Evans-Myron: Primer

Myron W. Evans's Einstein-Cartan-Evans (ECE) theory, his 2010 claim that Einstein's field equation breaks a "dual Bianchi identity", and how to check that claim yourself in one page of algebra.

Source tags: [read] = from a document in the project folder; [read: page image p.N] = equation read from a rendered page; [web] = Internet source, URL given; [memory] = recalled from training, with a confidence level; [calc] = computed by me and reproduced by example.py. Final verdicts come from arbiter.md. Page numbers are PDF page numbers unless marked "printed" (CEFE printed page = PDF page minus 8; Felker printed = PDF minus 6). This primer was revised after the quality check (qe.md) and the arbiter's "QE revision", and again after the arbiter's "Round 2" (a deep read of CEFE Ch.2 Secs 2.7-2.9, PDF pp.49-66); where verdicts changed, the revised verdict is used. Round 2 changed no verdict category but corrected and sharpened several reasons, chiefly in Step F and the Claim 3 and 7 rows. Every number in Section 5 was recomputed by me; run .venv/bin/python review/Evans-Myron/example.py to reproduce them.

Source gap (repeated from claims.md and arbiter.md). The folder holds only two books: Evans, Crothers, Eckardt and Pendergast, Criticisms of the Einstein Field Equation (CEFE, 2010, 548 pp.) and L. G. Felker, The Evans Equations of Unified Field Theory (2005, 373 pp., a popularisation not written by Evans). There are no Evans journal papers (the AIAS papers 63, 93, 94 and 122 that the books cite are absent), no device descriptions and no experimental data. Both books were summarised page by page; deep reads cover the CEFE preface, Ch.2 Secs 2.6-2.9 (Sec. 2.6 resonance; Secs 2.7-2.9 Faraday disk, Sagnac, g-factor, Lamb shift and the list of 36 "advances"), Crothers's Ch.3 (including his gravitational-wave section, Secs 3.5-3.11), the Ch.4 definitions and several of its tables, and Felker Chs 9-12. Not deep-read: most of the 62 Ch.4 tables, CEFE Ch.5 and most of Felker outside Chs 9-12. Everything below about critics and replications rests on web summaries (chiefly Wikipedia, "Einstein-Cartan-Evans theory") plus [memory]; where a statement rests on [memory] alone it says so. To go further, obtain AIAS papers 63 and 94 (Coulomb resonance, Bedini application), 93 (the Physica B submission), and the critiques by Hehl and Obukhov (2008), Bruhn (2008) and Rodrigues and Souza (2005).

---

1. The idea in one paragraph

Myron Evans is a Welsh chemical physicist who, from about 2003, developed what he calls Einstein-Cartan-Evans (ECE) theory, published mostly through his own Alpha Institute for Advanced Study (AIAS) website [read: CEFE preface; Felker p.11 ("about 700 published technical papers ... in physical chemistry"); CEFE Ch.5, pp.505-513, on his Harrison Prize and Meldola Medal]. His starting point is real and respectable: the mathematician Élie Cartan showed in the 1920s that a curved space can have two kinds of "shape", curvature (how a vector rotates when carried round a loop) and torsion (whether small parallelograms fail to close), and Einstein's general relativity uses only the first. Evans proposes that the second, torsion, is electromagnetism, so that gravity and electricity become two faces of one geometry. In the 2010 book he goes further: he says that ignoring torsion makes Einstein's equation break a rule of geometry (a "dual Bianchi identity"), and that a computer check of 62 known solutions proves it, so black holes, the Big Bang and gravitational waves are all wrong; one co-author wrote in 2010 that the gravitational-wave detector LIGO was "destined to detect nothing", five years before it did. CEFE Chapter 2 also claims that ECE gives the electron's magnetic moment without quantum electrodynamics. Felker's book and CEFE Chapter 2 add that a "spin connection" can make electric circuits resonate and draw power from spacetime, and that the Heisenberg uncertainty principle is refuted [read]. The mainstream answer, which this primer lets you check for yourself, is that the "violated identity" is a disguised form of Einstein's own equation for empty space, so the computer test found only that matter is present where matter is present. The torsion idea itself is legitimate physics (Einstein-Cartan theory); Evans's use of it is where things go wrong.

2. What you need to know first

This project lives in general relativity (GR), which the reader profile does not assume. The refreshers below build exactly the tools needed, starting from flat space in polar coordinates, which you already know.

Index notation and the summation convention. Write the coordinates of a point as x^μ, where the superscript μ (Greek "mu") is a label, not a power: in 2D polar coordinates x^1 = r, x^2 = θ. A repeated index, once up and once down, means "sum over it": A_μ B^μ = A_1 B^1 + A_2 B^2 + ... . An object with indices (A^μ, g_μν, R^ρ_σμν) is just a table of numbers or functions; its rank is the number of indices. A tensor is such a table whose entries change between coordinate systems by a definite rule (the chain rule applied once per index). That rule is all that "tensor" means here. Read: Schutz, A First Course in General Relativity, the chapters on vector analysis and tensor analysis in special relativity.

The metric. The metric g_μν tells you distance: ds² = g_μν dx^μ dx^ν. Flat 2D polar coordinates: ds² = dr² + r² dθ², so g_rr = 1, g_θθ = r², g_rθ = 0. The inverse metric g^μν is the matrix inverse (here g^rr = 1, g^θθ = 1/r²); it is used to "raise" an index: V^μ = g^μν V_ν. In special relativity, with x^0 = ct (c = speed of light, t = time), ds² = −(c dt)² + dx² + dy² + dz²; the minus sign makes time different from space.

