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Conformal Electromagnetic Relativity (2/2)

  1. Aug 4, 2010 #1
    Conformal Electromagnetic Relativity

    (...coming from "Conformal Electromagnetic Relativity (1/2)")

    4 Field Equations

    The corresponding field equations are:

    [tex](4.1)\ \ \ \ \delta g^{ij} \left. \right) \ \ \ \ \ \ \ \ \bar{\stackrel{\rightharpoonup}{\tilde{R}}}.(\bar{\stackrel{\rightharpoonup}{\tilde{R}}}_{ij} -{\tfrac{1}{4}} .\bar{\stackrel{\rightharpoonup}{\tilde{R}}}.g_{ij} )=\][/tex]

    [tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =k_{1} .(-(\hat{\stackrel{\rightharpoonup}{\tilde{R}}}_{i} ^{k} -{\tfrac{1}{2}} .\stackrel{\leftharpoonup}{\tilde{R}}_{i} ^{k} ).(\hat{\stackrel{\rightharpoonup}{\tilde{R}}}_{jk} -{\tfrac{1}{2}} .\stackrel{\leftharpoonup}{\tilde{R}}_{jk} )+{\tfrac{1}{4}} .(\hat{\stackrel{\rightharpoonup}{\tilde{R}}}^{kr} -{\tfrac{1}{2}} .\stackrel{\leftharpoonup}{\tilde{R}}^{kr} ).(\hat{\stackrel{\rightharpoonup}{\tilde{R}}}_{kr} -{\tfrac{1}{2}} .\stackrel{\leftharpoonup}{\tilde{R}}_{kr} ).g_{ij} )+\] [/tex]

    [tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \[+k_{2} .(-\partial \hat{\Gamma }_{i} ^{k} .\partial \hat{\Gamma }_{jk} +{\tfrac{1}{4}} .\partial \hat{\Gamma }^{kr} .\partial \hat{\Gamma }_{kr} .g_{ij} )+\] [/tex]

    [tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \[+{\tfrac{1}{2}} .k_{3} .(-(\hat{\stackrel{\rightharpoonup}{\tilde{R}}}_{i} ^{k} -{\tfrac{1}{2}} .\stackrel{\leftharpoonup}{\tilde{R}}_{i} ^{k} ).\partial \hat{\Gamma }_{jk} -(\hat{\stackrel{\rightharpoonup}{\tilde{R}}}_{j} ^{k} -{\tfrac{1}{2}} .\stackrel{\leftharpoonup}{\tilde{R}}_{j} ^{k} ).\partial \hat{\Gamma }_{ik} +{\tfrac{1}{2}} .(\hat{\stackrel{\rightharpoonup}{\tilde{R}}}^{kr} -{\tfrac{1}{2}} .\stackrel{\leftharpoonup}{\tilde{R}}^{kr} ).\partial \hat{\Gamma }_{kr} .g_{ij} )+[/tex]

    [tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +{\tfrac{1}{2}} .k_{4} .\left(-3.(\tilde{R}_{pi} ^{\ln } +\tilde{R}_{p} ^{n} _{i} ^{l} +\tilde{R}_{p} ^{\ln } _{i} ).(\tilde{R}^{p} _{j\ln } +\tilde{R}^{p} _{njl} +\tilde{R}^{p} _{\ln j} )+\right. \] [/tex]

    [tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \[+(\tilde{R}_{i} ^{r\ln } +\tilde{R}_{i} ^{nrl} +\tilde{R}_{i} ^{\ln r} ).(\tilde{R}_{jr\ln } +\tilde{R}_{jnrl} +\tilde{R}_{j\ln r} )+\] [/tex]

    [tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \[\left. +{\tfrac{1}{2}} .(\tilde{R}_{p} ^{r\ln } +\tilde{R}_{p} ^{nrl} +\tilde{R}_{p} ^{\ln r} ).(\tilde{R}^{p} _{r\ln } +\tilde{R}^{p} _{nrl} +\tilde{R}^{p} _{\ln r} ).g_{ij} \right)+\] [/tex]

    [tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \[-({\tfrac{1}{2}} .k_{1} +{\tfrac{3}{8}} .k_{3} +3.k_{4} ).\left(-(\tilde{R}^{m} _{i} ^{l} _{o} +\tilde{R}^{m} _{oi} ^{l} +\tilde{R}^{ml} _{oi} ).(\tilde{R}^{o} _{jlm} +\tilde{R}^{o} _{mjl} +\tilde{R}^{o} _{lmj} )+\right. \] [/tex]

    [tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \[\left. +{\tfrac{1}{4}} .(\tilde{R}^{mrl} _{o} +\tilde{R}^{m} _{o} ^{rl} +\tilde{R}^{ml} _{o} ^{r} ).(\tilde{R}^{o} _{rlm} +\tilde{R}^{o} _{mrl} +\tilde{R}^{o} _{lmr} ).g_{ij} \right)\] [/tex]


