Application to the scalar-tensor theories.

The general theory developed in the above sections is valid for any metric theory of gravity. Let us now examine the implications of Eqs.(65) and (66) within the scalar-tensor theories of gravity.

The class of theories that we consider here is described by the action

where R is the Ricci scalar curvature ($R=g^{\mu \nu}R_{\mu \nu}$), $\Phi$ is the scalar gravitational field, g is the determinant of the metric components $g_{\mu \nu}$, $\omega(\Phi)$ is an arbitrary function of the scalar field $\Phi$, and ${\cal{J}}_{m}$ is the matter action. We assume that ${\cal{J}}_{m}$ is a functional of the metric and of the matter fields $\psi_m$ only. This means that ${\cal{J}}_{m}$ does not depend explicitly upon the scalar field $\Phi$(it is the assumption of universal coupling between matter and metric).

We consider here the weak-field approximation only. So we assume that the scalar field $\Phi$ is of the form

$$\Phi = \Phi_0 + \phi$$ (69)

where $\phi$ is a first order perturbation of an averaged constant value $\Phi_0$. Consequently, the field equations deduced from (74) reduce to the following system:

$$R^{(1)}_{\mu \nu} = 8 \pi \Phi^{-1}_0 (T^{(0)}_{\mu \nu} -\f... ...0^{-1} (\phi_{,\mu \nu} + \frac{1}{2} \Box \phi \eta_{\mu \nu})$$ (70)

$$\Box \phi = \frac{8 \pi}{2\omega (\Phi_0) + 3} T^{(0)}$$ (71)

where $T^{(0)}_{\mu \nu}$ is the energy-momentum tensor of the matter fields $\psi_m$ at the lowest order, $T^{(0)}=\eta^{\alpha \beta}T^{(0)}_{\alpha \beta}$ and $\Box$ denotes the d'Alembertian operator on Minkowski spacetime: $\Box \phi = \eta^{\alpha \beta} \phi_{,\alpha \beta}$.

It is easily seen that any solution $h_{\mu \nu}$ to the field equations (76) is given by

$$h_{\mu \nu} = h^{E}_{\mu \nu} - \frac{\phi}{\Phi_0} \eta_{\mu \nu}$$ (72)

where $h^{E}_{\mu \nu}$ is a solution to the equations

$$R^{(1)}_{\mu \nu}(h^{E}_{\alpha \beta}) = 8 \pi \Phi^{-1}_0 (T^{(0)}_{\mu \nu} - \frac{1}{2} T^{(0)}\eta_{\mu \nu})$$ (73)

which are simply the linearized Einstein equations with an effective gravitational constant $G_{eff} = c^4 \Phi_0^{-1}$. Indeed, inserting (78) in (67) yields the following expression for the curvature tensor

from which one deduces the Ricci tensor

$$R^{(1)}_{\mu \nu}(h_{\alpha \beta}) = R^{(1)}_{\mu \nu}(h^{E... ...0^{-1} (\phi_{,\mu \nu} + \frac{1}{2} \eta_{\mu \nu} \Box \phi)$$ (74)

Then substituting for $R^{(1)}_{\mu \nu}(h_{\alpha \beta})$ from its expression (81) into the field equations (76) gives Eqs.(79), thus proving the proposition.

The decomposition (78) of the gravitational perturbation $h_{\mu \nu}$implies that each term contributing to $\dot{{\cal{N}}}/{\cal{N}}$ can be split into a part built from the Einsteinian perturbation $h_{\mu \nu}^{E}$ only and into an other part built from the scalar field $\phi$ alone. In what follows, we use the superscript ST for a functional of a solution $(h_{\mu \nu},\phi)$ to the field equations (76)-(77) and the superscript E for the same kind of functional evaluated only with the corresponding solution $h_{\mu \nu}^{E}$.