The connection (Christoffel symbols). Differentiating a vector field in curvilinear coordinates needs a correction because the basis vectors themselves change from point to point (in polar coordinates the unit vectors r̂ and θ̂ rotate as you move). The correction coefficients are the connection Γ^λ_μν ("Gamma"). The covariant derivative is D_μ V^ρ = ∂_μ V^ρ + Γ^ρ_μλ V^λ, where ∂_μ is the ordinary partial derivative ∂/∂x^μ. For a metric with no torsion the connection is fixed by the metric:

Γ^λ_μν = ½ g^λσ (∂_μ g_σν + ∂_ν g_σμ − ∂_σ g_μν).

This formula is symmetric in μ and ν by construction. For flat polar coordinates it gives Γ^r_θθ = −r and Γ^θ_rθ = Γ^θ_θr = 1/r, and all others zero [calc]. These are the same terms that give centripetal acceleration (−r θ̇²) and the Coriolis-like term (2ṙθ̇/r) in polar mechanics, which you met in classical mechanics. The connection is not a tensor: it is nonzero in polar coordinates and zero in Cartesian coordinates for the same flat plane. Read: Schutz, the chapter titled "Preface to curvature" (it does exactly this polar-coordinate example).

Geodesics. A free particle in curved spacetime follows d²x^λ/ds² + Γ^λ_μν (dx^μ/ds)(dx^ν/ds) = 0. In flat polar coordinates this reproduces straight-line motion written in r and θ. In GR, for slow motion in weak gravity, the term Γ^i_00 ≈ ∂_i Φ / c² (Φ = Newtonian gravitational potential) reproduces Newton's d²x/dt² = −∇Φ [memory, high]. Notice that only the symmetric part of Γ enters this equation, because (dx^μ/ds)(dx^ν/ds) is symmetric in μ and ν.

Torsion. Split any connection into symmetric and antisymmetric parts in its lower indices. The torsion tensor is twice the antisymmetric part:

T^λ_μν = Γ^λ_μν − Γ^λ_νμ.

Unlike Γ itself, T is a genuine tensor. Geometrically, nonzero torsion means that if you go a short distance along A then along B, and separately along B then along A, the two endpoints do not meet: infinitesimal parallelograms do not close. The connection used in GR (the Levi-Civita connection, the formula above) has T = 0. A connection with T ≠ 0 is perfectly allowed mathematically; Cartan introduced it in 1922-23 [memory, high].

Curvature: Riemann, Ricci, Kretschmann. Carry a vector around a tiny closed loop using the connection; if it comes back rotated, the space is curved. The Riemann tensor measures this:

R^ρ_σμν = ∂_μ Γ^ρ_νσ − ∂_ν Γ^ρ_μσ + Γ^ρ_μλ Γ^λ_νσ − Γ^ρ_νλ Γ^λ_μσ

(this is CEFE Eq. 4.16 [read], and the standard form). It is antisymmetric in its last pair (μν). Summing over the first and third index gives the Ricci tensor R_σν = R^ρ_σρν, and one more contraction gives the Ricci scalar R = g^σν R_σν. Ricci is only part of Riemann: Ricci can be zero while Riemann is not. A convenient single-number test for "really curved" is the Kretschmann scalar K = R_αβγδ R^αβγδ. Read: Carroll, Spacetime and Geometry, Chapter 3 ("Curvature").

Bianchi identities. These are theorems true for every torsion-free connection, whatever the metric. The first (cyclic) Bianchi identity is R^ρ_σμν + R^ρ_μνσ + R^ρ_νσμ = 0 (CEFE Eq. 4.12 [read]). The second is a similar cyclic sum of derivatives; contracted twice it says the Einstein tensor below has zero divergence, which is what makes energy-momentum conservation automatic in GR [memory, high]. With torsion, both identities pick up extra torsion terms; Cartan's form of the first is D∧T^a = R^a_b∧q^b (CEFE preface Eq. 14 [read: page image p.v]).

Tetrads, spin connection and forms (only needed to read Evans). A tetrad q^a_μ is a set of four little orthonormal rulers attached at each point (a = 0..3 labels which ruler, μ the coordinate direction); g_μν = q^a_μ q^b_ν η_ab, with η = diag(−1,1,1,1). The spin connection ω^a_b is the connection rewritten in the ruler basis. A differential form is an antisymmetric tensor written compactly; ∧ (the "wedge product") is the antisymmetric product; d∧ is the antisymmetrised derivative (it generalises curl). The Hodge dual (a tilde in Evans) swaps a 2-index antisymmetric object for its complement using the totally antisymmetric symbol ε_μναβ (in 3D the Hodge dual of the antisymmetric tensor ∂_i A_j − ∂_j A_i is the curl vector). You need these only to follow Evans's notation; the worked example uses ordinary index notation. Read: Carroll, the appendix on non-coordinate bases, or Schutz, Geometrical Methods of Mathematical Physics.

Einstein's field equation. G_μν ≡ R_μν − ½ R g_μν = (8πG/c⁴) T_μν, where G (no indices) is Newton's constant and T_μν is the energy-momentum tensor of matter (for still dust, T_00 = ρc², ρ = mass density). An equivalent form is R_μν = (8πG/c⁴)(T_μν − ½ T g_μν). In empty space T_μν = 0, so the vacuum equation is simply R_μν = 0. Read: Hartle, Gravity: An Introduction to Einstein's General Relativity (physics first, light on maths); Carroll Chapter 4.

Electrostatics and resonance (needed for Claim 4). The electric field E = −∇φ (φ = potential) obeys Gauss's law ∇·E = ρ_q/ε₀ (ρ_q = charge density, ε₀ = vacuum permittivity), so ∇²φ = −ρ_q/ε₀ (Poisson's equation). A driven, lightly damped oscillator responds strongly when driven at its natural frequency (resonance), but every joule in the oscillation is supplied by the driver. Read: Griffiths, Introduction to Electrodynamics, Chapter 2 (electrostatics); Taylor, Classical Mechanics, the chapter on oscillations.

