Variational Principle and Running Couplings

Variational Principle and Running Couplings

Alejandro Hernandez A.

Advisor:

PhD. Pedro Bargueno

Co-advisor:

PhD. Ernesto Contreras

A document submitted in fulfilment of the requirementsfor the degree of MSc. Physics.

1

Abstract:

The purpose of this work is to study the theory of scale-dependent gravity in (2 + 1)

and (3 + 1) dimensions. We do this by using generalized field equations derived from ap-

plying the variational principle to an effective action. In (2 + 1) dimensions this theory is

studied in the context of non-linear electrodynamics, making emphasis on the thermody-

namic properties of charged black holes. In (3 + 1) dimensions, we concentrate on studying

the cosmological consequences of this theory and compare our results with those of the

ΛCDM model. Regarding (3 + 1) dimensions, one of the most important results is that

when we consider the case of “true running”, in which the Newton “constant” G and the

dark energy density ρΛ are scale-dependent functions, namely G(t) and ρΛ(t), our model fits

well with the experimental data for the Hubble parameter H(z). This is due to a very slow

decay of G(t). This essentially makes our model indistinguishable from the ΛCDM model,

therefore requiring further test or more measurements in order to determine the advantages

(or disadvantages) of our model over the classical one.

2

Resumen:

El proposito de este trabajo es estudiar la teorıa de gravedad dependiente de escala en

dimension (2+1) y (3+1). Esto lo hacemos usando ecuaciones de campo generalizadas

derivadas de aplicar el principio variacional a una accion efectiva. En dimension (2+1)

se estudia esta teorıa en el contexto de electrodinamica no lineal, haciendo enfasis en las

propiedades termodinamicas de agujeros negros cargados. En dimension (3+1), nos con-

centramos en estudiar las consecuencias cosmologicas de dicha teorıa y comparar nuestros

resultados con los del modelo ΛCDM. Al respecto de dimension (3+1), uno de los resul-

tados mas importantes es que cuando consideramos el caso “true running”, en el cual la

“constante” de Newton G y la densidad de energıa oscura ρΛ son funciones dependiente de

escala, a saber, G(t) y ρΛ(t), nuestro modelo se ajusta muy bien a los datos experimentales

para el parametro de Hubble H(z). Esto se lo atribuımos a un decaimiento muy lento de

G(t). Lo anterior hace que nuestro modelo no difiera esencialmente de los resultados predi-

chos por ΛCDM, haciendo necesaria la adquisicion de nuevas mediciones o la aplicacion

de otras pruebas de nuestro modelo para poder establecer las ventajas (o desventajas) de

nuestro enfoque sobre el modelo clasico.

Contents

1 Introduction 5

2 Linear and non-linear electrodynamics and scale-dependent gravity in (2+1)

dimensions 7

2.0.1 Classical linear and non-linear electrodynamics in (2+1) dimensions 9

2.0.2 Einstein-Maxwell case . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1 Einstein-power-Maxwell case . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Scale dependent couplings and scale setting . . . . . . . . . . . . . . . . . 13

2.3 The null energy condition . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4 Scale dependence in Einstein-Maxwell theory . . . . . . . . . . . . . . . . 16

2.4.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.4.2 Asymptotic behaviour . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4.3 Horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4.4 Thermodynamic properties . . . . . . . . . . . . . . . . . . . . . 21

2.4.5 Total charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 Einstein-power-Maxwell scale dependence . . . . . . . . . . . . . . . . . . 25

2.5.1 Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5.2 Asymptotic behaviour . . . . . . . . . . . . . . . . . . . . . . . . 26

2.5.3 Horizons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.5.4 Thermodynamic properties . . . . . . . . . . . . . . . . . . . . . 30

2.5.5 Total charge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3 Cosmology and scale-dependent gravity in (3+1) dimensions 35

3

CONTENTS 4

3.1 ΛCDM and running–vacuum models . . . . . . . . . . . . . . . . . . . . 35

3.1.1 G(t) = G0 and ρΛ(t) 6= 0 . . . . . . . . . . . . . . . . . . . . . . 37

3.1.2 G(t) 6= 0 and ρΛ(t) = ξ . . . . . . . . . . . . . . . . . . . . . . . 38

3.1.3 G(t) 6= 0 and ρΛ(t) 6= 0 . . . . . . . . . . . . . . . . . . . . . . . 39

3.2 Scale–dependent couplings and scale setting . . . . . . . . . . . . . . . . 40

3.2.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2.2 G = G0 and ρΛ 6= 0 . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.2.3 G 6= 0 and ρΛ = ξ . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.2.4 G(t) 6= 0 and ρΛ(t) 6= 0. . . . . . . . . . . . . . . . . . . . . . . 44

4 Conclusions 46

4.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

A Appendix 49

A.1 Detailed calculations in (2+1) dimensions . . . . . . . . . . . . . . . . . . 49

Chapter 1

Introduction

A theoretical description of gravity can be obtained from the classical Einstein–Hilbert (EH)

action

Γ0[gµν ] =

∫d4x√−g

[1

2κ0

(R− 2Λ0

)+ LM

]. (1.0.1)

Varying respect to the metric field one obtains

Gµν + Λ0gµν = κ0Tµν , (1.0.2)

where Gµν is the Einstein tensor, R is the Ricci scalar, Tµν is the energy–momentum tensor,

Λ0 and G0 are the cosmological and Newton’s couplings, respectively, and κ0 = 8πG0 is

the so–called Einstein’s constant. Besides, the energy momentum tensor is defined in terms

of the matter lagrangian density LM as

Tµν = LMgµν − 2δLMδgµν

. (1.0.3)

Usually, the couplings in Einstein’s field equations, G0 and Λ0, are considered as fixed values.

However, some models which go beyond General Relativity consider that these quantities

are no longer constant. Dirac was the first one who considered the gravitational coupling

5

CHAPTER 1. INTRODUCTION 6

as a time dependent quantity [1]. After Dirac’s paper, other models appeared as candidates

to explain why and how these coupling “constants” depend on space and time [2, 3, 4]. In

those models, Einstein field equations are obtained by promoting G0,Λ0 to G(t),Λ(t)

directly in Eq. (1.0.2). Also, some measurements suggest that the coupling “constants”

G0 and Λ0 could, indeed, vary in time [5, 6]. Following those ideas, many models have

been made in order to incorporate time dependence of the of the coupling constants G0

and Λ0 (for an incomplete list see, [6, 7, 8, 9]). Usually the starting point is within the

Einstein field equations for time dependent couplings. However, those procedures have

the problem that most ad hoc modifications of the field equations lead to inconsistencies

with the symmetries of the underlying theory. Even though such models have proven to

fit successfully to experimental data of both couplings [10], it is necessary to provide a

self consistent treatment which give us effective equations from the variational principle

incorporating running couplings.

Those issues of self consistency can be avoided in a new approach, where the coupling con-

stants are treated as functions of a scale/field. This new approach has been proposed and

successfully applied to cosmology and black holes physics [11, 12, 13, 14, 15, 16, 17, 18, 19,

20]. It is inspired by the fact that the effective quantum action presents a scale dependent

behavior, which is very well known in the literature. Thus, one has to take into account

deviations from the classical (non scale dependent) solutions of general relativity. Similar

ideas of models with running gravitational coupling have been considered in the context of

grand unification theories [21].

In this work we will investigate several features of scale-dependent gravity in (2+1) and

(3+1) dimensions. In the first case, we will focus on studying linear and non-linear electro-

dynamics and some thermodynamical aspects of charged black holes. In the second case,

we will do a detailed analysis of the cosmological evolution of the matter-dominated era of

the universe taking into account several cases for the scale dependence of the couplings.

Chapter 2

Linear and non-linear electrodynamics

and scale-dependent gravity in (2+1)

dimensions

In recent years gravity in (2+1) dimensions has attracted a lot of interest for several rea-

sons. The absence of propagating degrees of freedom, its mathematical simplicity, and the

deep connection to Chern-Simons theory [22, 23, 24] are just a few of the reasons why to

study threedimensional gravity. In addition (2+1) dimensional black holes are a good test-

ing ground for the four-dimensional theory, because properties of (3+1)-dimensional black

holes, such as horizons, Hawking radiation and black hole thermodynamics, are also present

in their threedimensional counterparts.

