The phase diagram of Einstein-Weyl gravity S. Silveravalle12and A. Zuccotti3 1Universit a degli Studi di TrentoVia Sommarive 14 IT-38123 Trento Italy

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The phase diagram of Einstein-Weyl gravity

S. Silveravalle1,2and A. Zuccotti3

1Universit`a degli Studi di Trento,Via Sommarive, 14, IT-38123, Trento, Italy

2INFN - TIFPA,Via Sommarive, 14, IT-38123, Trento, Italy

3Ghent University, Technologiepark-Zwijnaarde 126, Be-9052 Gent, Belgium

Thanks to their interpretation as ﬁrst order correction of General Relativity at high energies,

quadratic theories of gravity gained much attention in recent times. Particular attention has been

drawn to the Einstein-Weyl theory, where the addition of the squared Weyl tensor to the action

opens the possibility of having non-Schwarzschild black holes in the classical spectrum of the theory.

Static and spherically symmetric solutions of this theory have been studied and classiﬁed in terms of

their small scales behaviour; however, a classiﬁcation of these solutions in terms of the asymptotic

gravitational ﬁeld is still lacking. In this paper we address this point and present a phase diagram of

the theory, where the diﬀerent types of solutions are shown in terms of their mass and the strength

of a Yukawa-like correction to the gravitational ﬁeld. In particular we will show that, in the case of

compact stars, diﬀerent equations of state imply diﬀerent Yukawa corrections to the gravitational

potential, with possible phenomenological implications.

I. INTRODUCTION

General Relativity is one of the most successful theory

of last century, however there is still no general consensus

on how gravity should be described at the quantum level.

It is known that the Einstein-Hilbert action of General

Relativity is not renormalizable within standard pertur-

bative methods, and modiﬁcations of such action are ex-

pected at high energies. The study of modiﬁed gravity

theories is largely used to address the high energy limit

of gravity while preserving General Relativity at lower

energies.

The ﬁrst corrections expected are quadratic terms in the

curvatures [1], which in 4 dimensions modify the action

SQG =Zd4x√−g[γR +βR2−αCµνρσCµνρσ +Lm].

(1)

Indeed such action has a long history: in [2] it is proved

that the action (1) in the vacuum is perturbatively renor-

malizable, in [3] it appears in the low energy limit of

string theory, and recently it has emerged in the frame-

work of the renormalization group ﬂow [4–6], Asymptot-

ically Safe program [7, 8] and fakeons theory [9, 10]. The

physical content of (1) can be resumed in the standard

massless graviton, plus a massive scalar mediator and a

massive spin two ghost mediator, which at the quantum

level implies the loss of unitarity. While various authors

proposed solutions to the ghost problem, in this work we

want to focus on the classical content of the quadratic ac-

tion. In particular we are interested in studying the case

of static spherically symmetric solutions without a cos-

mological constant in order to describe the gravitational

ﬁeld of isolated objects. Despite the classical solutions

of (1) have been largely investigated in recent works [11–

15], given the non linear nature of the ﬁeld equations,

an exact form of the general solution is lacking, and nu-

merical methods have to be used in order to understand

the link between the asymptotic ﬁeld and the physical

nature of the solution. In the following we consider the

Einstein-Weyl action

SEW =Zd4x√−g[γ R −α Cµνρσ Cµνρσ +Lm],(2)

i.e. the quadratic theory restricted to the β= 0 case.

Both the theories deﬁned in (1) and (2) present a large va-

riety of solution families in addition to the Schwarzschild

one, but while the eﬀect of the R2term has been largely

studied in astrophysical and cosmological contexts [16–

19], the C2term received relatively less attention un-

til recent times. The Einstein-Weyl theory indeed gives

interesting insight on the physical content of quadratic

gravity. It is proved that the solution space of Einstein-

Weyl gravity coincides with the one of (1) with the con-

straint R= 0, and a no-hair theorem presented in [20],

and corrected in [14], states that under certain condi-

tions the Ricci scalar must vanish. As main consequence

this theorem implies that all the asymptotically ﬂat black

hole solutions of quadratic gravity are present only in

the Einstein-Weyl restriction. Moreover, the results in

[14] show that, together with black holes, the main new

families of solutions appearing in quadratic gravity, also

appear in the Einstein-Weyl theory. Finally, it has also

been shown that the C2has a much stronger eﬀect than

the R2one in the properties of compact stars [21].

While attempts to link the diﬀerent families of solutions

to the asymptotic ﬁeld have been made in the full the-

ory [22, 23], the presence of two massive mediators in

the full quadratic case brings numerical issues when in-

tegrating the asymptotic ﬁeld, namely some non-linear

terms can be larger than linear ones, which aﬀects the

physical results obtained. Such numerical instabilities

have not been encountered in the Einstein-Weyl theory,

so we found sensible to restrict our results to this case

where the numerical procedure is much more reliable. In

this paper we use the analytical approximation found in

previous works, together with numerical shooting tech-

niques, in order to show the complete solution space of

the Einstein-Weyl gravity in form of phase diagram of

arXiv:2210.13877v1 [gr-qc] 25 Oct 2022

the theory; to the best of our knowledge this is the ﬁrst

time the link between the asymptotic ﬁeld and the fam-

ilies of solutions is shown with this level of robustness.

Building the phase diagram of the theory is a crucial

step to understand the physical content since it allows

to connect the type of solution with the observables at

large distances. In particular the results encoded in the

phase diagram gives new insight on the structure of the

solution space of quadratic gravity, showing that some of

the new families of vacuum solution expected in previous

work appear together with Schwarzschild black holes for

all positive mass values, while others are conﬁned in a

ﬁnite mass interval.

In what follows we ﬁrst recall the analytical approxima-

tion needed as boundary condition and we describe the

numerical method used to integrate the ﬁeld equations.

Then we list the vacuum solution families encountered in

the solution space, the behavior of their gravitational po-

tential, together with their causal structure. We present

the phase diagram of the theory, in which is indicated

the type of solution in function of the gravitational ﬁeld

at large distances. The case of a self-gravitating perfect

ﬂuid is studied, showing the corresponding gravitational

ﬁeld and mass-radius relation compared to their General

Relativity counterpart.

II. EQUATIONS OF MOTION, ANALYTICAL

APPROXIMATIONS AND NUMERICAL

INTEGRATION

The equations of motion of Einstein-Weyl gravity in

tensorial form are

Hµν =γRµν −1

2R gµν +

−4α∇ρ∇σ+1

2RρσCµρνσ =1

2Tµν ,

(3)

with the trace being

Hµ

µ=γR =1

2Tµ

µ.(4)

Given the condition ∇µHµν = 0 and Hφφ = sin2θHθθ,

only two of these equations are independent. As the

ansatz for the static spherically symmetric metric we

choose the one with Schwarzschild coordinates

ds2=−h(r)dt2+dr2

f(r)+r2dΩ2.(5)

The equations of motion result equivalent to a system of

two second order ordinary diﬀerential equations in h(r),

f(r), as it is shown in [14, 24], corresponding to

Hµµ=γR =1

2Tµ

µ,

Hrr +A(r)∂rHµµ−1

2Tµ

µ+

+B(r)Hµµ−1

2Tµ

µ2

+C(r)Hµµ−1

2Tµ

µ=1

2Trr ,

(6)

where A(r), B(r) and C(r) are combinations of the cou-

pling α, the metric functions h(r), f(r) and their ﬁrst

derivatives. We show these equations explicitly in the

vacuum case

4h(r)2rf0(r) + f(r)−1−r2f(r)h0(r)2

+rh(r)rf0(r)h0(r)+2f(r)rh00(r)+2h0(r)= 0,

αr2f(r)h(r)rf0(r)+3f(r)h0(r)2

+ 2r2f(r)h(r)2h0(r)αrf00(r) + αf0(r)−γr

+h(r)3r3αrf0(r)2−4αf0(r)+2γr

−2f(r)4α+ 2αr2f00(r)−2αrf0(r) + γr2

+ 8αf(r)2−αr3f(r)2h0(r)3= 0.

(7)

To our knowledge, no analytical solutions of such sys-

tem have been found, with the exception of the ones

present in General Relativity i.e. the Minkowski and the

Schwarzschild spacetime. Numerical methods and ana-

lytical approximations have to be used in order to study

the complete solution space.

A. Linearized solutions at large distance

Since we are interested in studying isolated objects

without a cosmological constant, that is we look for

asymptotically ﬂat solutions, we can describe the metric

at large distances using the weak ﬁeld limit. As described

in [14, 24, 25], we write the functions h(r) and f(r) as

h(r) = 1 +  V (r), f(r) = 1 +  W (r),(8)

and solve (3) at linear order in .

When imposing asymptotic ﬂatness and ﬁxing h(r)→

1 as r→+∞, it is possible to show that the solutions

result to be

h(r) = 1 −2M

r+ 2S−

e−m2r

f(r) = 1 −2M

r+S−

e−m2r

r(1 + m2r),

(9)

with m2

2=γ

2αbeing the mass of the spin-two ghost, and

Mthe ADM mass in Planck units. We note that in the

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ThephasediagramofEinstein-WeylgravityS.Silveravalle1;2andA.Zuccotti31UniversitadegliStudidiTrento,ViaSommarive,14,IT-38123,Trento,Italy2INFN-TIFPA,ViaSommarive,14,IT-38123,Trento,Italy3GhentUniversity,Technologiepark-Zwijnaarde126,Be-9052Gent,BelgiumThankstotheirinterpretationasrstordercorrectiono...

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The phase diagram of Einstein-Weyl gravity S. Silveravalle12and A. Zuccotti3 1Universit a degli Studi di TrentoVia Sommarive 14 IT-38123 Trento Italy

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