Comparing a prediction with a measurement (needed for Claim 7). A measured value is quoted as, for example, 2.0023193043718(75): the digits in brackets are the uncertainty in the last digits, here ±0.0000000000075 = ±7.5×10⁻¹². This uncertainty σ ("sigma", one standard deviation) is the typical size of the measurement error. A prediction that differs from the measurement by a few σ may be unlucky; one that differs by thousands of σ is ruled out. If the prediction itself depends on an input x with uncertainty σ_x, its own uncertainty is roughly |dg/dx|·σ_x (first-order error propagation: the derivative tells you how much the output moves per unit change of input). The comparison is fair only if both error bars are honest. Read: J. R. Taylor, An Introduction to Error Analysis (University Science Books), the chapters on propagation of uncertainties.

The electron g-factor (needed for Claim 7). A spinning charged particle acts like a small magnet. Its magnetic moment is μ = g (e/2m) S, where e is the electron's charge, m its mass, S its spin angular momentum and g a pure number, the g-factor. Dirac's 1928 relativistic equation predicts g = 2 exactly. Quantum electrodynamics (QED, the quantum theory of light and electrons) adds small corrections in powers of the fine-structure constant α = e²/(4πε₀ħc) ≈ 1/137: the first, found by Schwinger in 1948, is α/π, giving g ≈ 2.00232 [memory, high]. Higher terms (α², α³, ...) have been computed, and the full QED prediction matches the measured value to about one part in 10¹² [memory, high]. The Lamb shift is a related QED effect: a tiny energy difference (1057.8 MHz, about 4.4×10⁻⁶ eV) between two hydrogen levels that Dirac's equation predicts to be equal [memory, high]. Read: Griffiths, Introduction to Elementary Particles, the chapter on quantum electrodynamics (qualitative discussion of g−2).

The uncertainty relation (needed for Claim 5). For any normalised wavefunction ψ(x), with σ_x and σ_p the standard deviations of position and momentum, σ_x σ_p ≥ ħ/2 (ħ = reduced Planck constant). This is a theorem of Fourier analysis plus the Cauchy-Schwarz inequality (Kennard 1927, Robertson 1929) [memory, high]. Read: Griffiths, Introduction to Quantum Mechanics, the section on the generalised uncertainty principle.

3. The textbook view

Geometry. Given a metric, there are infinitely many connections, but exactly one is both metric-compatible (lengths are preserved when you carry a vector around, D_λ g_μν = 0) and torsion-free: the Levi-Civita connection, i.e. the Christoffel formula above. Any other metric-compatible connection is Levi-Civita plus a tensor (the contortion) built from the torsion [memory, high]. So torsion is an extra, independent field you may add; it is not hidden inside the metric, and setting it to zero is a choice, not a mistake.

General relativity. Einstein chose T = 0, so the metric alone describes gravity. Matter curves spacetime through G_μν = (8πG/c⁴)T_μν; free bodies follow geodesics. The vacuum solution around a spherical mass (Schwarzschild 1916) is

ds² = −(1 − r_s/r) c²dt² + dr²/(1 − r_s/r) + r²(dθ² + sin²θ dφ²), r_s = 2GM/c²,

with M the mass, fixed by matching to the Newtonian limit far away (or to an interior solution for the star) [memory, high]. It has R_μν = 0 everywhere outside the mass, but K = 12 r_s²/r⁶ = 48G²M²/(c⁴r⁶) ≠ 0 [calc], so the geometry is genuinely curved there: Ricci-flat does not mean flat. Inside matter R_μν ≠ 0. GR passes the perihelion precession of Mercury, light deflection, Shapiro delay, gravitational redshift, binary-pulsar orbital decay, frame dragging (Gravity Probe B, final result 2011, about 19% precision [web: https://en.wikipedia.org/wiki/Gravity_Probe_B, via arbiter.md]), and direct gravitational-wave detection (GW150914, recorded 14 September 2015, announced February 2016) [web: https://en.wikipedia.org/wiki/Tests_of_general_relativity; https://en.wikipedia.org/wiki/First_observation_of_gravitational_waves, both via arbiter.md].

Torsion in mainstream physics. Einstein-Cartan theory (Cartan; Sciama and Kibble in the 1960s) adds torsion sourced by the intrinsic spin density of matter. It agrees with GR wherever spin density is small, which is everywhere we can currently test, so it is a consistent, accepted, untested-in-practice extension [memory, high; review: Hehl, von der Heyde, Kerlick and Nester, Rev. Mod. Phys. 48, 393 (1976)]. Einstein himself spent 1928-31 on a "teleparallel" unified theory using tetrads and torsion (curvature zero, torsion nonzero) in the hope of geometrising electromagnetism; he abandoned it [memory, high]. Evans's program is in this tradition, and so are Weyl's 1918 and Kaluza-Klein's 1921/1926 attempts to make electromagnetism geometric [memory, high].

Electromagnetism. In potentials, E = −∇φ − ∂A/∂t and B = ∇×A, where A is the vector potential. Maxwell's equations follow, and QED (quantum electrodynamics) agrees with experiment to parts per billion or better (electron magnetic moment) [memory, high]. Electromagnetism in the Standard Model is a gauge theory: physics is unchanged by A → A + ∇χ, φ → φ − ∂χ/∂t for any function χ.

4. Where this researcher departs

The table puts each departure next to the textbook statement. CEFE equation numbers are from the preface [read: page images pp.iii-v] and Ch.2/Ch.4 [read].

The pattern in the first four rows is one move repeated: a legitimate generalisation (torsion may be nonzero) is turned into the claim that the special case (torsion is zero) is self-contradictory. A special case of a more general theory is not inconsistent; it is just special.

5. Worked example: running Evans's own test on two universes

Goal: compute, by hand and with example.py, the exact quantity CEFE Chapter 4 computes, and find out what it really measures. Everything here is [calc] unless tagged.

