Induced gravity
Induced gravity, proposed by Soviet physicist Andrei Sakharov in 1967, is a theoretical framework in quantum gravity positing that the dynamics of spacetime curvature in general relativity emerges as an effective low-energy phenomenon from quantum fluctuations of matter fields in a curved background, rather than being a fundamental interaction.[1] In this view, the "metrical elasticity" of space arises from the displacement of zero-point energy levels due to curvature, analogous to how macroscopic properties like viscosity emerge from microscopic particle interactions in hydrodynamics.[1] The core mathematical formulation derives the effective gravitational action at the one-loop level of quantum field theory on a Lorentzian manifold, yielding an expansion that includes a cosmological constant term, the Einstein-Hilbert action proportional to the Ricci scalar R (with Newton's constant G induced by integrating over matter field modes up to a high-energy cutoff near the Planck scale), and higher-derivative terms like R².[2] Sakharov identified the gravitational constant as arising from the spectrum of particle masses and a momentum cutoff k₀ ≈ 1 in natural units (G = ℏ = c = 1), linking it to the heaviest "maximon" particles and imposing fundamental limits on space and causality.[1] Nonlinear corrections, such as those from R² terms, become significant near singularities, as in Friedmann cosmological models, where logarithmic divergences scale with factors like ≈137 from quantum electrodynamics.[1] Since its inception, induced gravity has influenced modern quantum gravity approaches, including renormalization group methods where the effective action flows from ultraviolet fixed points, supersymmetric variants ensuring finiteness akin to Pauli's early ideas, and connections to emergent spacetime in string theory and holography.[2] Observational constraints tightly bound parameters, such as the induced cosmological constant satisfying |8πGΛ| ≲ 10⁻¹²⁰ Mₚ₄ (where Mₚ is the Planck mass) from cosmic microwave background data, and higher-derivative coefficients K ≲ 10⁺⁶⁴ from solar system tests.[2] The theory avoids direct quantization of gravity, instead treating it as a mean-field approximation, and continues to inspire research into unifying gravity with quantum mechanics without infinities.[2]Introduction
Definition and Core Concept
Induced gravity refers to a theoretical framework in which the effects of gravity, including spacetime curvature and its dynamics, arise as an emergent phenomenon from the quantum fluctuations of non-gravitational matter fields, rather than being a fundamental force. In this approach, the geometry of spacetime serves as a classical background, and the interactions among quantum fields—such as fermions and bosons—generate an effective action that induces gravitational behavior through a mean field approximation. This means that the collective, average effects of these microscopic quantum processes lead to macroscopic gravitational laws at low energies, without the need to quantize gravity itself.[3] Unlike traditional theories like general relativity, where gravity is postulated as a fundamental interaction mediated by the metric tensor, induced gravity treats it as a derived, low-energy effective description stemming from underlying quantum field theory dynamics. The Einstein-Hilbert action, which encodes the dynamics of spacetime curvature in general relativity, is not assumed a priori but instead emerges from one-loop quantum corrections to the matter sector. This perspective positions gravity as an induced effect, akin to how macroscopic properties in condensed matter physics arise from microscopic constituents.[3] A key analogy for induced gravity is the emergence of phonons in solids, where these quasiparticles represent collective vibrational modes arising from the interactions of individual atoms, rather than being fundamental entities. Similarly, gravitational effects in induced gravity manifest as collective excitations from quantum field fluctuations, providing a unified view within quantum field theory without invoking additional fundamental gravitational degrees of freedom. The concept was first proposed by Andrei Sakharov in 1967.[3]Significance in Theoretical Physics
