Vielbein formulation and wedge potentials

In the vielbein, or frame, formulation of gravity the basic variable is a frame field e: at each point of spacetime, a set of reference axes that are orthonormal by definition, from which the metric is built as a quadratic expression, g = eᵀηe. The program calls frames first-order variables and metrics, being quadratic in frames, second-order. Hinterbichler and Rosen showed that the ghost-free potentials coupling two spin-2 fields can be written as wedge products of the two frames: polynomials, with none of the matrix square roots that the same potentials need when written with metrics. The program’s tuning test passed only in this form: candidate mass potentials for the momentum graviton X built from metrics fail the ghost test, while the frame-built one lands in the ghost-free class. That the program’s potential takes the frame form was selected for minimality and consistency rather than derived, which is an open problem.

Status: Established — [T] — the Hinterbichler–Rosen construction is cited in the checkpoint as [12†], from model memory and not yet source-checked. The program’s use of it, the frame tier, is tagged “[D — structural, resting on T]” in #7 §3. No test of the verification battery covers the frame tier, and the class was selected rather than derived: Open: deriving the wedge sector from S_H (OP1).

Frames and metrics#

The frame field#

A frame, or vielbein, assigns to each point of spacetime d basis vectors, labelled by an index a, that are orthonormal with respect to a fixed constant matrix η_ab: the Minkowski metric in Lorentzian signature, the identity matrix in Euclidean signature [T]. This η_ab is the textbook symbol, unrelated to the program’s neutral metric η on phase space. In four dimensions the frame is also called a vierbein or tetrad; the German words mean “many legs” and “four legs”. Writing the frame components as a matrix e^a_μ, with a labelling the axes and μ the coordinates, the metric is

g_μν = e^a_μ η_ab e^b_ν          in matrix form:   g = eᵀ η e

The metric is quadratic in the frame. The frame carries more than the metric. Rotating the axes at each point, e → Λe with Λ a local Lorentz transformation (a local rotation in Euclidean signature), leaves g unchanged, so a frame is a metric together with a choice of axes, and that choice is a gauge freedom [T].

First-order and second-order, in the program’s usage#

Calc note #7 sorts the program’s geometric objects into tiers by this degree. The frame is first-order; the metric, quadratic in frames, is second-order: “J and E live on the same geometric tier (first-order structures on the tangent, like the Born connection), while H = EᵀE is second-order” (#7 §3). Here J is the polarization, E the frame of the phase-space metric H (see below), and the connection is the Born connection. The two halves of the tuning test follow the same split: “metric-tier (second-order, H-coupled)” potentials (#7 §2) against frame-tier ones.

In general relativity the phrase “first-order formalism” more often names actions in which the connection is varied independently of the frame or metric, as in the Palatini and Einstein–Cartan forms [T]. The program’s documents use “first-order” for the degree of a variable in the frame, not for the order of derivatives.

Massive and bimetric gravity in frame variables#

The square root in the metric form#

The ghost-free potentials of massive and bimetric gravity, in metric variables, are built from the matrix square root √(g⁻¹f) of the two metrics g and f: the potential is a linear combination, with constant coefficients βₙ, of the elementary symmetric polynomials eₙ of that matrix [T]. These are de Rham–Gabadadze–Tolley (dRGT) massive gravity, with f fixed, and Hassan–Rosen bimetric gravity, with f dynamical. The square root makes the metric form awkward: it is defined through the eigenvalues of g⁻¹f, and for Lorentzian metrics a real square root need not exist [T].

