13.1 on Robert Wald's "General Relativity" has a nice discussion of measurement theory in general relativity in terms of "families of observers" that agrees with what I say below on the distinction between local objective invariants (Platonic forms, Jungian Archetypes) and their observer-dependent subjective faithful representations and provides more details.
Diffeomorphisms are not physical, they are too general with tremendous gauge freedom redundancy. We only want their sub Lie group of isometries ("Killing" (man's name) vector fields) so that different observers can meaningfully compare their subjective data, put it into the machine and crank out the same local invariants from the theory. Their goal is each to get the same set of real numbers for observations of equivalent happenings.
"when (and only when) Diff(4) is an isometry, we can use [it] ... to map our original family of physical observers associated with the [tetrads e^a] into a new family of observers associated with [tetrads e^a'] ..." p. 343.
"Here upon we're all agreed all that we two will agree to. To entrust you in The Art, elemental, fundamental ..." W.S. Gilbert, Yeoman of the Guard
Given a set of tetrad 1-forms e^a then as spinors we have
e^AA' = e^a(Newman-Penrose)a^AA'
Each A index = 0,1 is 1 qubit
IT FROM QUBIT
e^BB' = I^BB(flat)' + A(warped)^BB'
Now we come to a strange miracle - the Bell quantum teleportation pair states appear.
Use qubit |A+> of Wald type (0,1;0,0) & qubit |A'->of Wald type (0,0;0,1) (p. 347) each in 2 complex dimensions.
Then, for special relativity, space-time AKA emerges from 2-qubit strings |A> & |A'>.
First take the local diagonal at the same local objective IT coincidence P = P', the space-time operators are entangled 2-qubit strings in 4 complex dimensions, i.e., spinor tensors of type (0,1;0,1) in Wald's notation Ch 13). Thus, we have the pair entangled qubit states
|t^AA'(P)> = (2)^-1/2[|A(P)+>|A'(P)+> + |A(P)->|A'(P)->]
Note |t^0^0'>, |t^0^1'>, |t^1^0'> |t^1^1'> span 4 complex dimensions. Similarly for
|x^AA'(P)> = (2)^-1/2[|A(P)+>|A'(P)-> + |A(P)->|A'(P)+>]
|y^AA'(P)> = (2)^-1/2[|A(P)+>|A'(P)-> - |A(P)->|A'(P)+>]
|z^AA'(P)> = (2)^-1/2[|A(P)+>|A'(P)+> - |A(P)->|A'(P)->]
We can generalize the above to the non-local pair states
|t^AA'(P,P')> = (2)^-1/2[|A(P)+>|A'(P')+> + |A(P)->|A'(P')->]
|x^AA'(P,P')> = (2)^-1/2[|A(P)+>|A'(P')-> + |A(P)->|A'(P')+>]
|y^AA'(P,P')> = (2)^-1/2[|A(P)+>|A'(P')-> - |A(P)->|A'(P')+>]
|z^AA'(P,P')> = (2)^-1/2[|A(P)+>|A'(P')+> - |A(P)->|A'(P')->]
We may need to put a path-dependent anholonomic propagator U(P<--->P') here, one for each homotopy equivalence class of paths <---> homotopic to each other from topological obstructions like black hole event horizons and observer dependent horizons in dark energy dominates deSitter space-time.
Below everything is local on the diagonal P = P' for Einstein's "local coincidences" defined by Rovelli in Ch II of his 'Quantum Gravity."
guv ---> gAA'BB' = EABE*A'B" (13.1.15) p. 349 Wald's GR
|EAB> = |A+>|B-> - |A->|B+> = -|EBA>
Choose the following canonical basis with components (A), (B) etc. The 2x2 "basis" matrices are
E(A)(B)11 = 0
E(A)(B)12 = 1
E(A)(B)21 = -1
E(A)(B)22 = 0
t^(A)^(A') = Pauli Spin Matrix(t) (with added factor -1/2^1/2)
x^(A)^(A') = Pauli Spin Matrix(x)
y^(A)^(A') = Pauli Spin Matrix(y)
z^(A)^(A') = Pauli Spin Matrix(z)
These 2x2 matrices do not commute, so we have a non-commutative geometry in spinor tensor space. They have the same formal Lie algebra of internal SU(2) of the weak force fiber even though they represent Minkowski spacetime base space.
Any globally flat Minkowski world vector first rank tensor can be represented as the second rank spinor (mod - 1/2^1/2) in a 2x2 matrix representation
V^A^A' = V^t Pauli Spin Matrix(t) + Vx Pauli Spin Matrix(t) + (13.1.29) p. 351 Wald
Lorentz transformations are SL(2C) similarity transformations in this notation.
