Minkowski Diagram

A Minkowski spacetime diagram is special kind of spacetime graph, one that represents phenomena in Minkowski space. In particular, the coordinate axes have the property that they are orthogonal with respect to the Minkowski inner product. The time axis is shown vertically; one or two of the spatial dimensions are suppressed, i.e., not shown.

Below is an example of a Minkowski diagram having only one space dimension. The grid represents the spacetime coordinates.

Three worldlines — Fig.4.1 Three timelike worldlines in Minkowski space and the straight null worldline of a flash of light.

Because light travels at such high speed, it is common to choose the units along the axes so that the path of a light ray is at a 45-degree angle, This may be accomplished by taking the light second (or any convenient fraction) as the unit along both the time axis and any space axis. We may then label the vertical axis with the ‘time’ coordinate $t = x^{0}$ and the horizontal axis with the space coordinate $x = x^{1}$ .

The red worldline stands for a stationary particle; it has no motion through space. For a stationary particle, the amount of proper time is equal to the amount of coordinate time.
The blue worldline is depicts a moving particle; the velocity in this example is taken to be: $v = Δ x / Δ t = 1 / 4$ , that is a quarter of the speed of light. The amount of proper time is about 96% of coordinate time; even at this rather high velocity, proper time is only slowed down very little.
The green worldline is for a particle moving at a variable speed, but always slower than the speed of light, which is the highest speed allowed only to photons and other massless particles.
The yellow straight line represents a flash of light in vacuum with velocity $c = 1$ .

Minkowski diagrams allow no curvature of spacetime itself, although particles and other massive objects can have curving paths in space. Any object represented with a curving path in the diagram is accelerating.

Events on the same horizontal line of the Minkowski diagram are simultaneous in this reference frame.
Given the units chosen for the above diagram, no worldline can tilt down more than 45 degrees, or else that object is moving faster than the velocity of light.