File:Bell observers expansion.png

Summary
This figure depicts the graph of the nonzero component of the expansion tensor of the Bell observers discussed in Bell's spaceship paradox, namely
 * $$\theta[\vec{X}]_{11} = \frac{k^2 \, T}{\sqrt{1 + k^2 \, T^2}}$$.

As shown in the figure, near time zero, this is asymptotic to $$k \, t$$ and as t grows without bound, it is asymptotic to the constant k.  Here T is the coordinate time of a suitable Cartesian chart, but the component is measured by the Bell observers themselves and represents the rate at which they are moving apart from their neighbors in the common direction of acceleration. Since this rate approaches a constant, their mutual distance increases, unlike what happens in Newtonian mechanics. This is the basis for Bell's spaceship paradox.

This png image was converted using eog from a jpeg image made using Maple by User:Hillman.