Elektrodynamikvorlesung von Prof. Dr. E. Schöll, PhD
{{#set:Urheber=Prof. Dr. E. Schöll, PhD|Inhaltstyp=Script|Kapitel=6|Abschnitt=2}}
Kategorie:Elektrodynamik
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Ziel: Ko- / Kontravariante Schreibweise der Elektrodynamik im Vakuum
Grund: Die klassische Elektrodynamik ist bereits eine Lorentz- invariante Theorie!!
Historisch gab die Maxwellsche Elektrodynamik und nicht die Mechanik den Anstoß zur Relativitätstheorie überhaupt!
Ladungserhaltung aus Kontinuitätsgleichung:
![{\displaystyle {\begin{aligned}&div{\bar {j}}+{\frac {\partial \rho }{\partial t}}={\frac {\partial {{j}_{x}}}{\partial x}}+{\frac {\partial {{j}_{y}}}{\partial y}}+{\frac {\partial {{j}_{z}}}{\partial z}}+{\frac {\partial c\rho }{\partial ct}}=0\\&0={\frac {\partial \rho }{\partial t}}+\sum \limits _{\alpha =1}^{3}{}{{\partial }_{\alpha }}{{j}^{\alpha }}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b89b84e07862e72fd1dacd6c689df7e0b3be2e2)
Somit gewinnen wir aber ebenfalls wieder einen Lorentz- Skalar, nämlich
![{\displaystyle {{\partial }_{\mu }}{{j}^{\mu }}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b9536762a23f7f47a88641ec0b38d329371935e)
in Viererschreibweise.
Die Vierer- Stromdichte ist
![{\displaystyle \left\{{{j}^{\mu }}\right\}=\left\{c\rho ,{\bar {j}}\right\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/410aea58aa3a36af7bc175f1fd26e71b38afc1cb)
ebenfalls ein kontravarianter Vierer- Vektor. Er heißt Vierer- Stromdichte.
Die Kontinuitätsgleichung ist gleich
![{\displaystyle {{\partial }_{\mu }}{{j}^{\mu }}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b9536762a23f7f47a88641ec0b38d329371935e)
Forderung:
Ladungserhaltung soll in allen Inertialsystemen gelten!
→
![{\displaystyle {{j}^{\mu }}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20853995760a4648a796f5321a0b856d36287c4a)
muss sich wie ein Vierervektor transformieren, damit das Skalarprodukt
![{\displaystyle {{\partial }_{\mu }}{{j}^{\mu }}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b9536762a23f7f47a88641ec0b38d329371935e)
Lorentz- invariant ist!:
![{\displaystyle {\begin{aligned}&{{x}^{0}}{\acute {\ }}=\gamma \left({{x}^{0}}-\beta {{x}^{1}}\right)\Leftrightarrow t{\acute {\ }}=\gamma \left(t-{\frac {v}{{c}^{2}}}{{x}^{1}}\right)\\&{{x}^{1}}{\acute {\ }}=\gamma \left({{x}^{1}}-\beta {{x}^{0}}\right)\Leftrightarrow {{x}^{1}}{\acute {\ }}=\gamma \left({{x}^{1}}-vt\right)\\&{{x}^{2}}{\acute {\ }}={{x}^{2}}\\&{{x}^{3}}{\acute {\ }}={{x}^{3}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2055dd4e36a752b4b46112f2c031aca073f6ec28)
Also gilt für Ladungs- und Stromdichten:
![{\displaystyle {\begin{aligned}&{{j}^{0}}{\acute {\ }}=\gamma \left({{j}^{0}}-\beta {{j}^{1}}\right)\Leftrightarrow \rho {\acute {\ }}=\gamma \left(\rho -{\frac {v}{{c}^{2}}}{{j}^{1}}\right)\\&{{j}^{1}}{\acute {\ }}=\gamma \left({{j}^{1}}-\beta {{j}^{0}}\right)\Leftrightarrow {{j}^{1}}{\acute {\ }}=\gamma \left({{j}^{1}}-v\rho \right)\\&{{j}^{2}}{\acute {\ }}={{j}^{2}}\\&{{j}^{3}}{\acute {\ }}={{j}^{3}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de84b750375d99d8fbec64491594960d92aca17f)
Merke: Es sollte kein Missverständnis geschehen: Ist ein Vektor in ein Lorentz- invariantes Skalarprodukt verwickelt, so ist es ein Vierervektor. Damit ist klar: Seine Komponenten transfornmieren nach der Lorentz- Trafo.
