Metric Invariance: UNIZOR.COM - Relativity 4 All - Galilean

Notes to a video lecture on http://www.unizor.com

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(X_A,Y_A,Z_A) and B(X_B,Y_B,Z_B)
equals to
d²(A,B) = (X_B−X_A)² +
+ (Y_B−Y_A)² + (Z_B−Z_A)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}.

Further assume that time is absolute and the same in both reference frames (t=T), that frame β at time t=T=0 coincides with α and then moves relative to α along a straight line with uniform velocity vector v(v_X,v_Y,v_Z), preserving parallelism of axes.
Then the Galilean transformation of coordinates from α to β, as specified in the previous lecture, is
t = T
x(t) = X(T) − v_X·T
y(t) = Y(T) − v_Y·T
z(t) = Z(T) − v_Z·T

Applied this transformation to coordinates of points A and B, we obtain
x_A(t) = X_A(T) − v_X·T
y_A(t) = Y_A(T) − v_Y·T
z_A(t) = Z_A(T) − v_Z·T
and
x_B(t) = X_B(T) − v_X·T
y_B(t) = Y_B(T) − v_Y·T
z_B(t) = Z_B(T) − v_Z·T

Now we are ready to calculate the distance between these two points in the reference frame β{t,x,y,z}.

d_β²(A,B) = (x_B−x_A)² +
+ (y_B−y_A)² + (z_B−z_A)²

As seen from the coordinate transformation above,
x_B−x_A = [(X_B(T)−v_X·T) −
− (X_A(T)−v_X·T)] =
= X_B(T)−X_A(T)
Analogously,
y_B−y_A = Y_B(T)−Y_A(T)
z_B−z_A = Z_B(T)−Z_A(T)

From this immediately follows
d_β²(A,B) = (x_B−x_A)² +
+ (y_B−y_A)² + (z_B−z_A)² =
= (X_B−X_A)² +
+ (Y_B−Y_A)² + (Z_B−Z_A)² =
= d_α²(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

The law of universal gravity between two objects states that there is a force of attraction F between any two objects that is equal to
F = G·m₁·m₂ /r²
where
F is the force of attraction,
G is the gravitational constant,
m₁ is the mass of object 1,
m₂ is the mass of object 2,
r is the distance between centers of the masses of these objects.

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

The Coulomb's law of electrostatic attracting or repelling force between two electrically charged objects states that there is a force F between any two charged objects that is equal to
F = k_e·q₁·q₂ /r²
where
F is the electrostatic force,
k_e is the Coulomb constant,
q₁ is the electric charge of object 1,
q₂ is the electric charge of object 2,
r is the distance between centers of the charges in these objects.

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.