Jack Sarfatti
Dec19-04, 07:18 AM
<jabberwocky><div class="vbmenu_control"><a href="jabberwocky:;" onClick="newWindow=window.open('','usenetCode','toolbar=no, location=no,scrollbars=yes,resizable=yes,status=no ,width=650,height=400'); newWindow.document.write('<HTML><HEAD><TITLE>Usenet ASCII</TITLE></HEAD><BODY topmargin=0 leftmargin=0 BGCOLOR=#F1F1F1><table border=0 width=625><td bgcolor=midnightblue><font color=#F1F1F1>This Usenet message\'s original ASCII form: </font></td></tr><tr><td width=449><br><br><font face=courier><UL><PRE>This problem arose from a student in the General Relativity course I am\nteaching. Comments would be helpful.\n\n"The Question is: What is The Question?" J.A. Wheeler\n\nThe {LC} connection field in GR curved spacetime is analogous to the EM\nvector potential A connection in internal space.\n\nThe tidal stretch-squeeze GCT tensor field = {LC} curl of itself is\nanalogous to the Maxwell EM Fuv field tensor.\n\n{LC} and A are both Cartan 1-forms. Curvature and Fuv are both Cartan\n2-forms.\n\nStudents have asked me if there is GR analog to the Maxwell decomposition\n\nF = -GradU + CurlA\n\nNote Cartan identities d^2 = 0\n\nCurlGradU = 0\n\nDivCurlA = 0\n\nGradU is a Cartan 1-form = dS\n\nCurl 1-form = d(1-form)\n\ndS is an exact 1-form\n\nCurlA =/= 0 if the 1-form A is NOT exact.\n\nCurlA =/= 0 is an exact 2-form = dA\n\nDivCurlA is d^2A = 0 vanishing 3-form\n\nAll vector fields in 3D are UNIQUELY defined if their circulation\ndensities (curl) and source densities (divergence) are given functions\nof the coordinates at all points in space, and if the totality of the\nsources, as well as the source density, is zero at infinity. Panofsky &\nPhillips "Classical Electricity and Magnetism" p. 2\n\nNote an infinite flat plate (vacuum domain wall) violates the required\nasymptotic flatness in the GR application.\n\nTherefore, The Question is, is there an analogous decomposition for GR\nat the level, not of the tidal curvature, but of the {LC} itself?\n\nThat is, thinking of {LC} as a "vector field" does it have a COORDINATE\nINDUCED (INERTIAL) divergence part from accelerating LNIF non-geodesic\nobservers + INTRINSIC GEOMETRY curl part?\n\nNow in the above analogy\n\n{LC} the connection of 1916 GR is a 1-form.\n\nTherefore, The Question is, is there a decomposition\n\nLC 1-form = d(Zero Form) + 1-form = exact 1-form + non-exact 1-form\n\nThe exact 1-form is the INERTIAL FIELD "coordinate-dependent" part.\n\nThe non-exact 1-form is the INTRINSIC GEOMETRY part.\n\nNote that the CURVATURE 2-form is\n\nd{LC} = d(non-exact 1-form)\n\nHowever, in non-Abelian GR\n\nd{LC} = {LC} covariant curl of itself!\n\nBack to EM\n\nU = Integral over a Green\'s function of a scalar density d = I(Gs)\n\nA = Integral over the same Green\'s function of a vector density B = I(GB)\n\nTherefore if\n\n{LC} = GradU + CurlA\n\nCurl{LC} = CurlCurlA\n\nBut self-consistency requires that\n\nB = Curl{LC}\n\nA = I(Gcurl{LC})\n\nTherefore, the CONJECTURE is, in analog form:\n\n{LC} = -GradI[Gs] + CurlI(GCurl{LC}) ?\n\nThe GLOBAL boundary conditions are in the Green\'s function G.\n\nINERTIAL FORCE coordinate part of {LC} = -GradI(Gs)\n\nINTRINSIC GEOMETRY PART of {LC} = CurlI(GCurl{LC})\n\nThe Curl and Grad are of course {LC} covariant operations so that this\nis all highly nonlinear. There is also the problem of the definition of\nthe Green\'s functions as well as the Integrals over the curved manifold.\n\nIt is not clear if this idea can be carried out.\n\nThere is also the side issue of whether a stationary homogeneous g-field\ni.e.\n\nmc^2{LC}^ztt + External Non-Gravity Force^z = 0\n\nIn the LNIF rest frame of a HOVERING non-geodesic observer at FIXED\nDISTANCE z from the FLAT PLATE source Tuv =/= 0, where\n\n{LC}^ztt = g/c^2\n\n{LC} Curl of {LC} = 0 where Tuv = 0\n\nCan be an EXACT SOLUTION of Einstein\'s Guv = (8piG/c^4)Tuv ?\n\nOne student in my relativity class claims Vilenken\'s vacuum domain wall\nis an example.\n\n</UL></PRE></font></td></tr></table></BODY><HTML>');"> <IMG SRC=/images/buttons/ip.gif BORDER=0 ALIGN=CENTER ALT="View this Usenet post in original ASCII form"> View this Usenet post in original ASCII form </a></div><P></jabberwocky>This problem arose from a student in the General Relativity course I am
teaching. Comments would be helpful.