On the other hand, the main motivation to study non-linear electrodynamics (NLED) was

to overcome certain problems of the standard Maxwell theory. In particular, non-linear

electromagnetic models are introduced in order to describe situations in which this field is

strong enough to invalidate the predictions provided by the linear theory. Originally, the

Born-Infeld nonlinear electrodynamics was introduced in the 30’s in order to obtain a finite

7

CHAPTER 2. LINEAR AND NON-LINEAR ELECTRODYNAMICS AND SCALE-DEPENDENTGRAVITY IN (2+1) DIMENSIONS 8

self-energy of point-like charges [25]. During the last decades this type of action reappears

in the open sector of superstring theories [26, 27] as it describes the dynamics of D-branes

[28]. Also, these kind of electrodynamics have been coupled to gravity in order to obtain,

for example, regular black holes solutions [29, 30, 31, 32], semiclassical corrections to the

black hole entropy [33] and novel exact solutions with a cosmological constant acting as

an effective Born-Infeld cutoff [34]. A particularly interesting class of NLED theories is the

so called power-Maxwell theory described by a Lagrangian density of the form L(F ) = F β,

where F = FµνFµν/4 is the Maxwell invariant, and β is an arbitrary rational number. When

β = 1 one recovers the standard linear electrodynamics, while for β = D/4 with D being

the dimensionality of spacetime, the electromagnetic energy momentum tensor is traceless

[35, 36]. In 3 dimensions the generic black hole solution without imposing the traceless con-

dition has been found in [37], while black hole solutions in linear Einstein-Maxwell theory

are given in [38, 39].

As mentioned in previous sections, scale dependence at the level of the effective action is

a generic result of quantum field theory. Regarding quantum gravity it is well-known that

its consistent formulation is still an open task. Although there are several approaches to

quantum gravity (for an incomplete list see e.g. [40, 41, 42, 43, 44, 45, 46, 47] and refer-

ences therein), most of them have something in common, namely that the basic parameters

that enter into the action, such as the cosmological constant or Newton’s constant, become

scale dependent quantities. Therefore, the resulting effective action of most quantum grav-

ity theories acquires a scale dependence. Those scale dependent couplings are expected to

modify the properties of classical black hole backgrounds.

The main objective of this chapter is to study the scale dependence at the level of the

effective action of threedimensional charged black holes in linear (Einstein-Maxwell) and

non-linear (Einstein-power-Maxwell) electrodynamics. We use the formalism and notation


of [16, 17] where the authors applied the same technique to the BTZ black hole [48, 49].

2.0.1 Classical linear and non-linear electrodynamics in (2+1) dimensions

In this section we present the classical theories of linear and non-linear electrodynamics.

Those theories will then be investigated in the context of scale dependent couplings. The

starting point is the so-called Einsteinpower-Maxwell action without cosmological constant

(Λ0 = 0), assuming a generalized electrodynamics i.e. L(F ) = c0|F |β which reads

Γ0 =

∫d3x√−g[

1

16πG0

R− 1

e2β0

L(F )

], (2.0.1)

where G0 is Einstein’s constant, e0 is the electromagnetic coupling constant, R is the Ricci

scalar, L(F ) is the electromagnetic Lagrangian density where c0 is a constant, F is the

Maxwell invariant defined in the usual way i.e. F = (1/4)FµνFµν and Fµν = ∂µAν − ∂νAν

is the electromagnetic field strength tensor. We use the metric signature (−,+,+), and

natural units (c = ~ = kB = 1) such that the action is dimensionless. Note that β is

an arbitrary rational number, which also appears in the exponent of the electromagnetic

coupling in order to maintain the action dimensionless. The special case β = 1 reproduces

the classical Einstein-Maxwell action, and thus the standard electrodynamics is recovered.

For β 6= 1 one can obtain Maxwell-like solutions. In the following we shall consider both

cases: first when β = 3/4, since it is this value that allows us to obtain a trace-free

electrodynamic tensor, precisely as in the four-dimensional standard Maxwell theory, and

second when β = 1 because is the usual electrodynamics in 2+1 dimensions. In both

cases one obtains the same classical equations of motion, which are given by Einstein’s field

equations

Gµν =κ0

e2β0

Tµν . (2.0.2)

The energy momentum tensor Tµν is associated to the electromagnetic field strength Fµν


through

Tµν = L(F )gµν − LFFµγFν γ, (2.0.3)

where LF = dL/dF . In addition, for static circularly symmetric solutions the electric field

E(r) is given by

Fµν = (δrµδtν − δrνδtµ)E(r). (2.0.4)

For the metric, circular symmetry implies

ds2 = −f(r)dt2 + g(r)dr2 + r2dφ2 (2.0.5)

Note that in the classical case one finds that g(r) = f(r)−1. Finally, the equation of motion

for the Maxwell field Aµ(x) reads

Dµ

(LFF µν

e2β0

)= 0. (2.0.6)

With the above in mind, for charged black holes one only needs to determine the set of

functions f(r), E(r). Using Einstein’s field equations (2.0.2) and (2.0.6) combined with

Eq. (2.0.4) and the definition of LF , one obtains the classical electric field as well as the

lapse function f(r). It is possible to determine the electric field as well as the lapse func-

tion without assuming a particular value for β for classical solutions, however we will focus

on two of them. First, the Einstein-Maxwell case is in itself interesting due to its relation

with the four-dimensional case. On the other hand, the Einstein-power-Maxwell case with

β = 3/4 is a desirable one due to a remarkable property: it has a null trace, which is also

present in the four-dimensional case.


2.0.2 Einstein-Maxwell case

The field equations (2.0.2) for β = 1 read

8πc0G0E(r)2

e0

+e0f

′(r)

r= 0,

e0f′′(r) =

8πc0G0E(r)2

e0

= 0,

(2.0.7)

and the Maxwell equations (2.0.6) reduces to

c0 (rE ′(r) + E(r))

e0r= 0. (2.0.8)

The set of equations (2.0.7), (2.0.8) are easily solved and the classical (2+1)-dimensional

Einstein-Maxwell black hole solution (β = 1) is given by

f0(r) = −M0G0 −1

2

Q20

e20

ln(r/r0), (2.0.9)

E0(r) =Q0e

20

r, (2.0.10)

where M0 is the mass and Q0 is the electric charge of the black hole and r0 stands for the

radius where the electrostatic potential vanishes. The apparent horizon r0 is obtained by

demanding that f0(r0) = 0, which reads

r0 = r0e− 2M0G0e

20

Q20 , (2.0.11)

and rewriting the lapse function using the apparent horizon one gets

f0(r) = −Q20

2e20

ln

(r

r0

). (2.0.12)

Since black holes have thermodynamic behaviour, it is interesting to study such a behaviour

for the β = 1 case. Here, the Hawking temperature T0, the Bekenstein-Hawking entropy


S0, as well as the heat capacity C0 are found to be

T0(r0) =1

4π

∣∣∣∣∣ Q20

2e20r0

∣∣∣∣∣, (2.0.13)

S0(r0) =AH(r0)

4G0

, (2.0.14)

C0(r0) = T∂S

∂T

∣∣∣∣∣Q

= −S0(r0). (2.0.15)

Note that AH(r0) is the horizon area which is given by

AH(r0) =

∮d2x√h = 2πr0. (2.0.16)

2.1 Einstein-power-Maxwell case

The field equations (2.0.2) and the Maxwell equation (2.0.6) are

4 4√

2πc0G0E(r)3/2

e3/20

+f ′(r)

r= 0,

f ′′(r) =8 4√

2πc0G0E(r)3/2

e3/20

,

c0 (rE ′(r) + 2E(r))

e3/20 rE(r)1/2

= 0.

