Solution to Einstein's Field Equations where T^uv not= 0?

Many of the widely-studied solutions to Einstein's field equations are taken
in vacuo, that is, at events where the energy momentum tensor T^uv=0. This
includes Schwarzchild and Kerr geometries, for example.

Have there been many exact solutions found where T^uv not= 0? I am speaking
of analytical solutions where the differential equations are solved exactly,
*not* numerical approximations.

I am especially interested in any exact solutions based on the usual Maxwell
energy tensor of electrodynamics T^u_v = (1/4pi) [F^ut F_vt - (1/4)
lambda^u_v F^st F_st]. I am interested in solutions both where F^uv_u=0
(free space) and also where F^uv_u=J^v (space with current sources).
Conditions of interest include static spherical symmetry in the nature of
Schwarzchild, and rotation with spherical symmetry about the z-axis in the
nature of Kerr.

To be clear, I am *not* looking for solutions where the metric is assumed to
be a Minkowski metric. Lots of analyses assume a flat-space background for
electrodynamics.

Rather, I am looking for *exact* solutions, to the extent that such
solutions are known, which derive a curved spacetime metric from the
electromagnetic field strength tensor, that is, which derive g_uv =
g_uv(F^uv) via the Maxwell tensor T^u_v, whereby T^u_v(g_uv, F_uv) simply
becomes T^u_v(F_uv) once the g_uv(F^uv) are found.

Thanks.

Jay R. Yablon

Email:

"Jay R. Yablon" wrote in message ...
Many of the widely-studied solutions to Einstein's field equations are taken
in vacuo, that is, at events where the energy momentum tensor T^uv=0. This
includes Schwarzchild and Kerr geometries, for example.

Have there been many exact solutions found where T^uv not= 0? I am speaking
of analytical solutions where the differential equations are solved exactly,
*not* numerical approximations.

I am especially interested in any exact solutions based on the usual Maxwell
energy tensor of electrodynamics T^u_v = (1/4pi) [F^ut F_vt - (1/4)
lambda^u_v F^st F_st]. I am interested in solutions both where F^uv_u=0
(free space) and also where F^uv_u=J^v (space with current sources).
Conditions of interest include static spherical symmetry in the nature of
Schwarzchild, and rotation with spherical symmetry about the z-axis in the
nature of Kerr.

To be clear, I am *not* looking for solutions where the metric is assumed to
be a Minkowski metric. Lots of analyses assume a flat-space background for
electrodynamics.

Rather, I am looking for *exact* solutions, to the extent that such
solutions are known, which derive a curved spacetime metric from the
electromagnetic field strength tensor, that is, which derive g_uv =
g_uv(F^uv) via the Maxwell tensor T^u_v, whereby T^u_v(g_uv, F_uv) simply
becomes T^u_v(F_uv) once the g_uv(F^uv) are found.


You might try posting this on sci.physics.research.

Martin Hogbin

You might try posting this on sci.physics.research.

Martin Hogbin

I sent it there three days ago, but nobody is minding the store right now.
Nothing new has posted there since Monday. I expect it will post there
soon.

Jay R. Yablon

Wikipedia seems to have a pretty good section on this:

http://en.wikipedia.org/wiki/Exact_s...ral_relativity

-- Ben

Thank you, that is helpful.

Jay.

--
_____________________________
Jay R. Yablon
Email:
"Ben Rudiak-Gould" wrote in message
...
Wikipedia seems to have a pretty good section on this:

http://en.wikipedia.org/wiki/Exact_s...ral_relativity

-- Ben

In sci.physics Ben Rudiak-Gould wrote:
Wikipedia seems to have a pretty good section on this:

http://en.wikipedia.org/wiki/Exact_s...ral_relativity

This looks pretty good. If you want more, the standard source is the
"exact solutions" book -- Stephani, Kramer, MacCallum, Hoenselaers,
and Herlt, _ Exact Solutions of Einstein's Field Equations_ (second
edition, 2003), Cambridge University Press. There's also an online
searchable database at http://www.maths.soton.ac.uk/staff/d...erno/database/
but this requires that you know a bit about what you're looking for.

Steve Carlip

Ben Rudiak-Gould wrote:
Wikipedia seems to have a pretty good section on this:

http://en.wikipedia.org/wiki/Exact_s...ral_relativity

-- Ben

Yes, that wiki on GR is quite good, given the
difficulty of the subject.
One can go to the Electrovacuum solution and
encounter the Rainich *issues*, but that doesn't
satisfy Jay's g_uv = g_uv(F_uv) question regarding
an exact solution.

There's more at,

http://arxiv.org/PS_cache/gr-qc/pdf/0202/0202092.pdf

which focuses on antisymmetrization to relate 4 and 5D.

In Tolman's "Relativity" book, pg.265 is the classical
treatment but that approach, as far as I know, has
been largely discredited because it depends on the
self energizing of a the fundamental charge, and that
effect is theoretically weak and not substantiated by
by any experiment, that I can find.

Hopefully someone else can provide an answer to
Jay's question, apart from,

http://www.vacuum-physics.com/KST/GR_Charge_Couple3.pdf

which provides a sort of bench-mark to attack the
problem.

Regards
Ken S. Tucker

Thank you for the suggestion re the "exact solutions" book. I browsed it on
Amazon, and it is just what I am looking for, so I placed an order for it.

Jay R. Yablon

--
_____________________________
Jay R. Yablon
Email:
wrote in message
...
In sci.physics Ben Rudiak-Gould wrote:
Wikipedia seems to have a pretty good section on this:

http://en.wikipedia.org/wiki/Exact_s...ral_relativity

This looks pretty good. If you want more, the standard source is the
"exact solutions" book -- Stephani, Kramer, MacCallum, Hoenselaers,
and Herlt, _ Exact Solutions of Einstein's Field Equations_ (second
edition, 2003), Cambridge University Press. There's also an online
searchable database at http://www.maths.soton.ac.uk/staff/d...erno/database/
but this requires that you know a bit about what you're looking for.

Steve Carlip