Geometric interpretation of the spacetime invariant

[fuzzytoad]

For a euclidean space, the interval between 2 events (one at the origin) is defined by the equation:
L^2=x^2 + y^2
The graph of this equation is a circle for which all points on the circle are separated by the distance L from the origin.

For space-time, the interval between 2 events is defined by the equation
S^2 = r^2 - (ct)^2
The graph of this equation is a hyperbola. In this case, what would S represent geometrically given that all events on the hyperbola share the same space-time invariant?

 

[Daverz]

For a euclidean space, the interval between 2 events (one at the origin) is defined by the equation:
L^2=x^2 + y^2
The graph of this equation is a circle for which all points on the circle are separated by the distance L from the origin.

For space-time, the interval between 2 events is defined by the equation
S^2 = r^2 - (ct)^2
The graph of this equation is a hyperbola. In this case, what would S represent geometrically given that all events on the hyperbola share the same space-time invariant?

Am I missing something? You seem to have answered your own question.

Seen as an active transformation, any (orthochronous) Lorentz transformation moves a point on the hyperboloid to another point on the same branch of the hyperboloid (a non-orthochronous LT would move the point to the other branch of the hyperboloid). Similarly, seen as an active transformation, any rotation on Euclidean space moves a point on a sphere to another point on the sphere.

 

[bcrowell]

Are you asking for a geometrical interpretation, or a physical one? Physically, if s² is less than zero, the interpretation is that $\sqrt{-s^2}$ is the time interval on a clock that moved uniformly from the origin to the event (t,r).