In order to perform the calculation of the integrals (65) and (66), we note that $K^{\mu} F_{,\mu}$ is the usual total derivative of the quantity F along the unperturbed ray path, which implies that

$$\int \limits_{-\infty}^{v_{obs}} K^{\mu} F_{,\mu} dv = F_{obs} - F_{(-\infty)}$$ (75)

The 4-vector $K^{\mu}$ (supposed here to be future oriented, i.e such that K⁰ > 0) gives the direction of propagation of the light coming from the observed source. For a given observer moving with the 4-velocity $U^{\mu}$, let us put

$$N^{\mu} = (\eta^{\mu \nu} - U^{\mu} U^{\nu}) \frac{K_{\nu}}{... ...\lambda})} = \frac{K^{\mu}} {(U^{\lambda}K_{\lambda})}- U^{\mu}$$ (76)

We have $\eta^{\mu \nu} N_{\mu}N_{\nu} = -1$. Since $N^{\mu}$ is orthogonal to $U^{\mu}$ by construction, $N^{\mu}$ can be identified to the unit 3-vector $\vec{N}$ giving the direction of propagation of the light ray in the usual 3-space of the observer.

Using (76), (77), (80) and the assumption $\phi_{(-\infty)} = 0$, it is easily seen that (65) and (66) can be respectively written as

$$\left. 2\frac{\dot{a}}{a} \right\vert^{ST}_{obs} = \left. 2... ...^{E}_{obs} + \left. \frac{\dot{\phi}}{\Phi_0}\right\vert _{obs}$$ (77)

and

where

$$\dot{\phi} = U^{\mu} \phi_{,\mu}$$ (78)

and

$$\vec{N}.\vec{\nabla} \phi=N^{\mu}\phi_{,\mu}$$ (79)

As a consequence, the rate of variation in the photon flux as received by the observer is given by the general formula

$$\left.\frac{\dot{{\cal{N}}}}{{\cal{N}}}\right\vert^{ST}_{obs... ...ac{1}{2 \Phi_0}(3 \dot{\phi} - \vec{N}.\vec{\nabla} \phi)_{obs}$$ (80)

In a vacuum $(T^{(0)}_{\mu \nu} = 0)$, the metric $h^{E}_{\mu \nu}$satisfies the linearized Einstein field equations (71) and Eq.(84) reduces to

$$2\left.\frac{\dot{a}}{a} \right\vert^{ST}_{obs} = \left. \frac{\dot{\phi}}{\Phi_0}\right\vert _{obs}$$ (81)

In Eq.(88), $({\dot{\cal{N}}}/{\cal{N}})^E_{obs}$ is reduced to the term given by Eq.(73), where $R^{(1)}_{\mu \rho \nu \sigma}$ is constructed with $h^{E}_{\mu \nu}$.

It follows from (89) that contrary to general relativity, the scalar-tensor theories (defined by (74)) predict the existence of a first-order geometrical scintillation effect produced by gravitational waves. This effect is proportional to the amplitude of the scalar perturbation. It should be noted that an effect of the same order of magnitude is also due to the change in the spectral shift.

To finish, let us briefly examine the case where the scalar wave $\phi$ is locally plane (it is a reasonable assumption if the source of gravitational wave is far from the observer). Thus we can put in the vicinity of the observer located at the point x_obs

$$\phi = \phi(u)$$ (82)

where u is a phase function which admits the expansion

$$u(x) = u(x_{obs}) + L_{\mu}(x^{\mu}-x^{\mu}_{obs}) + O(\vert x^{\mu}-x^{\mu}_{obs}\vert^2)$$ (83)

with

$$L_{\mu} = const.$$ (84)

It follows from Eq.(77) with T⁽⁰⁾ = 0 that $L_{\mu}$ is a null vector of Minkowski spacetime.

Replacing $K_{\mu}$ by $L_{\mu}$ in (83) defines the spacelike vector $P^{\mu}$, which can be identified with the unit 3-vector $\vec{P}$ giving the direction of propagation of the scalar wave in the 3-space of the observer. Then introducing the angle $\theta$ between $\vec{N}$ and $\vec{P}$, a simple calculation yields

$$\left.\frac{\dot{{\cal{N}}}}{{\cal{N}}}\right\vert^{ST}_{obs... ...\theta}{2}) \left. \frac{\dot{\phi}}{\Phi_0} \right\vert _{obs}$$ (85)

This formula shows that the contribution of the scalar wave to the scintillation cannot be zero, whatever be the direction of observation of the distant light source.