Global conservation laws require global symmetries.

Tuv^,v = 0

,v = ordinary global T4 partial derivative

Tuv = stress-energy current density local AFFINE tensor of non-gravity matter fields i.e. only for global Poincare group!

globally conserved integrals for Pu exist no problem in accord with usual ideas of energy-momentum conservation.

In 1915 GR above picture breaks down replaced by

Tuv^;v = 0

;v = T4(x) covariant derivative from bilinear forms of e^a and its gradients (e.g. Rovelli's eq 2.89 in Ch 2 of "Quantum Gravity")

All the trouble of Roger Penrose's "nonlocal gravity field energy" leading to wrong theories like the Yilmaz theory embraced by Eric Davis et-al, abandoned by Yilmaz himself it is alleged, come from

[Tuv(matter) + tuv(gravity-matter)]^,v = 0

Now tuv(gravity-matter) though still an AFFINE T4 tensor is not a GCT T4(x) tensor.

This is THE TROUBLE.

General relativity breaks, in particular, global T4 symmetry inducing general transformations between locally coincident observers Alice and Bob in arbitrary relative motion including accelerations, jerks et-al to all orders. This is different from simply painting new coordinate labels for a fixed local observer. I will call the latter non-physical "redundant gauge transformations" as distinct from the former physical "local frame transformations." This issue also arises in Yang-Mills internal symmetry field theory with global special relativity for the internal transformations of the electromagnetic, weak and strong forces. "Solved" for special relativity quantum field theory by Faddeev & Popov using Feynman path integrals - very complicated with ghost fields that violate spin-statistics connection in 3D+1 spacetime, i.e. spin zero fermion ghosts. Feynman tried to apply this to GR "quantum gravity" as well, but with a global flat background I think missing all important non-perturbative strong field phenomena.

The special relativity total 4-momentum Pu of any field or set of coupled fields are global integrals of local stress-energy density currents on 3D spacelike surfaces.

Because of causality they are not observables, only the local current densities are.

Total 4-momentum Pu are the Lie algebra "charges" of the 4-parameter global translation subgroup T4 of the 10-parameter global Poincare group that is the universal symmetry group defining 1905 Special Relativity that is a GLOBAL theory, i.e. inertial frames span the entire universe. See Julian Barbour on difficulties with this conception, but that need not concern us here.

Carmeli et-al in "Gravitation, L(2,C) Gauge Theory and Conservation Laws" Ch 4 gives historical summary

1. Einstein's local gravity field nonsymmetric non-tensor

2. Landau-Lifshitz's symmetric fix

Both above have the Bauer paradox, i.e. the non-physical repainting of coordinates for the same observer of the Galilean relativity form

x^0' = x^0

x^j' = f^j'(x^j)

ji,j = 1,2,3

change tuv(gravity, matter)

3. Moller's version

4. SL(2,C) local tensor

however none of the above are satisfactory

Note that Carmeli's SL(2,C) is incomplete. He is simply doing Utiyama 1956 local gauging of only 6-parameter Lorentz group. Doing that gives both disclination curvature from dilocation torsion gap - his gauge field. Locally gauging only 4-parameter T4 to GCT T4(x) gives only disclination curvature without torsion (H. Kleinert) This causes confusion in the literature from formalists who do not have the H. Kleinert world crystal lattice physics picture to guide their intuition. Tensors came from crystallography in the first place.

Note adding quantum gravity makes the world crystal lattice spacing a fractal "wavelet" ~ N^1/6Lp, N is the wavelet ZOOM parameter, i.e. if you measure a length L, the quantum uncertainty in L is

(1) &L = (Lp^2L)^1/3 from Wigner

However, from Bekenstein -> World Hologram

(2) L^2 ~ NLp^2

substitute (2) in (1) to get

(3) &L ~ N^1/6Lp

N = number of Bekenstein BITS on the dominating causal "horizon," e.g. FUTURE RETRO-CAUSAL dark energy de Sitter observer-dependent cosmological horizon ~ 10^122 for our pocket universe in the multiverse, which as a bonus explains "Arrow of Time" (2nd Law of Thermodynamics).

The physical local frame transformations of 1915 GR come from locally gauging ONLY T4 to T4(x) the compensating gauge potential is the piece of tetrad A^a where

e^a = e^audx^u = I^a + @A^a

ds^2 = guvdx^udx^v = e^aea is Einstein's 1916 GR

@ is a dimensionless renormalizable coupling constant

I^aIa = ds^2 for 1905 special relativity

I^au(x) is curvilinear in non-geodesic frames defined relative to Minkowski spacetime. Otherwise it is Kronecker delta in Global Inertial Frames GIFs.

e^a is a Minkowski 4-vector.

Let L^a'a be a Lorentz group transformation

e^a' = L^a'ae^a

However,

I^a' = L^a'aI^a + X Not a Minkowski tensor because of inhomogeneous term X

A^a' = L^a'aA^a - X/@

keeping total e^a a Minkowski 4-vector

formally the same as e.g, U(1) EM internal symmetry gauge transformations

When A^a = 0, then X = 0

i.e. I^a' = L^aaI^a is strictly only for global T4 frame transformations, X =/= 0 is only for local T4(x) transformations.

The dislocation torsion field 2-form is the formal Yang-Mills spin 1-field

T^a = de^a + w^abce^b/\e^c

The disclination curvature 2-form (that can include torsion parts) is

R^a^b = dS^a^b + S^ac/\S^c^b

S^a^b = - S^b^a = S^a^budx^u is spin connection 1-form where

S^a^b = w^a^bce^c

Note that when A^a = 0

R^a^b = 0

Einstein's vacuum field equations are Rovelli's (2.9)

{I,J,K,L}[e^I/\R^J^K + (Dark Energy Curvature)e^I/\e^J/\e^K] = 0

{I,J,K,L} = antisymmetric Minkowski tensor - this is a classical cubic Yang-Mills field equation that is t'Hooft renormalizable if quantized.

Note the substratum decomposition

e^a = I^a + @A^a

in my emergent gravity world hologram model

@ = 1/N^1/3

A^a = M^a^a diagonal elements of world hologram "destiny" (retro-causal) matrix of 8 vacuum ODLRO post-inflation Goldstone phases from 9 real Higgs spin 0 fields.

torsion field part of spin connection is

S^a^b = M^[a,b] = - S^b^a

in gr-qc/0602022