Step A. Can a connection be antisymmetric? Flat plane in polar coordinates. Metric g_rr = 1, g_θθ = r². Only ∂_r g_θθ = 2r is nonzero. Christoffel formula: - Γ^r_θθ = ½ g^rr(−∂_r g_θθ) = ½(1)(−2r) = −r. - Γ^θ_rθ = Γ^θ_θr = ½ g^θθ ∂_r g_θθ = ½(1/r²)(2r) = 1/r.

Torsion: T^θ_rθ = Γ^θ_rθ − Γ^θ_θr = 1/r − 1/r = 0. Riemann: all components zero (the plane is flat). Evans's Eq. 6 would require Γ^λ_μμ = 0 ("both the commutator and the connection vanish if μ is the same as ν" [read: preface p.iv]), but Γ^r_θθ = −r ≠ 0, and Γ^θ_rθ = +Γ^θ_θr rather than −Γ^θ_θr. Physically, Γ^r_θθ = −r is the centripetal term: a particle moving straight past the origin, described in polar coordinates, has r̈ = r θ̇². Make Γ antisymmetric and that term disappears, and straight-line motion can no longer be written in polar coordinates. Conclusion: the connection need not be, and in general cannot be, antisymmetric.

Step B. Evans's dual connection for a symmetric Γ. Preface Eq. 8: Γ̃^λ_μν = ½|g|^½ ε_μν^αβ Γ^λ_αβ. The symbol ε is antisymmetric under swapping α and β; a symmetric Γ^λ_αβ is unchanged by the swap. Pair each term (α,β) with (β,α): they are equal and opposite, so the sum is zero. Hence Γ̃ = 0, T̃ = 0, and R̃ (Eq. 12, built only from Γ̃) = 0. The "Cartan-Evans identity" D∧T̃ = R̃∧q becomes 0 = 0 for every GR solution. example.py checks this for a general symmetric Γ. So whatever Chapter 4 tests, it is not Eq. 13 applied to GR.

Step C. What Chapter 4 actually computes. CEFE Eq. 4.7-4.8 [read] raise the last two indices of the Riemann tensor and sum the second index against the third: E^κν ≡ R^κ_μ^μν = g^μα g^νβ R^κ_μαβ. Use the antisymmetry of the Riemann tensor in its first pair of indices, R_μκαβ = −R_κμαβ (standard for a metric-compatible connection [memory, high], and checked by example.py below):

g^μα R_κμαβ = −g^μα R_μκαβ = −R^α_καβ = −R_κβ (Ricci).

So E^κν = −Ric^κν: Evans's "curvature tensor" is minus the Ricci tensor, nothing more.

Check on a real universe. Flat FLRW (Friedmann-Lemaître-Robertson-Walker, the standard expanding-universe metric): ds² = −(c dt)² + a(t)²(dx² + dy² + dz²), where a(t) is the scale factor (how much distances have stretched). With x^0 = ct and a dot meaning d/dx^0, example.py gives:

Ric_00 = −3 ä/a, Ric_11 = a ä + 2ȧ², Evans E^00 = +3 ä/a, E^11 = −(a ä + 2ȧ²)/a⁴,

and E^κν + Ric^κν = 0 for all 16 components. The genuine first Bianchi identity (Eq. 4.12) has 0 violations. (This matches the arbiter's Calc 1 and the mathematician's FLRW check; the overall sign depends on index placement, which is why the reviewers quote ±3ä/a.)

The book's own printout agrees. You do not have to trust this algebra against the book: the book's Chapter 4 tables print the same thing. For FLRW its "charge density" is −3ä/a [read, per qe.md: digest pp.252-256], the Ricci component above (sign set by index placement). Every "charge density" and "current density" in the 62 tables is a component of minus Ricci (arbiter.md, QE revision). The clearest case is the Kasner metric, ds² = −dt² + t^(2p₁)dx² + t^(2p₂)dy² + t^(2p₃)dz² (here c = 1 and p₁, p₂, p₃ are constant exponents; it describes a universe that expands in some directions and contracts in others). The book prints [read: CEFE pp.276-282, per qe.md]

Ric₁₁ = p₁(p₁ + p₂ + p₃ − 1) t^(2p₁−2), J₁ = −p₁(p₁ + p₂ + p₃ − 1) t^(−2p₁−2),

and example.py Part F confirms that the book's J₁ is exactly Evans's E^11 = −Ric^11. Kasner is a vacuum solution when p₁ + p₂ + p₃ = 1 and p₁² + p₂² + p₃² = 1, and then J₁ = 0. The book says these conditions can "probably" be met only in trivial cases like (1, 0, 0). That is false: (−1/3, 2/3, 2/3) satisfies both (sum 1; squares 1/9 + 4/9 + 4/9 = 1). The book then plots p = (1, −1, 0), whose sum is 0, so that metric is not a vacuum solution at all and its nonzero "violation" is simply its nonzero Ricci [calc]. The book also says "several well known line elements failed" metric compatibility and the cyclic identity, but names none, and all 62 tables print "o.k." for both [read: p.144 and tables, per arbiter.md]. Both checks are identities for any Christoffel connection, so a failure could only have come from a coding or input error.

Now use the Einstein equation for dust: R_00 = (8πG/c⁴)(T_00 − ½T g_00) with T_00 = ρc² and T = g^μν T_μν = −ρc², so

R_00 = (8πG/c⁴)(ρc² − ½ρc²) = 4πGρ/c²,

which with R_00 = −3ä/a gives ä/a = −(4π/3)Gρ/c² (in x^0 units), the Friedmann acceleration equation: matter decelerates expansion. Therefore Evans's "charge density" E^00 = −4πGρ/c². It is zero if and only if ρ = 0. "Violation of the dual identity" means "the universe contains matter".

