Relativistic Effects. 1 Introduction

Transcription

1 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. 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 2 Summary of special relativity For a more complete summary of 4-vectors and Special Relativity, see Rybicki and Lightman, Radiative Processes in Astrophysics, or Rindler, Special Relativity High Energy Astrophysics: Relativistic Effects 2/93

3 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

4 x x vt t t y We put the space-time coordinates on an equal footing by putting ct. The the x tpart of the Lorentz transformation can be written: x 0 The reverse transformation is: y z z v - c Vx High Energy Astrophysics: Relativistic Effects 4/ c 2 x x x 0 x 0 x 0 x (1) (2)

5 i.e., x x + Vt y y z z t t Vx c 2 x x + x 0 x 0 x 0 + x (3) (4) High Energy Astrophysics: Relativistic Effects 5/93

6 2.2 Lorentz Fitzgerald contraction 2.3 Time dilation y y V S S x x High Energy Astrophysics: Relativistic Effects 6/93

7 y S y S V L 0 x x x 2 x 2 x 1 L 0 x 1 x 2 x 1 Vt 2 t 1 L 0 L L 1 L 0 High Energy Astrophysics: Relativistic Effects 7/93

8 Consider a clock at a stationary position in the moving frame which registers a time interval. The corresponding time T 0 interval in the lab frame is given by: T t 2 t 1 t 2 t 1 Vx 2 x 1 c 2 t 2 t 1 T 0 i.e. the clock appears to have slowed down by a factor of (5) High Energy Astrophysics: Relativistic Effects 8/93

9 2.4 Doppler effect P 1 V l Vt P 2 To observer d Vtcos D 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

10 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 P 1 to P 2. If is the emitted circular frequency of the radiation in the em rest frame, then t em and the time between the two events in the observer s frame is: (6) High Energy Astrophysics: Relativistic Effects 10/93

11 t t em (7) However, this is not the observed time between the events because there is a time difference involved in radiation emitted from P 1 and P 2. Let and D distance to observer from P 2 t 1 time of emission of radiation from P 1 t 2 time of emission of radiation from P 2 (8) (9) High Energy Astrophysics: Relativistic Effects 11/93

12 Then, the times of reception, and are: t 1 rec t 2 rec trec D + Vtcos 1 t c trec D 2 t c (10) Hence the period of the pulse received in the observer s frame is High Energy Astrophysics: Relativistic Effects 12/93

13 t 2 rec Therefore, trec D 1 t D + Vtcos t c c t 2 t 1 t1 V -- tcos c V -- cos c (11) High Energy Astrophysics: Relativistic Effects 13/93

14 obs em V -- cos c obs em V 1 -- cos c em cos (12) The factor is a pure relativistic effect, the factor 1 cos is the result of time retardation. In terms of linear frequency: obs em cos (13) High Energy Astrophysics: Relativistic Effects 14/93

15 The factor cos (14) 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

16 P 1 V l Vt P 2 To observer d Vtcos D l Vtsin High Energy Astrophysics: Relativistic Effects 16/93

17 Consider an object which moves from P 1 to P 2 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 P 1 and P 2 are given by: t rec t1 V -- cos c The apparent distance moved by the object is l Vtsin (15) (16) High Energy Astrophysics: Relativistic Effects 17/93

18 Hence, the apparent velocity of the object is: Vtsin V app V t1 -- cos c V app c V -- sin c V 1 -- cos c Vsin V 1 -- cos c (17) In terms of V c High Energy Astrophysics: Relativistic Effects 18/93

19 app V app c sin app cos The non-relativistic limit is just (18), 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. V -- c V app Vsin High Energy Astrophysics: Relativistic Effects 19/93

20 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 d This derivative is zero when 1 coscos sinsin cos 2 cos cos 2 cos (19) (20) High Energy Astrophysics: Relativistic Effects 20/93

21 At the maximum: sin 1 app cos (21) 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

22 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, ). High Energy Astrophysics: Relativistic Effects 22/93

23 0.98 app Plots of app of as a function of. for various indicated values High Energy Astrophysics: Relativistic Effects 23/93

24 The following images are from observations of 3C 273 over a period of 5 years from 1977 to They show proper motions in the knots C 3 and C 4 of mas/yr and mas/yr respectively. These translate to proper motions of h 1 c and h 1 c respectively. High Energy Astrophysics: Relativistic Effects 24/93

