Technical Notes 2006

I. GENERAL RELATIVITY AND THE SSCP

General Relativity takes the form:

R_μν – ½ g_μν R = 8π/c⁴ G T_μν

Where the left hand side is the Einstein tensor (curvature of spacetime) and T_μν is the stress-energy tensor (density and flux of mass/energy and momentum).  The equation says, in effect: the mass/energy of objects tells spacetime how to curve; spacetime curvature tells objects how to move.

According to the SSCP, the Newtonian gravitational constant G (= 6.67 x 10^-8 cm³/g sec²) in General Relativity must be replaced by the factor:

[Λ^1-D ]^Ψ G ,

where Λ = 5.2 x 10¹⁷ and D = 3.174 .

Equivalently, G → [Λ^-2.174 ]^Ψ G , since G α L³/MT² and Λ³/Λ^D Λ² = Λ^-2.174.

Ψ = {…,-2,-1,0,1,2,…} designates the infinite, discrete hierarchy of cosmological Scales, with the usual designation of  Ψ =  0 for the Stellar Scale. It can be seen that for Ψ =  0, [Λ^1-D ]^Ψ G = G, in agreement with observations for Stellar Scale gravitational interactions.

A revised form of General Relativity that incorporates discrete cosmological self-similarity (a form of discrete conformal invariance known as discrete dilation invariance) would be:

R_μν – ½ g_μν R = 8π/c⁴ [Λ^{1 - D}]^Ψ G T_μν

Instead of a single equation for General Relativity, we now have an infinite number of identical equations, one for each cosmological Scale.  Another way to express this is the following equation:

R_μν – ½ g_μν R = 8π/c⁴ G_Ψ T_{μν ,}

where G_Ψ ( =  [Λ^1-D ]^Ψ G) is the infinite series of correct SSCP coupling constants for gravitational interactions, one “constant” for each cosmological Scale.

II. GENERAL COVARIANCE AND DISCRETE SCALE RELATIVITY

Discrete Conformal Invariance of General Relativity and Electromagnetism

1.      General Relativity is covariant under conformal transformations (which preserve angles and length ratios, but do not involve absolute length scales), including dilation invariance, when masses and G are suitably scaled.

2.      Maxwell’s equations of Electromagnetism are likewise covariant under conformal transformations, including dilation invariance, when electric charges are suitably scaled.

From the point of view of the SSCP, the fact that GR and EM are both consistent with discrete global dilation invariance is exciting, since this form of discrete scale invariance fits very well with the conceptual properties of the SSCP and its discrete self-similar Scale transformation equations.

Extension of the Principle of General Covariance

The principle of general covariance states that the laws of physics should be fully independent of arbitrary choices of reference frames and coordinate systems.  From this one can infer that the laws of physics are independent of:

Spatial location
Time
Spatial orientation
State of Motion (inertial or accelerated)

Before the advent of the SSCP, size or scale (note small s) was the one unique thing that was not relative, but rather appeared to be absolute.  The hydrogen atom in its ground state was thought to have just one fixed set of scale values (m, r, t, …).  The SSCP, especially in the exact cosmological self-similarity form, changed things radically.

Within a cosmological Scale, one could still invoke absolute scale, but absolute scale no longer applied generally to nature’s infinite, discrete hierarchy of self-similar Scales.  If each Scale is exactly self-similar to any other Scale, then there are an infinite number of differently sized hydrogen atoms (one for each Scale).  Each of these hydrogen atoms has a unique and equally valid set of scale parameters (m_Ψ, r_Ψ, t_Ψ, …).

The bottom line is that General Covariance must now be expanded to include:

Location
Time
Orientation
State of motion
Discrete cosmological Scale

The latter addition reflects the idea that the laws of physics are independent of our arbitrary choice of a particular cosmological Scale as our reference system.  This is equivalent to saying that the fundamental laws of physics are identical on all cosmological Scales (for the case of exact cosmological self-similarity).

Discrete Scale Relativity

The discrete global dilation invariance discussed above is formally equivalent to invariance with respect to a discrete global transformation of length, time and mass units.

To understand this clearly, start with the assumption of exact cosmological self-similarity.  This means that each cosmological Scale is identical (except for scale) in terms of the objects comprising the Scale and their dynamical interactions.  Observers on any Scale would describe identical “observable universes”.

Fixed Units Approach: If we want to measure and compare populations of analogue systems from different Scales in terms of one fixed set of units (say the conventional Stellar Scale cm, g and sec that we are familiar with), then all length, time and mass parameters, as well as all dimensional “constants” scale according to the discrete self-similar Scale transformation equations of the SSCP (see “Main Ideas”).  For example, a Stellar Scale proton has a radius of about 4.2 x 10⁴ cm and the Atomic Scale proton has a radius of about 0.8 x 10^-13 cm, differing by the now familiar 5.2 x 10¹⁷ scale factor for lengths.

Relative Units Approach: Again we choose the cm, g and sec as our length, mass and time units, but now we acknowledge that there is a different set of cm, g and sec for each Scale of the infinite cosmological hierarchy.  Because the self-similarity is exact, each [cm_Ψ, g_Ψ, sec_Ψ] set is identical, equally fundamental (bye, bye reductionism!) and equally valid.  In this approach the discrete self-similar Scale transformation equations tell us the relative “sizes” of these units.  For example, the Stellar Scale proton and the Atomic Scale proton both have the same radius of about 4.2 x 10⁴ cm_Ψ, but the Atomic Scale centimeters are 5.2 x 10¹⁷ times smaller than the Stellar Scale centimeters.

With this discrete scaling of units, all fundamental physical laws and their constants are identical on each Scale.  This relativity of Scale can be called Discrete Scale Relativity, or more simply Scale Relativity so long as we remember that:

Scale ≡ cosmological Scale ≠ scale.

Scale Relativity is a further generalization of General Relativity, and the hierarchical subsuming of relativity theories can be symbolically expressed as follows,

{[(Special Relativity) General Relativity] Scale Relativity}.