We are all familiar with velocities. Velocities tell us how positions change with time. Velocities can not be assigned to individual objects, as they describe a relation between pairs of objects. We know this since Galileo Galilei. Yet, in common day language this profound fact is mostly ignored.

Make a statement like "The ball has a distance of 30 meters", and sure enough, people will frown and reply with a question for clarification: "Distance to what?". Make the equally silly statements "The ball has a velocity of 20 meters per second" and the same folks will nod to indicate they perfectly understand what you are saying.

Galilean Relativity

Relativity is commonly associated with Einstein, yet the first person to formulate a principle of relativity was Galileo Galilei. Core of Galilean relativity is the mere fact that velocities are relative. In the following, I will make this explicit by writing v_XY for the velocity of object X as seen from the perspective of object Y.

Imagine a railroad track. Velocities along this track are measured using real (positive or negative) numbers that denote the velocity expressed in terms of the speed of light. A train moves along the track. Travis is sitting still in the train. A housefly flies forward in the train and passes Travis. Travis (T) observes the fly (F) to have a velocity v_FT. The train with Travis and the fly passes a railway station. Stacey (S), who is standing still at the station, sees Travis passing by at a velocity v_TS. She doesn't notice the fly, but the fly does notice Stacey, and sees her passing by at a velocity v_SF. 

Galilean relativity tells us how all these velocities relate to each other. They simply add up to zero:

v_FT + v_TS + v_SF = 0

Notice the cyclic structure of the subscripts in this equation. This cyclicity is essential: swap the subscripts in one of the velocity terms, and the equation becomes invalid. Swap all three pairs of subscripts, and the equation is again cyclic, and again valid:

v_FS + v_ST + v_TF = 0

 In fact, the above equation is not just valid for Stacey, Travis and the Fly, but for any three objects moving along a line.

This velocity summation equation even holds true when applied to only two objects. Such a special case results when identifying one object to another. Let's say we ignore the fly and substitute the subscript T (Travis) for F (Fly). It follows that:

v_TS + v_ST + v_TT = 0

However, the velocity of an object seen from the object's own perspective is identical to zero:  v_TT = 0. Therefore:

v_TS + v_ST = 0

or

v_ST = -v_TS

In other words: if Stacey sees Travis in the train moving with a certain speed in one direction, Travis - when looking out of the train - sees Tracey moving with the same speed in the opposite direction. We are all familiar with this velocity reciprocity.

Let's go back to the first equation, and put in some numbers for the relative velocities. To make thing interesting, we assume a more than capable ultra-high-speed train and a very fast fly. Let's say the train moves relative to the station at 2/3 of the speed of light (v_TS = 2/3) and the fly propels itself through the train at 1/2 the speed of light (v_FT = 1/2). From above equation it follows that the fly sees Tracey moving backward at 7/6 times the speed of light (v_SF = -7/6). And hence, Tracey sees the fly moving forward at 7/6 times the speed of light (v_FS = 7/6). 

Lorentzian Relativity

Already as a teenager, Einstein was concerned about outcomes predicted by Galilean relativity. A fly (or anything else) overtaking a beam of light didn't make sense to him. Yet this is exactly what would happen in the above example if Tracey would shine a flashlight in the direction of movement of the train. How would the fly perceive the beam of light? The laws of electromagnetism which describe the propagation of light could not answer this question. Electromagnetism is not compatible with Galilean relativity. 

Anyone else would have tried to modify the equations of electromagnetism to render them compatible with Galilean relativity. Not so Einstein. His deep physical intuition and thorough understanding of electromagnetism told him that something else had to give. He became convinced that the concept of relativity of position and motion is in itself correct, but that Galilean relativity is rotten. So he modified the venerable 273 years old Galilean relativity to render it compatible with the laws of electromagnetism published by James Maxwell 40 years earlier.  

When Einstein was done, a new principle of relativity emerged. A beautiful principle of relativity that is compatible with particle motion as well as with electromagnetism. The mathematical equations for this form of relativity were already written down by Hendrik Lorentz, without him realizing what they really meant.  

