Re: Is general relativity incompatible with the Newtonian limit?


Juan R. wrote:
> When one ask by the Newtonian limit of GR one heard in Baez page
>
> http://math.ucr.edu/home/baez/RelWWW/wrong.html
>
> that "The theorem stating that gtr does indeed go over to Newtonian
> gravitostatics in the very weak field, very slow motion limit is proven
> in detail in almost every gtr textbook."
>
> Then one read one of those textbooks and discover like NG is 'obtained'
> from GR, but one finds that derivation is NOT rigorous.
>
> Then one takes a more rigorous textbook (Wald) and one discovers that,
> effectively, NG is NOT derived from GR in the linear regime.
[etc.]

In effect, you're looking for the solution to
lim c->infinity (The Theory of GR)
which would be a curved spacetime theory of gravity for Newtonian
spacetime. That's Galilean General Relativity. Wheeler developed this
early on, I believe, in the 1960's and in more recent times Jadczyk (an
occasional poster here) has touched on it, as well.

> Of course, if one takes limit c --> infinite then tGT --> tNG and Phy
> transforms into an instantaneous field (still is not a potential) but
>
> the Schwartzild metric
>
> g00 = (1 - 2Phy/c^2) --> 1
>
> and
>
> gRR = -(1 - 2Phy/c^2)^-1 --> -1

Things get a little subtle in the limit since the spacetime underlying
Minkowski is of signature (+++-), whereas the spacetime underlying
Newtonian physics is of signature (+++0), which might be considered as
the "hyperbolic" and "parabolic" cases of a general family that
includes as its "elliptical" case the Euclidean 4-space (++++).

For the signature (+++0), the metric splits into 2 independent
constructs. In Lorentzian spacetime, you can equivalently write the
metric as (+++-) as a "space-like" metric; or as (---+) as a measure of
proper time. This is the equivalence that splits up as you go from
(+++-) to (+++0).

The Riemannian spacetime of General Relativity goes over into a
spacetime that is NOT Riemannian. The main task of Wheeler's
development was to show what kind of spacetime that underlying Galilean
General Relativity is.

.

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