Democratic principle

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In discussions of so-called Mach principles in classical theories of gravitation, the democratic principle is an attempt to state one possible version of a Mach principle. Unfortunately, while this statement is fairly intuitive, it is not well-defined.

The essential idea behind Mach principles has been expressed by John Archibald Wheeler via the slogan mass-energy there influences inertia here. More precisely, the definition of inertial motion at some event in spacetime should be physically determined by the relative distribution and motion of all the mass in the universe. (In the context of classical relativistic field theories of gravitation, which usually employ curved Lorentzian manifolds as models of spacetime, "mass" should be understood as "mass-energy".)

The democratic principle is Wheeler's poor-man's explanation, in the absence of a well-defined principle, for the simplest possible notion of how this process might work.

1 Attempted mathematical statement
2 Objections
3 Relation to GEM formalism
4 See also
5 References

[edit] Attempted mathematical statement

For simplicity, we can pretend that the only mass-energy in the universe is in the form of particles of ordinary matter. To state the principle clearly (and to see what is wrong with it), we should work with flat spacetime.

Consider the past light cone of the origin, equipped with an affine null coordinate u. Add up the four-momentum vectors on this cone, weighted by a factor of 1/u. (If necessary, one can replace summation by integration.) Normalize the result and consider it as a reference four-velocity vector located at the origin.

Proceed similarly for other events very near the origin. Now one has locally defined a timelike unit vector field near the origin. The integral curves of this vector field should locally define a timelike congruence, representing the world lines of reference observers. Supposedly, these observers can be considered to be at rest (on average) with respect to the universe. These observers are assumed, more or less following Mach, to be in a state of inertial motion. We can now hope to compare the motion of a new observer to these reference observers to decide whether he too is in a state of inertial motion.

The "voting process" just described is held to represent the outcome of an "election" in which every particle in the universe votes on the question of how inertial motion at the origin should be defined, with the vote of each particle weighted by the product of its own four-momentum with the reciprocal of its "distance" from the origin. The name "democratic" is ironic, since not every particle has an equal role in determining the outcome!

Somewhat more generally, one can envision a similar weighted average over the past light cones of points near the origin of the stress tensor, representing the amount and motion of any matter, in addition to the amount of field energy of any nongravitational fields which may be present. This would be held to result in an imaginary "fluid" which is at rest (on average) with respect to the universe.

[edit] Objections

It is not clear that this statement gives a well-defined procedure even in flat spacetime, because the integral over the past light cone might not be well-defined (and might not converge).

Even worse, it is not clear how to fix this up to work in a curved spacetime. The most obvious obstacle is that the notion of distance from the origin becomes highly problematic in this context, since there are infinitely many ways of defining "spatial distance" even on a given Cauchy hyperslice, with none of them obviously holding a physically preferred status. In the context of flat spacetime, we were able to sweep this problem under the rug by appealing to the intuitive idea that only the ratio of distances should be relevant, so that in flat spacetime we can hope to achieve a covariant definition of our reference congruence by replacing spatial distances with an affine null coordinate.

A more subtle objection is that it is not clear that the alleged reference congruence is vorticity-free. If it is not, geometrically speaking, the world lines of the reference congruence are twisting about one another, which would appear inconsistent with the expectation that they are locally non-rotating.

In the extension to an imaginary fluid, there is no apparent reason to expect the resulting reference stress-tensor field near the origin to model a perfect fluid.

In addition, we haven't even attempted to describe how to compare a given world line with our reference congruence.

[edit] Relation to GEM formalism

Gravitoelectromagnetism (GEM) is a somewhat informal term which refers to set of formal analogies between the gravitational and electromagnetic field. While some of these analogies make sense even in strong fields, as a whole they find their most striking expression in the so-called GEM formalism, in which one reformulates weak-field general relativity in terms of a force law which is approximately valid for slowly moving test particles far from an isolated massive object. As such, the formalism is applicable to typical solar system tests of general relativity.

The GEM force law closely resembles the Lorentz force law. For a test particle with mass m and (small) velocity $\vec{v}$ , it can be written

$\vec{F} = -m \, \left( \vec{E} + 2 \, \vec{v} \times \vec{B} \right)$

Here, the so-called gravitoelectric vector field $\vec{E}$ is the gradient of Newton's scalar gravitational potential $Φ$ , and the gravitomagnetic vector field $\vec{B}$ is the curl of a new gravitational vector potential $\vec{A}$ . These potentials can be estimated by a process analogous to the statement of the democratic principle offered above via

$\Phi = \frac{M}{|\vec{x}|}, \; \; \; \vec{A} = \frac{\vec{J}}{|\vec{x}|^2} \times \frac{\vec{x}}{|\vec{x}|}$

However, these expressions are only approximately valid, far from the source of the field, which has mass M and angular momentum vector $\vec{J}$ , so while they can be taken to lend some support to the assumed form of the weighted average which is proposed in the democratic principle, GEM should not be confused with a Mach principle.

[edit] See also

[edit] References

Wheeler, John Archibald (1990). A journey into gravity and spacetime. W. H. Freeman. ISBN 0-7167-5016-3. See pp 232-233. This is a semi-popular book.

Ciufolini, Ignazio, and Wheeler, John Archibald (1995). Gravitation and Inertia. Princeton University Press. ISBN 0-691-03323-4. This engaging but ultimately exasperating book, which is difficult to classify as either an essay, a monograph, or a textbook, but contains elements one might expect to see in any of these, offers no mathematical statement of the democratic principle, but the version offered above seems close to what Wheeler might have in mind in Eq. (1.1.1). This book extensively discusses the gravitational vector potential and the gravitomagnetic vector field, unfortunately in a manner which encourages confusing GEM with a Mach principle.