The Lorentz transformation is one of the central mathematical tools of special relativity. It connects the coordinates
of the same physical event as measured in two inertial frames moving at constant relative speed along the x direction.
In classical mechanics, time is absolute and the Galilean transformation is enough. In relativity, however, the speed
of light is invariant, so space and time must mix together in a precise way. That mixing is exactly what the Lorentz
transformation describes.
Suppose frame S′ moves with speed v relative to frame S along the positive x axis. Then the standard forward
transformation is:
Forward transformation. Start with coordinates in S and compute the coordinates in S′.
\[
\begin{aligned}
x' &= \gamma \left(x - v t\right), \\
t' &= \gamma \left(t - \frac{v x}{c^2}\right), \\
y' &= y, \\
z' &= z,
\end{aligned}
\qquad
\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}.
\]
The factor γ is called the Lorentz factor. It stays close to 1 when v is much smaller than c, but it grows rapidly
when v becomes a large fraction of the speed of light. That is why relativistic effects are negligible at everyday
speeds but become essential for particle physics, high-energy astrophysics, and precision timing.
Inverse transformation. Start with coordinates in S′ and recover the coordinates in S.
\[
\begin{aligned}
x &= \gamma \left(x' + v t'\right), \\
t &= \gamma \left(t' + \frac{v x'}{c^2}\right), \\
y &= y', \\
z &= z'.
\end{aligned}
\]
Notice that the same Lorentz factor γ appears in both directions. The difference between the forward and inverse forms
is the sign of the mixed term. This sign change reflects the fact that when one frame sees the other moving in the
positive x direction, the reverse description sees the motion in the opposite sense.
The transverse coordinates y and z are unchanged because the relative motion is purely along x. Only the coordinate
parallel to the motion mixes with time. This is why the calculator leaves y and z alone while transforming x and t.
Why simultaneity changes
One of the most important consequences of the Lorentz transformation is the relativity of simultaneity. Two events that
happen at the same time in one frame do not generally happen at the same time in another. To see this clearly, subtract
the coordinates of two events:
Event separations. Compare two events with coordinate differences.
\[
\begin{aligned}
\Delta x' &= \gamma \left(\Delta x - v \Delta t\right), \\
\Delta t' &= \gamma \left(\Delta t - \frac{v \Delta x}{c^2}\right).
\end{aligned}
\]
If the two events are simultaneous in S, then Δt = 0. Substituting that into the time-separation formula gives:
Simultaneous in S. Set \(\Delta t = 0\) and simplify the transformed time difference.
\[
\begin{aligned}
\Delta t' &= \gamma \left(0 - \frac{v \Delta x}{c^2}\right) \\
&= -\,\gamma \frac{v \Delta x}{c^2}.
\end{aligned}
\]
Unless the two events also occur at the same position, Δt′ is not zero. That is the exact mathematical statement of
the relativity of simultaneity. The calculator’s second-event option is built specifically to show this effect.
Worked example
Take the sample input β = 0.8, x = 100 m, and t = 0 s in frame S. First compute the Lorentz factor:
Step 1. Compute γ from β.
\[
\begin{aligned}
\gamma &= \frac{1}{\sqrt{1-\beta^2}} \\
&= \frac{1}{\sqrt{1-(0.8)^2}} \\
&= \frac{1}{\sqrt{0.36}} \\
&= 1.667.
\end{aligned}
\]
Step 2. Transform the x coordinate.
\[
\begin{aligned}
x' &= \gamma \left(x - v t\right) \\
&= 1.667 \left(100 - v \cdot 0\right) \\
&= 166.7\ \mathrm{m}.
\end{aligned}
\]
Step 3. Transform the time coordinate.
\[
\begin{aligned}
t' &= \gamma \left(t - \frac{v x}{c^2}\right) \\
&= 1.667 \left(0 - \frac{0.8c \cdot 100}{c^2}\right) \\
&= -\,1.667 \cdot \frac{80}{c} \\
&\approx -\,4.45 \times 10^{-7}\ \mathrm{s} \\
&\approx -\,0.445\ \mu\mathrm{s}.
\end{aligned}
\]
So the event has coordinates \(x' = 166.7\ \mathrm{m}\) and \(t' \approx -0.445\ \mu\mathrm{s}\) in S′. The negative
value of \(t'\) means that, in frame S′, this event occurs before the S′-frame origin event at \(t' = 0\).
How to read the diagram
The calculator’s diagram uses x horizontally and c·t vertically. This is a standard trick in relativity because it
gives both axes the same unit, meters. The light-cone lines satisfy \(x = \pm c t\), or equivalently \(c t = \pm x\).
Any signal that moves at the speed of light lies on one of those diagonal lines.
The transformed axes are tilted relative to the original axes. The x′ axis represents \(t' = 0\), and the c·t′ axis
represents \(x' = 0\). The same event point can be read using either coordinate grid. That geometric picture makes it
easier to understand why different observers disagree about time intervals, lengths, and simultaneity while still
describing the same underlying physical reality.
Core formulas summary. These are the main relations used by the calculator.
\[
\begin{aligned}
x' &= \gamma \left(x - v t\right), \\
t' &= \gamma \left(t - \frac{v x}{c^2}\right), \\
\Delta t' &= \gamma \left(\Delta t - \frac{v \Delta x}{c^2}\right), \\
\gamma &= \frac{1}{\sqrt{1-\beta^2}}.
\end{aligned}
\]
At a more advanced university level, these formulas are often rewritten using four-vectors and the invariant interval.
That language is more compact and more powerful, but the event-by-event Lorentz transformation remains the clearest
starting point for building physical intuition.