Step D. Put in numbers. Take a matter-only (Einstein-de Sitter) toy universe with today's Hubble rate H₀ = 67.4 km/s/Mpc [memory, high; Planck 2018]: - H₀ = 67.4×10³ m/s ÷ 3.0857×10²² m = 2.184×10⁻¹⁸ s⁻¹. - Critical density ρ = 3H₀²/(8πG) = 3(2.184×10⁻¹⁸)² / (8π × 6.674×10⁻¹¹) = 8.53×10⁻²⁷ kg/m³ (about five hydrogen atoms per cubic metre). - Evans's quantity from the matter side: 4πGρ/c² = 4π(6.674×10⁻¹¹)(8.53×10⁻²⁷)/(2.998×10⁸)² = 7.96×10⁻⁵³ m⁻². - From the geometry side: for a ∝ t^(2/3), ä/a = −½H² (in ordinary time t), so |R_00| = 3·½H₀²/c² = 1.5(2.184×10⁻¹⁸)²/(2.998×10⁸)² = 7.96×10⁻⁵³ m⁻². The two sides agree, as the Einstein equation says they must. - Inside ordinary water (ρ = 1000 kg/m³): 4πGρ/c² = 9.33×10⁻²⁴ m⁻². The square root's inverse, 3.3×10¹¹ m (about two Earth-Sun distances), is the radius of curvature that a body of water's own density produces. Nonzero, and exactly what GR predicts.

Second universe, the vacuum case Evans calls "fortuitous": Schwarzschild. example.py finds E^κν = 0 for all components (Ricci = 0 outside the mass), yet K = 12r_s²/r⁶ ≠ 0. For Earth (M = 5.972×10²⁴ kg, R = 6.371×10⁶ m): r_s = 2GM/c² = 8.87 mm, and K at the surface = 12(8.869×10⁻³)²/(6.371×10⁶)⁶ = 1.41×10⁻⁴⁴ m⁻⁴. So the vacuum solution is curved (tides are this curvature) and still passes Evans's test, because the test is blind to everything except Ricci, i.e. except the local matter.

Step E (side check, Claim 4). Does a spin connection make Coulomb's law resonate? With E = −(∇ + ω)φ and ω radial, Gauss's law in spherical symmetry gives φ'' + (2/r + ω)φ' + (ω' + 2ω/r)φ = −ρ_q/ε₀ (matches CEFE Eq. 2.160 per mathematician.md). Evans's choice ω = −1/r gives φ'' + φ'/r − φ/r² = 0 in empty space. Try φ = r^s: s(s−1) + s − 1 = s² − 1 = 0, so s = ±1 and φ = C₁r + C₂/r. Power laws, no sine or cosine, no resonance for any real radius. Oscillation appears only after the substitution κr = exp(iκR), which puts the physical radius on a circle in the complex plane.

Bonus number (Claim 4, electrogravitic). Felker's E = φ(0) g/c² with g = 9.8 m/s² and an assumed φ(0) = 1 V (no value is given in the book [read]): E = 9.8/(2.998×10⁸)² = 1.09×10⁻¹⁶ V/m. And a static field with zero curl does zero work around any closed loop, so it cannot drive a circuit.

Step F (Claim 7). The electron g-factor, and how an error bar can hide a discrepancy. This step and the bonus number after it are reproduced by example.py Part G; the Kasner paragraph in Step C by Part F. CEFE Eq. 2.214 gives g(ECE) = 2(1 + α/4π)², which expands to Eq. 2.223, 2 + α/π + α²/(8π²) [read: page image p.55, per the Round 2 deep read; pp.55-56]. With α = 7.2973525693×10⁻³ [memory, high; CODATA 2018]: - α/π = 2.32282×10⁻³ and α²/(8π²) = 6.744×10⁻⁷, so g(ECE) = 2.0023234939. - Experiment (as the book quotes it): 2.0023193043718 ± 7.5×10⁻¹². - Difference: 4.19×10⁻⁶, which is 5.6×10⁵ times the experimental uncertainty. The book itself admits a difference of 0.000004.

The book's defence is that ECE's own value is uncertain, because α is "limited by the accuracy of h". Propagate the uncertainty (Section 2): dg/dα = 1/π + α/(4π²) ≈ 0.3185. - With the book's α = 0.007297(34), the relative error is 34×10⁻⁶ / 0.007297 = 4.7×10⁻³, so σ_g = 0.3185 × 7.297×10⁻³ × 4.7×10⁻³ ≈ 1.1×10⁻⁵. The gap is then 0.4σ and looks fine. - But the book's own inputs (its quoted e and h with their uncertainties) give α = 7.2973525546×10⁻³ with relative uncertainty 2.4×10⁻⁷ (α ∝ e²/h, so the relative errors add in quadrature as √((2×8.7×10⁻⁸)² + (1.66×10⁻⁷)²)). Then σ_g ≈ 5.6×10⁻¹⁰ and the gap is about 7,500σ [calc, example.py Part G; the arbiter's Round 2 relies on the deep read's figures].

The book's error bar is about 2×10⁴ times too big (arbiter Round 2; QE's earlier figure was about 10⁴), and that alone makes the discrepancy look acceptable. Its "(49)" on g, ±4.9×10⁻⁵, cannot be reproduced by any of three propagations (book's α error, true α error, h's error), and it is 6.5×10⁶ times the experimental error bar [calc, per arbiter Round 2]. Since the 2019 SI redefinition, h has an exact defined value [memory, high], so the "accuracy of h" argument no longer applies at all.