"The Question is: What is The Question?" J.A. Wheeler
The {LC} connection field in GR curved spacetime is analogous to the EM
vector potential A connection in internal space.
The tidal stretch-squeeze GCT tensor field = {LC} curl of itself is
analogous to the Maxwell EM Fuv field tensor.
{LC} and A are both Cartan 1-forms. Curvature and Fuv are both Cartan
2-forms.
Students have asked me if there is GR analog to the Maxwell decomposition
F = -GradU + CurlA
Note Cartan identities d^2 =
CurlGradU =
DivCurlA =
GradU is a Cartan 1-form = dS
Curl 1-form = d(1-form)
dS is an exact 1-form
CurlA =/= if the 1-form A is NOT exact.
CurlA =/= is an exact 2-form = dA
DivCurlA is d^{2A} = vanishing 3-form
All vector fields in 3D are UNIQUELY defined if their circulation
densities (curl) and source densities (divergence) are given functions
of the coordinates at all points in space, and if the totality of the
sources, as well as the source density, is zero at infinity. Panofsky &
Phillips "Classical Electricity and Magnetism" p. 2
Note an infinite flat plate (vacuum domain wall) violates the required
asymptotic flatness in the GR application.
Therefore, The Question is, is there an analogous decomposition for GR
at the level, not of the tidal curvature, but of the {LC} itself?
That is, thinking of {LC} as a "vector field" does it have a COORDINATE
INDUCED (INERTIAL) divergence part from accelerating LNIF non-geodesic
observers + INTRINSIC GEOMETRY curl part?
Now in the above analogy
{LC} the connection of 1916 GR is a 1-form.
Therefore, The Question is, is there a decomposition
LC 1-form = d(Zero Form) + 1-form = exact 1-form + non-exact 1-form
The exact 1-form is the INERTIAL FIELD "coordinate-dependent" part.
The non-exact 1-form is the INTRINSIC GEOMETRY part.
Note that the CURVATURE 2-form is
d{LC} =[/itex] d(non-exact 1-form)
However, in non-Abelian GR
d{LC} = {LC} covariant curl of itself!
Back to EM
U = Integral over a Green's function of a scalar density d = I(Gs)
A = Integral over the same Green's function of a vector density B = I(GB)
Therefore if
{LC} = GradU + CurlA
Curl{LC} = CurlCurlA
But self-consistency requires that
B = Curl{LC}
A = I(Gcurl{LC})
Therefore, the CONJECTURE is, in analog form:
{LC} = -GradI[Gs] + CurlI(GCurl{LC}) ?
The GLOBAL boundary conditions are in the Green's function G.
INERTIAL FORCE coordinate part of {LC} = -GradI(Gs)
INTRINSIC GEOMETRY PART of {LC} = CurlI(GCurl{LC})
The Curl and Grad are of course {LC} covariant operations so that this
is all highly nonlinear. There is also the problem of the definition of
the Green's functions as well as the Integrals over the curved manifold.
It is not clear if this idea can be carried out.
