Can Lorentz Transformations in Opposite Directions Cancel Each Other Out?

[john morrison]

1. Show that if we transform first in the x-direction and then in the minus x direction with the same speed (v), we end up with the original space-time coordinates. Note: For this problem you will need to apply the transformation equation twice. You will also need to apply the transformations to both x and t.

Homework Equations

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x' = \gamma(x-Vt)
y' = y
z' = z
t' = \gamma(t-(Vx)/c^2))

The Attempt at a Solution

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I attempted to plug in -x into the x for x' and then plug in x' whenever I found an x. However, this didn't get me far.

 

[Orodruin]

The problem is asking you to apply the Lorentz transformation once with velocity v and once to the resulting coordinates with velocity -v (you can call the coordinates resulting from this t'' and x''). Your task is then to show that x'' = x and t'' = t. The problem does not tell you to switch the directions of the x or t axes.