A Brief Diversion Into Imaginary Time

In our previous entry in our introduction to special relativity, we introduced the interval, or proper time, which provides a four-dimensional extension of length:

 And we noticed the oddity that this extension of squared length seems to have a negative term.

Of course, there is a way that squaring some element can give us a negative number, if time is an imaginary quantity.  Let's consider the case of one spatial dimension.  A general vector representing a separation between two events, Δ, with one time and one space component, might be written as:

This has some convenience, since it clearly separates space from time within the term in a manner similar to basis vectors.  We can express events using the complex plane, where the imaginary axis becomes the temporal axis and the real axis a spatial axis.

As discussed before, a boost can be viewed as a rotation in 4 dimensions.  Rotations in the complex plane can be described according to Euler's formula:

A rotation by the angle, ϕ, is a multiplication by this unit norm complex number.

   We therefore need only find a way to relate this unusual spacetime angle to velocity.  Keep in mind that if time is imaginary then so must be the time derivative:

 Remember that the tangent is the derivative dx/dτ: 

 So we can find the angle, with one tiny wrinkle, that imaginary unit.  When that appears, we need to invoke hyperbolic trig functions to move it outside the function:

Thanks to similar relationships, we can rewrite our sine and cosine terms above using their hyperbolic equivalents as below, where we'll make the substitution ϕ → iϕ to simplify our expression:

And thanks to some basic properties of hyperbolic trig functions:

 Demonstrating that we can quite easily rewrite our earlier matrix representation of the boost using this new angle we've introduced, which is termed the rapidity.  This is equivalent to the boost matrix from before:

 Rapidity is often useful as it allows for adding velocities easily.  Two successive boosts can be represented using the same formalism as:

  So rapidity has utility outside the niche application of imaginary time.  We can move between complex and matrix representations of a two dimensional system by the comparison:

And from this we'll finally express the Lorentz Boost in 2d using a purely complex representation:

 Plotting the complex rotation term we see that both the real (cosh) and imaginary (sinh) components explode as rapidity approaches ±10.  Note also that, plotting β in terms of rapidity, we'll see that it asymptotically approaches 1 well before rapidity reaches 10:

  The imaginary time formalism isn't very popular as it does not provide much in the way of insight beyond more traditional four-vectors.  And you know what's better than using an obscure formalism to derive a useful tool?  Jumping from it into an entirely new formalism!  After all, space isn't one dimensional, it's three dimensional.  To generalize complex spacetime, let's find a 3+1d representation.  Fortunately, exactly such a number system exists, quaternions.  Can we find a quaternion formalism that effectively describes special relativity?  Let's see.

Universal Speeding: The Lorentz Group

 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 long time. 
3.  You must break rule one.

Option 1 can be rejected as boring.  Option 2 I've already looked at in the form of generation ships.  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?  Let's 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.  A better introduction might show how γ can be derived from Galilean Relativity and Lorentz Invariance as we did in the previous post.

While this approach is useful and instructive for discussing ideas like time dilation and length contraction, and while it does demonstrate why we shouldn't be able to exceed the speed of light, it doesn't help us understand the source of these issues much less suggest a way to solve them.  

To do that, we'll need to introduce the formal language of four-vectors to discuss in detail how time and space are connected.

Universal Speeding: Using The Four-Force

 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 long time. 
3.  You must break rule one.

Option 1 is boring, so we'll ignore it.  Option 2 is something we're already looking at.  For now, 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?  Let's begin with some very basic concepts in Special Relativity, because we'll need them later.  We've already shown how the Lorentz Factor γ can be derived from Galilean Relativity and Lorentz Invariance.  While this approach is useful and instructive for discussing ideas like time dilation and length contraction, it only tells us that going faster than light should be strange, not why we simply can't get there.

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.

Airship Sails Addendum

A recent comment gave me reason to read back over my post on using sails on an airship and realize I didn't include any pictures of how the vectors I was talking about were oriented.  In that post I was so much more concerned with finishing and organizing my own thoughts I didn't do a very good job of expressing what I was talking about at all.

This picture shows the two most important vectors when talking about either a triangular sail or a wing.  Blue here represents the prevailing wind, or possibly the current direction of motion for the vehicle.  Green represents the lift, or the direction that the sail causes the vehicle to move in.  Wings convert forward motion into an upward force, while sails convert wind into a forward force.
On a ship the collected forces cause a net motion as shown below:

Here the wind exerts a net pressure (in orange) on the hull which pushes the vehicle in that direction.  This force should be fairly weak, but on an airship can't be compensated for by the keel as on a waterborne sailing ship.  Between the lift and pressure forces a net force creates some velocity (in yellow) which in turn creates a drag force (in red) on the ship.  This is what I calculated in the previous post, merely to demonstrate that such a vehicle could indeed generate a meaningful velocity.

There is a concern in this image that I didn't raise in the previous post.  What happens when the breeze is coming from the fore, rather than the aft, of the ship?  So long as the sail can be angled properly and the wind is coming from at least a little behind, or even directly to the side, of the desired direction then this image is roughly accurate.  When it comes from ahead, though, that pressure force that normally would be compensated for by the keel becomes a serious problem:

Here the pressure and lift create a net velocity not ahead, but to the side.  Obviously a sufficiently well sized sail and a properly sleek ship can somewhat mitigate this issue, but never eliminate it.  By beating to windward, however, this problem is significantly reduced.  Doing so causes the pressure and drag forces to be almost in line, meaning velocity is reduced but the direction is controlled:

This is a pretty complex maneuver used in real sailing, one that I can't possibly do justice to here. In short, it involved keeping the desired momentum more or less opposite the direction of the prevailing wind.  The ship constantly moves back and forth about that vector, so the wind is always coming slightly to either side.  In this case so long as the forward component of Lift is greater than the backwards components of Drag and Pressure then the ship will continue moving forwards.  Since the sideways components of the two orientations are opposite each other they cancel out over time and the drift is minimal.  Unfortunately the effect of pressure here will make this a much less effective maneuver than it is for a ship at sea, but it ensures the ship can maintain forward velocity regardless of the direction of wind.

I haven't bothered here with the particulars of design or calculation, the majority of which was done in the first post.  While the new issues raised here would impact any real attempt to construct such a vehicle we've also seen that such issues could be overcome by a sufficiently dedicated designer.  There is a combination of hull and sail design that would make this work.  That is enough for now, unless we want to start really building one.