    [tex](4.2)\ \ \ \ \delta \tilde{\Gamma }^{i} _{jk} \left. \right)\ \ \ \ \ \ \ \ \tilde{\Pi }_{i} (\bar{\stackrel{\rightharpoonup}{\tilde{R}}}.g^{jk} )-\delta _{i}^{k} .\tilde{\Pi }_{r} (\bar{\stackrel{\rightharpoonup}{\tilde{R}}}.g^{jr} )+2.\hat{\Gamma }^{k} _{ir} .(\bar{\stackrel{\rightharpoonup}{\tilde{R}}}.g^{jr} )= [/tex]

    [tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \[=-k_{1} .(\tilde{\Pi }_{i} (\hat{\stackrel{\rightharpoonup}{\tilde{R}}}^{jk} -{\tfrac{1}{2}} .\stackrel{\leftharpoonup}{\tilde{R}}^{jk} )-\delta _{i}^{k} .\tilde{\Pi }_{r} (\hat{\stackrel{\rightharpoonup}{\tilde{R}}}^{jr} -{\tfrac{1}{2}} .\stackrel{\leftharpoonup}{\tilde{R}}^{jr} )+2.\hat{\Gamma }^{k} _{ir} .(\hat{\stackrel{\rightharpoonup}{\tilde{R}}}^{jr} -{\tfrac{1}{2}} .\stackrel{\leftharpoonup}{\tilde{R}}^{jr} ))+\] [/tex]

    [tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \[-k_{2} .(\delta _{i}^{k} .(\tilde{\Pi }_{p} \partial \hat{\Gamma }^{pj} +\hat{\Gamma }^{j} _{pr} .\partial \hat{\Gamma }^{pr} )-\delta _{i}^{j} .(\tilde{\Pi }_{p} \partial \hat{\Gamma }^{pk} +\hat{\Gamma }^{k} _{pr} .\partial \hat{\Gamma }^{pr} ))+\] [/tex]

    [tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \[+k_{1} .\delta _{i}^{j} .(\tilde{\Pi }_{p} (\hat{\stackrel{\rightharpoonup}{\tilde{R}}}^{pk} -{\tfrac{1}{2}} .\stackrel{\leftharpoonup}{\tilde{R}}^{pk} )+\hat{\Gamma }^{k} _{rp} .(\hat{\stackrel{\rightharpoonup}{\tilde{R}}}^{rp} -{\tfrac{1}{2}} .\stackrel{\leftharpoonup}{\tilde{R}}^{rp} ))+\] [/tex]

    [tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \[-{\tfrac{1}{2}} .k_{3} .(\tilde{\Pi }_{i} \partial \hat{\Gamma }^{jk} -\delta _{i}^{k} .\tilde{\Pi }_{p} \partial \hat{\Gamma }^{jp} +2.\hat{\Gamma }^{k} _{ip} .\partial \hat{\Gamma }^{jp} )+\] [/tex]

    [tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \[-{\tfrac{1}{2}} .k_{3} .\delta _{i}^{k} .(\tilde{\Pi }_{p} (\hat{\stackrel{\rightharpoonup}{\tilde{R}}}^{pj} -{\tfrac{1}{2}} .\stackrel{\leftharpoonup}{\tilde{R}}^{pj} )+\hat{\Gamma }^{j} _{pr} .(\hat{\stackrel{\rightharpoonup}{\tilde{R}}}^{pr} -{\tfrac{1}{2}} .\stackrel{\leftharpoonup}{\tilde{R}}^{pr} ))+[/tex]

    [tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ +{\tfrac{1}{2}} .k_{3} .\delta _{i}^{j} .(\tilde{\Pi }_{p} (\hat{\stackrel{\rightharpoonup}{\tilde{R}}}^{pk} -{\tfrac{1}{2}} .\stackrel{\leftharpoonup}{\tilde{R}}^{pk} )+\hat{\Gamma }^{k} _{pr} .(\hat{\stackrel{\rightharpoonup}{\tilde{R}}}^{pr} -{\tfrac{1}{2}} .\stackrel{\leftharpoonup}{\tilde{R}}^{pr} ))+\] [/tex]

    [tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \[+{\tfrac{1}{2}} .k_{3} .\delta _{i}^{j} .(\tilde{\Pi }_{p} \partial \hat{\Gamma }^{pk} +\hat{\Gamma }^{k} _{rp} .\partial \hat{\Gamma }^{rp} )+\] [/tex]

    [tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \[+2.({\tfrac{1}{2}} .k_{1} +{\tfrac{3}{8}} .k_{3} +3.k_{4} ).(\tilde{\Pi }_{p} (\tilde{R}^{kjp} _{i} +\tilde{R}^{k} _{i} ^{jp} +\tilde{R}^{kp} _{i} ^{j} )+\hat{\Gamma }^{k} _{rp} .(\tilde{R}^{pjr} _{i} +\tilde{R}^{p} _{i} ^{jr} +\tilde{R}^{pr} _{i} ^{j} )+\] [/tex]

