why Is There Anything? which is available for the Kindle on Amazon.com.

sargasso

I am also currently collaborating on a multi-volume novel of speculative hard science fiction and futuristic deep-space horror called Sargasso Nova.  Publication of the first installment will be January 2015; further details will be released on Facebook, Twitter, or via email: SargassoNova (at) gmail.com.

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sazon

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/]

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