25 From Unwin et al., ApJ, 289, 109 High Energy Astrophysics: Relativistic Effects 25/93

26 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 x P 1 V 2 L d xcos To observer D l xsin High Energy Astrophysics: Relativistic Effects 26/93

27 Consider a rod of length L 1 L 0 (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 P 2 corresponds to when the leading end of the rod passes at time t 2. Equating the arrival times for photons emitted from P 1 and P 2 at times t 1 and t 2 respectively, High Energy Astrophysics: Relativistic Effects 27/93

28 When the trailing end of the rod reaches has to go a further distance secs. Hence, D + xcos D t t c c t 2 t 1 x xcos c P 2 (23) the leading end L which it does in t 2 t 1 High Energy Astrophysics: Relativistic Effects 28/93

29 x and the apparent projected length is L x Vxcos c L V 1 -- cos c (24) Lsin L app xsin (25) 1 cos cos L 0 This is another example of the appearance of the ubiquitous Doppler factor. L 0 High Energy Astrophysics: Relativistic Effects 29/93

30 2.7 Transformation of velocities The Lorentz transformation x x + Vt y y z z can be expressed in differential form: t t Vx dx dx + Vdt dy dy dz dz dt dt c 2 Vdx c 2 (26) (27) High Energy Astrophysics: Relativistic Effects 30/93

31 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 dt v x dx + Vdt Vdx dt v x + V Vv x c 2 c 2 dx V dt V dx c dt (28) High Energy Astrophysics: Relativistic Effects 31/93

32 For the components of velocity transverse to the motion of S, dy dy v dt y Vdx dt c 2 dz dz ---- v dt z Vdx dt c 2 v y Vv x c 2 v z Vv x c 2 (29) In invariant terms (i.e. independent of the coordinate system), take High Energy Astrophysics: Relativistic Effects 32/93

33 v v Component of velocity parallel to V Component of velocity perpendiciular to V (30) then v + V v v Vv c 2 v Vv c 2 (31) The reverse transformations are obtained by simply replacing V by V so that: v V v Vv c 2 v (32) High Energy Astrophysics: Relativistic Effects 33/93 v Vv c 2

34 and these can also be recovered by considering the differential form of the reverse Lorentz transformations. 2.8 Aberration S y v v v S y v v v x x High Energy Astrophysics: Relativistic Effects 34/93

35 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, tan v v v + V v (33) (The difference from the non-relativistic case is the factor of.) The most important case of this is when v ccos v csin vsin vcos+ V v ccos v csin v v c. We put (34) High Energy Astrophysics: Relativistic Effects 35/93

36 and V -- c (35) and the angles made by the light rays in the two frames satisfy: ccos csin ccos + V cos V cos c csin sin V cos c cos cos sin cos (36) High Energy Astrophysics: Relativistic Effects 36/93

37 Half-angle formula There is a useful expression for aberration involving half-angles. Using the identity, sin tan cos the aberration formulae can be written as: tan / 1 + tan (37) (38) High Energy Astrophysics: Relativistic Effects 37/93

38 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. S y sin S y V x x High Energy Astrophysics: Relativistic Effects 38/93

39 Consider a rays emitted at right angles to the direction of motion. This has --. The angle of these rays in S are given 2 by the transformation for sin, with 2. This gives: sin 1 -- (39) (40) High Energy Astrophysics: Relativistic Effects 39/93

40 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: Four vectors 3.1 Four dimensional space-time Special relativity defines a four dimensional space-time continuum with coordinates x 0 ct x 1 x x 2 y x 3 z (41) High Energy Astrophysics: Relativistic Effects 40/93

41 An event is the point in space-time with coordinates The summation convention where Wherever there are repeated upper and lower indices, summation is implied, e.g. A B The metric of space-time is given by ds A B dx dx x (42) (43) High Energy Astrophysics: Relativistic Effects 41/93

42 where Inverse Hence the metric ds 2 dx dx dx 2 2 +dx 3 2 (44) (45) (46) High Energy Astrophysics: Relativistic Effects 42/93

43 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 High Energy Astrophysics: Relativistic Effects 43/93 by (47) Indices are raised and lowered with, e.g. if is a vector, then ds 2 c 2 d 2 A A (48) This extends to tensors in space-time etc. Upper indices are referred to as covariant; lower indices as contravariant. A