    [tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \[+\tilde{\Pi }_{p} (\tilde{R}^{jpk} _{i} +\tilde{R}^{j} _{i} ^{pk} +\tilde{R}^{jk} _{i} ^{p} )+\hat{\Gamma }^{k} _{rp} .(\tilde{R}^{jrp} _{i} +\tilde{R}^{j} _{i} ^{rp} +\tilde{R}^{jp} _{i} ^{r} )+\] [/tex]

    [tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \[+\tilde{\Pi }_{p} (\tilde{R}^{pkj} _{i} +\tilde{R}^{p} _{i} ^{kj} +\tilde{R}^{pj} _{i} ^{k} )+\hat{\Gamma }^{k} _{rp} .(\tilde{R}^{rpj} _{i} +\tilde{R}^{r} _{i} ^{pj} +\tilde{R}^{rj} _{i} ^{p} ))+\] [/tex]

    [tex].\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \[-6.k_{4} .(\tilde{\Pi }_{p} (\tilde{R}_{i} ^{jpk} +\tilde{R}_{i} ^{kjp} +\tilde{R}_{i} ^{pkj} )+\hat{\Gamma }^{k} _{rp} .(\tilde{R}_{i} ^{jrp} +\tilde{R}_{i} ^{pjr} +\tilde{R}_{i} ^{rpj} ))\] [/tex]


    5 GR-Compatible Field Equations

    Consider the following connection and conformal gauge:

    [tex](5.1)\ \ \ \ \tilde{\Gamma }^{i} _{jk} \equiv \Gamma ^{i} _{jk} +\Theta ^{i} _{jk} +c_{1} .\delta _{j}^{i} .A_{k} [/tex]

    [tex](5.2)\ \ \ \ / \ \ \ \ \Theta ^{i} _{jk} \equiv \Theta ^{i} _{kj} \ \ \ \ \ \ \ \ ,\ \ \ \ \ \ \ \ \Theta _{i} ^{k} _{j} +\Theta ^{k} _{ij} -\delta _{i}^{k} .(\Theta _{j} ^{r} _{r} +\Theta ^{r} _{jr} ) \equiv 0[/tex]

    [tex](5.3)\ \ \ \ \bar{\stackrel{\rightharpoonup}{\tilde{R}}}\equiv 4.\lambda _{c}\ \ \ \ \ \ \ \ (metric\ \ conformal\ \ gauge) [/tex]

    The field equations then reduces to:

    [tex](5.4)\ \ \ \ R_{ij} -{\tfrac{1}{2}} .R.g_{ij} ={\tfrac{8.\pi .k_{g} }{c^{4} }} .T_{ij} -\lambda _{c} .g_{ij} +{\tfrac{8.\pi .k_{g} }{c^{4} }} .\left({\tfrac{1}{4.\pi }} .(-F_{i} ^{k} .F_{jk} +{\tfrac{1}{4}} .F^{kr} .F_{kr} .g_{ij} )\right) [/tex]

    [tex](5.5)\ \ \ \ T_{ij} \equiv {\tfrac{c^{4} }{8.\pi .k_{g} }} .\left(\bar{\stackrel{\rightharpoonup}{\Theta }}_{ij} -{\tfrac{1}{2}} .g^{kr} .\bar{\stackrel{\rightharpoonup}{\Theta }}_{kr} .g_{ij} \right) [/tex]

    [tex](5.6)\ \ \ \ / \ \ \ \ \bar{\stackrel{\rightharpoonup}{\Theta }}_{ij} \equiv -\nabla _{k} \Theta ^{k} _{ij} +\Theta ^{k} _{ir} .\Theta ^{r} _{jk} +{\tfrac{1}{2}} .(\nabla _{i} \Theta ^{k} _{jk} +\nabla _{j} \Theta ^{k} _{ik} )-\Theta ^{r} _{ij} .\Theta ^{k} _{rk}[/tex]

    [tex](5.7)\ \ \ \ \nabla _{i} g^{jk} =0[/tex]

    [tex](5.8)\ \ \ \ \nabla _{k} F^{jk} =0[/tex]

    [tex](5.9)\ \ \ \ / \ \ \ \ F_{ij} \equiv \partial _{i} A_{j} -\partial _{j} A_{i} \ \ \ \ \ \ \ \ \to \ \ \ \ \ \ \ \ \nabla _{k} F_{ij} +\nabla _{j} F_{ki} +\nabla _{i} F_{jk} =0 [/tex]


    6 References

    [1] "The meaning of Relativity" Albert Einstein
    Princeton University Press ISBN 84-239-6460-4

    [2] "Gravitation" Charles W. Misner, Kip S. Thorne and John A. Wheeler
    W.H. Freemand and Company ISBN 0-7167-0344-0

    [3] "On the History of Unified Field Theories" Hubert F. M. Goenner
    Max Planck Institute for Gravitational Physics
    Albert Einstein Institute, Germany
     
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