LECTURE NOTES The Relativistic Version of Maxwell s Stress Tensor

Transcription

1 UIUC Phsis 36 M ields & Sures II all Semester, 5 Let. Ntes 8.75 Prf. Steen rrede LCUR NOS 8.75 he Relatiisti Versin f Mawell s Stress ensr Despite the fat that we knw that the M energ densit um B and Pnting s Vetr, S B are nt Lrent inariant quantities, we ask: Is there a related entit, relatiisti in nature, frm whih we an understand the transfrmatin prperties f u and S, in ging frm ne IR(S) t anther IR(S )? M he answer is es the -dimensinal relatiisti generaliatin f the 3-dimensinal lassial eletrdnamis Mawell s stress tensr: ij!!! Reall that the lassial eletrdnamis 3-dimensinal Mawell stress tensr is, a 9-mpnent, rank tw 3 3 smmetri tensr (i.e. a matri) whse elements are: n.b. ij elements are smmetri: ij = ji is a smmetri rank- tensr. ij i j ij BB i j ijb where: i, j :3 if i j he 3-D Kreneker -funtin: ij if i j Phsiall, is the fre per unit area (r stress) ating n a surfae f interest. ij fre per unit area in i th diretin ating n an element f surfae in the j th diretin. hus:,,, phsiall represent pressures. SI units: N m And:,,,,, phsiall represent shears. SI units: N m has same SI units as energ densit In lassial eletrdnamis, the fre per unit lume (aka fre densit) is: J N m N 3 3 m m m Sr, t f r, t r, t where: Sr, t r, tbr, t = Pnting s Vetr t SI units f fre densit: 3 N m SI units: Watts m n.b. SI units f all same (= pressure): N - kg m s kg m m m-s Sr, t t f r t d r t d d t he ttal fre is therefre:,, Use the diergene therem n the st integral:, Sr, t t r t da d S t SI units: - / N kg m s Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-5. All Rights Resered.

2 UIUC Phsis 36 M ields & Sures II all Semester, 5 Let. Ntes 8.75 Prf. Steen rrede In ging frm the lassial eletrdnamis 3-D spatial ersin f Mawell s stress tensr a 3 3 = 9 element smmetri rank tw tensr ij i j ij i j ij BB i j ijb BB i j ij B i, j :3 t the relatiisti -dimensinal spae-time ersin f Mawell s stress tensr a = 6 element smmetri tensr, :3 we epet the new /additinal tempral mpnents f i.e. a new tp rw (rw # lumn # :3 ) and a new LHS lumn (rw # :3, lumn # ) t: a.) be smmetri, i.e. b.) hae the same phsial SI units ( N m = pressure/energ densit) as ij.) hae smething t d with the tempral aspets f M field energ flw d.) be related t the M field tensr, (r equialentl, G ) We define the relatiisti ersin f Mawell s stress tensr as: g g g g {n.b. impliit sum er repeated indies!} Where the flat spae-time metri tensr, g g g where: is defined as: fr g g fr,,3 fr Nte the similarities and differenes in the phsial appearane between definitins f the 3-D lassial M stress tensr ij and the -D relatiisti M stress tensr : + i j ij BB i j ij B ij g g n.b. smmetri stress tensrs, SI units: N kg m m s Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-5. All Rights Resered.

3 UIUC Phsis 36 M ields & Sures II all Semester, 5 Let. Ntes 8.75 Prf. Steen rrede he differenes arise frm ur (& Griffiths ) definitin f the anti-smmetri M field tensr, B B B B and: B B B B (See Griffiths prblem.5, p. 537) r larit s sake, here we present the results fr Appendi A f these leture ntes. Reall that Pnting s Vetr, S B Reall that M field linear mmentum densit: M B : and plae the (tedius!) alulatinal details in SI units: Watts J s J N-m N m m m -s m -s m-s M B B S S M S B B B SI units f M S : kg m kg but: N kg m m s s ms s kg m N N m J = pressure = = energ densit 3 3 m s m m m Reall als that the M field energ densit u M is defined as: um B B B B B J N m N SI units: = energ densit = pressure = fre per unit area 3 3 m m m = {linear} mmentum flu densit Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-5. All Rights Resered. 3

