The Duality of Time Theory

Galileo had already postulated that there is no privileged reference frames. Frames which are moving with constant speeds, relative to the observer, are equivalent. This principle is called Galileo’s principle of relativity. The Galilean transformations are used to transform between the coordinates of two reference frames which differ only by constant relative motion with velocity. For simplicity we consider that motion is only on theaxis, so that the other two axes are unaffected (,), whileandare related by the following equation:

In these equations, the primed parameters denote a reference frame that moves at constant speedrelative to the stationary unprimed frame. The frames overlap at time, and both frames agree on the value offorever.

Galileo formulated these concepts in his description of uniform motion. These transformations together with spatial rotations and translations in space and time form the inhomogeneous Galilean group. Without the translations in space and time the group is the homogeneous Galilean group. The Galilean group is the group of motions of Galilean relativity action on the four dimensions of space and time, forming the Galilean geometry.

When we combine these transformations with the principle of the constancy of the speed of light, we can derive the Lorentzian transformations which led to Special Relativity. By using Lorentz transformations (described in section 2.2), Einstein extended Galileo’s principle so that it accounted for the constant and invariant speed of light, and he also postulated that it holds for all the laws of physics, including both the laws of mechanics and electrodynamics.

Therefore, Galilean relativity is an approximation of Special Relativity that is valid for low speeds, while Special Relativity is an approximation of General Relativity that is valid for weak gravitational fields. As we shall see in section 3, General Relativity incorporates non-Euclidean geometry in order to represent gravitational effects as the geometric curvature of space-time, while Special Relativity is restricted to the flat space-time known as Minkowski space, described in section 2.3.