Induced gravity provides a conceptual framework for reconciling quantum mechanics with general relativity by positing that gravitational effects emerge from underlying quantum field theories rather than being a fundamental primitive force. This approach treats spacetime curvature as a macroscopic manifestation of quantum fluctuations in matter fields, analogous to how thermodynamic properties arise from microscopic particle interactions. By deriving the Einstein-Hilbert action from quantum corrections, it offers a pathway toward a full quantum theory of gravity without introducing new fundamental entities beyond those of the Standard Model.[3] A key significance lies in its potential to address the hierarchy problem, which concerns the vast disparity between the Planck scale (~10^{19} GeV) and electroweak scales (~100 GeV) in particle physics. In induced gravity models, loop corrections from quantum fields at an ultraviolet cutoff scale naturally generate the gravitational coupling, linking the Planck mass directly to particle physics dynamics without requiring unnatural fine-tuning of parameters. This mechanism transmutes quadratic divergences in the Higgs mass into curvature couplings, thereby stabilizing the electroweak scale against quantum corrections that would otherwise push it toward the Planck scale.[4][3] The theory also offers insights into the profound weakness of gravity compared to the other fundamental forces, attributing the small value of Newton's constant G to its origin as a cutoff-dependent effective coupling in quantum field theory. Specifically, G emerges inversely proportional to the square of the ultraviolet cutoff (typically the Planck scale) multiplied by logarithmic loop factors from matter fields, explaining why gravitational interactions are suppressed by ~10^{40} relative to electromagnetic forces at low energies. This emergent perspective, first intuited by Sakharov in 1967, underscores gravity's secondary role in a quantum framework.[3] Since the early 2000s, induced gravity has seen renewed interest due to its alignments with holographic principles and quantum entanglement in modern quantum gravity research. Holographic dualities, such as those in AdS/CFT correspondence, resonate with the idea of gravity emerging from boundary quantum theories, while recent derivations link entanglement entropy across causal diamonds to gravitational actions, including quadratic curvature terms. Recent extensions include applications to inflationary models and renormalization group flows, as of 2025. These connections have revitalized the paradigm, positioning it as a bridge between quantum information and gravitational dynamics.[3][5][6][7]Historical Development
Sakharov's 1967 Proposal
In the mid-1960s, amid the vibrant theoretical physics community in the Soviet Union, particularly at the Lebedev Physical Institute in Moscow, Andrei Sakharov turned his attention to the interplay between quantum field theory and general relativity, building on emerging ideas about vacuum fluctuations in curved spacetime.[8] As a prominent physicist known for his contributions to nuclear physics and cosmology, Sakharov sought to address the unification of gravity with quantum mechanics by examining how quantum effects could give rise to gravitational phenomena.[3] His work was influenced by contemporary developments in quantum electrodynamics and the renormalization group, where vacuum polarization effects modify classical fields.[9] Sakharov's seminal proposal appeared in a concise three-page paper published in 1967, titled "Vacuum Quantum Fluctuations in Curved Space and the Theory of Gravitation," in Doklady Akademii Nauk SSSR, volume 177, pages 70-71 (English translation: Soviet Physics Doklady 12, 1040-1041, 1968).[8] The core idea posits that gravity is not a fundamental force but an induced effect arising from quantum loops of massless fields—such as photons and other elementary particles—in a background curved spacetime.[3] These fluctuations generate an effective metric description, where the geometry of spacetime emerges as a mean field approximation from the underlying quantum matter fields, analogous to how macroscopic elasticity arises from atomic interactions.[3] Central to Sakharov's argument is the role of vacuum energy contributions, which he emphasized as the source of the induced gravitational interaction; the zero-point energy of quantum fields in curved space leads to a modification of the spacetime metric, effectively producing the Einstein field equations.[8] This perspective highlighted how the gravitational constant itself could be determined by the spectrum of particle masses and couplings in the quantum vacuum.[3] Notably, this proposal predated the development of full-fledged quantum gravity frameworks, such as loop quantum gravity in the 1980s, by offering an early emergent view of gravity rooted in quantum field theory.[3]Post-Sakharov Developments up to the 1980s
