You can’t get there from here

In this series of posts I’ve tried to get across several ideas:

  • By Einstein’s theories, in our Universe every possible combination of place and time is an event that can be identified with an “address” like (ct,x,y,z) where t is time, c is the speed of light, and x, y and z are spatial coordinates
  • The Pythagorean distance between two events is d=√[(x1-x2)2+(y1-y2)2+(z1-z2)2]
  • The Minkowski interval between two events is √[(ct1-ct2)2 d2]
  • When i=√(-1) shows up somewhere, whatever it’s with is in some way perpendicular to the stuff that doesn’t involve i

That minus sign in the third bullet has some interesting implications.  If the time term is bigger then the spatial term, then the interval (that square root) is a real number.  On the other hand, if the time term is smaller, the interval is an imaginary number and therefore is in some sense perpendicular to the real intervals.

We’ll see what that means in a bit.  But first, suppose the two terms are exactly equal, which would make the interval zero.  Can that happen?

Sure, if you’re a light wave.  The interval can only be zero if d=(ct1-ct2).  In other words, if the distance between two events is exactly the distance light would travel in the elapsed time between the same two events.

In this Minkowski diagram, we’ve got two number lines.  The real numbers (time) run vertically (sorry, I know we had the imaginary line running upward in a previous post, but this is the way that Minkowski drew it).  Perpendicular to that, the line of imaginary numbers (distance) runs horizontally, which is why that starscape runs off to the right.Minkowski diagram
The smiley face is us at the origin (0,0,0,0).  If we look upwards toward positive time, that’s the future at our present locationLooking downward to negative time is looking into the past.  Unless we move (change x and/or y and/or z), all the events in our past and future have addresses like (ct,0,0,0).

What about all those points (events) that aren’t on the time axis?  Pick one and use its address to figure the interval between it and us.  In general, the interval will be a complex number, part real and part imaginary, because the event you picked isn’t on either axis.

The two orange lines are special.  Each of them is drawn through all the events for which the distance is equal to ct.  All the intervals between those events and us are zero.  Those are the events that could be connected to us by a light beam.

The orange connections go one way only — someone in our past (t less than zero) can shine a light at us that we’ll see when it reaches us.  However, if someone in our future tried to shine a light our way, well, the passage of time wouldn’t allow it (except maybe in the movies).  Conversely, we can shine a light at some spatial point (x,y,z) at distance d=√(x2+y2+z2) from us, but those photons won’t arrive until the event in our future at (d,x,y,z).

space VennThe rest of the Minkowski diagram could do for a Venn diagram.  We at (0,0,0,0) can do something that will cause something to happen at (ct,x,y,z) to the left of the top orange line.  However, we won’t be able to see that effect until we time-travel forward to its t.  That region is “reachable but not seeable.”

Similarly, events to the left of the bottom orange line can affect us (we can see stars, for instance) but they’re in our past and we can’t cause anything to happen then/there.  The region is “seeable but not reachable.”

Then there’s the overlap, the segment between the two orange lines.  Events there are so far away in spacetime that the intervals between them and us are imaginary (in the mathematical sense).  To put it another way, light can’t get here from there.  Neither can cause and effect.

Physicists call that third region space-like, as opposed to the two time-like regions.  Without a warp drive or some other way around Einstein’s universal speed limit, the edge of “space-like” will always be The Final Frontier.

~~ Rich Olcott