Rankings Primer Arbiter Physicist Mathematician Contrarian QE Claims Sources

Evans-Myron: Mathematician's review (Step 2)

Reviewer role: PhD mathematician (derivations, notation, units, hidden assumptions).

Source gap (repeated from Step 1): only two books are in the folder: CEFE-print.pdf (2010, 548 pp.) and Evans_Equations_Rev3.pdf (Felker 2005, popularisation, 373 pp.). No Evans papers and no experimental data are present. I read the CEFE preface, TOC, Sec 2.6 (text and page images pp.39-40 [read: page image p.47-48 of PDF]), Ch.3 Secs 3.2 and 3.4, Ch.4 Secs 4.1-4.3, and Felker's sections on the Lemma, wave equation, Ch.10 (uncertainty) and Ch.14 (electrogravitic). I did NOT read the 62 metric tables, Crothers's gravitational-wave section, or the ECE derivations in Evans's own papers (AIAS papers 63, 93, 94, 122). Statements about those rest on [web]/[memory] and are flagged.

Tags: [read] from the books; [calc] my own SymPy computation (scratch scripts, reproducible); [web] URL; [memory] with confidence.

Corrections to Step 1

Step 1 tagged as "[memory] medium" the diagnosis that Evans conflates commutator labels with connection symmetry. This is now [read]: preface Eqs (3)-(6) argue that because D_{mu nu} = [D_mu, D_nu] is antisymmetric, "the connection therefore has the same symmetry as the commutator". The torsion definition T = Gamma^l_{mu nu} - Gamma^l_{nu mu} (Eq 5) is the antisymmetric part only; nothing forces Gamma itself to be antisymmetric. (Reasoning is mine; the quoted text is [read].)
Step 1 key_claim 2 says Eckardt's algebra "shows" a violation of the dual identity. Better: [calc] the quantity computed in CEFE Eq (4.8), R^{kappa mu}{mu nu}, is minus the mixed Ricci tensor, -Ric^kappa_nu. I verified this symbolically for a general FLRW metric (a(t)): R^{0}{}^{mu}{}{mu 0} = -3a''/a and spatial components -(a a'' + 2a'^2)/a^2, i.e. exactly minus Ric^kappa_nu; and the ordinary first Bianchi identity R^k_{[mnr]} = 0 holds identically (0 violations). So Eckardt's code is probably right; what it "found" is that Ricci is nonzero wherever matter is present. Step 1's one-line summary of Ch.4 ("each violates Eq. 4.1") should read "Eq. 4.1 with T=0 is equivalent to Ric = 0".
Step 1 cites "[memory] low: Hehl/Obukhov showed the dual identity does not hold". The [web] source (Wikipedia, ECE theory) only confirms that Hehl and Obukhov published two critical assessments, Rodrigues and Souza (2005) challenged the tetrad postulate, Bruhn (2008) attacked the O(3) law, and 't Hooft (editor, Foundations of Physics) stated ECE is absent from peer-reviewed literature. It does not say what Hehl/Obukhov specifically showed. Treat that point as unverified; my own critique (below) is independent of it.
Step 1 places the dual identity in "preface, Ch.2, Ch.4"; the identity (Eqs 1-2, 13, 15-16) is in the preface and Ch.4 intro (4.1). Minor.
Step 1 key claim "Maxwell-Heaviside equations are the zero-spin-connection limit" is fine; add that the spin connection omega is left a free function ("There is freedom of choice of the spin connection ... must ultimately be determined experimentally", CEFE p.40 [read: page image]), which matters for testability (claim 3/4).
Claims.md claim 4 attributes the E = phi(0) g/c^2 relation to "Felker Ch.14" correctly. Felker eq (7) as printed has a typo-like term on the right (see Claim 4).

