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Brans-Dicke Solutions of Stationary, Axially Symmetric Spacetimes

By Pınar Kirezli Uludağ

Submitted: June 19th 2019Reviewed: September 24th 2019Published: March 5th 2020

DOI: 10.5772/intechopen.89906

Downloaded: 22

Abstract

One of the most known alternative gravitational theories is Brans-Dicke (BD) theory. The theory offers a new approach by taking a scalar field ϕ instead of Newton’s gravitational constant G. Solutions of the theory are under consideration and results are discussed in many papers. Stationary, axially symmetric solutions become important because gravitational field of celestial objects can be described by such solutions. Since obtaining exact solutions of BD is not an easy task, some solution-generating techniques are proposed. In this context, some solutions of Einstein general relativity, such as black hole or wormhole solutions, are discussed in BD theory. Indeed, black hole solutions in BD theory are not fully understood yet. Old and new such solutions and their analysis will be reviewed in this chapter.

Keywords

  • Brans-Dicke
  • stationary symmetric

1. Introduction

Einstein’s theory of general relativity (GR), which is undoubtedly one of the greatest theories of the last century, is still being tried to be understood. Recently, the theory is supported by the observations of the gravitational waves which are observed by LIGO and Virgo collaboration [1]. On the other hand, GR may have some problems regarding defining gravity accurately at all scales. One of the problems that GR faced was that it could not fully describe the accelerated expansion of the universe [2, 3, 4] without unknown materials, i.e., dark matter and dark energy. Although, in order to understand the theory and satisfy the scientific cruosity, GR is modified with higher-order Ricci scalar [5, 6] soon after the theory is published, this modifications were not paid attention. The pioneer of studies on scalar-tensor theory were done by Brans and Dicke [7] by changing Newton’s gravitational constant Gwith a scalar field ϕ=1/G. In order to understand the BD theory, several experimental tests of GR are studied, and they are summarized in [8]. Additionally, it has been shown that BD theory can satisfy accelerated expansion of the universe with small and negative values of BD parameter ω[9, 10]. But these values of BD parameter cannot satisfy the solutions of our solar system and latest CMB datas. Extended BD theories which include a potential for the BD field are allowed to construct a number of analytic approximations [11]. Although, in the beginning, we construct our solutions with a potential, then we choose zero potential which is usually called as massless BD theory, in order to make it more simple.

Obtaining exact solutions of any theory is important in order to make comparison with observations or in order to obtain the results of the theory under consideration. Stationary, axially symmetric solutions are one of the important classes of these solutions, since the gravitational field of compact celestial objects such as stars, galaxies, and black holes can be represented by such solutions. Due to the complexity of the field equations, some solution-generating techniques are constructed. Obtaining Ernst BD equations is one of the most known of these techniques [12, 13, 14]. Also, Nayak and Tiwari [15] obtained vacuum stationary, axially symmetric BD solutions and generalized Maxwell field by Rai and Singh [16]. Their theory depends on finding out the relation between the field equations of BD and GR theories. After defining this relation, the corresponding BD solution of any known GR solution can be obtained. This method, which we call the Tiwari-Nayak-Singh-Rai (TNSR) method, is the most direct one. Instead of the one-parameter solution, which is called TNSR method, solutions with two parameters were constructed in [17]. A brief summary of [17] will be provided in the rest of the chapter.

The outline of the chapter will be as follow; in Section 2, we review the BD field equations and explain the Ernst equations and extended TNSR method. In Section 3, we study several solutions in order to understand how the extended TNSR method works. In addition, we mention the GR limit of the BD solutions.

2. Field equations of Brans-Dicke theory

In general a four-dimensional Brans-Dicke action with matter in Jordan frame is given by

S=d4xgϕRωϕϕgμνμϕνϕVϕ+SME1

where 8πG=c=1, ϕis scalar field, ωϕis BD coupling, Ris Ricci scalar, Vϕis a potential BD field, and SMis action of the matter.

In many cases and origin of the BD theory, BD coupling ωϕis choosen a constant as ω. On the other hand, for different coupling function has a different modified gravity theory. One gets variations with respect to metric gμνand scalar field ϕ

Rμν12gμνR=Tμνϕωϕϕ2ϕ,μϕ,ν12gμνϕ,λϕ,λ1ϕϕ;μ;νgμνϕgμνVϕϕE2
2ω+3ϕ=T2V+ϕdVϕ,λϕ,λϕE3

Tμνrepresents energy-momentum tensor of the matter and Tis its trace and is d’Alembert operator with respect to full metric. The abstract index notation (i.e., gμν,Tμν) is used in order to show component of what we concern with. Greek indices run over the spacetime manifold, starting with time component tand space components r,θ,φin this work. For example, if we want to figure out t,tcomponent of Einstein tensor (Gμν=Rμν12gμνR) from Eq. (2), we get

Gtt=Tttϕωϕϕ2ϕ,tϕ,t12gttϕ,λϕ,λ1ϕϕ;t;tgttϕgttVϕϕE4

where the repeated index λmeans summation, such as ϕ,λϕ,λ=ϕ,tϕ,t+ϕ,ρϕ,ρ+ϕ,θϕ,θ+ϕ,φϕ,φ.

