Electromagnetic Waves Propagation in Complex Matter Part 10

Tham khảo tài liệu 'electromagnetic waves propagation in complex matter part 10', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | General Relativity Extended 167 Theorem 5. att rep n Epv em Rpv em Ip att rep 2 Rem gpv em 16nG Tatt rep 5 h v em 1 - Ygrav g11 grav c 86 Proof. i-1 Vqi Vx by Equation 76 87 c-att rep c2 v k y q V E11 em i1 mg -mo0VQ x by Equation 46 88 c2 mQoJ V mqo I - _ -ỌV _ - g VQ x _ _ S i s byEquation 43 . T12 em att rep 11 em 89 90 where Tp emme and T1jem j 2 3 4 are respectively the energy-flow and the momentum densities. Thus att rep att rep Eem KemTem has 91 E att rep Kem 11 e T 1 i SII by Equations 76 90 92 em Tatt rep 1 r Q II II vyq - f but 3c2 SII 4nr3 mq oV by Equations 43 46 93 so 6 4 nr3. Km - rp z 4 - rh Q2mr cf Remark 8 95 3 . by the preceding Lemma 4 96 c 1 Ygravg11 gra 3c2 - V5 97 Ự Ygrav g11 grav J c Remark 9. Til ent i IISII has unit recallingfrom Equation 45 mf 2 98 second meter kilogram meter2 1 99 second2 second meter2 kihgrtm 100 second3 168 Electromagnetic Waves Propagation in Complex Matter so that inem Tlerm has unit G kilogram 101 c5 second3 meter3 second5 kilogram kilogram second2 meter5 second3 102 102 1 1 meter2 rk2 103 measuring the local curvatures of M4m. We emphasize that our T11 em represents energy flows in a specific direction across an area of square meter per second which is different from the common identification ofT11 em with stationary energy densities with unit joule meter see . 35 45 equation 1. Remark 10. We can now obtain a geometric union of gravitation and electromagnetism to arrive at p Í 1 Í _ 8nG 16nG att rep fie - Rpv 2 R fie c2 iụr grav T 1 _ -2 gu c5 iụv em 104 where for expository neatness we set rep _ rep . rep _ rep _2. mn g pv em Spv em fiụv 1 g11 em gu em . 105 rT-rep _ rep .J 1 Trep _ rep Ils -I Tpv em Tpv em fipv 1 T11 em T11 em IIs t . 106 Theorem 6. The set of Einstein Field Equations p Í 1p - 8- G 16n G att rep fie Rpv 2 R fie c2 Tpv grav T 1 _ Y-2gn c5 ipv em 107 has solutions with w and Rpv Rpv grav Rụv em 108 R Rgrav Rem 109 gụv att rep wgra-u gụv grav wem gpv ein 110 Igrav R and wem Rem 1 .