Claim 1: The Einstein field equation is wrong because T = 0 violates the Hodge-dual Bianchi identity; the connection must be antisymmetric

a. Mathematics consistent? No. Three separate defects. - (i) Non sequitur on connection symmetry. [read] Preface Eqs (3)-(6): the commutator [D_mu, D_nu] is antisymmetric in its two labels; Evans concludes Gamma^l_{mu nu} must be antisymmetric in mu, nu. But the commutator's labels are the derivative labels; the connection's lower pair (mu = direction of derivative, nu = index being differentiated) are different kinds of index. Torsion is the antisymmetric part; the symmetric part is free. Counter-test [calc-by-hand]: if Gamma^l_{mu nu} = -Gamma^l_{nu mu}, then the geodesic equation d2x^l/ds2 + Gamma^l_{mu nu} (dx^mu/ds)(dx^nu/ds) = 0 contains a symmetric contraction of an antisymmetric object, which is identically zero. Every free particle would move in straight coordinate lines in every coordinate system; gravity (and Newtonian limit, which needs the symmetric Gamma^i_{00} = d_i phi) would vanish. Also, the Christoffel formula Gamma = (1/2) g^{ls}(d_mu g_{s nu} + d_nu g_{s mu} - d_s g_{mu nu}) is symmetric by construction, and for a metric-compatible connection torsion is an independent extra datum (contortion). So "must be antisymmetric" contradicts the metric and the geodesic equation, which Evans also uses. - (ii) The dual identity is not tested by the computation. [read] Eqs (8)-(12) define dual connection and curvature from Gamma-tilde = (1/2) eps Gamma. For a symmetric Gamma, Gamma-tilde^l_{mu nu} = (1/2) sqrt|g| eps_{mu nu}^{alpha beta} Gamma^l_{alpha beta} = 0 identically, so T-tilde = 0 and R-tilde (Eq 12) = 0: the dual identity (13) then reads 0 = 0, no contradiction. Evans instead evaluates the ordinary Riemann tensor (CEFE 4.7-4.8), not R-tilde. The "test" and the "identity" are different objects. Further, Evans himself states the dual objects are not tensors (preface), so the Hodge dual identity is not coordinate-covariant as written. - (iii) The contracted identity is not a consequence of the cyclic identity. Eq (15) is a cyclic sum over three form indices; Eq (16) D_mu T^{a mu nu} = R^{a mu}{mu nu} is asserted to be "the same as" it. With T = 0 the true torsion-free Bianchi identity is R^k{[mnr]} = 0, which contracts to "Ricci is symmetric", always true. Evans's Eq (16)-(18) with T = 0 instead says Ric^k_nu = 0 [calc, shown above], i.e. he has turned the vacuum condition into an "identity", then complained that matter-filled solutions break it. A contraction that is not implied by the parent identity cannot be used as a test.

b. Consistent with experiments? The claim (no valid GR) conflicts with GR tests: perihelion precession of Mercury, light deflection, Shapiro delay, gravitational redshift, binary pulsar orbital decay, LIGO/Virgo gravitational-wave detections. [memory] high on existence; none were checked against this book. The preface itself asserts the Einstein equation is "wholly incorrect" but offers no experimental discrepancy.

c. Tested? The mathematical claim is checkable by calculation, not experiment: refuted above. External responses: [web] https://en.wikipedia.org/wiki/Einstein%E2%80%93Cartan%E2%80%93Evans_theory lists critical papers (Rodrigues and Souza 2005; Bruhn 2008; Hehl and Obukhov 2008) and the 't Hooft editorial. Dismissed without testing? No, it was criticised on substance in print per that source; but I did not read those critiques.

d. If true? GR, the Levi-Civita connection, the Newtonian limit, every solar-system test would have to be discarded, and a new geometry would have to reproduce them anyway (Einstein-Cartan with torsion, which is accepted as a consistent extension, reduces to GR where spin density vanishes [memory] high).

Verdict: Internally inconsistent (non sequitur on antisymmetry; test and identity mismatched; contraction not implied) and, for the "GR violates geometry" part, Refuted by calculation. A weaker, defensible variant ("spacetime may have torsion") is Plausible, untested (Einstein-Cartan theory), but that is not what the book claims.

