Do Headlight's Work At The Speed Of Light? - Relativistic Velocity Addition

Monday, October 01, 2018

Do Headlight's Work At The Speed Of Light? - Relativistic Velocity Addition

Relativistic Velocity Addition

Galilean relativity and the invariant speed of light seem to contradict.

Galilean relativity says that if 2 cars approach each other, each driving at 100km/h in opposite directions, their closing velocity is 200km/h. This closing velocity is gotten by adding the two car's speeds.

So what happens if you drive towards a light source at 100km/h? Will you measure the velocity of the light to be 100 + c?
Another harder question is what happens if we turn on headlights in a car while going at the speed of light? Think about it for a minute and you'll see why its interesting. Things with mass can't reach the speed of light but it is still interesting to think what would happen.

Firstly lets answer the simpler question and then hopefully we can apply the answer to our super fast car.

No the velocity measured is not 100 + c. The speed of light is invariant. No matter how fast you drive towards a light source; you will always measure it at speed c. A spectator watching you drive towards the light source will measure a closing velocity of your speed + c but for the person in the car this formula does not apply.

Clearly, combining velocities in special relativity must be different to what we expect. The formula must have 2 key features
Adding two velocities should never give a result bigger than the speed of light because nothing with mass can reach the speed of light.
The formula should show that Galilean velocity addition is simply an approximation for what really happens and is only accurate at low speeds.

To find the formulas we will need to use the Lorentz transformations. This framework will allow us to easily move between frames of reference and see what is happening from the different point of views. We will use the 1D transformations to simplify the problem.

The Hare And The Tortoise

Lets consider the hare and the tortoise. They have a race where the hare moves at speed w and the tortoise moves at speed v (the hare is faster than the tortoise).
From our perspective, the closing velocity is v - w (a negative quantity meaning they are actually separating) or the speed of separation is w - v.

We will take the start line to be the origin.
From our frame of reference
the hare has space-time coordinates of (t , wt) (at time t the hare is at position wt)
The tortoise is at (t , vt)

From the perspective of the tortoise things are different however. We can use the Lorentz transformations to go from our POV to the tortoise's.
t' = γ(t - vx/c²)
x' = γ(x - vt) (Note that we use v because this is the hare's POV)

We can plug in the position of the hare (x = wt) to tell us where the hare is at time t from the tortoise's POV.

The location of the hare in space is given by substituting wt for x in the position equation
x' = γ(x - vt)
x' = γ(wt - vt)
x' = γt(w - v) - The location of the hare from the tortoise's POV

The location in time is given by doing the same thing
t' = γ(t - vx/c²)
t' = γ(t - vwt/c²)
t' = γt(1 - vw/c²) - The location in time of the hare from the tortoise's POV

We know the location of the hare from the tortoise's POV - x'
We know how long it took the hare to get there - t'

This means we can work out the relative velocity of the hare by doing speed = distance/time

Relative velocity = x'/t'
= γt(w - v) / γt(1 - vw/c²)
= (w - v)/ (1 - vw/c²)

And this is our equation for how to add velocities w and v in 1 dimension.

This fits the requirements we set for the equation at the start.
At low speeds it is very very very close to what classical mechanics predicts.
This is because v * w is a small quantity which is made even smaller when divided by c². This means we can ignore it and the equation becomes (w - v)/1
= w - v

At high speeds it always insures that the relative velocity of an object is never above c.
For example if you drive towards a light source at speed v (note that we change the signs in the equation since velocity is a vector); we can show that you will always measure it at speed c.

(c + v)/ (1 + vc/c²)

= (c + v)/ (1 + v/c)

= (c + v)/ (c + v)/c

= c (c + v) / (c + v)

= c

Headlight's At The Speed Of Light

Firstly, nothing with mass can move at the speed of light so that's one way to answer the question.
But if we say that the car is made out of photon's then what will the photon's see? Well remember the sad life of a photon part 1, part 2 and part 3? Photon's do not experience distance or time. Velocity is a distance divided by a time. So velocity doesn't exist in a photon car and the question is meaningless.

The formula confirms that it is a meaningless question to ask do headlights work at the speed of light.
w = c and v = c
(c - c)/ (1 - c²/c²)
= 0/ (1 - 1)
= 0/0

It's probably best to not try and interpret what measuring a speed of 0/0 means...

Now we have a formula that can add velocities in 1D, we can try to extend it to one which can work in 3D. We will do this in the next post on special relativity. After we have these tools we will be ready to take on Einstein's famous equation - E = mc².

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