The Origin of Spin: A Consideration of Torque and Coriolis Forces in Einstein's Field Equations and Grand Unification Theory

Scientific Paper
Title	The Origin of Spin: A Consideration of Torque and Coriolis Forces in Einstein's Field Equations and Grand Unification Theory
Read in full	Link to paper
Author(s)	Nassim Haramein, Elizabeth A Rauscher
Keywords	Unification, Einstein, forces, spin, Equivalence Principle, torque, Coriolis force, torsion, Kerr-Newman metric
Published	2004
No. of pages	16
Pages	153-168

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Abstract

We address the nature of torque and the Coriolis forces as dynamic properties of the spacetime metric and the stress-energy tensor. The inclusion of torque and Coriolis effects in Einstein's field equations may lead to significant advancements in describing novae and supernovae structures, galactic formations, their center supermassive black holes, polar jets, accretion disks, spiral arms, galactic halo formations and advancements in unification theory as demonstrated in section five. We formulate these additional torque and Coriolis forces terms to amend Einstein's field equations and solve for a modified Kerr-Newman metric. Lorentz invariance conditions are reconciled by utilizing a modified metrical space, which is not the usual Minkowski space, but the U₄ space. This space is a consequence of the Coriolis force acting as a secondary effect generated from the torque terms. The equivalence principle is preserved using an unsymmetric affine connection. Further, the U₁ Weyl gauge is associated with the electromagnetic field, where the U₄ space is four copies of U₁. Thus, the form of metric generates the dual torus as two copies of U₁ x U₁, which we demonstrate through the S³ spherical space, is related to the SU₂ group and other Lie groups. Hence, the S⁴ octahedral group and the cuboctahedron group of the GUT (Grand Unification Theory) may be related to our U₄ space in which we formulate solutions to Einstein's field equations with the inclusion of torque and Coriolis forces.

Overview

This paper, delivered in the Noetic Press volume Beyond the Standard Model: Searching for Unity in Physics (2005), asks a question standard cosmology treats as settled: where does rotation come from? The authors' complaint is that "current standard theory assumes spin/rotation to be the result of an initial impulse generated in the Big Bang conserved over billions of years of evolution in a frictionless environment," whereas the observed universe is full of plasma viscosity and strong field interactions and is not frictionless at all. If rotation is not simply conserved, something must be driving it, and Haramein and Rauscher propose that the driver is torque built directly into the source term of the gravitational field equations.

The departure from the mainstream is therefore structural rather than observational. Standard General Relativity uses a symmetric stress-energy tensor and a symmetric (Christoffel) connection; torque and Coriolis terms are handled, when they appear at all, by choosing a non-rotating frame. The authors argue that this is a choice made for convenience which discards real physics: torque "is not included in Einstein's field equations in any manner," and they add it as an antisymmetric term coupled to the stress-energy tensor, with torsion appearing as the antisymmetric part of an unsymmetric affine connection. The resulting geometry they identify as U₄ rather than Minkowski space, and the resulting solution — an amended Kerr-Newman metric — they name the Haramein-Rauscher solution. They claim it can account for galactic structure, polar jets, accretion disks, spiral arms and haloes "without the need to resort to dark matter/dark energy constructs," and, in the final section, that the group theory of the resulting space connects gravitation to Grand Unification.

The argument

Torque, torsion and the Coriolis force

Section 2 rehearses standard rigid-body mechanics: angular momentum L = r × p, torque τ = dL/dt = r × F, and the conservation theorem that L is constant when τ vanishes. The Coriolis term, proportional to 2ω × v, is noted as the effect that rotates the plane of a Foucault pendulum, and is offered as the standing demonstration that such forces are physically measurable rather than artefacts of coordinates.

The authors then move to continuum mechanics, defining shear stress over shear strain as the shear modulus M, and taking over the engineering formula for the torsion of a circular shaft, in which the torque is proportional to the shear modulus, the fourth power of the radius and the twist angle, divided by the shaft length. Their key move is to relativise this expression: the shaft length is replaced by R, "the scalar curvature path in U₄ space over which torque acts," the shear modulus by a generalised modulus W for a fluid medium (citing Ellis 1971), and the result is treated as a tensor term Λ_μν to be added to the stress-energy tensor. Their slogan is that "torque acts as a force and torsion as a geometric deformation" — the first belonging to the source term, the second to the geometry.

