Focal Difference Compared
The same constant distance difference on an antenna hyperbola and a spacetime hyperbola, and what it means in each.
Comparison of the focal distance difference on an antenna hyperbola and on a spacetime hyperbola, with a draggable point and live distance readouts.
2.0
Antennas: the foci are real towers, fixed in place, and the difference k is a physical path difference (time difference of arrival × c). Changing k changes the curve's shape.
Spacetime: the meaningful quantity is the interval s = √(t² − x²), the time on a clock moving from the origin to the event. The focal difference 2s holds on paper, but those foci are just points of the drawing. Changing s rescales the curve without changing its shape.