Comptes Rendus Physique

José Lebreuilly

Iacopo Carusotto

Quantum simulation of zero-temperature quantum phases and incompressible states of light via non-Markovian reservoir engineering techniques

Alexis Deschamps

Franck Tancret

Imed-Eddine Benrabah

Frédéric De Geuser

Hugo P. Van Landeghem

Combinatorial approaches for the design of metallic alloys

Dossier Sommaire

Tullio Joseph Tanzi

Madhu Chandra

Tome 19 (2018) – N◦ 1–2 – janvier–février 
Cover illustration / Illustration de couverture : © Daniel de Fornel, Atelier Isatis, Dijon, France.
DOSSIER
Radio science for Humanity / Radiosciences au service de l’humanité
Journées scientifiques URSI-France 2017 – Campus SophiaTech, Sophia Antipolis, France,
1–3 February 2017 / 1er–3 février 2017
Coordinators / Coordinateurs : Tullio Joseph Tanzi, Madhu Chandra
• Foreword
Tullio Joseph Tanzi, Madhu Chandra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
• Conforming discretizations of boundary element solutions to the electroencephalography forward problem
Lyes Rahmouni, Simon B. Adrian, Kristof Cools, Francesco P. Andriulli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
• DEMETER observations of manmade waves that propagate in the ionosphere
Michel Parrot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
• Solar radio bursts as a tool for space weather forecasting
Karl-Ludwig Klein, Carolina Salas Matamoros, Pietro Zucca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
• The double Brewster angle effect
Laetitia Thirion-Lefevre, Régis Guinvarc’h . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
• From quantum physics to digital communication: Single sideband continuous phase modulation
Haïfa Farès, D. Christian Glattli, Yves Louët, Jacques Palicot, Christophe Moy, Preden Roulleau . . . . . . . . . . . . . . 54
• RFID: A key technology for Humanity
Yvan Duroc, Smail Tedjini . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
• Drone-borne GPR design: Propagation issues
Madhu Chandra, Tullio Joseph Tanzi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
Guide for authors: https://www.elsevier.com/journals/comptes-rendus-physique/1631-0705/guide-for-authors
Electronic Submission: http://ees.elsevier.com/crphysique/


Foreword

Karl-Ludwig Klein

Carolina Salas Matamoros

Pietro Zucca

Solar radio bursts as a tool for space weather forecasting

Claude Weisbuch

Erich Spitz

Aurélien David

Tome 19 (2018) – N◦ 3 – mars–avril 
Cover illustration: Left: time evolution of LED efficiency for different colors (from C. Weisbuch, this issue). 
Right: illustration of a generic object color space. The cylindrical coordinates are lightness (L), chroma (C) 
and hue (H). The dotted circles indicate colors of constant chroma and lightness, but varying hue. The space 
is bounded by the colors white at top and black at the bottom (maximum and minimum lightness, 
respectively). (From A. David and L. A. Whitehead, this issue.)
Illustration de couverture : À gauche : évolution temporelle de l’efficacité des LED pour différentes couleurs 
(d’après C. Weisbuch, ce numéro). À droite : illustration de l’espace des couleurs d’un objet générique. Les 
coordonnées cylindriques sont la clarté (L), la saturation (C) et la teinte (H). Les cercles en pointillés indiquent 
des couleurs de teinte et de saturation constantes, mais de clarté variable. L’espace est délimité par les couleurs 
blanc en haut et noir en bas (clarté maximale et minimale, respectivement). (D’après A. David et L. A. Whitehead, 
ce numéro.)
DOSSIER
LEDs: The new revolution in lighting / Les LED : la nouvelle révolution de l’éclairage
Coordinators / Coordinateurs : Claude Weisbuch, Erich Spitz, Aurélien David
• Foreword
Claude Weisbuch, Erich Spitz, Aurélien David . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
• Historical perspective on the physics of artificial lighting
Claude Weisbuch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
• Invention, development, and status of the blue light-emitting diode, the enabler of solid-state lighting
Daniel Feezell, Shuji Nakamura . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
• LED lighting efficacy: Status and directions
Paul Morgan Pattison, Monica Hansen, Jeffrey Y. Tsao . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
• LED advances accelerate universal access to electric lighting
Peter Alstone, Arne Jacobson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
• LED lighting, ultra-low-power lighting schemes for new lighting applications
Marc Fontoynont . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
• LED-based white light
Aurélien David, Lorne A. Whitehead . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Guide for authors: https://www.elsevier.com/journals/comptes-rendus-physique/1631-0705/guide-for-authors
Electronic Submission: http://ees.elsevier.com/crphysique/


