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 1forms
>
> dx^0 = cdt
> dx^1 = dr
> dx^2 = dtheta
> dx^3 = dphi
>
> With the Grassmann basis sort of "Clifford" algebra" of 2^4 = 16
> pforms, p = 0,1,2,3,4
>
> *pform = (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 LeviCivita 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 spacetime
>
> My original suggestion gthetatheta = gphiphi = 1 will not work here
> because physically we have a stretchsqueeze 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 GRAVITYMATTER 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 1forms
>
> 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 RiemannChristoffel tensor is now dimensionally selfconsistent,
> 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 nonspinning 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