Conservation of the energy-momentum tensor of the matter leads to

Tν;νμ=0.E5

Furthermore, we set BD coupling ωϕωand scalar potential Vϕ=0in the rest of the paper for simplicity and to easily obtain the field equations. Also, nonvanishing scalar potential and ωϕare mostly used for cosmological solutions.

2.1 BD solution with electromagnetic field

A four-dimensional general stationary, axially symmetric spacetime can be represented with a metric in cylindrical coordinates in the canonical form as

ds2=e2Udt2+A2+e2KUdρ2+dz2+e2UW2dφ2E6

where all the metric functions depend on the coordinates ρand z. We shall consider the field content described by Maxwell field such that the energy-momentum tensor is

Tμν=2FμαFνα14gμνFαβFαβE7

and we study on the potential one form which shares the symmetry of the metric (6) as

A=A0ρz+A3ρzE8

with Maxwell equation

F;νμν=0.E9

For simplicity we define BD field Eq. (2) as

Gdnμν=Gμνωϕ2ϕ,μϕ,ν12gμνϕ,αϕ,α1ϕϕ;μ;νgμνϕ2Tμνϕ=0E10

and some of the field equations of the metric (6) become

Gdnρρ+Gdnzz=2ϕW=0E11
Gdnφt=4AA02A0A3+We2U.e4UϕAW=0E12
Gdntt+Gdnφφ=2ωW2ϕ2+2ϕWWϕϕ2e4UA2+4W2ϕ22K4ϕ2W2WU+2ϕ2WWU2+2W=0E13
GdnttGdnφφ=e2UW2A02e2UA2A02A32AW.e4UϕAWe4UϕA2+W2.ϕWU.ϕW=0E14

where f=ρf,zf. The field equation of (3) becomes

2ω+3.Wϕ=0.E15

Using this result and Eq. (11) we obtain the last term in Eq. (14) vanishes except ω=32. Maxwell equation of (9) becomes

MEt=.e2UWA0+Ae2UWA3AA0=0E16
MEφ=.e2UWAA0A3=0E17

BD field and Maxwell equations for stationary, axially spacetime are more complicated to obtain exact or approximate solutions. Even for GR, some solution-generating techniques are used because of this complexity. Firstly, we introduce Ernst equations obtained from BD field equations.

2.2 Ernst equations

The original Ernst equations in GR with the presence of Maxwell field are;

ε+ε+Φ22ε=2ε+2ΦΦεE18
ε+ε+Φ22Φ=2ε+2ΦΦΦE19

where εand Φredefine potentials. Some exact solutions are obtained, integrating these equations. Our aim is to obtain BD-Maxwell field equations in this form. First, we rewrite the metric Ansatz:

ds2=αeΩ/2dt+A2+αeΩ/2dφ2+e2ναdρ2+dz2E20

which simplifies the forthcoming equations considerably. Metric (6) and metric (20) relations are given:

W=α,U=Ω4+12lnα,K=Ω4+14lnα+ν.E21

Defining an operator as f=ρf,zf, BD-Maxwell field equations become

EEρρ+EEzz=2ϕα=0E22
EEtφ=12.αϕeΩA2eΩ/2μ0AA02A0.A3=0E23
EEtt+EEφφ=18αϕΩ2+2αϕ2ν12αϕeΩA2122αϕ+α322ϕ+mϕϕ2=0E24
EEttEEφφ=2μ0eΩ/2A02A32eΩ/2A0212.αϕΩ+eΩαϕA2+A.eΩαϕA=0E25
3+2ω.αϕ=0E26

From the equation of (9), we can write Maxwell equations as

Et=eΩ/2A0+AeΩ/2A3AA0=0E27
Eφ=eΩ/2A3AA0=0E28

With a new potential from the last Eq. (28),

eφ×A˜3=eΩ/2A3AA0,
eφ×eφ×A˜3=eΩ/2eφ×A3Aeφ×A0,
.eφ×A3=.Aeφ×A0eΩ/2A˜3=0E29

With the new potential, the other Maxwell Eq. (27) is written:

eΩ/2A0+Aeφ×A˜3=0.E30

We define a new complex potential

Φ=A0+iA˜3E31

and Maxwell equations become

.eΩ/2ΦiAeφ×Φ=0E32

where the real part is equal to Eq. (27) and the complex part is equal to (28).

