The description of gravitational waves in geometric scalar gravity
Abstract
It is investigated the gravitational waves phenomena in the geometric scalar theory of gravity (GSG), a class of theories such that gravity is described by a single scalar field. The associated physical metric describing the spacetime is constructed from a disformal transformation of Minkowski geometry. In this theory, a weak field approximation gives rise to a description similar to that one obtained in general relativity, with the gravitational waves propagating at the same speed as the light, although they have a characteristic longitudinal polarization mode, besides others modes that are observer dependent. We also analyze the energy carried by the gravitational waves as well as how their emission affects the orbital period of a binary system. Observational data coming from Hulse and Taylor binary pulsar is then used to constraint the theory parameter.
1 Introduction
Although general relativity (GR) has been a very successful gravitational theory during the last century, many proposals for modification of Einstein original formulation appeared in the literature over the past decades. Most of these ideas come up within the cosmological scenario, where GR only works if unknown components, like dark matter or dark energy, are introduced. Such alternative descriptions are basically variations of Einstein’s theory, either assuming most general Lagrangians for the gravitational field or adding new fields together with the metric.
A complete theory can only be set if one defines the functions A and B, and also the Lagrangian of the scalar field. Then, a field equation, characterizing the theory, can be derived. We refer to this class of gravitational theories as geometric scalar gravity (GSG). In early communications on GSG, it was explored a specific set of those functions defining the theory, which shows that it is possible to go further in representing the gravitational field as a single scalar, giving realistic descriptions of the solar system and cosmology [1, 7]. An analysis of GSG within the so called parametrized postNewtonian formalism was also made and, although the theory is not covered by the formalism, a limited situation indicate a good agreement with the observations [8]. Intending to improve the understanding of how GSG deals with gravitational interaction, the present work develops the theoretical description and characterization of gravitational waves (GW).
The direct detections of GW by LIGO and Virgo collaborations initiated a new era of testing gravitational theories. It enables to construct constraints over a series of theoretical mechanisms associated with GW’s physics, but crucial point relies on the observed waveform and how a theory can reproduce it [9]. Notwithstanding, this is not the scope of this work. We are mainly focused in analyzing the GW fundamentals on the perspective of GSG, studying their propagation, polarization modes and defining an appropriated tensor to describe the energy and momentum carried by the waves. The velocity of propagation of GW has been measured with good precision indeed, but this data does not constraint GSG once the gravitational signal travels in vacuum with the same speed of light, as it will be shown later. However, the theory can be constrained by observational data from pulsars through its prediction for the orbital variation of a binary system that should be caused by the loss of energy due to gravitational radiation.
The paper is organized as follows. In Sect. 2 is presented a brief overview of GSG in order to introduce to the reader the main features of this theory. The following section describes the theory’s weak field approximation. In Sect. 4 the study of the propagation and vibration modes associated to gravitational waves is made. The definition of a energymomentum tensor for the linear waves is treated in Sect. 5. Generation of waves, including the computation of the orbital variation of binary systems due to the emission of GW, is discussed in Sect. 6 and the last section presents our concluding remarks. Also, two appendices were introduced in order to complement the middle steps of calculations present in Sect. 6.
2 Overview of geometric scalar gravity
3 Weak field approximation
3.1 The cosmological backgroung
4 Propagation and polarization of gravitational waves
4.1 Polarization states
The most general (weak) gravitational wave that any metric theory of gravity is able to predict can contain six modes of polarization. Considering plane null waves propagating in a given direction, these modes are related to tetrad components of the irreducible parts of the Riemann tensor, or the Newmann–Penrose quantities (NPQ): \(\varPsi _2, \varPsi _3, \varPsi _4\) and \(\varPhi _{22}\) (\(\varPsi _3\) and \(\varPsi _4\) are complex quantities and each one represents two modes of polarization) [12]. The others NPQ are negligible by the weak field approximation, or are described in terms of these four ones.

\(\textit{Class}\,\textit{II}_6\) If \(\varPsi _2 \ne 0\) , all the standard observers agree with the same nonzero \(\varPsi _2\) mode, but the presence or absence of the other modes is observerdependent.

\(\textit{Class}\,\textit{III}_5\) If \(\varPsi _2=0\) and \(\varPsi _3\ne 0\) , all the standard observers measure the absence of \(\varPsi _2\) and the presence of \(\varPsi _3\), but the presence or absence of all other modes is observer dependent.

\(\textit{Class}\,\textit{N}_3\) If \(\varPsi _2=\varPsi _3=0\,, \varPsi _4\ne 0\) and \(\varPhi _{22}\ne 0\) , this configuration is independent of observer.

\(\textit{Class}\,\textit{N}_2\) If \(\varPsi _2=\varPsi _3=\varPsi _2=0\) and \(\varPsi _4\ne 0\) , this configuration is independent of observer.