Dadurch aber ist die Trafo für seine Komponenten, die Beispielsweise Ladungs- und Stromdichten sind, gefunden.
4- Potenziale:
Die Potenziale
![{\displaystyle \Phi ,{\bar {A}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91e4d319ffc950a10062aedefd2d8d432c71f290)
sind in der Lorentz- Eichung
![{\displaystyle \nabla \cdot {\bar {A}}+{\frac {1}{{c}^{2}}}{\frac {\partial }{\partial t}}\phi =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4badf2e138c1d62bd76fc6093d4a41016ce60552)
Lösungen von
![{\displaystyle {\begin{aligned}&\Delta {\bar {A}}\left({\bar {r}},t\right)-{\frac {1}{{c}^{2}}}{\frac {{\partial }^{2}}{\partial {{t}^{2}}}}{\bar {A}}\left({\bar {r}},t\right)=-{{\mu }_{0}}{\bar {j}}\\&\#{\bar {A}}\left({\bar {r}},t\right)=-{{\mu }_{0}}{\bar {j}}\\&\#=-{{\partial }_{\mu }}{{\partial }^{\mu }}\\&{{\mu }_{0}}c={\frac {1}{{{\varepsilon }_{0}}c}}\\&\#{\bar {A}}\left({\bar {r}},t\right)=-{{\mu }_{0}}{\bar {j}}\Leftrightarrow {{\partial }_{\mu }}{{\partial }^{\mu }}c{{A}^{\alpha }}={\frac {1}{{{\varepsilon }_{0}}c}}{{j}^{\alpha }}\\&\alpha =1,2,3\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f09dd4f818892d589411e8465314d833356ae73b)
![{\displaystyle {\begin{aligned}&\Delta \phi \left({\bar {r}},t\right)-{\frac {1}{{c}^{2}}}{\frac {{\partial }^{2}}{\partial {{t}^{2}}}}\phi \left({\bar {r}},t\right)=-{\frac {\rho }{{\varepsilon }_{0}}}=-{{\mu }_{0}}{{c}^{2}}\rho \\&\#\phi \left({\bar {r}},t\right)=-{\frac {\rho }{{\varepsilon }_{0}}}\Leftrightarrow {{\partial }_{\mu }}{{\partial }^{\mu }}\phi ={\frac {1}{{{\varepsilon }_{0}}c}}{{j}^{0}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1117395941a441cab490d03c65f217a0db73744a)
Zusammen:
![{\displaystyle {\begin{aligned}&-\#{{\Phi }^{\mu }}={{\partial }_{\alpha }}{{\partial }^{\alpha }}{{\Phi }^{\mu }}={{\mu }_{0}}{{j}^{\mu }}\\&{{\Phi }^{0}}:=\phi \\&{{\Phi }^{i}}:=c{{A}^{i}}\quad i=1..3\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1bdefd08e993ba322437ac5fbc299aea4f0bfe77)
Da
![{\displaystyle {{j}^{\mu }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0dd289a64e6fac97b479534448f4569c0c0ebb0)
Vierervektoren sind (wie Vierervektoren transformieren), muss auch
![{\displaystyle {{\Phi }^{\mu }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e547d9f82dfc9eedfc6c174f4f0d5a8ceb7eb474)
wie ein Vierervektor transformieren.
