Can the two assumptions of Deser's classical D = 4 selection argument — completeness of the second-order gravitational action and uniqueness of the stable vacuum — be derived from deeper principles rather than postulated?

Deser (2020) showed that two classical postulates about gravity in $D$ spacetime dimensions jointly force $D=4$: (A) the action must include, with nonzero coefficients, all local curvature invariants whose field equations are second order in derivatives of the metric — the Einstein term, the cosmological term, and the full Lovelock tower; and (B) acceptable theories must possess exactly one stable constant-curvature vacuum. Since the vacua of a complete Lovelock action in $D\geq 5$ are the roots of a polynomial that always has both $\Lambda=0$ and at least one $\Lambda\neq 0$ root, assumption B excludes all $D>4$; and $D=2,3$ are excluded by degenerate dynamics, leaving $D=4$ uniquely. Deser himself emphasizes that both postulates may be 'too Draconian' — lifting either permits any dimension — and that the argument leaves the spacetime signature undetermined, treats quantum corrections only as perturbative vertices, and does not integrate matter into the selection principle. The open problem is whether assumptions A and B have a deeper physical status: can they be derived from more fundamental principles (for example, effective-field-theory consistency, quantum-mechanical vacuum selection, or holography), or replaced by equally natural assumptions that still select $D=4$, or are they arbitrary choices with no principled foundation? Resolving the status of A and B is the open core.

Background

Einstein's general relativity made the metric a dynamical field, but the number of spacetime dimensions $D$ remained an external input. Nordström's 1914 five-dimensional theory, Kaluza–Klein compactification, and modern string theory ($D=10$ or $11$) all treat dimensionality as a model-building choice or as a dynamical variable whose observed value must be explained by compactification, anthropic selection, or some selection principle. The question 'why is observed spacetime $3+1$-dimensional?' has a long literature: anthropic arguments hold that only $3+1$ dimensions support stable planetary orbits, stable atoms, and non-degenerate wave propagation (Tegmark 1997); string-gas cosmology (an early-universe scenario in which a gas of string winding and momentum modes selects which dimensions expand) and other dynamical mechanisms were surveyed by Durrer, Kunz and Sakellariadou (2005); several quantum-gravity programs exhibit dimensional reduction at short distances rather than selection (Carlip 2017); and gravitational-wave observations bound the number of large extra dimensions observationally (Pardo, Fishbach, Holz and Spergel 2018, using GW170817, the 2017 binary-neutron-star merger observed in both gravitational waves and light). Deser (2020), following his 2004 survey 'The many dimensions of dimension', added a purely classical-gravity entry to this list, and it is the status of that argument's premises which is at issue here.

The technical background is the Lovelock classification. Lovelock's theorem (Lovelock 1971) states that in $D$ dimensions the only symmetric, divergence-free, second-rank tensors built from the metric and its first two derivatives are the metric, the cosmological term, the Einstein tensor, and — for $D>4$ — a finite tower of 'Lovelock' curvature polynomials. Equivalently, the most general generally covariant action whose Euler–Lagrange equations are second order in metric derivatives is the Einstein–Hilbert action plus a cosmological term plus these Lovelock terms. The quadratic Lovelock term is the Gauss–Bonnet invariant, a combination of curvature-squared terms that is topological (contributes no field equations) in $D=4$; higher terms first contribute in still higher $D$. Boulware and Deser (1985) showed that the sum of the Einstein and quadratic Lovelock terms in $D=5$ has two distinct constant-curvature vacuum solutions — flat space and an (anti-)de Sitter branch (a constant-curvature vacuum with nonzero effective cosmological constant) on which the two terms cancel — and Deser and Tekin (2003) showed both branches are stable, so stability cannot eliminate either. In general, with all Lovelock couplings $a_n\neq 0$, the constant-curvature vacua are the roots of the polynomial $\sum_{n=1}^{\lfloor(D-1)/2\rfloor}a_n\Lambda^{n}=0$, in which, by assumption A, the bare cosmological term is included as an $a_1\Lambda^{1}$ contribution because its contribution to the field equations has no derivatives; since the sum starts at $\Lambda^{1}$, $\Lambda=0$ (flat space) is always a root, and for $D\geq 5$ there is at least one further root. Related facts: 'critical gravity' theories — higher-curvature theories in $D>3$ whose couplings sit at values where the would-be massive spin-2 mode degenerates with the massless graviton — (Deser, Liu, Lü, Pope, Şişman and Tekin 2011) are excluded from the scheme for having multiple vacua; the spectra and unitarity of Lovelock vacua were analyzed by Şişman, Güllü and Tekin (2012); and topologically massive gravity (Deser, Jackiw and Templeton 1982), a $D=3$ theory with third-order but effectively second-order dynamics, is a borderline case excluded by the letter of the second-order requirement.