(2.1.1)

Solving the previous equations, the lapse function f(r) and the electric field E(r) are found

to be

f0(r) =4G0Q

20

3r−G0M0, (2.1.2)

E0(r) =Q0

r2. (2.1.3)

It is worth mentioning that, unlike in the previous section, the solutions here considered

do not contain the electromagnetic coupling. This is due to the fact that a dimensional

analysis on the action (2.0.1) for β = 3/4 reveals that the electric charge is dimensionless


in this case. As a consequence, we can set the electromagnetic coupling to unity without

affecting the classical action. At classical level a horizon is present, and it is computed by

requiring that f(r0) = 0, which reads

r0 =4

3

Q20

M0

. (2.1.4)

Expressing the mass M0 in terms of the horizon one obtains

f0(r) =4

3G0Q

20

[1

r− 1

r0

]. (2.1.5)

Classical thermodynamics plays a crucial role since it provides us with valuable information

about the underlying black hole physics. The Hawking temperature T0, the Bekenstein-

Hawking entropy S0 as well as the heat capacity C0 are given by

T0(r0) =1

4π

∣∣∣∣∣M0G0

r0

∣∣∣∣∣, (2.1.6)

S0(r0) =AH(r0)

4G0

, (2.1.7)

C0(r0) = −AH(r0)

4G0

, (2.1.8)

In agreement with the notation in the previous section, AH(r0) is the so-called horizon area.

2.2 Scale dependent couplings and scale setting

This section summarizes the equations of motion for the scale dependent Einstein-Maxwell

and Einstein-powerMaxwell theories. The notation follows closely [50] as well as [16, 17].

The scale dependent couplings of the theories are i) the gravitational coupling Gk, and ii)

the electromagnetic coupling 1/ek. Furthermore, there are three independent fields, which

are the metric gµν(x), the electromagnetic four-potential Aµ(x), and the scale field k(x).


The effective action for the non-linear electrodynamics reads

Γ[gµν , k] =

∫d3x√−g[

1

2κkR− 1

e2βk

L(F )

], (2.2.1)

The equations of motion for the metric gµν(x) are given by

Gµν =κk

e2βk

T effecµν , (2.2.2)

with

T effecµν = T EM

µν −e2βk

κk∆tµν . (2.2.3)

Note that T EMµν is given by Eq. (2.2.3), κk = 8πGk is the Einstein constant and the

additional object ∆tµν is defined as follows

∆tµν = Gk

(gµν−∇µ∇ν

)G−1k . (2.2.4)

The equations of motion for the four-potential Aµ(x) taking into account the running of ek

are

Dµ

(LFF µν

e2βk

)= 0. (2.2.5)

It is important to note that since the renormalization scale k is actually not constant any

more, this set of equations of motion do not close consistently by itself. This implies that

the stress energy tensor is most likely not conserved for almost any choice of functional

dependence k = k(r). This type of scenario has largely been explored in the context of

renormalization group improvement of black holes in asymptotic safety scenarios [51, 52,

53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65]. The loss of a conservation laws comes

from the fact that there is one consistency equation missing. This missing equation can be


obtained from varying the effective action (2.2.1) with respect to the scale field k(r), i.e.

d

dkΓ[gµν , k] = 0, (2.2.6)

which can thus be understood as variational scale setting procedure [11, 13, 66, 50, 67].

The combination of (2.2.6) with the above equations of motion guarantees the conservation

of the stress energy tensors. A detailed analysis of the split symmetry within the functional

renormalization group equations, supports this approach of dynamic scale setting [68].

The variational procedure (2.2.6), however, requires the knowledge of the exact beta func-

tions of the problem. Since in many cases the precise form of the beta functions is unknown

(or at least unsure) one can, for the case of simple black holes, impose a null energy condi-

tion and solve for the couplings G(r),Λ(r), e(r) directly [16, 17, 14, 15]. This philosophy

of assuring the consistency of the equations by imposing a null energy condition will also be

applied in the following study on Einstein-Maxwell and Einstein-power-Maxwell black holes.

2.3 The null energy condition

The so-called Null Energy Condition (hereafter NEC) is the less restrictive of the usual

energy conditions (dominant, weak, strong, and null), and it helps us to obtain desirable

solutions of Einstein’s field equations[69, 70]. Considering a null vector `µ, the NEC is

applied on the matter stress energy tensor such as

Tmµν`µ`ν ≥ 0. (2.3.1)

The application of such a condition was appropriately implemented in Ref. [16] inspired

by the Jacobson idea [70]. Note that in proving fundamental black hole theorems, such as

the no hair theorem [71] and the second law of black hole thermodynamics [72], the NEC


is, indeed, required. For scale dependent couplings, one requires that the aforementioned

condition is not violated and, therefore, the NEC is applied on the effective stress energy

tensor for a special null vector `µ = f−1/2, f 1/2, 0 such as

T effecµν `µ`ν = (T EM

µν − α∆tµν)`µ`ν ≥ 0. (2.3.2)

In addition, the left hand side (LHS) is null as well as T EMµν `

µ`ν = 0 and the condition reads

∆tµν`µ`ν = 0 (2.3.3)

One should note that Eq. (2.3.3) allows us to obtain the gravitational coupling G(r) easily

by solving the differential equation

G(r)d2G(r)

dr2− 2

(dG(r)

dr

)2

= 0 (2.3.4)

which leads to

G(r) =G0

1 + εr(2.3.5)

The NEC allows us to decrease the number of degrees of freedom, and thus it becomes an

important tool for scale dependent black hole problems.

Let us begin by presenting the general equations of the scale dependent (2+1) gravity

coupled to a power-Maxwell source for an arbitrary β.

2.4 Scale dependence in Einstein-Maxwell theory

In order to get insight into non-linear electrodynamics regarding the running of couplings,

one first has to discuss the effects of scale dependence in linear electrodynamics. With this

in mind, one also needs to determine the set of four functions G(r), E(r), f(r), e(r)2


which are obtained by combining Einstein’s effective equations of motion with the NEC

taking into account the EOM for the four-potential Aµ.

2.4.1 Solution

The solution for this scale dependent black hole is given by

G(r) =G0

1 + εr,

E(r) =Q0

re(r)2,

f(r) = − G0M0

(rε+ 1)2− Q2

0

2e20

(ln(r/r0) + rε)

(rε+ 1)2, (2.4.1)

e(r)2 = e20

[1

(1 + rε)3+ 4

rε

(1 + rε)3− (4M0G0 − 5Q2

0 + 2Q20 ln(r/r0))

r2ε2

(1 + rε)3

],

where the integration constants are chosen such as the classical Einstein-Maxwell (2+1)-

dimensional black hole is recovered according to [39]. It is relevant to say that the gravi-

tational coupling G(r) is obtained by taking advantage of NEC, hile the electric field E(r)

is given by the covariant derivative (2.2.5), which depends on the electromagnetic coupling

constant e(r). Besides, the lapse function f(r) and the coupling e(r) are directly obtained

by using Einstein’s effective field equations combined with the solutions for E(r) and G(r).

In addition, our solution reproduces the results of the classical theory in the limit ε→ 0, i.e.

limε→0

G(r) = G0,

limε→0

E(r) =Q0e

20

r,

limε→0

f(r) = −G0M0 −Q2

0

2e20

ln(r/r0), (2.4.2)

limε→0

e(r)2 = e20.

which justifies the naming of the constants aforementioned G0,M0, Q0, e0 in terms of their

meaning in the absence of scale dependence [16], as it should be. Besides, the parameter ε


controls the strength of the new scale dependence effects, and therefore it is useful to treat

it as a small expansion parameter as follows

G(r) ≈ G0

[1− rε+O(ε2)

](2.4.3)

E(r) ≈ Q0e20

r

[1 + εr +O(ε2)

], (2.4.4)

f(r) ≈ f0(r) +

[2G0M0 −

Q20

2e20

+Q2

0

e20

ln(r/r0)

]rε+O(ε2) (2.4.5)

e(r)2 ≈ e20

[1 + εr +O(ε2)

](2.4.6)

In Figure 2.1 the lapse function f(r) is shown for different values of ε in comparison to

the classical (2+1)- dimensional Einstein-Maxwell solution. The figure shows that the

scale dependent solution for small εr values is consistent with the classical case. However,

when εr becomes sufficiently large, a deviation from the classical solution appears. The

electromagnetic coupling e(r) is shown in Figure 2.2 for different values of ε. Note that

when ε is small the classical case is recovered, but when ε increases the electromagnetic

coupling tends to decrease until it is stabilized.

