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CGR_JAMA
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Conformal Electromagnetic Relativity
1 Abstract
A Lagrangian based on geometric variables (Riemann metric and generic affine connection)
is proposed which maintains compatibility with General Relativity.
Compatibility means that some solutions of G.R. are also solutions of the resulting theory
since this last generates under the proper conditions the known dynamics for
Gravitation and Electromagnetism along with some sort of continuous stress-energy
matter tensor coming from geometry itself.
The theory is developed over the tangent space of a four-dimensional real manifold
(usual space-time) where a conformal symmetry can be defined acting on the metric tensor,
like in Weyl's theory.
The electromagnetic four-vector potential is identified with the
space-time's torsion trace and the stress-energy matter tensor is derived from the
symmetric part of the connection minus the standard Christoffel one.
Since this theory is far too long for being posted on this forum due to size limits
the following condensation was extracted for presenting the first ideas.
Invitation is opened for downloading the full article from the following URL:
"[PLAIN Gauge Relativity.pdf"]http://ConformalGaugeRelativity.cubi.ca/Files/Conformal Gauge Relativity.pdf[/URL]
Comments about this extract or other topics in the article will be followed on this thread.
* ! Due to LaTex rendering, upper-lower index alignement may not be correct: refer to the original document for having the accurate expressions. !*
2 Geometrical Objects
2.1 Metric and Generic Connection
Consider a 4-dimensional real manifold with a Riemann metric [tex] g^{ij} [/tex] and
a generic affine connection [tex]\tilde{\Gamma }^{i} _{jk} [/tex] , non-symmetric in its lower index pair
which may be expressed as the sum of a symmetric connection and a torsion tensor:
[tex](2.1.1)\ \ \ \ \tilde{\Gamma }^{i} _{jk} \equiv \bar{\Gamma }^{i} _{jk} +\hat{\Gamma }^{i} _{jk}[/tex]
[tex](2.1.2)\ \ \ \ \ \ \ \ \ \ \ \ /\ \ \ \ \bar{\Gamma }^{i} _{jk} =\bar{\Gamma }^{i} _{kj}\ \ \ \ ,\ \ \ \ \hat{\Gamma }^{i} _{jk} =-\hat{\Gamma }^{i} _{kj}[/tex]
The usual standard connection is the one derived from the metricity condition:
[tex](2.1.3)\ \ \ \ \nabla _{k} g^{ij} \equiv 0\ \ \ \ \ \to \ \ \ \ \Gamma ^{k} _{ij} ={\tfrac{1}{2}} .g^{kr} .(\partial _{i} g_{rj} +\partial _{j} g_{ir} -\partial _{r} g_{ij} )[/tex]
2.2 Notation
The following tensors and notations will be used:
[tex](2.2.1)\ \ \ \ \hat{\Gamma }_{i} \equiv \hat{\Gamma }^{k} _{ik}\ \ \ \ ,\ \ \ \ \stackrel{\rightharpoonup}{\tilde{\Gamma }}_{i} \equiv \tilde{\Gamma }^{k} _{ik}[/tex]
[tex](2.2.2)\ \ \ \ \partial \hat{\Gamma }_{ij} \equiv \partial _{i} \hat{\Gamma }_{j} -\partial _{j} \hat{\Gamma }_{i}[/tex]
[tex](2.2.3)\ \ \ \ \tilde{R}^{i} _{jkl} \equiv \partial _{k} \tilde{\Gamma }^{i} _{jl} -\partial _{l} \tilde{\Gamma }^{i} _{jk} +\tilde{\Gamma }^{i} _{rk} .\tilde{\Gamma }^{r} _{jl} -\tilde{\Gamma }^{i} _{rl} .\tilde{\Gamma }^{r} _{jk}[/tex]
[tex](2.2.4)\ \ \ \ \stackrel{\rightharpoonup}{\tilde{R}}_{jl} \equiv \tilde{R}^{k} _{jkl}\ \ \ \ ,\ \ \ \ \stackrel{\leftharpoonup}{\tilde{R}}_{kl} \equiv \tilde{R}^{j} _{jkl}[/tex]
