Energy Momentum Tensor for Photonic System

Transcription

1 018 IJSST Volume 4 Issue 10 Prin ISSN : Online ISSN : X Themed Seion: Siene and Tehnology Energy Momenum Tensor for Phooni Sysem ampada Misra Ex-Gues-Teaher, Deparmens of Eleronis, Vidyasagar Universiy, Midnapur, Wes Bengal, India ABSTACT Differen expressions for energy momenum ensor in Caresian o-ordinaes under differen ondiions have been obained by aking ino onsideraion he ransformaion of o-ordinaes from a phooni sysem. In doing so, he fundamenal ensor, he Chrisofell s hree index symbol, he iemann-chrisofell urvaure ensor, salar urvaure have been alulaed whih leads o he expression of he energy-momenum ensor. The lue for ransforming he main expression o hose under differen ondiions has been menioned in his work. Keywords : Transformaion marix, Energy-momenum ensor, Chrisofell hree index symbol, Spae-ime urvaure ensor. I. INTODUCTION We know ha a ligh beam onsiss of phoons. Eah phoon possesses energy. I, also, possesses linear momenum direed along he beam axis normal o he wave fron. Again, eah one has a spin angular momenum aligned in and opposie o he direion of propagaion. Thus, a phoon is a omplex sysem as i possesses parile and wave naure and, also, has spin and linear moions simulaneously [1]. Again, here is osillaion of he impulse of he ligh wave. I is noe worhy ha osillaion perpendiular o he direion of propagaion gives rise o linear as well as angular momenum as also o polarizaion of he waves. Thus, a phoon has linear veloiy, a roaion and an osillaion [, 3]. Now, i is well known ha he energy momenum ensor is of fundamenal imporane in he sudy of graviaional physis. This parameer is relaed o he haraerisis of graviy, as seen from Einsein s Field Equaion (negleing osmologial onsan) 1 8 G g T K T 4 0 (1) In his equaion he erms on he lef represen geomery of spae-ime while ha on he righ is energy- momenum ensor whih is a measure of he maer energy densiy [4]. I is o be menioned ha for empy spae 0. Hene, in absene of maer T 0 [5]. I is o be remembered ha differen ypes of expressions for energy-momenum ensor were pu forward by differen workers. The fa is ha any one of he expressions ould be used for differen works [6]. I is eviden ha he energy momenum ensor for phooni sysem would be a bi differen from ha of ohers. So, if one an alulae he exa energy momenum ensor in he sysem of phoon hen Einsein s equaions and Einsein s ensor ould be easily obained. On solving hese equaions one may obain he haraerisis of phoon wave [7]. If we onsider a generalized field equaion for an arbirary onservaive field where K is he oupling IJSST eeived : 07 O 018 Aeped : 0 O 018 Sepember-Oober -018 [ 4 (10) : ] 33

2 ampada Misra e al. In. J. S. es. Si. Tehnol. 018 Sepember-Oober-018; 4(10) : onsan appropriae for he onerned field hen he energy- momenum ensor in he sysem of phoon ould be obained using fundamenal ensor of i. So, in his work we shall ry o find ou Chrisofell s hree index symbol from he fundamenal ensor and, hen, alulae he omponens of energy-momenum ensor in Caresian o-ordinae sysem under differen ondiions. Menion mus be made o he fa ha long expressions are no shown in he ex when here is he ase of mahemaial alulaion only. Bu hese are shown in he appendix and hins are given in he ex so ha one an obain he full form of he expression if he is ineresed abou i. II. Energy-Momenum Tensor From (1) we an wrie he energy-momenum ensor o be 1 1 T ( g ) K () Due o he symmery of energy-momenum ensor and for he ase of phoon T, insead of T, will suffie o desribe he phoon. Hene, he energymomenum ensor ould be expressed in he marix form as From (5) he deerminan of g beomes T T T T T (3) The elemens of energy-momenum ensor ould be wrien as 1 1 T ( g ) K (4) We may find ou he expression of g as in [8, 9],, for he sysem of phoon under differen ondiions and an have he values of T from (4) as shown in he following seions. Seion II.A. Using Caresian o-ordinaes Considering he haraerisis of phoon he fundamenal ensor g in Caresian o-ordinaes would be given by [10] g 1 Sin 0 0 ( x Sin y Sin ) x Sin ( ) 0 x Sin y Sin x Sin x Sin x Cos x y Cos y Sin (6) Also, we shall have 11 x Cos x y Cos y g, Sin (7) x Sin 33 1 g, g 1, g Using (5) and (7) we an alulae he neessary Chrisofell s hree index symbols o be g x ( g), g y ( g), 11 g ( g11 ), (8) g x ( g) 41, 4 g y ( g) 4, g ( g) (5) Inernaional Journal of Sienifi esearh in Siene and Tehnology ( 34

