Skip to main content

Spheres, Special Relativity, and Rotations

I've been working though a set of ideas centered around explaining special relativity as rotations in four dimensional space.  +Jonah Miller 's excellent post on Minkowski space this week spurred me into finally capturing some of my nascent ideas here.  I should caution that what follows are the starting points for directions I intend to explore and are hypothetical thought processes, not (until they're mathematically verified or refuted), necessarily the way things actually work.  First though, we'll start out with a little bit of well established background on how special relativity does work.

Time Dilation and Four Velocity
Two different sources led to my ideas.  First, there's the concept of a four velocity[2].  The first place I saw this concept described was in a Brian Greene book[3].  The basic idea is that everything in the universe moves at a constant speed, the speed of light, in a four dimensional space, (hence the term four velocity), that consists of the three space directions, and the fourth 'direction', time.  We can't change our total speed, all we can do is change our direction.  So, we can move more quickly in a space direction, but that means we slow down in the time direction.  Normally, most of our velocity points in the time direction, so we don't feel like we're moving at the speed of light.

If we steer a significant portion of our velocity into the space direction, we start to see one of the effects of special relativity, time dilation.  For us, time slows down.  If we speed away at a significant portion of the speed of light, turn around and speed back to our original location, we'll find that more time has passed four our friends that stayed behind than the has passed for us.  Why?  It's because we slowed our velocity in the time direction when we sped up in the space direction.

This phenomenon is called time dilation.  It's often taught in physics courses using something called the twin paradox.  More about the paradox and it's resolution in a future post.  For now, I'll just leave you with a simplistic diagram of what the twin did when he sped away and returned (picture 1).

At the start, the two twins are moving along path a.  As I said, most of their velocity is in the time direction.  Suddenly, one of the twins uses some of his velocity to go in the space direction and consequently slows down in the time direction.  Meanwhile, his sister has been cooking along at the same, unchanged, higher, speed in the time direction.  After a bit, the brother turns turns around and returns at the same faster space velocity.  When he arrives home, he finds that his twin sister has aged more than he did during his trip because her velocity in the time direction was faster than his was when he steered a portion of his velocity into the space direction.

Even Einstein Got it Wrong
Our second bit of background is something called the Lorentz contraction.  Special relativity specifies that as something increases its speed in the space direction, not only will time move more slowly for it, it will appear shorter.  What does this mean for someone watching a sphere fly by at reltivistic speeds?  For a long time, the general conception was that sphere would look like an elipse.  In other words, you'd see the circular outline of the sphere turn into an eliptical outline (picture 2).

However, starting in 1959, a number of authors including Terrell[4] and Penrose[5] showed that this was in fact not the case.  The outline of a relativistically moving sphere looks like the outline of a sphere that is moving at the same speed as you.  This may not have even been apparent to Einstein are Terrell points out (picture 3).

Consider the following three dimensional analog.  If a straight line rotated away from you, you'd perceive it as being shorter.  If a sphere rotated away from you, you'd still see a perfect sphere.

Given the background sections above, first, that we can think of moving relativistically as 'steering' our speed into and out of the time dimension, and second, the fact that a sphere still looks like a sphere at relativistic speeds, is it possible to describe all of special relativity merely as rotations in a four dimensional space?  Physicists already talk about special relativity in terms of Lorentz transforms which are mathematically called rotations.  However, as demonstrated by the sphere example above, we stop short of considering these rotations to be common every day rotations as opposed to formal mathematical constructs.  If it is possible, what do the mathematical constructs that describe these rotations look like?  What simplifications and conceptual enlightenment can be gained from looking at the theory from this perspective?




3.  Check out pp. 47-48

4.  Invisibility of the Lorentz Contraction

Phys. Rev. 116, 1041 (1959) - Invisibility of the Lorentz Contraction

5.  The apparent shape of a relativistically moving sphere


Jonah said…
Great post! The fact that a sphere always looks the same after Lorentz transform always puzzled me. Thanks for explaining it so nicely!