These single qubit "potential" spinors map to null vectors on the light cone in globally flat Minkowski spacetime. The qubit is the square root of a null world vector V^^a^a', i.e.
|V^A^A'*> = |A+>|A'+>*
One can also make 2nd rank antisymmetric null world field tensors (AKA null bivectors).
F^A^A'^B^B' = |A>||B>E*^A'^B' + |A'>*||B'>*E^A^B
F^2 = 0
FV = 0
(indices understood) p. 352 Wald.
This null F defines the Penrose "null flag".
To include T4 translations we must extend SL(2,C) to ISL(2,C) covering the 10-parameter Poincare group whose local gauging in all non-gravity field actions i.e. equivalence principle gives curvature + torsion i.e. independent 16 non-trivial tetrad components and 24 antisymmetric spin connection components packaged into 4 tetrad 1-forms e^a and 6 spin-connection 1-forms W^a^b in tangent vector/cotangent form fibers.
Lie derivatives of spinor fields are only definable for Killing isometries. You cannot do it for a general Diff(4) p. 353 Wald.
Newman-Penrose components are using the 16 tetrad components
(Sigma)^aAA' = e^a0tAA' - e^a1xAA' - e^a2yAA' - e^a3zAA'
so this shows how to generalize to curved and torsioned spacetime!
"The Question is: What is The Question?" John Archibald Wheeler
The total energy of the universe is obviously not conserved in the standard model that fits observations to ~ 1% precision.
The total w = -1 dark energy repulsive cosmic antigravity field accelerating the 3D space expansion of the universe is obviously not conserved. The total w = -1 dark energy content of our universe from Rocky Kolb's first slide above scales as a(t)^3.
The total w = 0 on-mass shell finite rest mass matter-energy is conserved scaling as a(t)^0 = 1.
The total zero rest mass w = +1/3 on light cone radiation energy is not conserved, it scales as a(t)^-1 -> zero from the cosmic redshift. Note that the chemical potential of radiation is zero, therefore there is no general global conservation law for radiation.
The general point here is that global space-time conservation laws are not fundamental unlike the local versions, which are satisfied e.g.
Tuv(matter)^;v = 0
locally in 1915 GR with the (LC) connection covariant partial derivative ;v.
Global conservation laws require
[Tuv(Matter) + tuv(Matter-Gravity)]^,v = 0
where ,v is the ordinary partial derivative.
This cannot be done in general in curved space-time with tuv(Matter-Gravity) as a kosher localized T4 tensor. tuv(Matter-Gravity) is a pseudo-tensor because in LNIF's the energy-momentum of the non-geodesic detectors powered by non-gravity forces makes a contribution to the vacuum gravity field that, by the equivalence principle, cannot be locally distinguished from the "real gravity field." Garbage in -> garbage out. Ask a stupid question, get a stupid answer. Trying to globally conserve total energy, trying to conserve total linear and angular momentum in a generally curved and torsioned spacetime is a stupid thing to try to do. We already know this from Noether's theorem. This is simply because the global Poincare group is locally gauged and all we can hope for is to locally conserve the total stress-energy current densities, which in fact is the case.
((-detguv)^1/2Tuv(matter))^,v + (-detguv)^1/2(LC(observer))u^v^wTvw(matter) = 0
Actual observations given above show that the total energy of the universe is not conserved. It's time to slay that Sacred Cow.
What the pure mathematicians, who lack physical understanding do not get, is that any representation of curved space-time guv is observer dependent i.e. relative to any conceivable network of ideal local observers on arbitrary worldlines inside the local light cone field. For example, in the non-rotating black hole outside the event horizon rs/r < 1
g00 = -1/g11 = 1 - rs/r
is only for that special class of static LNIF "shell" (J.A. Wheeler's term) observers at fixed r without orbital angular momentum. They need to fire rockets to stay in place. Note that warp drive "UFO" observers see a different metric field representation. What "Diff(4)" (local T4) frame shifts do is to connect different networks of local observers. That's the physical meaning of the abstract math missed by many of the formalists in the field. The local relation between objective reality and the observer's experience is
ds^2(objective observer invariant) = guv(observer dependent)dx^udx^v
Space-time physics is local only because curved space-time is emergent from a local vacuum ODLRO world hologram Higgsian field with several coherent Goldstone phases that encode the 10^122 bits of our universe retrocausally from far future Omega to past Alpha at The Creation in our deSitter Universe.
"If we knew what it was we were doing, it would not be called research, would it?"
- Albert Einstein