Editing Special relativity (section)

=== Graphical representation of the Lorentz transformation ===
{{multiple image|perrow = 2|total_width=400
| image1 = Spacetime diagram development A.svg |width1=535|height1=535
| image2 = Spacetime diagram development B.svg |width2=535|height2=535
| image3 = Spacetime diagram development C.svg |width3=535|height3=535
| image4 = Spacetime diagram development D.svg|width4=535|height4=535
| footer = Figure 3-1. Drawing a Minkowski spacetime diagram to illustrate a Lorentz transformation.}}

Spacetime diagrams ([[Minkowski diagram]]s) 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.<ref name=Morin2007/>

To draw a spacetime diagram, begin by considering two Galilean reference frames, S and S′, in standard configuration, as shown in Fig.&nbsp;2-1.<ref name=Morin2007/><ref name="Mermin1968">{{cite book |last1=Mermin |first1=N. David |title=Space and Time in Special Relativity |url=https://archive.org/details/spacetimeinspeci0000merm |url-access=registration |date=1968 |publisher=McGraw-Hill |isbn=978-0881334203}}</ref>{{rp|155–199}}

'''Fig.&nbsp;3-1a'''. Draw the <math>x</math> and <math>t</math> axes of frame S. The <math>x</math> axis is horizontal and the <math>t</math> (actually <math>ct</math>) axis is vertical, which is the opposite of the usual convention in kinematics. The <math>ct</math> axis is scaled by a factor of <math>c</math> 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 <math>t = 0.</math> The slope of these worldlines is 1 because the photons advance one unit in space per unit of time. Two events, <math>\text{A}</math> and <math>\text{B},</math> have been plotted on this graph so that their coordinates may be compared in the S and S' frames.

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

'''Fig.&nbsp;3-1c'''. Units in the primed axes have a different scale from units in the unprimed axes. From the Lorentz transformations, we observe that <math>(x', ct')</math> coordinates of <math>(0, 1)</math> in the primed coordinate system transform to <math> (\beta \gamma, \gamma)</math> in the unprimed coordinate system. Likewise, <math>(x', ct')</math> coordinates of <math>(1, 0)</math> in the primed coordinate system transform to <math>(\gamma, \beta \gamma)</math> in the unprimed system. Draw gridlines parallel with the <math>ct'</math> axis through points <math>(k \gamma, k \beta \gamma)</math> as measured in the unprimed frame, where <math> k </math> is an integer. Likewise, draw gridlines parallel with the <math>x'</math> axis through <math>(k \beta \gamma, k \gamma)</math> as measured in the unprimed frame. Using the Pythagorean theorem, we observe that the spacing between <math>ct'</math> units equals <math display=inline>\sqrt{(1 + \beta ^2)/(1 - \beta ^2)}</math> times the spacing between <math>ct</math> units, as measured in frame S. This ratio is always greater than 1, and ultimately it approaches infinity as <math>\beta \to 1.</math>

'''Fig.&nbsp;3-1d'''. Since the speed of light is an invariant, the ''worldlines'' of two photons passing through the origin at time <math>t' = 0</math> still plot as 45° diagonal lines. The primed coordinates of <math>\text{A}</math> and <math>\text{B}</math> 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.