3 ampada Misra e al. In. J. S. es. Si. Tehnol. 018 Sepember-Oober-018; 4(10) : Oher Chrisofell symbols are no required for our purpose. Here x represen differeniaion of a funion wih respe o x and so on. Using (8) we shall obain he iemann-chrisofell urvaure ensor from ( ) ( ) o be x x x( 14 ) ( 11 ) ( 14) y ( ) ( ) 0 ( ) ( ) x y 41 4 So, he salar urvaure [ g ] omes ou o be g [ ( ) ( ) ( ) ] g [ ( ) ( ) ] x y 4 4 g [ ( ) ( ) ] x y 41 4 Using (9) and (10) we migh have he elemens of energy-momenum ensor from (4) by subsiuing required values from APPENDIX. Seion II. A.1. Using Caresian o-ordinaes wih ω = π/. In his ase y x ( g ) / y x 0 ( x y ) The deerminan of he above marix beomes. This would lead one o 1 11 y x 33 1 g ( 1), g ( 1), g 1, g (1) Wih he use of (1) we obain 1 x y 4 ( y ), ( x ), 11 0, (13) 4 x 4 4 y , 4 4 All oher Chrisofell symbols are zero and are no required in he presen ase. Using (13) we obain he spae-ime urvaure ensors o be 6 x y x y 11 ( 1), ( 1), 33 0, (14) 4 The salar urvaure ensor omes ou o be 4 ( x y ) 4 (15) 4 Now, we shall have he elemens of he energy-momenum ensor as shown below (11) (9) (10) Inernaional Journal of Sienifi esearh in Siene and Tehnology ( 35

4 ampada Misra e al. In. J. S. es. Si. Tehnol. 018 Sepember-Oober-018; 4(10) : y T11 [ 1] K 1 x T [ 1] K (16) 1 T33 [ ( x y ) ] K 4 1 x y 4 4 T [4 ( x y ) 3( x y )] K We may pu hese in he form of (3) if we like so. I is o be poined ou ha all hese relaions ould be obained easily by subsiuing / in he orresponding relaions of seion II. A. Seion II. A.. Using Caresian o-ordinaes wih ω = 0. This ase is ha of a plane polarized phoon beam. In his ase we shall have Sin 0 0 x Sin Cos ( g ) 0 (17) x Sin Cos 0 0 x Cos This will lead o Sin (18) Now, we shall have he following 11 x Cos g, g 1, g 1, g (19) Sin The hree index Chrisofell symbols ould be easily obained as x Cos x Cos 4 Sin, 0, 11, Sin (0) 3 4 x Cos x Sin 14 41, 4 0 4, Wih he help of (17) and (0) we obain he iemann-chrisofell urvaure ensor o be 4 x Cos x Sin 11 [ Cos Cos ], 0, 4 (1) x Cos x Cos 33 0, Co [1 ] 4 Hene, he salar urvaure ensor beomes x Co Cos x 4 x Co Cos Co Cos x Co Sin Cos x Cos Co 4 4 Sin Co () Inernaional Journal of Sienifi esearh in Siene and Tehnology ( 36

