Relativistic Effects

1 Introduction

The radio-emitting plasma in AGN contains electrons with relativistic energies. The Lorentz factors of the emitting electrons 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 properties of the emitted radiation – principally through the effects 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 transformation 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

L0 x x 2 1 x  – x  = L0 2 1 x  – x  = x2 – x1 – Vt2 – 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 – Vx2 – 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 effects of relativistic motion in astrophysics. The effect is the

High Energy Astrophysics: Relativistic Effects 9/93 combination of both relativistic time dilation and time retardation. 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 because 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 = t1 – --- 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 linear 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 energy astrophysics and which is analysed in a very similar way to the Doppler effect, relates to the apparent transverse velocity 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 = Vtsin

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 difference between the time of receptions of photons emitted at P1 and P2 are given by: V t = t1 – --- cos (15) rec c The apparent distance moved by the object is

l = Vtsin (16)

High Energy Astrophysics: Relativistic Effects 17/93 Hence, the apparent velocity of the object is: Vtsin Vsin V ==------app V V t1 – --- 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 consequence 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 apparent velocity is a maximum:

dapp 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 velocities 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 Vxcos 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. This can be used to relate velocities in the 2 frames via dx ------+ V dx dx + Vdt dt ------==------dt Vdx V dx dt + ------1 + ------2 c2 dt c (28) vx + V  vx = ------Vvx 1 + ------c2

High Energy Astrophysics: Relativistic Effects 31/93 For the components of velocity transverse to the motion of S ,

dy dy vy ------==v ------=------dt y Vdx dt + ------Vvx c2 1 + ------c2 (29) dz dz vz ----- ==v ------=------dt z Vdx dt + ------Vvx c2 1 + ------c2 In invariant terms (i.e. independent of the coordinate system), take

High Energy Astrophysics: Relativistic Effects 32/93 v|| = Component of velocity parallel to V (30) v = Component of velocity perpendiciular to V then

v|| + V v v = ------v = ------(31) || 2  2 1 + Vv||  c 1 + Vv||  c The reverse transformations are obtained by simply replacing V by –V so that:

v|| – V v v  = ------v  = ------(32) || 2  2 1 – Vv||  c 1 – Vv||  c High Energy Astrophysics: Relativistic Effects 33/93 and these can also be recovered by considering the differential form of the reverse Lorentz transformations.

2.8 Aberration

y y S v S v v v    v  v|| || x x

High Energy Astrophysics: Relativistic Effects 34/93 Because of the law of transformation of velocities, a velocity vector makes different angles with the direction of motion. From the above laws for transformation of velocities,

v v vsin tan ==------=------(33) v|| v|| + V vcos + V (The difference from the non-relativistic case is the factor of .) The most important case of this is when vv== c . We put

v|| = ccos v = csin (34) v|| = ccos v = csin

High Energy Astrophysics: Relativistic Effects 35/93 and V  = --- (35) c and the angles made by the light rays in the two frames satis- fy: ccos + V cos +  ccos ==------ cos ------V 1 + cos 1 + --- cos c (36) csin sin csin ==------ sin ------V 1 + cos 1 + --- cos c

High Energy Astrophysics: Relativistic Effects 36/93 Half-angle formula There is a useful expression for aberration involving half-angles. Using the identity,  sin tan--- = ------(37) 2 1 + cos the aberration formulae can be written as:  1 –  12/  tan--- = ------tan---- (38) 2 1 +  2

High Energy Astrophysics: Relativistic Effects 37/93 Isotropic radiation source Consider a source of radiation which emits isotropically in its rest frame and which is moving with velocity V with respect to an observer (in frame S ). The source is at rest in S which is moving with velocity V with respect to S .

y 1 y sin–1 --- S  S

x x

High Energy Astrophysics: Relativistic Effects 38/93 Consider a rays emitted at right angles to the direction of mo-  tion. This has  = --- . The angle of these rays in S are given 2 by the transformation for sin , with =    2 . This gives: 1 sin = --- (39)  (40)

High Energy Astrophysics: Relativistic Effects 39/93 These rays enclose half the light emitted by the source, so that in the reference frame of the observer, half of the light is emitted in a forward cone of half-angle 1   . This is relativistic 1 beaming in another form. When  is large:  --- . 