Frames remove the square root#

Give each metric a frame: e for g and, following calc note #8’s notation, f for the second metric (#8 §1 uses the letter f for both the second frame and its metric):

g_μν = e^a_μ η_ab e^b_ν           f_μν = f^a_μ η_ab f^b_ν

If the frames satisfy the symmetry condition that e^a_μ η_ab f^b_ν is symmetric in μ and ν, then g⁻¹f = (e⁻¹f)², with e⁻¹f the matrix product of the inverse first frame and the second frame [T]. The square root is then a product of frames:

√(g⁻¹f) = e⁻¹f          (when e^a_μ η_ab f^b_ν is symmetric in μν)

Wedge products#

With the frames written as one-forms, e^a = e^a_μ dx^μ, the potential can be written as a sum of wedge products. In four dimensions the n-th term contracts 4 − n copies of e and n copies of f with the totally antisymmetric symbol ε_abcd [T]:

ε_abcd e^a ∧ e^b ∧ e^c ∧ e^d,    ε_abcd e^a ∧ e^b ∧ e^c ∧ f^d,    …,    ε_abcd f^a ∧ f^b ∧ f^c ∧ f^d

Each wedge product equals a constant times det(e) eₙ(e⁻¹f) d⁴x. Under the symmetry condition, det(e) is ±√|g| and e⁻¹f = √(g⁻¹f), so the n-th wedge product is, up to a constant, the n-th term √|g| eₙ(√(g⁻¹f)) of the metric potential [T]. The wedge products are polynomials in the components of the two frames. Calc note #7: “In vielbein variables the ghost-free potentials are polynomial — wedge products ε(e ∧ … ∧ f ∧ …) of two frames (Hinterbichler–Rosen) [T] — with no square roots and no tuning to miss” (#7 §3).

Ghost-free by construction#

Hinterbichler and Rosen built interacting spin-2 theories from such wedge products of frames [12†]; for two frames they are the frame form of the dRGT and Hassan–Rosen potentials [T]. Every wedge term is admissible, so no coefficient has to be tuned to keep the Boulware–Deser ghost, a mode whose energy has the wrong sign, out of the theory (the Fierz–Pauli mass term and the Boulware–Deser ghost). Calc note #7 summarizes the class as “ghost-free by construction, Fierz–Pauli automatic, dRGT structure inherited rather than tuned” (#7 §3). The metric-language counterpart is calc note #6’s criterion: “Massive spin-2 coupled to gravity has exactly one known consistent structure: the Fierz–Pauli tuning at quadratic order completed by a dRGT/Hassan–Rosen-class potential [T: HR 2011]” (#6 §3).

The checkpoint’s entry for the work (#11, reference list):

  • [12]† K. Hinterbichler, R. A. Rosen, “Interacting spin-2 fields,” JHEP 1207, 047 (2012), arXiv:1203.5783.

Calc note #7 cites the construction as [T] and #8 names it, neither with a reference entry; #11 supplies the entry and marks it †, cited from model memory. Checking every † reference against its source is open problem 10 of #11 §14, the † literature battery (OP10).

The program’s frame tier#

The symplectic vielbein and the polarization frame#

The program’s phase-space metric H, a point of the Siegel domain Sp(2d, ℝ)/U(d), has a frame of its own: “the symplectic vielbein E, the coset representative with H = EᵀE” (#7 §3). It is the Euclidean form of g = eᵀηe, with the identity in place of η. The polarization J, the real structure (an involution, J² = 1) that the vacuum selects (the order parameter: J, η and K; the structures ω, η, H), supplies a second frame, the polarization frame E₀[J] (#11 §7.2). The documents name E₀[J] without writing out its form.

The tier argument#

Calc note #7 argues from the tiers to the coupling:

J and E live on the same geometric tier (first-order structures on the tangent, like the Born connection), while H = EᵀE is second-order. The geometrically minimal coupling of the polarization to the modular sector is therefore E-linear — wedge polynomials of (E, E₀[J]) — which is precisely the Hinterbichler–Rosen class: ghost-free by construction, Fierz–Pauli automatic, dRGT structure inherited rather than tuned. (#7 §3)

The checkpoint restates it: “Frame-tier couplings — wedge polynomials of the symplectic vielbein E (H = EᵀE) with the polarization frame E₀[J] — are the Hinterbichler–Rosen class [12†], ghost-free by construction; J, E, and the Born connection share the first-order tier, so this is the geometrically minimal formulation” (#11 §7.2). The argument classifies the objects by tier and then takes the minimal coupling at the frame tier. That the program’s potential is this minimal coupling is a choice, made for minimality and consistency (#8 §5).

