A geometrized scalar-tensor theory of gravitation

A very interesting class of generalizations of Einstein's theory is based on the replacement of the gravitational constant $G$ by a variable scalar field [69,70,71]. A motivation for this assumption is the explanation of the extremely small value of $G$ when measured in atomic units (Dirac's large numbers problem). According to these theories, the value of the scalar field that replaces $G$ is determined by the distribution of matter in the universe, in agreement with the ideas of Mach [103] on the influence of very far celestial bodies on the locally observed phenomena. The best known Lagrangian scalar-tensor theory of this kind has been proposed by Brans and Dicke [71] and theories including torsion have been discussed in ref. [104].

We think that it is important to give to this scalar field a geometric interpretation, by writing it as a function of the structure coefficients, in agreement with the idea that all the structure coefficients should have a dynamical relevance. In the discussion of this problem we also clarify some concepts introduced in Chapters 2 and 3 and we introduce some ideas useful for the developments of Chapter 7.

We have seen in Section 5.2 that one can include the coupling constants of a Yang-Mills theory in the structure constants $F_{ab}^c$ of the gauge group. If the coupling constant becomes a variable [105], these structure coefficients acquire a dynamical role. In a similar way, one can try to include the gravitational constant $G$, or the variable field that replaces it, in the structure constants $F_{[ik][jl]}^{[mn]}$ and $F_{[ik]j}^l$ of the Poincaré group. A theory of this kind, equivalent, if spin is neglected, to the Brans-Dicke theory [71], has been discussed in ref. [5]. Here we give a slightly different treatment.

We define the scalar field $\phi$ by means of the formula

$$\phi = (12)^{-1} g^{kj} F_{[ik]j}^i.$$ (6.1)

Note that if $F_{[ik]j}^i = \hat F_{[ik]j}^i$, we obtain $\phi = 1$. Since we want to preserve the large amount of physical information contained in Einstein's theory, we consider a minimal modification of the Lagrangian form given by eqs. (5.1) and (5.2). First of all, we replace the constants $k, k_1,\ldots, k_4$ by suitable functions of $\phi$ indicated by the same symbols. Note that adding a constant to the functions $k_1$ or $k_2$ results in adding to the Lagrangian an irrelevant exact form, that leaves the field equations unchanged.

Moreover, we have to add new terms to the Lagrangian, in order to obtain the field equations that determine the field $\phi$ starting from the matter distribution in the universe and from suitable initial conditions. It seems that these new terms should contain the derivatives $A_i \phi$, but we want to avoid the appearance in the Lagrangian of derivatives of the structure coefficients. We try to use for the same purpose suitable functions of the structure coefficients, increasing in this way their physical role, in agreement with our program.

The structure coefficients that appear in the forms $d \omega^{\alpha}$ are not sufficient and we have to introduce some other expressions $\chi_i$. After several attempts, one finds that it is convenient to put

$$\chi_i = F_{ik}^k$$ (6.2)

and to add to the Lagrangian the expression

$$\lambda^{\chi} = k_5 \chi_i \epsilon^i{}_{kjl} d\omega^k \wedge \ome... ...^i{}_{kjl} g_{mn} \omega^{[km]} \wedge \omega^n \wedge \omega^j \wedge \omega^l$$

$$+ k_5 \chi_i \chi^i \eta, \qquad k_5 \neq 0,$$ (6.3)

which contributes to both $\lambda^H$ and $\lambda^A$. It may look strange to involve the torsion tensor in the description of the variable gravitational coupling, but we shall see that this procedure works.

The derivatives of $\lambda$ that appear in the normal field equation (4.28) contain two contributions, one coming from the forms $d \omega^{\alpha}$ and the other originated by the dependence on the quantities $\phi$ and $\chi_i$, namely we have

$$\frac{\partial \lambda}{\partial F_{\epsilon \zeta}^{\eta}} ... ...} \frac{\partial \chi_i}{\partial F_{\epsilon \zeta}^{\eta}},$$ (6.4)

where

$$\sigma_{[ik]} = 2 (k - k_1) \epsilon_{ikjl} \, \omega^j \wedge \omega^l - 2 k_2 \epsilon_{ikjn} g_{lm} \omega^{[jl]} \wedge \omega^{[mn]},$$

$$\sigma_i = 2k_1 \epsilon_{ikjl} \, \omega^{[kj]} \wedge \omega^l + k_5 \chi_k \epsilon^k{}_{ijl} \omega^j \wedge \omega^l.$$ (6.5)

The first contribution satisfies the normal field equation automatically and the second contribution gives the condition

$$\left( \frac{\partial \phi}{\partial F_{\epsilon \zeta}^{\et... ... \right) \wedge \frac{\partial \lambda}{\partial \chi_i} = 0.$$ (6.6)

By using the definitions of $\phi$ and $\chi_i$ and choosing the indices $\epsilon, \zeta, \eta, \theta$ in a proper way, we obtain, for all the values of $[jk]$

$$\omega^{[jk]} \wedge \frac{\partial \lambda}{\partial \phi} ... ...ga^{[jk]} \wedge \frac{\partial \lambda}{\partial \chi_i} = 0.$$ (6.7)

For a given value of $[jk]$, the condition $\omega^{[jk]} \wedge \alpha = 0$ implies that the form $\alpha$ is proportional to $\omega^{[jk]}$. If we require that this condition holds for all the six values of $[ik]$, either $\alpha$ vanishes or it is a form of degree not smaller than 6. In conclusion, we obtain the conditions

$$\frac{\partial \lambda}{\partial \phi} = 0,$$ (6.8)

$$\frac{\partial \lambda}{\partial \chi_i} = 0,$$ (6.9)

that are equivalent to the normal field equations.

