Universal Speeding: An Introduction to Relativity

There are two important, well-known rules about interstellar travel:

1.  Nothing can travel faster than light.
2.  If you want to get anywhere interesting in a reasonable amount of time then you need to go faster than light.

This leads to three options:

1.  You just can't go anywhere. 
2.  You can go someplace nearby, but you have to stay on your ship for a very, very long time. 
3.  You must break rule one.

Option 1 is boring, so we'll ignore it.  I've already spoken at some length about Option 2 and generation ships, and probably will speak about it again.  For now, though, let's talk about Option 3: If the speed of light is the universal speed limit then let's start speeding. 

Before we can exceed the speed limit we need to know what the speed limit is, and why it's the limit.  So why can't anyone go faster than the speed of light?  Well, why not have a look?

Let's begin with some very basic concepts in Special Relativity, because we'll need them later.  Typically this is introduced in introductory classes at the university and high school level by arbitrarily introducing something called the Lorentz Factor, γ.  Let's start by trying to understand where this magical factor comes from and why it stops us from going faster than light.

  Einstein's theory of Special Relativity is required by two postulates: Galilean Relativity and Lorentz Invariance.

The First Postulate: Galilean Relativity

 Let's start with Classical or Galilean Relativity, our intuitive experience of the world.  Galileo introduced the first of two postulates needed to describe relativity: that the laws of physics are the same in all reference frames.  As Galileo explained, imagine standing below decks on a ship that is moving at some steady speed in calm waters.  Any objects moving around your deck would behave exactly the same (from your point of view) as if you were standing on dry land.

You Could Play Tennis On A Boat: Source

  Someone standing on the shore and observing would see something different but you should be able to relate your observations in your respective frames of reference to each other.  Let's explore the most general way of doing so.

  Imagine two reference frames, F and F', each with its own (x,y,z) coordinate system and its own clock and an observer standing in each frame.  Frame F' and its observer are moving at some velocity, v, away from observer F. 

 Observer F' tosses a ball.  Observers F and F' both measure the distance covered and time of flight.  Since their relative velocity (the velocity of F' as seen by F) is entirely in the x direction, they will measure the same height, y.  Their measurements of the distance covered, x, however will be different.

 Observer F measures a distance, x, and time of flight, t; observer F' measures x' and t'. A general set of equations relating the two sets of measurements should look like:

 Inverting the relations to express x and y in terms of x' and t' they are:

  Where the coefficients a,b,d, and f are functions of velocity and relate the flight time and distance as measured in the two frames.  If A is simply holding the ball then observer F' does not measure any distance, x' = 0, and observer F measures only the distance covered due to the primed frame's velocity, x = vt, yielding:

 From the perspective of F', it is F who is moving away at a velocity of -v.  From her point of view, F moves away a distance of x' = -vt'.  So, using the relation for x in terms of x' and t' with x = 0 and our above solution for b:

Substituting in these coefficients:

 

But there shouldn't be any reason for the coefficients of these two sets of equations to be different.  Again, from the point of view of F' it is F that is moving.  The relation relating x to x' and t' should be described by the same equation as related x' to x and t with the sign of velocity reversed.

So we set these two different equations for x in terms of x' and t' equal to each other:

  In the familiar, Galilean transform, where the velocity is much less than the speed of light we understand that t = t' which yields a = 1 (d = 0) and thus x' = x - vt which is in line with our everyday intuition.

The Second Postulate: Lorentz Invariance

In the more general case, where t'≠ t, meaning that if both observers have clocks in their respective frames they will not necessarily observe the same times.

The second of our two postulates, known as Lorentz Invariance, holds that the speed of light is the same in all reference frames.  No matter the what the relative velocity between our two observers, both will report that a photon is traveling at about 299,792,458 m/s.

  Let's replace the ball in our previous example with a photon.

 Because of Lorentz Invariance, the fixed speed of light, c, relates distance covered and time between measurements in both frames:

 Let's use this fact, along with our original above set of equations for x' and t' in terms of x and t, to find a concrete definition for the coefficients a and d in terms of only v and c:

 And so our final result is:

Returning to our two observers:

By Maschen - Own work, CC0, Link

 We introduced the Lorentz Factor γ, to simplify our text.  We see that for very small v, γ approaches 1 and reproduces conventional expectations.  

 Consider the two clocks, one in each frame.  Observer F' is moving away at some significant fraction of the speed of light and measures the stationary F clock and notes both the time on the stationary frame and the time in her own frame at two different moments.  Since the clock doesn't move, Δx = 0, and the intervals measured in each frame, Δ​​t and Δt', are related by:

 Now imagine that Observer F holds up a ruler and Observer F' attempts to measure its length as she moves past.  Within her frame, length is measured in one moment (Δt' = 0) but this is not necessarily true of the time passed in frame F (Δt ≠ 0) the photons reaching her eyes from the two ends of the ruler may not have been emitted at the same moment.  The relationship between Δx and Δx' is:

 As the velocity of observer F' approaches the speed of light, she sees lengths from outside (the frame of observer F) shrink while the passage of time in his frame seems to accelerate.  That is, she sees his ruler distort to become shorter and the clock seems to tick much faster.  Looking from the other direction, observer F sees objects in the F' frame stretch out and slow down; to him, her ruler seems to distort to grow longer and her clock moves slower.

 From the perspective of a photon, by definition traveling at the speed of light, distances shrink to nothing and all moments happen simultaneously.  This already boggles the mind, but it gets worse if we try to imagine objects moving at yet higher velocity, if v > c then γ is imaginary, which seems absurd!  

 Already it looks like getting anywhere is impossible and this is only the start of our problems.  Beyond considering the seeming absurdity that comes with exceeding the speed of light, in the next post we'll encounter the impossibility of even accelerating an object to that speed in the first place!