Special Relativity shifts from spatial group (sine-cosine matrix) to space-time group (hyperbolic sine-cosine matrix), another antitonic connection.

In pre-relativistic physics, the subject is 3-D space, separate from 1-D time. Space is whatever remains invariant under the circular tranformation:

			---                  ---
                        |   cos q     - sin q  |
                        |                      |
                        |__ sin q       cos q__|

This condition acts on [x,y,z], [x',y',z'] to satisfy Galilean relativity, linking translation along a line of two inertial frames, F, F' by the Galilean velocity transformations, for a "body" moving with velocity v along the x axis:

		x' = x - vt, y' = y. z' = z, t' = t

From this, we can find the acceleration transformations, a_x = D²_t²x, etc., for constant velocity (in the inertial frame) and same time in both frames:

		a'x = ax, a'y = ay, a'z = az

However, Galilean relativistic transformation does not satisfy transmission of light. So another tranformation must be sought.

Let a light signal emanate from the origin of the coordinate system. The light spreads out in all directions in the form of a sphere with radius r = ct, vacuum light speed . Then, for the two frames, we have the spherical equations:

		x2 + y2 + z2 = c2t2, x'2 + y'2 + z'2 = c2t'2

Modeling on the Galilean velocity transformations, but allowing for "difference in time between the two frames:

		x' = g(x - vt), y' = y, z' = z, t' = g(t + d)

We substitute the last equations into the spherical equation:

	g2(x2 - 2vxt + t2) + y2 + z2 = c2g2(t2 + 2dt + d2)

In order for this last result to equation the sperical equation in the origin frame, the extra terms on the left (above) must equal the extra terms on the right:

		-g22vxt = t2g22dt Þ d = vx/c2

Note from this that d has the dimension of time, also it goes to zero as v/c goes to zero, so that the equation satisfies Galilean Relativity.

In the spherical equation, subsituting the determined value for d, and collecting on the factors of x², t², we find:

	x2g2(1 - v2/c2) + y2 + z2 = c2t2g2(1 - v2/c2)

This becomes the spherical equation if:

	g2(1 - v2/c2) = 1 Þ g = 1/(1 - v2/c2)

Note, from this, that g is dimensionless (as desired) and goes to 1 as v/c goes to zero, recovering Galilean Relativity.

Subsituting this in our suggested transformation above (which we put into the spherical equation), we have the Lorentz Transformation:

x' = 1/(1-v2/c2)(x-vt), y' = y, z' = z, t' = 1/(1-v2/c2)(t-vx/c)

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We began with a (rotation-type) circular (matrix) transformation which conserves in space, hence, will work under Galilean Relativty, wherein we assume the same time for two differential inertial frames. But the Lorentz Transformations show we can't assume this. We need something to correlate time in different inertial frames. We wish (say, in simplified spatial 1-D) x² - c²t² = x'² - c²t'².

This can be achieved by

• writing t = it and t' = it', where i = -1;
• and relating the frames as:

  	x' = x cos q + ct sin q
  	ct' = -x sin q + ct cos q

Since t is imaginary, then q must be also. We can achieve this by writing q = if for real f, and similarly in the other frame. But we also know that cos (if) = cosh f, and sin (if) = i sinh f.

So we have a space-time which is invariant under the (rotation-type) hyperbolic transformation:

			---                    ---
                        |   cosh f     - sinh f  |
                        |                                                                          |
                        |__ sinh f       cosh f__|

We have the homology:

		space: space-time:: circular: hyperbolic

These are hyperbolic functions. and we recall that the antitonic process has a hyperbolic prototype. Hence, the antitonic pattern persists.

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ANTITONE IN GENERAL RELATIVITY

Einstein's Principle of Equivalence finds no difference between the acceleration due to gravity and local acceleration (say, uniform in an elevator). This is similar to an "inertial frame".