# On a Set of Conform-Invariant Equations of the Gravitational Field

01 Jan 1953-Vol. 10, Iss: 01, pp 16-20

About: The article was published on 1953-01-01 and is currently open access. It has received 15 citations till now. The article focuses on the topics: Conformal field theory & Conformal anomaly.

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TL;DR: In this paper, the field equations following from a Lagrangian L(R) were deduced and solved for special cases, and it was shown that these equations are of fourth order in the metric.

Abstract: The field equations following from a Lagrangian L(R) will be deduced and solved for special cases. If L is a non-linear function of the curvature scalar, then these equations are of fourth order in the metric. In the introduction, we present the history of these equations beginning with the paper of H. Weyl from 1918, who first discussed them as alternative to Einstein's theory. In the third part, we give details about the cosmic no hair theorem, i.e. the details of how within fourth order gravity with L= R + R2, the inflationary phase of cosmic evolution turns out to be a transient attractor. Finally, the Bicknell theorem, i.e. the conformal relation from fourth order gravity to scalar-tensor theory, will be shortly presented.

200 citations

### Cites background from "On a Set of Conform-Invariant Equat..."

...…theories (i.e., the affinity is always presumed to be Levi–Civita) and want only to mention here that fourth order field equations following from a variational principle can be formulated in scalar-tensor theories, theories with independent affinity, and other theories alternative to GRT as well....

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TL;DR: In this article, the initial value problem of metric and Palatini f(R) gravity is studied by using the dynamical equivalence between these theories and Brans-Dicke gravity.

Abstract: The initial value problem of metric and Palatini f(R) gravity is studied by using the dynamical equivalence between these theories and Brans–Dicke gravity. The Cauchy problem is well formulated for metric f(R) gravity in the presence of matter and well posed in vacuo. For Palatini f(R) gravity, instead, the Cauchy problem is not well formulated.

78 citations

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TL;DR: In this paper, the authors derived the master equation describing the quasinormal radiation by using a relation between the Schwarzschild-anti de Sitter black holes and Weyl solutions, and also the conformal invariance property of the Weyl action.

Abstract: The recent reported gravitational wave detection motivates one to investigate the properties of different black hole models, especially their behavior under (axial) gravitational perturbation. Here, we study the quasinormal modes of black holes in Weyl gravity. We derive the master equation describing the quasinormal radiation by using a relation between the Schwarzschild-anti de Sitter black holes and Weyl solutions, and also the conformal invariance property of the Weyl action. It will be observed that the quasinormal mode spectra of the Weyl solutions deviate from those of the Schwarzschild black hole due to the presence of an additional linear r-term in the metric function. We also consider the evolution of the Maxwell field on the background spacetime and obtain the master equation of electromagnetic perturbations. Then, we use the WKB approximation and asymptotic iteration method to calculate the quasinormal frequencies. Finally, the time evolution of modes is studied through the time-domain integration of the master equation.

15 citations

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TL;DR: In this article, a lower bound on the event horizon radius of a nearly extreme conformal de Sitter black hole is derived for scalar perturbations, which corresponds to an upper bound of the frequency of the quasinormal frequencies.

Abstract: Stability criteria of nearly extreme black holes in the perturbations level is one of the interesting issues in gravitational systems. Considering the nearly extreme conformal--de Sitter black holes, in this paper, we obtain an exact relation for the quasinormal modes of scalar perturbations. As a stability criteria, we find a lower bound on the event horizon radius which is corresponding to an upper bound on the value of the quasinormal frequencies. In addition, we show that the asymptotic behavior of quasinormal modes gives highly damped modes, which is important due to the possible connection between their real part and the Barbero-Immirzi parameter. We also obtain the Lyapunov exponent and the angular velocity of unstable circular null geodesics. Finally, we examine the validity of the relation between calculated quasinormal modes and unstable circular null geodesics.

14 citations

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TL;DR: In this article, the authors examined the critical behavior of conformal gravity in an extended phase space in which the cosmological constant should be interpreted as a thermodynamic pressure and the corresponding conjugate quantity as thermodynamic volume.

Abstract: {We examine the critical behaviour i. e. $P-V$ criticality of conformal gravity~(CG) in an extended phase space in which the cosmological constant should be interpreted as a thermodynamic pressure and the corresponding conjugate quantity as a thermodynamic volume.} The main potential point of interest in CG is that there exists a {non-trivial} \emph{Rindler parameter ($a$)} in the {spacetime geometry. This geometric parameter has an important role to construct a model for gravity at large distances where the parameter "$a$" actually originates}. We also investigate the effect of the said parameter on the {black hole~(BH) \emph{thermodynamic} equation of state, critical constants, Reverse Isoperimetric Inequality,} {first law of thermodynamics, Hawking-Page phase transition and Gibbs free energy} for this BH. We speculate that due to the presence of the said parameter, there has been a deformation {in the shape} of {the} isotherms in the $P-V$ diagram in comparison with {the} charged-AdS~(anti de-Sitter) BH and {the} chargeless-AdS BH. Interestingly, we find {that} the \emph{critical ratio} for this BH is $\rho_{c} = \frac{P_{c} v_{c}}{T_{c}}= \frac{\sqrt{3}}{2}\left(3\sqrt{2}-2\sqrt{3}\right)$, which is greater than the charged AdS BH and Schwarzschild-AdS BH {i.e.} $\rho_{c}^{CG}:\rho_{c}^{Sch-AdS}:\rho_{c}^{RN-AdS} = 0.67:0.50:0.37$. The symbols are defined in the main work. Moreover, we observe that \emph{{the} critical ratio {has a constant value}} and {it is} independent of the {non-trivial} \emph{Rindler parameter ($a$)}. Finally, we derive {the} \emph{reduced equation of state} in terms of {the} \emph{reduced temperature}, {the} \emph{reduced volume} and {the} \emph{reduced pressure} respectively.

12 citations

### Cites background or methods from "On a Set of Conform-Invariant Equat..."

...Later Buchdahl (1953)18 considered a particular case µ = q = 0, δ = 1 by using Eq....

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...(18) where the area of the BH is given by Ai = 4πr(2) i ....

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##### References

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TL;DR: In this paper, it was shown that two invariants are inactive in the formation of field equations and thus may be omitted from the integrand of the action principle, i.e., I, = Ra, 6Ra# and 12 = R2.

Abstract: Introduction. If the geometry of nature is Riemannian and the field equations of this geometry are controlled by a scale-invariant action principle, there are four a priori possible and algebraically independent invariants which may enter in the integrand of the action principle. This abundance of invariants hampers the mathematical development and the logical appeal of the theory. The present paper shows that two of these invariants are inactive in the formation of field equations and thus may be omitted. Only the two invariants I, = Ra,6Ra# and 12 =R2

641 citations