Interactive Minkowski Diagrams

Click to add an event or select an existing event or world-line.

Shift-click to add a connected event, connect to an existing event, or disconnect from a connected event.

Drag to move an event.

Delete key to delete an event or world-line.

Event colour: #

Line colour: #

Axis colour: #

Boost to velocity:c

How to use this diagram

The diagram is initially blank, except for a time and a space axis, and a grid. The scale is not labelled, but natural units have been used so a light-like path will always appear at 45°. If you interpret the horizontal scale as one light second per grid square, the vertical scale is one second per grid square.

Click on the Minkowski diagram to add an event, select an existing event, or select an existing world-line. Shift click to add or select an event and add a world-line connecting this event to the currently selected event (marked by a grey circle), if any. Events and world-lines are added with the colours specified in the Event colour and Line colour controls to the right of the Minkowski diagram.

You can remove a world-line by selecting the event at one end, then shift-clicking on the other end. You can also remove the selected world-line or selected event (and any attached world-lines) by pressing the delete key.

The Minkowski diagram shows the current time and space axes. If you click the Keep axis button, the current axis will also be drawn in the colour specified by the Axis colour control, and will remain when a boost is performed. Tick marks on the axis correspond to the main grid as it would appear if boosted to the frame defined by these axes.

You can perform a Lorentz boost in two ways. One way is to enter a velocity (positive or negative) as a fraction of the speed of light in the Boost to velocity control and click the Boost button. The other is to select a time-like world-line and click on the Boost to selected line rest frame button. This button is only active when a time-like world-line is selected.

You can reset the diagram to its initial reference frame by clicking the Rest to initial conditions button, and you can clear all events, world-lines and axes by clicking the Clear diagram button.

If you like your diagram and want to keep it, you can save it as a PNG file by clicking the Save diagram button.

A note on rounding error

Events remember their coordinates in the frame that was in use when they were created or dragged. They also remember the velocity of this frame relative to the initial frame. This methodology has the advantage that rounding errors do not build up. It is possible that rounding error will cause a null-separated pair of events to be non-null separated after a boost, but the effect can be undone by reversing the boost.

This methodology does, however, mean that there are likely to be issues around boosting to very high speeds relative to the initial frame. Depending on the sequence of boosts, you may find that a further boost does not change the velocities, or that the new velocity is at or above lightspeed. In this case, returning to the intial conditions is probably advisable.

Some custom Minkowski diagrams

Twin paradox

The twin paradox is a classic scenario in which one twin stays on Earth while another travels at relativistic velocities to a distant planet and returns younger than the stay-at-home. This version has the traveller travelling at 0.6c, and shows the world-line of the stay-at-home in green, the traveller in blue, and of the destination planet in purple. There are a number of variants.

Ladder and barn paradox

The ladder and barn is another classic scenario in which a ladder that is too long to fit in a barn can be made to fit because of length-contraction. The world-lines of the ends of the barn are shown in red, and of the rod in green.

Einstein's train

Einstein's derivation of the Lorentz transforms was based on a train being struck by lightning at both ends, simultaneously according to an observer on a nearby embankment. The ends and middle of the train are shown as green lines, while the ends and middle of the embankment are shown in blue. Orange light pulses are emitted simultaneously, in the embankment frame, with the middle of the train passing the middle of the embankment.

Compass roses

This does not represent any particular physical system, but just generates compass rose-like groups of world-lines (space-like lines in blue, time-like in green, and null in orange). Boosting gives a feel for how different interval types transform under Lorentz boosts.

Grid

Another diagram that does not represent a physical system, this just creates a grid of world-lines overlaying the axis grid. Boost to a different frame to see how the grid would transform.

Hyperbolae

A third diagram that does not represent a physical system, this creates an array of (straight-line approximations to) hyperbolae which are self-similar under Lorentz transformation. That is, each point on a hyperbola will move to another point on that same hyperbola when Lorentz transformed. Additionally, the two null paths through the origin, which are also self-similar under Lorentz transformation, are shown in yellow. Note that the green hyperbolae, the ones that cross the x-axis, are the world-lines of observers accelerating at a constant proper acceleration. There are two versions of this example.