Metric Invariance: UNIZOR.COM - Relativity 4 All - Galilean
Channel: Zor Shekhtman on UNIZOR Education
Date: 2023-05-25

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


One of the most important characteristic of space we live in is distance between two points.

Classical approach to Physics prior to Theory of Relativity was that our space is Euclidian and, therefore, the square of a distance between two points
A(XA,YA,ZA) and B(XB,YB,ZB)
equals to
d²(A,B) = (XB−XA)² + (YB−YA)² + (ZB−ZA)²

Let's analyze how this distance transforms by Galilean transformation.

Assume, we have two reference frames α{T,X,Y,Z} and β{t,x,y,z}.

t = T
x(t) = X(T) − vX·T
y(t) = Y(T) − vY·T
z(t) = Z(T) − vZ·T


dβ²(A,B) = (xB−xA)² + (yB−yA)² + (zB−zA)²

As seen from the coordinate transformation above,
xB−xA = XB(T)−XA(T)
yB−yA = YB(T)−YA(T)
zB−zA = ZB(T)−ZA(T)

From this immediately follows
dβ²(A,B) = (xB−xA)² + (yB−yA)² + (zB−zA)² =
= (XB−XA)² + (YB−YA)² + (ZB−ZA)² = dα²(A,B)

This means that the distance between two points in Euclidian space, as measured in two different inertial reference frame, is the same.
In other words, the Euclidian metrics are invariant relative to Galilean transformation of coordinates.

Consequently, a circle in one inertial frame will look like a circle of the same radius in another, a square in one inertial frame will look exactly the same in another etc.


Newton's Law of Universal Gravitation

F = G·m1·m2 /r²

Observed from two different inertial reference frames, objects are the same, their masses are the same and, as proven above, the distance between them is the same as well.

Therefore, the force of gravity between them is also the same in both reference frames and we can say that the Law of Universal Gravitation is invariant relative to the Galilean transformation.


Coulomb's Law of Electrostatic Force

F = ke·q1·q2 /r²

Observed from two different inertial reference frames, objects are the same, their electric charges are the same and, as proven above, the distance between them is the same as well.

Therefore, the electrostatic force between them is also the same in both reference frames and we can say that the Coulomb's Law of Electrostatic Force between two electrically charged objects is invariant relative to the Galilean transformation.