Lorentzian (aka 'special') relativity modifies the cyclic velocity sum equation by adding another cyclic term to it, a term that is the product of the relative velocities:

v_FT + v_TS + v_SF + v_FT . v_TS . v_SF = 0

   The result is a modified velocity composition equation that no longer guarantees the three velocities to add to zero, but that remains compatible with velocity reciprocity. This reciprocity can be derived fromthe above equation by again ignoring the fly (F) and substituting the subscript T (Travis) for F (Fly):

v_TT + v_TS + v_ST + v_TT . v_TS . v_ST = 0

Using v_TT = 0 the product term drops out of the equation, and the velocity reciprocity relation follows:  v_TS + v_ST = 0.

More importantly, the Lorentzian velocity composition relation is fully compatible with electromagnetism. This can be seen by eliminating the fly (F), and replacing it with the light from a flashlight (F) that Travis shines in the forward direction of the train. Travis sees the flashlight photons traveling forward at the speed of light (v_FT = 1) and, according to the laws of electromagnetism, Stacey sees the same flashlight photons also moving forward at the speed of light (v_FS = 1). Using reciprocity (v_SF = -v_FS) it follows that v_SF = -1. Substituting the values for v_FT and v_SF in the above Lorentzian velocity composition equation it follows that 

1  + v_TS -1  -  v_TS  = 0

This equation is valid for any relative velocity v_TS between Travis and Stacey. In other words: no matter how fast the train moves, both Travis in the train and Stacey outside the train see the photons move at the same velocity. Exactly as expected from the equations of electromagnetism.

Now let's investigate again the situation of the fly moving through the train. How does the Lorentzian velocity formula change the outcomes when the fly moves at half the speed of light relative to the train, and the train moves at 2/3 of the speed of light relative to Stacey? Substituting v_TS = 2/3 and v_FT = 1/2, it follows that v_SF = -7/8. (Check for yourself.) Again using reciprocity, it follows that v_FS = 7/8. In other words: based on the modified equation Stacey sees the fly moving at less than the speed of light. A result compatible with the fact that nothing, no train, no fly, no particle, no neutrino (!?) can overtake an observer with a speed larger than the speed of light.

Lost Elegance?

The Lorentzian velocity equation has an additional term compared to the corresponding Galilean equation. This triple velocity product term is negligible compared to the other terms when all relative velocities are small compared to the speed of light. Yet, it still is an extra term. Somehow it seems that Lorentzian/special relativity is less elegant than Galilean relativity.

This is no more than an appearance. An appearance that disappears when one looks deeper into the concepts playing a role in both relativity theories. Galilean relativity places a lot of emphasis on velocities, not so Lorentzian relativity. Einstein's theory that leads to Lorentzian relativity is based on a spacetime description in which velocities are represented as slopes. Would the time and space dimensions behave similarly, velocities relative to an observer would be the tangent of angles with the time-axis relevant to that observer. However, time behaves in a way subtly different from the way spatial dimensions behave, and as a result velocity is measured by the hyperbolic tangent of an angle with the time-axis. This angle is referred to as rapidity. And the point is: these angles (rapidities) do add up to zero. That is to say: a Lorentzian rapidity composition formula would take the shape of three rapidities adding to zero. A Galilean rapidity composition formula, on the other hand, would be ugly. In terms of rapidities, Lorentzian relativity is the most elegant form of relativity.

Moreover, in contrast to velocities, rapidities can be determined as the sum of subsequent accelerations experienced by an object. Arguably, rapidities in Lorentzian relativity are the closest thing to the velocities that appear in Galilean relativity.  Small rapidities are indistinguishable from velocities, but the rapidity of light is infinite. 

Also rapidities are directly observable as the change in 'tone' of light due to Doppler shift. So when you are stopped by a police officer who accuses you of speeding, ask him how he established your velocity. If his answer contains the words "radar gun", you politely point out that radar Doppler measurements yield rapidities and not velocities. And you finish the conversation with an empathic "Surely you must have learned at the police academy that rapidities are always larger than velocities". 

By the way: next time you meet folks who believe in tachyonic neutrinos: ask them what rapidities these neutrinos attain...