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## Main Question or Discussion Point

Hi,

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

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!

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!