Modifying the field equations

Section 3 argues that torque has been removed from general relativity by two conventions, both traced to Misner, Thorne and Wheeler's Gravitation: the assumption of a symmetric stress-energy tensor T_μν = T_νμ, and the practice of tying the observer's frame to a non-rotating gyroscope. Fermi-Walker transport is rejected as "the incorrect transport equation because it is formulated in a rotating frame that eliminates torque." The authors instead require dL/dt ≠ 0, so that L is not orthogonal to the four-velocity, and build their frame on a torus rather than a sphere — their topological argument being the hairy-ball theorem: a vector field on a sphere must have a discontinuity ("a part in the hairs"), whereas one curling around the short axis of a torus need not, so "all the vectors of the space obey invariance conditions" and absolute parallelism is maintained.

Torsion enters as the antisymmetric part of the affine connection. The authors note the component counting: the connection has 4³ = 64 components, of which 40 are symmetric and 24 belong to the torsion tensor (antisymmetric in the first two indices, giving six independent components times four for the third). The 24 will matter later. A variational principle δ∫(R + L)√(−g) d⁴x = 0 yields field equations in which a second-rank antisymmetric potential Λ_μν plays the role of a torsion potential, with gauge invariance Λ_μν → Λ_μν + ∂_[μξ_ν], and the total source becomes T*_μν = T_μν + KΛ_μν, K being a second coupling constant alongside κ = 8πG/c⁴. The general form given is

R_μν − ½g_μνR + Λg_μν = κT_μν + KΛ_μν,

with a non-zero cosmological term that the authors suggest "may yield correct approximations for the universal cosmological acceleration of distant objects." To make the units work they introduce a "fundamental force" F = c⁴/G in dynes, note that the Planck length can be written ℓ = (ħc/F)^1/2, and rewrite the torque contribution in terms of F so that it can be added to the geometric side, whose units are length squared.

The Coriolis and centrifugal terms are then obtained not from the source but from the coordinate transformation itself: writing a general rotation-plus-translation x^j = A^jkx_k + a^j, the cross terms in the transformed connection give the Coriolis force, the AᐟA terms the centrifugal force, and the second time derivative of the translation the inertial force. The authors stress that if these higher-order terms vanish, no Coriolis or centrifugal contribution survives — hence the necessity, on their account, of a frame that is allowed to rotate.

The Haramein-Rauscher solution

Section 4 starts from Schwarzschild, then Kerr-Newman with mass M, charge q and spin parameter a = s/M, and appends a torque term to the line element. Since a is built from angular momentum and torque is the time derivative of angular momentum, the authors argue that Kerr-Newman is the natural place to introduce it. The added term is proportional to Wr⁴θ/R, with a precession factor cos θ, expressed in Planck units. The authors note the standard black-hole condition M² ≥ q² + a² and are interested in the near-equality case, where "under imminent collapse, near r_s, centrifugal forces and/or electrostatic and plasma electromagnetic repulsion will be delayed, or halt and collapse, and become balanced." The Coriolis contributions are described as higher-order and smaller than the other terms "but still significant."

The claimed payoff is qualitative and broad: a spacetime that torques itself is said to correlate with "black holes, galactic topology, supernova formation, stellar plasma dynamics and planetary science such as ring formation and the Coriolis structure of atmospheric dynamics," and to explain the observed early formation of mature spiral galaxies reported in the Gemini Deep Deep Survey.

The group-theoretical unification

Section 5 is a chain of identifications. The 24 independent components of the torsion tensor are related to the 24-element octahedral group; the octahedron is dual to the cube under S₄; the 24-element group through S² yields the cuboctahedral group; and the cuboctahedron's cover group is said to generate the dual torus, U₁ × U₁ crossed with U₁ × U₁, which the authors call the Harameinian topology. U₄ is presented as four copies of U₁, hence a dual torus. Drawing heavily on the work of Saul-Paul Sirag, they write C[O] ≅ U₁ × U₁ × SU₂ × SU₃, assign U₁ to the photon and to the spin-two graviton, SU₂ to the weak interaction, U₁ × SU₂ to the electroweak force and SU₃ to the colour force, and connect the whole to SU(2,2/1), conformal supergravity, the Penrose twistor and Kaluza-Klein five-space. The conclusion drawn is that "the modification of Einstein's field equations with the inclusion of torque and Coriolis terms yields a group theoretical basis in the U₄ metrical space that forms a possible unification of the gravitational force with the strong, weak, and electromagnetic forces."

Assessment

The paper's opening question is a good one, and its central technical instinct is respectable. Torsion in gravitation is not a fringe idea: Cartan proposed it to Einstein in the correspondence the authors cite, and Einstein-Cartan theory is a well-developed extension of general relativity in which intrinsic spin sources an antisymmetric part of the connection, the equivalence principle survives, and the theory reduces to Einstein's in the absence of spin. Insisting that a rotating frame is a legitimate frame, and that the Foucault pendulum shows Coriolis effects are physical, is also fair. And the observation that fitting torque into a symmetric formalism requires an unsymmetric connection is correct as far as it goes.

