Gravitoelectromagnetism. II. Speed of Light in Gravitational Field
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1 Zbigniew Osiak aitoeletomagnetism. II. May 9, 8 aitoeletomagnetism. II. peed of Light in aitational Field Zbigniew Osiak zbigniew.osiak@gmail.om Abstat Fom fou Maxwell-Hetz equations fo the auum and two modified mateial equations, two equations desibing the popagation of the eletomagneti wae with the slowe speed, the stonge the gaitational field, wee obtained. Keywods: Maxwell-Hetz equations, mateial equations, Minkowski spae-time, hwazshild spae-time, onfomally flat spae-time, wae equation, Blak Hole Uniese.. Intodution In the theoy of elatiity, the speed of is detemined by the disappeaane of the squae of the diffeential of spae-time distane. The geneal theoy of elatiity examines the defomations of spae-time though the masses. Fom the geneal fom of the meties of suh spae-times it follows that the speed of is the smalle the stonge the gaitational field and the speed of is only onstant in the onfomally flat spae-times. Due to the simpliity in meti equations onsideed in this wok we will limit ouseles to only two aiables: spatial and tempoal. The symbol will mean the standad alue of the speed of. Fom the fou Maxwell-Hetz equations fo the auum and two modified mateial equations, we get two equations desibing the popagation of the eletomagneti wae with the slowe speed, the stonge the gaitational field.. peed of in Minkowski spae-time ( ds) ( ) ( ) ( ds)
2 Zbigniew Osiak aitoeletomagnetism. II. May 9, 8 3. peed of in onfomally flat spae-time ( ) [ ] ( ) ( ) Φ (x,t) ds ( ) ds 4. peed of in the gaitational field aoding to Einstein Albet Einstein in the papes [,, 3, 4, 5] disussed the poblem of the definition of the speed of in the gaitational field, poposing the following expession: Φ Φ gaitational potential 5. peed of in hwazshild spae-time hwazshild meti is sometimes witten [6] in the following fom: x ds dy y dz z ( ) M, yz dydz xz dz xy dy, x, y z, d, dy dz ( ) ( ) ( ) d ds ( ) ds M d M
3 Zbigniew Osiak aitoeletomagnetism. II. May 9, 8 6. Idem pe idem Below we gie an example of inoet detemination of the speed of based on the hwazshild meti. ( ds) ( d) ( ) d ( ds) M d M The aboe equation is an algebai equation of the fouth degee with espet to. 4 4M d 4 M 7. Maxwell-Hetz equations and modified mateial equations in the pesene of a gaitational field in a auum B ote dib D oth did D B E H E eto of eleti field intensity D eto of eleti indution B eto of magneti indution H eto of magneti field intensity auum pemittiity auum pemeability o elatie eleti pemittiity of the gaitational field elatie magneti pemeability of the gaitational field desibes the eleti popeties of the auum desibes the magneti popeties of the auum desibes the eleti popeties of the gaitational field desibes the magneti popeties of the gaitational field Fom fou Maxwell-Hetz equations and two modified mateial equations, assuming that,, 3
4 Zbigniew Osiak aitoeletomagnetism. II. May 9, 8 afte toublesome tansfomations, two equations an be obtained whose left sides will be in the fom of the wae equation. Wae equation fo eto E E E t Wae equation fo eto H H ( gad ) gad ( E gad) H H t E ( gad) gad ( H gad ) Fom the aboe two equations it follows that: o In the ase of the hwazshild spae-time in a auum: hwazshild adius distane fom the ente of the soue mass 9 On the sufae of the Eath:,5. In the Blak Hole Uniese [7, 8]: R distane fom the Blak Hole Uniese R adius of the Blak Hole Uniese 8. Final emaks I poposed that gaity "ente" the auum mateial equations though the elatie eleti pemittiity of the gaitational field and elatie magneti pemeability of the gaitational field. In [9] I hae gien whih physial ontents ontain geneally oaiant Maxwell-Hetz equations, witten in a modified thee-dimensional fom. Refeenes [] Albet Einstein: Übe das Relatiitätspinzip und die aus demselben gezogenen Folgeungen. Jahbuh de Radioaktiität und Elektonik 4 (97)
5 Zbigniew Osiak aitoeletomagnetism. II. May 9, 8 [] Albet Einstein: Übe den Einfluß de hwekaft auf die Ausbeitung des Lihtes. Annalen de Physik 35, (9) [3] Albet Einstein: Lihtgeshwindigkeit und tatik des aitationsfeldes. Annalen de Physik 38, 7 (9) [4] Albet Einstein und Mael ossmann: Entwuf eine eallgemeineten Relatiitätstheoie und eine Theoie de aitation. Zeitshift fü Mathematik und Physik 6, 3 (93) 5-6. [5] Albet Einstein und Adiaan Daniël Fokke: Die odstömshe aitationstheoie om tandpunkt des absoluten Diffeentialkalküls. Annalen de Physik 44, (94) [6] Zbigniew Osiak: Ogólna Teoia Względnośi (eneal Theoy of Relatiity). elf Publishing (), IBN: , [7] Zbigniew Osiak: Anti-gaity. ixa:6.6 (96) [8] Zbigniew Osiak: Blak Hole Uniese and peed of Light. ixa:85.88 (98) [9] Zbigniew Osiak: aitoeletomagnetism. I. aitational Faday s Law and aitational Ampèe s Law. ixa:85.57 (8) 5
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