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Relativistic Effects Relativistic Effects 1 Introduction The radio-emitting plasma in AGN contains electrons with relativistic energies. The Lorentz factors of the emitting elec- trons are of order 1026– . We now know that the bulk motion of the plasma is also moving relativistically – at least in some regions although probably “only” with Lorentz factors about 10 or so. However, this has an important effect on the prop- erties of the emitted radiation – principally through the ef- fects of relativistic beaming and doppler shifts in frequency. This in turn affects the inferred parameters of the plasma. Relativistic Effects ©Geoffrey V. Bicknell 2 Summary of special relativity For a more complete summary of 4-vectors and Special Rel- ativity, see Rybicki and Lightman, Radiative Processes in As- trophysics, or Rindler, Special Relativity High Energy Astrophysics: Relativistic Effects 2/93 2.1 The Lorentz transformation y S y S V x x The primed frame is moving wrt to the unprimed frame with a velocity v in the x–direction. The coordinates in the primed frame are related to those in the unprimed frame by: High Energy Astrophysics: Relativistic Effects 3/93 Vx x= xvt– t= t – ------ c2 y = y z = z (1) v 1 = -- = ------------------- c 1 – 2 We put the space-time coordinates on an equal footing by putting x0 = ct. The the xt– part of the Lorentz transforma- tion can be written: x= x – x0 x0= x0 – x (2) The reverse transformation is: High Energy Astrophysics: Relativistic Effects 4/93 x = x + Vt yy= zz= Vx (3) t = t + -------- c2 i.e., x = x+ x0 x0 = x0+ x (4) High Energy Astrophysics: Relativistic Effects 5/93 2.2 Lorentz–Fitzgerald contraction 2.3 Time dilation y y V S S x x High Energy Astrophysics: Relativistic Effects 6/93 y y S S V L0 x x 2 1 x – x = L0 2 1 x – x = x2 – x1 – Vt2 – t1 –1 L0 = L L = L0 High Energy Astrophysics: Relativistic Effects 7/93 Consider a clock at a stationary position in the moving frame which registers a time interval T0. The corresponding time interval in the “lab” frame is given by: 2 Tt==2 – t1 t2 – t1 – Vx2 – x1 c (5) ==t2 – t1 T0 i.e. the clock appears to have slowed down by a factor of High Energy Astrophysics: Relativistic Effects 8/93 2.4 Doppler effect P1 V lV= t P2 T o o bse dV= tcos rv D er The Doppler effect is very important when describing the ef- fects of relativistic motion in astrophysics. The effect is the High Energy Astrophysics: Relativistic Effects 9/93 combination of both relativistic time dilation and time retar- dation. Consider a source of radiation which emits one period of radiation over the time t it takes to move from P1 to P2. If em is the emitted circular frequency of the radiation in the rest frame, then 2 t = ---------- (6) em and the time between the two events in the observer’s frame is: High Energy Astrophysics: Relativistic Effects 10/93 2 t ==t---------- (7) em However, this is not the observed time between the events be- cause there is a time difference involved in radiation emitted from P1 and P2. Let D = distance to observer from P2 (8) and t1 = time of emission of radiation from P1 (9) t2 = time of emission of radiation from P2 High Energy Astrophysics: Relativistic Effects 11/93 rec rec Then, the times of reception, t1 and t2 are: DV+ tcos trec = t + -------------------------------- 1 1 c (10) D trec = t + ---- 2 2 c Hence the period of the pulse received in the observer’s frame is High Energy Astrophysics: Relativistic Effects 12/93 D DV+ tcos trec – trec = t + ---- – t + -------------------------------- 2 1 2 c 1 c V = t – t – --- tcos (11) 2 1 c V = t1 – --- cos c Therefore, High Energy Astrophysics: Relativistic Effects 13/93 2 2 V ----------- = ---------- 1 – --- cos obs em c (12) em em ==---------------------------------- --------------------------------- obs V 1 – cos 1 – --- cos c The factor is a pure relativistic effect, the factor 1 – cos is the result of time retardation. In terms of lin- ear frequency: em = --------------------------------- (13) obs 1 – cos High Energy Astrophysics: Relativistic Effects 14/93 The factor 1 = --------------------------------- (14) 1 – cos is known as the Doppler factor and figures prominently in the theory of relativistically beamed emission. 