Following Andrei Sakharov's 1967 proposal, Yakov Zeldovich extended the concept of induced gravity by exploring vacuum energy contributions from elementary particles, positing that the cosmological constant arises from zero-point fluctuations of quantum fields, thereby linking vacuum effects to gravitational phenomena. In the early 1970s, Zeldovich's collaboration with Alexei Starobinsky further advanced these ideas through analyses of vacuum polarization in curved spacetimes, demonstrating how quantum fields in anisotropic gravitational fields produce effective stress-energy contributions that mimic induced gravitational terms. A pivotal refinement came in Steven Weinberg's 1979 examination of ultraviolet divergences in quantum gravity theories, where he connected induced gravity to renormalization group flows, showing that the effective gravitational coupling evolves with energy scale due to quantum corrections from matter fields, providing a framework for asymptotic safety in gravity.[10] This work highlighted how induced terms could resolve infinities in perturbative quantum gravity. During the late 1970s supergravity boom, following the formulation of N=1 supergravity in 1976 and higher extensions by 1978, interest in induced gravity resurged as an alternative quantum approach, with researchers exploring its compatibility with supersymmetric matter sectors to address unification challenges. Specific developments included incorporating induced gravity into gauge theories, as in Anthony Zee's 1979 model where gravity emerges as a Goldstone mode from spontaneous symmetry breaking in a quantum field theory framework. Early attempts to include fermions appeared in Stephen Adler's 1980 calculation, deriving the induced gravitational constant from quantum loops involving Dirac fields in a symmetry-broken theory. By the 1980s, connections to Kaluza-Klein theories emerged, with D.J. Toms' 1983 analysis showing how quantum corrections in higher-dimensional spacetimes induce an effective four-dimensional Einstein-Hilbert action, addressing the mass hierarchy problem through compact extra dimensions.[11]Theoretical Foundations
Quantum Fluctuations and Mean Field Approximation
In induced gravity, quantum fluctuations arise from virtual particle-antiparticle pairs that permeate the vacuum of quantum fields in curved spacetime. These fluctuations, governed by quantum field theory, respond to the geometry of spacetime, generating contributions to the stress-energy tensor that mimic gravitational effects. Unlike classical vacuum energy, which is uniform, the presence of curvature disturbs these virtual processes, leading to a non-zero expectation value for the energy-momentum tensor even in the absence of real particles. This mechanism, first proposed by Sakharov, posits that the elasticity of spacetime emerges from the collective response of these quantum vacuum modes to metric perturbations. The mean field approximation provides a framework to interpret these quantum effects at macroscopic scales. In this approach, the rapid, fluctuating quantum fields are averaged over appropriate spacetime scales, yielding an effective classical background metric that incorporates the averaged influence of the fluctuations. This averaging process transforms the microscopic quantum corrections into a coherent gravitational field, where the metric acts as the mean field variable, and higher-order fluctuations are neglected. Such an approximation is particularly suited to semiclassical gravity, where matter fields are quantized but the gravitational sector remains classical, allowing the induced terms to dynamically couple curvature to the vacuum. A key aspect involves the conformal anomaly arising from massless quantum fields, such as photons or gravitons in certain approximations. In conformally invariant theories, the trace of the stress-energy tensor vanishes classically, but quantum effects introduce a trace anomaly proportional to curvature invariants like the square of the Weyl tensor or the Euler density. This anomaly breaks scale invariance at the quantum level, inducing terms in the effective action that couple to spacetime curvature and effectively generate gravitational dynamics from otherwise traceless fluctuations. The coefficients of these anomaly terms depend on the number and type of quantum fields involved, providing a natural link between particle physics content and gravitational strength. These vacuum fluctuations are inherently ultraviolet divergent due to contributions from arbitrarily high-energy modes, necessitating regularization techniques such as momentum cutoffs or dimensional regularization. Intriguingly, the natural scale for this cutoff aligns with the Planck length, where quantum gravity effects become dominant, thereby tying the induced gravitational constant to fundamental high-energy physics without introducing arbitrary parameters. This regularization preserves the renormalizability of the theory while ensuring that the induced effects remain finite and physically meaningful at low energies.Relation to Effective Field Theories