I didn't explain this in my post on the topic, because I felt it was too technical for my blog, but there's a transformation called a Wick rotation we use to go between Lorentzian and Euclidian spacetimes.

The idea is that you rotate through the complex plane to change the lorentzian signature to a Euclidean one. You've probably seen a matrix for Lorentz transforms that takes the form

[ cosh(phi) -sinh(phi) ]
[ -sinch(phi) cosh(phi) ],

where phi is called the boost parameter, and

phi = cosh^{-1}(gamma).

The secret is that the hyperbolic trig functions are just trig functions of an imaginary angle.

cosh(x) = cos(ix)
sinh(x) = -isin(ix)

So you're absolutely right that Lorentz transforms are a rotation. The secret is that if the norm square of timelike vectors is negative, those vectors are complex. Furthermore, the angle of "rotation" for a Lorentz boost is imaginary.
Hamilton Carter said…
Thanks Jonah!

I've seen the Wick rotation a lot over on the Casimir side of my work. Both Sommerfeld, in the second volume of his physics course, and Moller have excellent examples where they treat the complex rotation angle as real. In Sommerfeld's case, he shows very nicely how the E and B fields rotate into each other as the velocity of a travelling charge is changed.

Popular posts from this blog

Cool Math Tricks: Deriving the Divergence, (Del or Nabla) into New (Cylindrical) Coordinate Systems

The following is a pretty lengthy procedure, but converting the divergence, (nabla, del) operator between coordinate systems comes up pretty often. While there are tables for converting between common coordinate systems, there seem to be fewer explanations of the procedure for deriving the conversion, so here goes!

What do we actually want?

To convert the Cartesian nabla

to the nabla for another coordinate system, say… cylindrical coordinates.

What we’ll need:

1. The Cartesian Nabla:

2. A set of equations relating the Cartesian coordinates to cylindrical coordinates:

3. A set of equations relating the Cartesian basis vectors to the basis vectors of the new coordinate system:

How to do it:

Use the chain rule for differentiation to convert the derivatives with respect to the Cartesian variables to derivatives with respect to the cylindrical variables.

The chain rule can be used to convert a differential operator in terms of one variable into a series of differential operators in terms of othe…

The Valentine's Day Magnetic Monopole

There's an assymetry to the form of the two Maxwell's equations shown in picture 1.  While the divergence of the electric field is proportional to the electric charge density at a given point, the divergence of the magnetic field is equal to zero.  This is typically explained in the following way.  While we know that electrons, the fundamental electric charge carriers exist, evidence seems to indicate that magnetic monopoles, the particles that would carry magnetic 'charge', either don't exist, or, the energies required to create them are so high that they are exceedingly rare.  That doesn't stop us from looking for them though!

Keeping with the theme of Fairbank[1] and his academic progeny over the semester break, today's post is about the discovery of a magnetic monopole candidate event by one of the Fairbank's graduate students, Blas Cabrera[2].  Cabrera was utilizing a loop type of magnetic monopole detector.  Its operation is in concept very simpl…

Unschooling Math Jams: Squaring Numbers in their own Base

Some of the most fun I have working on math with seven year-old No. 1 is discovering new things about math myself.  Last week, we discovered that square of any number in its own base is 100!  Pretty cool!  As usual we figured it out by talking rather than by writing things down, and as usual it was sheer happenstance that we figured it out at all.  Here’s how it went.

I've really been looking forward to working through multiplication ala binary numbers with seven year-old No. 1.  She kind of beat me to the punch though: in the last few weeks she's been learning her multiplication tables in base 10 on her own.  This became apparent when five year-old No. 2 decided he wanted to do some 'schoolwork' a few days back.

"I can sing that song... about the letters? all by myself now!"  2 meant the alphabet song.  His attitude towards academics is the ultimate in not retaining unnecessary facts, not even the name of the song :)

After 2 had worked his way through the so…