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VI.  An Account of Gravitation

As mentioned above, we insisted so much in detailing the application of Stoka Theorem only in order to make its point among very known and physically most treasured facts. We also pointed out that it sometimes happens that physically we know the SL (2, R) realizations we have to work with. One such case is that of the stationary gravitational field, to be considered in this section. We have chosen this case first of all because it is, again, among the most treasured in Physics and positive results here would mean a plus in making the point. However the main course of action is directed towards showing what unexplored possibilities (even technological ones) are still waiting for us, if we only accept a slight change in the angle of consideration of things.

We are now talking of gravitation in the framework of the General Theory of Relativity. The reason is that we want to extract as much as possible towards a mathematical philosophy of Physics in first place. And here, for the first time in history, the field concept, inasmuch as its omnipresence and permanence were considered, has been submitted by Einstein to a logical analysis based on the idea of particle, insofar as the point of space and the moment of time aspects of this last concept are involved. The result of this analysis is the well-known Einstein - or gravitational - field equations. The vacuum and electromagnetic vacuum gravitational field equations have a nice solution amenable to a notable 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, for reasons that 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

(68)

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

(69)

take the form of the system of nonlinear partial differential equations

(70)

The explanation of 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

(71)

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) ≡ (A_γ) describes the electromagnetic field whose intensities are given by its covariant rotor:

(72)

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

(73)

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

(74)

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

(75)

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

(76)

the round parentheses denoting symmetrization with respect to indices concerned.

As mentioned, F. J. Ernst (Ernst 1968) introduced the complex potential ε 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 1972)), but only the stationarity of the metric field (independence of time). In spite of this, however, 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. Actually, it is well known, this problem has never been solved in that spirit. The deep reason is quite simple: 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 (69). However, the right hand side of these equations contains the energy tensor whose construction, while accounting for interaction properties, requires the apriori knowledge of the metric tensor. This problem has repeatedly been 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 gravitational field (Born, Infeld 1934, Misner, Wheeler 1957, Fock 1964, Einstein 1967, Appendix II). In particular it is to be noticed the conclusion that the Einstein’s nonsymmetric field theory is entirely equivalent to the Born-Infeld nonlinear electrodynamics (Born, Infeld 1934) provided the electromagnetic field is defined as the antisymmetric part of the metric tensor (Chernikov 1977). Indeed, the metric tensor (g_αβ) may not be necessarily symmetric, although in the metric we cannot have but its symmetric part by the very algebraic nature of the expression of metric. Thus, if we take a general non-symmetric metric tensor and accept that it is compatible with the connection of space-time, then the conclusion comes out that the antisymmetric part of the metric is identical in nature with a nonlinear Born-Infeld electromagnetic field. There is much to say about this fascinating problem of the contemporary Theoretical Physics but, again, we limit ourselves for now to what the introduction of the Ernst complex potential has just made possible, revealing thus what this potential means for the measurement process.

It has been noticed that the problem of gravitational field could be solved if the logic is taken a little out of the usual line so to speak, in the sense that the space metric g should be allowed to be arbitrary, therefore conveniently chosen. Indeed, Israel and Wilson observe (Israel, Wilson 1973) that equations (76) are to be taken as compatibility conditions between a selected space metric and the complex fields ε and . In the particular case of a flat space the compatibility conditions amount to a single linear relation

(77)

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

(78)

By equation (77) 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 a 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 (77) is a mark of the measurement process, by showing its relevance for the case of harmonic oscillator. 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 (Ernst 1972) he proved that the above theory yielding equations (70) and (76), when applied for the case of the pure gravitational field is amenable to the variational principle

(79)

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

(80)

In other words, only in cases where the gravitational field defines an electromagnetic field by a linear relation like (77), the gravitational field is described solely by the complex potential.

We will restrict the present work to this last case of vacuum gravitational field. The approach just presented opens an unexpected way to solution for the problem of vacuum fields, because the variational principle (80) 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 (80) 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 (80) describes a harmonic map from the Euclidean space to SL (2, R). This is much more palpable if, instead of Ernst potential ε we use the field variable h ≡ iε, so that the equation (80) becomes

(81)

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,

(82)

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

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 (75) for the case of null electromagnetic field (pure gravitational field) gives

(83)

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

(84)

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 (81) - the ‘Ernst equation’ of the problem - takes the form

(85)

and has an immediate solution as

(86)

with a real. Thus here the solution of the stationary gravitation field problem is also reduced to that of the Laplace equation in regular space. However, by equation (86) the gravitational potential has a particular meaning, which is almost obvious for the diligent reader. For, there is not too much to add here, in order to see in the equation (86) an ensemble of oscillators of the kind needed in order to reduce a Lagrangian to a perfect square (see equation (27) and its preceding explanation). Here however, the oscillators have a geographical distribution: their Stoler parameter is constant on harmonic surfaces in space. As we have seen the Stoler parameter is also a time of evolution, but this is a ‘cosmic’ time. Indeed, here we can add that this is a kind of wave front evolution like those used for the geometrization of space-time (Fock 1964), for it is represented by surfaces in space. At this stage of the theory there is no apparent relation between the time of local dynamics - embedded in the phase α in equation (86) - and this ‘cosmic’ time. Notice that the result explicitly confirms Jaynes’ principle referred to in the Introduction, for the circumstances left unspecified in this problem of gravitational field are related to its oscillator structure.

Physically, the invariant metric (82) opens a possibility to describe the interaction between different oscillators. The SL (2, R) structure involved here allows us to say that, as long as we consider harmonic oscillators in gravitational field, it is this field that determines their parameters via Stoka Theorem. Indeed, the above invariance of the gravitational field equations corresponds to a differential realization of SL (2, R) structure whose infinitesimals are the known Beltrami operators (Stoka 1968), which can be written at will, either as

(87)

or as

(88)

Mention should be made that this realization of the structure is a measurable one (Stoka 1968), the group measure being given, up to a multiplicative constant, by

(89)

where ‘Λ’ denotes exterior multiplication of differential forms. Again this is a physically attractive point of the theory - even though the realization is not a compact one - especially for stochastic constructions.

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