Marc Barthelemy

Transitions in spatial networks

Louis M. Shekhtman

Michael M. Danziger

Dana Vaknin

Shlomo Havlin

Robustness of spatial networks and networks of networks

Seyed Hamid Reza Fazlollahi

C. R. Physique 19 (2018) 337–340Contents lists available at ScienceDirect
Comptes Rendus Physique
www.sciencedirect.com
Plausible black hole solutions in higher-derivative gravity
Possibles solutions à trous noirs en gravité étendue
Seyed Hamid Reza Fazlollahi
Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran, 19839, Iran
a r t i c l e i n f o a b s t r a c t
Article history:
Available online 12 October 2018
Keywords:
Lichnerowicz-type theorem
Ricci scalar
No-hair theorem
Non-Schwarzschild solution
Mots-clés :
Thèorème de type Lichnerowicz
Scalaire de Ricci
Théorème « sans cheveux »
Solution non schwarzschildienne
Here we obtain explicit black hole solutions in Extension Gravity models with high-order 
derivative terms, while the Lichnerowicz-type theorem simplifies our analysis by vanishing 
Ricci’s scalar curvature. We find out two explicit static, spherical solutions that satisfy the 
presented action: the first one is the same usual Schwarzschild solution and the other one 
is the new non-Schwarzschild solution. It means that Schwarzschild’s solution following 
the no-hair theorem can describe any black hole object on each gravity theory. Without 
considering the first law of thermodynamics for it, we show that the non-Schwarzschild 
solution is depending on its set of constants, and then we consider its entropy and other 
thermodynamic parameters for specific values of the constants.
© 2018 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
r é s u m é
Dans le cadre de modèles de gravité étendue comportant des dérivées d’ordre supérieur, 
nous obtenons des solutions explicites représentant des trous noirs. Nous trouvons 
deux solutions sphériques statiques qui vérifient les équations : la première est la 
solution de Schwarzschild bien connue, alors que l’autre est une solution nouvelle non 
schwarzschildienne. Ainsi, la solution de Schwarzschild, qui résulte du théorème « sans 
cheveux », peut décrire un trou noir dans l’une ou l’autre des théories de la gravité. Sans 
partir de la première loi de la thermodynamique, nous montrons que la solution non 
schwarzschildienne dépend de certaines constantes, et nous évaluons son entropie et les 
autres grandeurs thermodynamiques pour des valeurs spécifiques des constantes.
© 2018 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
The study of string and of some other effective gravities show that the usual Einstein–Hilbert action only describes a 
low-energy system, so one can assume Einstein’s gravity to be the effective low-energy theory that needs correction with 
some terms built from the different powers of the curvature tensor and its derivative to illustrate high-energy systems [1]. 
Adding these terms to the usual Einstein–Hilbert action allows us to dissolve the non-renormalizability of gravity theory, 
albeit at the price of introducing ghost models [2].
E-mail address: shr.fazlollahi@hotmail.com.https://doi.org/10.1016/j.crhy.2018.09.009
1631-0705/© 2018 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved.
338 S.H.R. Fazlollahi / C. R. Physique 19 (2018) 337–340Studying Einstein’s gravity with added curvature terms is worthwhile to shed light on some facts about the gravity 
