Originally Posted by

**milli360**
Consider a thought experiment identical to the one Einstein describes in

**§3** of the 1905 paper, but in which the "mirror" which reflects observer k's light signal is at spatial co-ordinates (0,d,0) - that is, it's

**d** units above the moving observer, at right angles to the direction in which he's moving.

From his perspective the light signal moves vertically up to the mirror, is reflected, and moves vertically back down to him - a round trip of

**2d** units. The time for the round trip by his clock we will call

**Δť**. Since the speed of light is

**c**:

**cΔť = 2d ..... Equation 1**
Now let's look at the same events from the point of view of the stationary K observer.

During the light signal's round trip, his clock registers a time of

**Δt**. In this interval, the k observer moves a distance of

**vΔt**. This means that the light signal does not move vertically up and down, but diagonally along the arms of a lambda-shape: Λ

By simple geometry,

**l**, the length of one of these arms, is:

**l = √[d²+(½uΔt)²]**
Since the speed of light is the same for both observers (the condition that footnote is referring to):

**cΔt = 2l**
=>

**cΔt = 2√[d²+(½uΔt)²] ..... Equation 2**
Solving

**Equation 1** for

**d**, we get:

**d = ½cΔť **
Substituting this in

**Equation 2**, we get:

**cΔt = 2√[(½cΔť)² +(½uΔt)²]**
Solving this for Δt, we get:

**Δt = Δť/√[1-(u²/c²)]**
This is the time-dilation factor. A similar thought experiment (which I won't rehearse), in which the light pulse is emitted horizontally in the same direction as k's motion, allows us to derive the Fitzgerald-Lorentz-contraction equation:

**l = ľ√[1 - (u²/c²)]**
It's fairly easy to derive the Lorentz transformation equations once you are equipped with these two relations.