The Lorentz Transformation
Welcome! This page introduces the physics behind the Lorentz Transformation interactive diagram. The material is aimed at pre-university physics students meeting special relativity for the first time.
Why Do We Need Special Relativity?
In everyday life we combine velocities by simple addition. If you walk at $2\;\text{m/s}$ on a train travelling at $30\;\text{m/s}$, a person on the platform sees you move at $32\;\text{m/s}$. This is the Galilean velocity addition:
$$u = u' + v$$
However, experiments in the late 19th century—most famously the Michelson–Morley experiment (1887)—showed that the speed of light is the same for every observer, regardless of their motion. This contradicts Galilean addition: if you shine a torch on a train, the light does not move at $c + v$ relative to the ground. It moves at exactly $c$.
In 1905, Albert Einstein resolved this by publishing the theory of special relativity, built on two postulates:
- Principle of Relativity: The laws of physics are the same in every inertial (non-accelerating) reference frame.
- Invariance of $c$: The speed of light in a vacuum, $c \approx 3.00 \times 10^{8}\;\text{m/s}$, is the same for all inertial observers.
The mathematical framework that makes these two postulates consistent is the Lorentz transformation.
The Lorentz Transformation Equations
Consider two inertial reference frames, $S$ (the "rest frame") and $S'$ (the "moving frame"). Frame $S'$ moves at velocity $v$ along the positive $x$-direction relative to $S$. If an event has coordinates $(x,\,t)$ in $S$, then in $S'$ its coordinates $(x',\,t')$ are given by:
$$x' = \gamma\,(x - v\,t)$$ $$t' = \gamma\!\left(t - \frac{v\,x}{c^{2}}\right)$$
where the Lorentz factor $\gamma$ is defined as:
$$\gamma = \frac{1}{\sqrt{1 - \dfrac{v^{2}}{c^{2}}}}$$
Natural units: In the interactive diagram, we measure distances in units where $c = 1$. This means that one unit on the $x$-axis equals one light-second, and one unit on the $t$-axis equals one second. Setting $c = 1$ simplifies the equations to:
$$x' = \gamma\,(x - v\,t), \qquad t' = \gamma\,(t - v\,x)$$
$$\gamma = \frac{1}{\sqrt{1 - v^{2}}}$$
Key Properties of $\gamma$
- When $v = 0$: $\gamma = 1$ and the transformation reduces to the identity—both frames agree.
- As $v \to c$: $\gamma \to \infty$, and relativistic effects become extreme.
- $\gamma$ is always $\geq 1$ for any physical velocity $\lvert v \rvert < c$.
| $v / c$ | $\gamma$ |
|---|---|
| $0$ | $1.000$ |
| $0.10$ | $1.005$ |
| $0.50$ | $1.155$ |
| $0.80$ | $1.667$ |
| $0.90$ | $2.294$ |
| $0.95$ | $3.203$ |
| $0.99$ | $7.089$ |
Consequences of the Lorentz Transformation
1. Time Dilation
A clock at rest in $S'$ (i.e. at a fixed $x'$) ticks more slowly as seen from $S$. If the clock measures a proper time interval $\Delta t_0$ between two events at the same location, the time interval measured in $S$ is:
$$\Delta t = \gamma\,\Delta t_0$$
Since $\gamma > 1$, moving clocks run slow: $\Delta t > \Delta t_0$. This is not an illusion—it is a real, measurable effect confirmed by experiments with fast-moving muons and precision atomic clocks on aircraft.
Try it in the app: Place two events at the same $x'$ position (on the same moving-frame worldline) and read off their $t$ and $t'$ coordinates. You will see that the time difference in the rest frame is larger than in the moving frame by a factor of $\gamma$.
2. Length Contraction
A rod at rest in $S'$ with proper length $L_0$ appears shorter when measured in $S$:
$$L = \frac{L_0}{\gamma}$$
The rod is contracted along its direction of motion. Like time dilation, this is a real physical effect that follows directly from the Lorentz transformation.
Try it in the app: Place two events at the same $t$ (simultaneously in the rest frame) separated by some distance $\Delta x$. Now look at their $x'$ separation. The moving frame measures a different spatial separation because the events are not simultaneous in $S'$.
3. Relativity of Simultaneity
Two events that are simultaneous in one frame ($\Delta t = 0$) are generally not simultaneous in another frame ($\Delta t' \neq 0$), unless they also occur at the same location. From the transformation:
$$\Delta t' = \gamma\!\left(\Delta t - v\,\Delta x\right)$$
If $\Delta t = 0$ but $\Delta x \neq 0$, then $\Delta t' = -\gamma\,v\,\Delta x \neq 0$.
