Consistency of physical dimensions in General Relativity
OK we hit a temporary snag solved below. In elementary physics first rule is to check your units and physical dimensions. Don't mix apples with oranges etc. Yet GR theorists do that nonchalantly and sloppily even in text books.
For example, the SSS metric is typically written as
gtt = -(1 - 2GM/c^2r) [dimensionless]
grr = (1 - 2GM/c^2r)^-1 [dimensionless]
But hold on
gthetatheta = r^2 [area]
gphiphi = r^2sin^2theta [area]
Where we have the incommensurate basis set of Cartan 1-forms
dx^0 = cdt
dx^1 = dr
dx^2 = dtheta
dx^3 = dphi
With the Grassmann basis sort of "Clifford" algebra" of 2^4 = 16 p-forms, p = 0,1,2,3,4
*p-form = (4 - p)-form, when N = 4.
This gives an incommensurate set of Levi-Civita connection field components in the hovering LNIFs
(LC)^001 = [2(1 - 2GM/c^2r)]^-1 (1 - 2GM/c^2r),r [1/length]
(LC)^122 = -r(1 - 2GM/c^2r) [length]
(LC)^233 = -sinthetacostheta [dimensionless]
(LC)^100 = (1/2)(1 - 2GM/c^2r),r(1 - 2GM/c^2r) [1/length]
(LC)^133 = -(rsin^2theta)(1 - 2GM/c^2r) [length]
(LC)^313 = (LC)^212 = 1/r [1/length]
(LC)^111 = (1/2)(1 - 2GM/c^2r)(1 - 2GM/c^2r)^-1,r [1/length]
(LC)^323 = cottheta [dimensionless]
all other (LC) identically & globally zero in this FRAME BUNDLE of hovering LNIFs all over this toy model 4D space-time
My original suggestion gthetatheta = gphiphi = 1 will not work here because physically we have a stretch-squeeze tidal curvature that requires the theta dependence in addition to the radial dependence.
Nevertheless we MUST use commensurate infinitesimal basis sets for our local frames and the (LC) components MUST all be of the same physical dimension in order to define consistent Diff(4) covariant derivatives.
Au;v = Au,v - (LC)uv^wAw
The GRAVITY-MATTER MINIMAL COUPLING SUM (LC)uv^wAw must have physically commensurate (LC) components because Au is arbitrary! For example, Au can be the Maxwell EM vector potential, and all the components of Au have same physical dimensions.
Therefore ALL the (LC) MUST obey [LC] = 1/length
So, how to we accomplish this?
Simple, use engineering dimensional analysis and introduce a scale L.
What is L? Is L = Lp = (hG/c^3)^1/2 or is L = GM/c^2 or?
For now let's call it "L".
Therefore the SSS metric is now the physically commensurate dimensionless array
gtt = -(1 - 2GM/c^2r)
grr = (1 - 2GM/c^2r)^-1
gthetatheta = (r/L)^2
gphiphi = (r/L)^2sin^2theta
Where we NOW have the commensurate set of basic 1-forms
dx^0 = cdt
dx^1 = dr
dx^2 = Ldtheta
dx^3 = Ldphi
,0 = (1/c),t
,1 = ,r
,2 = (1/L),theta
,3 = (l/L),phi
Therefore, all the (LC) are now [1/length]
LC)^001 = [2(1 - 2GM/c^2r)]^-1 (1 - 2GM/c^2r),r
(LC)^122 = -(r/L^2)(1 - 2GM/c^2r)
(LC)^233 = -(1/L)sinthetacostheta
(LC)^100 = (1/2)(1 - 2GM/c^2r),r(1 - 2GM/c^2r)
(LC)^133 = -(rsin^2theta/L^2)(1 - 2GM/c^2r)
(LC)^313 = (LC)^212 = 1/r
(LC)^111 = (1/2)(1 - 2GM/c^2r)(1 - 2GM/c^2r)^-1,r
(LC)^323 = (1/L)cottheta [dimensionless]
The Riemann-Christoffel tensor is now dimensionally self-consistent, i.e. 1/Area
Note that L cancels out of the frame invariant
ds^2 = guvdx^udx^v
and it must cancel out of any local physical quantity.
In particular it must cancel out of the geodesic equation and the tidal geodesic deviation.
It's pretty obvious that L will be physically locally unobservable. It's a bit like the Weyl gauge parameter.
Note that the geodesic equation for a non-spinning point test particle is
D^2x^u/ds^2 = d^2x^u/ds^2 - (LC)^uvw(dx^v/ds)(dx^w/ds) = 0
So the 1/L's in the (LC)s cancel the L's in x2 & x^3
Similarly with geodesic deviation
d(x^u - x'^u)/ds = R^uvwl(x^v - x'^v)(dx^w/ds)(dx^l/ds)
Note that (LC)^uvw and R^uvwl are NEVER MEASURED DIRECTLY in isolation. What is measured is
d(x^u - x'^u)/ds