4 UIUC Phsis 36 M ields & Sures II all Semester, 5 Let. Ntes 8.75 Prf. Steen rrede pliitl reminding the reader that the 3-D lassial eletrdnamis ersin f Mawell s stress tensr is defined as: Clumn # j = :3 = = 3= Rw # i = :3 j i ij = BB i j ij B 3 Rw # Clumn # (:3) (:3) if i ij if i j j B B BB BB BB B B BB BB BB B B B B B BB BB BB B B B B BB BB BB B B B Nte that the ij elements are smmetri, i.e. ji ij Phsiall: i, j :3 are pure spae-spae mpnents. ij ij th i mpnent f fre arss unit area perpendiular t j th diretin.,, represent pressures n enlsing surfaes in the,, diretins, respetiel. ii,, represent shear stresses n enlsing surfaes in the,, ij r diretins. But nte als that phsiall: = e f the rate f flw f the i th mpnent f M field linear mmentum p thrugh unit area ij whse nrmal is in the j th diretin, i.e. ij is the i th mpnent f the linear mmentum flu densit transprted in the j th diretin b the M fields. Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-5. All Rights Resered.

5 UIUC Phsis 36 M ields & Sures II all Semester, 5 Let. Ntes 8.75 Prf. Steen rrede hen, the relatiisti -D spae-time ersin f Mawell s stress tensr defined as: g gies: Clumn # Rw # Clumn # =t = = 3= Rw # :3 :3 3 = t tt t t t 3 = t = 3 = = t = t Nw while we might hpe that the {, : 3} pure spae-spae elements f the relatiisti -D spae-time stress tensr wuld be idential t that f the 3-D lassial eletrdnamis / Mawell s stress tensr ij, beause f ur definitins fr and what we instead btain is:, =:3 ij i, j:3 hus, we see that the pure spae-spae {, :3} mpnents f the relatiisti stress tensr phsiall represent M field linear mmentum flu densities, the negatie f whih phsiall rrespnds t stresses/shears n bunding surfaes! he determinatin f the indiidual = 6 elements f the relatiisti -D spae-time stress tensr is arried ut in Appendi A at the end f these Phsis 36 Leture ntes. he tempral mpnents f are: st lumn : um B B B BB B B B B B 3 B B B B Realling that Pnting s Vetr: S B hus: S Sˆ ˆ ˆ ˆ ˆ ˆ S S B B B B B B S B BB Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-5. All Rights Resered. 5

6 UIUC Phsis 36 M ields & Sures II all Semester, 5 Let. Ntes 8.75 Prf. Steen rrede st rw : S B B S B B 3 S B B hus, we (epliitl) see that: S S S 3 3 r: S B hus the mpnents f the relatiisti -D/spaetime ersin f Mawell s stress tensr are { smmetri tensr}: g 3 tt t t t tt t t t 3 t t 3 t t t t 3-D spae-spae mpnents f lassial eletrdnamis Mawell s stress tensr ij um B S S S B B BB BB BB B B BB BB BB B B 6 Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-5. All Rights Resered.

7 UIUC Phsis 36 M ields & Sures II all Semester, 5 Let. Ntes 8.75 Prf. Steen rrede Phsiall: M field energ -flw f M -flw f M -flw f M densit, u M field energ, S field energ, S field energ, S flu f M field flw f flw f flw f mmentum densit, M field linear M field linear M field linear mmentum densit, mmentum densit, mmentum densit, flu f M field flw f flw f flw f mmentum densit, M field linear M field linear M field linear mmentum densit, mmentum densit, mmentum densit, flu f M field flw f flw f flw f mmentum densit, M field linear M field linear M field linear mmentum densit, mmentum densit, mmentum densit, SI units f J N-m N Pasals pressure m m m : energ densit = 3 3 u B B M using: B B B B S B B B B S B B B B S 3 3 S B B B S B B B S B B B Nte that while g 3 3 and/r are nt Lrent-inariant quantities, g and/r are indeed Lrent-inariant quantities. Hene, additinall nte the alue f the trae f this frm f Mawell s relatiisti stress tensr (a Lrent-inariant quantit): 33 r r g u u 3u M M M Intimatel nneted t fat that the phtn mass m must be alid in an/all IR s!! Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-5. All Rights Resered. 7

8 UIUC Phsis 36 M ields & Sures II all Semester, 5 Let. Ntes 8.75 Prf. Steen rrede As a simple eample f the use f the -D/relatiisti ersin f Mawell s stress tensr, let us nsider the purel eletrstati prblem f a parallel-plate apaitr at rest in the lab frame IR(S), with large area plates t the - plane as shwn in the figure belw: Inside the plates { d, }: ˆ elsewhere B eerwhere hen: u S S S Onl the diagnal elements f = are nn-er (here) M u +e nerg Densit (J/m 3 ) M +e M pressure = +e M pressure = e M pressure = 33 in ˆ -diretin!!! in ŷ -diretin!!! in ẑ -diretin {n.b. ˆ }!!! Plates f apaitr attrated t eah ther net attratie fre ating n bttm/tp plates: ensin: w ˆ Q ˆ, w ˆ Q ˆ bt tp bt 8 Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-5. All Rights Resered.