3 Four vectors

3.1 Four dimensional space-time Special relativity defines a four dimensional space-time con- tinuum with coordinates

x0 = ct x1 = x x2 = y x3 = z (41)

High Energy Astrophysics: Relativistic Effects 40/93 An event is the point in space-time with coordinates x where  = 0123. The summation convention Wherever there are repeated upper and lower indices, summation is implied, e.g. 3   AB =  AB (42)  = 0 The metric of space-time is given by 2   ds = dx dx (43)

High Energy Astrophysics: Relativistic Effects 41/93 where

–1 000 –1 000  =0 100 Inverse == 0 100(44)  0 010 0 010 0 001 0 001 (45) Hence the metric

ds2 = –dx0 2 +++dx1 2 dx2 2 dx3 2 (46)

High Energy Astrophysics: Relativistic Effects 42/93 This metric is unusual for a geometry in that it is not positive definite. For spacelike displacements it is positive and for timelike displacements it is negative. This metric is related to the proper time  by

ds2 = –c2d2 (47)  Indices are raised and lowered with  , e.g. if A is a vector, then  A = A (48) This extends to tensors in space-time etc. Upper indices are referred to as covariant; lower indices as contravariant.

High Energy Astrophysics: Relativistic Effects 43/93 3.2 Representation of a Lorentz transformation A Lorentz transformation is a transformation which preserves ds2. We represent a Lorentz transformation by:    x = x (49)    =    That is, a Lorentz transformation is the equivalent of an or- thogonal matrix in the 4-dimensional space time with indefi- nite metric. Conditions: • det = 1 – rules out reflections (xx – )

High Energy Astrophysics: Relativistic Effects 44/93 0 •  0  0 – isochronous For the special case of a Lorentz transformation involving a boost along the x –axis

–00  –  00   = (50) 0010 0001

High Energy Astrophysics: Relativistic Effects 45/93 3.3 Some important 4-vectors The 4-velocity This is defined by

dx dx0 dxi u ==------ ------(51) d d d The zeroth component

High Energy Astrophysics: Relativistic Effects 46/93 dx0 dt dt ------==c----- c------d d dt2 + –c–2dx1 2 ++dx2 2 dx3 2 c (52) ==------c v2 1 – ----- c2 Note that we use  for the Lorentz factor of the transformation and  for particles. This will later translate into  for bulk motion and  for the Lorentz factors of particles in the rest- frame of the plasma. The spatial components:

High Energy Astrophysics: Relativistic Effects 47/93 dxi dxi dt ------==------vi (53) d dt d so that

u = c vi (54) The 4-momentum The 4-momentum is defined by E p ==m u mc mvi =---  pi (55) 0 c where

High Energy Astrophysics: Relativistic Effects 48/93 Ec==2p2 + m2c4 mc2 (56) is the energy, and i i p = m0v (57) is the 3-momentum. Note the magnitude of the 4-momentum   0 2 1 2 2 2 3 2 p p = –p +++p p p (58) E 2 ==–--- + p2 –m2c2 c

High Energy Astrophysics: Relativistic Effects 49/93 3.4 Transformation of 4-vectors Knowing that a 4-component quantity is a 4-vector means that we can easily determine its behaviour under the effect of a Lorentz transformation. The zero component behaves like x0 and the x component behaves like x . Recall that:

x = x+ x0 x0 = x0+ x (59) Therefore, the components of the 4-velocity transform like

U0= U0 – U1 (60) U1= – U0 + U1 Hence,

High Energy Astrophysics: Relativistic Effects 50/93 v1 c== c– v1 1 – ----- c v1==– c + v1 v1 v1 – c (61) v2= v2 v3= v3 Transformation of Lorentz factors Putting v1 = vcos gives v = 1 – -- cos (62) c

High Energy Astrophysics: Relativistic Effects 51/93 This is a useful relationship that can be derived from the pre- vious transformations for the 3-velocity. However, one of the useful features of 4-vectors is that this transformation of the Lorentz factor is easily derived with little algebra. Transformation of 3-velocities Dividing the second of the above transformations by the first:

v1 – c v1 – c v1 – V v1 ===------(63) v1 v1 Vv 1 – ----- 1 – ----- x c c 1 – ------c2 Dividing the third equation by the first:

High Energy Astrophysics: Relativistic Effects 52/93 v2 v2 v2 ==------(64) v1 Vv 1 – ----- x c 1 – ------c2 and similarly for v3 . These are the equations for the transformation of velocity components derived earlier.