Claim 2: Computer algebra on ~62 metrics shows all matter solutions violate the identity, so black holes, GWs, Big Bang, dark matter are invalid; Schwarzschild was misderived (Crothers)

a. Mathematics consistent? No. - Ch.4: see Claim 1. [calc] R^{kappa mu}{mu nu} = -Ric^kappa_nu for the FLRW case; same algebra holds for any Levi-Civita metric (it is a contraction of the Riemann tensor). So 'nonzero for every matter solution' is a restatement of G{mu nu} = k T_{mu nu} with T != 0. Ch.4 also says several well-known line elements "failed" the cyclic test (4.12) and metric compatibility (4.13). For any metric, Levi-Civita satisfies both identically ([calc] FLRW 0 violations), so failures indicate input transcription or code errors, not failure of GR. I did not check which metrics. - Ch.3 [read]: Crothers's Eq (3.2) with R = (r^3 + alpha^3)^(1/3) is Schwarzschild's 1916 coordinate. This is a monotonic relabeling of the same metric; with alpha = 2m the metric in R is identical to the standard form. The argument in Sec 3.4 that e^lambda, e^beta > 0 "cannot change sign" mistakes a derivation-region assumption for a global constraint: the signs are fixed in the exterior region where the ansatz is solved; the solution is analytically continued, and the horizon at r = 2m is a coordinate artifact removable by Eddington-Finkelstein or Kruskal coordinates, in which the metric is regular. The curvature invariant (Kretschmann scalar) 48 m^2/r^6 is finite at r = 2m [memory] high (not recomputed here). Crothers's observation that Schwarzschild's own constant alpha is a priori undetermined is true (it is an integration constant); it is fixed to 2m by the Newtonian limit g_00 -> 1 - 2GM/(c^2 r). That "Ric = 0 contains no mass" is the same as saying Laplace's equation in empty space contains no source, yet the solution phi = -GM/r carries a constant M determined by boundary conditions or by matching to an interior source; the vacuum solution outside a star (Birkhoff) is exactly this. The self-contradiction charge is therefore a hidden assumption about what integration constants mean. - Point-mass singularity prohibition and gravitational wave claim (Crothers Secs 3.5, 3.8): not read.

b. Experiments: GW150914 and later LIGO/Virgo events, the Event Horizon Telescope image of M87, orbits of S2 around Sgr A, CMB and nucleosynthesis for the Big Bang. [memory] high for existence; no URLs fetched (not needed for the verdict, which rests on calculation).

c. Tested? The Maxima code itself was tested only against hand results by Evans's group [read]. No independent replication found [web: Wikipedia summary]. Experimental GW detections are by LIGO collaboration [memory high].

d. If true: Cosmology, relativistic astrophysics and GPS corrections (which use GR [memory high]) would need rederivation.

Verdict: Refuted by calculation for the "violation" computation (it measures Ricci, not a violation) and for Crothers's black-hole "impossibility" argument as read in Secs 3.2/3.4 (sign-change and relabeling errors). Not assessed: Crothers's GW section (unread).

Claim 3: Electromagnetism is spacetime torsion (A = A(0) q, F = A(0) T); unified ECE theory; Maxwell-Heaviside and gauge theory incomplete

a. Mathematics consistent? Mixed; I did not read the full derivations. - [read] Dimensions: A(0) in V s/m times dimensionless tetrad gives potential; the ansatz is dimensionally fine. But the tetrad q^a_mu carries a Lorentz index a, so A^a_mu has an internal index that ordinary electromagnetism does not have. To recover the Maxwell field one has to contract or choose a particular a; the book's route is to say the Maxwell-Heaviside equations are the zero-spin-connection limit with extra indices suppressed. The relation F = A(0) T = A(0) (D ^ q) is a definition, not a derivation: one can multiply any geometric 2-form by a constant. Predictive content depends entirely on omega (the spin connection), which is left free (see Claim 4). - The tetrad postulate derivations were contested by Rodrigues and Souza 2005 [web, as above]. I did not check the content.