Induced gravity aligns closely with the effective field theory (EFT) paradigm in quantum field theory, where general relativity emerges as a low-energy description rather than a fundamental theory. In this framework, the Einstein-Hilbert action is viewed as an irrelevant operator generated by integrating out high-energy degrees of freedom from underlying matter fields, such as fermions and gauge bosons, at scales above the Planck mass.[3] This integration yields an effective action for the metric tensor on a fixed background manifold, capturing gravitational dynamics as an approximate, long-wavelength phenomenon without requiring gravity to be quantized at the outset. A distinctive feature of induced gravity within EFTs is the treatment of the gravitational coupling constant G, which becomes a running parameter dependent on the ultraviolet cutoff scale \Lambda. Unlike the fixed G in classical general relativity, this running arises from logarithmic corrections in the renormalization group flow, reflecting the scale at which quantum fluctuations of matter are integrated out, typically setting \Lambda \sim M_{\rm Pl}.[3] Such scale dependence underscores the emergent nature of gravity, where $1/G scales quadratically with \Lambda in leading approximations.[3] The induced gravitational terms specifically originate at the one-loop level in the EFT expansion, primarily from matter self-energy diagrams in curved spacetime. These contributions, computed via techniques like the heat kernel method, generate the Ricci scalar R term in the effective action, alongside higher-order curvature invariants and a cosmological constant, without any tree-level gravitational input.[3] This one-loop mechanism ensures that Einstein gravity appears dynamically from quantum matter effects, providing a natural ultraviolet completion probe within the EFT validity range below \Lambda. This EFT perspective sharply distinguishes induced gravity from fundamental general relativity, where the metric tensor serves as a primitive dynamical variable in the action from the outset. In contrast, induced gravity posits the metric as a secondary, collective degree of freedom derived from matter quantum fluctuations on a non-dynamical background, rendering gravity "elastic" in a metrical sense rather than intrinsically geometric.[3] This emergent viewpoint facilitates connections to broader quantum field theory principles while avoiding the non-renormalizability issues of treating gravity as fundamental at all scales.Mathematical Formulation
Derivation of the Induced Einstein-Hilbert Action
In induced gravity, the effective action for the gravitational field emerges from quantum corrections due to matter fields propagating in a fixed curved spacetime background. The starting point is the path integral formulation of quantum field theory, where the effective action S_{\text{eff}} for the metric g_{\mu\nu} is obtained by integrating out the matter fields \phi:S_{\text{eff}} = -i \hbar \ln \int \mathcal{D}\phi \, \exp\left( i S_{\text{matter}}[\phi, g]/\hbar \right),
with the background metric g treated as fixed.[2] This approach posits that spacetime curvature and its dynamics arise as a mean field approximation from these quantum fluctuations.[2] At one-loop order, which dominates in Sakharov's original conception under the assumption of no tree-level gravitational action, the effective action simplifies for quadratic matter Lagrangians. For a generic set of matter fields (scalars, spinors, vectors), the one-loop contribution is proportional to the functional determinant of the field operator in curved space:
S_{\text{ind}} = \frac{i}{2} \sum_f \eta_f \operatorname{Tr} \ln \left( -\nabla^2 + m_f^2 + \xi_f [R](/page/R) \right),
where the sum runs over field species f, \eta_f = +1 for bosons and \eta_f = -1 for fermions (accounting for statistics), [R](/page/R) is the Ricci scalar, and \xi_f is the curvature coupling.[2] This trace logarithm encodes the quantum corrections, with divergences regulated to extract finite gravitational terms. To handle ultraviolet divergences, standard methods such as the heat kernel expansion or zeta-function regularization are employed. In the heat kernel approach, the trace is expressed via the proper-time representation:
\operatorname{Tr} e^{-s (-\nabla^2 + m^2 + \xi R)} = \int d^4x \, \sqrt{-g} \, \frac{1}{(4\pi s)^2} \sum_{n=0}^\infty a_n(g) s^n,
