Relativistic effects

1. Relativistic effects High Energy Astrophysics 2009

2. Beaming hypothesis 3C279 CGRO EGRET F>100MeV Doppler boosting

3. The compactness problem • t ~ 1-10 ms Compact sources R0 c t ~ 3 107 cm • Cosmological sources (D~3 Gpc) L~ fD2~1052 erg/s R of our galaxy ~ 30 kpc: extragalactic objects 1 2≥(mec2)2  e+e- ~ pR0nT~TL/4R0c~1015  ~ 1 MeV p fraction of photons above the thereshold of pair production T=6.25x10-25cm2 Optical depth  ( e+e-) >>1

4. Implications: The fireball How can the photons escape the source? Relativistic motion: A plasma of e+ e-, a Fireball, which expands and accelerate to relativistic velocities, optical depth reduced by relativistic expansion with Lorentz factor  The reasons: In the comoving frame  below the thereshold for pair production ’=/ ’1 ’2≤(mec2)2 Number of photons above the threshold reduced by 2( -1) (~2 high-energy photon index); Emitting region has a size of 2R0 reduced by a factor 2+2  6.  < 1 for  ≥100

5. Special relativity The two basic postulates: The relativity principle: the laws of physics are the same for all inertial observers The constancy of the speed of light: c is the same for all inertial observer, independent of their velocity or motion relative to the source of light. Two main insights at the base of the theory (Einsten, Poincare, Lorentz): Electromagnetism: electromagnetic waves in terms of motion of an aether. But, if there is an aether, there would be a preferred observational frame of reference. So if all physical laws should hold for observers in all inertial frame one should abandon the idea of an aether (and introduce the idea of a field). A new concept of space and time. Although space and time are different, they can no longer be considered independent, the quantities one measures depend on the speed at which he/she is traveling.

6. Implications of the constancy of c It is possible to derive all special relativity from these two postulates. And the two postulate together tells that Newton’s laws are incomplete. If one accepts the two postulate, there is no choice, but to replace Newton’s law with new rules that are consistent with them. The new rules must resemble the Newton’s ones when applied to objects moving slowly. In Newtonian mechanics speeds are simply added together. But if the speed of light is constant a beam launched by a train running at 0.5 c, runs always at c. Quite a shocking results! How can it be? The explanation is in the new concept of space and time. If they are linked, it is only a combination of the two which appears the same to two observers who move relative to one another.

7. d c s Time dilatation Clock of a woman on a rocket Same clock seen by a men at rest v

8. 0 L0 non moving rod t0‘ t0 0 t’ v(t-t’) L moving rod Lorentz contraction For the man at rest the star is indeed 4 light years and the rocket takes 4.6 yr to reach it. But the woman on the rocket thinks she is at rest and the star is rushing by her at 0.866c. Because of the Lorentz contraction, what is 4 light year for the man at rest is 2 light years for the women on the rocket. Both man and woman are right, according to SR any inertial observer has the right to consider himself at rest. If the length of a rod in its rest frame is L0 and if the rod moves along its length at speed v with respect to an observer, than the observer sees a length L=L0/ vt’

9. Lorentz contraction

10. Lorentz transformations

11. Lorentz transformations

12. Lorentz transformations

13. Lorents transformations

14. Doppler effect The effect is the combination of both relativistic time dilation and time retardation. Consider a a source of radiation which emits one period of radiation over the time t it takes to move from P1 to P2

15. Doppler effect

16. Four dimensional space-time Indices are raised and lowered with :

17. Representation of Lorentz transformations For the special case of a Lorentz transformation involving aboost along the x-axis:

18. Lorentz invariance Under Lorentz transformations, the dot product of two four vectors, xy = xμyμ is preserved. In terms of Λ, this means: Our first LORENTZ INVARIANT is the PROPER TIME  of an event, which is just the square root of the scalar product of the space-time 4-vector with itself:

19. Phase space number density

20. Energy density and specific intensity

21. βc θ 

22. Beaming relativistico - Moto superluminale Per θ = arccos(β) si ottiene: Quindi se β~1: