Remark: I have noticed that sometimes Firefox does not upload the applets. Apologies if you do not see them in the panels. Google Chrome, however, seems to be OK every time.

It is time to finally show how the addition of relativistic velocities can be constructed geometrically. It should not be a surprise that it can be achieved with the menhir (explained in the previous weeks). Before I show the details (in the next segment) let me present a nice property that plays some role in the discovery of this construction.

Butterfly Porism: Given a circle and a line and four points on the line. If there exists a quadrilateral inscribed in the circle through the four points then there exists an infinitely many such quadrilaterals.

The applet below explains the situation:

Move the red point on the circle to see that the fourth point (blue) on the line does not move!

You may also move the yellow points to change the position of the line.

Move the red points on the line to change the butterfly's setup.

Some challenges:

1. Prove the above property
2. Redraw with some of the four points possibly outside the disk
3. What does it have to do with relativity?...

This will be a short entry. The methods of menhirs presented in the previous blog allows one for a simple geometric construction of the composition of two velocities in more than 1 dimension. Here it the result:

Move $\mathbf v_1$ and $\mathbf v_2$ by dragging their tips (red points)

Exercise: Check out a number of configurations with $\mathbf v_1\bot \mathbf v_2$. You should notice something interesting; check it against the first formula of the last blog.

The meaning: two consecutive boosts, first by velocity $\mathbf v_1$ denoted $B(\mathbf v_1)$, followed by boost $B(\mathbf v_2)$ can be replaced by a single boost along certain velocity denoted $\mathbf v_1 \oplus \mathbf v_2$ and called the composition of the two velocities. In the Galilean physics, we have simply $\mathbf v_1 \oplus \mathbf v_2 = \mathbf v_1 + \mathbf v_2$, and the case is closed.

In the relativity, the velocity $\mathbf v_1 \oplus \mathbf v_2$ is not anymore a linear sum of two vectors. Moreover, two boosts can be replaced by a single boost followed by a rotation. $$ B(\mathbf v_2) \ \circ \ B(\mathbf v_1) \quad = \quad R \ \circ \ B(\mathbf v_1 \oplus \mathbf v_2) $$ The applet above shows both $\mathbf v_1 \oplus \mathbf v_2$ as a vector and the rotation angle marked by an arc on the circle. Have fun exploring different configurations of the velocities.

To see better the difference between the relativistic case and simple sum of vectors (classical Galilean composition), play with the version of the applet on which the regular sum is shown too (red arrow)

Move $\mathbf v_1$ and $\mathbf v_2$ by dragging their tips (red points).

Exercise: check when the discrepancy between $\mathbf v_1+ \mathbf v_2$ and $\mathbf v_1\oplus\mathbf v_2$ is the greatest / smallest.

And that's it for now! The explanation of how and why it works and what does it have to do with the cromlech and menhirs will come soon.

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:

Which stars do not move?

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?

What if you remember the positions of only two stars from the previous evening?

What if you remember the position of only one star?

Would you be able to reconstruct v if two consecutive boosts by two different velocities happened?

Here is a truly amazing puzzle. The idea is to put together three identical metal pieces into a 3-dimensional cross. That it can be done is shown below.

The task seems rather simple yet it is unexpectedly difficult. Some of my friends who are versed in such puzzles are still baffled.

This toy may serve as a great metaphor for a problem the essence of which lies not in complexity but rather in the invisibility of the solution. I am sure you know many examples from the history of science.

Questions: Who is the inventor? Is it available for purchase? Is it described anywhere? If you know anything about it, please let me know.

PS. My copy is a gift from my friend Tomasz Kasprowicz. It bears a logo of a bank in Poland (Centralny Dom Maklerski).

Adding two velocities v and w (in 1 dimension) in the relativistic mechanics needs to be done with the following formula:
$$v\oplus w = \frac{v+w}{1+vw}$$
The speed of light is assumed c=1. The interval $(-1,1)$ together with $\oplus$ forms a group. Here is an example of such "addition":
$$\frac{1}{2}\oplus\frac{1}{2}= \frac{1/2 + 1/2}{ 1+\frac{1}{2}\frac{1}{2}} = \frac{4}{5} $$
The following diagram is sort of a slide rule for space travelers (like Ijon Tichy).
The horizontal line is the real axis for the values of velocities.
Point labelled "v+w" in the figure represents the relativistic sum of velocities v and w. Move "v" or "w" to see how the result changes.

You may even experiment with tachyons! Slide velocity v to a value greater than 1 and see how adding a small velocity w slows it down.