New Concepts from the Evans Unified Field Theory. Part One

In the paper [1] (also contained in [2, Chap.14]) under review M.W. Evans derives so-called
"Evolution Equations" for the scalar curvature R and the contracted
energy momentum tensor T to show the *bounded* longtime behavior of the
universe. However, this evolution equation is merely a (trivial) differential
*identity* which does not specify any type of solution. While
M.W. Evans believes that only bounded solutions are possible,
the identity admits *arbitrary* unbounded solutions as well.

M.W. Evans' derivation of his useless "Evolution Equations" is superfluous,
the chain rule of calculus would be sufficient. Nevertheless, the derivation
contains serious errors referring to two "Conventions" allegedly by Einstein
and Cartan the latter of which conventions is even wrong. By neglecting the rules of
tensor calculus M.W. Evans uses these conventions for the resolution
of a linear equation by a method that is as surprising as wrong.

The last section of M.W. Evans' article is based on the decomposition
of the tetrad matrix (q_{μ}^{a}) into its symmetric and antisymmetric parts.
This decomposition is well-known, but both symmetric and antisymmetric part
don't behave Lorentz covariantly, and thus symmetric and antisymmetric parts
cannot have any physical meaning while M.W. Evans draws far reaching
physical conclusions from this decomposition.

The page numbers of the web copy mentioned in [1] start with 1 instead of 139 (= 138+1).
Equations from M.W. Evans' article [1] appear with equation labels [1, (nn)] in the left margin.
Quotations from Evans' article appear in **black**.

The author M.W. Evans bases far reaching conclusions about the evolution of the universe (quoted below) on his "equations for the evolution of R and T" [1, p.138+1] where R is the scalar curvature and T is the contracted energy-momentum tensor.

**The abstract of [1]:**

The Evans field equation is solved to give the equations governing the evolution
of scalar curvature R and contracted energy-momentum T.
These equations show that R and T are always analytical, oscillatory,
functions without singularity and apply to all radiated and matter
fields from the sub-atomic to the cosmological level.
One of the implications is that all radiated and matter
fields are both causal and quantized, contrary to the Heisenberg
uncertainty principle. The wave equations governing this
quantization are deduced from the Evans field equation.
Another is that the universe is oscillatory without singularity,
contrary to contemporary opinion based on singularity theorems.
The Evans field equation is more fundamental than, and leads to,
the Einstein field equation as a particular example,
and so modifies and generalizes the contemporary Big Bang model.
The general force and conservation equations
of radiated and matter fields are deduced systematically from
the Evans field equation. These include the field equations
of electrodynamics, dark matter, and the unified or hybrid field.

Keywords: Evans field equation; equations of R; oscillatory universe;
general field and force equations; causal quantization

**The equations for the evolution of R and T:**

[1, (4)]
^{1}/_{R} ∂^{μ} R =
__+__ R ∂^{μ} (^{1}/_{R}) ,

[1, (5)]
^{1}/_{T} ∂^{μ} T =
__+__ T ∂^{μ} (^{1}/_{T}) ,

and with specified sign (− instead of __+__) at [1, p.138+9]:

[1, (42)]
^{1}/_{R} ∂^{μ} R =
− R ∂^{μ} (^{1}/_{R}) ,

or, specifically, the time component

[1, (43)]
^{1}/_{R} ^{∂R}/_{∂t} =
− R ^{∂}/_{∂t} (^{1}/_{R}) ,

then a solution of Eq.(43) is seen to be

[1, (44)]
R = R_{o} e^{iωt},

with a real part

[1, (45)]
Re(R) = R_{o} cos ωt.

The cosine function is bounded by plus or minus unity and never goes to infinity. . . .