[tex](2.2.5)\ \ \ \ \ \ \bar{\stackrel{\rightharpoonup}{\tilde{R}}}_{ij} \equiv {\tfrac{1}{2}} .(\stackrel{\rightharpoonup}{\tilde{R}}_{ij} +\stackrel{\rightharpoonup}{\tilde{R}}_{ji} )\ \ \ \ ,\ \ \ \ \hat{\stackrel{\rightharpoonup}{\tilde{R}}}_{ij} \equiv {\tfrac{1}{2}} .(\stackrel{\rightharpoonup}{\tilde{R}}_{ij} -\stackrel{\rightharpoonup}{\tilde{R}}_{ji} )[/tex]
[tex](2.2.6)\ \ \ \ \bar{\stackrel{\rightharpoonup}{\tilde{R}}}\equiv g^{kr} .\bar{\stackrel{\rightharpoonup}{\tilde{R}}}_{kr}[/tex]
2.3 Variational Derivative
Define the variational derivative as the following operator:
[tex](2.3.1)\ \ \ \ \tilde{\Pi }_{k} V^{i} \equiv \tilde{\nabla }_{k} V^{i} -(\stackrel{\rightharpoonup}{\tilde{\Gamma }}_{k} -\Gamma _{k} ).V^{i} [/tex]
3 Conformal Electromagnetic Gravity Action
The unification action proposed is the following:
[tex](3.1)\ \ \ \ \[I_{CEMG} \equiv k_{0} .\int _{D}\sqrt{-\left|g_{\bullet \bullet } \right|} .g^{kr} .g^{sl} .\left(\bar{\stackrel{\rightharpoonup}{\tilde{R}}}_{kr} .\bar{\stackrel{\rightharpoonup}{\tilde{R}}}_{sl} +k_{1} .(\hat{\stackrel{\rightharpoonup}{\tilde{R}}}_{ks} -{\tfrac{1}{2}} .\stackrel{\leftharpoonup}{\tilde{R}}_{ks} ).(\hat{\stackrel{\rightharpoonup}{\tilde{R}}}_{rl} -{\tfrac{1}{2}} .\stackrel{\leftharpoonup}{\tilde{R}}_{rl} ).+\right.[/tex]
[tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \[+k_{2} .\partial \hat{\Gamma }_{ks} .\partial \hat{\Gamma }_{rl} +k_{3} .(\hat{\stackrel{\rightharpoonup}{\tilde{R}}}_{ks} -{\tfrac{1}{2}} .\stackrel{\leftharpoonup}{\tilde{R}}_{ks} ).\partial \hat{\Gamma }_{rl} +\] [/tex]
[tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \[+k_{4} .g^{mn} .g{}_{op} .(\tilde{R}^{o} _{ksm} +\tilde{R}^{o} _{mks} +\tilde{R}^{o} _{smk} ).(\tilde{R}^{p} _{r\ln } +\tilde{R}^{p} _{nrl} +\tilde{R}^{p} _{\ln r} )+\] [/tex]
[tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \[\left. \mathop{}\limits^{} -({\tfrac{1}{2}} .k_{1} +{\tfrac{3}{8}} .k_{3} +3.k_{4} ).(\tilde{R}^{m} _{kso} +\tilde{R}^{m} _{oks} +\tilde{R}^{m} _{sok} ).(\tilde{R}^{o} _{rlm} +\tilde{R}^{o} _{mrl} +\tilde{R}^{o} _{lmr} )\right).d\Omega \] [/tex]
[tex](3.2)\ \ \ \ \ \ \ \ \ \ \ \ /\ \ \ \ 3.k_{2} +k_{3} \equiv {\tfrac{32.k_{g} .\lambda _{c} }{3.c^{4} .c_{1} ^{2} }} \ne 0 [/tex]
[tex]Where:\ \ \ \ \ \ \ \ k_{1} ,k_{2} ,k_{3} ,k_{4} \ \ \ \ \ \ \ \ \ \ \ \ Lagrangian\ \ constants[/tex]
[tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k_{g} \ \ \ \ \ \ \ \ \ \ \ \ \ \ Newton's\ \ gravitational\ \ constant[/tex]
[tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \lambda _{c} \ \ \ \ \ \ \ \ \ \ Cosmological\ \ constant[/tex]
[tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ c \ \ \ \ \ \ \ \ \ \ \ \ Speed\ \ of\ \ light[/tex]
[tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ c_{1} \ \ \ \ \ \ \ \ \ \ \ \ \ \ Electromagnetic\ \ coupling\ \ constant[/tex]
This Lagrangian is conformally invariant on the metric only on a four-dimensional manifold:
[tex](3.3)\ \ \ \ \ \ \ \ L(g^{ij} )=L(e^{\phi } .g^{ij} ) \ \ \ \ \ \ \ \ \leftrightarrow \ \ \ \ \ \ \ \ n=4[/tex]
(continues on "Conformal Electromagnetic Relativity (2/2)"...)