\(\textit{Class}\,\textit{O}_1\) If \(\varPsi _2=\varPsi _3=\varPsi _4=0\) and \(\varPhi _{22}\ne 0\) , this configuration is independent of observer.

\(\textit{Class}\,\textit{O}_0\) If \(\varPsi _2=\varPsi _3=\varPsi _4=\varPhi _{22}=0\) , this configuration is independent of observer.
Thus, the description of GW by GSG carries a substancial distinction from GR, as it predicts the presence of a longitudinal polarization mode. Up to now, the recent detections of GW can not exclude the existence of any one of the six modes of polarization [17, 18]. But in the future, with the appropriated network of detectors, with different orientations, this information can be used to restrict gravitational theories.
5 Energy of the gravitational wave
6 Orbital variation of a binary system
Data from PSR 1913+16 [23] adapted to the present notation
Parameter (units)  Value 

e  0.6171340(4) 
\(m_{1}\) (solar masses)  1.438(1) 
\(m_{2}\) (solar masses)  1.390(1) 
T (days)  0.32997448918(3) 
\(\dot{T}\)  \(2.398 (4)\times 10^{12}\) 
It is worth to note that the orbital parameters of the binary system are extracted from the timing pulsar observations in a theoryindependent way, but the determination of the masses of the pulsar and its companion are model dependent [24]. The mass values in Table 1 are from GR but its usage here is reasonable due to the satisfactory agreement of GSG in the Solar System tests at the postNewtonian level. However, any modification on these values will lead to a distinct estimation of \(\lambda \) but not an invalidation of GSG by pulsars data.
7 Concluding remarks
We have presented a discussion on gravitational waves (GW) in the context of the geometric scalar gravity (GSG), a class of theories describing the effects of gravity as a consequence of a modification of spacetime metric in terms of a single scalar field. GSG overcomes the serious drawbacks present in all previous attempts to formulate a scalar theory of gravity. Its fundamental idea rests on the proposal that the geometrical structure of the spacetime is described by a disformal transformation of a conformal flat metric. The model analyzed here has already showed several advances within the scalar gravity program, featuring a good representation of the gravitational phenomena both in the solar system domains as well as in cosmology.
Initially it was shown the procedure used to construct the weak field limit in GSG considering an expansion of the scalar field over a background cosmological solution. Within this approximation scheme the scalar dynamical equations assumes oscillatory solutions that represent GW in the spacetime structure propagating with light velocity, which is in agreement with recent data from GW and gammaray burst detections from the merge of a binary neutron star system.
An important distinction appears in the polarization states of the waves, which is characterized by the presence of a longitudinal mode in GSG. Within the E(2)classification of gravitational theories, GSG is of the type \(II _{6}\), since \(\varPsi _{2}\ne 0\). This is the most general class, where the detection of all the other five polarization modes depend on the observer. The detection of extra polarization states (or the absence of them) shall be a decisive test to alternative theories of gravity [25]. Modelindependent tools that allow to see how polarization modes affect the response function in GW detectors has been recently developed and must be also applied in GSG [26]. This procedure should be considered in the future.
It was also discussed how to define an energymomentum tensor for the linear GW, following a field theoretical point of view. An ambiguity emerges since GSG fundamental equation includes a non trivial interaction between matter/energy and the scalar field, leading to nonunique expression for the approximated gravitational energymomentum tensor. This freedom is encoded in the constant parameter \(\lambda \), which has directly influence in the energyloss rate when emitting gravitational waves. Consequently, GSG prediction for the orbital variation of a binary system can be used to constraint the theory’s parameter with observational data coming from PSR 1913+16. This numerical computation was performed using GR mass values as a first estimation since GSG is in agreement with classical tests and should not present strong deviations on these values. It is then expected that, after analyzing the so called postKeplerian parametrization of the theory to extract the mass values of a binary system according to GSG,^{4} the theory can be more properly constrained. This task will be addressed in a future work.
Footnotes
 1.
The nonlinearity of the field must be specifically in the kinetic term of the Lagrangian, namely w. Thus, the Lagrangian density of the scalar field can be described as \(L=F_1(\varPhi ,w)w+F_2(\varPhi )\), with the condition that \(F_1\) can not be a constant.
 2.
The index \(_0\) in the Hubble parameter is used to indicate a background quantity.
 3.
Although symbol R was previously used as the Ricci scalar in Sect. 4, we draw the reader’s attention that in this section it is indicating the point where the gravitational field is being calculated.
 4.
A phenomenological parametrization for binary pulsars introduced by Damour [27], where the Keplerian and postKeplerian parameters can be read off.
Notes
Acknowledgements
I wish to thank J.C. Fabris and T.R.P. Caramês for dedicated reviews of this work. This research is supported by FAPES and CAPES through the PROFIX program.
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