M.W. Evans' so-called evolution equations are simply the result of applying the
chain rule of calculus to ^{1}/_{R} (and ^{1}/_{T} respectively).
This yields by time differentiation

^{∂}/_{∂t}^{1}/_{R} =
− ^{1}/_{Rē}
^{∂R}/_{∂t}

or, multiplied by R,

^{1}/_{R} ^{∂R}/_{∂t}
= − R ^{∂}/_{∂t}^{1}/_{R} ,

satisfied for *arbitrary* differentiable functions R = R(t).
Therefore there is no reason for Evans' far reaching conclusions at [1, p.138+9]:

. . . Therefore there can no singularity in the scalar curvature R.
It follows from Eq. (18) that there is never a singularity in the metric
g_{μν} or Ricci tensor R_{μν}.
In other words, the universe evolves without
a singularity, and it follows that the well-known singularity theorems
built around the Einstein field equation do not have any physical meaning.
These singularity theorems are complicated misinterpretations. In
other words, general relativity must always be a field theory that is everywhere
analytical [17]. Similarly, the older Newton theory must be
everywhere analytical. There are no singularities in nature. Equation
(45) shows that the universe can contract to a dense state, but then
re-expands and re-contracts. Apparently we are currently in a state of
evolution where the universe is on the whole expanding. This does not
mean that every individual part of the universe is expanding. Some
parts may be contracting or may be stable with respect to the laboratory
observer.

**No singularity?**

Why not? Consider e.g. the *unbounded* function R = ^{1}/_{t} for t > 0
or R = e^{t} which both fulfil the "evolution equation" [1, (43)] as well.

2. Evolution of Fields

[1, (11)]
R_{μν} − ½ R g_{μν} = k T_{μν}

and his Evans field equation

R_{μ}^{a} − ½ R q_{μ}^{a} =
k T_{μ}^{a}
(1)

as displayed in the central box on [1, p.138+5] and in Evans' flowchart (see Evans Field Equation).

Evans asserts at [1, p.138+5]:

The Einstein field equation can be deduced [1-10] as a special case of the Evans field equation.

This is only partially true: The Einstein field equation [1, (11)]
is no *special* case of the Evans field equation (1),
but both equations are (almost trivially) *equivalent*:
Evans provides the following tools correctly [1, p.138+5]:

[1, (13)]
R_{μν} = R_{μ}^{a} q_{ν}^{b} η_{ab},

[1, (14)]
T_{μν} =
T_{μ}^{a} q_{ν}^{b} η_{ab},

[1, (15)]
g_{μν} =
q_{μ}^{a} q_{ν}^{b} η_{ab}.

Inserting these relations into the Einstein field equation yields

(R_{μ}^{a} −
½ R q_{μ}^{a}) q_{ν}^{b} η_{ab} =
T_{μ}^{a} q_{ν}^{b} η_{ab},

and, since the matrices (q_{ν}^{b}) and (η_{ab}) are both invertible,
this is equivalent to the Evans field equation

R_{μ}^{a} − ½ R q_{μ}^{a} = T_{μ}^{a} .

However, Evans does not remark this simple equivalence and goes astray. He asserts at [1, p.138+6]

[1,(16)]
G_{μ}^{a} = − ¼ R q_{μ}^{a},

[1,(17)]
T_{μ}^{a} = ¼ T q_{μ}^{a},

where the definition of G_{μ}^{a} can be found in the central box
of [1, p.138+3] or
Evans' flowchart.
Both equations [1,(16-17)] are *wrong* in general as claiming *proportionality* of
the matrices
(G_{μ}^{a}), (T_{μ}^{a}) and
(q_{μ}^{a}).
Evans' embarrassing error can be seen below from Evans' attempt of giving a proof:
He starts with the (correct) definitions of the quantities R and T at [1, p.138+6]:

[1, (18)]
R = R_{μν} g^{μν} ,
T = T_{μν} g^{μν}

the "Einstein convention"

[1, (19)]
g^{μν} g_{μν} = 4,

and the "Cartan convention"

[1, (20)]
q_{μ}^{a} q_{a}^{μ} = 1,

the latter of which is clearly wrong (=4 would be correct), and both "conventions"
are unknown under these names. Evans' unspecified citation [1, [12]] refers to S.M. Carroll's
Lecture Notes [3] where neither an "Einstein convention" nor a "Cartan convention" can be found.
Therefore Evans' unspecified citation [1, [12]] is *wrong*.

Then Evans tries to express R by tetrad-related quantities [1, p.138+6].

R = g^{μν} R_{μν}

[1, (21)]
= q_{a}^{μ} q_{b}^{ν} η^{ab}
R_{μ}^{a}
q_{ν}^{b}η_{ab},

= (η^{ab}η_{ab})(q_{b}^{ν}
q_{ν}^{b})(q_{a}^{μ} R_{μ}^{a}) =
4 q_{a}^{μ} R_{μ}^{a} .