We assume that $\lambda^M$ does not depend on $\chi_i$ and that its dependence on $\phi$ is given by eq. (5.45). The normal equation (6.9) involves only the additional Lagrangian $\lambda^{\chi}$ and it is equivalent to the conditions

$$F_{[ik]j}^l =\phi \hat F_{[ik]j}^l, \qquad F_{[ik][jl]}^m = 0.$$ (6.10)

In order to obtain simpler and more consistent results, we choose the functions $k_1$, $k_2$ and $k_3$ in such a way that

$$k'_3 = \phi (k' + k'_1), \qquad k'_2 = 0, \qquad k' - k'_1 \neq 0,$$ (6.11)

where $k'$ indicates the derivative of $k$ with respect to $\phi$. By taking into account the formula (5.45), the normal equation (6.8) gives the conditions

$$F_{[jl][mn]}^{[ik]} = \phi \hat F_{[jl][mn]}^{[ik]},$$ (6.12)

$$2 (k' - k'_1) F_{ik}^{[ik]} + k'_4 + k'_5 \chi_i \chi^i = 0,$$ (6.13)

$$4 k'_1 \phi (F_{jk}^i + \delta^i_j F_{kl}^l - \delta^i_k F_{... ...elta^i_j) - 4 (k' - k'_1) \phi F_{l[jk]}^{[il]} = T_{[jk]}^i.$$ (6.14)

From eqs. (6.10) and (6.12) and the special case $J^i_{[jk][mn]l} = 0$ of the generalized Jacobi identity (1.55), we see that

$$A_{[ik]} \phi = 0,$$ (6.15)

namely that $\phi$ is indeed a scalar field.

The eqs. (5.5), (5.6) and (5.7) are still valid and the explicit equations (5.8) and (5.9) contain additional terms proportional to $k_5$ and to the derivatives of $\phi$ and $\chi_i$. By performing the calculations, if the 10-momentum of matter has the local form (4.68), from the vertical tangential equation we obtain first of all the condition

$$4 k (F_{jk}^i + \delta^i_j F_{kl}^l - \delta^i_k F_{jl}^l) - 2 k_5 \phi (\chi_j \delta^i_k - \chi_k \delta^i_j)$$

$$- 4 (k' - k'_1)(A_k \phi \delta^i_j - A_j \phi \delta^i_k) = T^i_{[jk]},$$ (6.16)

which is compatible with the normal equation (6.14) only if

$$k_1' \phi = k,$$ (6.17)

$$A_k \phi \delta^i_j - A_j \phi \delta^i_k = \phi F_{l[jk]}^{[il]}.$$ (6.18)

Continuing the analysis of the vertical tangential equation, we see that it is compatible with eq. (6.10) only if we choose

$$k_3 = \phi k.$$ (6.19)

If we take into account all the conditions found up to now, from the horizontal tangential equation we obtain

$$\phi F_{i[jk]}^{[pq]} = A_i \phi (\delta^p_j \delta^q_k - \delta^q_j \delta^p_k),$$ (6.20)

$$4 k (- F_{jk}^{[ik]} + 2^{-1} \delta^i_j F_{lk}^{[lk]}) - 2 k'_5 (\chi^i A_j \phi - \chi^k A_k \phi \delta^i_j)$$

$$- 2 k_5 (A_j \chi^i - A_k \chi^k \delta^i_j) + (k_4 - k_5 \chi_k \chi^k) \delta^i_j = T^i_j$$ (6.21)

and eq. (6.18) follows as a consequence.

By using the conditions (6.10), from the generalized Jacobi identity (1.55) we obtain the formula

$$J^i_{[jk]li} = A_{[jk]} \chi_l - F_{[jk]l}^i \chi_i + A_j \phi g_{lk} - A_k \phi g_{lj} + \phi F_{q[jk]}^{[pq]} g_{lp} = 0,$$ (6.22)

which combined with eq. (6.18) gives the condition

$$A_{[jk]} \chi_l = F_{[jk]l}^i \chi_i,$$ (6.23)

showing that $\chi_l$ is a vector field.

Note that, if we put $k_5 = 0$ from eqs. (6.13) and (6.21) we have the condition

$$T_i^i = 4k_4 - 2 k k'_4 (k' - k'_1),$$ (6.24)

which is too restrictive, at least if massive particles are present. This means that the introduction of the term (6.3), proportional to $k_5$, in the Lagrangian cannot be avoided

Attempts to generalize the above discussed model by introducing anisotropic features of the gravitational field are discussed in ref. [108].