## 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

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

where

In many cases and origin of the BD theory, BD coupling *ϕ*

where the repeated index

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

Furthermore, we set BD coupling

### 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

where all the metric functions depend on the coordinates

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

with Maxwell equation

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

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

where

Using this result and Eq. (11) we obtain the last term in Eq. (14) vanishes except

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;

where

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

Defining an operator as

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

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

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

We define a new complex potential

and Maxwell equations become

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

Eq. (23) is written in the form

where

and the equation of (23) becomes

A new function is defined

and (25) is obtained:

If one introduces the complex function

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

The last term of Eq. (39) becomes zero from the field equation of (22). Additionally, this field equation permits us to choose

### 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

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

When we choose the Einstein and BD field share the same Maxwell field which means

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

## 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

where

and

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

where

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

### 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

which correspond to mass of KTN solution which is

This solution was obtained by a sigma-model theory in [18] where the parameters have a relation like

### 3.3 BD solution of magnetized Kerr-Newman solution

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

where

where

where

where

## 4. GR limit of the solutions

According to the common belief, since BD parameter

## 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.