Author 
Message 
Jack Sarfatti science forum Guru
Joined: 29 Apr 2005
Posts: 487

Posted: Fri May 13, 2005 3:28 pm Post subject:
"Weyl" type hidden gauge invariance 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 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 

Back to top 


Ken S. Tucker science forum Guru
Joined: 30 Apr 2005
Posts: 1230

Posted: Sun May 15, 2005 1:13 am Post subject:
Re: "Weyl" type hidden gauge invariance in General Relativity



Jack Sarfatti wrote:
Quote:  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
Quote:  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 


Back to top 


Google


Back to top 



The time now is Sat Feb 25, 2017 10:45 pm  All times are GMT

