International Journal of Advanced Research in Physical Science (IJARPS) Volume 6, Issue 7, 2019, PP 27-29 ISSN No. (Online) 2349-7882 www.arcjournals.org

Energy-Momentum Density’s Conservation Law of Electromagnetic Field in Rindler Space-time

Sangwha-Yi* Department of Math, Taejon University 300-716 *Corresponding Author: Sangwha-Yi, Department of Math, Taejon University 300-716 .

Abstract: We find the energy-momentum density of electromagnetic field by energy-momentum tensor of electromagnetic field in Rindler space-time. We find the energy-momentum density’s conservation law of electromagnetic field in Rindler spacetime. Keywords: The general relativity theory, The Rindler spacetime, Energy-momentum density, Conservation law PACS: 04,04.90.+e, 41.20

1. INTRODUCTION Our article’s aim is that we find the energy-momentum density of electromagnetic field by energy- momentum tensor of electromagnetic field in Rindler space-time. We find the energy-momentum density’s conservation law of electromagnetic field in Rindler space-time.

In inertial frame, the energy-momentum tensor T  of the electromagnetic field is

 1   1   T  (F F   F F ) (1) 4c 4   In this time, in inertial frame, Faraday tensors F ,F  are

 0 E x E y E z   0  E  E  E     x y z  E 0 B B   x z  y  , E 0 B  B  (2) F    F  x z y  E  B 0 B  E  B 0 B   y z x   y z x   E B B 0  E B B 0   z y  x   z y  x  0  Hence, the energy density p f and the momentum density p f of electromagnetic field are 2 2   0 00 E  B 0i  E B T  p f  , T  p  ,i  1,2,3 8c f 4c   E  E ,B  B (3) In inertial frame, the energy-momentum conservation law of electromagnetic field is by Noether theorem,  00 0i T , T ,0 T ,i ,   1  E 2  B 2  E B  ( )  ( ) 0 (4) c t 8c 4c

2. ENERGY-MOMENTUM DENSITY'S CONSERVATION ELECTROMAGNETIC FIELD IN RINDLER SPACETIME Rindler space-time is a  1 1 1 d 2  (1 0 )(d 0 )2  [(d 1 )2  (d 2 )2  (d 3 )2 ]   g d  d  (5) c 2 c 2 c 2  In Rindler space-time, the energy-momentum tensor of the electromagnetic field is

International Journal of Advanced Research in Physical Science (IJARPS) Page | 27 Energy-Momentum Density’s Conservation Law of Electromagnetic Field in Rindler Space-time

 1   1   T  (F F  g F F ) (6) 4c 4   In this time, in Rindler space-time, Faraday tensor F  is[2]

 0 E 1 E 2 E 3       1 1  a 0 a 0   E 1 0 (1 )B 3  (1 )B 2   c 2  c 2      (7) F  1 1   a 0 a 0   E 2  (1 )B 3 0 (1 )B 1   c 2  c 2    a  1 a  1   0 0   E 3 (1 )B 2  (1 )B 1 0    c 2  c 2   In Rindler space-time, Faraday tensor F is[2]  

1 1 1  a 0 a 0 a 0   0  (1 )E 1  (1 )E 2  (1 )E 3   c 2  c 2  c 2    a  1  (1 0 )E 0 B B (8)   2 1  3   2  F   c    1  a 0  (1 )E 2  B 3 0 B 1  c 2       a 1  0  (1 )E 3 B 2  B 1 0   c 2     0  Hence, the energy density p and the momentum density p of electromagnetic field are in Rindler f  f space-time. 2 2 0 1 E  B T 00  p    (9)  f a  1 8c (1 0 ) c 2    E B T 0i p   , i  1,2,3 (10)  f   4c E   E  ,B   B  (11)

In Rindler space-time, the energy-momentum conservation law of electromagnetic field is by Noether theorem,

 00 0i 0 T ; T ;0 T ;i T ; , i  1,2,3

0 T 0   0    T   T (12) x  In this time, affine connections are in Rindler space-time

1 a 0 1 a 0  00  (1  ) , 0 0 1 a 0 (13) c 2 c 2  10   01  a  1 c 2 (1 0 ) c 2 Hence, in Rindler space-time, the energy-momentum conservation law of electromagnetic field is

0 T 0 01   3 01T x 

International Journal of Advanced Research in Physical Science (IJARPS) Page | 28 Energy-Momentum Density’s Conservation Law of Electromagnetic Field in Rindler Space-time

  E 2 B 2  E B 1 1        1 a 0 1  0  ( )  ( ) 3 (E 3 B 2  E 2 B 3 ) a  1 c  0 8c  4c a  1 c 2 4c     (1 0 ) (1 0 ) c 2 c 2       ( , , )   1  2  3   E   E  ,B   B  (14)

3. CONCLUSION We find the energy-momentum density’s conservation law of electromagnetic field in Rindler space- time. REFERENCES [1] S.Yi, “Electromagnetic field equation and Lorentz Gauge in Rindler Space-time”,The African Review of Physics,11,33(2016)-INSPIRE-HEP [2] S.Yi, “Einstein’s Notational Equation of Electro-Magnetic Field Equation in Rindler space-time”, International Journal of Advanced Research in Physical Science,6,5(2019)pp 4-6 [3] S.Weinberg,Gravitation and Cosmology(John wiley & Sons,Inc,1972) [4] W.Rindler, Am.J.Phys.34.1174(1966) [5] P.Bergman,Introduction to the Theory of Relativity(Dover Pub. Co.,Inc., New York,1976),Chapter V [6] C.Misner, K,Thorne and J. Wheeler, Gravitation(W.H.Freedman & Co.,1973) [7] S.Hawking and G. Ellis,The Large Scale Structure of Space-Time(Cam-bridge University Press,1973) [8] R.Adler,M.Bazin and M.Schiffer,Introduction to General Relativity(McGraw-Hill,Inc.,1965) [9] A.Miller, Albert Einstein’s Special Theory of Relativity(Addison-Wesley Publishing Co., Inc., 1981) [10] W.Rindler, Special Relativity(2nd ed., Oliver and Boyd, Edinburg,1966) [11] J.W.Maluf and F.F.Faria,”The electromagnetic field in accelerated frames”:Arxiv:gr-qc/1110.5367v1(2011) [12] Massimo Pauri, Michele Vallisner, "Marzke-Wheeler coordinates for accelerated observers in special relativity":Arxiv:gr-qc/0006095(2000)

Citation: Sangwha-Yi, (2019). Energy-Momentum Density’s Conservation Law of Electromagnetic Field in Rindler Space-time. International Journal of Advanced Research in Physical Science (IJARPS) 6(7), pp.27- 29, 2019.

Copyright: © 2019 Authors, this is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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