🚀 go-pugleaf

In article <21125@ucdavis.ucdavis.edu> carlip@landau.ucdavis.edu (Steve Carlip) writes:
>In article <1993Jan7.085151.13461@nuscc.nus.sg> scip2124@nusunix1.nus.sg (MARC PAUL JOREF VAN LOO) writes:

:  There exists no decent derivation of the
:  gravitational radiation of a binary pulsar.
:  Yu's objection to this derivation is correct.
:  But I found out recently that it is not new:
:  Read the 16th lecture of Buchwald's book
:  17 simple lectures on GR. (1982).

>Lecture 15, probably.  Buchdahl argues --- rather convincingly, I
>think --- that the linear approximation "at best...might be taken
>as a heuristic guide," for several reasons:

>(1) it treats the motion of the sources inconsistently (the paths are
>approximated as Keplerian, while actually they should be independent
>straight lines at this order of approximation);

	I think that this refers to the linearized equation

	D^2[h_ij] = k*T_ij

	where D^2 is the appropriate differential operator, giving
something like

	hb_ij,k^k - hb_ik,j^k - hb_jk,i^k + hb_kl^,kl*eta_ij

	where

	hb_ij = h_ij - 1/2*eta_ij*h_k^k

	Here, h_ij = g_ij - eta_ij, where eta_ij is the flat-spacetime
metric. Raising an index corresponds here to multiplying by eta^ij.

	In the de Donder gauge, hb_ik^,k = 0.

	The left side satisfies

	(D^2[h^ij])_,j = 0

	which implies that

	T^ij_,j = 0

	which is valid only for flat spacetime.

	The true equation should be

	D^2 h_ij = k*T_ij + O(h^2) + O(h*k*T)

	which _is_ self-consistent.

	The O() terms are relatively small compared to k*T_ij, so the
linearized approximation is OK for the first order.

	I might add that it successfully passes a range of tests:

	Newtonian order: O(h) for g_(time,time): Solar System, binary
stars, stellar pulsations, etc.

	Post-Newtonian order: O(h) for g_(space,space) and O(h^2) for
g_(time,time):

	Deflection of light and delay of radio signals in the Solar
system [the first of these]

	Periapsis advance in excess over the Newtonian value: [both of
these]

	Gravity Probe B, if it is ever launched, will allow tests of
g_(space,time), which is unambiguously predicted, to within gauge
changes, to order O(h).

	A question about this criticism is why linearized GR would
give the right results for these Newtonian and post-Newtonian effects
but not for G-waves.

>(2) fields are strong inside the sources, so the linear approximation
>is not valid everywhere;

	That is a legitimate objection only for neutron stars and
black holes, and it certainly does not hold for the Solar System and
similar un-condensed domains.

	Furthermore, the consistency question can be evaded by doing
field matching over some suitable boundary where the compact object's
field is relatively weak.

>(3) it has not been shown that the linear approximation is the first
>term in a convergent series, and, if so, what the characteristics of
>the full series are; and
>(4) it has not been shown that higher order corrections are small.

	If one wishes to be rigorous, that may well be the case.

	However, there is the question of why it manages to work to
Newtonian and post-Newtonian order.

	[On more rigorous work...]

	That would be welcome to get around the inelegancy that some
perceive of the "standard" linearization approximation.
-- 
/Loren Petrich, the Master Blaster
/lip@s1.gov