2.4.2 Asymptotic behaviour

In this subsection a few invariants need to be revisited. In particular we will focus on the

Ricci scalar R and the Kretschmann scalar K. Both of them are relevant in order to check

if some additional divergences appear. For the static and circularly symmetric metric we

have considered, the Ricci scalar is given by

R = −f ′′(r)− 2f ′(r)

r(2.4.7)


Figure 2.1: Lapse function f(r) for ε = 0 (black solid line), ε = 0.04 (blue dashed line), ε = 0.15 (dottedred line) and ε = 1 (dotted dashed green line). The values for the rest of the parameters have been takenas unity.

Figure 2.2: Electromagnetic coupling e(r)2 for ε = 0 (black solid line), ε = 0.0025 (short dashed blueline), ε = 0.007 (dotted red line), ε = 0.02 (dotted dashed green line), ε = 0.08 (long dashed orange line)and ε = 0.5 (double dotted dashed purple line). The other values have been taken as unity.

or more precisely

R =Q2

0

2r2e20(1 + rε)4

− 8M0G0e20 + 4Q2

0 ln(r/r0)

2r2(1 + rε)4e20

rε+4M0G0e

20 − 7Q2

0 + 2Q20 ln(r/r0)

2r2(1 + rε)4e20

(rε)2.

(2.4.8)


We require that classically the Ricci scalar reads

R0 =1

2

Q20

e20r

2(2.4.9)

Considering r values close to zero one obtains

R ≈ R0

[1−

(8M0G0e

20

Q20

+ 4 + 4 ln(r/r0)

)rε+O(r2)

](2.4.10)

Thus, upon comparing Eq. (2.4.9) with Eq. [?] we observe that the scale dependent effect

strongly distorts this invariant. Despite that, for small values of r the standard case R0 is

recovered. In the same way, one expects that ε should be small, therefore one can expand

the Ricci scalar around ε = 0 but the solution is exactly the same reported for r 1.

Regarding the Kretschmann scalar, it is computed to be

K = RµναβRµναβ. (2.4.11)

Thus, when ε is small the Kretschmann scalar reads

K ≈ 3Q40

4r4

[1−

(16M0G0e

20

3Q20

+8 ln(r/r0)

3

)rε

]+O(ε2). (2.4.12)

Note that the classical result for this invariant is indeed K0 = 3Q40/r

4, which coincides with

our solution when ε→ 0.

The other regime of asymptotic behaviour can be studied in a large radius expansion r →

∞. In this limit the lapse function f(r) decays as r−1 which disagrees with the classical

result shown in Eqs. (2.1.2). On the other hand, the electromagnetic coupling e(r) also

tends to zero as r−1 in contrast with the expected result, e0. Finally, one obtains that

E(r) ∼ r−2, R ∼ r−4 and K ∼ r−6, all of them going to zero as expected. However, it

can be shown that these functions decay faster than those corresponding to the classical


solutions. In fact, in absence of running coupling, a straightforward calculation reveals that

E(r) ∼ r−1, R ∼ r−2 and K ∼ r−4.

2.4.3 Horizons

The apparent horizon occurs when the lapse function vanishes, i.e. f(rH) = 0. Thus, this

Einstein-Maxwell black hole solution represents a non trivial deviation from the classical

solution which is manifest when we compare our solution with the corresponding black hole

solution without the scale dependence. Here, the horizon read

rH =1

εW

(εe− 2G0M0e

20

Q20

)(2.4.13)

where W () is the so-called Lambert-W function, which is a set of functions, namely the

branches of the inverse relation of the function Y (rε) = rεerε with rε being a complex

number. In particular, Eq (2.4.13) is also the principal solution for rε. In Figure 3 the scale

dependent effect on horizon is shown. We can see that the deviation from the classical case

is also evident for small M0 values.

In addition, one can expand the horizon around ε = 0 obtaining the classical solution plus

corrections i.e.

rH ≈ r0

[1− εr0 +O(ε2)

](2.4.14)

2.4.4 Thermodynamic properties

After having gained experience on the horizon structure one can now move towards the

usual thermodynamic properties associated with our solution shown at Eq. (2.4.1). Thus,

the Hawking temperature of the black hole assuming the ansatz (2.0.5) is given by

TH(rH) =1

4π

∣∣∣∣∣ limr→rH

∂rgtt√−gttgrr

∣∣∣∣∣, (2.4.15)


Figure 2.3: Black hole horizons rH as a function of the mass M0 for ε = 0 (black solid line), for ε = 1(blue dashed line), for ε = 2.5 (dotted red line) and for eε = 6 (dotted dashed green line). The values ofthe rest of the parameters have been taken as unity.

i.e.

TH(rH) =1

4π

∣∣∣∣∣ Q20

2rH(1 + εrH)e20

∣∣∣∣∣. (2.4.16)

Taking advantage of the fact that the integration constant ε should be small, one can expand

around ε = 0 to get the well-known Hawking temperature (at leader order) i.e.

TH(rH) ≈ T0(r0)∣∣∣1 + εr0 +O(ε2)

∣∣∣. (2.4.17)

In Figure 4 we show the effective temperature which takes into account the running coupling

effect.

Moreover, the Bekenstein-Hawking entropy for his black hole is

S =AH(rH)

4G(rH)= S0(rH)(1 + εrH) (2.4.18)


Figure 2.4: The Hawking temperature TH as function of the classical mass M0 for ε = 0 (black solid line),ε = 750 (blue dashed line), ε = 1800 (dotted red line) and ε = 3000 (dotted dashed green line). The othervalues of the rest of the parameters have been taken as unity. Note that the vertical axis is scaled OJO1 : 170.

and assuming small values of ε one can expand to get

S ≈ S0(r0)

[1− 1

2(εr0)2 +O(ε3)

](2.4.19)

In Figure 5 below we show the entropy for our (2+1)-dimensional Einstein-Maxwell scale

dependent black hole. It is clear that the running effect is dominant when ε is not small,

while for large values of M0 the effect is practically zero.

Finally, the heat capacity is computed in the usual way i.e.:

CQ = T∂S

∂T

∣∣∣∣∣Q

(2.4.20)

which read

CQ = −S0(rH)(1 + εrH). (2.4.21)


Figure 2.5: The Bekenstein-Hawking entropy S as function of classical mass M0 for ε = 0 (black solidline), ε = 200 (blue dashed line), ε = 600 (dotted red line) and ε = 1000 (dotted dashed green line). Theother values have been taken as unity.

The classical case is, of course, recovered in the ε→ 0 limit. Due to a weak ε dependence

it was necessary to plot the figure with very large values of ε in order to generate a visible

effect. The scale dependent effect is notoriously small for those quantities.

2.4.5 Total charge

The electric field is parametrized through the total charge Q, but in our previous discussion

Q0 only denotes an integration constant which coincides with the charge of the classical

theory. In general, we need to compute the total charge by the following relation [73]

Q =

∫ √−gdΩ

(LFFµνe2βk

)nµσν , (2.4.22)

where nµ and σν are the unit spacelike and timelike vectors normal to the hypersurface

of radius r, and they are given by nµ = (f−1/2, 0, 0) and σν = (0, f 1/2; 0) as well as


√−gdΩ = rdφ. Making use of these we obtain

Q = 2πQ0, (2.4.23)

which is proportional to the classical value and has no ε dependence.

2.5 Einstein-power-Maxwell scale dependence

This section is devoted to the study of a (2+1) scale dependent gravity coupled to a

power-Maxwell source. As mentioned before, the case β = 3/4 leads to a dimensionless

electromagnetic coupling which was set to the unity in the section 2.1. However, if one

considers a scale dependent gravity, the electromagnetic coupling has a non-trivial scale

dependence. Therefore, in this section we shall hold the electromagnetic coupling depen-

dence of the action (2.0.1). In this way, the solution consists of a set of four functions

G(r), E(r), f(r), e(r)3, which are obtained by combining Einstein’s effective equations

of motion with the NEC taking advantage of the EOM for the four-potential Aν . In what

follows we shall obtain the solutions of the system in terms of the functions mentioned

above.