1 Abstract
A Lagrangian based on geometric variables (Riemann metric and generic affine connection)
is proposed which maintains compatibility with General Relativity.
Compatibility means that some solutions of G.R. are also solutions of the resulting theory
since this last generates under the proper conditions the known dynamics for
Gravitation and Electromagnetism along with some sort of continuous stress-energy
matter tensor coming from geometry itself.
The theory is developed over the tangent space of a four-dimensional real manifold
(usual space-time) where a conformal symmetry can be defined acting on the metric tensor,
like in Weyl's theory.
The electromagnetic four-vector potential is identified with the
space-time's torsion trace and the stress-energy matter tensor is derived from the
symmetric part of the connection minus the standard Christoffel one.
Since this theory is far too long for being posted on this forum due to size limits
the following condensation was extracted for presenting the first ideas.
Invitation is opened for downloading the full article from the following URL:
"[PLAIN Gauge Relativity.pdf"]http://ConformalGaugeRelativity.cubi.ca/Files/Conformal Gauge Relativity.pdf[/URL]
Comments about this extract or other topics in the article will be followed on this thread.
* ! Due to LaTex rendering, upper-lower index alignement may not be correct: refer to the original document for having the accurate expressions. !*
2 Geometrical Objects
2.1 Metric and Generic Connection
Consider a 4-dimensional real manifold with a Riemann metric [tex] g^{ij} [/tex] and
a generic affine connection [tex]\tilde{\Gamma }^{i} _{jk} [/tex] , non-symmetric in its lower index pair
which may be expressed as the sum of a symmetric connection and a torsion tensor:
[tex](2.1.1)\ \ \ \ \tilde{\Gamma }^{i} _{jk} \equiv \bar{\Gamma }^{i} _{jk} +\hat{\Gamma }^{i} _{jk}[/tex]
[tex](2.1.2)\ \ \ \ \ \ \ \ \ \ \ \ /\ \ \ \ \bar{\Gamma }^{i} _{jk} =\bar{\Gamma }^{i} _{kj}\ \ \ \ ,\ \ \ \ \hat{\Gamma }^{i} _{jk} =-\hat{\Gamma }^{i} _{kj}[/tex]
The usual standard connection is the one derived from the metricity condition:
[tex](2.1.3)\ \ \ \ \nabla _{k} g^{ij} \equiv 0\ \ \ \ \ \to \ \ \ \ \Gamma ^{k} _{ij} ={\tfrac{1}{2}} .g^{kr} .(\partial _{i} g_{rj} +\partial _{j} g_{ir} -\partial _{r} g_{ij} )[/tex]
2.2 Notation
The following tensors and notations will be used:
[tex](2.2.1)\ \ \ \ \hat{\Gamma }_{i} \equiv \hat{\Gamma }^{k} _{ik}\ \ \ \ ,\ \ \ \ \stackrel{\rightharpoonup}{\tilde{\Gamma }}_{i} \equiv \tilde{\Gamma }^{k} _{ik}[/tex]
[tex](2.2.2)\ \ \ \ \partial \hat{\Gamma }_{ij} \equiv \partial _{i} \hat{\Gamma }_{j} -\partial _{j} \hat{\Gamma }_{i}[/tex]
[tex](2.2.3)\ \ \ \ \tilde{R}^{i} _{jkl} \equiv \partial _{k} \tilde{\Gamma }^{i} _{jl} -\partial _{l} \tilde{\Gamma }^{i} _{jk} +\tilde{\Gamma }^{i} _{rk} .\tilde{\Gamma }^{r} _{jl} -\tilde{\Gamma }^{i} _{rl} .\tilde{\Gamma }^{r} _{jk}[/tex]
[tex](2.2.4)\ \ \ \ \stackrel{\rightharpoonup}{\tilde{R}}_{jl} \equiv \tilde{R}^{k} _{jkl}\ \ \ \ ,\ \ \ \ \stackrel{\leftharpoonup}{\tilde{R}}_{kl} \equiv \tilde{R}^{j} _{jkl}[/tex]