5 ampada Misra e al. In. J. S. es. Si. Tehnol. 018 Sepember-Oober-018; 4(10) : We an easily find ou he elemens of energymomenum ensor by using (17), (1) and () in (4). Of ourse, all he expressions ould be easily obained from hose in seion A by subsiuing 0. expression for energy-momenum ensor gives us informaion abou he haraerisis of he medium and he sae of polarizaion of he phoon wave. This is learly seen espeially from seion II. A. 3. ( Seion II. A. 3. Using Caresian o-ordinaes wih boh ω = π/ and ω = 0 simulaneously. Here, he siuaion is ha of a polarized wave. We shall have g ) /, 0 This leads o 3 and g 1 g g, g (3) (4) I ould be easily shown ha all he hree index Chrisofell symbols as well as all he urvaure ensors are zero. Hene, all he elemens of energymomenum ensor are zero. Now, aording o Einsein s law of graviaion he iemann urvaure ensor is zero for empy spae. So, he resul signifies ha here is absene of maer in he spae onsidered and also he spae-ime is fla. The idal fore is, also, absen in his ase [11, 1]. So, he ondiion / and 0 means absene of maer in he spae-ime oninuum. III. CONCLUSION I is obvious from his work ha energy-momenum ensor ould be easily obained from he priniple of o-ordinae ransformaion in phooni sysem. From he main expression for a pariular o-ordinae sysem, expressions for oher ases ould be obained by simple subsiuion of he relevan ondiion in he appropriae expression for energy-momenum ensor. Las bu no he leas, we may onlude ha he IV. EFEENCES [1]. M. C. Das,. Misra (014), Lorenz Transformaion in Super Sysem and in Super Sysem of Phoon, Inernaional Leers of Chemisry, Physis and Asronomy, 19 (1), p []. He Li, Huan Li, Mo Li (016), Opohemial Measuremen of Phoon Spin Angular Momenum and Opial Torque in Inegraed Phooni Devies, Siene Advanes,, e [3]. S. J. Van Enk, G. Nienhuis (1994), Spin and Orbial Angular Momenum of Phoons, Europhysis Leers, 5 (7), p-47. [4]. J. B. Harle (009), Graviy: An Inroduion o Einsein s General elaiviy, Pearson Eduaion. [5]. Saya Prakash (1995), elaivisi Mehanis (Theory of elaiviy), Pragai Prakashan, Meeru, India, p-379. [6]. S. W. Hawking, G.F.. Ellis (1973), The Large Sale Sruure of Spae-ime, Cambridge Universiy Press. [7].. Misra, M. C. Das (018), Soluion of Einsein Equaion in ase of Phoon, Paper Communiaed for Publiaion. [8]. Chandru Iyer, G. M. Prabhu (007), Comparison of Two Lorenz Booss hrough Spaial and Spae-ime oaions. Journal of Physial and Naural Sienes, 1 (), p-. [9]. Candru Iyer, G. M. Prabhu (007), Lorenz Transformaions wih Arbirary Line of Moion, European Journal of Physis, 8, p [10].. Misra, M. C. Das, (018), Some Forms of Fundamenal Tensor in Phooni Sysem, Paper Communiaed for Publiaion. Inernaional Journal of Sienifi esearh in Siene and Tehnology ( 37

6 ampada Misra e al. In. J. S. es. Si. Tehnol. 018 Sepember-Oober-018; 4(10) : [11]. S. P. Puri (013), General Theory of elaiviy, Pearson, Delhi, India, p-95. [1]. K. D. Krori (010), Fundamenals of Speial and General elaiviy, PHI Learning Pv. Ld., p- 13. Appendix Equaion (8) of he ex [ x Cos Sin x Cos x y Cos Sin x Sin Cos x y Cos x y Cos x y Sin x y Cos y Cos x Sin x Cos y Cos ] [ x Sin Cos x y Sin x Cos y ] 4 4 x Sin x Cos y Cos 4 11 Sin, 14 41, 3 4 x Cos y 4 4 x ( ) Sin x y Sin 4 4, The urvaure ensors in equaion (9) of he ex are given below x Sin x Cos 11 [ Sin Cos Cos 3 y Cos x Sin x y Sin Sin 3 3 x y Cos x Sin x Sin x y Sin Cos ] [ x Cos y x y Cos ], 33 0, [3x Cos Sin 3x Cos x y Cos Sin 4 3 { x y Cos y Cos y Sin x Cos Sin x Cos x y Cos x y Cos 3x y Cos x y Sin y Cos x ( Sin Cos ) y Cos } 1 1 { x ( Sin Cos ) y Cos }] [ x Cos y ] 4 [ x Sin ][ x Cos y ]. Inernaional Journal of Sienifi esearh in Siene and Tehnology ( 38