Limitation of Applicability of Einstein's Energy-Momentum Relationship

Scientific Paper
Title	Limitation of Applicability of Einstein's Energy-Momentum Relationship
Read in full	Link to paper
Author(s)	Koshun Suto
Keywords	Special Theory of Relativity, Einstein?fs energy-momentum relationship, Klein-Gordon equation, Dirac equation.
Published	2005
Journal	General Science Journal
No. of pages	12

Read the full paper here

Abstract

When a particle moves through macroscopic space, for an isolated system, as its velocity increases, the kinetic energy and hence total energy of the particle will increase. However, according to classical quantum theory, when the momentum and kinetic energy of an electron inside a hydrogen atom increases, total energy decreases. From this truth, it is evident that the equation for Einstein's energy-momentum relationship does not hold true inside a hydrogen atom.

Overview

Koshun Suto's paper argues that Einstein's relation E² = c²p² + E₀² is a statement about isolated particles in free space and fails inside a bound system. His observation is that for a free particle, increasing momentum increases total energy, whereas for an electron falling to a lower level of a hydrogen atom the momentum and kinetic energy increase while the total energy decreases. He concludes that the sign of the momentum term must be reversed for a bound electron, and proposes

(E₀ + E_n)² + c²p_n² = E₀²  (n = 1, 2, ···, E_n < 0),

where E_n is the usual (negative) Bohr level and E₀ + E_n is what he calls the electron's total energy "defined in absolute terms," measured from the free electron at rest rather than from zero at infinity.

Having obtained this relation, Suto quantises it as Einstein's relation is quantised to give the Klein–Gordon equation, and follows Paul Dirac's route of factorising the second-order operator. He finds a set of 4×4 coefficient matrices differing from Dirac's, and offers them not as a refutation but as "another form of Dirac's equation." The paper is thus a claim about the domain of a relativistic identity rather than an attack on relativity as such: Suto explicitly says he does not disagree with quantum mechanics.

The argument

Why the standard derivation does not carry over

Suto begins from the textbook route to E² = c²p² + E₀² (he cites A. P. French's Special Relativity), whose key step is dE = v dp. That step relies on the work–energy theorem dK = F dx = (dp/dt) dx = v dp, together with the assumption that the total energy and the kinetic energy increase together, dE = dK.

Inside the atom, he argues, the second premise fails. If the potential energy of a hydrogen atom falls by ΔV(r), energy conservation gives −ΔV(r) = ΔK + ħω: half the released potential energy raises the electron's kinetic energy and half leaves the atom as a photon. Hence ΔK = −ΔV(r)/2 and ΔE = ΔV(r)/2, so

dE = −dK, and therefore −dE = v dp.

Appendix B supplies the classical backing from the circular Bohr orbit: mv²/r = e²/4πε₀r² gives mv²/2 = e²/8πε₀r = −V(r)/2, so E = K + V = −K = V/2.

Integrating the reversed relation

Combining p = mv with m = E/c² gives E = c²p/v. Multiplying this by −dE = v dp yields E dE = −c²p dp, which integrates to E² = −c²p² + const. Suto notes that the constant "should normally be determined through experimentation," but takes it, "from the analogy" with Einstein's relation, to be E₀²:

E² + c²p² = E₀².

He then argues that the E appearing here must be an absolute quantity including the rest energy. The conventional Bohr energy E_n = −(1/n²)(m_ee⁴/2(4πε₀ħ)²) is measured from zero at infinite separation and is negative; but an electron at rest at infinity "should have rest mass energy E₀." He therefore defines E_ab,n = E₀ + E_n and arrives at the paper's headline result, equation (4.4).

Quantisation and the coefficient matrices

Section 5 applies the substitutions E → iħ∂/∂t, p → −iħ∇. Applied to Einstein's relation these give the Klein–Gordon equation; applied to Suto's relation they give the same wave operator with the sign of the spatial derivatives reversed. Following Dirac, he writes a first-order equation with unknown coefficients α_i and β, squares the operator, and matches. The conditions he obtains are the familiar anticommutation relations α_iα_j + α_jα_i = 0, α_iβ + βα_i = 0, β² = 1 — but with α_i² = −1 in place of Dirac's α_i² = +1. He exhibits a 4×4 solution (his equation 5.8) differing from Dirac's standard set (5.9) by factors of i, and a four-component wave function, and declines to discuss the significance of the conditions further.