b. Experiments: Where omega = 0 the theory equals Maxwell, so it matches all of Maxwell's successes by construction. Where omega != 0 no value is given. The photon "mass" and B(3) (Felker Ch.11) conflict with limits: photon mass < 1e-18 eV [web] https://pdg.lbl.gov/2025/listings/rpp2025-list-photon.pdf (search result states PDG limit m_photon < 1e-18 eV; I read only the search summary, not the PDF).

c. Tested? Not on record in the folder. No independent test found; [web] Wikipedia notes no accepted experimental evidence.

d. If true: Gauge principle of the Standard Model (U(1), SU(2), SU(3)) would not be fundamental; Maxwell would be a limit, and the photon would acquire geometry-related properties.

Verdict: Not testable as stated (free function omega, no quantitative prediction that differs from Maxwell); the unification-in-principle part is Plausible, untested as a formal idea (torsion-electromagnetism proposals exist) but as formulated it is also partly Internally inconsistent where it equates a Lorentz-indexed tetrad field with a U(1) potential without a reduction map (my reading of limited text; low-to-medium confidence).

Claim 4: Spin connection makes the Coulomb/Poisson equation resonant (Tesla, Bedini), allowing circuits to draw power from spacetime; electrogravitic E = phi(0) g/c^2

a. Mathematics consistent? No, in the derivation I could check. [read: page images] CEFE pp.38-40 (PDF pp.46-48). - Setup: E = -(grad + omega) phi gives Eq (2.151) div E = rho/eps0: laplacian phi + omega . grad phi + (div omega) phi = -rho/eps0. [calc] For radial omega_r, this is phi'' + (2/r + omega_r) phi' + (1/r^2)(2 r omega_r + r^2 omega_r') phi = -rho/eps0 (matches Eq 2.160). - Eq (2.155)-(2.156): omega_r = -1/r gives phi'' + phi'/r - phi/r^2 = -rho/eps0 [calc: checked]. Its homogeneous solutions are phi = C1/r + C2 r [calc, SymPy dsolve]. There are no oscillatory solutions in real r. - "Euler transform" Eq (2.158): kappa r = exp(i kappa R). With r d/dr = d/dx for r = e^x/kappa, the equation becomes phi_xx - phi = -r^2 rho/eps0; setting x = i kappa R gives Eq (2.159) (I verified this algebra). So the oscillation arises only because r has been continued onto the complex unit circle |kappa r| = 1; real R then corresponds to complex radius. The physical radius r is real and positive; the "resonance in R" is not a resonance in any measurable variable. This is a hidden assumption, and the call to "take Real(...)" is applied after a complex substitution. - Eqs (2.161)-(2.165): [calc] with omega_r = omega0^2 r - 4 beta ln r - 4/r the coefficient of phi' is 2/r + omega_r = omega0^2 r - 4 beta ln r - 2/r (not the constant 2 beta of Eq 2.162), and the coefficient of phi is not omega0^2 either (SymPy: -8 beta ln r / r - 4 beta/r + 3 omega0^2 - 4/r^2). So Eq (2.162) does not follow from (2.160) and (2.161). Eq (2.164) says omega_r = 2(beta - 1/r) and kappa0^2 = (4/r)(beta - 1/r) + omega_r'; that makes kappa0^2 = 4 beta/r - 2/r^2, a function of r, not a "Hooke's-law wave number" constant; and (2.165) differs from (2.164). Also two coefficient conditions are imposed on a single function omega_r: an overdetermined system, so "constant coefficients" is not achievable for generic constants. Also ln r of a dimensioned r, and "beta = 1/r" (Eq 2.166), a "friction coefficient" equal to a coordinate-dependent quantity. - Free omega: since omega is any chosen function, any desired ODE for phi can be manufactured; this is curve-fitting, not a law. Spin connection (a "gauge-like" field) is "freedom of choice" per [read] p.40. - Energy: nothing in Sec 2.6 contains an energy budget. A resonance in an ODE solution amplifies the response to a driving term; it does not create energy. The claim "take power from space-time" is an assertion with no calculation. - Electrogravitic [read] Felker Ch.14, Eqs (4)-(9): E = (1/c^2) grad(phi(0) phi) = phi(0) g/c^2. Units: V x (m/s^2)/(m^2/s^2) = V/m, consistent. But it is obtained by postulating that the electric potential is a constant multiple of the gravitational potential, which forces charge density to be proportional to mass density everywhere (both obey Poisson sources). Eq (7) as printed, "laplacian (phi(0) phi) = 4 pi G (phi(0) rho phi)", has an extra factor phi (potential) on the right compared to Eq (5), a dimensional inconsistency in the source text (possibly a typo). Worked number: take phi(0) = 1 V (Felker does not give a value; he calls it "a fundamental voltage"), g = 9.8 m/s^2: E = 9.8 / (8.99e16) = 1.09e-16 V/m. Such a static, curl-free field can power nothing: the work around a closed loop is zero.

