Gravity energy and entropy is nonlocal
The "Euclidean space" E is the tangent fiber space of a metric space that should not be confounded with local coordinate patches in the base space.
"Riemann's surfaces provided the first instances of the general notion of a manifold, which is a space that can be thought of as 'curved' in various ways, but where LOCALLY (i.e. IN A SMALL ENOUGH NEIGHBORHOOD OF ANY OF ITS POINTS) it looks like a piece of ordinary Euclidean space." 8.1 p. 138 Penrose RR
In the case of spacelike 3D you cannot use the global form r^2 = x^2 + y^2 + z^2 where r reaches between 2 arbitrary points of the manifold with a general metric guv(P) specified. You can do so only when the metric field is globally flat. (In answer to Zielinski)
In general, the Taylor series expansion of the "foreign" metric field IMPOSED on the pre-metrical manifold is
guv(P + &P) = guv(P) + guv(P),w(&P)^w + (1/2!)guv(P),w,l(&P)^w(&P)^l
The special LIF geodesic "normal coordinates" correspond to the CHOICE OF LOCAL REPRESENTATION in which guv(P),w = 0 - different for each P in the general curved metric field.
That is, we have a FUNCTION FIBER SPACE of gauge equivalent metric fields at each P and we look for a CRITICAL POINT in that function space.
So that, where dx^w --> (&P)^w
ds^2 = guv(P + &P)dx^udx^v ~ guv(P)dx^udx^v + terms of 4th order in smallness.
Now for the LOCAL field equation
Ruv - (1/2)Rguv = 8piTuv
G = c = h = kB = 1 absolute Planck units convention
The LOCAL RICCI MATERIAL CURVATURE information is in the second order (1/2!)guv(P),w,ldx^wdx^l
Apparently the NONLOCAL Weyl conformal VACUUM curvature information is in all the other terms!
The Ricci flat vacuum Ruv = 0 means zero local Ricci curvature everywhere-when.
Note that zero point DARK ENERGY /\zpfguv counts as a "material source" of LOCAL Ricci curvature. This zero point energy induces ZERO GRAVITY ENTROPY if we use Penrose's conjecture, i.e. conformally flat with zero gravity entropy.
The event horizon in the Ricci FLAT Kerr solution i.e. Ruv = 0 is pure NONLOCAL CONFORMAL CURVATURE and it has NONLOCAL GRAVITY ENTROPY and NONLOCAL ENERGY-MOMENTUM - the NONLOCALITY is key here! This is what is implied by Penrose's CONJECTURE. It shows why the Yilmaz idea is fundamentally wrong if you believe Penrose's conjecture - it is clear why the Ricci flat gravity energy-momentum is NONLOCAL i.e. from derivatives in the metric field higher than second order!
Classically, partial differential equations higher than second order are known to be NONLOCAL.
Jack Sarfatti wrote to Zielinski:
They are being sloppy that's all. There is NOTHING in the quote you sent that requires the metric equation
r^2 = x^2 + y^2 + ... (1)
That's an INDEPENDENT postulate like Euclid's 5th!
The point is that you cannot have global (1) and the generic
ds^2 = guvdx^udx^v (2)
together in same problem unless you severely constrain allowed guv ad hoc so that 4th rank spatial curvature vanishes like K = 0 FLRW metric for example.