I have been wondering if there is a Lorentz-invariant quantity that satisfies the definition of a metric for space-time.

The space-time interval

*s*

^{2}=

*t*

^{2}-

**r**^{2}[where

*is the vector (x,y,z)] does not satisfy the requirement for a metric*

**r***m*that

*m(t*= 0 if and only if

_{1},**r**, t_{1}_{2},**r**)_{2}*(t*=

_{1},**r**)_{1}*(t*. For instance, ANY two points on the path of a beam of light have a space-time interval of 0. Also, the space-time interval can be either positive or negative, which violates one of the conditions of a mathematical metric.

_{2},**r**)_{2}From what I've been able to dig up, there's some way to map 4-dimensional space-time coordinates to 5-dimensional hyperbolic space and then there is a Lorentz-invariant metric of hyperbolic space, but I couldn't figure out or find what that mapping is (and this might not even be correct in the first place). If anyone could provide a formula for a Lorentz-invariant metric of space-time or point me in that direction, I'd be much obliged!