2.5.1 Solution

The integration constants have been chosen such as the scale dependent solution reduces

to the classical NLED case when the appropriate limit is taken. Thus, our solution reads

G(r) =G0

1 + εr,

E(r) =Q0

r2

(e(r)

e0

)3

, (2.5.1)

f(r) =4G0Q

20

3r(rε+ 1)3− M0G0 (r3ε2 + 3r2ε+ 3r)

3r(rε+ 1)3,

e(r)3 = e30

[(2rε(3rε+ 2) + 1)

(rε+ 1)4− M0r

3ε2(rε+ 4)

4Q20(rε+ 1)4

].

In the limit ε→ 0 we obtain

limε→0

G(r) = G0,

limε→0

E(r) =Q0

r2, (2.5.2)

limε→0

f(r) =4G0Q

20

3r−G0M0,

limε→0

e(r)3 = e30.

Note that if we set e0 = 1, the classical solution in section 2.1 is recovered. Even more, if

one demands that G0 = 1 (which is the standard lore) then we are in complete agreement

with the classical solution given at Ref. [35].

2.5.2 Asymptotic behaviour

The asymptotic behaviour of this solution can be studied by computing geometrical invari-

ants i.e. the Ricci scalar, which for our solution is

R = −4G0ε

[M0 + 4Q2ε

r(rε+ 1)5

], (2.5.3)


where the classical case (with a null cosmological constant) is clearly R0 = 0. For r → 0

one obtains

R ≈ −4G0ε

[M0 + 4Q2

0ε

r

]+O(r). (2.5.4)

We observe that the Ricci scalar is altered in presence of scale dependent coupling. In

addition, one note that an unexpected r6 divergence appears, which is controlled by ε.

Another geometrical invariant is the Kretschmann scalar K which is given by

K = RµναβRµναβ. (2.5.5)

For r → 0 one can obtain the first terms which are

K ≈ 32G20Q

40

3r6

[1−

(M0

Q20

ε+ 4ε2

)r2

]+O(r−3). (2.5.6)

Taking into account that the ε should be small we have

K ≈ 32G20Q

40

3r6

[1− M0r

2

Q20

ε+O(ε2)

], (2.5.7)

where the standard value K0 has been obtained demanding that ε goes to zero. Classically,

the Ricci scalar for null cosmological constant is identically zero, however in presence of scale

dependent couplings it exhibits a singularity. The Kretschmann scalar exhibits a singularity

at r → 0 for both the classical and the scale dependent case. On the other hand, the

opposite regime of asymptotic behaviour is studied in the large radius expansion r → ∞

both for the Ricci and the Kretschmann scalar. The Ricci scalar as well as the Kretschmann

scalar are asymptotically close to zero. Regarding the limit r →∞ the lapse function goes

as r−1 in agreement with the asymptotic behaviour of the classical solution. In addition,

note the unusual behaviour of the electromagnetic coupling in the light of scale dependent

framework in Fig. 2.7. Starting from e30 the electromagnetic coupling decays softly and it


stabilizes when

limr→∞

e(r)3 = −(

1

3r0ε

)e3

0, (2.5.8)

instead of reach the classical value. The electric field tends to zero as expected but slowly

compared with the classical case. In fact, E(r) behaves as r−1 in clearly deviation with

respect to the result shown in Eq. (2.1.3). Finally, the curvature and Kretschmann scalars

hold the same asymptotic behaviour of the results obtained in absence of running, i.e.

R ∼ r−4 and K ∼ r−6.

Figure 2.6: Lapse function f(r) for ε = 0 (black solid line), ε = 0.04 (blue dashed line), ε = 0.15 (dottedred line) and ε = 1 (dotted dashed green line). The values for the rest of the parameters have been takenas unity.


Figure 2.7: Electromagnetic coupling e(r)3 for ε = 0 (black solid line), ε = 0.25 (dashed blue line),ε = 0.45 (dotted red line), ε = 1 (dotted dashed green line). The other values have been taken as unity.

2.5.3 Horizons

Applying the condition f(rH) = 0 one obtains the scale dependent horizon which reads

rH = −1

ε

[1−

[1 + 3 εr0

]1/3], (2.5.9)

r± = −1

ε

[1 +

1

2(1± i

√3)

[1 + 3εr0

]1/3]. (2.5.10)

where r0 is the classical value given by Eq. (2.1.4). Note that one obtains three horizons,

out of which one is real (physical horizon) and two r± are complex (nonphysical).

In addition, since the scale dependence of coupling constants is usually assumed to be weak,

it is reasonable to consider the dimensionful parameter ε as small compared to the other

scales and, therefore, one can expand around ε close to zero, which gives us

rH ∼= r0

[1− εr0 +

5

3(εr0)2 + · · ·

]. (2.5.11)

One should note that when ε tends to zero the classical case is recovered. Besides, although


Figure 2.8: Black hole horizons rH as a function of the mass M0 for ε = 0 (black solid line), for ε = 1(blue dashed line), for ε = 1 (dotted red line) and for eε = 2 (dotted dashed green line). The values of therest of the parameters have been taken as unity.

ε could take positive or negative values, here in order to obtain desirable physical results we

require that ε > 0. In our set of solutions G(r), E(r), f(r), e(r) we can expand around

zero for small values of ε, i.e.

G(r) ≈ G0

[1− rε+O(ε2)

], (2.5.12)

E(r) ≈ E0(r) +O(ε2), (2.5.13)

f(r) ≈ f0(r) +

[2G0M0 −

4G0Q20

r

]rε+O(ε2), , (2.5.14)

e3(r) ≈ e30

[1 +O(ε2)

], (2.5.15)

2.5.4 Thermodynamic properties

Using the horizon structure and the lapse function (which is given by Eq. (2.5.1)) one can

calculate the Hawking temperature of the corresponding scale dependent black hole. At the


outer horizon this temperature is given by the simple formula

TH =1

4π

∣∣∣∣∣ limr→rH

∂rgtt√−gttgrr

∣∣∣∣∣, (2.5.16)

which reads in term of the horizon radius

TH =1

4π

∣∣∣∣∣ M0G0

rH(1 + εrH)

∣∣∣∣∣. (2.5.17)

Figure 2.9: Hawking temperature TH as a function of the classical mass M0 for ε = 0 (black solid line),ε = 20 (blue dashed line), ε = 50 (dotted red line) and ε = 100 (dotted dashed green line). The values ofthe rest of the parameters have been taken as unity.

In order to recover the classical result we expand around ε = 0 and upon evaluating at the

classical horizon we obtain

TH(rH) ≈ T0(r0)

∣∣∣∣∣1 +1

3(εr0)2 +O(ε3)

∣∣∣∣∣, (2.5.18)

where it is clear that ε→ 0 coincides with Eq. (2.1.6) as it should be.


In addition, the Bekenstein-Hawking entropy obeys the well-known relation heritage of

Brans-Dickey theory applied to the (2+1)-dimensional case

S =1

4

∮d2x

√h

G(x), (2.5.19)

where hij is the induced metric at the horizon. For the present circularly symmetric solution

this integral is trivial because the induced metric for constant t and r slices is ds = rdφ

and moreover G(x) = G(rH) is constant along the horizon. Using these facts, the entropy

for this solution is found to be [16, 17]

S =AH

4G(rH)= S0(rH)(1 + εrH), (2.5.20)

while for small values of ε one obtains

S ≈ S0(r0)

[1− 1

3(εr0)2 +O(ε3)

](2.5.21)

which, of course, coincides with the classical results in the limit ε → 0. In addition, the

heat capacity (at constant charge) CQ can be calculated by

CQ = T∂S

∂T

∣∣∣∣∣Q

. (2.5.22)

Combining Eq. (2.5.17) with (2.5.20) one obtains the simple relation

CQ = −1

8

M0

TH= −S0(rH)(1 + εrH) (2.5.23)

Note that the black hole is unstable since CQ < 0, and it coincides with the classical result

in the limit ε→ 0.


Figure 2.10: The Bekenstein-Hawking entropy S as a function of the classical mass M0 for ε = 0 (blacksolid line), ε = 20 (blue dashed line), ε = 50 (dotted red line) and ε = 100 (dotted dashed green line). Theother values have been taken as unity.

