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An alternative presentation of a relativistic theory of gravitation, equivalent to general relativity, is given. It is based upon the restriction of the Lorentz invariance of special relativity from a global invariance to a local one. The resulting expressions appear rather simple as we consider the transformations of a local set of pseudo-orthonormal coordinates and not the geometry of a 4-dimension hyper-surface described by a set of curvilinear coordinates. This is the major difference with the usual presentations of general relativity but that difference is purely formal. The usual approach is most adequate for describing the universe on a large scale in astrophysics and cosmology. The approach of this paper, derived from particle physics and focused on local reference frames, underlines the formal similarity between gravitation and the other interactions inasmuch as they are associated to the restriction of gauge symmetries from a global invariance to a local one.

In the usual presentations of general relativity [

for example in the absence of other external forces than gravitation, a particle follows the minimum path

between two space-time points A and B. In the presence of a gravitation field there is no global transformation of the coordinates allowing the

We give here an alternative presentation of a relativistic theory of gravitation, equivalent to general relativity, in which gravitation is introduced as a gauge field associated to restricting the global Lorentz invariance of special relativity to a local one.

The presentation of gravitation as a gauge field has been first introduced by Lanczos in the context of a variational approach to general relativity [

That restriction from a global invariance to a local one, associated to the emergence of a force field, is indeed a deep similarity between gravitation and the other interactions. However the Lorentz group is isomorphic to the special complex linear group

In the present work, we assume that the Lorentz group invariance is not a global symmetry of space-time but a local symmetry of a 4-dimension hyper-surface, which can be thought of as embedded in a space with a larger number of dimensions and we consider the transformations of a local set of pseudo-orthonormal coordinates and not the geometry of the 4-dimension hyper-surface described by a set of curvilinear coordinates. This is a major difference with other presentations of relativistic gravitation; its interest is to lead to facially simpler expressions although that difference is essentially formal.

Let us consider the 4-dimension space-time of special relativity and a pseudo-orthonormal base

Any infinitesimal transformation R of the Lorentz group can be written as

where

We introduce the 2 anti-symmetric tensors

and

Each component

More generally, any transformation of the Lorentz group can be figured by

We now assume that the Lorentz group invariance is not a global symmetry of space-time but a local symmetry of a 4-dimension hyper-surface, which can be thought of as embedded in a space with a larger number of dimensions, for example 10 as it is envisaged in many unification theories. If we consider the tangent plane to this hyper-surface in any point M, it is possible to define in this plane a pseudo-orthonormal reference frame, and in fact an infinite set of similar frames deduced from each other by a Lorentz transformation or a rotation; in the close vicinity of M the laws of physics are invariant under the Lorentz group. We can in another point

Let us first perform on the surface an infinitesimal displacement

But if

Comparing the two expressions Equation (8) and Equation (9) above, we see that

with

Now considering some function

is in a similar way transformed into

i.e.

The impulsion

is the infinitesimal generator of space-time translations. Equation (14) above means that

The orbital angular momentum anti-symmetric tensor

can be written as

where

can be written as

That expression allows to evidence a gauge invariance property: it is possible to add to

We have here above considered the transformations of a local set of pseudo-orthonormal coordinates and not the geometry of the 4-dimension hyper-surface described by a set of curvilinear coordinates. This is the major difference with other presentations of relativistic gravitation and notably with general relativity but that difference is purely formal. As a consequence, many mathematical expressions look simpler; for example, the inva-

riant 4-dimension volume element

We now consider a scalar particle of mass m (but the procedure can be straightforwardly generalized to a particle of any spin, be it massive or not) and the Lagrangian density

with

We perform the transformation

where we have introduced the effective metrics

with

We have written

Applying the Lagrange equations to the

gives

From the expression of

Equation (26) shows that actually

The wave equation so appears as the wave equation of a free particle in which the original Minkowski metrics

The gravitation field is thus described by a modification of the geometry of space-time by replacing the Minkowski metrics

We now assume for the gravitation field itself a Lagrangian density term quadratic in

or

[N.B.: as

The full Lagrangian density of the (field + particle) system is

where

Applying the Lagrange equations to

with

These are the equations of the gravitation field and their nonlinear character is obvious. The term on the right- hand side

is proportional to the energy-impulsion density tensor of the particle; it is the source term of the gravitation field. In the classical, i.e. non quantum, limit, the correspondence

So Equation (32) becomes

For the sake of commodity, we will re-write it in a different, more workable way. Let us introduce the two quantities

Combining Equations (35) and (36) we finally get after some manipulations

In the case of a weak gravitation field, the quadratic terms in the field equations can be neglected. In the absence of matter, the linearized equations take the form of propagation like equations:

If matter is present, there is a source term:

with

i.e.

In the non-relativistic limit

hence

Moreover if the field is slowly varying with time, the time derivatives on the left hand side of Equation (40) vanish and those equations become:

The above expression for

where

On the other hand, the dynamical equation of a massive particle is given by Equation (27)

or

Let us put

so that Equation (44b) becomes

or

Let us also put

We proceed by successive iterations; replacing

or

In the non-relativistic limit

Moreover in the weak field limit

If we compare the above expression with the non-relativistic expression

we see that the particle is undergoing an effective gravitation potential

Since

Since from Equation (42a)

By comparison with the expression of the classical gravitation potential

If we had retained the

From the expressions Equation (42) we derive in the weak, slowly varying field limit

so that

In addition, Equation (48b) gives

This has to be compared with the Newtonian expression at the lowest order in

or

Introducing the Planck length

The usual approach of general relativity is most adequate for describing the universe on a large scale in astrophysics and cosmology. The approach of this paper, derived from particle physics and focused on local reference frames, underlines the formal similarity between gravitation and the other interactions inasmuch as they are associated to the restriction of a global symmetry to a local one.

In a 10-dimension space-time as it is considered in certain unification theories, gravitation is linked to the geometry of the 4 usual dimensions whereas the other fundamental interactions can be associated to the geometry of the 6 additional ones; in that approach extra fields (which eventually may account for dark matter and/or dark energy) naturally come out by regarding the geometry of the full 10-dimension set.