Denn: Der d´Alembert- Operator ist Lorentz- invariant:
![{\displaystyle {{\partial }_{\alpha }}{{\partial }^{\alpha }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/14df3e6814c4a7815dfc09f8bb252ee5c2327d0b)
lorentz- invariant!:
![{\displaystyle {\begin{aligned}&{{\Phi }^{0}}{\acute {\ }}=\gamma \left({{\Phi }^{0}}-\beta {{\Phi }^{1}}\right)\quad bzw.\quad \Phi {\acute {\ }}=\gamma \left(\Phi -v{{A}^{1}}\right)\\&{{\Phi }^{1}}{\acute {\ }}=\gamma \left({{\Phi }^{1}}-\beta {{\Phi }^{0}}\right)\quad bzw.\quad A{{\acute {\ }}^{1}}=\gamma \left({{A}^{1}}-{\frac {v}{{c}^{2}}}\Phi \right),{{A}^{{\acute {\ }}2}}={{A}^{2}},A{{\acute {\ }}^{3}}={{A}^{3}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0a4b7ea0752fcb354d8b35bc8551a46d78a97d0d)
Nun: Lorentz- Eichung:
![{\displaystyle \nabla \cdot {\bar {A}}+{\frac {1}{{c}^{2}}}{\frac {\partial }{\partial t}}\phi =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4badf2e138c1d62bd76fc6093d4a41016ce60552)
Lorentz- Eichung ↔ Lorentz- Invarianz
![{\displaystyle {{\partial }_{\mu }}{{\Phi }^{\mu }}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0c1a3a2a93d0ba0d74925bb3fc327c3dc92089f)
(Gegensatz zur Coulomb- Eichung)
![{\displaystyle {{\partial }_{\mu }}{{\Phi }^{\mu }}=0\Leftrightarrow \nabla \cdot {\bar {A}}+{\frac {1}{{c}^{2}}}{\frac {\partial }{\partial t}}\phi =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/334f7caff3a7290e80bf4c19e7bb151376d49b56)
Umeichung:
![{\displaystyle {\begin{aligned}&{\tilde {\bar {A}}}={\bar {A}}+\nabla F\\&{\tilde {\phi }}=\phi -{\frac {\partial }{\partial t}}F\\&\Leftrightarrow \\&c{{\tilde {A}}^{\alpha }}=c{{A}^{\alpha }}+{{\partial }_{\alpha }}cF=c{{A}^{\alpha }}-{{\partial }^{\alpha }}cF\\&{{\tilde {\Phi }}^{0}}={{\Phi }^{0}}-{{\partial }_{0}}cF={{\Phi }^{0}}-{{\partial }^{0}}cF\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e037a144c8f70dc05dbd8b00c638da9b9af70b5)
Also:
![{\displaystyle {{\tilde {\Phi }}^{\mu }}={{\Phi }^{\mu }}-{{\partial }^{\mu }}cF}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c22017bd86c42b80c05133cd602e08211c515c72)
Felder E und B:
![{\displaystyle {\begin{aligned}&{\bar {E}}=-grad\phi -{\frac {\partial }{\partial t}}{\bar {A}}\\&\Rightarrow {{E}^{\alpha }}=-{{\partial }_{\alpha }}\phi -{\frac {1}{c}}{\frac {\partial }{\partial t}}c{{A}^{\alpha }}=-{{\partial }_{\alpha }}{{\Phi }^{0}}-{{\partial }_{0}}{{\Phi }^{\alpha }}={{\partial }^{\alpha }}{{\Phi }^{0}}-{{\partial }^{0}}{{\Phi }^{\alpha }}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1268ce37c387d5a8eb17da983ae216eec9357000)
![{\displaystyle {\begin{aligned}&{\bar {B}}=\nabla \times {\bar {A}}\\&\Rightarrow c{{B}^{1}}={{\partial }_{2}}c{{A}^{3}}-{{\partial }_{3}}c{{A}^{2}}={{\partial }_{2}}{{\Phi }^{3}}-{{\partial }_{3}}{{\Phi }^{2}}={{\partial }^{3}}{{\Phi }^{2}}-{{\partial }^{2}}{{\Phi }^{3}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/645fd2f83b467d0c7a11de9e7f10c5db34dc8342)
Die anderen Komponenten gewinnt man durch zyklische Vertauschung:
![{\displaystyle {\begin{aligned}&c{{B}^{2}}={{\partial }^{1}}{{\Phi }^{3}}-{{\partial }^{3}}{{\Phi }^{1}}\\&c{{B}^{3}}={{\partial }^{2}}{{\Phi }^{1}}-{{\partial }^{1}}{{\Phi }^{2}}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59219689350ae0ca68e662e649303c97b215f587)