b. Experiments: Standard electrostatics (Coulomb's law to high precision; Poisson equation has no resonance in a source-free static setting) holds; Evans's reply that Coulomb's law is "tested off resonance" is unfalsifiable as stated. No test data are given in the book for the Bedini motor claim (papers 63, 94 not in folder). Energy conservation (Noether, time-translation invariance) is tested to extreme precision [memory] high.

c. Tested? Not in the documents. Whether Bedini-type motors have been independently tested: not found in sources I read; I did not search (outside my verdict dependency). Say unknown.

d. If true: A usable external energy source would mean time-translation symmetry (energy conservation) fails or that spacetime is a hidden reservoir; Maxwell's equations would acquire an extra free field.

Verdict (split): - Resonance derivation (Sec 2.6): Internally inconsistent (Eqs 2.161-2.165 do not produce 2.162/2.163; complex-radius substitution). - "Free power from spacetime" (strong form): Never properly tested (no neutral, controlled test on record; not dismissed without testing from what I read, but no data are offered), and Refuted by calculation only in the sense that no energy budget is supplied and a static curl-free field yields zero cycle work; I do not claim a device test was run. - Electrogravitic E = phi(0) g/c^2: Not testable as stated (phi(0) unspecified; Felker concedes "difficult to test"); for phi(0) of order 1 V the effect is about 1e-16 V/m at Earth's surface.

Claim 5: Heisenberg uncertainty principle refuted and replaced by (box + kT) q = 0; photon has B(3) field and mass; electroweak masses without Higgs

a. Mathematics consistent? No, in the pieces I read. - Lemma: [read] Felker Ch.7 and Ch.10: box q^a_mu = R q^a_mu, "a geometrical identity based on the tetrad postulate", and "an eigenequation". As a statement about a function q, box q = R q is a definition of R(x) = (box q)/q when q != 0; it constrains nothing until R is separately fixed. Combined with R = -kT (the trace of Einstein's equation) it gives (box + kT) q = 0, an equation whose effective "mass squared" is kT, a field-dependent quantity. To get the Klein-Gordon equation (box + (mc/hbar)^2) psi = 0 one must set R = (mc/hbar)^2 by hand ("Principle of Least Curvature", Felker Ch.14 p.257). That is relabeling m^2 as a curvature, not deriving it. Also the q^a_mu components are used as a wavefunction though a Lorentz-vector-valued one-form has no probability interpretation; Born's rule is replaced by "volume density" with no calculation shown. The Lemma's derivation was criticised [web: Rodrigues and Souza 2005, per Wikipedia]; I did not read their argument. - Uncertainty replacement: [read] Felker Eq (8): x^a ^ p^b -> hbar (wedge product of position and momentum "densities", defined by dividing by volumes V, V0). The standard relation is an inequality: sigma_x sigma_p >= hbar/2 for any square-integrable wavefunction, a theorem of Fourier analysis (Kennard 1927; Robertson 1929 for general operators: sigma_A sigma_B >= (1/2)|<[A,B]>|) [memory] high. It follows from the Cauchy-Schwarz inequality and the Fourier-pair property of the wavefunction, with no interpretation. An equation x ^ p -> hbar with a wedge (antisymmetric: x^a ^ p^a = 0) is a different kind of statement, dimensionally dependent on the defined volumes, so it neither contradicts nor replaces an inequality. Quote [read]: Felker's footnote cites Evans: the principle "has been refuted experimentally by the Croca group in Lisbon". - Mass of photon and boson masses: no derivation read; Felker ch.11-12 not read in detail.

