Density-Metric Unimodular Gravity: Vacuum Spherical Symmetry

Scientific Paper
Title	Density-Metric Unimodular Gravity: Vacuum Spherical Symmetry
Read in full	Link to paper
Author(s)	Amir M Abbassi
Keywords	Unimodular gravity, Modified gravity.
Published	2008
Journal	ArXiv
Volume	0706
Number	.0451v3
No. of pages	16

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Abstract

We analyze an alternative theory of gravity characterized by metrics that are tensor density of rank (0, 2) and weight −1 2 . The metric compatibility condition is supposed to hold. The simplest expression for the action of gravitational field is used. Taking the metric and trace of connections as dynamical variables, the field equations in the absence of matter and other kinds of sources are derived. The solutions of these equations are obtained for the case of vacuum static spherical symmetric spacetime. The null geodesics and advance of perihelion of ellipses are discussed. We confirm a subclass of solutions are regular for r > 0 and there is no event horizon while it is singular at r = 0.

Overview

The paper is arXiv:0706.0451v3 [gr-qc], by Amir H. Abbassi of Tarbiat Modares University and Amir M Abbassi of the University of Tehran. It belongs to the unimodular tradition — the alternative theory Einstein considered in 1919, in which the determinant of the metric is not a dynamical variable and the cosmological constant emerges as a constant of integration. The authors' complaint about that tradition is that the unimodular constraint is imposed by hand: "The reason for this very feature of the theory is obscure… it is often not invoked by any decisive evidence or physical interpretation." Their aim is to make the constraint emerge rather than be assumed.

The device is a change in what kind of object the metric is. In special relativity every transformation has unit Jacobian, so any tensor may equally be regarded as a tensor density of any weight; the choice only becomes consequential when one passes to general coordinate transformations. The authors propose that the metric be taken as a symmetric tensor density of rank (0,2) and weight −½, so that its determinant is a scalar which may without loss of generality be normalised to one. The unimodular condition then follows from the weight assignment instead of being postulated. They state plainly that "this proposal for the role of the metric is novel in our approach and has no acquaintance in the literature."

The construction

Metric density, connection and the undetermined trace

Writing the density metric as g̃_μν, the three conditions are that it be a symmetric density of weight −½, that its determinant equal one, and that metric compatibility ∇_λg̃_μν = 0 hold. Compatibility now reads ∂_λg̃_μν − Γ^ρ_λμg̃_ρν − Γ^ρ_λνg̃_μρ + ½Γ^ρ_ρλg̃_μν = 0, the extra term being the density weight's contribution. Cyclic permutation and contraction give the connection in the usual Christoffel form plus a piece built from the trace Γ^ρ_ρλ — and crucially that trace is not fixed by the metric and its derivatives. It survives as an independent field. Unit determinant also forces g̃^μνδg̃_μν = 0, "which is a required condition for the unimodular relativity" — the constraint the authors wanted to derive rather than impose.

Riemann and Ricci tensors are defined in the ordinary way from this connection and are ordinary tensors, of weight zero. Contracting Ricci with the inverse metric, itself a density of weight +½, gives a curvature scalar density of weight +½.

Why the action must be quadratic

This is the structurally most interesting consequence. Because the volume element d⁴x is a density of weight −1 and the curvature scalar has weight +½, the Einstein–Hilbert combination is no longer a scalar; the determinant of the metric is unavailable to fix the weight, since it has been normalised to one. The simplest scalar action is therefore I = ∫κR̃²d⁴x — quadratic in the curvature scalar, not linear. Treating g̃_μν and Γ^ρ_ρλ as independent dynamical variables, in the manner of the Palatini formalism, gives two field equations. Variation with respect to the connection trace yields ∇_λR̃ = 0; variation with respect to the metric, with the unimodular condition imposed by a Lagrange multiplier, yields the traceless equation R̃_μν − ¼g̃_μνR̃ = 0. Both are checked as consistent with the Bianchi identity.