The difficulties begin with the paper's characterisation of what it is departing from. The symmetry of the stress-energy tensor is not a convention adopted "so as to make the torque vanish"; in general relativity it is a consequence of local Lorentz invariance, and the Belinfante-Rosenfeld procedure shows precisely how a canonical tensor with an antisymmetric spin part is related to the symmetric one, with angular momentum conservation preserved rather than discarded. The paper does not engage this, and it does not compare its construction with Einstein-Cartan theory at all — which is the crucial omission, because Einstein-Cartan already tells us how strong the effect is. Torsion there couples with the same gravitational constant, so spin-spin contact effects become comparable to gravity only at densities of order 10⁵⁴ g/cm³ for nucleons. At galactic densities the effect is smaller than the Newtonian term by tens of orders of magnitude. A paper proposing that spacetime torque shapes spiral arms and drives polar jets owes a calculation showing why its coupling constant K escapes this bound, and none is given: K is introduced, never evaluated, and never constrained.

More generally, the paper's claims are qualitative throughout. It never computes a galactic rotation curve, never derives a jet velocity, never obtains a value for the cosmological constant, and never produces a number that could be compared with an observation. The assertion that the model dispenses with dark matter is therefore unsupported by any calculation, and it must contend with several independent measurements that the paper does not mention: flat rotation curves are only one line of evidence, and the others are harder for a modified-gravity approach to absorb — the offset between the lensing mass and the X-ray gas in the Bullet Cluster (1E 0657-56), the relative heights of the acoustic peaks in the CMB power spectrum, which fix the baryon and cold-dark-matter densities separately, and the light-element abundances from Big Bang nucleosynthesis, which cap the baryon density well below the total matter density. Similarly, the suggestion that the torque term may supply the cosmological acceleration is offered without a magnitude to compare against the Type Ia supernova distance-modulus data from which that acceleration was measured.

Several individual statements are simply wrong. Fermi-Walker transport is rejected on the ground that it "acts at the center of mass so that I, the moment of inertia, is zero" — the moment of inertia of an extended body about an axis through its centre of mass is not zero, and Fermi-Walker transport is a statement about transporting vectors along a worldline, which has nothing to say about moments of inertia. The claim that "SU₂ acts on a two dimensional real space" is incorrect; SU(2) acts on a two-dimensional complex space, as the paper itself says earlier when it defines the group as unit-determinant 2 × 2 complex matrices acting on C². The statement "SU₅ = SU₂ ⊗ SU₃" is false: in the Georgi-Glashow model SU(3) × SU(2) × U(1) is a subgroup of SU(5), and the two have different dimensions (24 against 11).

The group-theoretical section is the weakest part, and its central move is an equivocation on the symbol U₄. In the torsion literature — including the Hammond papers the authors cite — U₄ denotes a four-dimensional Riemann-Cartan manifold: a spacetime with metric and torsion. It is not the unitary group U(4), and it is not "four copies of U₁." U(1)⁴ is the maximal torus of U(4), a four-dimensional abelian subgroup of a sixteen-dimensional group, and the two are not interchangeable. Everything in section 5 rests on treating them as the same object. The chain that follows compounds the problem by trading on a numerical coincidence: the torsion tensor has 24 independent components and the octahedral group has 24 elements, but a count of tensor components is not a group order, and no homomorphism between them is exhibited — the paper says only that they "can be related." Likewise SU(5) is called "a 24 element group," conflating the dimension of a Lie algebra with the order of a finite group. Since the unification claim is the paper's headline result, and since it is assembled entirely from such identifications rather than from any dynamical calculation, the claim does not stand.

There is also a structural oddity worth noting for readers: the paper's own two halves are barely connected. Nothing in section 5's group theory uses the modified Kerr-Newman line element of section 4, and nothing in section 4 uses the octahedral symmetry of section 5. What links them is the number 24 and the word "torque." Taken narrowly, as an argument that rotation deserves a dynamical rather than an initial-conditions explanation and that torsion is where one might look for it, the paper raises a legitimate question. Taken as offered — as a modified field equation, a new black-hole solution, and a route to unification — it asserts far more than it derives.

See also

• Nassim Haramein
• Elizabeth A Rauscher
• General Relativity
• Black Hole
• Spin
• Equivalence Principle
• Dark Matter
• Dark Energy
• Cosmological Constant
• Gravity
• Vacuum
• Supernova