KAHIBARO
Table of Contents
Why New Transformations Are Needed
In everyday life, we often connect measurements made in two moving reference frames by using simple subtraction of speeds. That idea belongs to Galilean relativity, which has its own chapter. Lorentz transformations replace that older rule when motion involves speeds comparable to the speed of light.
The key reason is that space and time cannot be transformed independently if the speed of light has the same value in all inertial frames. If one observer measures a flash of light moving with speed $c$, then every other inertial observer must also measure the same value $c$. To make this possible, coordinates of space and time must mix together.
The Basic Setup
Consider two inertial frames, $S$ and $S'$. Frame $S'$ moves with constant speed $v$ in the positive $x$ direction relative to $S$. We choose the axes so that they are parallel, and we let the origins coincide at time $t = t' = 0$.
An event is something that happens at a definite place and time, such as a flash of light or a particle decay. In frame $S$, the event has coordinates $(x, y, z, t)$. In frame $S'$, it has coordinates $(x', y', z', t')$.
Lorentz transformations tell us how these coordinates are related.
The Lorentz Transformation Equations
For motion along the $x$ axis, the Lorentz transformations are
$$
x' = \gamma (x - vt)
$$
$$
t' = \gamma \left(t - \frac{vx}{c^2}\right)
$$
$$
y' = y
$$
$$
z' = z
$$
where
$$
\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}
$$
The factor $\gamma$ is called the Lorentz factor. It becomes very close to $1$ when $v \ll c$, so the relativistic formulas then approach the classical ones.
Important Lorentz transformation for standard motion along the $x$ axis:
$$
x' = \gamma (x - vt), \qquad
t' = \gamma \left(t - \frac{vx}{c^2}\right)
$$
with
$$
\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}
$$
The crucial new feature is that $t'$ depends on both $t$ and $x$. Time and space are linked.
The Inverse Transformations
If we want to go from $S'$ back to $S$, we reverse the relative velocity, replacing $v$ by $-v$. This gives
$$
x = \gamma (x' + vt')
$$
$$
t = \gamma \left(t' + \frac{vx'}{c^2}\right)
$$
$$
y = y'
$$
$$
z = z'
$$
The symmetry is important. Neither frame is preferred. Each observer can regard the other as moving.
Meaning of the Lorentz Factor
The Lorentz factor
$$
\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}
$$
controls how strong relativistic effects are. Its value increases rapidly as $v$ approaches $c$.
| $v/c$ | $\gamma$ |
|---|---|
| 0.1 | 1.005 |
| 0.5 | 1.155 |
| 0.8 | 1.667 |
| 0.9 | 2.294 |
| 0.99 | 7.089 |
At low speeds, $\gamma \approx 1$, so Lorentz transformations nearly match ordinary intuition. At high speeds, the difference becomes large.
How Light Leads to These Equations
Suppose a pulse of light is emitted from the common origin when $t = t' = 0$. In frame $S$, the light satisfies
$$
x^2 + y^2 + z^2 = c^2 t^2
$$
In frame $S'$, it must also satisfy
$$
x'^2 + y'^2 + z'^2 = c^2 t'^2
$$
The Lorentz transformation is constructed so that both equations are true. This preserves the same light speed for all inertial observers.
This is the central mathematical difference from classical coordinate changes.
Spacetime Mixing
In classical physics, changing frames changes position, but time stays the same. In relativity, the formulas show something deeper:
$$
x' \text{ depends on } x \text{ and } t
$$
$$
t' \text{ depends on } x \text{ and } t
$$
This means that two observers moving relative to each other generally disagree about both where and when an event happens. This idea supports later topics such as time dilation, length contraction, and relativity of simultaneity, which are treated in their own chapters.
Relativity rule:
A change of inertial frame does not just shift space coordinates. It also changes time coordinates.
So, in special relativity, space and time are not separate. They form spacetime.