Within this framework, Deser's argument runs: (A) 'what is not forbidden is compulsory' — in any $D$, keep every term of the second-order-complete (Lovelock-plus-Einstein-plus-cosmological) action with nonzero coefficient; (B) accept only theories with a single stable constant-curvature vacuum. Then $D=2$ is excluded (the Einstein term is topological and produces no field equations), $D=3$ is excluded (only the Einstein term is allowed and it is flat-only, returning to a rigid background), and every $D\geq 5$ is excluded because a complete Lovelock action necessarily has at least two vacuum branches. Uniquely in $D=4$, the Lovelock tower collapses to the topological Gauss–Bonnet invariant, the only surviving $\Lambda$-terms are the bare cosmological term and the Einstein term's own constant, and a single vacuum requires either flat space or a tuning of the two $\Lambda^{1}$ coefficients to reproduce cosmological general relativity. Both assumptions are load-bearing: dropping (A) — for example keeping a single Lovelock term, as in the pure-Lovelock and 'Lovelock unique vacuum' theories studied by several groups — allows single-vacuum theories in any sufficiently high $D$; dropping (B) trivially allows all $D$. Deser himself defends both postulates — the second-order requirement is 'a sacred classical Newtonian one, that needs no justification', (A) is 'hardly arguable', and (B)'s 'hidden purpose ... is to get our desired dimension; however, it is a reasonable physical requirement on its own' — while conceding that the two conditions 'may perhaps be too Draconian'. Deser explicitly notes that the argument is purely classical, that higher-derivative quantum corrections are relegated to 'vertex' rather than fundamental status, that matter is only required to be '$D=4$-compatible' rather than integrated into the selection, and that the metric signature is 'not yet in our ambit'. The open question is whether the two premises are physically derivable — necessary consequences of deeper consistency conditions — or merely sufficient-condition postulates chosen because they yield the desired answer.

Problem Statement

The problem arising from the caveats Deser (2020) attached to his own argument is to determine the physical status of the two postulates that jointly force spacetime dimension $D=4$ in classical gravity:

(1) Assumption A ('what is not forbidden is compulsory'): the gravitational action in any dimension must include, with nonzero coefficients, all local curvature invariants whose field equations are second order in derivatives of the metric — the Einstein term, the cosmological term, and the full Lovelock tower.

(2) Assumption B: acceptable theories must possess exactly one stable constant-curvature vacuum.

The core question: can A and B be derived from deeper, independently motivated principles — candidates include consistency of gravity as an effective field theory (a framework treating gravity as the lowest-order term of a perturbative expansion in curvatures), quantum-mechanical vacuum selection, or holographic consistency — or shown to be necessary in some precise sense, rather than postulated because they produce the desired answer? Equivalently, a resolution may exhibit such a derivation or necessity argument; or exhibit replacement assumptions — motivated no less soundly than A and B, and not reverse-engineered from a presupposed dimension — that still single out $D=4$; or establish the negative result that A and B follow from no deeper principle, whether because they are underivable from first principles yet remain independently reasonable — the position Deser takes when he calls B 'a reasonable physical requirement on its own' — or because they are arbitrary, carrying no independent justification beyond their yielding the desired dimension. A resolution must settle the status of both A and B, since lifting either alone admits any $D$.

Deser attaches scope caveats to the argument, as parenthetical remarks rather than as open questions or completing steps of a program: the signature determination is 'not yet in our ambit'; the presence of normal, positive matter sources does not affect the reasoning 'except that of course they must be $D=4$-compatible'; and higher-derivative quantum corrections are 'to be treated only as vertex, rather than "propagator", contributions, in the sense of effective theory'. The paper's Discussion raises only the status of A and B — which 'may perhaps be too Draconian', though 'both are needed: lifting either allows for any $D$' — while the classical result itself is presented as a sufficient condition. These caveats delimit the reach of that sufficient condition; the open core is the principled status of the two assumptions. An answer that resolves the status of A and B — positively by a derivation, or by replacement with assumptions that are no less independently motivated and still single out $D=4$, or negatively by a principled counter-construction — settles the open core.

References

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