Spin-2 fields on The Broken-Born program

General relativity and the graviton

Mon, 01 Jan 0001 00:00:00 +0000

General relativity is Einstein’s theory of gravity. A single field, the spacetime metric g_μν, fixes distances and durations; its curvature is gravity; and a freely falling body follows the straightest path the metric allows, feeling no force. Small ripples of the metric about flat spacetime are the graviton, a massless spin-2 field whose gauge symmetry is the linearized form of a change of coordinates (a diffeomorphism), and a single such field that couples consistently to itself and to matter rebuilds general relativity. The program keeps general relativity as the long-range gravity of ordinary matter; its second, massive spin-2 field reaches ordinary matter only at gravitational strength and over a short range. It uses the graviton as the template for that second field, the momentum graviton X, whose local symmetry is a second copy of the same linearized diffeomorphisms, and its conjugate special relativity reads the equivalence principle as the relativity principle of the momentum polarization.

Planck units and the two Planck masses

Mon, 01 Jan 0001 00:00:00 +0000

Planck units are the units of length, time, mass and energy built from three constants of nature: Planck’s constant ℏ, the speed of light c and Newton’s constant G. The Planck length √(ℏG/c³) and the Planck energy √(ℏc⁵/G) mark the scale at which the quantum behaviour of gravity is expected to matter, and their ratio, c⁴/G, is a force that contains no ℏ. The Planck mass √(ℏc/G) also measures the strength of gravity: a spin-2 field such as the graviton couples to matter with a strength inversely proportional to it, so the Planck mass appears wherever the coupling of gravity or the expansion rate of the universe enters a formula. It is written in two conventions, M_Pl = G^{−1/2} in units with ℏ = c = 1 and the reduced M̄_Pl = M_Pl/√(8π), which describe the same physics provided that each formula keeps one of them throughout. The program’s conjugate special relativity sets its Born length equal to the Planck length, which makes its maximum force c⁴/G. The review that retired the program’s soliton dark-matter candidate found that the candidate’s reported factor of about 3 between the two conventions came from mixing them.

The equivalence principle: free fall and local inertial frames

Mon, 01 Jan 0001 00:00:00 +0000

The equivalence principle is the statement that gravity acts on every body in the same way. In its weak form, the universality of free fall, a body released at a given place with a given velocity follows the same path whatever it is made of. In Einstein’s form, a small enough freely falling laboratory is an inertial frame, one in which a body on which no force acts moves in a straight line at constant speed, and the laws of special relativity hold in it. Equivalently, a laboratory accelerating uniformly through empty space cannot be told apart from one held at rest in a uniform gravitational field. General relativity builds the principle in by coupling all matter and energy to one spacetime metric. The program postulates it for both of its kinds of matter: postulate P4 couples ordinary matter and conjugate matter χ to the same geometry, so χ falls, orbits and bends light like any other dark matter. Separately, the paper on conjugate special relativity (#13) reads the principle as the relativity principle of the momentum polarization, the program’s mirror description in which momentum plays the part of position. That reading is a claim of the program, alive but unreviewed, with its own page: the equivalence principle as dual relativity.

Yukawa forces and their range

Mon, 01 Jan 0001 00:00:00 +0000

A Yukawa force is the force carried by a field whose quanta have a mass M. Between two sources its potential falls as e^(−Mr)/r: at short distances it is the inverse-square force of a massless field, but beyond a distance of about ℏ/(Mc), the mediator’s reduced Compton wavelength, it dies away exponentially, so the mass of the mediator sets the range of the force. In momentum the same law is the factor 1/(q² + M²), and that is the form in which such a force enters scattering. The program uses it in several places. Its momentum graviton X is massive, so the force X exerts on ordinary matter is “Yukawa-cut at 1/M_X”, which the program gives as the reason experiments see no sign of it (safety by range). The standard self-interacting dark-matter models that the program’s scattering predictions are measured against scatter through a light Yukawa mediator, and the program’s own amplitude for conjugate matter keeps the Yukawa factor in the momentum transfer and multiplies it by a velocity gate. Conjugate special relativity reads X’s short range as a range in momentum, which the checkpoint it cites does not state. The physics is textbook. The same name also covers a different object, the Yukawa coupling that gives particles their masses, which the program’s toy models use as an analogy.

