General Galilean invariance

The following is generalized from the explanation of Galilean invariance here.

Chorocosm (inertial frames)

Among the axioms from Newton’s theory are:

(1) There exists an original inertial frame in which Newton’s laws are true. An inertial frame is a reference frame in uniform motion relative to the original inertial frame.

(2) All inertial frames share a common time.

Chorocosmic invariance

Consider two inertial frames O and O′. A physical event in S will have location coordinates r = (rx, ry, rz) and time t; similarly for O′. By the second axiom above, one can synchronize the rod-clocks in the two frames so that t = t′. Suppose O′ is in uniform motion relative to O with velocity v. Consider a point object whose location is given by r = r(t) in O. Then

r′(t) = r(t) − vt.

This transformation of variables between two inertial frames is called the (chorocosmic) Galilean transformation. The velocity of the particle is given by the time derivative of the position:

u′(t) = d/dt r′(t) = d/dt r(t) − v = u(t) − v.

Another differentiation gives the acceleration in the two frames:

a′(t) = d/dt u′(t) = d/dt u(t) − 0 = a(t).

This result implies (chorocosmic) Galilean relativity. Assuming that mass is invariant in all inertial frames, the above equation shows that Newton’s laws of mechanics, if valid in one inertial frame, must hold for all frames. By axiom (1) they hold in one inertial frame, therefore (chorocosmic) Galilean relativity holds in all inertial frames.


Chronocosm (facilial frames)

Among the axioms from the general Newton’s theory are:

(1) There exists an original facilial frame in which the Newton’s laws are true. A facilial frame is a reference frame in uniform motion relative to the original facilial frame.

(2) All facilial frames share a common stance.

Chronocosmic invariance

Consider two facilial frames O and O′. A physical event in O will have chronation coordinates t = (tx, ty, tz) and stance s; similarly for O′. By the second axiom above, one can synstancize the rod-clocks in the two frames so that s = s′. Suppose O′ is in uniform motion relative to O with lenticity w. Consider a point object whose chronation is given by t = t(s) in O. Then

t′(s) = t(s) − ws.

This transformation of variables between two facilial frames is called the chronocosmic Galilean transformation. The lenticity of the tempicle is given by the stance derivative of the chronation:

ℓ′(s) = d/ds t′(s) = d/ds t(s) − w = (s) − w.

Another differentiation gives the relentment in the two frames:

b′(s) = d/ds ℓ′(s) = d/ds (s) − 0 = b(s).

This result implies chronocosmic Galilean relativity. Assuming that vass is invariant in all facilial frames, the above equation shows that Newton’s laws of mechanics, if valid in one facilial frame, must hold for all frames. By axiom (1) they hold in one facilial frame, therefore Galilean relativity holds in all facilial frames.