field’s behavior in the high-energy system. Black holes, among the interesting objects in general relativity, are among the 
best candidates for this goal. Hereupon in this paper, we study black hole solutions with high-order curvature terms added 
to the usual Einstein–Hilbert action. We suppose that the general action is given by
I =
∫
d4x
√−g(αR − βCμνρσ Cμνρσ + γ R2) (1)
where α, β , and γ are constants and Cμνρσ is the Weyl tensor, which is a trace-free part of the Riemann tensor. We shall 
work in units where α = c = G = 1. The equations of motion following (1) are then
Rμν − 1
2
Rgμν − 4βBμν + 2γ R
(
Rμν − 1
4
Rgμν
)
+ 2γ (gμνR − ∇μ∇ν R) = 0 (2)
where Bμν = (∇ρ∇σ + 12 Rρσ )Cμνρσ represents the Bach tensor. The equation of motion (2) can describe massive spin-2 
with mass-squared m22 = 1/(2β), and massive spin-0 or massless spin-2 with mass-squared m20 = 1/(6γ ).
We will now attempt to find black hole solutions explicitly for the equations of motion (2) when, without loss of 
generality, a static, spherically symmetric metric is given by
ds2 = − f (r)dt2 + dr
2
f (r)
+ r2(dθ2 + sin2 θ dϕ2) (3)
With tedious calculation, field equation for non-vanishing components, with γ set to zero for the reasons discussed in [3], 
becomes
Gμν + 2β
(
5
3
gμνR − RννCμνμν
)
= 0 (4)
where Gμν is usual Einstein tensor. Obviously, for β = 0, one gets the usual Schwarzschild solution for static, spherically 
symmetric ansatz (3). Therefore, when β = 0, one has:
I. G00 + 2β
(
−5
3
f (r)R − RiiC0i0i
)
= 0, i = 1,2,3
II. G11 + 2β
(
5
3
R
f (r)
− R11C1i1i
)
= 0, i = 2,3
III. G22 + 2β
(
5
3
r2R − R33C2323
)
= 0
and
IV. G33 = G22 sin2 θ = 0 (5)
For θ = 0, substituting (IV) into the (III) gives
R = 3
5
1
r2
R33C2323 (6)
so that the explicit forms of I and II are
G00 − 2β
[
R11C0101 + R22C0202 + R33
(
f (r)
r2
C2323 + C0303
)]
= 0
G11 − 2β
[
R22C1212 + R22C0202 + R33
( −1
r2 f (r)
C2323 + C1212
)]
= 0 (7)
Solving each one of Eqs. (7) gives two different metric function f (r),
f1(r) = 1 + c1
r
f2(r) = c2r + c3r2 1
2β
[−3 ln(r)r2 + 3r3 + 2β] (8)
where c1, c2 and c3 are constants. Comparing f1(r) with the usual Schwarzschild sheds light on the fact that, for c1 = −2 M , 
Schwarzschild’s solution again could represent the black hole structure in high-order energy, which describes the equations 
of motion (2). On the other hands, the metric function f2(r) represents the general non-Schwarzschild solution that is 
S.H.R. Fazlollahi / C. R. Physique 19 (2018) 337–340 339Fig. 1. Comparison of the geodesy of a falling physical particle for three different values of c2 and c3 when β = 1, r0 = 10.5 into the horizon area.
shown in [3]; therefore, we can consider it as a metric function describing the space–time structure around black holes. 
One obtains some plausible metric functions. For instance, when c2 = 0, c3 = −3/(2r20) and β ≡ r20 , the metric function 
f2(r) becomes
f2(r) = 1 − 3
2
(
r
r0
)2[
(r − 1) − (r − 1)
2
2
+ (r − 1)
3
3
− . . .
]
(9)
where we use Taylor series for the natural logarithm. An approach similar to (9) but with different coefficients has been 
investigated in ref. [3]. As shown in Fig. 1, the behavior of a physical particle falling in non-Schwarzschild black holes can 
be much different. For example, if c2 = c3 = 0, with respect to other constants of the model, the particle goes to singularity 
faster than in the usual Schwarzschild solution, while for two other cases when at least one of the constants is opposite 