9 UIUC Phsis 36 M ields & Sures II all Semester, 5 Let. Ntes 8.75 Prf. Steen rrede Appendi A: Calulatin f the lements f the Relatiisti Versin f Mawell s Stress ensr g g g g SI units f J N-m N Pasals pressure m m m : energ densit = 3 3 flat spae-time metri tensr: fr g g fr fr, r 3 i.e. g g Anti-smmetri M field tensr, And: Clumn # Rw # : 3 3 B B 3 B B B B g means that we plae a minus () sign in frnt f the tempral mpnents, i.e. elements in the first ertial lumn ( ) f g means that we plae a minus () sign in frnt f the tempral mpnents, i.e. elements in the first hrintal rw ( ) f g g means that we plae a minus () sign in frnt f bth f the Clumn # 3 Rw # 3. tempral mpnents, elements in the first ertial lumn ( ).AND. the tempral mpnents, elements in the first hrintal rw ( ) f.. g means that we plae a minus () sign in frnt f the tempral mpnents, i.e. elements in the first hrintal rw ( ) f. Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-5. All Rights Resered. 9

10 UIUC Phsis 36 M ields & Sures II all Semester, 5 Let. Ntes 8.75 Prf. Steen rrede hus: B B g B B n.b. has n epliit B B smmetr neither smmetri nr antismmetri! B B B B B B B B g n.b. has n B B epliit smmetr B B neither smmetri nr antismmetri! B B B B B B B B g g g B B B B B B n.b. is nb.., anti-smmetri! B B B B B B g B B n.b. has n B B epliit smmetr neither smmetri nr antismmetri! B B g B B B B Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-5. All Rights Resered.

11 UIUC Phsis 36 M ields & Sures II all Semester, 5 Let. Ntes 8.75 Prf. Steen rrede irst, we want t epliitl alulate: where: rw # lumn # B B B B B B B B B B B B B B B B B B B B hen: g g B See als Griffiths prblem.5 a.) Net, we need t alulate whih is a = 6 element smmetri tensr hus: sine S S, then nl ut f the 6 elements f S are unique. n.b. repeated indies are summed er :3 fr eah element f S We epliitl alulate all 6 elements f the tensr: S B B B B S B B B B B B B B Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-5. All Rights Resered.

12 UIUC Phsis 36 M ields & Sures II all Semester, 5 Let. Ntes 8.75 Prf. Steen rrede ( st lumn) f S :3 : S S B B B B S B B B B S ( nd lumn) f B B B B S :3 : S B B B B S BBBB B B S BBBB B B S B B B B B B Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-5. All Rights Resered.

13 UIUC Phsis 36 M ields & Sures II all Semester, 5 Let. Ntes 8.75 Prf. Steen rrede (3 rd lumn) f S :3 : S B B B B S B B B B B B S BBBB B B S ( th lumn) f B B B B B B S :3 : S B B B B S BBBB B B S BBBB B B S BBBB B B 33 Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-5. All Rights Resered. 3

14 UIUC Phsis 36 M ields & Sures II all Semester, 5 Let. Ntes 8.75 Prf. Steen rrede Clleting ur results fr 6 elements the {intermediate} tensr: S S S S S S S S S S S S S S S S S S B B B B B B B B B B BB BB B B BB B B B B B B BB BB B B Nte that S is indeed a smmetri matri, i.e. S S hen: g S B where: g + g + + S Nte that the g B B B B B B B B B B BB BB B B BB B B B B B B BB BB B B term in ntributes nl t the diagnal elements f S g B B B B B um, whih are: Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-5. All Rights Resered.