2.5 Apparent transverse velocity Derivation A relativistic effect which is extremely important in high en- ergy astrophysics and which is analysed in a very similar way to the Doppler effect, relates to the apparent transverse veloc- ity of a relativistically moving object. High Energy Astrophysics: Relativistic Effects 15/93 P1 V lV= t P2 To ob se dV= tcos rv D er l = Vtsin High Energy Astrophysics: Relativistic Effects 16/93 Consider an object which moves from P1 to P2 in a time t in the observer’s frame. In this case, t need not be the time between the beginning and end of a periodic wave. Indeed, in practice, t is usually of order a year. As before, the time dif- ference between the time of receptions of photons emitted at P1 and P2 are given by: V t = t1 – --- cos (15) rec c The apparent distance moved by the object is l = Vtsin (16) High Energy Astrophysics: Relativistic Effects 17/93 Hence, the apparent velocity of the object is: Vtsin Vsin V ==------------------------------------ ------------------------------ app V V t1 – --- cos 1 – --- cos c c V (17) --- sin Vapp c ----------- = ------------------------------ c V 1 – --- cos c In terms of = Vc High Energy Astrophysics: Relativistic Effects 18/93 Vapp V = ----------- = --- app c c (18) sin = ------------------------ app 1 – cos The non-relativistic limit is just Vapp = Vsin, as we would expect. However, note that the additional factor is not a con- sequence of the Lorentz transformation, but a consequence of light travel time effects as a result of the finite speed of light. High Energy Astrophysics: Relativistic Effects 19/93 Consequences For angles close to the line of sight, the effect of this equation can be dramatic. First, determine the angle for which the ap- parent velocity is a maximum: dapp 1 – cos cos – sin sin -------------- = --------------------------------------------------------------------------------- d 1 – cos 2 (19) cos – 2 = -------------------------------- 1 – cos 2 This derivative is zero when cos = (20) High Energy Astrophysics: Relativistic Effects 20/93 At the maximum: sin 1 – 2 ====------------------------ ----------------------- ------------------- (21) app 1 – cos 2 1 – 1 – 2 If » 1 then 1 and the apparent velocity of an object can be larger than the speed of light. We actually see such effects in AGN. Features in jets apparently move at faster than light speed (after conversion of the angular motion to a linear speed using the redshift of the source.) This was originally used to argue against the cosmological interpretation of qua- High Energy Astrophysics: Relativistic Effects 21/93 sar redshifts. However, as you can see such large apparent ve- locities are an easily derived feature of large apparent velocities. The defining paper on this was written by Martin Rees in 1966 (Nature, 211, 468-470). High Energy Astrophysics: Relativistic Effects 22/93 0.98 0.95 app 0.9 0.8 Plots of app for various indicated values of as a function of . High Energy Astrophysics: Relativistic Effects 23/93 The following images are from observations of 3C 273 over a period of 5 years from 1977 to 1982. They show proper mo- tions in the knots C3 and C4 of 0.79 0.03 mas/yr and 0.99 0.24 mas/yr respectively. These translate to proper mo- tions of 5.5 0.2h–1c and 6.9 1.7h–1c respectively. High Energy Astrophysics: Relativistic Effects 24/93 From Unwin et al., ApJ, 289, 109 High Energy Astrophysics: Relativistic Effects 25/93 2.6 Apparent length of a moving rod The Lorentz-Fitzgerald contraction gives us the relationship between the proper lengths of moving rods. An additional factor enters when we take into account time retardation. P P1 V x 2 L To ob se d = xcos rv D er l = xsin High Energy Astrophysics: Relativistic Effects 26/93 Consider a rod of length –1 L = L0 (22) in the observer’s frame. Now the apparent length of the rod is affected by the fact that photons which arrive at the observer at the same time are emitted at different times.P 1 corresponds to when the trailing end of the rod passes at time t1 and P2 corresponds to when the leading end of the rod passes at time t2. Equating the arrival times for photons emitted from P1 and P2 at times t1 and t2 respectively, High Energy Astrophysics: Relativistic Effects 27/93 D + xcos D t + ----------------------------- = t + ---- 1 c 2 c (23) xcos t – t = ------------------- 2 1 c When the trailing end of the rod reaches P2 the leading end has to go a further distance xL– which it does in t2 – t1 secs. Hence, High Energy Astrophysics: Relativistic Effects 28/93 Vxcos xL– = ----------------------- c L (24) x = ------------------------- V 1 – --- cos c and the apparent projected length is Lsin L0 L ==xsin ------------------------ =--------------------------------- =L (25) app 1 – cos 1 – cos 0 This is another example of the appearance of the ubiquitous Doppler factor. High Energy Astrophysics: Relativistic Effects 29/93 2.7 Transformation of velocities The Lorentz transformation x = x + Vt yy= zz= Vx (26) t = t + -------- c2 can be expressed in differential form: dx = dx + Vdt dy= dy dz= dz Vdx (27) dt = dt + ----------- c2 High Energy Astrophysics: Relativistic Effects 30/93 so that if a particle moves dx in time dt in the frame S then the corresponding quantities in the frame S are related by the above differentials.
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