Special relativity
Graphical representation of the Lorentz transformation

Spacetime diagrams (Minkowski diagrams) are an extremely useful aid to visualizing how coordinates transform between different reference frames. Although it is not as easy to perform exact computations using them as directly invoking the Lorentz transformations, their main power is their ability to provide an intuitive grasp of the results of a relativistic scenario.

To draw a spacetime diagram, begin by considering two Galilean reference frames, S and S', in standard configuration, as shown in Fig. 2-1.

Fig. 3-1a. Draw the x {\displaystyle x} and t {\displaystyle t} axes of frame S. The x {\displaystyle x} axis is horizontal and the t {\displaystyle t} (actually c t {\displaystyle ct}) axis is vertical, which is the opposite of the usual convention in kinematics. The c t {\displaystyle ct} axis is scaled by a factor of c {\displaystyle c} so that both axes have common units of length. In the diagram shown, the gridlines are spaced one unit distance apart. The 45° diagonal lines represent the worldlines of two photons passing through the origin at time t = 0. {\displaystyle t=0.} The slope of these worldlines is 1 because the photons advance one unit in space per unit of time. Two events, A {\displaystyle {\text{A}}} and B, {\displaystyle {\text{B}},} have been plotted on this graph so that their coordinates may be compared in the S and S' frames.

Fig. 3-1b. Draw the x ′ {\displaystyle x'} and c t ′ {\displaystyle ct'} axes of frame S'. The c t ′ {\displaystyle ct'} axis represents the worldline of the origin of the S' coordinate system as measured in frame S. In this figure, v = c / 2. {\displaystyle v=c/2.} Both the c t ′ {\displaystyle ct'} and x ′ {\displaystyle x'} axes are tilted from the unprimed axes by an angle α = tan − 1 (β), {\displaystyle \alpha =\tan ^{-1}(\beta),} where β = v / c. {\displaystyle \beta =v/c.} The primed and unprimed axes share a common origin because frames S and S' had been set up in standard configuration, so that t = 0 {\displaystyle t=0} when t ′ = 0. {\displaystyle t'=0.}

Fig. 3-1c. Units in the primed axes have a different scale from units in the unprimed axes. From the Lorentz transformations, we observe that (x ′, c t ′) {\displaystyle (x',ct')} coordinates of (0, 1) {\displaystyle (0,1)} in the primed coordinate system transform to (β γ, γ) {\displaystyle (\beta \gamma,\gamma)} in the unprimed coordinate system. Likewise, (x ′, c t ′) {\displaystyle (x',ct')} coordinates of (1, 0) {\displaystyle (1,0)} in the primed coordinate system transform to (γ, β γ) {\displaystyle (\gamma,\beta \gamma)} in the unprimed system. Draw gridlines parallel with the c t ′ {\displaystyle ct'} axis through points (k γ, k β γ) {\displaystyle (k\gamma,k\beta \gamma)} as measured in the unprimed frame, where k {\displaystyle k} is an integer. Likewise, draw gridlines parallel with the x ′ {\displaystyle x'} axis through (k β γ, k γ) {\displaystyle (k\beta \gamma,k\gamma)} as measured in the unprimed frame. Using the Pythagorean theorem, we observe that the spacing between c t ′ {\displaystyle ct'} units equals (1 + β 2) / (1 − β 2) {\textstyle {\sqrt {(1+\beta ^{2})/(1-\beta ^{2})}}} times the spacing between c t {\displaystyle ct} units, as measured in frame S. This ratio is always greater than 1, and ultimately it approaches infinity as β → 1. {\displaystyle \beta \to 1.}

Fig. 3-1d. Since the speed of light is an invariant, the worldlines of two photons passing through the origin at time t ′ = 0 {\displaystyle t'=0} still plot as 45° diagonal lines. The primed coordinates of A {\displaystyle {\text{A}}} and B {\displaystyle {\text{B}}} are related to the unprimed coordinates through the Lorentz transformations and could be approximately measured from the graph (assuming that it has been plotted accurately enough), but the real merit of a Minkowski diagram is its granting us a geometric view of the scenario. For example, in this figure, we observe that the two timelike-separated events that had different x-coordinates in the unprimed frame are now at the same position in space.

While the unprimed frame is drawn with space and time axes that meet at right angles, the primed frame is drawn with axes that meet at acute or obtuse angles. This asymmetry is due to unavoidable distortions in how spacetime coordinates map onto a Cartesian plane, but the frames are actually equivalent.
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