15 UIUC Phsis 36 M ields & Sures II all Semester, 5 Let. Ntes 8.75 Prf. Steen rrede S g B B B B B B B B B B B B B B B B B B B B B B B B B B B B rm the abe elements 33 S, S, S and their li permutatin smmetries we see that: B B B B B and: 33 B B B B B hen fr all ther remaining elements f the relatiisti ersin f Mawell s stress tensr,, we see that sine: S hus: g then: B um B B B B B B B B B B B B BB B B B B B B B B B B B B BB B B Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-5. All Rights Resered. 5

16 UIUC Phsis 36 M ields & Sures II all Semester, 5 Let. Ntes 8.75 Prf. Steen rrede Pnting s etr is: S B ˆ ˆ ˆ B B B B ˆ ˆ ˆ B B B B B B hus: S ˆ ˆ B B B B B B ˆ he 3-D relatiisti linear mmentum densit is: S r: S. Hene, we see that: u S S S 3 3 M S S S 3 3 hus: g g g g gies: um S S S B B BB BB B B B B B B B B BB B B 6 Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-5. All Rights Resered.

17 UIUC Phsis 36 M ields & Sures II all Semester, 5 Let. Ntes 8.75 Prf. Steen rrede he lassial eletrdnamis 3-D fre densit is: where: f fˆ fˆ fˆ But:, =:3 ij i, j:3 and: S S f t t S t ˆ ˆ ˆ ˆ ˆ 3 ˆ S S S he relatiisti eletrdnamis -etr fre densit is:, f f f (SI units: he 3-D spatial etr mpnent f the fre densit is: f JB he erth/tempral/salar mpnent f the -etr f is: f J net B J B 3 N m ) Magneti fres d n wrk! u J M t SI units: 3 N m he relatiisti etr fre densit f J. But we als hae the relatin: f 3 3 where: he ttal relatiisti -etr fre an likewise be btained b nting that the 3-D spatial lassial eletrdnamis ersin f the 3-D ttal fre etr is: S S f d d t t hus: f d d d where:, he -D spae-time lume has lume element d dt ddd and: ˆ ˆ ˆ is the ariant -D gradient peratr. We epliitl shw that: f J Mawell s relatiisti stress tensr is: g g and: g, hen: g g hus: g. Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-5. All Rights Resered. 7

18 UIUC Phsis 36 M ields & Sures II all Semester, 5 Let. Ntes 8.75 Prf. Steen rrede B the hain rule f differentiatin: and: Mawell s inhmgeneus equatin is: But: J g g J J g g J g g J J. But and are dumm indies. Let and. hen: J. Nw: beause the M field strength tensr is anti-smmetri. J. Hene: J J f f hen, nting that and are dumm indies, we replae and, thus re-writing the braketed term in the abe epressin as: Mawell s hmgeneus equatin (in ariant frm) is: It an be shwn (ia mre tensr manipulatin) that Mawell s hmgeneus equatin in its ntraariant frm is:. hus, we an see that the last tw terms in the RHS braket abe: Hene: f f But nte that the term:, i.e.. Hweer, the duble-ntratin f an antismmetri tensr is smmetri with respet t ehange f the indies with a smmetri tensr S S r the smmetri prtin f a rank-3 tensr is er! Nte the li permutatins f,, Hene: f r: f J 8 Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-5. All Rights Resered.

19 UIUC Phsis 36 M ields & Sures II all Semester, 5 Let. Ntes 8.75 Prf. Steen rrede If ne wishes t Lrent bst these results defined in ne IR(S) t anther IR(S') ming with relatie elit, there are tw equialent methds t amplish this task: Methd I: irst, prided at least ne M field mpnent is parallel t the bst ais alng, Lrent transfrm the and B fields ia use f the matri relatin: B B where: n.b. peratr matri See/read P36 Leture Ntes 9 fr further details f this methd. hen mpute the new, and {.. nb } and the new S in IR(S'): g, f, f d d his methd has the adantage that all quantities, e.g., B, S,,, S,, f and are epliitl knwn/alulated in IR(S'). Methd II: Lrent transfrm diretl, sine the Lrent transfrmatin f in IR(S) t in IR(S') ming with elit relatie t IR(S) is gien b: hen: n.b. in matri frm: sine is smmetri. Sine: g S g hen: S g But: B = relatiisti inariant i.e. = same alue in an/all IR S!!! Define: S S and: g g Als nte that and, are relatiistiall inariant salar quantities (same in all IR s) hen: S g One is btained, then alulate: f and: f d d Prfessr Steen rrede, Department f Phsis, Uniersit f Illinis at Urbana-Champaign, Illinis 5-5. All Rights Resered. 9