The conformal transformation’s controversy: what are we missing?

Quirós, Israel; García-Salcedo, Ricardo; Madriz-Aguilar, Jose Edgar; Matos, Tonatiuh

doi:10.1007/s10714-012-1484-7

Cited by 65 publications

(85 citation statements)

References 81 publications

(338 reference statements)

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“…Mathematically this result is the well known transformation rule for Christoffel symbols under the conformal transformation [3,23], or the definition corresponding to Weyl-integrable geometry [13,25]. But here the point is simply thatΓ λ µν remains invariant under the transformations (2) and (3).…”

Section: B Equations Of Motion In Terms Of the Invariantsmentioning

confidence: 92%

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Invariant quantities in the scalar-tensor theories of gravitation

et al. 2015

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We consider the general scalar-tensor gravity without derivative couplings. By rescaling of the metric and reparametrization of the scalar field, the theory can be presented in different conformal frames and parametrizations. In this work we argue, that while due to the freedom to transform the metric and the scalar field, the scalar field itself does not carry a physical meaning (in a generic parametrization), there are functions of the scalar field and its derivatives which remain invariant under the transformations. We put forward a scheme how to construct these invariants, discuss how to formulate the theory in terms of the invariants, and show how the observables like parametrized post-Newtonian parameters and characteristics of the cosmological solutions can be neatly expressed in terms of the invariants. In particular, we describe the scalar field solutions in Friedmann-Lemaître-Robertson-Walker cosmology in Einstein and Jordan frames, and explain their correspondence despite the approximate equations turning out to be linear and non-linear in different frames.

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Section: B Equations Of Motion In Terms Of the Invariantsmentioning

confidence: 92%

“…From an alternative point of view, also letting the units to rescale inversely with the metric neutralizes the effect of conformal transformation [9,12], and the question of physical frame becomes superfluous. This can be interpreted by generalizing the underlying geometry from Riemann into Weyl-integrable [13].…”

Section: Introductionmentioning

confidence: 99%

Invariant quantities in the scalar-tensor theories of gravitation

et al. 2015

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“…Usage can be generally separated into two camps (see for example [6], who identify a number of works to that date with one camp or the other, and [8] which sets out clear definitions of "equivalence"): it can be taken to imply that the physics in both frames is identical, or it can be taken to imply that a system set up in the Jordan frame can be solved in the Einstein frame as long as it is transformed back to the Jordan frame for interpretation. The former case relies on us clearly stating what "physical equivalence" means; in [7] (and, similarly [59]) the authors take the reasonable definition that the observables should remain the same, so long as the correct length and time-scales are employed.…”

Section: The Equivalence Between the Framesmentioning

confidence: 99%

“…If instead we were motivated, as in coupled quintessence, by exotic couplings between a scalar field and the matter sector, then the Einstein frame would be the physical frame. Particularly clear discussions of this issue are found in [1,6] and for some time it seems generally agreed that the "equivalence" is mathematical in nature, but since it occasionally reappears in the literature (see for instance [7,8,[59][60][61][62][63][64][65][66][67][68] for some examples since 2004) we briefly re-address the question here.…”

Section: The Equivalence Between the Framesmentioning

confidence: 99%

“…In the Einstein frame, as in standard GR, the theory is second-order -a significant simplification. Aspects of the transformation have been controversial for some time (see for example [1,[6][7][8] and further references in §III). The discussion has centred upon the nature of the equivalence, and authors can be separated [1,6] into two camps: those who feel the equivalence is "physical" and observables can be calculated in either frame, and those who feel the equivalence is mathematical in nature and that observables should be calculated in a chosen "physical frame".…”

Section: Introductionmentioning

confidence: 99%

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Gauge issues in extended gravity andf(R) cosmology

Brown¹,

Hammami²

2012

J. Cosmol. Astropart. Phys.

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We consider issues related to the conformal mapping between the Einstein and Jordan frames in f (R) cosmology. We consider the impact of the conformal transformation on the gauge of a perturbed system and show that unless the system is written in a restricted set of gauges the mapping could produce an inconsistent result in the target frame. Newtonian gauge lies within the restricted group but synchronous gauge does not. If this is not treated carefully it could in principle contaminate numerical calculations. I. INTRODUCTIONExtended gravity theories, where the Einstein-Hilbert Lagrangian density L EH = R is replaced by a more general function including terms of higher-order in derivatives of the metric (R 2 , R µν R µν , R αβµν R αβµν . . .) and couplings to new dynamical degrees of freedom, have long been of interest in relativity. The last few decades have seen increasing applications of these models to cosmology. See [1,2] and their references for an overview and further details on these models and their applications to cosmology.1 In the last decade attention has focused on exploiting extended gravity to model dark energy without the need to introduce exotic particle species. A model frequently employed in this context is f (R) gravity (see for example [4,5] for recent reviews), where the Einstein-Hilbert Lagrangian density is replaced with an arbitrary function of the Ricci scalar, L = f (R), while the matter couples minimally to the metric and follows its geodesics in free motion. This representation of an extended gravity model is known as the "Jordan frame".The action can be transformed to a variety of different forms. One of the most common transformations is into the so-called "Einstein frame" where the action is manipulated to isolate a Ricci scalar of a new metric. The residual terms can be interpreted as an effective scalar field to which matter is non-minimally coupled and deflected from the geodesics of the metric. The field equations are otherwise those of standard general relativity. We review the model in the Jordan and Einstein frames in §II.Transforming from the Jordan frame to the Einstein frame is extremely useful in the study of f (R) gravity. We employ the metric formulation, in which the action is varied with respect to the metric alone, and in which the field equations are fourth-order. In the Einstein frame, as in standard GR, the theory is second-order -a significant simplification. Aspects of the transformation have been controversial for some time (see for example [1,[6][7][8] and further references in §III). The discussion has centred upon the nature of the equivalence, and authors can be separated [1,6] into two camps: those who feel the equivalence is "physical" and observables can be calculated in either frame, and those who feel the equivalence is mathematical in nature and that observables should be calculated in a chosen "physical frame". We briefly discuss this issue in §III.Modern cosmology is the study of Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes, perturbed to...

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Paving the Way for Transitions—A Case for Weyl Geometry

Scholz

2017

Towards a Theory of Spacetime Theories

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This paper presents three aspects by which the Weyl geometric generalization of Riemannian geometry, and of Einstein gravity, sheds light on actual questions of physics and its philosophical reflection. After introducing the theory's principles, it explains how Weyl geometric gravity relates to Jordan-Brans-Dicke theory. We then discuss the link between gravity and the electroweak sector of elementary particle physics, as it looks from the Weyl geometric perspective. Weyl's hypothesis of a preferred scale gauge, setting Weyl scalar curvature to a constant, gets new support from the interplay of the gravitational scalar field and the electroweak one (the Higgs field). This has surprising consequences for cosmological models. In particular it leads to a static (Weyl geometric) spacetime with "inbuilt" cosmological redshift. This may be used for putting central features of the present cosmological model into a wider perspective.

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The conformal transformation’s controversy: what are we missing?

Cited by 65 publications

References 81 publications

Invariant quantities in the scalar-tensor theories of gravitation

Invariant quantities in the scalar-tensor theories of gravitation

Gauge issues in extended gravity andf(R) cosmology

Paving the Way for Transitions—A Case for Weyl Geometry

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