Equivalence of the Einstein and Jordan frames

Abstract

No experiment can measure an absolute scale: every dimensionfull quantity has to be compared to some fixed unit scale in order to be measured, and thus only dimensionless quantities are really physical. The Einstein and Jordan frame are related by a conformal transformation of the metric, which amounts to rescaling all length scales. Since the absolute scale cannot be measured, both frames describe the same physics, and are equivalent. In this article we make this explicit by rewriting the action in terms of dimensionless variables, which are invariant under a conformal transformation. For definiteness, we concentrate on the action of Higgs inflation, but the results can easily be generalized. In addition, we show that the action for f(R)-gravity, which includes Starobinsky inflation, can be written in a frame-independent form.

arXiv commentpublished version

References in corpus (28)

• Planck 2013 results. XVI. Cosmological parameters
• The Standard Model Higgs boson as the inflaton
• Cosmological viability of f(R)-gravity as an ideal fluid and its compatibility with a matter dominated phase
• Standard Model Higgs boson mass from inflation: two loop analysis
• Running Inflation in the Standard Model
• Inflation scenario via the Standard Model Higgs boson and LHC
• Toward an Understanding of Foreground Emission in the BICEP2 Region
• Phantom scalar dark energy as modified gravity: understanding the origin of the Big Rip singularity
• Superconformal Symmetry, NMSSM, and Inflation
• Higgs inflation at the critical point
• Higgs inflation still alive
• BICEP2 II: Experiment and Three-Year Data Set
• Conformal Transformations with Multiple Scalar Fields
• Einstein and Jordan frames reconciled: a frame-invariant approach to scalar-tensor cosmology
• Inflation and non-minimal scalar-curvature coupling in gravity and supergravity
• Extended Slow-Roll Conditions and Rapid-Roll Conditions
• Quantum corrections in Higgs inflation: the real scalar case
• Conformal invariance of curvature perturbation
• The conformal transformation's controversy: what are we missing?
• Curvature perturbation in multi-field inflation with non-minimal coupling
• Gauge invariant cosmological perturbations for the nonminimally coupled inflaton field
• The Conformal Transformation in General Single Field Inflation with Non-Minimal Coupling
• Inflation, Symmetry, and B-Modes
• Frame independent cosmological perturbations
• Non-linear curvature perturbation in multi-field inflation models with non-minimal coupling
• The Gravitational Wave Background and Higgs False Vacuum Inflation
• Conformal Inflation Coupled to Matter
• Conformal transformations and the conformal equivalence principle

Cited by in corpus (106)