Eq. (23) is written in the form

.eΩϕαA2eφ×ImΦΦ=0E33

where 1/μ0=1. From the last equation, we can define a new potential like

eφ×h=eΩϕαA2eφ×ImΦΦ
eφ×eφ×h=eΩϕαeφ×A2eφ×eφ×ImΦΦ
eφ×A=eΩϕαh+2ImΦΦE34

and the equation of (23) becomes

.eΩϕαh+2ImΦΦ=0.E35

A new function is defined

f=eΩ/2αϕE36

and (25) is obtained:

fαϕ.αϕff.ff2αϕ2αϕ=2fΦ.Φh+2ImΦΦ2E37

If one introduces the complex function

ε=fΦ2+ihE38

field equations of (23) and (25) and Maxwell Eqs. (27) and (28) can be writen as

ε+ε+Φ21αϕ.αϕε=ε+2ΦΦ.ε+Re2ε+Φ22αϕαϕE39
ε+ε+Φ21αϕ.αϕΦ=ε+2ΦΦ.Φ.E40

The last term of Eq. (39) becomes zero from the field equation of (22). Additionally, this field equation permits us to choose αϕ=ρwhich reduces the field equations to Ernst equations of (18) and (19). Besides, BD-Maxwell Ernst equations do not include the field equations of (24) and (26). For obtaining solutions, the chosen appropriate physical potentials for εand Φwhich satisfy the Ernst equations are not sufficent. They must also satisfy the field equations of (24) and (26). For that reason, we introduce another method for solution of BD-Maxwell equations of stationary, axially symmetric spacetimes.

2.3 Extended Tiwari-Nayak-Rai-Singh method

In this subsection, we try to analyze how to obtain BD-Maxwell solution from a known Einstien-Maxwell solution for stationary, axially symmetric spacetime. We start with writing a metric as

ds2=e2Uedt+Ae2+e2Ke2Uedρ2+dz2+We2e2Uedφ2E41

where the subscript refers to Einstein metric functions. The first of the field equations in GR is obtained:

Gρρ+Gzz=2We=0,E42
Gdtϕ=Wee4UeAeWe+2e2UeA0A3AeA02.E43

When we choose the Einstein and BD field share the same Maxwell field which means A0and A3are the same for GR and BD, Eqs. (11) and (12) become more similar to Eqs. (42) and (43). From the first equations, we can write We=ϕW, and it permit us to choose W=Wekand ϕ=W1k. In the next step, we take the metric function Aof BD and Aeof GR are the same which satisfy the relation between U=Ue12lnϕ. Finally, by using the equation Gdntt+GdnφφGdtt+Gdφφ=0, we obtain K=Ke+142ω1k2ω+3lnϕ. Previous studies took the metric functions Ke=Kwhere k=2ω12ω+3[15]. We can summarize that, if there is a GR solution in the form of metric (6), it has a corresponding BD solution with transformation of the metric function as

A=Ae,W=Wek,ϕ=We1k,E44
U=Ue12lnϕ,E45
K=Ke+142ω1k2ω+3lnϕ.E46

Additionally, if the seed solution is in the form of Eq. (20), the corresponding BD solution may be obtained by the transformation as

α=αe,ϕ=αe1k,Ω=ΩeE47
A=Aeν=νe+2ωk2ω+34lnϕ.E48

3. Examples of BD solutions with extended TNSR method

3.1 BD solution of Kerr-Taub-NUT metric

We also know Kerr-Taub-NUT (KTN) vacuum solution is

ds2=1ρ2Δa2sin2θdt2+2ρ2Δαaρ2+sin2θdtdϕ+1ρ2ρ2+2sin2θdϕ2+ρ2dr2Δ+dθ2E49

where

Δ=r22Mr+a2n2,ρ2=r2+n+acosθ2α=asin2θ2ncosθ

and nis NUT parameter, ais rotational parameter, and Mis the mass. By doing a coordinate transformation like r=eR+M+M2a2+n24eR, Kerr-Taub-NUT metric becomes

ds2=1ρ2Δa2sin2θdt2+2ρ2Δαaρ2+sin2θdtdϕ+1ρ2ρ2+2sin2θdϕ2+ρ2dR2+dθ2.E50

which is similar to metric (6) and the functions are

Δ=L22ML+a2n2,ρ2=L2+n+acosθ2

where L=eR+M+M2a2+n24eR. From this metric we can easily find that

e2Ue=1ρ2Δa2sin2θAe=aρ2+2sin2θΔαΔa2sin2θe2Ke=Δa2sin2θWe=Δsinθ

From Eqs. (44) to (46) and by doing a coordinate transformation again, the solution looks

ds2=Δsinθk1[Δa2sin2θρ2dt22ρ2aρ2+sin2θΔαdtdϕ+ρ2+2sin2θΔα2dϕ2]+Δsinθ1k21+2ωk2ω+3ρ2dr2Δ+ρ2dθ2E51