zero, the particle follows a strange path. Note that Fig. 1 shows only the particle’s behavior in the event horizon, and so the 
metric function f2(r) does not comply with the no-hair theorem, since this theory discusses classical properties of black 
holes out of the event horizon [9].
Entropy and the first law of thermodynamic are most interesting issues to investigate black hole solutions. Hereupon, 
we consider entropy in the following of our discussion. Since we are working with higher-order derivative theory, generally 
entropy is not one-quarter of the area of the event horizon but, as mentioned in [4], the entropy of black holes, when the 
latter are described with a usual Schwarzschild solution, is the same as in Bekenstein–Hawking’s entropy formula. For a 
non-Schwarzschild solution that is illustrated with the metric function f2(r), we need to use the formula that was derived 
by Wald [5,6]. This has been evaluated for the ansatz (3) in quadratic curvature gravities in [7] and for f (R) gravities in [8]. 
In our case, with γ = 0 and β = r20 , entropy is given by (AH is area of event horizon)
S = AH
4
[
1 − 4
3r0
c2 − 16
3
c3 + 8
r20
(
1
3
+ ln(r0)
)]
(10)
and the Hawking temperature is
T = 1
4π
[
c2 + 2r0c3 + 3
2
(
1 − 2 ln(r0)
)]
(11)
So, the Noether charge with respect to Eqs. (10), (11) writes
Q = −2AH
[
1 − 4
3r0
c2 − 16
3
c3 + 8
r20
(
1
3
+ ln(r0)
)][
c2 + 2r0c3 + 3
2
(
1 − 2 ln(r0)
)]
(12)
and, finally, the mass of the black hole for the metric function f2(r) is
M = 1
3
1
r20r
2
[−36c2c3r20r2 ln(r) − 16c23r20r3 − 6c2r20r2 ln(r) + 48c3r0r2(ln(r))2 − 12c3r20r3 − 36r3(ln(r))2
− 39c2r0r2 ln(r) + 18r0r3 ln(r) + 10c22r20r − 132c3r0r3 + 198r3 ln(r) − 27r0r3 − 4c2r20 − 234r3 − 24r0r
]
(13)
Generally, by choosing different values of the constants, we get a different behavior for each thermodynamic parameter, 
but from the slope of M(T ), it can be seen that the specific heat C = ∂M/∂T is negative for both usual Schwarzschild and 
non-Schwarzschild black holes.
Following the mentioned discussion, investigating the entropy and other thermodynamic parameters needs considering 
each case of constants. Here, we only consider the first law of thermodynamics. Analytically, we found that for c2 = 0, 
340 S.H.R. Fazlollahi / C. R. Physique 19 (2018) 337–340entropy, the Hawking temperature, and the mass of black holes will be complex. Hereupon, we choose c2 = 0 and c3 = −3/2
as in Eq. (9). We have the freedom to add a constant multiple of the Gauss–Bonnet invariant to the Lagrangian, which shifts 
the entropy by a parameter independent constant without affecting the equations of motion. Using this approach allows us 
to find the mass and the temperature of these non-Schwarzschild black holes as a function of entropy:
M ≈ −2.6784 + 0.147668 S − 0.0045 S2 + 0.00010 S3
T ≈ 0.14668 − 0.00890 S − 0.0004 S2 + 1.96 10−7 S3 (14)
It can be seen that ∂M/∂ S ≈ 0.147668 − 0.0090S , which is very close to the expression for the temperature; thus, the 
non-Schwarzschild black holes are seen to obey the first law of thermodynamics with high precision; the non-Schwarzschild 
solution satisfies the first law, dM = T dS .
As mentioned above, the slope of M(T ) is negative; to prove it for the above fixed set of constants, we have
C ≈ −453.09 − 2601.8T − 8255.7T 2 (15)
In this paper, we considered mathematically Einstein’s gravity with some quadratic curvature invariants to find plausible 