Two more checks from the same page: - Is ECE's extra term an improvement? Schwinger's one term, 2 + α/π = 2.0023228195, is 3.5×10⁻⁶ above experiment. ECE adds +α²/(8π²) = +6.7×10⁻⁷, which pushes the gap up to 4.19×10⁻⁶. The true second-order QED term is about −3.5×10⁻⁶ in g, the opposite sign; standard QED through α², α³ and α⁴ lands at −3.0×10⁻⁸, +1×10⁻¹⁰ and −1×10⁻¹¹ from experiment [calc, per arbiter Round 2; QED coefficients memory, high for the first three]. Adding the ECE term makes the answer worse. - Is NIST's α inconsistent? The book says its α = 0.007297 disagrees with NIST's 7.297352560×10⁻³ "in the eighth decimal place" [read: p.57]. Unrounded, the book's own inputs give 7.2973525546×10⁻³, which differs from that NIST value by 5×10⁻¹², 0.2 of NIST's quoted σ (2.4×10⁻¹¹). The "inconsistency" is the rounding [calc, example.py Part G]. The book also says α "drops to 1/128" at high energy; in fact α rises (1/α falls to about 128 near the Z mass) [memory, high, per arbiter].

The Lamb shift: a fitted number that looks like a prediction. The measured 1057.8 MHz, divided by c in cm/s, is 1057.8×10⁶ / 2.998×10¹⁰ = 0.03528 cm⁻¹. The book's ECE formula (Eq. 2.242) is ΔE = (1/(16π^(3/2)))(α/a)(ħ/mc)(1/r), with a = Bohr radius (5.29×10⁻¹¹ m), ħ/mc = 3.86×10⁻¹³ m (the reduced Compton wavelength) and r a length. The book states the value "Here: r = 1.69×10⁻⁷ m" (Eq. 2.243) [read: p.59] but gives no derivation and no physical meaning for it. (Earlier versions of this primer, following qe.md, said r was unstated; that was wrong.) Put in the numbers: - With r = 1.69×10⁻⁷ m: ΔE = 0.0354 cm⁻¹, matching the book's printed 0.0353. The r that gives the measured value exactly is 1.694×10⁻⁷ m [calc, example.py Part G]. - But 1.69×10⁻⁷ m is about 3,190 Bohr radii, far outside a 2s orbit (mean radius about 3.2×10⁻¹⁰ m). If r really were the electron-proton distance, r = 4a gives 28 cm⁻¹ and r = a gives 113 cm⁻¹, 800 to 3,200 times the measured shift [calc, example.py Part G].

So the formula has one free length, and that length was set to the value that returns the data: it works as a fit, not a prediction [arbiter Round 2]. The formula also drops the angular dependence (cos θ = 1) and, at fixed r, has none of the real Lamb shift's dependence on n and l [memory, high, per arbiter]. Smaller slips on the same pages: Eq. 2.207 labels h as ħ; Eq. 2.249 prints 3.48×10⁻¹³ m where its own formula gives 3.47×10⁻¹⁴ m (the book's remark "about ten times the classical radius" fits the correct figure); Eqs. 2.231 and 2.240 give two different coefficients (1/(4π) and 1/(2π^(3/2))) for the same quantity, and Eq. 2.253 is half of Eq. 2.242 [calc, per arbiter Round 2].

Bonus number (Felker Ch.9, a units-and-arithmetic check). Felker's minimum particle volume is V₀ = kħ²/(mc²), with k = 8πG/c² = 1.866×10⁻²⁶ m/kg [read: pp.214-219]. For the electron (m = 9.109×10⁻³¹ kg): V₀ = 1.866×10⁻²⁶ × (1.0546×10⁻³⁴)² / (9.109×10⁻³¹ × 8.988×10¹⁶) = 2.5×10⁻⁸¹ m³. The book prints 1×10⁻⁷⁹ m³, 40 times larger, and pairs it with a radius of about 4×10⁻¹³ m; a sphere of 10⁻⁷⁹ m³ has radius 2.9×10⁻²⁷ m [calc]. Felker labels the electron-shape speculation his own [read: p.216 n.46]. The relation mV₀ = constant holds by definition, so it predicts nothing.

6. Evidence table

Verdicts are the arbiter's final verdicts (arbiter.md). "Tested?" means a controlled test of this specific claim.

7. What a learner should take away

What is solid. - Torsion is real mathematics, and a torsionful spacetime is a legitimate physical hypothesis (Einstein-Cartan). Evans is right that GR's T = 0 is an assumption rather than a theorem. - The program of making electromagnetism geometric has a serious history (Weyl, Kaluza-Klein, Einstein's teleparallel theory). Wanting it is not crankish. - Eckardt's computer algebra appears to be correct: by the book's own account it reproduced known Christoffel, Riemann and Ricci results [read, Ch.4.2], and every table reviewers saw printed the real Bianchi identity as "o.k." The numbers are right; the interpretation is wrong. - Crothers's narrow point that "r" in Schwarzschild is an areal coordinate, not a measured radial distance, is textbook GR. His complaint that popular accounts describe a black hole by an "escape velocity" is a fair point about wording, though GR defines the horizon causally. - The Sagnac section in CEFE Sec. 2.7 ends at the standard result 4ΩA/c², though only because that result is its input (see the Section 4 row "Faraday disk and Sagnac"); it is a consistent restatement, not a new derivation [calc, per arbiter Round 2]. Pendergast's Ch.5 accepts light bending, gravitational lensing and the stellar orbits around Sgr A* [read, per qe.md], so the book does not reject every standard observation.