High Energy Astrophysics: Relativistic Effects 53/93 4 Distribution functions in special relativity

In order to properly describe distributions of particles in a rel-

pz Distribution of momenta in momentum space. p py

High Energy Astrophysics: Relativistic Effects 54/93 ativistic context and in order to understand the transformations of quantities such as specific intensity, etc. we need to have relativistically covariant descriptions of statistical distributions of particles. Recall the standard definition of the phase space distribution function: No of particles within an elementrary volume fd3xd3p = (65) of phase space

High Energy Astrophysics: Relativistic Effects 55/93 4.1 Momentum space and invariant 3-volume The above definition of fxt p is somewhat unsatisfacto- ry from a relativistic point of view since it focuses on three dimensions rather than four. Covariant analogue of d3p The aim of the following is to replace d3p by something that makes sense relativistically. Consider the space of 4-dimensional momenta. We express the components of the momentum in terms of a hyperspherical angle  and polar angles  and  .

High Energy Astrophysics: Relativistic Effects 56/93 p0 = mccosh p1 = mcsinhsin cos  (66) p2 = mcsinhsin sin p3 = mcsinhcos The Minkowksi metric is also the metric of momentum space and we express the interval between neighbouring momenta   as dp dp . In terms of hyperspherical angles:

 dpdp = –+dmc2  (67) + m2c2d 2 + sinh2d2 + sin2d2

High Energy Astrophysics: Relativistic Effects 57/93 This is proved in Appendix A. The magnitude of the 3-dimensional momentum is pmc= sinh (68) A particle of mass m is restricted to the mass shell m = constant. This is a 3-dimensional hypersurface in momentum space. From the above expression for the metric, it is easy to read off the element of volume on the mass shell:

d = mc 3 sinh2sindd (69) This volume is an invariant since it corresponds to the invar- iantly defined subspace of the momentum space, m = constant.

High Energy Astrophysics: Relativistic Effects 58/93 On the other hand, the volume element d3p refers to a subspace which is not invariant. The quantity

d3pdp= 1dp2dp3 (70) depends upon the particular Lorentz frame. It is in fact the projection of the mass shell onto p0 = constant . However, it is useful to know how the expression for d3p is expressed in terms of hyperspherical coordinates. In the normal polar coordinates:

d3pp= 2 sindpdd (71) Putting pmc= sinh in this expression,

High Energy Astrophysics: Relativistic Effects 59/93 d3pmc==3 sinh2coshsin ddd coshd (72) That is, the normal momentum space 3-volume and the invariant volume d differ by a factor of cosh .

4.2 Invariant definition of the distribution function The following invariant expression of the distribution function was first introduced by J.L. Synge who was one of the in- fluential pioneers in the theory of relativity who introduced geometrical and invariant techniques to the field. We begin by defining a world tube of particles with momentum p (4-velocity u).

High Energy Astrophysics: Relativistic Effects 60/93 The distribution function fx p is defined by:

t u y

World tube of particles with 4- n velocity u . The cross-sectional 3-area of the tube is d0 .

d x d0

High Energy Astrophysics: Relativistic Effects 61/93 The three-area of the world tube, d0 is the particular 3-area that is normal to the world lines in the tube. Using d0 , we define the distribution function, fx p , by the following definition: Number of world lines within   the world tube with momenta = fx p d0d (73) within d

This is expressed in terms of a particular 3-area, d0 .

High Energy Astrophysics: Relativistic Effects 62/93 Now consider the world lines intersecting an arbitrary 3-area (or 3-volume) d that has a unit normal n . The projection relation between d and d0 is –1  d0 = d  –c u n (74) Proof of last statement First, let us define what is meant by a spacelike hypersurface. In such a hypersurface every displacement, dx , is spacelike.   That is, dx dx  0 . The normal to a spacelike hypersurface is timelike. The square of the magnitude of a unit normal is -1:

High Energy Astrophysics: Relativistic Effects 63/93   n n = –1 (75) Example: The surface t = constant (76) is spacelike. Its unit normal is:

n = 1000 (77) We can also contemplate a family of spacelike hypersurfaces in which, for example t = variable (78)

High Energy Astrophysics: Relativistic Effects 64/93 This corresponds to a set of 3-volumes in which x1 x2 and x3 vary, that are swept along in the direction of the time-axis. As before the unit normal to this family of hypersurfaces is n = 1000 and the corresponding 4-velocity is dt u ===c----- 0 0 0 c000 cn (79) d In the present context, we can consider the set of 3-spaces d0 corresponding to each cross-section of a world tube as a family of such spacelike hypersurfaces. Each hypersurface is defined as being perpendicular to the 4-velocity so that, in

High Energy Astrophysics: Relativistic Effects 65/93  general, u0  c000 . However, it is possible to make a Lorentz transformation so that in a new system of coordinates   n0 = 1000 and u0 = c000 . The significance of d What is the significance of a surface d as indicated in the figure? This is an arbitrary surface tilted with respect to the original cross-sectional surface d0 .This surface has its own    unit normal n and 4-velocity, u = cn .

