Relativistic effects High Energy Astrophysics 2009

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

The compactness problem  t ~ 1-10 msCompact sources R 0  c  t ~ cm Cosmological sources (D~3 Gpc) L  ~ fD 2 ~10 52 erg/s   ~  p R 0 n   T ~  T L  /4  R 0 c   ~10 15   ~ 1 MeV  e + e -   1   2 ≥ (m e c 2 ) 2  e + e -  p fraction of photons above the thereshold of pair production (  e + e - ) Optical depth   (  e + e - ) >>1 T =  T = 6.25x cm 2 R of our galaxy ~ 30 kpc: extragalactic objects

Implications: The fireball 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  How can the photons escape the source? The reasons: 1.In the comoving frame  below the thereshold for pair production  ’  =   /   ’  1  ’  2 ≤ (m e c 2 ) 2 2.Number of photons above the threshold reduced by  2(  -1) (  ~2 high-energy photon index); 3.Emitting region has a size of  2 R 0.   reduced by a factor  2+2   6.   < 1 for  ≥100

Special relativity The two basic postulates: 1.The relativity principle: the laws of physics are the same for all inertial observers 2.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): 1.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). 2.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.

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.

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

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 L 0 and if the rod moves along its length at speed v with respect to an observer, than the observer sees a length L=L 0 /  0 L 0 non moving rod t 0 ‘ t0t0 L moving rod 0 t’ v(t-t’) vt’

Lorentz contraction

Lorentz transformations

Lorents transformations

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 P 1 to P 2

Doppler effect

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

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

Lorentz invariance Under Lorentz transformations, the dot product of two four vectors, x  y = x μ y μ 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:

Phase space number density

Energy density and specific intensity

βcβc θ 

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