What is wrong, and why the mistake is instructive. - Confusing a part with the whole. Torsion is the antisymmetric part of Γ. Concluding that Γ must be antisymmetric is like concluding a matrix must be antisymmetric because you defined A − Aᵀ. Splitting objects into symmetric and antisymmetric parts is a tool you will use all through physics; this is a clean example of misusing it. - Turning a field equation into an "identity". An identity holds for every geometry. Eq. 4.1 with T = 0 holds only when Ricci = 0, i.e. only in vacuum. Whenever a proposed identity holds for some cases and fails for others, it is an equation, not an identity. Checking a claim against a simple case (flat polar coordinates, empty FLRW) is the fastest way to catch this. - A circular test. The Ch.4 test cannot distinguish "GR is wrong" from "there is matter here". A test that gives the same answer whether the theory is right or wrong tests nothing. Ask of any test: what result would have counted against my claim? - Free functions are not predictions. With the spin connection ω chosen freely, any desired equation for φ can be manufactured. A theory with a free function per problem fits everything and predicts nothing until the function is fixed by a rule. - Resonance is not a source. A resonance amplifies a response to a drive that is already supplying energy. Any free-power claim needs an energy budget; Sec. 2.6 has none. - A prediction that was tested later. Crothers's 2010 statement that LIGO was "destined to detect nothing" was a clear, dated, falsifiable prediction, and GW150914 falsified it five years later. Making such a prediction is good science; the lesson is what happens to an argument (here, "no two-body solution, so no waves") when its prediction fails. - Error bars that hide a discrepancy. An error bar about 2×10⁴ times too large turned a 7,500σ miss into a 0.4σ "agreement" (Step F). When a theory agrees with experiment "within errors", check where the errors came from. - A fitted constant is not a prediction. The Lamb-shift formula matches the measurement because its one free length, r = 1.69×10⁻⁷ m, has the value that makes it match; any physically meaningful atomic distance gives an answer hundreds to thousands of times too large (Step F). Ask of any "agreement": how many numbers were chosen, and could they have been chosen to fit? - Adding a term can make a result worse. ECE's α²/(8π²) has the wrong sign against the real second-order QED term, so it moves g away from experiment. Comparing each order of a series against data is how physicists tell a real correction from a decoration. - Rounding can manufacture a contradiction. The book's "inconsistency in NIST" disappears when α is not truncated to four significant figures (Step F). - Numbers asserted, not derived. The W and Z masses (78.6 and 89.3 GeV) are stated as results with no calculation, from an equation whose input W(0) is never given, and they do not even match the measured values. A number with no visible route to it cannot count as a prediction. - Arithmetic you can check. The electron volume is off by a factor of 40 from its own formula, and one sphere volume is computed without cubing the radius (Felker p.217). Redoing a book's arithmetic is a quick and reliable test. - Complex substitutions. Putting r = e^(iκR)/κ makes sines appear, but r is a real, positive distance. A change of variables that leaves the physical domain changes the problem.

What is genuinely open (not credit to Evans). - Whether spacetime carries torsion at small scales or high spin densities is not settled; current bounds are tight but not zero [memory, medium]. - Dark matter has strong gravitational evidence but no particle detection; modified-gravity alternatives are a live research area [memory, high]. - Whether anyone has ever measured a Bedini-type device under neutral, calorimetric conditions: no such test is on record in the sources here. It would be cheap to do (see the Bedini-John primer). - What exactly Hehl and Obukhov and Bruhn proved, and whether any critique addresses the 2007-09 "dual identity" specifically: unread by every reviewer. - Whether a derivation of the stated "vacuum radius" r = 1.69×10⁻⁷ m exists anywhere in Evans's papers, which would turn the Lamb-shift formula into a real prediction; and whether any paper gives the W(0) values behind 78.6 and 89.3 GeV. Neither is in the folder. - Eckardt's torsion simulations of galaxy rotation curves are cited but not in the folder, so that claim cannot be judged here.

8. Check yourself

1. In 2D polar coordinates, compute Γ^r_θθ from the Christoffel formula. Then explain in one sentence why a connection that is antisymmetric in its lower indices could not describe a ball rolling in a straight line on a flat table, written in polar coordinates.
2. For the FLRW metric, Evans's quantity E^00 is +3ä/a (x^0 = ct). Using the Einstein equation for dust, what is it in terms of the mass density ρ? Evaluate it for air (ρ ≈ 1.2 kg/m³).
3. The Schwarzschild metric passes Evans's test. Does that mean spacetime near Earth is flat? What quantity tells you, and what is it at Earth's surface?
4. Evans says that since T_μν = 0 is assumed in the vacuum equation, the mass M cannot appear in the solution. Give the electrostatics analogue that shows why this reasoning fails.
5. Solve φ'' + φ'/r − φ/r² = 0 for real r > 0. Could a circuit described by this equation resonate? What did Evans do to get oscillating solutions?
6. CEFE gives g(ECE) = 2 + α/π + α²/(8π²). Evaluate it with α = 7.2973525693×10⁻³ and compare it with g(exp) = 2.0023193043718 ± 7.5×10⁻¹². How many experimental standard deviations apart are they? Then check whether the Kasner exponents (2/3, 2/3, −1/3) and (1, −1, 0) satisfy p₁ + p₂ + p₃ = 1 and p₁² + p₂² + p₃² = 1.

9. Further reading

Mainstream (learning the background). - B. F. Schutz, A First Course in General Relativity (Cambridge). Gentlest route to index notation; the "Preface to curvature" chapter does polar-coordinate Christoffel symbols. - J. B. Hartle, Gravity: An Introduction to Einstein's General Relativity (Addison-Wesley). Physics first; good on Schwarzschild, cosmology and tests. - S. M. Carroll, Spacetime and Geometry (Addison-Wesley). Chapter 3 ("Curvature") defines torsion, Riemann and the Bianchi identities exactly as CEFE quotes them; the appendix on non-coordinate bases covers tetrads and spin connections. - F. W. Hehl, P. von der Heyde, G. D. Kerlick, J. M. Nester, "General relativity with spin and torsion: Foundations and prospects", Rev. Mod. Phys. 48, 393 (1976) [memory, high]. The standard review of Einstein-Cartan theory. - D. J. Griffiths, Introduction to Electrodynamics, Chapter 2 (electrostatics) and Chapter 7 (electrodynamics, including the Faraday disk); Introduction to Quantum Mechanics for the uncertainty theorem. - K. Schwarzschild (1916), "On the gravitational field of a mass point according to Einstein's theory", English translation: https://arxiv.org/abs/physics/9905030 [web, via contrarian.md]. - L. A. Rozema et al., "Violation of Heisenberg's measurement-disturbance relationship by weak measurements" (2012), https://arxiv.org/abs/1208.0034 [web, via mathematician.md]: shows what a real refinement of Heisenberg looks like (error-disturbance, not σ_xσ_p ≥ ħ/2).

• B. P. Abbott et al. (LIGO Scientific Collaboration and Virgo Collaboration), "Observation of Gravitational Waves from a Binary Black Hole Merger", Phys. Rev. Lett. 116, 061102 (2016) [memory, high]; summary at https://en.wikipedia.org/wiki/First_observation_of_gravitational_waves [web, via arbiter.md]. The test of Crothers's prediction.
• C. W. F. Everitt et al., "Gravity Probe B: Final Results of a Space Experiment to Test General Relativity", Phys. Rev. Lett. 106, 221101 (2011) [memory, high]; summary at https://en.wikipedia.org/wiki/Gravity_Probe_B [web, via arbiter.md].
• J. R. Taylor, An Introduction to Error Analysis (University Science Books): propagation of uncertainties, needed for Step F.
• D. J. Griffiths, Introduction to Elementary Particles (Wiley): a readable account of QED, the g-factor and the W and Z bosons.

Critiques of ECE. - J. G. Giglio and W. A. Rodrigues Jr., "One More Assessment of 'ECE Theory'", https://arxiv.org/abs/1109.5403 [web, via arbiter.md; abstract only read]. - Hehl and Obukhov (2008), Bruhn (2008), Rodrigues and Souza (2005), and the 't Hooft editorial in Foundations of Physics (2008), as listed at https://en.wikipedia.org/wiki/Einstein%E2%80%93Cartan%E2%80%93Evans_theory [web]. Not read by any reviewer.

The researcher's own works. - M. W. Evans, S. J. Crothers, H. Eckardt, K. Pendergast, Criticisms of the Einstein Field Equation: The End of 20th Century Physics (2010) [read, in folder: book/CEFE-print.pdf]. Read the preface (pp.iii-v) side by side with Section 5 of this primer. - L. G. Felker, The Evans Equations of Unified Field Theory (Rev. 3, 2005) [read, in folder: book/Evans_Equations_Rev3.pdf]. The friendliest statement of what ECE intends. Felker says Evans "guided and permitted free use of his writings but did not correct" the book, and that the speculative parts are his own [read, per qe.md: pp.20-21], so it is not always Evans's own view. - For the claims added after QE, read CEFE Ch.3 Secs 3.5-3.9 (Crothers on gravitational waves, PDF pp.110-136), Secs 2.7-2.9 (Faraday disk and Sagnac, PDF pp.49-54; g-factor and Lamb shift, PDF pp.54-61, with r stated on p.59; the list of 36 "advances", PDF pp.61-66) and the Kasner table (PDF pp.276-282). - AIAS papers ("UFT" series) 63, 93, 94, 122 at www.aias.us, cited in CEFE [read]; not in the folder.

---

Answers

1. Γ^r_θθ = ½ g^rr(∂_θ g_rθ + ∂_θ g_rθ − ∂_r g_θθ) = ½(1)(0 + 0 − 2r) = −r. In the geodesic equation the connection is multiplied by the symmetric product (dx^μ/ds)(dx^ν/ds), so an antisymmetric Γ contributes nothing; the equations would become r̈ = 0 and θ̈ = 0, which describe a spiral, not a straight line (a straight line in polar coordinates needs r̈ = rθ̇², the −Γ^r_θθ θ̇² term).
2. E^00 = −Ric^00 = −Ric_00 = −4πGρ/c². For air: 4π(6.674×10⁻¹¹)(1.2)/(2.998×10⁸)² = 1.12×10⁻²⁶ m⁻², so E^00 ≈ −1.1×10⁻²⁶ m⁻². Nonzero, as it must be wherever there is mass: this is the Einstein equation, not a violation of geometry.
3. No. Ricci is zero outside Earth but the Riemann tensor is not. The Kretschmann scalar K = 12r_s²/r⁶ ≈ 1.4×10⁻⁴⁴ m⁻⁴ at the surface (r_s = 8.87 mm). Tidal forces are the physical face of this curvature.
4. Outside a point charge, ∇²φ = 0 (no charge there), yet the solution φ = q/(4πε₀r) contains q. The constant enters through the boundary condition (or by matching to the region that contains the charge), not through the source term in the empty region. The vacuum Einstein equation works the same way, with M fixed by the Newtonian limit or by matching to an interior solution.
5. φ = r^s gives s² − 1 = 0, so φ = C₁r + C₂/r: no oscillation, so no resonance at any real radius. Evans substituted κr = exp(iκR), i.e. he made the radius complex (|κr| = 1), and the oscillation lives in the new variable R, not in any physical distance.
6. α/π = 0.0023228195 and α²/(8π²) = 0.0000006744, so g(ECE) = 2.0023234939. The difference from experiment is 4.19×10⁻⁶, and 4.19×10⁻⁶ / 7.5×10⁻¹² ≈ 5.6×10⁵ standard deviations, so the formula is ruled out. (2/3, 2/3, −1/3): sum 1, squares 4/9 + 4/9 + 1/9 = 1, so it is a vacuum Kasner universe with zero Ricci and passes Evans's test. (1, −1, 0): sum 0, squares 2, so it is not a vacuum solution, and the "violation" the book plots for it is just its nonzero Ricci.