Diese Gleichungen werden zusammengefasst durch den antisymmetrtischen Feldstärketensor:
![{\displaystyle {\begin{aligned}&\left\{{{F}_{\mu \nu }}\right\}=\left\{{{\partial }_{\mu }}{{\Phi }_{\nu }}-{{\partial }_{\nu }}{{\Phi }_{\mu }}\right\}=\left({\begin{matrix}0&{\frac {1}{c}}{{E}_{x}}&{\frac {1}{c}}{{E}_{y}}&{\frac {1}{c}}{{E}_{z}}\\-{\frac {1}{c}}{{E}_{x}}&0&-{{B}_{z}}&{{B}_{y}}\\-{\frac {1}{c}}{{E}_{y}}&{{B}_{z}}&0&-{{B}_{x}}\\-{\frac {1}{c}}{{E}_{z}}&-{{B}_{y}}&{{B}_{x}}&0\\\end{matrix}}\right)\\&{{F}^{\mu \nu }}=\left\{{{\partial }^{\mu }}{{\Phi }^{\nu }}-{{\partial }^{\nu }}{{\Phi }^{\mu }}\right\}=\left({\begin{matrix}0&-{\frac {1}{c}}{{E}_{x}}&-{\frac {1}{c}}{{E}_{y}}&-{\frac {1}{c}}{{E}_{z}}\\{\frac {1}{c}}{{E}_{x}}&0&-{{B}_{z}}&{{B}_{y}}\\{\frac {1}{c}}{{E}_{y}}&{{B}_{z}}&0&-{{B}_{x}}\\{\frac {1}{c}}{{E}_{z}}&-{{B}_{y}}&{{B}_{x}}&0\\\end{matrix}}\right)\\&\Leftrightarrow {{F}^{\mu \nu }}=\left\{{{\partial }^{\mu }}{{\Phi }^{\nu }}-{{\partial }^{\nu }}{{\Phi }^{\mu }}\right\}=\left({\begin{matrix}0&-{{E}^{1}}&-{{E}^{2}}&-{{E}^{3}}\\{{E}^{1}}&0&-c{{B}^{3}}&c{{B}^{2}}\\{{E}^{2}}&c{{B}^{3}}&0&-c{{B}^{1}}\\{{E}^{3}}&-c{{B}^{2}}&c{{B}^{1}}&0\\\end{matrix}}\right)\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8bab5487c88d3b1175b6a7b12229c447b6694047)
Wegen der Antisymmetrie hat
![{\displaystyle {{F}^{\mu \nu }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4dbaf2f754b6a921b256656137027007c33f0b6d)
nur 6 unabhängige Komponenten!
Das bedeutet, die Raum- Raum- Komponenten entsprechen
![{\displaystyle rot{\bar {A}}={\bar {B}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49d3d0f38c56b4738d7c782940ebbdb933627161)
während die Raum- zeit- Komponenten:
![{\displaystyle {\bar {E}}=-grad\phi -{\frac {\partial }{\partial t}}{\bar {A}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a691924eef0e289286aaea9acbb83449a4c2e53c)
erfüllen.
Lorentz- Trafo der Felder:
Der Feldstärketensor ist kovariant und transformiert demnach über die inverse Lorentz- Transformation.
Das heißt: Für die Transformation in ein in x- Richtung mit konstanter Geschwindigkeit
![{\displaystyle {\bar {v}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ba1d09340f8f6c1979330c2f23e514e38f243a3b)
bewegtes System K´ gilt:
![{\displaystyle {{F}_{}}{{\acute {\ }}^{\mu \nu }}={{U}^{\mu }}_{\lambda }{{U}^{\nu }}_{\kappa }{{F}^{\lambda \kappa }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/860580dc744def57f2908009fe7c9ddb2db71778)
![{\displaystyle {{U}^{i}}_{k}=\left({\begin{matrix}{\frac {1}{\sqrt {1-{{\beta }^{2}}}}}&{\frac {-\beta }{\sqrt {1-{{\beta }^{2}}}}}&0&0\\{\frac {-\beta }{\sqrt {1-{{\beta }^{2}}}}}&{\frac {1}{\sqrt {1-{{\beta }^{2}}}}}&0&0\\0&0&1&0\\0&0&0&1\\\end{matrix}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19c642bf2517f89a564441eca4745baa190c5079)
Damit läßt sich nun das uns unbekannte Transformationsverhalten der Felder
und ![{\displaystyle rot{\bar {A}}={\bar {B}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49d3d0f38c56b4738d7c782940ebbdb933627161)
berechnen, die auch kovariant transformieren müssen. Dabei sollte keinesfalls die Summation über die Indices auf der rechten Seite vergessen werden!!
![{\displaystyle {\begin{aligned}&E{{\acute {\ }}^{1}}=F{{\acute {\ }}^{10}}={{U}^{1}}_{\lambda }{{U}^{0}}_{\kappa }{{F}^{\lambda \kappa }}=-\beta \gamma {{U}^{0}}_{\kappa }{{F}^{0\kappa }}+\gamma {{U}^{0}}_{\kappa }{{F}^{1\kappa }}={{\left(\beta \gamma \right)}^{2}}{{F}^{01}}+{{\gamma }^{2}}{{F}^{10}}=\\&={{\gamma }^{2}}\left(1-{{\beta }^{2}}\right){{F}^{10}}={{E}^{1}}\\&{{\gamma }^{2}}\left(1-{{\beta }^{2}}\right)=1\\&\\&E{{\acute {\ }}^{2}}=F{{\acute {\ }}^{20}}={{U}^{2}}_{\lambda }{{U}^{0}}_{\kappa }{{F}^{\lambda \kappa }}={{U}^{0}}_{\kappa }{{F}^{2\kappa }}=\gamma {{F}^{20}}-\beta \gamma {{F}^{21}}=\gamma \left({{E}^{2}}-v{{B}^{3}}\right)\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efd9f2127b6f10657b6a261b21ea4c70c9396dbf)
![{\displaystyle E{{\acute {\ }}^{3}}=F{{\acute {\ }}^{30}}={{U}^{0}}_{\kappa }{{F}^{3\kappa }}=\gamma {{F}^{30}}-\beta \gamma {{F}^{31}}=\gamma \left({{E}^{3}}+v{{B}^{2}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/24866a354f5bd28dbad998bff7afaedb521832b7)
![{\displaystyle {\begin{aligned}&B{{\acute {\ }}^{1}}={\frac {1}{c}}F{{\acute {\ }}^{32}}={\frac {1}{c}}{{U}^{3}}_{\lambda }{{U}^{2}}_{\kappa }{{F}^{\lambda \kappa }}={\frac {1}{c}}{{F}^{32}}={{B}^{1}}\\&B{{\acute {\ }}^{2}}={\frac {1}{c}}F{{\acute {\ }}^{13}}={\frac {1}{c}}{{U}^{1}}_{\lambda }{{U}^{3}}_{\kappa }{{F}^{\lambda \kappa }}={\frac {1}{c}}{{U}^{1}}_{\kappa }{{F}^{\kappa 3}}=-{\frac {\beta \gamma }{c}}{{F}^{03}}+{\frac {\gamma }{c}}{{F}^{13}}=\gamma \left({{B}^{2}}+{\frac {v}{{c}^{2}}}{{E}^{3}}\right)\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/377617cd4d864c7a4c19b0dca414823af81a2be1)
![{\displaystyle B{{\acute {\ }}^{3}}=\gamma \left({{B}^{3}}-{\frac {v}{{c}^{2}}}{{E}^{2}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6447a83d08a9c199a53c1a5fe50723d0df4b87a)
Zusammenfassung
![{\displaystyle {\begin{aligned}&{{E}^{1}}{\acute {\ }}={{E}^{1}}\\&{{E}^{2}}{\acute {\ }}={\frac {1}{\sqrt {1-{{\beta }^{2}}}}}\left({{E}^{2}}-v{{B}^{3}}\right)\\&{{E}^{3}}{\acute {\ }}={\frac {1}{\sqrt {1-{{\beta }^{2}}}}}\left({{E}^{3}}+v{{B}^{2}}\right)\\&{{B}^{1}}{\acute {\ }}={{B}^{1}}\\&{{B}^{2}}{\acute {\ }}={\frac {1}{\sqrt {1-{{\beta }^{2}}}}}\left({{B}^{2}}+{\frac {v}{{c}^{2}}}{{E}^{3}}\right)\\&{{B}^{3}}{\acute {\ }}={\frac {1}{\sqrt {1-{{\beta }^{2}}}}}\left({{B}^{3}}-{\frac {v}{{c}^{2}}}{{E}^{2}}\right)\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7898feac43e5bbb01fdf93558e6b9862c17213c6)
Elektrische und magnetische Felder werden beim Übergang zwischen verschiedenen Inertialsystemen ineinander transformiert!
Umeichung:
![{\displaystyle {{\tilde {\Phi }}^{\mu }}={{\Phi }^{\mu }}+{{\partial }^{\mu }}\phi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb8d42be7af95a367a4736dc66e54b0590b94a40)
Somit:
![{\displaystyle {\begin{aligned}&{{\tilde {F}}^{\mu \nu }}={{\partial }^{\mu }}{{\tilde {\Phi }}^{\nu }}-{{\partial }^{\nu }}{{\tilde {\Phi }}^{\mu }}={{\partial }^{\mu }}\left({{\Phi }^{\nu }}+{{\partial }^{\nu }}\phi \right)-{{\partial }^{\nu }}\left({{\Phi }^{\mu }}+{{\partial }^{\mu }}\phi \right)\\&={{\partial }^{\mu }}{{\Phi }^{\nu }}-{{\partial }^{\nu }}{{\Phi }^{\mu }}+{{\partial }^{\mu }}{{\partial }^{\nu }}\phi -{{\partial }^{\nu }}{{\partial }^{\mu }}\phi ={{F}^{\mu \nu }}\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/059928583d5bf518c6ec873c83dc1cf749b3894c)
Homogene Maxwell- Gleichungen
![{\displaystyle {\begin{aligned}&\nabla \cdot {\bar {B}}={{\partial }_{1}}{{B}^{1}}+{{\partial }_{2}}{{B}^{2}}+{{\partial }_{3}}{{B}^{3}}=0\\&\Rightarrow {{\partial }_{1}}{{F}^{32}}+{{\partial }_{2}}{{F}^{13}}+{{\partial }_{3}}{{F}^{21}}=0\\&\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/549585465881ffa4c470c667e17e69df0a8ed8f4)
Mit
![{\displaystyle {\begin{aligned}&{{\partial }_{1}}=-{{\partial }^{1}}\\&{{F}^{32}}=-{{F}^{23}}\\&\Rightarrow {{\partial }^{1}}{{F}^{23}}+{{\partial }^{2}}{{F}^{31}}+{{\partial }^{3}}{{F}^{12}}=0\\&\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f62e8ac2d2bf77845d91f0c79102031ff9d668df)
+ zyklisch in (123)
innere Feldgleichung für E- Feld
![{\displaystyle \nabla \times {\bar {E}}=-{\frac {\partial }{\partial t}}{\bar {B}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6fa7f0ed4f800ee3a9f829ea9bffc2c446d51c0)
- Komponente
![{\displaystyle {{\partial }_{2}}{{E}^{3}}-{{\partial }_{3}}{{E}^{2}}+{\frac {\partial }{\partial t}}{{B}^{1}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/23d1206aaac3c9ba4625bf872773baaebf6348dd)
![{\displaystyle \Rightarrow {{\partial }^{0}}{{F}^{23}}+{{\partial }^{2}}{{F}^{30}}+{{\partial }^{3}}{{F}^{02}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dd5f859e391a7b96b52b4621776c0fceb2957c1f)
und zyklisch (023)
zyklische Permutation 1 → 2 → 3 → 1 und mit
![{\displaystyle {{F}^{ik}}=-{{F}^{ki}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4b83b5cd50ccbed0a99f11d5765909bc349f8304)
liefert:
![{\displaystyle {\begin{aligned}&\Rightarrow {{\partial }^{0}}{{F}^{13}}+{{\partial }^{3}}{{F}^{01}}+{{\partial }^{1}}{{F}^{30}}=0\quad zyklisch(013)\\&\Rightarrow {{\partial }^{0}}{{F}^{12}}+{{\partial }^{1}}{{F}^{20}}+{{\partial }^{2}}{{F}^{01}}=0\quad zyklisch(012)\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cc7e42625eb17de03ed315cbdaf628e561b4dbfb)
Zusammenfassung der homogenen Maxwellgleichungen
![{\displaystyle {{\varepsilon }^{\kappa \lambda \mu \nu }}{{\partial }_{\lambda }}{{F}_{\mu \nu }}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/90586431d5a593faa7bbf01bd93e8ed21af06625)
![{\displaystyle {{\varepsilon }_{\kappa \lambda \mu \nu }}{{\partial }^{\lambda }}{{F}^{\mu \nu }}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/48e45c52ce1bcc8503fdd5714eb696917f7b47b3)
Die "4- Rotation" des Feldstärketensors verschwindet!
Levi- Civita- Tensor:
+1 für gerade Permutation von 0123
-1 für ungerade Permutation von 0123
0, sonst
Bemerkungen
- Levi- Civita ist vollständig antisymmetrisch (per Definition).
![{\displaystyle {{\varepsilon }^{\kappa \lambda \mu \nu }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b719be8dc3c35985729eac1745d45fa982eab44f)
- transformiert unter Lorentz- Trafo
![{\displaystyle {\begin{aligned}&{{\varepsilon }^{\kappa \lambda \mu \nu }}{\acute {\ }}={{U}^{\kappa }}_{\alpha }{{U}^{\lambda }}_{\beta }{{U}^{\mu }}_{\gamma }{{U}^{\nu }}_{\delta }{{\varepsilon }^{\alpha \beta \gamma \delta }}\\&=\left|{\begin{matrix}{{U}^{\kappa }}_{0}&{{U}^{\kappa }}_{1}&{{U}^{\kappa }}_{2}&{{U}^{\kappa }}_{3}\\{{U}^{\lambda }}_{0}&{{U}^{\lambda }}_{1}&{{U}^{\lambda }}_{2}&{{U}^{\lambda }}_{3}\\{{U}^{\mu }}_{0}&{{U}^{\mu }}_{1}&{{U}^{\mu }}_{2}&{{U}^{\mu }}_{3}\\{{U}^{\nu }}_{0}&{{U}^{\nu }}_{1}&{{U}^{\nu }}_{2}&{{U}^{\nu }}_{3}\\\end{matrix}}\right|=\left(\det U\right)\cdot {{\varepsilon }^{\kappa \lambda \mu \nu }}\\&\left(\det U\right)=\pm 1\\\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8e1d5addcadf5d17ea85b5d37941c4e311ef6bd)
Damit nun der Levi- Civita- Tensor invariant unter Lorentz- Trafos wird, also
,
muss vereinbart werden, dass die Transformation lautet
![{\displaystyle {{\varepsilon }^{\kappa \lambda \mu \nu }}{\acute {\ }}=\left(\det U\right){{U}^{\kappa }}_{\alpha }{{U}^{\lambda }}_{\beta }{{U}^{\mu }}_{\gamma }{{U}^{\nu }}_{\delta }{{\varepsilon }^{\alpha \beta \gamma \delta }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0bb83b6c7b2febf0af3030b6a5297fba21bc106)
Damit ist der Tensor aber ein Pseudotensor!
Insgesamt ist die vierdimensionale Schreibweise die gleiche Formalisierung wie im Dreidimensionalen:
![{\displaystyle {{\left(\nabla \times {\bar {A}}\right)}_{\alpha }}={{\varepsilon }^{\alpha \beta \gamma }}{{\partial }_{\beta }}{{A}_{\gamma }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9240c8f2b8ad0f25524dcf94c52a6d01cbf289d3)
Mit Pseudovektor
![{\displaystyle {{\left(\nabla \times {\bar {A}}\right)}_{\alpha }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/268e50a4eab5e64f73af60014492516d729aa460)