The compactification of a spatial dimension, say $x^1$ given by the identification $x \sim x^1 + 2\pi R$ is said to be related to the lightlike compactification by a Lorentz boost :
$$ \left( \begin{array}{c} x^0 \\ x^1 \\ \end{array} \right) \sim \left( \begin{array}{c} x^0 \\ x^1 \\ \end{array} \right) +2\pi \left( \begin{array}{c} -R \\ R \\ \end{array} \right) $$
What exactly is the relationship between them? How can I see this? And what does it mean from a physics point of view?