2.5.5 Total charge

As in the previous case, the total charge Q needs to be computed by the relation [73]

Q =

∫ √−gdΩ

(LFFµνe2βk

)nµσν . (2.5.24)

In this case we obtain

Q =Q0

2e3/20

(2.5.25)

which also is proportional to the classical value and does not have ε dependence.

2.6 Discussion

Scale dependent gravitational couplings can induce nontrivial deviations from classical Black

Holes solutions. We have studied two cases, first Einstein-Maxwell and second Einstein-


power-Maxwell case. Both of them have a common feature: the lapse function tends to

zero when r!→∞, characteristic which is absent in the classical solutions. In addition, the

total charge is modified as a consequence of our scale dependent framework. Moreover, we

have found that, for the same value of the classical black hole mass, the apparent horizon

radius (and the Bekenstein-Hawking entropy) decreases when the strength of the scale

dependence increases. This is in agreement with the findings in [51, 52, 53, 54, 55, 56, 57,

58, 59, 60, 61, 62, 63, 64, 65]. On the other hand, the Hawking temperature increases with

ε. Please, note that the effect of the scale dependence in the Einstein-power-Maxwell case

is stronger than the Eintein-Maxwell case. The behaviour of the electromagnetic coupling

e(r) depends on the choice of the electromagnetic Lagrangian density. While e(r) goes to

zero in the limit r →∞ for a Maxwell Lagrangian density, it approaches a constant value for

the power-Maxwell case. Finally, it is well known that a black hole (as a thermodynamical

system) is locally stable if its heat capacity is positive [74]. In both scale dependent cases

it is found that these black holes are unstable (CQ < 0), like their classical counterparts.

Chapter 3

Cosmology and scale-dependent

gravity in (3+1) dimensions

In this chapter we deduce the effective Einstein field equations assuming scale (temporal)

dependent couplings for a FLRW background. After that, we solve the corresponding

effective Einstein field equations in some relevant cases and compare the solutions with the

previously obtained solutions reported in Ref. [75, 10]) and with observations in case this is

available. Throughout the manuscript we will be interested in one of the main cosmological

functions related to the scale factor of the Universe, which is the Hubble parameter as a

function of the redshift, H(z). Even more, given the large number of both measurements

and theoretical computations on H(z) (see [10, 76] and references therein) in a matter

dominated epoch, we focus our attention only in this era.

3.1 ΛCDM and running–vacuum models

In this section we briefly review both ΛCDM and some running-vacuum cosmological models

which have received considerable attention given the concordance between them and recent

measurements [10]. We first start by revisiting the classical ΛCDM model which corresponds

35

CHAPTER 3. COSMOLOGY AND SCALE-DEPENDENT GRAVITY IN (3+1) DIMENSIONS 36

to constant couplings, namely usual General Relativity [77]. Specifically, both the Newton’s

constant, G0, and the vacuum energy density, ρΛ0 ≡ Λ0/(8πG0), are taken as constants.

As an additional constraint, this model takes advantage of the covariant conservation of the

energy–momentum tensor, namely

∇µTµν = 0. (3.1.1)

Besides, it assumes a perfect fluid such that Tµν = diag(ρm, pm, pm, pm). Using Eq. 3.1.1

for a perfect fluid it is obtained that

ρm + 3H (ρm + pm) = 0. (3.1.2)

The unknowns are the scale factor a(t), the matter density ρm(t) and the pressure pm(t).

In the simplest case, one can assume dust (pm(t) = 0) and one just needs to find the other

two functions. The classical Friedmann equations are sufficient for determining those two

unknown functions.

Regarding the running cases, the simplest model consists in working with the usual Fried-

mann equations but replacing G0,Λ0 with G(t),Λ(t). The equations are given by

H2 =1

3κ(t)

[ρ(t) +

Λ(t)

κ(t)

]− K

a2, (3.1.3)

H +H2 = −1

6κ(t)

[ρ(t) + 3p(t)

]+

1

3Λ(t), (3.1.4)

where, as usual, κ(t) ≡ 8πG(t). Nevertheless, these equations are not enough to solve

for the unknowns in general, as it will be shown in the following. In this sense, extra

information is needed to account for a complete description of the cosmological model

under consideration.


3.1.1 G(t) = G0 and ρΛ(t) 6= 0

The key assumption lying at the heart of this case is promoting only Λ0 → Λ(t) in Einstein’s

equations, namely

Gµν + Λ(t)gµν = κ0Tµν . (3.1.5)

By using ∇µGµν = 0, one obtains

ρΛ + ρm + 3H (ρm + pm) = 0. (3.1.6)

In this case, an exchange of energy between matter and vacuum takes place. The unknowns

are a(t), ρm(t) and Λ(t). Therefore, Eqs. (3.1.3), (3.1.4) and (3.1.6) allow to solve for the

unknowns provided an extra input on ρΛ. A common choice is [78]

ρΛ(t) =Λ(t)

8πG0

= m0 +m1H(t) +m2H(t)2, (3.1.7)

where m0,m1 and m2 are coupling constants and the Hubble paramenter H(t) is defined

according to H(t) ≡ a(t)/a(t). As pointed out recently [10], the above expression has been

suggested in the literature from the quantum corrections of QFT in curved spacetime. The

running parameter m0 is of order 10−3M4, while m1 ∼ 10−4M3, and m2 ∼ 10−2M2 where

M is an average (large) mass scale (see, for example, [79, 80, 81] and references therein).

Although the present case is considered in Refs. [82, 83], in these references the authors

do not present an explicit solution for H(t), which is the observable we are interested in, as

previously commented. The analytical solution we find in this case is

H(t) =2M

8πm2 − 3tan(M (2c1 + t)

)− 4πm1

8πm2 − 3, (3.1.8)


and the scale factor is given by the simple relation

a(t) = C2 exp

(−2 (log (cos (M (2c1 + t))) + 2πm1t)

8πm2 − 3

). (3.1.9)

This can be written as

a(t) = C2e− 4πm1

8πm2−3tcos

23−8πm2

(M(2c1 + t)

), (3.1.10)

where we have defined the auxiliary function

M =

√2πm0

(8πm2 − 3

)−(2πm1

)2. (3.1.11)

The remaining unknowns Λ(t) and ρm(t) are easily obtained from the previous expressions.

3.1.2 G(t) 6= 0 and ρΛ(t) = ξ

In this case, we will consider the two quantities G(t),Λ(t) keeping the vacuum energy

density as a constant value. Thus, from the covariance of the Einstein’s tensor we obtain

[3H(p+ ρ) + ρ

]+χ(ρ+ ρΛ) = 0. (3.1.12)

where χ ≡ G(t)/G(t).

In this case, the modified Einstein field equations given by Eqs. (3.1.3) and (3.1.4), combined

with Eq. (3.1.12), are not enough to solve for the four unknowns a(t), ρm(t), Λ(t), G(t).

In addition we can not use the ansatz (3.1.7) since ρΛ is constant, thus we need to get

the extra information from some additional assumptions. One possibility suggested in this

context can be expressed as [84]

ρm(t) = m4a(t)−3+ε, (3.1.13)

where ε signals a possible deviation from ΛCDM. We note that a partial solution for this


case is developed in [84]. With this, one gets

G(t) =

(m4a(t)ε−3 +m0

m0 +m4

)ε

3−ε (3.1.14)

Nevertheless, an analytical expression for a(t) is not presented here because it is too lengthy.

3.1.3 G(t) 6= 0 and ρΛ(t) 6= 0

This is the most general case within this framework. In addition to Eqs. (3.1.3) - (3.1.4)

and the covariance of the Einstein tensor, the covariant conservation of Tµν is assumed [78],

giving

ρΛ + χ(ρ+ ρΛ) = 0, (3.1.15)

which can be solved, using some ansatz, for instance, for ρΛ(t) (see Eq. (3.1.7)). Interest-

ingly, the function G(H) can be obtained analytically, as

G(H) =G0

1 +m2 ln (H2/H20 ), (3.1.16)

where G0 and H0 are the Newton’s constant and the Hubble parameter at a given time

(usually taken as the present time), respectively.