Spherical symmetry and gauge fixing

Starting from the usual ansatz ds² = B(r)dt² − A(r)dr² − r²dΩ² and forming g̃_μν = g_μν/|g̃|^1/4, the components acquire explicit sin^1/2θ factors. The resulting Ricci components depend on θ, which is inconsistent with spherical symmetry — unless the free trace components are chosen as Γ^ρ_ρt = Γ^ρ_ρφ = 0, Γ^ρ_ρθ = cot θ and Γ^ρ_ρr a function of r alone. With that choice the symmetry is manifest, and the tt, rr and θθ components of the field equation reduce to two independent relations, the φφ component adding nothing and ∇_λR̃ = 0 being satisfied automatically.

Section 4 supplies the justification for going further and setting Γ^ρ_ρr = 2/r. The authors count degrees of freedom in the Taylor expansion of a coordinate change, in the standard fashion: sixteen numbers in the first Jacobian suffice to bring the metric to canonical form with six left over for the Lorentz group; of the forty second-derivative parameters, thirty-six kill the first derivatives of the metric and the remaining four kill the four components of Γ^ρ_ρλ. Under a diffeomorphism the density metric transforms as δg̃_μν = ∇_μξ_ν + ∇_νξ_μ − ½g̃_μν∇_λξ^λ and the trace as δΓ^ρ_ρμ = ∂_μ∂_λξ^λ, leaving the Ricci tensor invariant; this residual gauge freedom is fixed by Γ^ρ_ρμ = 0, which in polar coordinates takes the form (0, 2/r, cot θ, 0). The spacetime is noted to have the same four Killing vectors as Schwarzschild.

The solution

With the gauge fixed, the two equations for A and B combine into (A′/A + B′/B)′ = (3/8)(A′/A + B′/B)² − (4/r)(A′/A + B′/B), which integrates directly. Setting y = AB and imposing A = B = 1 at infinity gives AB = (1 + C/8r³)^−8/3. Writing C′ = C/8, the general solution for B contains three integration constants, and requiring the Newtonian limit B → 1 − 2GM/r relates two of them; the remaining constant α has dimensions of inverse length squared and "may be called the cosmological constant" Λ. Setting Λ = 0 for simplicity,

B(r) = (1 + C′/r³)^−2/3[1 − (2GM/r)(1 + C′/r³)^−1/3],

with A(r) the corresponding inverse expression. C′ = 0 returns Schwarzschild exactly; with Λ ≠ 0 the solution is asymptotically de Sitter–Schwarzschild.

The central claim follows from positivity. A and B remain non-negative over the whole range of r provided r³ ≥ (2GM)³ − C′, which holds everywhere if C′ ≥ (2GM)³. Under that condition, ∂_t stays timelike everywhere and there is no event horizon. Radial null geodesics confirm it: substituting R = (r³ + C′)^1/3 reduces the integral to ±dt = RdR/(R − 2GM), whose integral "shows no sign of singularity" over the whole range of r when C′ > (2GM)³.

Perihelion bound and the nature of the origin

Section 6 uses the classical test to constrain C′. With the conserved energy and angular momentum from the two Killing vectors and ε = 1 for massive particles, the orbit equation to lowest order becomes d²x/dφ² − 1 + x = (3G²M²/L² − 3GMC′/L⁴ + 4E²C′GM/L⁴)x², giving a perihelion advance

Δφ = [6πGM/(1 − e²)a][1 + C′/3G²M²(1 − e²)a],

after dropping a third term smaller by a factor GM/a. Since this "shows a severe dependence on C′," agreement with observation requires roughly C′ < G²M²a.

The conclusions examine r = 0. Unlike Schwarzschild, where the curvature scalar and R_μνR^μν vanish everywhere in vacuum, here the asymptotic forms of A and B near the origin give a curvature scalar density that, converted to Cartesian coordinates, is constant and negative, with R̃_μνR̃^μν constant and positive. The Riemann-squared density is therefore computed, and behaves as sin θ/r⁶ for C′ > (2GM)³ and sin θ/r⁴ for C′ = (2GM)³; converted to Cartesian coordinates these go as 1/r⁸ and 1/r⁶. "This is enough to convince us that r = 0 represents an actual singularity." The authors add that Λ < 0 would be expected to produce an event horizon at cosmological distances of order |Λ|^−1/2, and that the version with matter remains to be investigated.