Try it in the app: Place two events at the same $t$ but different $x$ values. Increase the velocity $v$ and watch their $t'$ coordinates diverge—demonstrating that simultaneity depends on the observer's frame of reference.
The Minkowski Diagram
The interactive diagram is a Minkowski diagram (also called a spacetime diagram). It plots position $x$ on the horizontal axis and time $t$ on the vertical axis.
Rest Frame Axes
The rest frame $S$ has the standard perpendicular axes $x$ and $t$. Gridlines of constant $x$ are vertical; gridlines of constant $t$ are horizontal.
Moving Frame Axes
When the moving frame $S'$ has velocity $v$ relative to $S$, its axes appear tilted on the diagram:
- The $t'$ axis (worldline of $x' = 0$) has the equation $x = v\,t$—it tilts towards the light cone.
- The $x'$ axis (line of $t' = 0$) has the equation $t = v\,x$—it also tilts towards the light cone.
As $v$ increases, both primed axes rotate towards the 45° light-cone line. They can never reach or cross it because nothing can travel at or beyond $c$.
The Light Cone
The dashed diagonal lines at 45° represent the paths of light rays passing through the origin ($x = \pm\,c\,t$, or simply $x = \pm\,t$ in natural units). These lines divide spacetime into regions:
- Inside the cone (above and below): timelike separated events. A massive particle can travel between them. Events above the origin are in its future; below, in its past.
- On the cone: lightlike (null) separated events. Only light can connect them.
- Outside the cone (left and right): spacelike separated events. No signal can travel between them. Different observers may disagree on their time ordering.
Key insight: The light cone is invariant under Lorentz transformations. Both frames agree on which events are inside, on, or outside the light cone. This is because the spacetime interval
$$s^{2} = c^{2}\,\Delta t^{2} - \Delta x^{2}$$
is the same in every inertial frame.
Relativistic Velocity Addition
If an object moves at velocity $u$ in frame $S$, what velocity $u'$ does an observer in frame $S'$ measure? The answer is not simple subtraction. The correct formula is:
$$u' = \frac{u - v}{1 - \dfrac{u\,v}{c^{2}}}$$
In natural units ($c = 1$):
$$u' = \frac{u - v}{1 - u\,v}$$
This formula ensures that no combination of sub-light velocities ever yields a result $\geq c$. For example, if $u = 0.9\,c$ and $v = 0.9\,c$:
$$u' = \frac{0.9 - 0.9}{1 - 0.81} = 0$$
The object is at rest in the moving frame, exactly as expected! And if $u = c$:
$$u' = \frac{c - v}{1 - v/c} = \frac{c\,(1 - v/c)}{1 - v/c} = c$$
Light always travels at $c$ in every frame—consistent with Einstein's second postulate.
Try it in the app: In the Velocity Transformation section, turn on "Show Velocity". The green line shows velocity $u$ in the rest frame, and the orange line shows the transformed velocity $u'$ in the moving frame. Adjust both $v$ and $u$ to see how the relativistic formula differs from simple subtraction.
Events on the Diagram
An event is something that happens at a specific place and time—it has definite coordinates $(x, t)$. On the Minkowski diagram, each event is marked with a rocket icon.
In the app, each event is plotted twice:
- A solid rocket at $(x, t)$—its rest-frame coordinates (plotted on the $S$ grid).
- A faded rocket at $(x', t')$—its moving-frame coordinates (also plotted on the $S$ grid to show the numerical values from the Lorentz transformation).
As you change $v$, the faded (moving-frame) rockets shift to show how the same physical event has different coordinates in a different frame. The coordinates table below the diagram updates in real time.
Reading Coordinates with Projection Lines
Click (or tap) any rest-frame event to select it. When an event is selected, the app draws dashed guide lines from the event showing how to read its coordinates in both frames:
- Green dashed lines project vertically down to the $x$-axis and horizontally across to the $t$-axis, giving the rest-frame coordinates $(x, t)$.
- Blue dashed lines project parallel to the primed axes: one line runs parallel to $t'$ until it meets the $x'$-axis (giving the $x'$ reading), and another runs parallel to $x'$ until it meets the $t'$-axis (giving the $t'$ reading).
This is the standard technique for reading coordinates on a Minkowski diagram: in the rest frame you project perpendicular to the axes, but in the moving frame you project parallel to the primed axes—not perpendicular to them.
