Relativity, Stonehenge and celestial sphere (Part 1)

Back to relativity!  The question is: how to add velocities in more than one dimension. Here is the formula suggested by advanced books on relativity (recalled just for a temporary intimidation).
$$\mathbf v\oplus \mathbf w =
\frac{  \left(1 +  \frac{(1-\sqrt{1-|\mathbf v|^2})\,\mathbf v\cdot \mathbf w}{|\mathbf v|^2}  \right) \mathbf v  +  \sqrt{1-|\mathbf v|^2} \mathbf w    }{1 + \mathbf v\cdot \mathbf w}$$ It looks very discouraging and devoid of insight. But here is the good news: you may forget this formula! The problem may be solved in a rather elegant simple geometric way. The result is really based on the homomorphism of Lie groups
$${\rm SO}_o(1,n) \ \cong \ {\rm SU}_{(1,1)}( \mathbb F) \ \cong \ {\rm Rev}(n)$$ where the first is the Lorenz group of $(1+n)$-dimensional space-time (``o'' means the connected piece of the pseudo-orthogonal group $SO(1,n)$)  and the second is the unitary group of two-by-two matrices and the third is a group of certain geometric reversions of a sphere $S^n$. $$  n=1, \ \mathbb F = \mathbb R \qquad
 n=2, \ \mathbb F = \mathbb C \qquad
 n=4, \ \mathbb F = \mathbb H $$ (real numbers, complex numbers, quaternions).

For now, we will ignore the algebraic content. Instead, I invite you to  a hypothetical cosmic adventure. Let's pretend we are a group of scouts on a planetoid, lost in space.  For simplicity, let the planetoid be flat, not rotating or accelerating. Something like this.

Suppose it acquires some velocity (boost) v  (again, for simplicity: in the direction lying in the plane). How can you detect it?  What about two velocities: how do you add them?  (definitely not as vectors!).  Answer: build your own Stonehenge and observe stars.

Aberration - the apparent shift of stars in the celestial sphere

If the planetoid undergoes a boost, the stars will make a drift on the horizon to new positions. This is called a star aberration in astronomy.  Somewhat surprisingly for some, the stars are regrouped towards the direction of the boost. You might have seen it in some Sci-fi movies.  Here is an counter-example from Spaceballs:  the stars look as if they were trees you pass while running through forest.

This is wrong (keeping science right in Spaceballs would be disappointing!). What would really happen is presented in the applet below. The yellow dots represent the stars as seen on the horizon. You are located in the center.  Now move the red spot to a new position to choose a velocity. You will notice how the stars' apparent positions on the horizon change. At the speed of light they all gather in a point in front of you.  Except the star that is directly behind you.

Move the red dot from the center to choose a velocity and see how the stars' apparent positions on the horizon change. (Aberration)

Megalithic vocabulary

Here are two terms drawn from celtic languages:

cromlech -- a  circle of stones, like the one below, left.  Click to see a better resolution.
menhir  -- a single stone, typically prolonged and set vertically.  See below right

Cromlech

Menhir

Build a cromlech

Back to our planetoid.  We will build a cromlech. Start with making a circle (with a string: every scout must have one) of considerable radius.  It will be our unit. Set stones on the circle at the points at which your favorite stars are visible on the horizon.  This is the cromlech, our private Stonehenge. It represents the celestial sphere (oe rather circle, in our 2D setup).  Any aberration of stars in the sky will be represented on the cromlech=circle.

From now on, we shall picture the planetoid and the cromlech as seen from above.  For example, the previous applet should be understood exactly in this way.


Predicting the aberration

Now, suppose the planetoid undergoes a boost.  That is, it acquires a velocity.  Can we predict how the stars on the horizon are going to shift?  In other words, what are the new positions in the cromlech? Here is a simple geometric construction how we can do it.  First of all, instead of the velocity, we shall use its reduced version.  First, draw on the ground the velocity vector  v  (1=speed of light) and construct a point that is somewhat closer to the origin, as shown below.  Mark it with a stone -- we shall call it the menhir of the velocity.

Every relativistic velocity (arrow with the green tip) defines a special point that we call "menhir" (red dot). Move the green point (velocity vector) to see how the menhir changes.

A simple challenge left to the reader:  what is the algebraic relation for the map:

velocity   $\longleftrightarrow$   menhir

Once we have the menhir set, the shift of the stars-stones in the cromlech due to the acquired velocity can be easily predicted by the construction shown below.

The red point is the menhir. The yellow point represents a star on the horizon. The orange point -- the star's new position due to aberration. You may move the star  (yellow point). You may move the menhir  (red point).

Play with the applet and answer some questions. For instance:

1. Which stars do not move? 
2. If the acceleration happened while you slept at night, can you reconstruct the velocity in the morning by inspecting what happened to the star positions? How?
3. What if you remember the positions of only two stars from the previous evening? 
4. What if you remember the position of only one star? 
5. Would you be able to reconstruct v if two consecutive boosts by two different velocities happened?

... to be continued ....