Hi,

The weak field or linear approximation to GR is based on writing the field
as a small perturbation (h_mu nu) from the Minkowski metric and expanding
Einstein's field equations to linear order in h. The resulting field
equations along with the linearized equation for Geodesic motion more or
less constitute a Lorentz covariant theory of gravitation.

Does this special-relativistic treatment of gravity give the same
predictions as GR ? I think at least for the classical tests (perihelion of
Mercury, bending of
starlight, gravitational redshift) it does since those results are
calculated to linear order in GM/r (that is, linear order in h). If so, then
what are the experimental or theoretical reasons
for ruling this out (as an exact theory of gravity rather than an
approximation)?

It's presumed that the source of gravity is momentum-energy current and that
the gravitational field also carries energy-momentum. If both assumptions
are true, that rules out a linear field equation - is that it or is there a
more compelling reason?

"Joe" wrote in message
...
Hi,

The weak field or linear approximation to GR is based on writing the field
as a small perturbation (h_mu nu) from the Minkowski metric and expanding
Einstein's field equations to linear order in h. The resulting field
equations along with the linearized equation for Geodesic motion more or
less constitute a Lorentz covariant theory of gravitation.

Does this special-relativistic treatment of gravity give the same
predictions as GR ? I think at least for the classical tests (perihelion
of
Mercury, bending of
starlight, gravitational redshift) it does since those results are
calculated to linear order in GM/r (that is, linear order in h). If so,
then
what are the experimental or theoretical reasons
for ruling this out (as an exact theory of gravity rather than an
approximation)?

Sure - check out
http://www.pupress.princeton.edu/sam...i/chapter3.pdf
See section 3.5.

Also it is well known that the gauge invariance of the linear equation is
infinitesimal coordinate transformations. It does not make much physical
sense for a theory to be invariant for infinitesimal transformations and not
finite ones - indeed one could arguer that finite ones could be built form a
large number of infinitesimal ones (of course one must always be careful of
such hand uses of infinitesimals). Once that is admitted then GR
immediately follows - see Chapter 6 - Gravitation and Space-time by Ruffini.

Thanks
Bill


It's presumed that the source of gravity is momentum-energy current and
that
the gravitational field also carries energy-momentum. If both assumptions
are true, that rules out a linear field equation - is that it or is there
a
more compelling reason?

Joe wrote:
The weak field or linear approximation to GR is based on writing the field
as a small perturbation (h_mu nu) from the Minkowski metric and expanding
Einstein's field equations to linear order in h. The resulting field
equations along with the linearized equation for Geodesic motion more or
less constitute a Lorentz covariant theory of gravitation.

It is not Lorentz covariant, at least in the usual way. Clearly if the
sun is at rest in this inertial frame, then it is not at rest in any
inertial frame related to the first by a boost.


Does this special-relativistic treatment of gravity give the same
predictions as GR ? I think at least for the classical tests (perihelion of
Mercury, bending of
starlight, gravitational redshift) it does since those results are
calculated to linear order in GM/r (that is, linear order in h).

For all weak-field tests of GR this approximation yields the same
predictions. In fact, for many tests this is what is used to compute the
GR prediction.


If so, then
what are the experimental or theoretical reasons
for ruling this out (as an exact theory of gravity rather than an
approximation)?

First and foremost it is NOT a generally covariant theory. One of the
foundational principles of modern physics is that any valid theory of
physics must be independent of coordinates, aka generally covariant. The
reason for this is clear and inescapable: coordinates are mere figments
of human imaginations, and clearly no coordinates are involved in ANY
natural phenomena.

But also, it is UGLY. The equations are long and complicated, and not at
all intutive. For instance, I believe there is no Lagrangian from which
it can be derived. By contrast, the Einstein field equation is
simplicity itself (in structure, not in solutions (:-)), and comes from
a simple and straightforward Lagrangian.


It's presumed that the source of gravity is momentum-energy current and that
the gravitational field also carries energy-momentum. If both assumptions
are true, that rules out a linear field equation - is that it or is there a
more compelling reason?

In GR there is no "gravitational field" that locally "carries" energy or
momentum, but in terms of the metric or the connection the field
equation is most definitely nonlinear.

If one assumes that gravitation is described by a spin-2 tensor field,
and that the field itself gravitates, one is led to GR in the classical
limit.


Tom Roberts

Joe wrote:
Hi,

The weak field or linear approximation to GR is based on writing the field
as a small perturbation (h_mu nu) from the Minkowski metric and expanding
Einstein's field equations to linear order in h. The resulting field
equations along with the linearized equation for Geodesic motion more or
less constitute a Lorentz covariant theory of gravitation.

Does this special-relativistic treatment of gravity give the same
predictions as GR ? I think at least for the classical tests (perihelion of
Mercury, bending of
starlight, gravitational redshift) it does since those results are
calculated to linear order in GM/r (that is, linear order in h). If so, then
what are the experimental or theoretical reasons
for ruling this out (as an exact theory of gravity rather than an
approximation)?

It's presumed that the source of gravity is momentum-energy current and that
the gravitational field also carries energy-momentum. If both assumptions
are true, that rules out a linear field equation - is that it or is there a
more compelling reason?

Good fire-works Joe...

Let's assume the Kronecker delta

g^u_v = g^uw g_vw = 1, 0 as u=v, u=/=v respectively.

as the basis of tensor analysis.

Then I'll write the pertubation from Minkowski's "e_uv" as,

g_uv = e_uv + h_uv and inner multiply with g^uw gives,

g^w_v = e^w_v + h^w_v .

Watch carefully, we should agree the Kroneker g^0_1 =0 then,

e^0_1 = - h^0_1

is antisymmetrical, for example,

g_uv = s_uv + a_uv

where s_uv is symmetrical and a_uv is antisymmetical.

Sustaining the e_uv to be symmetrical and orthogonal,
we'll need to split h_uv into symmetric nonorthogonal
"H_uv" components and antisymmetric "f_uv"
components respectively to get,

h_uv = H_uv + f_uv

and find,

g_uv = e_uv + H_uv + f_uv.

The sum (e_uv + H_uv) is equivalent to the " s_uv " above,
and is equivalent to the classical GR solution,

s_uv = e_uv + H_uv == classical GR metric,

where we've ripped off the "h_uv" and settled that to H_uv
in classical GR to be the gravitational effects and the "f_uv"
that are equivalent to the "a_uv" above and are the EM-effects.

I'll stop here pending replies.
Ken S. Tucker