But if you have a particle that is at rest (in whatever reference frame), then aren't the time-like and space-like directions at 90 degrees? |

I'm not sure if that even makes sense.

Take ordinary 2-dimensional space. If we stretch out the x-axis, say rescale it so that what was one unit of length becomes 2 units, then angles change -- assuming that we don't also stretch out the y-axis. So, in some sense, the magnitude of angles is an artifact of how we measure them.

In the case of spatial directions, we normally require rotational symmetry. And if we require rotational symmetry, we cannot stretch out the x-axis without also stretching out the y-axis.

As far as I know, we cannot rotate things from a space-like direction to a time-like direction. So we don't have something like rotational symmetry to normalize our way of measuring. So I think that unavoidably leaves measurements of angles between space-like and time-like directions to be dependent on our arbitrarily chosen standards.

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