Time dilation and length contraction are two of the best-known consequences of special relativity. They appear when
two observers move relative to one another at a speed that is a substantial fraction of the speed of light. In that
regime, classical intuition fails: moving clocks do not tick at the same rate as clocks at rest, and moving objects
do not keep the same measured length along the direction of motion. The calculator models both effects with the
Lorentz factor \(\gamma\), which controls how strongly relativistic corrections appear.
The key quantity is the speed fraction \(\beta = v/c\), where \(v\) is the relative speed and \(c\) is the speed of
light. From this, the Lorentz factor is:
Lorentz factor. This is the common scaling factor used in both time dilation and length contraction.
\[
\begin{aligned}
\gamma &= \frac{1}{\sqrt{1-\beta^2}} \\
&= \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}.
\end{aligned}
\]
When \(v\) is small compared with \(c\), \(\gamma\) is very close to 1, so relativistic effects are tiny. As
\(v\) approaches \(c\), \(\gamma\) grows rapidly, which makes the effects dramatic.
Proper time and dilated time
Proper time, written as \(\Delta t_0\), is the time interval measured by a clock that is at rest with the process
being observed. For example, if a muon exists in its own rest frame, the lifetime measured there is its proper
lifetime. An observer who sees that same muon moving at high speed measures a larger interval:
Time dilation formula. The moving clock appears to run slow, so the outside observer measures a longer interval.
\[
\begin{aligned}
\Delta t &= \gamma \Delta t_0.
\end{aligned}
\]
This does not mean the moving observer feels anything unusual. In the moving clock’s own rest frame, that clock
behaves normally. The difference comes from comparing measurements made in different inertial frames.
Proper length and contracted length
Proper length, written as \(L_0\), is the length of an object measured in the frame where that object is at rest.
If the object moves relative to an observer, then the measured length along the direction of motion becomes shorter:
Length contraction formula. Only the component parallel to the motion is contracted.
\[
\begin{aligned}
L &= \frac{L_0}{\gamma}.
\end{aligned}
\]
The contraction is not a physical crushing of the object in its own rest frame. Instead, it is the length obtained
by an observer in another frame who must determine the positions of both ends at the same time in that observer’s
frame. That simultaneity condition is essential.
Worked muon example
A standard introductory example uses the muon, whose proper lifetime is about \(2.2\ \mu\text{s}\). Suppose the muon
moves at \(\beta = 0.99\). Then:
Step 1. Compute the Lorentz factor.
\[
\begin{aligned}
\gamma &= \frac{1}{\sqrt{1-(0.99)^2}} \\
&= \frac{1}{\sqrt{1-0.9801}} \\
&= \frac{1}{\sqrt{0.0199}} \\
&\approx 7.09.
\end{aligned}
\]
Step 2. Compute the dilated lifetime seen in the Earth frame.
\[
\begin{aligned}
\Delta t &= \gamma \Delta t_0 \\
&= 7.09 \cdot 2.2\ \mu\mathrm{s} \\
&\approx 15.6\ \mu\mathrm{s}.
\end{aligned}
\]
So a muon that survives only \(2.2\ \mu\text{s}\) in its own rest frame can appear to survive about
\(15.6\ \mu\text{s}\) in the Earth frame when moving at \(0.99c\). This is why fast muons produced high in the
atmosphere can still be detected much lower down.
Step 3. If a rod has proper length \(L_0 = 10\ \mathrm{m}\), then the observed moving length is:
\[
\begin{aligned}
L &= \frac{L_0}{\gamma} \\
&= \frac{10}{7.09} \\
&\approx 1.41\ \mathrm{m}.
\end{aligned}
\]
This numerical contrast is useful because it shows that the same Lorentz factor makes time intervals larger and
lengths smaller when viewed from the external frame.
Twin paradox preview
The calculator also includes a twin-paradox style interpretation. On any single inertial leg of motion, the traveler
can be assigned a proper elapsed time \(\Delta t_0\), while the stay-at-home frame assigns the larger interval
\(\Delta t = \gamma \Delta t_0\). That means the traveler ages less during that leg. However, a complete twin paradox
analysis requires more than one inertial frame because the turnaround breaks the symmetry. At a more advanced level,
the relativity of simultaneity is what resolves the apparent contradiction.
How to interpret the animation
The animation compares an Earth-frame clock with a moving clock and also compares a proper rod with its contracted
version. The moving clock accumulates less proper time over the same Earth-frame interval, while the moving rod is
drawn shorter along the motion direction. The side strip summarizes the traveler-versus-lab elapsed times so that the
time-dilation effect is easy to see visually.
Core summary formulas. These are the main relations used by the calculator.
\[
\begin{aligned}
\beta &= \frac{v}{c}, \\
\gamma &= \frac{1}{\sqrt{1-\beta^2}}, \\
\Delta t &= \gamma \Delta t_0, \\
L &= \frac{L_0}{\gamma}, \\
d &= v \Delta t.
\end{aligned}
\]
Together, these relations show the main pattern of special relativity for one-dimensional motion: moving clocks are
observed to run slow, moving lengths are observed to shrink, and both effects are governed by the same Lorentz
factor. At university level, these ideas connect naturally to Lorentz transformations, spacetime intervals, and the
relativity of simultaneity.