There is also the side issue of whether a stationary homogeneous g-field
i.e.
mc^2{LC}^ztt + External Non-Gravity [itex]Force^z =
In the LNIF rest frame of a HOVERING non-geodesic observer at FIXED
DISTANCE z from the FLAT PLATE source Tuv =/= 0, where
{LC}^ztt = g/c^2
{LC} Curl of {LC} = where Tuv =
Can be an EXACT SOLUTION of Einstein's Guv = (8piG/c^4)Tuv ?
One student in my relativity class claims Vilenken's vacuum domain wall
is an example.
teaching. Comments would be helpful.
"The Question is: What is The Question?" J.A. Wheeler
The {LC} connection field in GR curved spacetime is analogous to the EM
vector potential A connection in internal space.
The tidal stretch-squeeze GCT tensor field = {LC} curl of itself is
analogous to the Maxwell EM Fuv field tensor.
{LC} and A are both Cartan 1-forms. Curvature and Fuv are both Cartan
2-forms.
Students have asked me if there is GR analog to the Maxwell decomposition
F = -GradU + CurlA
Note Cartan identities d^2 =
CurlGradU =
DivCurlA =
GradU is a Cartan 1-form = dS
Curl 1-form = d(1-form)
dS is an exact 1-form
CurlA =/= if the 1-form A is NOT exact.
CurlA =/= is an exact 2-form = dA
DivCurlA is d^{2A} = vanishing 3-form
All vector fields in 3D are UNIQUELY defined if their circulation
densities (curl) and source densities (divergence) are given functions
of the coordinates at all points in space, and if the totality of the
sources, as well as the source density, is zero at infinity. Panofsky &
Phillips "Classical Electricity and Magnetism" p. 2
Note an infinite flat plate (vacuum domain wall) violates the required
asymptotic flatness in the GR application.
Therefore, The Question is, is there an analogous decomposition for GR
at the level, not of the tidal curvature, but of the {LC} itself?
That is, thinking of {LC} as a "vector field" does it have a COORDINATE
INDUCED (INERTIAL) divergence part from accelerating LNIF non-geodesic
observers + INTRINSIC GEOMETRY curl part?
Now in the above analogy
{LC} the connection of 1916 GR is a 1-form.
Therefore, The Question is, is there a decomposition
LC 1-form = d(Zero Form) + 1-form = exact 1-form + non-exact 1-form
The exact 1-form is the INERTIAL FIELD "coordinate-dependent" part.
The non-exact 1-form is the INTRINSIC GEOMETRY part.
Note that the CURVATURE 2-form is
d{LC} =[/itex] d(non-exact 1-form)
However, in non-Abelian GR
d{LC} = {LC} covariant curl of itself!
Back to EM
U = Integral over a Green's function of a scalar density d = I(Gs)
A = Integral over the same Green's function of a vector density B = I(GB)
Therefore if
{LC} = GradU + CurlA
Curl{LC} = CurlCurlA
But self-consistency requires that
B = Curl{LC}
A = I(Gcurl{LC})
Therefore, the CONJECTURE is, in analog form:
{LC} = -GradI[Gs] + CurlI(GCurl{LC}) ?
The GLOBAL boundary conditions are in the Green's function G.
INERTIAL FORCE coordinate part of {LC} = -GradI(Gs)
INTRINSIC GEOMETRY PART of {LC} = CurlI(GCurl{LC})
The Curl and Grad are of course {LC} covariant operations so that this
is all highly nonlinear. There is also the problem of the definition of
the Green's functions as well as the Integrals over the curved manifold.
It is not clear if this idea can be carried out.
There is also the side issue of whether a stationary homogeneous g-field
i.e.
mc^2{LC}^ztt + External Non-Gravity [itex]Force^z =
In the LNIF rest frame of a HOVERING non-geodesic observer at FIXED
DISTANCE z from the FLAT PLATE source Tuv =/= 0, where
{LC}^ztt = g/c^2
{LC} Curl of {LC} = where Tuv =
Can be an EXACT SOLUTION of Einstein's Guv = (8piG/c^4)Tuv ?
One student in my relativity class claims Vilenken's vacuum domain wall
is an example.