as s \to 0^+, where a_n are the Seeley-DeWitt coefficients depending on the metric curvature invariants.[2] The integrated effective action then involves integrals over s, with UV divergences appearing as poles or logarithms after cutoff introduction (e.g., \kappa^{-2} for momentum scale \kappa). Alternatively, zeta-function regularization analytically continues the spectral zeta function \zeta(s) = \operatorname{Tr} (-\nabla^2 + m^2 + \xi R)^{-s} to s=0, yielding equivalent results for the determinant via \ln \det = -\zeta'(0). The structure of the induced action follows from these expansions, taking the form
S_{\text{ind}} = \frac{1}{2} \int d^4x \, \sqrt{-g} \left( c_0 + c_1 R + c_2 R_{\mu\nu} R^{\mu\nu} + c_3 R^2 + \cdots \right),
where the coefficients c_i depend on the matter content (field multiplicities, masses m_f, and couplings \xi_f) and the regularization scheme.[2] Specifically, the Einstein-Hilbert term \int \sqrt{-g} \, R arises from the a_1 heat kernel coefficient, which is linear in curvature: a_1 \sim (1/6 - \xi) R + \cdots, leading to a quadratic divergence \sim \kappa^2 R in the effective action, with subleading logarithmic terms \sim m^2 \ln(\kappa^2 / m^2) R depending on regularization. The higher-derivative terms like R^2 arise from logarithmic divergences associated with the a_2 coefficient. The coefficient c_1 is proportional to \sum_f \eta_f (1/6 - \xi_f) \left[ \kappa^2 - m_f^2 \ln(\kappa^2 / m_f^2) \right], ensuring the term is induced by quantum effects.[2]
Computation of the Induced Gravitational Constant
The computation of the induced gravitational constant G in induced gravity models involves extracting the coefficient of the Ricci scalar R from the one-loop effective action generated by quantum fluctuations of matter fields on a curved background. This coefficient determines the low-energy Einstein-Hilbert term, linking microscopic quantum field theory to the observable strength of gravity. The calculation requires regularizing the ultraviolet divergences in the functional determinant of the field operators, typically using a momentum cutoff Λ, with the result depending on the particle mass scale m and the spectrum of fields. Seminal calculations show that the net contribution is finite and positive for realistic field content when Λ is near the Planck scale, ensuring consistency with observed gravity.[12] The derivation starts with the one-loop effective action for the matter sector, given by \Gamma = \frac{1}{2} \sum_f \eta_f \Tr \log D_f , where the sum is over all matter fields f, η_f = +1 for bosons and -1 for fermions (accounting for the loop sign), and D_f is the covariant kinetic operator for field f in the metric g_μν (e.g., D = -∇² + m² + ξ R for a scalar with non-minimal coupling ξ). To obtain the induced gravity term, expand Γ to linear order in the Ricci scalar R (or more generally in curvature tensors, but the leading low-energy term is proportional to R). This expansion uses the heat kernel method or momentum space integration, yielding the effective Lagrangian term proportional to R after integrating out the high-momentum modes up to the cutoff Λ. Using a hard cutoff regularization in Euclidean space, the contribution to the coefficient of R arises from the trace over the Seeley-DeWitt coefficients in the heat kernel expansion of Tr e^{-s D_f}, integrated as ∫ ds/s Tr e^{-s D_f}. The linear-in-R term stems from the a_1 coefficient in the expansion, which is (1/6 - ξ) R for scalars (adjusted for spin), leading to an integral ∫ d^4k /(2π)^4 [field-specific factor] / (k² + m²). Evaluating this yields a quadratically divergent term ~ Λ² R plus subleading terms, but the physically relevant part in induced gravity is the quadratically divergent contribution ~ Λ² R, often with logarithmic enhancement log(Λ/m) to capture the scale separation between the UV cutoff and particle masses. The full expression for the inverse gravitational constant, normalized such that the effective action includes (1/(16π G)) ∫ R √g d^4x, involves field-dependent prefactors times Λ² log(Λ/m), with typical forms like G^{-1} \propto \frac{1}{16\pi^2} \sum_f \eta_f k_f \Lambda^2 \log\left(\frac{\Lambda}{m}\right), where k_f are spin-dependent factors.[12][3] The coefficient depends on the field content through the supertrace (str = ∑ η_f d_f Tr), where d_f is the number of degrees of freedom (e.g., 1 for real scalar, 4 for Dirac fermion). The spin-specific factors k_s (from the a_1 heat kernel coefficient, normalized relative to the scalar case) enter as str (k_s /6), yielding distinct contributions:| Field Type | Degrees of Freedom (d_f) | Sign (η_f) | k_s Factor | Contribution to Prefactor (per field) |
|---|---|---|---|---|
| Real scalar | 1 | +1 | 1 | +1/120 |
| Dirac fermion | 4 | -1 | -12/11 | -7/360 |
| Massive vector | 3 | +1 | 12/5 | +31/180 |