Review of the FoPL paper [1]

New Concepts from the Evans Unified Field Theory. Part One

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.

1. The Evolution Equations

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

                ^∂/_∂t¹/_R = − ¹/_Rē ^∂R/_∂t

or, multiplied by R,

                ¹/_R ^∂R/_∂t = − R ^∂/_∂t¹/_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^bη_ab,
                                    = (η^abη_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_μ^a = 4 (q_a^μq_μ^a) R_μ^a =_[1,(20)] 4 R_μ^a     Þ     [1,(22)]

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).

Thus, symmetric and antisymmetric parts of the tetrad matrix Q cannot have a physical meaning.