b. Experiments: The relevant tests: preparation uncertainty (Kennard) is confirmed in single-slit diffraction, electron and neutron interferometry, etc. [memory] high. Separately, the 2012 experiments (Rozema et al.) showed Heisenberg's original noise-disturbance heuristic could be violated while Ozawa's corrected relation held [web] https://arxiv.org/pdf/1208.0034 (search summary read, not the full paper). This is likely what the "refuted" statements refer to in general discussions, but it is a refinement of a measurement-disturbance statement, not a refutation of the theorem sigma_x sigma_p >= hbar/2; Evans's 2004-2005 text predates it, and he cites Croca (nonlinear quantum mechanics) [read], whose results I did not check. Photon mass < 1e-18 eV [web] https://pdg.lbl.gov/2025/listings/rpp2025-list-photon.pdf (via search result summary). Higgs boson 2012 and W/Z masses: [memory] high.

c. Tested? By Croca and co. per Evans; replication not found in sources. Not independently replicated per [web] Wikipedia summary.

d. If true: Fourier analysis itself would need to fail for quantum wavefunctions, or the wavefunction concept would have to change; the Standard Model Higgs mechanism would be redundant; the photon mass limit would be violated.

Verdict (split): - Heisenberg's uncertainty relation is false: Refuted by calculation (it is a mathematical theorem for the preparation form) plus Tested, not supported as experiments agree with it; the measurement-disturbance variant is a separate matter (Historically superseded but instructive: Heisenberg's heuristic error-disturbance form needs the Ozawa correction). - The Evans wave equation unifies GR and QM and gives Dirac/Klein-Gordon/Schroedinger: Internally inconsistent / Not testable as stated (Lemma is a definition of R; mass inserted by hand). - B(3) photon field and photon mass; electroweak masses without a Higgs: Not testable as stated on the text I read (no calculation shown); photon mass nonzero at significant size is Tested, not supported (bound 1e-18 eV).

Claim 6: Academic physics has ignored or abused these results ("belief retention")

a. Mathematics: not applicable; this is sociological. Note the internal logic: a rejection is read as evidence of belief retention, while a wide readership of AIAS web pages is read as evidence of acceptance; neither is a mathematical argument and the pattern makes the claim unfalsifiable as stated. b-d. No physics to compare. [web] 't Hooft editorial (via Wikipedia): ECE is not in standard archives or peer-reviewed journals, and published critiques exist (Rodrigues and Souza, Bruhn, Hehl and Obukhov), which means the work was criticised on substance, at least in part, not merely ignored. The "crude personal abuse" over Physica B rejection (paper 93) is Evans's own account [read] and I found no independent record.

Verdict: Not testable as stated (sociological claim; partially contradicted by existing published critiques).

Overall mathematical assessment

The core mathematical argument (Claim 1) fails on three grounds: antisymmetry is a non sequitur, the tested object is not the identity, and the contracted identity is equivalent to Ric = 0 when T = 0, so every matter solution "violates" it automatically. The Sec 2.6 resonance derivation has internal algebra errors and a hidden complex-radius step. Where the books make verifiable statements they either reduce to standard results (omega = 0) or are inconsistent; the testable-by-experiment content (free energy, uncertainty refutation) is unsupported by data in the folder.

Reproducibility: scratch scripts used SymPy (first Bianchi identity and R^{kappa mu}_{mu nu} for FLRW; dsolve and coefficient check for Eqs 2.156/2.161-2.165). Rendered page images in /tmp/ev (read-only source untouched).