where E s and E t represents the energy at the space time-points s and t respectively. With the notation E s = m s c
2 , one might identify the energy at a space-time
point s with the mass m s . One can see that the matrix Eq. (3.15) is commensurate
with the metric given by Eq. (3.12). For the case of photons (zero-rest-mass) one
obtains for κ r
ð Þ ≠ 1 ̸ 2 that ds = 0 and
cAdτ
− iBdx⃗
− iBdx⃗ − cAdτ
= cAdτ
1
− i
− i − 1
→ 2cAdτ
0 1
0 0
ð3:19Þ
which is consistent with Eq. (3.6′). One notes that the energy formula for
photons becomes
m 1 − 2κ r
ð Þ
ð
Þ= p ̸ c
in correspondence with the law of gravitational light bending. Furthermore at
κ r
ð Þ = 1 ̸ 2 the metric is conditionally singular for both mass- and massless particles.
In summary we have demonstrated that the a priori choice of conjugate relationships, like energy-time, impart simple relations, commensurate with the theory
of relativity, including the emergence of mathematical singularities at the precise
physical conditions commensurate with constraints like the limit velocity of light,
the associated loss of longitudinal spatial dimension and the emergence of a black
hole entity trapping energetic particles. Although not emphasized here, see also
footnotes 8 and 11, the matrix to the left of Eq. (3.13) displays a direct analogy with
the self-referential enigma exhibited by the Gödel’s inconsistency theorem(s),
12 see
e.g. Refs. [9, 11, 13, 16]. In the next section we will show how to extend the
degenerate formulation to any order including its interpretation in relation to the
Paradigm of Evolution.
4 The Correlated Dissipative Structure
As we have maintained above, time-energy and space-momentum provide a priori
observables of primary descriptions of the scientific discourse. In this portrayal of
processes in the natural world, one preferred option of a distinct constituent
involves by definition its conjugate companion. Within the theory of evolution,
time is the key variable with a fundamental spatial significance. To appreciate this
idea one needs to understand the emergence of spatio-temporal scales from the
12
The common inconsistency in the vernacular is also known as the liar’s paradox and it has
recently been discussed under the name of The Pinocchio Paradox [23].
A Simple Communication Hypothesis: The Process …
391
2 , one might identify the energy at a space-time
point s with the mass m s . One can see that the matrix Eq. (3.15) is commensurate
with the metric given by Eq. (3.12). For the case of photons (zero-rest-mass) one
obtains for κ r
ð Þ ≠ 1 ̸ 2 that ds = 0 and
cAdτ
− iBdx⃗
− iBdx⃗ − cAdτ
= cAdτ
1
− i
− i − 1
→ 2cAdτ
0 1
0 0
ð3:19Þ
which is consistent with Eq. (3.6′). One notes that the energy formula for
photons becomes
m 1 − 2κ r
ð Þ
ð
Þ= p ̸ c
in correspondence with the law of gravitational light bending. Furthermore at
κ r
ð Þ = 1 ̸ 2 the metric is conditionally singular for both mass- and massless particles.
In summary we have demonstrated that the a priori choice of conjugate relationships, like energy-time, impart simple relations, commensurate with the theory
of relativity, including the emergence of mathematical singularities at the precise
physical conditions commensurate with constraints like the limit velocity of light,
the associated loss of longitudinal spatial dimension and the emergence of a black
hole entity trapping energetic particles. Although not emphasized here, see also
footnotes 8 and 11, the matrix to the left of Eq. (3.13) displays a direct analogy with
the self-referential enigma exhibited by the Gödel’s inconsistency theorem(s),
12 see
e.g. Refs. [9, 11, 13, 16]. In the next section we will show how to extend the
degenerate formulation to any order including its interpretation in relation to the
Paradigm of Evolution.
4 The Correlated Dissipative Structure
As we have maintained above, time-energy and space-momentum provide a priori
observables of primary descriptions of the scientific discourse. In this portrayal of
processes in the natural world, one preferred option of a distinct constituent
involves by definition its conjugate companion. Within the theory of evolution,
time is the key variable with a fundamental spatial significance. To appreciate this
idea one needs to understand the emergence of spatio-temporal scales from the
12
The common inconsistency in the vernacular is also known as the liar’s paradox and it has
recently been discussed under the name of The Pinocchio Paradox [23].
A Simple Communication Hypothesis: The Process …
391