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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,

Tre−s(−∇2+m2+ξR)=∫d4x−g

(4πs)21∑n=0∞an(g)sn,
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
$S_{\text{ind}} = \frac{1}{2} \int d^4x \, \sqrt{-g} \left( c_0 + c_1 R + c_2 R_{\mu\nu} R^{\mu\nu} + c_3 R^2 + \cdots \right),$

(c0+c1R+c2RμνRμν+c3R2+⋯),
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 $\int \sqrt{-g} \, R$

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

These factors are derived from the trace over Dirac matrices and gauge indices in the expansion of log D_f, with the scalar as the baseline (k_s = 1 for minimal coupling ξ=0). For N_s real scalars, N_f Dirac fermions, and N_v massive vectors, the total prefactor is (N_s /120 - 7 N_f /360 + 31 N_v /180), multiplied by the loop integral (Λ² log(Λ/m))/(16 π²) to match the normalization after combining multiplicities. The negative fermion contribution partially cancels the positive bosonic one, but the net is positive for balanced spectra.^[12] For the Standard Model field content (with N_s = 4 from the Higgs doublet, N_f = 24 from 18 quark Dirac fermions (6 flavors × 3 colors) + 3 charged lepton Dirac + 3 Dirac neutrinos, N_v = 12 from gauge bosons: 8 gluons + 3 W/Z + 1 photon, adjusted for ghosts and mixing), the net coefficient yields a positive and finite G when Λ ≈ M_Pl ≈ 1.22 × 10^{19} GeV, matching the observed value 1/G ≈ 1.19 × 10^{27} m^{-2} (or M_Pl² ≈ 1.22 × 10^{38} GeV² in natural units ℏ = c = 1) within logarithmic factors of order 30–40 from log(Λ/m) with m ≈ 100 GeV. This finiteness arises because the cutoff is tied to the scale where gravity becomes strong, avoiding runaway divergences.^[12]^[3] The dependence on the number of fields N_f (total effective bosonic degrees, dominated by vectors and scalars in extensions) scales the prefactor linearly, such that for large N_f, G weakens (smaller G, stronger 1/G suppression relative to unity couplings in the UV theory). This reflects more quantum fluctuations contributing to the gravitational polarization, diluting the effective strength at low energies. Conversely, G is small because the high UV cutoff Λ ≈ M_Pl greatly suppresses the induced term compared to dimensionless couplings (O(1)) in the underlying quantum field theory; the large log(Λ/m) ≈ 40 enhances it just enough to match observations without fine-tuning.^[3]

Modern Interpretations and Extensions

Induced Gravity in String Theory and Extra Dimensions

In string theory, the concept of induced gravity manifests through the dynamics of D-branes, where four-dimensional Einstein-Hilbert gravity emerges on the brane worldvolume from quantum loops of open strings. These open strings, ending on the D-branes, generate effective gravitational interactions via one-loop diagrams in the low-energy effective action, leading to an induced Planck mass that scales with the string tension and the number of brane intersections. For instance, in type IIB orientifold models with coincident D3-branes, the gravitational coupling is induced by the backreaction of these open string modes, providing a natural mechanism for localizing gravity in higher-dimensional spacetimes. This framework echoes Sakharov's original idea by treating gravity as an emergent effect from underlying quantum string fluctuations.^[13] A related development arises in the AdS/CFT correspondence, where gravity in the anti-de Sitter bulk emerges holographically from a conformal field theory (CFT) on the boundary, inducing an effective gravitational theory at the asymptotic boundary. In this duality, the bulk Einstein-Hilbert action is reproduced by the large-N limit of the boundary CFT partition function, with quantum corrections from CFT operators generating curvature terms that mimic four-dimensional gravity. Specific calculations demonstrate that the induced gravitational constant at the boundary arises from the conformal anomaly of the CFT, linking the emergent gravity to holographic renormalization techniques. A modern perspective on these ideas, including their holographic implications, is provided by analyses connecting induced gravity to boundary dynamics in asymptotically AdS spacetimes.^[14] In extra-dimensional models, such as the Randall-Sundrum (RS) framework, four-dimensional gravity is localized on a brane embedded in a five-dimensional anti-de Sitter bulk, where the effective Einstein term is induced by quantum corrections from bulk fields. The warped geometry of the RS model exponentially suppresses Kaluza-Klein (KK) modes from the bulk, allowing the zero-mode graviton to dominate at low energies and induce a finite four-dimensional gravitational constant from the five-dimensional Planck scale. These KK modes, arising from the compactification or warping of the extra dimension, contribute higher-order curvature terms to the effective action, modifying short-distance gravitational interactions while preserving the long-range Newtonian potential. This induced localization resolves the hierarchy problem between the Planck and electroweak scales, with seminal work highlighting the role of brane-induced terms in stabilizing the effective four-dimensional theory.^[13]

