Answer to Question #100648 in Mechanics | Relativity for Kishwer

Question

What is minkowskian geometry in relativity ?

What is invariants distance vs the interval i special relativity. Explain in details

Answer 1

Minkowskian geometry in relativity:

Minkowskian geometry in relativity is an extension of the Euclidean three-space to a quasi-Euclidean four-space that included time i.e.  Minkowski gave an alternative formulation that used a real time coordinate instead of an imaginary one, representing the four variables (x, y, z, t) of space and time in coordinate form in a four dimensional real vector space.

Invariant distance vs the interval in special relativity:

The term invariant means non-changing

In Galilean relativity, distance $\Delta$r and temporal separation between two events $\Delta$t are independent invariants, the values of which do not change when observed from different frames of reference.

In special relativity, however, the interweaving of spatial and temporal coordinates generates the concept of an invariant interval, denoted as ${\Delta}s^2$

${\Delta}s^2=c^2{\Delta}t^2-({\Delta}x^2+{\Delta}y^2+{\Delta}z^2)$

The interweaving of space and time revokes the implicitly assumed concepts of absolute simultaneity and synchronization across non-comoving frames.