44 3.2 Representation of a Lorentz transformation A Lorentz transformation is a transformation which preserves ds 2. We represent a Lorentz transformation by: x x (49) That is, a Lorentz transformation is the equivalent of an orthogonal matrix in the 4-dimensional space time with indefinite metric. Conditions: det 1 rules out reflections ( x x) High Energy Astrophysics: Relativistic Effects 44/93

45 0 0 0 isochronous For the special case of a Lorentz transformation involving a boost along the x axis (50) High Energy Astrophysics: Relativistic Effects 45/93

46 3.3 Some important 4-vectors The 4-velocity This is defined by u The zeroth component dx d dx dxi d d (51) High Energy Astrophysics: Relativistic Effects 46/93

47 dx c dt dt c d d dt 2 + c 2 dx dx 2 2 +dx 3 2 c (52) c 1 v c 2 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 restframe of the plasma. The spatial components: High Energy Astrophysics: Relativistic Effects 47/93

48 so that dx i d The 4-momentum u dx i dt v dt d i The 4-momentum is defined by c v i (53) (54) where p m 0 u mc mv i E -- p c i (55) High Energy Astrophysics: Relativistic Effects 48/93

49 is the energy, and E c 2 p 2 + m 2 c 4 mc 2 p i m 0 v i (56) (57) is the 3-momentum. Note the magnitude of the 4-momentum p p p p p p 3 2 E -- c 2 + p2 m 2 c 2 (58) High Energy Astrophysics: Relativistic Effects 49/93

50 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 x 0 and the x component behaves like x. Recall that: x x + x 0 x 0 x 0 + x Therefore, the components of the 4-velocity transform like (59) Hence, U 0 U 0 U 1 U 1 U 0 + U 1 (60) High Energy Astrophysics: Relativistic Effects 50/93

51 c c v v1 c v 1 c + v 1 v 1 v 1 c v 2 v 2 v 3 v 3 Transformation of Lorentz factors (61) Putting v 1 vcos gives 1 v c - cos (62) High Energy Astrophysics: Relativistic Effects 51/93

52 This is a useful relationship that can be derived from the previous 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: v 1 v 1 c v c v v1 c c Dividing the third equation by the first: V Vv x c 2 v 1 (63) High Energy Astrophysics: Relativistic Effects 52/93

53 v 2 v v1 c v Vv x c 2 (64) and similarly for. These are the equations for the transformation of velocity components derived earlier. v 3 High Energy Astrophysics: Relativistic Effects 53/93

54 4 Distribution functions in special relativity p z p In order to properly describe distributions of particles in a relp y Distribution of momenta in momentum space. p x High Energy Astrophysics: Relativistic Effects 54/93

55 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: fd 3 xd 3 p No of particles within an elementrary volume (65) of phase space High Energy Astrophysics: Relativistic Effects 55/93

56 4.1 Momentum space and invariant 3-volume The above definition of fx t p is somewhat unsatisfactory from a relativistic point of view since it focuses on three dimensions rather than four. Covariant analogue of d 3 p The aim of the following is to replace d 3 p 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

57 p 0 mccosh p 1 mcsinhsincos p 2 mcsinhsinsin p 3 mcsinhcos (66) 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) + m 2 c 2 d 2 + sinh 2 d 2 + sin 2 d 2 High Energy Astrophysics: Relativistic Effects 57/93

58 This is proved in Appendix A. The magnitude of the 3-dimensional momentum is p mcsinh (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 sinh 2 sindd (69) This volume is an invariant since it corresponds to the invariantly defined subspace of the momentum space, m constant. High Energy Astrophysics: Relativistic Effects 58/93

59 refers to a sub- On the other hand, the volume element d 3 p space which is not invariant. The quantity d 3 p dp 1 dp 2 dp 3 (70) depends upon the particular Lorentz frame. It is in fact the projection of the mass shell onto is useful to know how the expression for terms of hyperspherical coordinates. In the normal polar coordinates: p 0 constant. However, it d 3 p is expressed in Putting p mcsinh d 3 p p 2 sindpdd in this expression, (71) High Energy Astrophysics: Relativistic Effects 59/93