Appendix C

The final appendix takes up Gasiorowicz's relativistic scalar treatment of the bound electron, the operator version of (E − V)² = c²p² + E₀². Suto notes that if E is read as the conventional bound-state energy, then E − V = (K + V) − V = K, which would require K² > E₀² — an inequality that "should normally not be possible." Reading E instead as E₀ − K repairs it and returns (E₀ + K)² = c²p² + E₀², which he offers as "strong evidence to validate" his absolute definition of total energy.

Assessment

The paper is careful, modest in tone, and makes its assumptions visible — including the one it cannot justify, where the constant of integration is fixed "from the analogy" rather than from anything derived. The physical observation that opens it is correct and worth stating: for a Coulomb-bound electron the virial theorem gives E = −K, so tighter binding really does mean more momentum and less total energy, and a reader who imports the free-particle intuition will get the sign wrong. Appendix B's derivation of that fact is textbook-correct.

The arithmetic also works. Putting the ground state into the paper's own equation (4.4), with E₀ = 510999 eV and E₁ = −13.606 eV, gives cp₁ = √(E₀² − (E₀+E₁)²) = 3729 eV, against the Bohr value E₀α = m_ec²/137.036 = 3728 eV — agreement to about one part in 10⁴. The relation therefore does reproduce the Bohr momentum.

But that agreement is not evidence for the equation, because it is an identity. Expanding (E₀+E_n)² + c²p_n² = E₀² gives c²p_n² = −2E₀E_n − E_n², i.e. p_n²/2m_e = |E_n| − E_n²/2E₀. To leading order this is exactly the virial statement K = −E_n that Appendix B started from, dressed in relativistic notation. The whole content of the new equation, apart from a term of order (E_n/E₀)², is the non-relativistic p² = 2m'K — put in at the start and recovered at the end.

Worse, the paper gives that second-order term in two mutually contradictory forms. Equation (4.4) yields c²p² = 2E₀K − K². Appendix C's equation, offered as confirmation of the same scheme, is (E₀ + K)² = c²p² + E₀², which yields c²p² = 2E₀K + K². The two differ in the sign of the only term that distinguishes the proposal from ordinary Bohr theory, and the appendix presented as corroboration in fact reproduces Einstein's relation unchanged, with the standard total energy E₀ + K. The internal case for the reversed sign therefore does not close.

The derivation has a further equivocation. dK = v dp is the work–energy theorem along a mechanical trajectory; dE = −dK is a relation between two different stationary states connected by the emission of a photon. Treating the latter as a differential along a continuous path in (E, p) space, and integrating it, silently converts a discrete radiative cascade into a smooth mechanical process. Nothing in the paper justifies that step, and it is the step that produces the reversed sign.

Set against measurement, the proposal is under-determined rather than wrong: equation (4.4) contains only the principal quantum number n, so it assigns one energy to each shell and predicts no fine structure at all. The Dirac Equation with a Coulomb potential, by contrast, gives the n,j dependence that matches the observed 2P_3/2–2P_1/2 splitting of 10 969 MHz in hydrogen, and the residual 1057 MHz Lamb shift between 2S_1/2 and 2P_1/2 is the classic confirmation of Quantum Electrodynamics. A relation with no angular-momentum label cannot address either. The framing question is also arguably a category error: Einstein's relation connects a free particle's energy and momentum, and the standard treatment of a bound electron does not apply it to the bound-state energy but embeds the potential in the wave equation, precisely as Appendix C's Gasiorowicz form does.

Finally, the quantised version carries a cost the paper does not weigh. Requiring α_i² = −1 means the α_i cannot be Hermitian, so the resulting Hamiltonian is not Hermitian; energies need not be real and probability need not be conserved. And E² + c²p² = E₀² has no real solutions for cp > E₀, imposing a hard ceiling p ≤ m_ec on any electron the equation can describe. Calling the result "another form of Dirac's equation" understates how much has changed. The paper is honest and readable, and its opening observation about the sign of dE inside an atom is sound; the construction built on it recovers a known identity and contradicts itself on the one point where it says something new.

See also

• Koshun Suto
• Hydrogen Atom
• Electron
• Dirac Equation
• Paul Dirac
• Niels Bohr
• Atom
• Mass
• Spin
• Fine Structure Constant
• Quantum mechanics
• Quantum Electrodynamics
• Relativity