The verdict of the tuning test#

Potentials built at the metric tier failed twice (the metric tier fails twice): at quadratic order they give the graviton a mass with no Fierz–Pauli trace term (battery test T4), and their nonlinear structure lies outside the linear-in-eₙ family (#7 §2; battery test T7 checks the power sum p₄). The frame tier passes. Calc note #7: “The rescue is not a tuning but a tier change” (#7 §3). The lab notebook states the verdict as “the theory passes iff formulated in first-order variables, which is the formulation the geometry prefers anyway.” (#1 ‘Addendum (v0.5)’), and the handoff repeats it (HANDOFF §2). See the verdict: a conditional pass.

What the pass delivers#

A pass did not yet fix the potential’s coefficients: “What a pass does not yet deliver: the βₙ coefficients” (#7 §3). Calc note #8 writes the frame-tier sector as a Hassan–Rosen potential for the two frames, “The Hinterbichler–Rosen sector for (e, f = E₀[J]) is V = 2m²M² Σₙ₌₀⁴ βₙ eₙ(𝕏), 𝕏 = √(g⁻¹f), with the exchange identity Sₙ(g,f) = S₄₋ₙ(f,g)” (#8 §1), and takes Born reciprocity to act on it as an exchange of the two frames, which forces the palindrome βₙ = β₄₋ₙ. Because J carries only compact moduli, the second frame cannot range over all metrics: the result is “Hassan–Rosen bimetric gravity with the second frame constrained to a compact moduli space Λ(d)” (#7 §4), with Λ(d) the Lagrangian Grassmannian; this is constrained bimetric gravity.

Status of the program’s use#

  • Tag. The frame tier is “[D — structural, resting on T]” (#7 §3). The checkpoint tags its §7.2 [D][V], but the machine checks there are the block algebra (battery T3) and the metric-tier failures (T4, T7); no test covers the frame tier, so the wiki tags it [D] (conflicts between sources; the verification battery).
  • Selected, not derived. Calc note #8 lists among Debt 1’s open residue “(i) first-principles reduction of the symplectic-vielbein wedge sector from S_H (the class was selected by minimality + consistency, not derived line-by-line)” (#8 §5). It is item 1 of the checkpoint’s open problems: “First-principles reduction of the frame-tier wedge sector from S_H (class selected, not derived) [O]” (#11 §14). Here S_H is the program’s geometric action (the geometric action S_H). The open problem has its own page: Open: deriving the wedge sector from S_H (OP1).

Standard references#

  • [12]† K. Hinterbichler, R. A. Rosen, “Interacting spin-2 fields,” JHEP 1207, 047 (2012), arXiv:1203.5783 — cited by #11.
  • C. W. Misner, K. S. Thorne, J. A. Wheeler, Gravitation (1973)†: orthonormal frames (tetrads) in general relativity. Added by the wiki from model memory.

Used by#

Sources#

  • papers/broken-born-checkpoint-v2.md §7.2, §14 item 1, ref [12] — the frame-tier couplings as the Hinterbichler–Rosen class; the open derivation; the citation, marked †.
  • notes/debt1a-iii-tuning-test.md §2, §3, §4 — the metric tier; wedge potentials, the symplectic vielbein, the tier argument and the verdict; the constrained bimetric picture.
  • notes/debt1a-iv-beta-coefficients.md §1, §5 — the wedge sector as a Hassan–Rosen potential; the class selected, not derived.
  • notes/debt1a-ii-X-classification.md §3 — the one known consistent structure for a massive spin-2 field.
  • notes/broken-born-theory.md ‘Addendum (v0.5)’ — the verdict as first recorded.
  • HANDOFF.md §2 — the verdict in the one-screen state.
Backward Elementary symmetric polynomials Partially massless gravity Forward