Relativity Metric: UNIZOR.COM - Relativity 4 All - Einstein View
Channel: Zor Shekhtman on UNIZOR Education
Date: 2023-07-06

UNIZOR.COM - Physics through a Looking Glass of Mathematician
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Relativity Metric


Metric is one of the most important characteristic of space we live in, and it defines a distance between two points.

Prior to Theory of Relativity our space was considered three-dimensional Euclidian and, therefore, the square of a distance between two points
A(XA,YA,ZA) and B(XB,YB,ZB)
was equal to
d²(A,B) = (XB−XA)² + (YB−YA)² + (ZB−ZA)²

We have analyzed how this distance transformed by Galilean transformation of coordinates from one inertial system into another, moving relative to the first, and came up with its invariance relative to this transformation.

Indeed, the Galilean transformations from inertial reference frame α{X,Y,Z} into inertial reference frame β{x,y,z} that moves relative to α with speed v along its X-axis are
t = T
x = X − v·T
y = Y
z = Z
where T=t is absolute time in both reference frames.

Then the distance between points A and B in β-frame is
dβ²(A,B) = (xB−xA)² + (yB−yA)² + (zB−zA)² =
= (XB−v·T−XA+v·T)² + (YB−YA)² + (ZB−ZA)² =
= dα²(A,B)

With a progress of the Theory of Relativity we came up with Lorentz Transformation of coordinates that takes time into account.
This type of transformation seemed to better correspond to our theories and experimental data, but a concept of length in our traditional sense was no longer invariant (see Length Transformation lecture in this part of a course).

An interesting approach would be to find a different definition of metric in our space that takes time into account and is invariant relative to Lorentz Transformation analogously to classic definition of Euclidean metric being invariant relative to Galilean transformation (see Metric Invariance lecture in the Galilean View part of a course).

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