I am working in relationship between General Relativity and Newton
theory.

My work consists of two parts. The first part addresses the common
claim that geodesic equation of General Relativity reduces to Newton law
of motion in the linear limit, i.e. g_ab = \eta_ab + h_ab when ||h|| is
small enough to ignore quadratic and higher orders.

Take for example Carroll lecture notes available online:

http://lanl.arXiv.org/abs/gr-qc/9712019v1

Carroll starts from geodesic equation (4.9) and derives equation (4.19),
which he identifies with Newton equation (4.4) after using (4.20).

During the derivation Carroll uses the linear constraint (4.13).

A more rigorous analysis does not support Carroll conclusions.

Carroll is not computing the linear geodesic equation of motion but
'inventing' a non-geometrical equation [see below].

The equation of motion in the linear limit is a = 0. Therefore, the
equation of motion of General Relativity does not coincide with that
from Newtonian gravity in the linear regime.

For both the zeroth and the linear regimes of General Relativity bodies
have to move on straight lines.

"Juan R." González-Álvarez wrote on Thu, 01 May 2008 13:25:06 -0600:

Mistake. I did mean that Carroll is assuming that left hand side may be
approximated by

"Juan R." González-Álvarez wrote on Thu, 01 May 2008 13:25:06 -0600:

As said in original message, conventional derivations of the Newtonian
limit of General Relativity (GR) are not correct.

If one takes consistently the linear limit of GR, one rigorously obtains
the final equation

a = 0

instead the incorrect equation showed in textbooks and lecture notes.

I have discussed this part of my work with an expert on curved spacetime
equations of motion, Eric Poisson [1]. Eric confirms that a = 0 in the
linear regime of GR:

(\blockquote
Since the energy-momentum tensor is already of first-order, in the
linearized theory the conservation equations must be written down with
the Minkowski metric, and this implies that the matter cannot have
gravitational interactions. Or as you point out, particles would have to
move on straight lines.
)

Thus spake Juan R. González-Álvarez canonicalscience.com>

Oh No wrote on Tue, 06 May 2008 05:38:53 -0600:

Do not know that zero acceleration mean?

Thus spake Juan R. González-Álvarez canonicalscience.com>

Oh No wrote on Tue, 06 May 2008 10:05:12 -0600:

If you have no serious argument then I will reply this irrelevant and
false one. I will copy and paste fragments of my communication with him:

"Several textbooks state that linearized General Relativity"

(\blockquote
But the condition \partial^\mu T_{\mu\nu} = 0 implies that test
bodies move on geodesics of the flat metric n_{\mu\nu}; i.e, if one stay
consistently within the linear approximation, one predicts that test
bodies are unaffected by gravity.
)

"... specific issue of linearized equations of motion."

"I have tried to derive that test bodies are unaffected by gravity within
the linear approximation ..."

Thus spake Juan R. González-Álvarez canonicalscience.com>

It is not much help. Incoherent and does not make your case.

Oh No wrote on Wed, 07 May 2008 08:55:12 -0600:

It would be if you read them. Fragments contain the terms "linear" or
"linearized" about ten times. That would be enough to understand i am
computing the linear limit of the geodesic equation of motion.

The title of this thread (chosen by me of course) also says "linear
regime". Four authors of a total of four understood i was speaking about
linearized GR. That seems to indicate i am writing fine and your
accusation had no basis.

Thus spake Juan R. González-Álvarez canonicalscience.com>

You keep saying that. But the limit you are calculating is still not the
Newtonian limit, and nor is it the weak field limit, so you have no
basis on which to say the text books are wrong.

Regards

--
Charles Francis
moderator sci.physics.foundations.
charles (dot) e (dot) h (dot) francis (at) googlemail.com (remove spaces and
braces)

http://www.teleconnection.info/rqg/MainIndex