3.2 BD solution of Kerr-Newman-Taub-NUT metric

The solution for Kerr-Newman-Taub-NUT (KNTN) metric is the same as Eq. (49), but the functions are

Δ=r22Mr+a2n2+Q2,ρ2=r2+n+acosθ2α=asin2θ2ncosθ

which correspond to mass of KTN solution which is M=MQ2/2r. By doing the same procedure for KTN solution, we can easily obtain KNTN solution, but there are some differences such as for metric transformation, r=eR+M+M2a2+n2Q24eRis used. Brans-Dicke solution of KNTN becomes

ds2=Δsinθk1[Δa2sin2θρ2dt22ρ2aρ2+sin2θΔαdtdϕ+ρ2+2sin2θΔα2dϕ2]+Δsinθ1k21+2ωk2ω+3ρ2dr2Δ+ρ2dθ2.E52

This solution was obtained by a sigma-model theory in [18] where the parameters have a relation like α=1k2ω+3/4. Moreover, analyses of the charged particle geodesics around this spacetime were discussed in [17] analytically.

3.3 BD solution of magnetized Kerr-Newman solution

Magnetized Kerr-Newman solution was found by Gibbons and his friends [19] as

ds2=Hfdt2+Σdr2Δ+dθ2+Ssin2θHΣβdt2E53
A=Φ0dt+Φ3βdt.E54

where

Σ=r2+a2cos2θΔ=r2+a22mr+p2+q2f=ΔΣSS=r2+a22a2Δsin2θβ=a2mrp2q2SE55

where ais rotational parameter, pis magnetic charge, and qis electric charge of rotating black hole. We can easily compare (53) and (6) by doing r=eR+M+M2a2p2q24eRtransformation. GR metric functions become

e2Ue=H2fΣSβ2sin2θHΣAe=sin2θH2fΣSβ2sin2θE56
We=Δsinθe2Ke=H2fΣSβ2sin2θ.E57

where

Σ=L2+a2cos2θΔ=L2+a22mL+p2+q2f=ΔΣSS=L2+a22a2Δsin2θβ=a2mLp2q2SE58

where L=eR+M+M2a2p2q24eR. Metric functions of BD solution of magnetized KN are obtained from Eqs. (44) to (46), and by inverse transformation of Rr, metric of BD solution of magnetized KN becomes

ds2=Δsinθk1Hf+Sβ2sin2θHΣdt22sin2θHΣdtdφ+Ssin2θHΣdφ2Δsinθ1k21+2ωk2ω+3HΣdr2Δ+dθ2.E59

4. GR limit of the solutions

According to the common belief, since BD parameter ω, the BD solutions reduce to corresponding GR ones. Contrary to this belief, several counter examples were presented in the literature [20, 21, 22, 23, 24]. In our study, the GR limit of the BD solutions is out of complexity. When k1, BD transformation equations of (44)(46) reduce to seed GR metric functions for any finite ωsince scalar field ϕbecomes constant. It is obvious from the given examples that, as k1, BD metrics reduce the corresponding GR ones.

5. Conclusion

In this section, we have studied to obtain corresponding BD or BD-Maxwell solution from any known solution of the Einstein or the Einstein-Maxwell theory for stationary, axially symmetric spacetimes in Jordan frame. First we present that, although several field equations of BD are not included by Ernst equations, BD field equations can be written in the form of Ernst Eqs. BD solutions can be obtained by selecting the appropriate physical potentials or by integrating Ernst equations, but it should be remembered that the equations which are not included in the Ernst equations should be provided.

In order to obtain BD solutions, we have constructed two parameter solution-generating techniques. It was seen that, in previous works, it was studied with one parameter. From any given seed GR solution of Eqs. (6) or (20), the corresponding BD solution can be obtained by the two parameter solution-generating techniques. In order to show how this method works, we have constructed several known solutions and also some new solutions for BD theory. We have also discussed the GR limit of these solutions.

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Pınar Kirezli Uludağ (March 5th 2020). Brans-Dicke Solutions of Stationary, Axially Symmetric Spacetimes [Online First], IntechOpen, DOI: 10.5772/intechopen.89906. Available from:

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