Review of the FoPL paper [1]

New Concepts from the Evans Unified Field Theory. Part One

Gerhard W. Bruhn, Darmstadt University of Technology


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 (qμa) 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.

1. The Evolution Equations

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/Rμ R = + R ∂μ (1/R) ,

[1, (5)]                 1/Tμ T = + T ∂μ (1/T) ,

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

[1, (42)]                 1/Rμ 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 = Ro eiωt,

with a real part

[1, (45)]                 Re(R) = Ro 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/ ∂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 gμν or Ricci tensor Rμν. 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 = et which both fulfil the "evolution equation" [1, (43)] as well.

2. Evolution of Fields

In Section 2 [1, p.138+5] Evans tries to discuss the relation between the Einstein field equation

[1, (11)]                 Rμν − ½ R gμν = k Tμν

and his Evans field equation

                                Rμa − ½ R qμa = k Tμa                                 (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)]                 Rμν = Rμa qνb ηab,

[1, (14)]                 Tμν = Tμa qνb ηab,

[1, (15)]                 gμν = qμa qνb ηab.

Inserting these relations into the Einstein field equation yields

                (Rμa − ½ R qμa) qνb ηab = Tμa qνb ηab,

and, since the matrices (qνb) and (ηab) are both invertible, this is equivalent to the Evans field equation

                Rμa − ½ R qμa = Tμa .

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

[1,(16)]                 Gμa = − ¼ R qμa,

[1,(17)]                 Tμa = ¼ T qμa,

where the definition of Gμa 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 (Gμa), (Tμa) and (qμa). 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 = Rμν gμν ,                 T = Tμν gμν

the "Einstein convention"

[1, (19)]                 gμν gμν = 4,

and the "Cartan convention"

[1, (20)]                 qμa qaμ = 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 = gμν Rμν
[1, (21)]                       = qaμ qbν ηab Rμa qνbηab,
                                    = (ηabηab)(qbν qνb)(qaμ Rμa) = 4 qaμ Rμa .

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

                                R = gμν Rμν = qaμ ηab qbν Rμν = qaμ ηab Rμb = qaμ Rμa .

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 qμa to obtain

[1, (22)]                 Rμa = ¼ R qμa,

[1, (23)]                Gμa = Rμa − ½ R qμa = − ¼ R qμa

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 = qaμ Rμa for the 16 quantities Rμa, which, of course, is impossible. He erroneously concludes from the wrong eq. [1,(21)] by wrong multiplication

                R = 4 qaμ Rμa | · qμa     Þ     R qμa = 4 (qaμqμa) Rμa =[1,(20)] 4 Rμa     Þ     [1,(22)]

yielding proportionality(!!!) of the matrices (Rμa), (Gμa), (Tμa) to (qμa), 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)].

3. Decomposition of Fields

Quotation from [1, p.138+12]

The unified potential field is the tetrad or vector-valued one-form qμa , 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)]                 qμa = qμa(S) + qμa(A),

[1, (64)]                 Aμa = A(o) qμa = Aμa(S) + Aμa(A),

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

                Q(S) = (qμa(S)) = ½ (Q + QT),                 Q(A) = (qμa(A)) = ½ (Q − QT)

where QT 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 QT is

                Q'T = (Q Λ)T = ΛT QT


                Q'(S) = ½ (Q' + Q'T) = ½ [Q Λ + (Λ Q)T] = ½ [QΛ + ΛT QT],

                Q'(A) = ½ (Q' − Q'T) = ½ [Q Λ − (Λ Q)T], = ½ [Q Λ − ΛT QT],

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.

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.


[1] M.W. Evans, New Concepts from the Evans Unified Field Theory. Part One: . . .,
      FoPL 18 139-155 (2005)

[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