In Fig. (3.1) we show the comparison between experimental data (see text for details) and

the computations under the ΛCDM and the running–vacuum model previously discussed

for H(z). The values of the parameters m0, m1, m2 were fixed as in [83]. Given the

experimental uncertainty, all the considered models reproduce fairly well the observations.

Therefore, H(z) does not provide an stringent test and other observables are needed. At

this point it is worth mentioning the recent work by Sola et al. [10], in which the authors

constrain possible running–vacuum models using the cosmological observables SN Ia + BAO

+ H(z) + LSS + BBN + CMB. However, as we will show in the following section(s), H(z)


is enough to discriminate between ΛCDM, running–vacuum and variational running–vacuum

models.

Figure 3.1: Behavior of H(z) for the running vacuum cases considered in the present section. See text fordetails.

3.2 Scale–dependent couplings and scale setting

In the present section we will introduce the general scale–dependent framework. From now

on, we will refer to it as the variational running vacuum model (VRV model hereafter).

The notation follows closely Ref. [50] as well as Refs. [16, 17, 18, 19, 85]. The classical

couplings are the Newton and the cosmological constant, both of them being promoted to

scale–dependent couplings, namely, G0,Λ0 → Gk,Λk which, in general, depend on the

renormalization scale, k. In addition, there are two independent fields, which are the metric

gµν(x) and the scale field k(x). The effective action is given by

Γ[gµν , k] =

∫d4x√−g

[1

2κk

(R− 2Λk

)+ LM

]. (3.2.1)


The effective Einstein’s field equations are

Gµν + Λkgµν = κkTeffecµν , (3.2.2)

and the effective energy–momentum tensor is defined according to

κkTeffecµν ≡ κkTµν −∆tµν , (3.2.3)

where ∆tµν is a tensor which takes into account the scale–dependence of the gravitational

coupling. This term is given by the following expression

∆tµν = Gk

(gµν−∇µ∇ν

)G−1k . (3.2.4)

Note that, since the renormalization scale k is actually not constant anymore, the corre-

sponding set of equations of motion is not closed, which means that the energy momentum–

tensor could be not conserved for an arbitrary choice of k. This kind of problem has

been investigated through renormalization group improvement of black holes in asymptotic

safety scenarios [86, 87, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65]. It is however

preferable to maintain the classical conservation laws and to formulate a closed system of

equations. This problem can be solved by a variational scale–setting procedure according

to [11, 13, 66, 50, 67]. Taking the renormalization scale k as an additional field k(x) one

obtains the algebraic equation

δΓ

δk= 0. (3.2.5)

With the help of Eq. (3.2.5) combined with Eq. (3.2.2), we guarantee the conservation of

the stress–energy tensor. Despite of that, a problem is still present: it is necessary to know

the explicit form of beta functions which encode the dependence of a coupling parameter

on the energy scale. The explicit computation is achievable by different mathematical


techniques which, in general, do not produce the same solution. Since our interest is the

variation of the cosmological quantities in time, and not with respect to k(t), we assume

that Ok → O(t).

Since, as we mentioned before, this ansatz is insufficient to determine the system we are

going to introduce an additional ansatz for ρΛ(H) in order to have enough information for

the system of equations to be solved. Finally, it is worth mentioning that, in general, a

solution for a scale–dependent model allows for recovering the classical limit when certain

running parameter is turned off [16, 17, 18, 19]. However, as it will be shown later, this

is not the case for certain VRV cases because either the running parameter can not be

identified due to the numerical nature of the solutions or the classical solution is forbidden

(see Sects. 3.2.3 and 3.2.4 for details).

3.2.1 The model

As usual, Tµν stands for the energy-momentum tensor which is defined as follows

Tµν ≡−2√−g

δ(√−gLM)

δgµν= gµνLM − 2

δLMδgµν

, (3.2.6)

and for a perfect fluid we have

Tµν =(ρ+ p

)uµuν + pgµν , (3.2.7)

where uµ = (1, 0, 0, 0) is the velocity vector field and gµν is the metric tensor corresponding

to the Friedmann–Lemaıtre–Robertson–Walker geometry, namely

ds2 = −dt2 + a(t)2

[1

1−Kr2dr2 + dΩ2

]. (3.2.8)


Assuming a flat universe (i.e. K = 0) and considering dust matter (pm(t) = 0), the

generalized Friedmann equations coming from (3.2.2) are

H2

[1− χ

H

]=

1

3κ(t)

[ρ+ ρΛ

], (3.2.9)

H +H2

[3

2− χ

H

]− 1

2

[χ− χ2

]=

1

2κ(t)ρΛ. (3.2.10)

Here we have defined the parameter χ ≡ ˙G(t)/G(t) which encodes the variability of the

Newton parameter. Further, we are able to get complementary information by the compu-

tation of the covariant derivative of the generalized Einstein equations (3.2.2)

(H2 + H)χ+Hχ2 +1

3κ(t)

[3H(p+ ρ) + (ρ+ ρΛ) + χ(ρ+ ρΛ)

]= 0. (3.2.11)

In addition, in some special cases will be useful to assume the conservation of the matter

stress–energy tensor in the classical sense, i.e. when the couplings are constant, which reads

from (3.2.11) and assuming p = 0

ρ+ 3Hρ = 0. (3.2.12)

In the rest of the present chapter we will explore some particular cases of these VRV models,

trying to constrain their validity by computing H(z) and comparing both with previous

running–vacuum models and experimental data. When possible, analytical solutions will be

given.

3.2.2 G = G0 and ρΛ 6= 0

If the gravitational Newton coupling G is constant, this implies that the tensor ∆tµν = 0 and

the variational principle leads to the same Equations corresponding to subsection (3.1.1).


3.2.3 G 6= 0 and ρΛ = ξ

In this case, we assume the same ansatz for the baryonic energy density as we used in

Section 3.1.2, that is

ρm(t) = m4a(t)−3+ε (3.2.13)

in order to have enough information to solve the system. We use this ansatz in order to

follow as close as possible the protocol that Sola uses to solve his equations in each one

of the cases. Even more, since we are introducing the running parameter ε, we interpret

it as a perturbation parameter with respect to the ΛCDM model, therefore, its numerical

value will be small. Taking into account the above, the system of differential equations to

solve consist of just the generalized Friedmann equations, and in this particular case they

are given by

a2 (3H(H − χ)−m0κ(t)) = m4κ(t)aε−1, (3.2.14)

m0κ(t)− 2H + 2Hχ− 3H2 + χ− χ2 = 0, (3.2.15)

After an appropriate manipulation of equations (3.2.14), (3.2.15) we get

G(t) =3H2

8π

(m4aε−3(H−(ε−2)H2)

H+2H2 +m0

) . (3.2.16)

But an analytical solution for the scale factor a(t) was not obtained due to the complexity

of the equation.

3.2.4 G(t) 6= 0 and ρΛ(t) 6= 0.

This is the most general case within the VRV framework. In addition to Friedmann equa-

tions, we assume the covariant conservtions of the energy momentum tensor Tµν in order


to have enough equations to solve for the unknowns a(t), G(t) and ρm(t). We make this

assumption since in this general case, we need to have the usual conservation of energy-

momentum in order to properly compare the results of this case to the ΛCDM model.

Therefore, the corresponding equations in this case are

H2

[1− χ

H

]=

1

3κ(t)

(ρ+ ρΛ

)(3.2.17)

H +H2

[3

2− χ

H

]− 1

2

[χ− χ2

]=

1

2κ(t)ρΛ, (3.2.18)

3Hρ+ ρ = 0 (3.2.19)

where, again, the first two equations correspond to the generalized Friedmann equations and

the last one is the usual conservation of the energy momentum tensor. After manipulating

the previous equations, introducing the dependences G(H(t)), Λ(H(t)), considering the

ansatz for ρΛ(t) given in Eq. (3.1.7), and doing the change of variables t→ z, we can solve

numerically for the function H(z).