Ghosts and perturbative instabilities

Mon, 01 Jan 0001 00:00:00 +0000

A ghost is a mode of a field whose kinetic term, the part of its energy that comes from the field’s rate of change, has the wrong sign, so that its excitations carry negative energy or, in the quantum theory, negative norm. On its own a ghost oscillates like any other field. Coupled to ordinary fields it leaves the energy with no lower bound, and the vacuum can decay into ghosts and ordinary particles without violating energy conservation, so a theory in which a ghost interacts with ordinary matter is not considered physical. The other instabilities of small perturbations about a solution are different: a tachyon, a mode of negative mass squared, grows at long wavelengths, and a gradient instability, a negative squared propagation speed, grows fastest at short wavelengths. Under strong coupling, which is not an instability, a mode’s kinetic term is so small that linear perturbation theory cannot decide its fate. Whether a mode is a ghost is read from the signs of the kinetic terms that remain once constraints have removed the components that do not propagate. Ghost freedom decides what the program’s momentum graviton X can be: it excludes an X without its gauge symmetry, and the tuning test asks whether the potential that gives X its mass carries the Boulware–Deser ghost of generic massive gravity. It also fixes the kinetic term of the real structure in model I, the program’s coset toy model, and a ghost field is one route to the phantom side of the dark-energy divide, which the program’s dark-energy field, an ordinary scalar, cannot reach.

The Fierz–Pauli mass term and the Boulware–Deser ghost

Mon, 01 Jan 0001 00:00:00 +0000

The Fierz–Pauli mass term is the only mass term that gives a free spin-2 field a mass without a ghost, a mode whose energy has the wrong sign. In matrix notation it is proportional to Tr(h²) − (Tr h)², where h is the field; any other relative weight of the two terms lets an extra mode propagate, and that mode is a ghost. When the theory is made nonlinear, a generic completion brings the extra mode back, the Boulware–Deser ghost, and keeping it out takes the special de Rham–Gabadadze–Tolley (dRGT) and Hassan–Rosen potentials. A related standard fact, the signature obstruction, says that a ghost-free two-derivative kinetic term for a symmetric tensor field must have the gauge symmetry of the massless graviton; the Fierz–Pauli theory keeps that kinetic term and breaks the symmetry only through its mass term. The program uses all three facts to decide what its second field, the momentum graviton X, can be. Its tuning test checks X’s mass term against the Fierz–Pauli form, and the version built from metrics fails that check.

The signature obstruction

Mon, 01 Jan 0001 00:00:00 +0000

The signature obstruction is the standard result that a symmetric tensor field in spacetime, such as the perturbation h_μν of the metric, has no ghost-free kinetic term with two derivatives unless that kinetic term has the gauge symmetry of the massless graviton. A ghost is a mode whose energy has the wrong sign. The obstruction comes from the signature of spacetime, in which time enters with the opposite sign to space: in a contraction such as h_μν h^μν the components h₀ᵢ, with one time index and one space index, enter with the opposite sign to the others, and in the simplest kinetic term built from such contractions that sign becomes the sign of their kinetic energy. The gauge symmetry avoids the problem by keeping those components from propagating: the kinetic term of linearized general relativity contains no time derivatives of h₀₀ and h₀ᵢ, which are constrained rather than propagated. The Fierz–Pauli theory of a massive spin-2 field keeps that kinetic term and breaks the symmetry only through its mass term. The program uses the result to close one option for its second symmetric tensor, the momentum graviton X: without its gauge symmetry X would carry a ghost, and with the symmetry unbroken X would be a second massless graviton interacting with the metric, which the multi-graviton no-go theorem excludes, so X must have a mass. Calc note #6 tags the statement as standard literature without citing a source, and no script checks it.

The multi-graviton no-go theorem

Mon, 01 Jan 0001 00:00:00 +0000

The multi-graviton no-go theorem, proved by Boulanger, Damour, Gualtieri and Henneaux, states that several massless spin-2 fields cannot interact with one another consistently. Under its assumptions (local couplings, at most two derivatives, no ghosts) no coupling can mix them. Each massless graviton can be made consistent on its own, as general relativity is, but a theory in which two of them couple to each other cannot. Massive spin-2 fields fall outside the theorem, which is how bimetric gravity, with one massless and one massive spin-2 field, avoids it. In the program the theorem closes one route for the momentum graviton X: X has the gauge symmetry of a second massless graviton and must couple to the metric and to conjugate matter, so it cannot stay massless. Together with the signature obstruction, this leaves a mass as the only consistent option for X.

dRGT massive gravity

Mon, 01 Jan 0001 00:00:00 +0000

dRGT massive gravity, after de Rham, Gabadadze and Tolley, is a theory of a single massive graviton whose nonlinear potential is built so that the Boulware–Deser ghost never appears; it completes the Fierz–Pauli mass term to all orders. Its potential is a linear combination, with constant coefficients, of the elementary symmetric polynomials of the matrix square root √(g⁻¹f), where g is the dynamical metric and f a fixed reference metric. With its two-metric extension by Hassan and Rosen, it is the one known consistent structure for a massive spin-2 field coupled to gravity. The program adopted this class as the health test for its momentum graviton X: X’s sector is healthy if and only if the potential that gives X its mass lands in it. Built from metrics, X’s potential misses the class; built from frames, it lands in it.

