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# Un-Minkowski Diagram -- Overview

#### Links to the Animated Versions

(or go to the static versions)

 Overview Hyperbolic View
 T2 minus X2 equals (spacetime distance)2?  I don't get it. Although the Minkowski Diagram is a powerful tool for visualizing the differences in the way spacetime is split into the appearance of space and time by different observers, it has some drawbacks: It is not apparent from looking at this kind of diagram why the invariant distance of Einstein's (flat) spacetime is obtained by subtracting X2 from T2 rather than by adding. In fact, from the appearance of the diagram, it would seem that the distance could be obtained by normal vector addition. We cannot read the amount of time and space dilation directly off the axes without use of a conversion factor. * Minkowski diagram
 So show me. In the diagrams on the following pages, we sacrifice the appearance of symmetry between space and time axes to which we have become accustomed in the Minkowski diagrams. In return, we obtain a quantitatively accurate, visual interpretation for the relationships between X, T, and τ. In the first view of the new diagram, we also sacrifice a fixed position for Δτ on the diagram. This allows us to keep the practice of positioning all observers at the origin O.
 But where is the hyperbola in hyperbolic spacetime? The second view of this diagram sets the stage for us to reveal the hyperbola implied by X2 - T2 = (invariant). This version restores τ to a fixed position. All that we give up is the ability to place every observer at (0,0). The origin is now reserved solely for observers in the rest frame of the event. (Beyond the small, but possibly finite, risk of hurting the egos of observers in motion, the sacrifice seems justified.) Constant Velocity Diagram View 2
 Hyperbolic View By superimposing many such graphs and then animating them, we can provide a fairly dramatic revelation of the hyperbola.

* This conversion factor can be shown geometrically--and by using hyperbolas, at that--but it is not particularly obvious how this should be done. But see, for example, G. Naber's The Geometry of Minkowski Spacetime: An Introduction to the Mathematics of the Special Theory of Relativity, 1992, Springer-Verlag, New York, pp. 32-6.

The constant velocity and hyperbolic graphs were created in Pacific Tech Graphing Calculator 3.0.
Microsoft Word was used to create the special annotation within the graphs and to render the graphs as gifs.
The Minkowski diagram was created in Word.