Try it in the app: Click an event, then increase $v$. Watch the blue projection lines tilt with the primed axes. The solid value pills at the axis intersections update in real time, so you can see exactly how the Lorentz transformation remaps the coordinates.
Try it in the app: Add two or more events, then slowly drag the velocity slider. Watch how the faded (moving-frame) rockets shift. Events that were simultaneous may no longer be, and time orderings can reverse for spacelike-separated events.
Using the App
-
Velocity Slider ($v$)
- Controls the velocity of frame $S'$ relative to $S$. The value is displayed with a $c$ suffix (e.g. $0.50c$).
- Ranges from $-0.95c$ to $+0.95c$.
- At $v = 0$, both frames are aligned and the primed axes overlap the unprimed axes.
-
Rest Frame Grid / Moving Frame Grid
- Toggle the gridlines for each frame on or off.
- In the Controls panel, row 1 contains rest-frame toggles and row 2 contains moving-frame toggles.
- The rest frame grid is always perpendicular. The moving frame grid shears as $v$ increases.
-
Velocity Transformation
- Turn on "Show Velocity" to enable the velocity addition visualisation.
- The green line represents an object moving at velocity $u$ in the rest frame.
- The orange line represents the same object's velocity $u'$ in the moving frame.
- The slider below controls $u$, displayed with a $c$ suffix (e.g. $0.30c$).
-
Adding & Selecting Events
- Click "Add Event" to place a new event (labelled A, B, C, ...) at a random position.
- Drag events to reposition them on the diagram.
- Click (or tap) an event to select it—a highlight ring appears and projection lines are drawn.
- Click the same event again, or click empty space, to deselect.
- Right-click (or long-press on mobile) an event to remove it.
- "Clear Events" removes all events from the diagram.
-
Projection Lines
- When an event is selected, dashed guide lines automatically show how to read coordinates.
- Projection values are shown in solid color pills so they remain visible over other visual elements.
- Green lines project perpendicular to the rest-frame axes, labelled $(x, t)$.
- Blue lines project parallel to the primed axes, labelled $(x', t')$.
-
Set v = 0
- This button sits beside "Relative Velocity, v" and resets $v$ to zero so both frames coincide.
-
Pan the Diagram
- Click (or touch) an empty area and drag to pan the view.
Suggested Exercises
- Verify time dilation: Place two events at $(0,\,0)$ and $(0,\,3)$ (same $x$, different $t$). The events are co-located in $S$, so the proper time is $\Delta t_0 = 3$. Set $v = 0.60\,c$ ($\gamma = 1/\sqrt{1 - 0.36} = 1.25$). Read off $\Delta t'$ from the app and confirm that it equals $\gamma\,\Delta t_0 = 1.25 \times 3 = 3.75$.
- Explore simultaneity: Place events at $(-2,\,3)$ and $(2,\,3)$. They share the same $t$, so they are simultaneous in $S$. Now increase $v$ and observe that $t'$ values diverge.
- Velocity addition: Enable velocity transformation. Set $u = 0.80\,c$ and $v = 0.60\,c$. Calculate $u'$ by hand using the formula and verify it matches the value displayed.
- Symmetry check: Compare $v = +0.5\,c$ with $v = -0.5\,c$. Notice how the axes tilt in opposite directions, but $\gamma$ is the same.
- Light speed invariance: Set $u = 0.95\,c$ in the velocity transformation and vary $v$. Notice that $u'$ never reaches $c$. Then consider: what would happen if $u = c$?
Summary of Key Equations
| Concept | Equation |
|---|---|
| Lorentz factor | $\gamma = \dfrac{1}{\sqrt{1 - v^{2}/c^{2}}}$ |
| Position transformation | $x' = \gamma\,(x - v\,t)$ |
| Time transformation | $t' = \gamma\!\left(t - \dfrac{v\,x}{c^{2}}\right)$ |
| Time dilation | $\Delta t = \gamma\,\Delta t_0$ |
| Length contraction | $L = L_0/\gamma$ |
| Velocity addition | $u' = \dfrac{u - v}{1 - uv/c^{2}}$ |
| Spacetime interval | $s^{2} = c^{2}\Delta t^{2} - \Delta x^{2}$ |
A note on notation: This page uses $t_0$ and $L_0$ (subscript zero) for proper time and proper length, following the convention used in the IB Diploma Programme Physics course and data booklet. Some university-level textbooks instead use primed symbols ($\Delta t'$, $L'$) for the same quantities. Both notations are equivalent—the subscript-zero convention emphasises that these are the values measured in the object's own rest frame.
Happy exploring!