Chapter 4

Conclusions

4.1 Conclusions

In the case of (2+1) dimensions, we have studied the scale dependence of charged black holes

in three-dimensional spacetime both in linear (Einstein-Maxwell) and non-linear (Einstein-

power-Maxwell) electrodynamics. In the second case we have considered the case where the

electromagnetic energy momentum tensor is traceless, which happens for β = 3/4. After

presenting the models and the classical black hole solutions, we have allowed for a scale

dependence of the electromagnetic as well as the gravitational coupling, and we have solved

the corresponding generalized field equations by imposing the ”null energy condition” in

three-dimensional spacetimes with static circular symmetry. Horizon structure, asymptotic

spacetimes and thermodynamics have been discussed in detail.

With respect to (3+1)-dimensions we have introduced a scale–dependent cosmological

model in a matter–dominated era. In the first part we have extensively reviewed the so–

called running vacuum models, incorporating some points not previously considered in the

literature. In all these cases the model fits well with experimental data for H(z) by consid-

ering an standard equation of state for dark energy pΛ = −ρΛ. Despite of that, it could be

more satisfactory, for a theoretical point of view, to have a running vacuum model coming

46

CHAPTER 4. CONCLUSIONS 47

from a variational principle. In this spirit, starting with an effective action based on the usual

Einstein–Hilbert one, we have promoted G0, ρΛ0 to scale–dependent couplings, Gk, ρΛk

to implement this scale–setting procedure into a cosmological context (variational running

vacuum model) for a matter–dominated era in the second part. Specifically, and for the

sake of comparison with other running vacuum models, we have limited our study to the

case of time–dependent couplings. It is worth mentioning that we have also assumed the

same ansatz for ρΛ(H) which were considered in the original running vacuum case. Note

that, since this ansatz comes from QFT considerations in a FLRW background, we keep

it as a suitable feature also for the variational running vacuum model. However, different

parametrizations for ρΛ(H) coming from other approaches to quantum gravity could be

assumed.

Within the variational procedure, we have studied several cases and a brief summary of the

key points in each one of them is shown in Table 4.1.

Table 4.1: Several cases of the VRV model.

Case Key point Description

1. G(t) = cte., ρΛ(t) = cte. Classical GR, that is, ΛCDM.

2. G(t) = cte., ρΛ(t) 6= 0 ∇µTµν 6= 0 and it is the same as Sola’s case.

3. G(t) 6= 0, ρΛ(t) = cte. ∇µTµν 6= 0 and we assume ρm(t) ∝ a(t)−3+ε .

4. G(t) 6= 0, ρΛ(t) 6= 0 True running, in which we assume ∇µTµν = 0.

First, if ρΛ is taken as a constant value, an analytic solution for the modified Einstein’s

field equations (including G(t)) is obtained. It is worth mentioning that this solution does

not correspond to a true running since the classical solution can not be recovered by turning

off of any parameter. Therefore, once the running case has been considered, the classical

case is automatically discarded. This feature is remarkable due in previous scale depen-

dent problems we always can recover the classical solution under certain election of the

CHAPTER 4. CONCLUSIONS 48

so–called running parameter of the theory. Second, a complete numerical solution has been

obtained when both ρΛ and G(t) are taken into account into the model. Although this

case corresponds to a true running, due to the numerics involved, we have not been able

to find any parameter which could control the strength of the scale–dependence for both

ρΛ and G(t). Third, regarding the comparison of these two models with the experimental

data for H(z), there are not significantly differences with respect to the ΛCDM model. A

possible explanation could be given in terms of the behavior of the Newton’s coupling. As

the classical solution implies a constant value, G0, for this coupling, we should expect the

decay to be slow so that its variation, with respect to the classical case, is not appreciable.

This turn out to be exactly the case, therefore the behavior of the Newton’s coupling with

respect to the constant coupling’s case could be an important point to take into account in

order to test the viability of any extension of general relativity, regardless of the fact that G

is not an observable of the model. In addition, we note that a slower decay of G(t) in the

corresponding running vacuum case gives place to an excellent agreement with experimental

data. Beyond that, it is necessary to say that what we did in the (3+1)-dimensional case

is only a preliminary test to establish whether or not the scale dependent gravity theory

fits well or not to the experimental data available for the time evolution of the Hubble

factor, and although this preliminary test is satisfied by our VRV model, further test are

required in order to decide if this model reproduces well the behavior of our universe at all

cosmological scales or not. Specifically, we can consider solar system test of our model or

linear perturbations of both the metric and the energy densities. Those and other aspects

of scale–dependent cosmology are currently under study and will eventually be the object

for future publication.

Appendix A

Appendix

A.1 Detailed calculations in (2+1) dimensions

In this appendix we study some features of the scale dependent (2 + 1) gravity coupled to

a power-Maxwell source for an arbitraty β. For this system the action is given by

S =

∫d3x√−g[

R

16πG(r)− L(F )

e(r)2β

], (A.1.1)

where G(r) and e(r) are the gravitational and the the electromagnetic scale-dependent

couplings, R is the Ricci scalar, L(F ) = Cβ|F |β is the electromagnetic Lagrangian density,

F = (1/4)FµνFµν is the Maxwell invariant, and C is a dimensionless constant which de-

pends on the choice of β. Metric signature (−,+,+) and natural units (c = ~ = kB = 1)

are used in our computations.

Variations of the Eq. (A.1.1) with respect to the metric field lead to the modified Einstein’s

equations

Rµν −1

2gµνR = 8π

G(r)

e2β(r)Tµν −∆tµν , (A.1.2)

49

APPENDIX A. APPENDIX 50

where Tµν stands for the power–Maxwell energy momentum tensor and

∆tµν = G(r)(gµν−∇µ∇ν)1

G(r). (A.1.3)

is the non-material energy momentum tensor which arises as a consecuence of the scale

dependence of the gravitational coupling. On the other hand, after variations of the action

Eq. (A.1.1) with respect to the electromagnetic four–potential, Aµ, one obtains the modified

Maxwell equations

Dµ

(F µ

νLFe(r)2β

)= 0. (A.1.4)

Henceforth, only static and spherically symmetric solutions will be considered. Therefore

we shall assume the ansatz

ds2 = −f(r)dt2 + f(r)−1dr2 + r2dΩ2 (A.1.5)

Fµν = (δtµδrν − δrµδtν)E(r) (A.1.6)

for the metric and the electromagnetic tensor, respectively. With the former prescription is

straightforward to prove, from Eq. (A.1.4), that the electric field is given in terms of the

electromagnetic coupling by

E(r) =2β−12β−1A−

β2β−1Q

12β−1

0 e(r)2β

2β−1

β1

2β−1 r1

2β−1

(A.1.7)

Please, note that setting β = 1 and A = 1 the electric field reported in Eq. (2.4.1) is

recovered

E(r) =Q0

re(r)2 (A.1.8)

APPENDIX A. APPENDIX 51

In the same way, for β = 3/4 and A = 27/33−43 e2

0Q2/3 one obtain

E(r) =Q0

r2

(e(r)

e0

)3

(A.1.9)

in complete agreement with Eq. (2.5.1). It is worth noting that, even in the general case the

electric field depends on an specific power of the charge as a consequence of the non–linear

electrodynamics, in the cases β = 1 and β = 3/4, this behaviour is not observed due to a

particular setting of A.

If the null energy condition is imposed on the solutions, we obtain that the scale–dependent

gravitational coupling reads

G(r) =G0

1 + εr, (A.1.10)

where G0 is Newton’s constant and ε is the running parameter. Note that the classical

limit is recovered in the limit ε → 0. Finally, Eq. (A.1.2) reduces to a pair of differential

equations for f(r), e(r) given by

π23−7β1−2β (2β − 1)G0rA

−αQ2αe(r)2α + β2αr2α ((2rε+ 1)f ′(r) + 2εf(r)) = 0, (A.1.11)

π23−7β1−2βG0A

−αQ2αe(r)2α − β2αr2α ((rε+ 1)f ′′(r) + 2εf ′(r)) = 0, (A.1.12)

where α = β2β−1

. It can be checked by the reader that, in the case β = 3/4, the solutions

of the set of equations (A.1.11), (2.5.13), (A.1.10) and (A.1.7) coincide with those listed

in Eq.(2.5.1) after an appropriate choice of the integration constants.

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Variational Principle and Running Couplings

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