explicit black hole solutions. These black holes may involve condensations of massive scalar, spin-2 and spin-1 modes and 
massless spin-2 graviton.
We focused our discussion on the static spherically symmetric black hole solutions. Attempting to find the explicit met-
ric function f (r) shows that, for ansatz (3), one has only two explicit black hole solutions, one of which is the usual 
Schwarzschild solution. Thus, the usual Schwarzschild solution allows one to investigate the black hole structure in high-
order energy. In fact, the quantization of usual Schwarzschild solution allows us to explore the behavior of a physical particle 
in a quantum-gravity medium without loss of generality. The other metric function implies a non-Schwarzschild solution 
that has not been considered in literature. We applied Wald’s formalism and derived an explicit entropy formula. Then the 
Hawking temperature, the Noether charge, and the mass of a non-Schwarzschild solution are given. We show that, for each 
set of constants, the non-Schwarzschild solution presents different behaviors. Thus, we use {c2, c3} = {0, −2/3}, in order to 
investigate the first law of thermodynamics to show that the non-Schwarzschild solution obeys the first law. Naturally, the 
non-Schwarzschild solution has wider classes than those considered here. We wish that our work will contribute to the 
future research in this direction.
We are very grateful to A. H. Fazlollahi for fruitful philosophical and physical discussions on this topic.
References
[1] G. ’t Hooft, M.J.G. Veltman, One loop divergences in the theory of gravitation, Ann. Inst. Henri Poincaré A, Phys. Théor. 20 (1974) 69.
[2] K.S. Stelle, Renormalization of higher derivative quantum gravity, Phys. Rev. D 16 (1977) 953.
[3] H. Lu, A. Perkins, C.N. Pope, K.S. Stelle, Black holes in higher-derivative gravity, Phys. Rev. Lett. 114 (2015) 171601.
[4] N. Li, X.-L. Li, S.-P. Song, An exploration of the black hole entropy via the Weyl tensor, Eur. Phys. J. C 76 (2016) 111.
[5] R.M. Wald, Black hole entropy is the Noether charge, Phys. Rev. D 48 (1993) 3427, arXiv:gr-qc /9307038.
[6] V. Iyer, R.M. Wald, Some properties of Noether charge and a proposal for dynamical black hole entropy, Phys. Rev. D 50 (846) (1994), arXiv:gr-qc /
9403028.
[7] Z.Y. Fan, H. Lü, Thermodynamical first laws of black holes in quadratically-extended gravities, arXiv:1501.00006 [hepth].
[8] J. Li, K. Meng, L. Zhao, Near horizon symmetry and entropy of black holes in gravity and conformal gravity, Gen. Relativ. Gravit. 47 (2015) 116.
[9] N. Gürlebeck, No-hair theorem for black holes in astrophysical environments, Phys. Rev. Lett. 114 (2015) 151102.


Plausible black hole solutions in higher-derivative gravity

K.P. Santhosh

Indu Sukumaran

Comparative study of the alpha decay of Hg isotopes using different forms of nuclear potentials

Not a member yet

Quantum simulation of zero-temperature quantum phases and incompressible states of light via non-Markovian reservoir engineering techniques

Combinatorial approaches for the design of metallic alloys

Dossier Sommaire

Foreword

Solar radio bursts as a tool for space weather forecasting

Foreword

Transitions in spatial networks

Robustness of spatial networks and networks of networks

Plausible black hole solutions in higher-derivative gravity

Comparative study of the alpha decay of Hg isotopes using different forms of nuclear potentials

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