Assessment

This is a competently executed piece of technical work, and it is worth saying so plainly, because much of the alternative-gravity literature is not. The calculations are done rather than gestured at: the modified compatibility condition is solved, the connection components are listed in full, the Ricci components are given, the degree-of-freedom counting in the gauge-fixing section is carried out explicitly, and the field equations reduce correctly to Schwarzschild in the limit C′ → 0. The motivating observation is also a genuine one. In special relativity the distinction between a tensor and a tensor density is invisible because all Jacobians are unity, so the question of which object to promote when passing to general covariance really is a choice, and it really is one that is normally made without comment. Deriving the unimodular constraint g̃^μνδg̃_μν = 0 from the weight assignment, rather than imposing it, is exactly the improvement the introduction promises.

Two structural consequences deserve more attention than they get. The first is that the action must be quadratic in curvature. The authors present ∫κR̃²d⁴x as "the simplest expression," which it is given their weight bookkeeping, but a curvature-squared action is not a minor variation on Einstein–Hilbert: theories of that family generically carry extra propagating degrees of freedom and, when quantised, ghost states with negative norm. The paper does no perturbative analysis and does not discuss stability, so nothing is known here about whether the theory has a healthy spectrum. The second is that the connection trace Γ^ρ_ρλ is a genuinely new dynamical field, which in the spherically symmetric sector is disposed of by two separate arguments — first by demanding manifest symmetry of the Ricci tensor, then by a gauge choice — and never allowed to do anything. Whether Γ^ρ_ρλ is pure gauge in general, or carries physical content in less symmetric situations, is left open, and it determines whether the theory is genuinely equivalent to a metric theory.

The headline result should be read carefully. The absence of an event horizon is not a general feature of the theory; it holds only in the subclass C′ ≥ (2GM)³, and the authors' own perihelion analysis then bounds C′ from above by about G²M²a. For the Sun those two conditions do leave a wide window, since the gravitational radius is a few kilometres while Mercury's semi-major axis is of order 10¹¹ metres, so the result is not self-defeating. But it is a constraint that depends on the central mass and on the orbit used to test it, and C′ is introduced as an integration constant, not a universal parameter — the paper never says whether C′ is fixed once for all or varies from source to source. If it is universal, the solar bound must be reconciled with compact objects, where 2GM is comparable to the source size; if it is not, the theory has an undetermined free function of the source and correspondingly little predictive power.

The deeper difficulty is what replaces the horizon. Removing the horizon while retaining a curvature singularity at r = 0 — which the authors demonstrate rather than deny, since the Riemann-squared density diverges as 1/r⁸ in Cartesian coordinates — produces a naked singularity visible to distant observers. That is the configuration cosmic censorship was formulated to exclude, and it is a considerably more serious feature than the one it removes; the paper presents it as "a novel feature of this work" without discussing the consequences for predictability. It is also striking that in this theory the vacuum curvature scalar is not zero near the origin, unlike Schwarzschild, which means the exterior solution is not Ricci-flat in the usual sense and the interpretation of "vacuum" differs from the general-relativistic one.

Against measurement, the exposed claims are the strong-field ones, and the paper predates most of what now constrains them. The perihelion test it uses is weak-field and was chosen because it is easy; the modern constraints on horizon structure are the Event Horizon Telescope images of M87* and Sgr A*, whose shadow diameters match the Kerr prediction to within roughly ten per cent, and the ringdown phases of binary black hole mergers observed by LIGO and Virgo, which test the quasinormal mode spectrum of a horizon directly. A horizonless compact object of the kind this solution describes would generically produce late-time echoes in the ringdown, and searches for those have found none. The authors could not have addressed these in 2008, but any assessment now must note that the paper's central positive claim is the one that subsequent measurement has pressed hardest. Within its own scope, though — deriving the unimodular constraint from a weight assignment, obtaining the field equations, and solving the static spherically symmetric vacuum case exactly — the paper does what it sets out to do and is honest about what it has left undone.

See also

• Amir M Abbassi
• General Relativity
• Albert Einstein
• Cosmological Constant
• Black Hole
• Gravity
• Gravitational Lensing