On Maxwell's Equations and Gravitation[Part II]

[Anamitra Palit]

This is with reference to a previous posting "On Maxwell's equations
and Gravitation"[Dated:25th March 2010]Link:https://groups.google.com/
group/sci.physics.foundations/browse_thread/thread/100f9008271a9fd8?
hl=en

Lets examine a typical GR metric:

ds^2=g(00)dt^2-g(11)dx^2-g(22)dy^2-g(33)dz^2

The "d" going with ds has its correct meaning when the path is
specified that is wrt a one dimensional manifold[remembering that ds
is the proper time interval which will depend on path]

The physical distance[spatial] between two points along the x-axis
between the points A and B is given by:
integtra g(11)dx from A to B and not by integral dx in curved space

Infinitesimal separation between points on the x axis are given by
g(11)dx and not by dx

Now in Maxwell's equations in the covariant form we have quantities
like delta-x,delta-y etc which are meaningful only in the
Euclidean[rather in the flat space-time Lorentzian] context.But
Maxwell's equations in the covariant form refer to curved space
time[wrt to strongly curved spacetime also]. Are these quantities
[delta-x,delta-y etc] expected to retain their physical significance
in curved space-time?
Better we could write[locally]:
ds^2=dT^2-dX^2-dY^2-dZ^2
Where
dT=g(00)dt
dX=g(11)dx
dY=g(22)dy
dZ=g(33)dz
[The "d" going with T,X,Yand Z are as justified as the d going with s]

Locally we have,

ds^2=dT^2-dX^2-dY^2-dZ^2
Therefore locally we have the same form of Maxwell's equations--
Maxwell's equations in the traditional form!

I would request the audience to review the above statements with
respect to the comments of Tom roberts in the third posting of "On
Gravitation and Electromagnetism" given in the link above.
[The coordinates t,x,y and z are simply labels in the context of
curved space-time]

[Anamitra Palit]

> respect to the comments of Tom Roberts in the third posting of "On

> Gravitation and Electromagnetism" given in the link above.
> [The coordinates t,x,y and z are simply labels in the context of
> curved space-time]

Though the form of Maxwell’s equations [traditional form being
referred to here] remains unchanged locally the values of the
individual variables may change, preserving the traditional form of
Maxwell’s equations [in the local inertial frames].
We may consider a pair of local labels x and x+dx.. The
physical distance between them along the x-axis is g(1,1)dx. If the
metric changes ,say due to the advance of a heavy mass or a high
density mass distribution, the physical intervals dX,dY etc will
change. To preserve the form of the equation the values of E,B ,j etc
should also change..
So gravity can change the magnitudes of E ,B etc. If one thinks in the
cosmological direction the curvature of space-time was very strong in
the remote past and gradually it weakened casting a heavy influence
on the values of the electric and the magnetic fields.
Anamitra

[Daryl McCullough]

On Wednesday, February 8, 2012 2:16:43 AM UTC-5, Anamitra Palit wrote:

> The physical distance[spatial] between two points along the x-axis
> between the points A and B is given by:

> integral g(11)dx from A to B and not by integral dx in curved space

That's not completely correct. The formula for proper distance ds is:

ds^2 = g_00 dt^2 + ...

In the special case of displacement in the x-direction, we have:

ds^2 = g_11 dx^2

Take the square-root, and you have:

ds = square-root(|g_11|) dx

So the relevant scaling factor is square-root(|g_11|), not g_11.

[Anamitra Palit]

Thanks for the correction. I make this mistake quite often --due to
inadvertence of course.

[Tom Roberts]

On 2/8/12 2/8/12 - 1:16 AM, Anamitra Palit wrote:
> [...]

It is rather difficult to cast the Maxwell's equations into their correct form
on a curved Lorentzian manifold using differentials and such.

It is MUCH easier to use the language of differential forms. The four vacuum
Maxwell's equations become two equations:
dF = 0
*d*F = J
Where F is the Maxwell 2-form, d is the exterior derivative (NOT at all a
differential, but the concepts are distantly related), J is the current 1-form.
and * is the Hodge dual operator.

The Lorentz force equation for a pointlike particle is:
f^i = F_ij V^j
The {f^i} are the components of the 4-force on a pointlike particle, the {F_ij}
are the components of the 2-form F, and the {V^j} are the components of the
particle's 4-velocity.

The 2-form F, when projected onto locally Minkowski coordinates, has six
non-zero components corresponding to the traditional Ex,Ey,Ez,Bx,By,Bz; it is
antisymmetric in its 2 indexes. The 1-form J, when projected onto the same
Minkowski coordinates, has four non-zero components corresponding to the
traditional \rho,j_x,j_y,j_z.

This works in curved manifolds and in non-Minkowski coordinates because the
exterior derivative includes the appropriate connection terms.


Tom Roberts

[Ken S. Tucker]

On Feb 8, 3:36 pm, Tom Roberts <tjrob...@sbcglobal.net> wrote:
> On 2/8/12 2/8/12 - 1:16 AM, Anamitra Palit wrote:
>
> > [...]
>
> It is rather difficult to cast the Maxwell's equations into their correct form
> on a curved Lorentzian manifold using differentials and such.
>
> It is MUCH easier to use the language of differential forms. The four vacuum
> Maxwell's equations become two equations:
> dF = 0
> *d*F = J
> Where F is the Maxwell 2-form, d is the exterior derivative (NOT at all a
> differential, but the concepts are distantly related), J is the current 1-form.
> and * is the Hodge dual operator.
>
> The Lorentz force equation for a pointlike particle is:
> f^i = F_ij V^j

To get a force, usually that is usually written,

f_i = q F_ij V^j

so as to include test charge "q" and balance the tensor.
Regards
Ken S. Tucker
...