Preludes to a relativistic theory of gravitation

November 9, 2010 § 1 Comment

In a time of universal deceit – telling the truth is a revolutionary act ~ George Orwell

The special theory of relativity had its origin in the development of electrodynamics. The mechanical theory of gravitation and the field theory of electrodynamics were based on the same concepts of space and time. The two theories were also fundamentally different and thus did not contradict each other.

Sadly, they were no longer compatible when Maxwell’s theory eliminated action at a distance from the realms of electrodynamics, and the Lorentz transformations ruled out action at a distance from the whole of physics, by depriving time and space of their absolute character.

Newton’s theory of gravitation is co-variant with respect to Galilean transformations, but not with respect to Lorentz transformations.

The theory of gravitation had to be changed into a relativistic field theory.

Principle of equivalence ~ weak form

According to Newton’s theory of gravitation, “inertial mass” and “gravitational mass” of the same body are always equal.

What if this approximate equality is accidental, and an accurate determination shows the two kinds of mass to be different?

This can readily be tested and proven “wrong”, and thus the theory is a proper open theory.

What can be done is to finding out whether the acceleration of all bodies is the same in the same gravitational field. It is impossible to measure time intervals accurately enough, but that can be measured indirectly. “Inertial acceleration” is independent of the mass of an accelerated body. Referring the motions of a body to a non-inertial frame of reference, accelerations do not correspond to real forces acting on that body. The found forces are reflections of the accelerations of the frame of reference to some inertial system.

What if we can make an experimental set-up in which we can observe bodies under the combined influence of inertial forces and gravitational forces, the direction of the result for a particular body depends on the inertial mass/gravitational mass? Will it be the same for all bodies tested?

Nature gives us a sensitive test system. The Earth may not be an inertial system but it rotates around its axis with a constant angular velocity. Bodies at rest relative to Earth is subject to gravitational attraction by earth and to “centrifugal force”. Except for on the equator, the two accelerations are not parallel, and the direction of the resultant is a measure of the sought ratio between inertial and gravitational mass. Eötvös did such experiments.

And what if electromagnetism does have something to do with it?

The physical property of mass has two distinct aspects, gravitational mass and inertial mass. The weight of a particle depends on its gravitational mass. According to the weak form of the equivalence principle, the gravitational and inertial masses are equivalent. But, we show here that they are correlated by a dimensionless factor, which can be different of one . The factor depends on the electromagnetic energy absorbed or emitted by the particle and of the index of refraction of the medium around the particle. This theoretical correlation has been experimentally proven by means of a very simple apparatus, presented here ~ Correlation Between Gravitational and Inertial Mass: Theory and Experimental Test.

Reformulating Newton’s theory

Action at a distance must be eliminated, for sure. The gravitational attraction of one body with mass m by several other ones can be represented by the sum of the gravitational potentials of these other bodies, and this sum represents the potential energy Um of the first body divided by its mass m.

The experienced force by that body is the negative gradient of its potential energy.

Introducing a “gravitational field strength” we find, just as in electrostatics, that the lines of force do not originate or terminate outside of masses, and that in a mass M = 4πκρ. This equation was first formulated by Poisson.

Poisson’s equation is not Lorentz invariant. Wherever ρ vanishes, we can assume that the (three dimensional) Laplacian operator 2 has to be replaced by its four dimensional analogue ηρσ. And this changes time.

Mass density ρ is not a scalar, but a component of the tensor Pμν. We can replace ρ by the Lorentz invariant scalar ημνPμν or replace the non-relativistic scalar G by a world tensor G μν

Inertial systems

I can be (relatively) short on this one. According to classical mechanics all real forces represent the interaction of bodies with one another. That means that a body is not subjected to forces if it is sufficiently far removed from all other bodies. But in a relativistic theory we must try to eliminate all concepts involving finite spatial distances.

There is a major difference between electromagnetic and gravitation fields. We can choose test bodies that are uncharged and unpolarised! The effects of a gravitational field on a test body cannot be eliminated, because its acceleration is supposedly independent of its mass.

The action of a gravitational field cannot be distinguished from “inertial accelerations”. Both are independent of the characteristics of the test body. We simply cannot separate the two and find an inertial system.

From this perspective, inertial systems are not a particular class of coordinate system; there is no real difference between a supposed inertial frame of reference with a gravitational field and a non-inertial frame of reference.

Einstein’s elevator

“Einstein’s Elevator” has provided a good straight forward description of the principle of equivalence (pdf).

Nature of the gravitational field

With all of the above, it may seem as if gravitational fields are not real. They could be nothing more than “inertial forces”. This goes against my instinct. This cannot be true.

It is impossible to introduce a Lorentzian coordinate system by the Riemannian character of space, and if it is impossible to introduce coordinate systems in which the components of the metric tensor assume constant pre-assigned values, the metric tensor becomes part of the field. There must be field equations as boundary conditions, restricting and determining functional dependence of gμν on the four world coordinates.

In the presence of a gravitational field, the gμν are the potentials which determine the accelerations of free bodies. The gμν are the potentials of the gravitational field.

Any theory of gravitation will have to deal with spaces that are not “quasi-Euclidian” ~ in which no inertial system can be introduced. Something like Riemannian space geometry is required. Some mathematical criterium that helps us determine whether a space is Euclidian or not.

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