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 P: n/a Jack Sarfatti wrote: > 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. Agreed, I'd rather call this thread *units in tensor analysis*, that can be ambiguous, let's do a primitive example. 1)Draw a straight Line on a blank piece of paper. 2)Trace that Line using 1 inch graph paper. 3)Trace using 1 cm graph paper. All 3 lines are equal, in all CS's (independant of the graph paper) and so the Line is invariant. Let's set X^1 = inches and x^1 = cm's then X^1 = 2.54 x^1 , ie. 1 inch = 2.54 cm's, is the tranformation. The invariant *Line* is given by, Line = E_1 X^1 = e_1 x^1 where E_1 = 1/inch in direction X^1, e_1 = 1/cm in direction x^1 . Please note Line has no units on the blank paper (1) above. It can only have a measurement in (2) or (3) or other graph paper. The "E_1" and "e_1" are refered to as covariant basis vectors. We'll follow the usual convention and define the "metric tensor" by scalar (dot) product like, g_UV = E_U.E_V and g_uv = e_u.e_v . It follows g_UV and g_uv have units of 1/area, i.e. 1/inch^2 and 1/cm^2 respectively, so that Line is invariant (has no units in any CS). We then define the usual "norm" by Line^2 = g_UV X^U X^V = g_uv x^u x^v = invariant. You can test that using the transformation, (check out a handy 30cm ~ 12 inch ruler), and find the invariant to a *unitless* scalar as all scalars must be. Since 1 second =~ 3*10^5 km by international agreement, extending the above to 4D SpaceTime CS's is straightfoward. Below are some issues Jack highlights... Regards Ken S. Tucker > 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. > > 1 > > dx^u > > dx^u/\dx^v > > dx^u/\dx^v/\dx^w > > dx^u/\dx^v/\dx^w/\dx^l > > 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. > > For example > > 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 > > Note that > > ,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^2x^u/ds^2 > > and > > d(x^u - x'^u)/ds