Relativistic kinematics

Transcription

1 Relativistic kineatics 4-oentu or a article o ass : (/c, x, y, z ) where total energy: γc and γv γβc ( cdt) ( dx) ( dy) ( dz The line eleent is an invariant I 4-vectors transor like ds, the scalar roduct o theselves is invariant under Lorentz transorations: P {P t,p x,p y,p z } P -P x -P y -P z P t or or 4-vectors PQ -P x Q x -P y Q y -P z Q z P t Q t For the energy-oentu 4-vector: In the rest rae c P (c,0) P c. This is the sae value it has in any re syste: P (/c) - γ c γ v γ c (1 v /c ) c Hence the total energy is ds c c 4 ) 1/19/06 13

2 Lorentz transoration o energy-oentu Given a article o energy and oentu, the 4-oentu is P(,) And P (c1) The velocity o the article is β / The energy and oentu viewed ro the rae oving with velocity β is * * γ γ β γ β γ, * T T is the coonent arallel to β And is the orthogonal one 1/19/06 14

3 CM and laboratory systes We have in a certain rae (laboratory) articles with 4-oenta P 1 and P What is the CM energy? Let s consider 3 invariants P 1 1 P [P 1 P or (P 1 ±P ) ] P 1 (ε 1, 1 ) In CM: * 1 * 0 Hence P* 1 P* (ε* 1 ε*, 0) (*, 0) * (ε* 1 ε* ) (P* 1 P* ) (P 1 P ) I M and P are the total ass and energy-oentu P (ε, ) P (P 1 P ) M * and since it is an invariant P (ε 1 ε ) - ( 1 ) Any 4-vector can be written as vγ and ε γ Laboratory syste So or the tot oentu and energy P Mβγ and Mγ (we assue c1) β CM / and γ CM /M Collisions o articles 1 and at an angle o θ one resect to the other: * P [(ε 1 ε ) -( 1 ) ] 1/ [ 1 ε 1 ε (1-β 1 β cosθ)] 1/ In a e e - collider 1, 1 <<, β 1 β 1 θ 180º * ~ In a ixed target exerient: M >> 1 and β 0, M * ~ (M) >>M 1/19/06 15 θ

4 1/19/06 16 xales 0 4 Proble: suose we ove with article 1, which is or us the energy o article? In the rest rae o 1 (laboratory rae): 1 0 and ε 1 1 P 1 P invariant ε 1 ε 1 1 ε So the energy o article in reerence 1 is: 1 ε P 1 P / 1 that is also an invariant ) ( PP These exressions are invariant and can be evaluated in any re syste

5 xales -body decay Conservation o energy and oentu: M M 1 1 M M 1 M ( ) ( ) 1 M 1 M 1 1 [M ( 1 )][M ( 1 )] M g and 0 ( ) M 4 ( 1 ) M ( 1 ) 4M 4 1/19/06 17

6 Reaction thresholds nergy o rojectile to roduce articles in the inal state at rest, t, t s c c1 True in any reerence syste tot kin t M 1 M...M n tot s M tot tot M M M kin ( t kin, ) ( t ) kin, ( t ) kin, kin kin kin tot t kin, kin kin, ( t ) kin, ( t ) kin, (t ) t kin, 1/19/06 18 M kin In the re rae where the target is at rest ( t ) t

7 Threshold or GZK cut-o [Greisen 66; Zatsein & Kuzin66] Threshold or -γ Δ N.73 K γ k, M ( t ) ( N ) t γ γ Δ Δ n 0 γ γ 145MeV 150MeV in rae where is at rest nergy o CMB hotons: 3k B T eective energy or Planck sectru And their energy in the laboratory rae (rest rae o roton) is γ ~ γ εγ 150MeV γ and the threshold energy o the roton is then ~ γ 10 0 ev Integrating over Planck sectru,th ~ ev 1/19/06 19

8 Transorations o velocity I a oint has velocity u in the rae K the velocity u in K is given by x γ(x'vt') dx γ(dx'vdt') u t γ t' vx' dt γ dt' vdx' x dx dt γ(dx'vdt') γ dt' vdx' u' v x 1 vu' x c c c c dy dy' u y dy dt dy' u' dz dz' γ dt' vdx' y γ 1 vu' x K K u z dz dt 1/19/06 0 v θ u x x c c c dz' u' γ dt' vdx' z γ 1 vu' x c

9 Transorations o velocity Hence the generalization o these equations to an arbitrary velocity v not necessarily along x can be stated in ters o coonents o u arallel and erendicular to v: u x dx dt γ(dx'vdt') γ dt' vdx' u' v x 1 vu' x u u' v u' u c 1 vu' c γ 1 vu' The directions o the velocities in the raes are related by the aberration orula tanθ u u 1 vu' u' c γ 1 vu' u' v c c u'sinθ' γ(u'cosθ'v) u y dy dt u z dz dt c dy' u' γ dt' vdx' y γ 1 vu' x c c c dz' u' γ dt' vdx' z γ 1 vu' x c And the aberration o light is obtained or u c tanθ csinθ' γ(c cosθ'v) sinθ' γ(cosθ'v / c) 1/19/06 1

10 Aberration and beaing Aberration is the aarent change in the direction o a oving object when the observer is also oving sinθ' For θ / tanθ γ eitted erendicular to v in K γ(cosθ'v / c) c γv sinθ tanθ 1tan θ c γv 1 c γv c γ v c 1 γ β 1 1 β β 1 β 1 β 1 γ For highly relativistic seed γ>>1 and θ becoes sall and θ 1/γ In K hotons are concentrated in the orward direction. Very ew hotons are eitted with θ>>1/γ 1/19/06

11 Aberration and beaing I we are at rest in the sacecrat we see light coing ro every direction ro stars, but i the sacecrat travels at relativistic seeds the whole ield o view Shrinks and even hotons coing ro behind, look as coing ro the orward direction. I the shi travels towards Orion 1/19/06 3