Re: Analogy Electromagnetism vs.Gravitation

From: Ken S. Tucker (dynamics_at_vianet.on.ca)
Date: 01/02/05 

Date: 2 Jan 2005 14:54:08 -0800

carlip-nospam@physics.ucdavis.edu wrote:
> tetrahedron <jarynth@yahoo.com> wrote:
> > Hi everyone. I was wondering
> [...]
>
> > 2) why it's impossible to extend the equations for gravitation in
> > analogy with Maxwell's equations for the electromagnetic field,
> > thus proving the existence of a gravitational "magnetic" field,
> > since Coulomb's and Newton's laws have essentially the same form
> [...]

Dr. Carlips initial post was very good, (IMHO),
a few brief comments follow.

> There are two basic differences between electrostatic and static
> gravitational forces: their sources transform differently under
> Lorentz transformations, and the energy of the gravitational field
> is itself a source of gravity.
>
> Start with electromagnetism. The source of an electric field is
> charge density, that is, charge per unit volume. It is observably
> true that charge is invariant under Lorentz transformations, while
> volume shrinks (the standard Lorentz contraction), so a moving
> observer will see charge density increase by a factor of gamma
> (where gamma = 1/sqrt{1-v^2/c^2}).

Here, an explicit definition of the volume is helpful,

1) dV = dx dy dz (works spatially)

2) dV = dx dy dz ds (works in spacetime)

3) dv = dx dy dz dt (appears to have a gamma^2 in SR).

>This is enough to show that
> charge density must be a component of a four-vector, and should
> therefore couple to a four-vector potential. It is also observably
> true that an electromagnetic field itself carries no charge, so the
> equations should be linear. Add these two facts to the observed
> form of Coulomb's law, and you are led to Maxwell's equations.

Steve, are you fairly confident Ampere's Law and magnetism
are consistent with your argument. (I am, it's not a trick
question, it's what I find too).

> Now compare gravity. In Newton's theory, the source of a
gravitational
> field is mass density. But it is observably true that the field does
> not just get a contribution from rest mass, but from energy as well.
> So the real source is energy density, that is, energy per unit
volume.
> Under Lorentz transformations, volume again shrinks, but now, from
> standard special relativity, energy grows by another factor of gamma,

> so a moving observer will see energy density increase by a factor of
> gamma^2. This is enough to show that energy density must be a
component
> of a rank two tensor (the stress-energy tensor), and should therefore

> couple to a rank two tensor field.

Yes

> It is also observably true that a gravitational field carries energy,

> and that gravitational energy is itself a source of gravity. (Even
> before the latter observation, it was necessary from self-consistency
> to assume that gravitational energy was a source of gravity;
otherwise
> you would get into trouble with conservation laws.)

At this point, we'll need to consider the LIGO results.
As I understand, the concept of storing gravitational
energy in a field - apart from the contributing masses -
is done independant of those radiating masses, by gravitational
radiation.
As I understand the problem, we should be able to prove
an independant existance of energy residing in the g-field,
and the radiation of that.

>That means that
> unlike Maxwell's equations, the gravitational field equations must be
> nonlinear -- a gravitational field can be its own source. Add these
> facts to the observed form of Newton's law, and you are led almost
> uniquely to the Einstein field equations. You'll find a discussion
> of this in the Feynman Lectures on Gravitation; for a detailed proof,
> look at http://arxiv.org/abs/gr-qc/0411023 (an online version of a
1970
> paper by Deser).

> There's another key difference: in electrostatics, like opposite
charges
> attract and like charges repel, while in gravity, like masses attract
> (and we know of no repulsive masses). This is enough to imply that
> electromagnetism must be described by a field of odd spin (that is,
by
> a tensor of odd rank), while gravitation must be described by a field

> of even spin. You could try to reverse things by flipping the sign
in
> Coulomb's law to make like charges attract, but if you did so, you
would
> find that a side effect was that electromagnetic waves carried
negative
> energy. This is physically unacceptable -- it would make virtually
every
> configuration of matter unstable against radiating away
negative-energy
> "light" and gaining energy. For details on this, see another paper
by
> Deser, http://arxiv.org/abs/gr-qc/0411026.
>
> Steve Carlip

Great post Steve,
Ken S. Tucker

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