Connections to Emergent Gravity Paradigms

Induced gravity shares conceptual foundations with modern emergent gravity paradigms, which posit that gravitational interactions arise from underlying quantum or informational structures rather than being fundamental. A prominent example is Erik Verlinde's entropic gravity proposal, introduced in 2010, where gravity emerges as an entropic force driven by gradients in the entanglement entropy of microscopic degrees of freedom.^[15] In this framework, the holographic principle underpins the emergence of spacetime, with gravitational attraction resulting from the statistical tendency to maximize entropy in a system of entangled quantum bits, analogous to thermodynamic forces in gases. Verlinde's later work in 2017 further elaborates this by deriving modified gravitational laws from entropy displacements in bipartite quantum states, yielding an effective force that mimics dark matter effects without invoking new particles.^[16] This entropic perspective connects directly to Sakharov's induced gravity through the role of quantum fluctuations, which can be reinterpreted as manifestations of entanglement in many-body quantum systems. In induced gravity, the Einstein-Hilbert action arises from vacuum fluctuations of matter fields, effectively averaging over microscopic quantum entanglements to produce macroscopic curvature. Recent analyses, such as those deriving Friedmann equations from entropic forces, highlight this similarity: spacetime emerges as a mean-field approximation from underlying microscopic degrees of freedom, mirroring how entanglement entropy gradients induce effective gravitational terms in quantum field theories. Condensed matter systems provide concrete analogs for these induced emergent effects, particularly in the 2013–2020s period. For instance, strained graphene sheets exhibit curvature-induced gauge fields and effective metrics that mimic gravitational phenomena, as described by tight-binding models with varying hopping parameters equivalent to a gauge theory with emergent gravity. In these setups, elastic deformations in the graphene lattice induce pseudogauge fields analogous to those in curved spacetime, allowing simulations of quantum gravitational effects from underlying electronic degrees of freedom—echoing Sakharov's vision of gravity as an induced response in defect-ridden crystals.^[17] These analogs underscore how microscopic lattice vibrations and entanglements can collectively produce long-range forces resembling induced gravity. Recent extensions include non-local formulations of induced gravity in cosmological models as of 2023.^[18] Overall, these paradigms frame gravity as a thermodynamic force, where the gravitational constant

G

emerges from the collective entropy of microscopic constituents, such as quantum fields or condensed matter excitations, rather than being a primordial parameter. This view aligns induced gravity with broader information-theoretic approaches, emphasizing entropy maximization as the driver of spacetime dynamics.

Applications and Implications

Role in Black Hole Thermodynamics

In induced gravity, black hole thermodynamics emerges from quantum fluctuations of matter fields near the event horizon, providing a microscopic basis for the Bekenstein-Hawking entropy formula

S = \frac{A}{4G}

, where

A

is the horizon area and

G

is the induced gravitational constant. This approach posits that gravity itself is not fundamental but arises as an effective low-energy description from integrating out high-energy quantum modes, leading to an Einstein-Hilbert action that governs black hole behavior. Quantum corrections in the vicinity of the horizon induce this effective general relativity, with the entropy arising from the entanglement of quantum fields across the horizon surface, effectively counting the degrees of freedom in these induced modes.^[19] A pivotal development in the 1990s came from work by Ted Jacobson, who demonstrated that conformal field fluctuations on the horizon generate the Bekenstein-Hawking entropy through the renormalization of the gravitational constant via quantum entanglement. In this framework, the entanglement entropy of quantum fields bifurcated by the horizon yields a universal contribution proportional to the area, with a natural ultraviolet cutoff at the Planck scale ensuring finiteness; this entropy matches