The Spacetime Interval
A very important quantity is the spacetime interval between two events. For differences
$$
\Delta x = x_2 - x_1, \quad
\Delta y = y_2 - y_1, \quad
\Delta z = z_2 - z_1, \quad
\Delta t = t_2 - t_1
$$
the interval is
$$
\Delta s^2 = c^2 \Delta t^2 - \Delta x^2 - \Delta y^2 - \Delta z^2
$$
Lorentz transformations preserve this quantity:
$$
\Delta s'^2 = \Delta s^2
$$
This is similar to how ordinary rotations preserve distance in geometry. Lorentz transformations can be thought of as spacetime rotations, but with the minus signs that reflect the structure of spacetime.
Invariant spacetime interval:
$$
\Delta s^2 = c^2 \Delta t^2 - \Delta x^2 - \Delta y^2 - \Delta z^2
$$
This quantity has the same value in all inertial frames.
Comparison with Galilean Transformations
It helps to compare the old and new coordinate rules.
| Quantity | Galilean transformation | Lorentz transformation |
|---|---|---|
| $x'$ | $x - vt$ | $\gamma (x - vt)$ |
| $t'$ | $t$ | $\gamma \left(t - \frac{vx}{c^2}\right)$ |
| Light speed invariant? | No | Yes |
| Space and time mixed? | No | Yes |
The formula for $t'$ is the clearest sign that relativity changes our picture of time.
A Simple Example
Suppose an event happens in frame $S$ at position
$$
x = 600 \, \text{m}
$$
and time
$$
t = 4.0 \times 10^{-6} \, \text{s}
$$
Let frame $S'$ move at
$$
v = 0.6c
$$
relative to $S$.
First compute the Lorentz factor:
$$
\gamma = \frac{1}{\sqrt{1 - 0.6^2}} = \frac{1}{\sqrt{0.64}} = 1.25
$$
Now transform the position:
$$
x' = \gamma (x - vt)
$$
Since $c = 3.0 \times 10^8 \, \text{m/s}$,
$$
v = 0.6c = 1.8 \times 10^8 \, \text{m/s}
$$
Then
$$
vt = (1.8 \times 10^8)(4.0 \times 10^{-6}) = 720 \, \text{m}
$$
So
$$
x' = 1.25(600 - 720) = 1.25(-120) = -150 \, \text{m}
$$
Now transform the time:
$$
t' = \gamma \left(t - \frac{vx}{c^2}\right)
$$
Compute
$$
\frac{vx}{c^2}
= \frac{(1.8 \times 10^8)(600)}{(3.0 \times 10^8)^2}
= 1.2 \times 10^{-6} \, \text{s}
$$
Thus
$$
t' = 1.25(4.0 \times 10^{-6} - 1.2 \times 10^{-6})
= 1.25(2.8 \times 10^{-6})
= 3.5 \times 10^{-6} \, \text{s}
$$
So the same event has coordinates
$$
x' = -150 \, \text{m}, \qquad t' = 3.5 \times 10^{-6} \, \text{s}
$$
in frame $S'$.
Low Speed Limit
Lorentz transformations must agree with classical physics when speeds are small compared with $c$. If $v \ll c$, then
$$
\gamma \approx 1
$$
and
$$
\frac{vx}{c^2}
$$
is extremely small. Then the transformations become approximately
$$
x' \approx x - vt
$$
$$
t' \approx t
$$
which are just the classical transformation rules.
This is why Newtonian mechanics works so well for ordinary everyday motion.
Geometric View
In ordinary geometry, rotating axes changes coordinates but preserves distance. In relativity, changing inertial frames changes coordinates but preserves the spacetime interval.
This analogy is useful, even though Lorentz transformations are not ordinary spatial rotations. They act in spacetime, not just in space.
The lines $x = \pm ct$ represent the motion of light. Lorentz transformations preserve these light paths.
What Lorentz Transformations Tell Us
Lorentz transformations give the mathematical rule for translating event coordinates between inertial frames in special relativity. Their most important message is that measurements of space and time depend on the observer's state of motion, but they do so in a precise and consistent way.
They preserve the speed of light and the spacetime interval, and they reduce to classical formulas at low speeds. Because of this, they form the foundation for the rest of special relativity.
Core ideas of Lorentz transformations:
$$
x' = \gamma (x - vt), \qquad
t' = \gamma \left(t - \frac{vx}{c^2}\right)
$$
They preserve the speed of light.
They preserve the spacetime interval.
They reduce to Galilean transformations when $v \ll c$.