San José State University
Thayer Watkins
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& Tornado Alley

The Derivation of the Lorentz Transformations
of Spacetime Observationsbetween Two Coordinate
Systems in Constant Motion with respect to One Another

Consider two events identified by their point in space and time of occurrence. In one system their spacetime coordinates might be

(x1, y1, z1, t1)
(x2, y2, z2, t2)

The spacetime interval σ between the two events is defined by

σ² = c²(t2−t1)² − (x2−x1)² − (y2−y1)² − (z2−z1

In another coordinate system the two events may have the spacetime coorinates

(x'1, y'1, z'1, t'1)
(x'2, y'2, z'2, t'2)

Let the second coordinate system be moving at a constant velocity with respect to the first. It is a property of spacetime that the spacetime interval separating the two events is the same in the two coordinate systems; i.e.

σ² = c²(t2−t1)² − (x2−x1)² − (y2−y1)² − (z2−z1
           = c²(t'2−t'1)² − (x'2−x'1)² − (y'2−y'1)² − (z'2−z'1

Now the question is what transformation between the intervals (Δx', Δy', Δz', Δt') of the second coordinate system will give the intervals (Δx, Δy, Δz, Δt) of the first coordinate system and maintain the constantcy of the spacetime interval between the two events.

In order to answer this question it is expedient to reduce the problem to manageable size. Let the coordinate systems' axes coincide at time zero and differ only in that the origin for the second system is moving along the x axis of the first system at a constant velocity v.

The constancy of the spacetime interval is now

σ² = (Δt)² − (Δx)² = (Δt')² − (Δx')²

The transformation we are looking for is linear and can be represented as

Δx = a·Δx' + b·Δt'
Δt = d·Δx' + f·Δt'

where a, b,d and f are functions.

The quest can be reduced significantly by making use of the assumption that the origins of the two coordinate systems coincide at t=0. Then the origin for the first system has x=0 but the origin of the second system has the coordinate x=vt in the first system

Then putting these values into the equation x=ax' + bt gives

0 = a(vt) + bt
which implies
b =−av

This means that the first equation of the Lorentz transformation is

x = a(x' − vt')

Events connected by a ray of light have a zero spacetime interval. Therefore any event, on the x axis, connected to the origin by a light ray

c²t² = x²
c²t'² = x'²

Replacing x' with a(x−vt) and t' with dx+ft in the second of the above two equations gives

a²(x−vt)² = c²(dx + ft)²
which expanded is
a²(x² − 2xvt + v²t²) = c²(d²x² + 2dfxt + f²t²)
and with like terms collected together
(a² − c²d²)x² − 2(va² + c²df)xt = (c²f² − a²v²)t²

But it also holds that

x² = c²t²

Therefore the coefficients of x², t² and xt must be the same in the two equations. This means that

a² − c²d² = 1
c²f² − a²v² = c²
va² + c²df = 0

Note that the second equation is equivalent to

f² −a²(v/c)² = 1

The first task is to eliminate d. The third equation gives

d = − va²/(c²f)
and hence
d² = v²a4/(c4f²)

When this expression for d² is substituted into the first equation the result is

a² − ((v/c)²/f²)a4 = 1
or, equivalently
((v/c)²/f²)(a²)² − a² + 1 = 0

There are now two equations in two unknowns, a² and f²; i.e.,

(v/c)²(a²)² = f²(a² − 1)
f² = a²(v/c)² + 1

When the expression for f² in the second equation is substituted into the first equation the result is:

(v/c)²(a²)² = (v/c)²(a²)² − a²(v/c)² + a² − 1
which reduces to
a²(v/c)² − a² + 1 = 0
and further to
(1 − (v/c)²)a² = 1


a = 1/(1 − (v/c)²))½


f²=a²(v/c)² +1
f² = (v/c)²/(1 − (v/c)²) + 1 = (((v/c)² + 1 − (v/c)²)/(1 − (v/c)²)
f² = 1/(1 − (v/c)²)

Thus f is equal to a.

Finally, from above,

d = − a²v/(c²f) = − av/c²
and hence
d = −(v/c²)/(1 − (v/c)²))½

Let 1/(1 − (v/c)²))½ be denoted as γ. Then the Lorentz transformations may be expressed as

x =γ(x' − vt')
t = γ(t' − x'v/c²)

With these formulas for the Lorentz transformation one can derive the relativistic formula for the addition of velocities. That is to say, if there is a second coordinate system moving with a velocity of v with respect a first coordinate system and in the second coordinate system an object is moving with a velocity of u the velocity of the object with respect to the first system is given by

u' = (v + u)/(1 + vu/c²)

This formula then implies the constantcy of the speed of light. If the object is a photon and u=c then

u' = (v + c)/(1 + vc/c²) = (v + c)/(1 + v/c)
and multiplying numerator
and denominator by c
u' = c(v+c)/(c+v) = c

Historically the intellectual development went the other way around. After Albert Michelson and Edward Morley found the constantcy of the speed of light in 1887, mathematical physicists looked for a coordinate transformation that would imply the constantcy of the speed of light. The Lorentz transformation is what they found.

(To be continued.)

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