Line Element in Minkowski Space: Geometric Meaning

[Joe Cool]

Hello,
how can you imagine the geometrically meaning of the minus sign in ds²=-dx₀²+dx₁²+dx₂²+dx₃², maybe similar to ds²=x₁²+dx₂² is the length in a triangle with the Pythagoras theorem?

 

[Dale]

Hello,
how can you imagine the geometrically meaning of the minus sign in ds²=-dx₀²+dx₁²+dx₂²+dx₃², maybe similar to ds²=x₁²+dx₂² is the length in a triangle with the Pythagoras theorem?

Geometrically it means that surfaces of constant spacetime interval form hyperboloids instead of spheres. This is important because it clearly distinguishes spacelike intervals which form hyperboloids of one sheet from timelike intervals which form hyperboloids of two sheets (future and past)

 

[Ibix]

@bcrowell's GR book, downloadable from http://www.lightandmatter.com/, includes some postulates (in the manner of the postulates of Euclidean geometry) for Minkowski geometry.

 

[Joe Cool]

Thanks a lot for your answers!