S = \frac{A}{4G_{\text{ren}}}

, where

G_{\text{ren}}

incorporates fluctuations from massless or conformal fields. The mechanism relies on the effective action derived from the trace anomaly or one-loop determinants, linking the horizon's geometry directly to thermodynamic properties without invoking ad hoc assumptions.^[19] Further elaboration by Frolov, Fursaev, and Zelnikov provided a statistical mechanical interpretation, showing that the black hole entropy originates from the microstates of ultraheavy quantum fields (with Planckian masses) that induce the gravitational interaction. By employing the conical singularity method to compute the off-shell effective action, they derived the entropy as the difference between the statistical entropy of these heavy constituents and a divergent term regularized by the horizon, resulting in

S = \frac{A}{4G}

where

G

is determined by the masses of the inducing fields. This counting of induced modes on a thin layer near the horizon resolves the statistical origin of black hole entropy in a general relativistic context, independent of specific ultraviolet completions like strings or loop quantum gravity, and aligns with the second law of black hole mechanics.^[20] This induced gravity perspective thus offers a unified explanation for black hole temperature (proportional to surface gravity) and entropy, treating them as emergent from quantum field theory rather than fundamental attributes, and highlights the horizon as the locus where quantum effects manifest thermodynamically.^[19]^[20]

Influence on Cosmological Models

In induced gravity theories, the cosmological constant arises as an induced term from the zero-point energies of quantum fields, where vacuum fluctuations contribute to an effective Λ through loop corrections in the effective action. This perspective, building on Sakharov's original framework, posits that the Einstein-Hilbert action and the cosmological term emerge from integrating out matter fields, but the resulting Λ is plagued by the same fine-tuning issues as in standard quantum field theory, exacerbating the cosmological constant problem due to the enormous mismatch between predicted (Planck-scale) and observed values.^[3]^[21] A key historical insight came from Yakov Zeldovich in the late 1960s, who suggested that vacuum energy dominance could play a crucial role in the early universe dynamics, where zero-point contributions from all quantum fields lead to an effective cosmological constant of order m⁴ (with m the field mass), potentially driving rapid expansion before matter domination.^[22] In this context, induced gravity amplifies such effects, as the gravitational coupling itself emerges from these fluctuations, linking vacuum energy directly to spacetime curvature without a bare cosmological term.^[23] Induced gravity finds specific application in inflationary models, where non-minimally coupled scalar fields generate induced terms that drive the exponential expansion of the early universe. For instance, in models where the gravitational constant is tied to the scalar field's vacuum expectation value, the potential energy of the scalar provides the necessary slow-roll conditions, yielding predictions consistent with cosmic microwave background observations while avoiding fine-tuning in the inflaton sector.^[24]^[25] The dynamic nature of the gravitational constant G in induced gravity—arising from the evolving scalar field—offers a mechanism for late-time cosmic acceleration without invoking a separate dark energy component. As the scalar field rolls toward its minimum, variations in G can mimic an effective cosmological constant, leading to accelerated expansion that aligns with supernova and large-scale structure data, though this requires careful tuning of the scalar potential to match observations.^[26]^[27]

Challenges and Open Questions

Renormalization and Ultraviolet Completion

In the framework of induced gravity, the Einstein-Hilbert action emerges as a one-loop effective term from the quantum fluctuations of massless matter fields propagating in a fixed gravitational background, and this contribution can be rendered ultraviolet finite by imposing supertrace vanishing conditions on the spectrum of matter fields, such as str(I) = 0 and str(m²) = 0, ensuring divergences in the effective action are canceled without introducing unphysical structures.^[3] This finiteness arises because the divergences in the effective action can be absorbed into the bare gravitational couplings without introducing new structures beyond the expected local terms.^[3] At higher loop orders, however, perturbative expansions reveal divergences that necessitate an infinite number of counterterms, establishing induced gravity as a non-renormalizable effective field theory valid only below some ultraviolet cutoff scale related to the Planck mass. These counterterms grow in complexity with each loop level, reflecting the power-counting non-renormalizability inherent to general relativity coupled to matter, where the dimensionful gravitational coupling leads to increasingly severe divergences. A detailed analysis of these renormalization properties in induced gravity models, including the structure of loop corrections and the role of conformal invariance, is provided in the comprehensive review by Buchbinder, Odintsov, and Shapiro. A key challenge in this approach is the ultraviolet cutoff dependence of the induced gravitational constant

G

, which is determined by the coefficient of the Ricci scalar term in the effective action and scales inversely with the loop integral up to the cutoff momentum, implying that

G

cannot be unambiguously defined without specifying the high-energy completion of the theory.^[3] This sensitivity underscores the requirement for a ultraviolet-complete framework, such as asymptotic safety, where the renormalization group flow of the dimensionless coupling

g = G \mu^2

(with

\mu

the energy scale) is predicted to approach a Gaussian or non-Gaussian fixed point, rendering physical observables independent of the cutoff. While asymptotic safety draws an analogy to asymptotic freedom in quantum chromodynamics, where the beta function drives the coupling to zero in the ultraviolet, the one-loop beta function for gravity in the Einstein-Hilbert truncation exhibits the opposite sign, causing the coupling to grow at high energies and potentially leading to a Landau pole unless higher-derivative terms or fixed-point mechanisms intervene to stabilize the flow. This sign mismatch complicates the ultraviolet behavior in induced gravity models, as the emergent nature of the gravitational action inherits these perturbative issues, motivating non-perturbative renormalization group investigations to explore viable fixed points.

Prospects for Experimental or Observational Tests

In induced gravity frameworks, the effective gravitational constant

G

arises from quantum vacuum polarization effects of matter fields and exhibits running behavior with energy scale in the low-energy effective field theory regime. This scale dependence could manifest in high-energy processes, such as those probed by particle colliders or astrophysical events involving strong gravitational fields, where deviations from the constant-

G

predictions of general relativity might emerge. However, the magnitude of the running is suppressed at energies accessible to current experiments, rendering direct detection challenging without reaching near-Planckian scales.^[28] Observational bounds on

G

variation offer indirect constraints on such models. Cosmological datasets, including measurements from the cosmic microwave background and big bang nucleosynthesis, restrict relative changes in

G

from the epoch of recombination to the present day to less than 5%, consistent with induced gravity predictions only if the underlying scale

\xi

exceeds approximately three times the Hubble radius. Solar system tests further limit the present-day rate of change to

|\dot{G}/G| < 10^{-12}

per year, providing no evidence for significant running at low energies but leaving room for ultraviolet effects.^[28] As of 2025, no direct experimental or observational tests have uniquely validated or ruled out induced gravity, though indirect probes via potential violations of the equivalence principle in quantum regimes hold promise. Tabletop experiments with entangled massive particles, such as the proposed Bose-Marletto-Vedral setup, seek to detect gravity-induced entanglement through phase shifts in superposed states,

\delta\phi = Gm^2 t / \hbar d

. A null result would favor

\psi

-incomplete interpretations like Sakharov's induced gravity, where emergent curvature couples preferentially to non-quantum beables rather than full quantum superpositions, thereby testing the foundational assumptions of gravity's induction from quantum fields.^[29]^[30]

History

Induced gravity

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Notes

Notes

Days in Chronicle

Induced gravity

Induced gravity

Introduction

Definition and Core Concept

Significance in Theoretical Physics

Historical Development

Sakharov's 1967 Proposal

Post-Sakharov Developments up to the 1980s

Theoretical Foundations

Quantum Fluctuations and Mean Field Approximation

Relation to Effective Field Theories

Mathematical Formulation

Derivation of the Induced Einstein-Hilbert Action

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