Question

Suppose a 2-dimensional space is spanned by the coordinates (v, x) and that the line
element is dened by:
ds² = -xdv² + 2dvdx

i) Assuming that the nature and properties of the spacetime line element still hold,
show that a particle in the negative x-axis can never venture into the positive
x-axis. That is, show that a particle becomes trapped if it travels into the
negative x-axis.

ii) Draw a representative light cone that illustrates how a particle is trapped.

Answer 1

The world line of a particle is trapped inside the light cone. And in the diagram below, it is shown that for x < 0, the light cones are always directed towards the negative x axis and away from the positive x axis. Hence a physical particle in the negative x axis can never venture towards the positive x axis. On the other hand a particle in the positive x axis can move to the region in the negative x axis because the light cones for x > 0 are directed towards the negative x axis. And once it moves in the x < 0 region, it becomes trapped.