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III. An Account of Inertia

We are now talking of gravitation in the framework of the General Theory of Relativity. Here, for the first time in history, the field concept, inasmuch as omnipresence and permanence of this concept are considered, has been submitted by Einstein to a logical analysis based on the idea of particle, to the extent where the point of space and the moment of time characteristics of this last concept are involved. The result of this analysis is the well-known Einstein – or gravitational – field equations (Einstein, 1967) offering, in principle, the metric tensor of space-time once we know the energy tensor of matter. It turned out that it is not possible to solve the problem of gravitation in the spirit Einstein put it for the first time (Einstein, 1967), i.e. given the energy tensor to find uniquely the metric tensor. The deep reason seems simple, and has been noticed many times: the field is deprived here of what we would like to call universality condition, the universality being defined as the presence of the field within any interaction in space. More to the point, in order to find a solution for the gravitational potential (the metric) we need to solve the Einstein equations. However, these equations contain the energy tensor whose construction, while accounting for interaction properties, requires the apriori knowledge of the metric tensor. This problem has been repeatedly brought about in Theoretical Physics in a way or another and among the attempts to solve it there are a few remarkable contributions to common knowledge of the nature of the problem of gravitational field (Born, Infeld 1934, Misner, Wheeler 1957, Fock 1964, Einstein 1967, Appendix II). There are however cases where the energy tensor does not involve the metric, or vice versa. The most obvious one is the case of vacuum – free space – where the energy tensor is trivially zero. This case is of particular interest here.

    The vacuum and electromagnetic vacuum gravitational field equations have a nice solution amenable to a noticeable form in the stationary case. It was Ernst (Ernst, 1968) who first revealed this form in the axially symmetric case, and later on Israel and Wilson (Israel, Wilson, 1973) treated the general stationary case. We will follow this last work in drawing the essential equations, first because it seems a little more explicit for our purpose and, secondly, because it has apparently a fresh hint of circumventing the indeterminacy related to the metric tensor with profitable outcome. We follow nonetheless the general idea of the original Ernst’s work to connect the gravitational field problem with a variational principle. The reasons for this will become clear shortly.

    The contention of Israel and Wilson’s work (Israel, Wilson, 1973) is that for a stationary space-time metric, conveniently written in the form

(3.01)

where the summation convention over repeated alternating indices is used, the Einstein field equations for the electromagnetic field in vacuum:

(3.02)

take the form of the system of nonlinear partial differential equations

(3.03)

Here symbol Ñ denotes the gradient with respect to the 3D metric (γ). The explanation of the other symbols is now in order: the Greek indices run from 1 to 4 while Latin indices run from 1 to 3 representing Space indices. The Space-Time metric tensor is defined by

(3.04)

and the 3D metric (γ_mn) is used to raise and lower the indices in space coordinates operations. All these components do not depend on time coordinate. A potential 4 - vector (, A₄) º (A_μ) describes the electromagnetic field whose intensities are given by its covariant rotor:

(3.05)

This electromagnetic field contributes to the only energy tensor of the problem

(3.06)

and G_αβ is the Einstein tensor of the metric field defined by

(3.07)

with R_αβ the Ricci tensor of the 4D metric and R the scalar invariant of the space-time curvature. In terms of these symbols we then have

(3.08)

where Φ is a magnetic potential and f is an arbitrary function. Once we know the functions e, Φ and f we are able to construct the 3D Ricci tensor corresponding to metric (γ_mn) by

(3.09)

the round parentheses denoting symmetrization with respect to indices concerned.

    As mentioned, F. J. Ernst (Ernst, 1968) introduced the complex potential e for the special case of the axially symmetric gravitational field. It turned out later that the Space symmetry is not a necessary condition for the existence of such a potential – see especially (Ernst, 1971) – but only the stationarity of the metric field (independence of time). Even so, the solution is complicated by the fact that equation (3.09) appears to be an identity that the solution must satisfy. For our limited interest here, we only notice that the problem of gravitational field can be nicely solved if the logic is taken a little out of the usual line, in the sense that the space metric (γ) is allowed to be arbitrary, therefore apt of convenient choice. Indeed, Israel and Wilson observe (Israel, Wilson, 1973) that equations (3.09) are to be taken as compatibility conditions between a selected space metric and the complex fields e and Y. In the particular case of a flat space the compatibility conditions amount to a single linear relation

(3.10)

and the whole construction comes down to solving the Laplace equation:

(3.11)

By equation (3.10) the gravitational field determines an electromagnetic field. This electromagnetic field is however not a transition field as we usually know it, but only reflects the omnipresence and permanence of the gravitational field. In their classical work (Misner, Wheeler, 1957) Misner and Wheeler admirably captured these attributes of the gravitational field (actually of the Space itself as an acting physical entity) and studied in depth their meaning for Physics. Here we are interested in making clear that the equation (3.10) is a mark of the measurement process, by showing its relevance for the case of planetary motion. Ernst himself (Ernst 1968, the second work) noticed the fact that a functional relation between the pure gravitational and pure electromagnetic complex potentials solves the problem of gravitational field. In 1971 (Ernst, 1971) he proved that the theory leading to the equations (3.03) and (3.09) above, when applied for the case of the pure gravitational field is obtainable from the variational principle

(3.12)

where R(γ ) is the scalar curvature of the metric γ. We can see now that only in a Ricci flat space this principle involves exclusively the complex Ernst potential

(3.13)

In other words, only in cases where the gravitational field defines an electromagnetic field by a linear relation like (3.10), this gravitational field can be described exclusively by the complex Ernst potential.

    We will restrict the present work to this last case of vacuum gravitational field, for the simple reason that this field can be assimilated to the action of Space itself, and therefore can be considered as an inertial field. Thus, the approach just presented opens an unexpected way to solution for the problem of inertial fields, because the variational principle (3.13) can be constructed in relation with the SL (2, R) continuous group we are considering here. Namely, it has been noticed by Matzner and Misner (Matzner, Misner, 1966) that the variational principle (3.13) is actually an answer to what in more contemporary terms is the problem of harmonic maps, a fact explicitly recognized later on by Misner (Misner, 1978). From this point of view equation (3.13) describes a harmonic map from the Euclidean space to SL (2, R). This is much more palpable if, instead of Ernst potential e , we use the field variable h º ie, so that the equation (3.13) becomes

(3.14)

Obviously this variational equation describes a harmonic map between the ordinary flat space of metric (γ_mn) and the complex half plane possessing the Poincaré metric,

(3.15)

known to be invariant metric of SL (2, R). This is the main idea contained in the Ernst approach and the reason we have followed this approach.

    The complex potential h is closer to the way this SL (2, R) geometry is built from many points of view. The most important one is its possible physical meaning. Indeed, equation (3.08) for the case of null electromagnetic field (pure inertial field) gives

(3.16)

so that the real part of the potential is arbitrary, while the imaginary part

(3.17)

has always the nice quality to be positive and has a fixed point unity (light speed), features required by the geometry of the upper half plane in the Poincaré representation.

    By this very fact the Poincaré metric is physically legitimated. Another attractive theoretical point of this potential is that the differential equation corresponding to variational principle (3.14) – the ‘ Ernst equation ‘ of the problem – takes the form

(3.18)

and has an immediate solution

(3.19)

with α real. Thus here the solution of the stationary inertial field problem is reduced to that of the Laplace equation in regular space. However, by equation (3.19) the inertial potential has a particular meaning. First, it can be seen that h represents a characteristic of the type (2.20) of a Kepler motion, for e º tanhψ and α º ω. Then, we will show in the next section how this physical interpretation can be taken into consideration.

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