Already this calculation is beyond any rules of tensor calculus. The correct calculation is

R = g^{μν} R_{μν}
=
q_{a}^{μ} η^{ab} q_{b}^{ν}
R_{μν}
=
q_{a}^{μ} η^{ab} R_{μb}
=
q_{a}^{μ} R_{μ}^{a} .

The next erroneous step follows immediately [1, p.138+7] and is even much worse than the calculation quoted before.

Multiply either side of Eq. (21) by q_{μ}^{a} to obtain

[1, (22)]
R_{μ}^{a} = ¼ R q_{μ}^{a},

[1, (23)]
G_{μ}^{a} =
R_{μ}^{a} − ½ R q_{μ}^{a} =
− ¼ R q_{μ}^{a}

which is Eq.[1,(16)]. Similarly, we obtain Eq. [1,(17)].

Evidently the author Evans uses his *own* rules of tensor calculus.
Under standard rules
the instruction "multiply" is not feasible since the indices μ and a are *not free*.
Evans' purpose is to resolve the (one) equation
R = q_{a}^{μ} R_{μ}^{a}
for the 16
quantities R_{μ}^{a}, which, of course, is impossible.
He erroneously concludes from the *wrong* eq. [1,(21)] by *wrong*
multiplication

R = 4 q_{a}^{μ} R_{μ}^{a}
| **· q _{μ}^{a}**
Þ
R q

yielding proportionality(!!!) of the matrices (R_{μ}^{a}),
(G_{μ}^{a}), (T_{μ}^{a}) to
(q_{μ}^{a}), a result *as remarkable as wrong*, attainable only
by applying *three* rules of Evans' New Math.

So in traditional Math there is no logical way to the aimed equations [1,(22)] and [1,(23)] which are identical with Eq.[1,(16)].

The unified potential field is the tetrad or vector-valued one-form
q_{μ}^{a} ,
which is in general an asymmetric square matrix. The latter can always
be written as the sum of symmetric and antisymmetric component
square matrices, components that are physically meaningful potential
fields of nature:

[1, (63)]
q_{μ}^{a} = q_{μ}^{a(S)} + q_{μ}^{a(A)},

[1, (64)]
A_{μ}^{a} =
A^{(o)} q_{μ}^{a} =
A_{μ}^{a(S)} + A_{μ}^{a(A)},

Indeed, the decomposition of a square matrix
Q := (q_{μ}^{a}) into its symmetric part
(q_{μ}^{a(S)}) and its antisymmetric part (q_{μ}^{a(A)})
is well-known:

Q^{(S)} = (q_{μ}^{a(S)}) = ½ (Q + Q^{T}),
Q^{(A)} = (q_{μ}^{a(A)}) = ½ (Q − Q^{T})

where Q^{T} is the transposition of the matrix Q.

The objection is that the matrices Q^{(S)} and Q^{(A)} *don't behave
Lorentz covariantly* since both summands transform in a different way:
Let the Lorentz transform of Q be

Q' = Q Λ

Then according to the rules of matrix calculus the Lorentz transform of Q^{T} is

Q'^{T} = (Q Λ)^{T} = Λ^{T} Q^{T}

hence

Q'^{(S)} = ½ (Q' + Q'^{T})
= ½ [Q Λ + (Λ Q)^{T}]
= ½ [QΛ + Λ^{T} Q^{T}],

Q'^{(A)} = ½ (Q' − Q'^{T})
= ½ [Q Λ − (Λ Q)^{T}],
= ½ [Q Λ − Λ^{T} Q^{T}],

which neither is a *covariant* transform of Q^{(S)} nor of Q^{(A)}.

The same holds for the subsequent decompositions [1, (64-68)] that have no physical meaning as well. Hence Evans far reaching consequences at [1, p.138+12 ff.] from his decompositions are obsolete.

References

[1]
M.W. Evans, New Concepts from the Evans Unified Field Theory. Part One: . . .,

FoPL 18 139-155 (2005)

http://www.aias.us/documents/uft/a12thpaper.pdf

[2] M.W. Evans, Generally Covariant Unified Field Theory, the geometrization of physics; Arima 2006

[3] S.M. Carroll, Lecture Notes on General Relativity, 1997

http://xxx.lanl.gov/pdf/gr-qc/9712019