[tex](2.2.5)\ \ \ \ \ \ \bar{\stackrel{\rightharpoonup}{\tilde{R}}}_{ij} \equiv {\tfrac{1}{2}} .(\stackrel{\rightharpoonup}{\tilde{R}}_{ij} +\stackrel{\rightharpoonup}{\tilde{R}}_{ji} )\ \ \ \ ,\ \ \ \ \hat{\stackrel{\rightharpoonup}{\tilde{R}}}_{ij} \equiv {\tfrac{1}{2}} .(\stackrel{\rightharpoonup}{\tilde{R}}_{ij} -\stackrel{\rightharpoonup}{\tilde{R}}_{ji} )[/tex]
[tex](2.2.6)\ \ \ \ \bar{\stackrel{\rightharpoonup}{\tilde{R}}}\equiv g^{kr} .\bar{\stackrel{\rightharpoonup}{\tilde{R}}}_{kr}[/tex]
2.3 Variational Derivative
Define the variational derivative as the following operator:
[tex](2.3.1)\ \ \ \ \tilde{\Pi }_{k} V^{i} \equiv \tilde{\nabla }_{k} V^{i} -(\stackrel{\rightharpoonup}{\tilde{\Gamma }}_{k} -\Gamma _{k} ).V^{i} [/tex]
3 Conformal Electromagnetic Gravity Action
The unification action proposed is the following:
[tex](3.1)\ \ \ \ \[I_{CEMG} \equiv k_{0} .\int _{D}\sqrt{-\left|g_{\bullet \bullet } \right|} .g^{kr} .g^{sl} .\left(\bar{\stackrel{\rightharpoonup}{\tilde{R}}}_{kr} .\bar{\stackrel{\rightharpoonup}{\tilde{R}}}_{sl} +k_{1} .(\hat{\stackrel{\rightharpoonup}{\tilde{R}}}_{ks} -{\tfrac{1}{2}} .\stackrel{\leftharpoonup}{\tilde{R}}_{ks} ).(\hat{\stackrel{\rightharpoonup}{\tilde{R}}}_{rl} -{\tfrac{1}{2}} .\stackrel{\leftharpoonup}{\tilde{R}}_{rl} ).+\right.[/tex]
[tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \[+k_{2} .\partial \hat{\Gamma }_{ks} .\partial \hat{\Gamma }_{rl} +k_{3} .(\hat{\stackrel{\rightharpoonup}{\tilde{R}}}_{ks} -{\tfrac{1}{2}} .\stackrel{\leftharpoonup}{\tilde{R}}_{ks} ).\partial \hat{\Gamma }_{rl} +\] [/tex]
[tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \[+k_{4} .g^{mn} .g{}_{op} .(\tilde{R}^{o} _{ksm} +\tilde{R}^{o} _{mks} +\tilde{R}^{o} _{smk} ).(\tilde{R}^{p} _{r\ln } +\tilde{R}^{p} _{nrl} +\tilde{R}^{p} _{\ln r} )+\] [/tex]
[tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \[\left. \mathop{}\limits^{} -({\tfrac{1}{2}} .k_{1} +{\tfrac{3}{8}} .k_{3} +3.k_{4} ).(\tilde{R}^{m} _{kso} +\tilde{R}^{m} _{oks} +\tilde{R}^{m} _{sok} ).(\tilde{R}^{o} _{rlm} +\tilde{R}^{o} _{mrl} +\tilde{R}^{o} _{lmr} )\right).d\Omega \] [/tex]
[tex](3.2)\ \ \ \ \ \ \ \ \ \ \ \ /\ \ \ \ 3.k_{2} +k_{3} \equiv {\tfrac{32.k_{g} .\lambda _{c} }{3.c^{4} .c_{1} ^{2} }} \ne 0 [/tex]
[tex]Where:\ \ \ \ \ \ \ \ k_{1} ,k_{2} ,k_{3} ,k_{4} \ \ \ \ \ \ \ \ \ \ \ \ Lagrangian\ \ constants[/tex]
[tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ k_{g} \ \ \ \ \ \ \ \ \ \ \ \ \ \ Newton's\ \ gravitational\ \ constant[/tex]
[tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \lambda _{c} \ \ \ \ \ \ \ \ \ \ Cosmological\ \ constant[/tex]
[tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ c \ \ \ \ \ \ \ \ \ \ \ \ Speed\ \ of\ \ light[/tex]
[tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ c_{1} \ \ \ \ \ \ \ \ \ \ \ \ \ \ Electromagnetic\ \ coupling\ \ constant[/tex]
This Lagrangian is conformally invariant on the metric only on a four-dimensional manifold:
[tex](3.3)\ \ \ \ \ \ \ \ L(g^{ij} )=L(e^{\phi } .g^{ij} ) \ \ \ \ \ \ \ \ \leftrightarrow \ \ \ \ \ \ \ \ n=4[/tex]
(continues on "Conformal Electromagnetic Relativity (2/2)"...)
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