High Energy Astrophysics: Relativistic Effects 66/93 In the coordinate system in which the normal to d0 has  components n0 = 1000 , let us assume that the 4-ve-  locity of d is u = c v . That is, d represents a surface that is moving with respect to d0 at the velocity v with Lorentz factor,  . The unit normal to d is  n =  (80) Relation between volumes in the two frames Let d be the primed (moving) frame. The element of volume of d is

High Energy Astrophysics: Relativistic Effects 67/93 d = dx1dx2dx3 (81) At an instant of time in the primed frame denoted by dt = 0

dx1 ==dx1 + vdt dx1 (82) dx2 = dx2 dx3 = dx3 Hence, 1 2 3 1 2 3 d0 ==dx dx dx dx dx dx =d (83) Expression of the Lorentz factor in invariant form Consider the invariant scalar product

High Energy Astrophysics: Relativistic Effects 68/93  nn0  == –1000 – (84)  –1  Dropping the  subscript on n and using n0  = c u we have –1   = –c u n (85) Hence, –1  d0 = –c u n d (86) Note that the “projection factor” is greater than unity, perhaps counter to intuition.

High Energy Astrophysics: Relativistic Effects 69/93 Number of world lines in terms of d We defined the distribution function by: Number of world lines within   the world tube with momenta = fx p d0d (87) within d Hence, our new definition for an arbitrary d : Number of world lines within   –1  the world tube crossing d = fx p –c u n dd with momenta within d

High Energy Astrophysics: Relativistic Effects 70/93 Counting is an invariant operation and all of the quantities ap- pearing in the definition of f are invariants, therefore

fx p = Invariant (88)

4.3 An important special case Take the normal to  to be parallel to the time direction in an arbitrary Lorentz frame. Then

n0 = 1000 (89) and  0 0 0 –un ==––u n u (90)

High Energy Astrophysics: Relativistic Effects 71/93 Also

d = d3x (91) Now

p0 = mccosh u0 = cosh (92) Therefore,

fx p –un dd = fx p cosh d3xd  (93) = fx p d3xd3p

High Energy Astrophysics: Relativistic Effects 72/93 Our invariant expression reduces to the noninvariant expression when we select a special 3-volume in spacetime. Thus the usual definition of the distribution is Lorentz-invariant even though it does not appear to be.

5 Distribution of photons 5.1 Definition of distribution function We can treat massless particles separately or as a special case of the above, where we let m  0 and cosh   in such a h way that mccosh  ------. In either case, we have for pho- c tons,

High Energy Astrophysics: Relativistic Effects 73/93 No of photons within d3x fx p d3xd3p = (94) and momenta within d3p and the distribution function is still an invariant.

5.2 Relation to specific intensity From the definition of the distribution function, we have Energy density of photons ==hfd3phfp2dpd (95) within d3p

High Energy Astrophysics: Relativistic Effects 74/93 The alternative expression for this involves the energy density per unit frequency per unit solid angle, u . We know that

I u  = ----- (96)  c Hence, the energy density within  and within solid angle d is –1 udd = c Idd (97) Therefore,

High Energy Astrophysics: Relativistic Effects 75/93 h 2 h hfp2dpd ==hf------d------d c–1I dd c c  (98) c2 I  f = ------h43 This gives the very important result that, since f is a Lorentz invariant, then

I ------= Lorentz invariant (99) 3 Thus, if we have 2 relatively moving frames, then

High Energy Astrophysics: Relativistic Effects 76/93 I I  3 ------= ------ I = ---- I (100)  3 3    Take the primed frame to be the rest frame, then 3 I =  I (101) where  is the Doppler factor.