• Dark Matter and Dark Energy Interactions: Theoretical Challenges, Cosmological Implications and Observational Signatures
• Higgs inflation
• Question of quantum equivalence between Jordan frame and Einstein frame
• Dynamically Induced Planck Scale and Inflation
• Higgs inflation from Standard Model criticality
• Higgs inflation with loop corrections in the Palatini formulation
• Invariant quantities in the scalar-tensor theories of gravitation
• Frame-dependence of higher-order inflationary observables in scalar-tensor theories
• Planck scale black hole dark matter from Higgs inflation
• A frame independent classification of single field inflationary models
• Coleman-Weinberg linear inflation: metric vs. Palatini formulation
• Higgs inflation in the Palatini formulation with kinetic terms for the metric
• New universal attractor in nonmininally coupled gravity: Linear inflation
• Higgs inflation at the hilltop
• Slow-roll inflation in f(R,T) gravity and a modified Starobinsky-like inflationary model
• Scalar induced gravitational waves from primordial black hole Poisson fluctuations in $f(R)$ gravity
• Quantum equivalence of $f(R)$ gravity and scalar-tensor theories
• UV (in)sensitivity of Higgs inflation
• Scale-Invariant Quadratic Gravity and Inflation in the Palatini Formalism
• Gravitational radiation from compact binary systems in screened modified gravity
• Scalar-torsion theories of gravity III: analogue of scalar-tensor gravity and conformal invariants
• Frame Covariant Nonminimal Multifield Inflation
• Frame-Covariant Formulation of Inflation in Scalar-Curvature Theories
• Palatini frames in scalar-tensor theories of gravity
• Quantum corrections in Higgs inflation: the Standard Model case
• Dynamical Emergence of Scalaron in Higgs Inflation
• Exact Solutions in Chiral Cosmology
• Unimodular $f(R,T)$ Gravity
• Frame-invariant approach to higher-dimensional scalar-tensor gravity
• Frame Covariance in Quantum Gravity
• Frame (In)equivalence in Quantum Field Theory and Cosmology
• Invariant slow-roll parameters in scalar-tensor theories
• Meaning of the field dependence of the renormalization scale in Higgs inflation
• Post-Newtonian parameters and cosmological constant of screened modified gravity
• Critical point Higgs inflation in the Palatini formulation
• Tree-level unitarity in Higgs inflation in the metric and the Palatini formulation
• A minimal model of inflation and dark radiation
• Analytic Solution of the Starobinsky Model for Inflation
• Reconstruction of extended inflationary potentials for attractors
• Non-minimal (self-)running inflation: metric vs. Palatini formulation
• Inflationary predictions of double-well, Coleman-Weinberg, and hilltop potentials with non-minimal coupling
• Post-Newtonian limit of scalar-torsion theories of gravity as analogue to scalar-curvature theories
• Distinguishing Jordan and Einstein frames in gravity through entanglement
• Equivalence of cosmological observables in conformally related scalar tensor theories
• On the number of $e$-folds in the Jordan and Einstein frames
• Beyond the Poles in Attractor Models of Inflation
• Multiple point criticality principle and Coleman-Weinberg inflation
• Inflation from High-Scale Supersymmetry Breaking
• Crossing the phantom divide line as an effect of quantum transitions
• Hidden and explicit quantum scale invariance
• Minimal plateau inflationary cosmologies and constraints from reheating
• Linearized modified gravity theories with a cosmological term: advance of perihelion and deflection of light
• Brans-Dicke inflation in light of the Planck 2015 data
• On the Trans-Planckian Censorship Conjecture and the Generalized Non-Minimal Coupling
• Novel Aspects of the Extended First Law of Entanglement
• Cosmological Evolution of Two-Scalar fields Cosmology in the Jordan frame
• A Non-minimally Coupled Potential for Inflation and Dark Energy after Planck 2015: A Comprehensive Study
• Unified $f(R)$ gravity at local scales
• Post-Newtonian parameters $\gamma$ and $\beta$ of scalar-tensor gravity for a homogeneous gravitating sphere
• Cosmology in Minkowski space
• Do metric fluctuations affect the Higgs dynamics during inflation?
• Cosmology from pregeometry
• Integrability from Point Symmetries in a family of Cosmological Horndeski Lagrangians
• Higgs field in cosmology
• Constraints on non-minimal coupling from quantum cosmology
• Covariant singularities in Quantum Field Theory and Quantum Gravity
• Revisiting Isocurvature Bounds on the Minimal QCD Axion
• Comparing geometric and gravitational particle production in Jordan and Einstein frames
• Imprints of screened dark energy on nonlocal quantum correlations
• Dynamical probe of thermodynamical properties in three-dimensional hairy AdS black holes
• Primordial flat frame -- a new view on inflation
• Scalar-tensor black holes in an expanding universe
• Global Portraits of Nonminimal Inflation: Metric and Palatini
• Covariant formulation of refracted gravity
• Black branes in four-dimensional conformal equivalent theories
• Gravitational lensing in the strong field limit for Kar's metric
• On the Cosmological Frame Problem
• Conformal Standard Model and Inflation
• Euclid preparation. Simulations and nonlinearities beyond $\Lambda$CDM. 1. Numerical methods and validation
• Hyperbolic inflation in the Jordan frame
• Conformal invariant cosmological perturbations via the covariant approach
• Einstein and Jordan frame correspondence in quantum cosmology: Expansion-collapse duality
• Spontaneous scale symmetry breaking at high temperature
• Mimetic gravity: mimicking the dynamics of the primeval universe in the context of loop quantum cosmology
• Pseudo-Periodic Natural Higgs Inflation
• Non-Minimal M-flation
• Dynamical system analysis in multiscalar-torsion cosmology
• Vacuum $f(R)$ thick brane solution with a Gaussian warp function
• Crossing the Big Bang singularity
• Field transformations and invariant quantities in scalar-teleparallel theories of gravity
• Inflation constraints for classes of f(R) models
• Slow-roll inflation in $f\left(R,T,R_{ab}{T}^{ab}\right)$ gravity
• Top-Goldstone coupling spoils renormalization of Higgs inflation
• Fifth forces and frame invariance
• Gravitational relaxation of electroweak hierarchy problem
• Hubble Diagrams in the Jordan and Einstein Frames
• Bouncing and collapsing universes dual to late-time cosmological models
• Energy momentum tensor of a non-minimally coupled scalar from the equivalence of the Einstein and Jordan frames
• $\beta$-function reconstruction of Palatini inflationary attractors
• Disformal transformations as a change of units
• Aspects of Quantum Cosmology
• Cosmology in a Globally U(1) Symmetric Scalar-Tensor Gravity
• Conformal invariant cosmological perturbations via the covariant approach: multicomponent universe
• $\nu$-Inflaton Dark Matter
• Non-perturbative approach for scalar particle production in Higgs-$R^2$ inflation
• What can we learn from the conformal noninvariance of the Klein-Gordon equation?