Hassan–Rosen bimetric gravity

Mon, 01 Jan 0001 00:00:00 +0000

Hassan–Rosen bimetric gravity is a theory of two dynamical metrics, g and f, each with its own Einstein–Hilbert term, coupled through a potential that keeps the Boulware–Deser ghost out. The potential is a sum of five terms with constant coefficients β₀ to β₄, the n-th term built from the elementary symmetric polynomial eₙ of the matrix √(g⁻¹f). An identity of these polynomials makes the exchange of the two metrics reverse the order of the terms, so the list of coefficients is read backwards. About proportional backgrounds the theory has one massless and one massive spin-2 field. The program’s geometric sector, the home of its momentum graviton X, lands in this theory with the second frame confined to a compact space, and Born reciprocity, acting as the exchange of the two frames, then forces the list of coefficients to read the same in both directions, the palindrome βₙ = β₄₋ₙ.

Elementary symmetric polynomials

Mon, 01 Jan 0001 00:00:00 +0000

The elementary symmetric polynomials of a list of numbers are the sums of all products of distinct entries taken n at a time: e₁ is the sum, e₂ the sum of all pairwise products, and so on up to the product of all the entries, with e₀ = 1. Applied to the eigenvalues of the matrix √(g⁻¹f), built from the two metrics of bimetric gravity, they are the building blocks of the ghost-free bimetric potential, which is a linear combination of them with constant coefficients. Power sums beyond the first, such as p₄, the sum of fourth powers, can be written in terms of the eₙ by Newton’s identities, but only as nonlinear polynomials, so they lie outside that family. The program’s verification battery checks two facts of this algebra: the linear coefficients of the eₙ near the identity matrix, which give the tadpole of the program’s potential (test T6), and that p₄ is not a linear combination of the eₙ (test T7).

Vielbein formulation and wedge potentials

Mon, 01 Jan 0001 00:00:00 +0000

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.

Partially massless gravity

Mon, 01 Jan 0001 00:00:00 +0000

Partially massless gravity is a spin-2 field on de Sitter space, the spacetime of a universe expanding at a constant rate under a positive cosmological constant, whose mass takes the one value at which an extra gauge symmetry appears and removes the field’s helicity-0 polarization [T]. That value is the edge of the Higuchi bound, below which a massive spin-2 field on de Sitter space carries a ghost. Bimetric gravity has a parameter point, studied as partially massless bimetric gravity, at which the theory has de Sitter solutions in which the two metrics are proportional, with a ratio that the equations leave undetermined, and its potential is symmetric under exchange of the two metrics. The point matters to the program because at α = 1, which in the literature’s usual notation means equal Planck masses for the two metrics, it lies inside the palindromic branch that Born reciprocity selects. That raises the open question whether reciprocity is the origin of the partially massless symmetry, and it means the program’s vacuum-energy identity must be compared with this literature before it can be called new.

Bimetric cosmology and its instabilities

Mon, 01 Jan 0001 00:00:00 +0000

Bimetric cosmology studies expanding-universe solutions of Hassan–Rosen bimetric gravity, in which both metrics take the homogeneous and isotropic form of standard cosmology. Published analyses of its solution branches find gradient or ghost instabilities in the perturbations when the massive spin-2 field is about as light as today’s Hubble rate and the two metrics have comparable Planck masses, though not in every model. A heavy spin-2 field has been shown to be linearly stable at late times when the second metric’s Planck mass is much smaller than the first. Separately, a massive spin-2 field on de Sitter space must satisfy the Higuchi bound m² ≥ 2H², which a heavy field meets trivially. For the program the known stable regime is out of reach: Born reciprocity sets the two Planck masses equal, so the cosmological stability of a heavy momentum graviton X is an open question.

The cosmological-constant problem

Mon, 01 Jan 0001 00:00:00 +0000

The cosmological-constant problem is the mismatch between the energy density of empty space that quantum field theory predicts, from the zero-point fluctuations and loops of matter fields, and the far smaller value that the accelerating expansion of the universe indicates. No accepted mechanism explains the gap, and bimetric gravity is no better placed than general relativity to explain it. For the program this page marks a boundary. The program’s vacuum-energy identity sets to zero only the vacuum energy of the spin-2 interaction potential at the aligned point; it does not by itself address the vacuum energy of matter loops. The program assigns the observed dark energy to a separate field, τ.