5.3 Transformation of emission and absorption coefficients Consider the radiative transfer equation:

High Energy Astrophysics: Relativistic Effects 77/93 dI j ------= S – I S = ------(102) d  Obviously, the source function must have the same transformation properties as I . Hence

S ------= Lorentz invariant (103) 3 Emission coefficient The optical depth along a ray passing through a medium with absorption coefficient  is, in the primed frame

High Energy Astrophysics: Relativistic Effects 78/93 In S In S

l l  

l  = ------(104) sin The optical depth in the unprimed frame is

High Energy Astrophysics: Relativistic Effects 79/93 l  = ------(105) sin and is identical. The factor e– counts the number of photons absorbed so that  is a Lorentz invariant. Hence

l sin ------= 1 (106) l sin The aberration formula gives sin sin ==------sin (107) 1 – cos

High Energy Astrophysics: Relativistic Effects 80/93 and the lengths l and l are perpendicular to the motion, so that ll= . Hence,

  ------==–1  ------1 (108)   i.e.  is a Lorentz invariant. The emission coefficient

S j j ------==------------ –1 =Lorentz invariant (109) 3 3 2     

High Energy Astrophysics: Relativistic Effects 81/93 2 Hence, j   is a Lorentz invariant.

5.4 Flux density from a moving source The flux from an arbitrary source is given by

V D

d Observer 1 F ==I cosd  I d ------j dV (110)      2     D V High Energy Astrophysics: Relativistic Effects 82/93 Now relate this to the emissivity in the rest frame. Since

j ------= Lorentz invariant 2 (111)  2 j ==---- j 2j     Therefore,

1 2 F ==------2j dV ------j dV (112)  2   2   D V D V The apparent volume of the source is related to the volume in the rest frame, by

High Energy Astrophysics: Relativistic Effects 83/93 dV = dV (113) This is the result of a factor of  expansion in the direction of motion and no expansion in the directions perpendicular to the motion. Hence the flux density is given in terms of the rest frame parameters by:

1 3 F ==------2j dV ------j dV (114)  2   2   D V D V Effect of spectral index For a power-law emissivity (e.g. synchrotron radiation),

High Energy Astrophysics: Relativistic Effects 84/93  – j ==j ---- j (115)    Therefore,

3 +  F = ------j dV (116)  2   D V This gives a factor of 3 +  increase for a blue-shifted source of radiation, over and above what would be measured in the rest frame at the same frequency.

High Energy Astrophysics: Relativistic Effects 85/93 Example: Consider the beaming factor in a  = 5 jet viewed at the angle which maximises the apparent proper motion.

The maximum app occurs when cos =  . 1  = 5   ==1 –------0.9798 52 (117) 1 1  ===------ 1 – cos 1 – 2 Hence,

High Energy Astrophysics: Relativistic Effects 86/93 3 +  ==53.6 330 (118) for a spectral index of 0.6

High Energy Astrophysics: Relativistic Effects 87/93 5.5 Plot of 

Plot of the Doppler factor as a function of view- ing angle. 



High Energy Astrophysics: Relativistic Effects 88/93 Appendix A Line element in momentum space in terms of hyperspherical angles We have the hyperspherical angle representation of a point in momentum space:

p0 = mccosh p1 = mcsinhsin cos  (119) p2 = mcsinhsin sin p3 = mcsinhcos We can write:

High Energy Astrophysics: Relativistic Effects 89/93 0 dp dmc 1 dp = A d (120) dp2 d dp3 d where

cosh mcsinh 0 0 sinh mccosh mcsinh –mcsinh A = sincos sincos  coscos  sincos (121) sinh mccosh mcsinh mcsinh  sinsin  cossin  cossin  sincos sinhcos mccoshcos –0mcsinhsin

High Energy Astrophysics: Relativistic Effects 90/93 Hence

dmc   † d dp dp = dmcd d d A A (122) d d where

High Energy Astrophysics: Relativistic Effects 91/93 sinh sinh –cosh sinhcos sincos  sinsin mccosh mccosh –mcsinh mccoshcos A† = sincos  cossin (123) mcsinh mcsinh 0 –mcsinhsin  coscos  cossin –mcsinh mcsinh 0 0  sincos  sincos On matrix multiplication we obtain:

  2 2 2 2 2 2 dp dp = – dmc++d sinh d + sin d (124)

High Energy Astrophysics: Relativistic Effects 92/93 –01 0 0 01 0 0 A†A = 00sinh2 0 00 0 sinh2sin2

High Energy Astrophysics: Relativistic Effects 93/93