60 d 3 p mc 3 sinh 2 coshsinddd (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 influential pioneers in the theory of relativity who introduced geometrical and invariant techniques to the field. p u We begin by defining a world tube of particles with momentum (4-velocity ). coshd High Energy Astrophysics: Relativistic Effects 60/93

61 The distribution function fx p is defined by: t n u y World tube of particles with 4- velocity u. The cross-sectional 3-area of the tube is d 0. d 0 d x High Energy Astrophysics: Relativistic Effects 61/93

62 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, we define the distribution function, definition: Number of world lines within the world tube with momenta within d fx p, by the following This is expressed in terms of a particular 3-area, d 0. fx p d 0 d 0 (73) High Energy Astrophysics: Relativistic Effects 62/93

63 Now consider the world lines intersecting an arbitrary 3-area (or 3-volume) d that has a unit normal. The projection relation between d and d 0 is n d 0 d c 1 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

64 Example: The surface n n 1 is spacelike. Its unit normal is: t constant (75) (76) 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

65 This corresponds to a set of 3-volumes in which x 1 x 2 and x 3 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 u and the corresponding 4-velocity is c----- dt c 000 cn d (79) 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

66 general, u 0 c 000. However, it is possible to make a Lorentz transformation so that in a new system of coordinates n and u 0 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

67 In the coordinate system in which the normal to d 0 has components n , let us assume that the 4-velocity 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 be the primed (moving) frame. The element of vol- d is Let d ume of High Energy Astrophysics: Relativistic Effects 67/93

68 d dx 1 dx 2 dx 3 At an instant of time in the primed frame denoted by (81) dt 0 Hence, dx 1 dx 1 + vdt dx 1 dx 2 dx 2 dx 3 dx 3 (82) d 0 dx 1 dx 2 dx 3 dx 1 dx 2 dx 3 d (83) Expression of the Lorentz factor in invariant form Consider the invariant scalar product High Energy Astrophysics: Relativistic Effects 68/93

69 n n (84) Dropping the subscript on n and using n 0 we have Hence, c 1 u c 1 u n (85) d 0 c 1 u n d (86) Note that the projection factor is greater than unity, perhaps counter to intuition. High Energy Astrophysics: Relativistic Effects 69/93

70 Number of world lines in terms of We defined the distribution function by: d Number of world lines within the world tube with momenta fx p d 0 d (87) within d Hence, our new definition for an arbitrary d: Number of world lines within the world tube crossing d with momenta within d fx p c 1 u n dd High Energy Astrophysics: Relativistic Effects 70/93

71 Counting is an invariant operation and all of the quantities appearing 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 and n u n u 0 n 0 u 0 (89) (90) High Energy Astrophysics: Relativistic Effects 71/93

72 Also d d 3 x (91) Now p 0 mccosh u 0 cosh (92) Therefore, fx p u n dd fx p coshd 3 xd fx p d 3 xd 3 p (93) High Energy Astrophysics: Relativistic Effects 72/93

73 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 photons, c High Energy Astrophysics: Relativistic Effects 73/93

74 fx p d 3 xd 3 p No of photons within d 3 x and momenta within d 3 p and the distribution function is still an invariant. (94) 5.2 Relation to specific intensity From the definition of the distribution function, we have Energy density of photons within d 3 p hfd 3 p hfp 2 dpd (95) High Energy Astrophysics: Relativistic Effects 74/93

75 The alternative expression for this involves the energy density per unit frequency per unit solid angle, u. We know that u I ---- c (96) Hence, the energy density within is Therefore, u dd c 1 I dd and within solid angle d (97) High Energy Astrophysics: Relativistic Effects 75/93

76 hfp 2 dpd hf h d h d c 1 c c I dd f (98) This gives the very important result that, since f is a Lorentz invariant, then c I h 4 3 I Lorentz invariant Thus, if we have 2 relatively moving frames, then (99) High Energy Astrophysics: Relativistic Effects 76/93

77 I I I I Take the primed frame to be the rest frame, then (100) I 3 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

78 di j S d I S (102) Obviously, the source function must have the same transformation properties as. Hence I Emission coefficient S Lorentz invariant (103) 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

79 In S In S l l The optical depth in the unprimed frame is l sin (104) High Energy Astrophysics: Relativistic Effects 79/93

80 l sin (105) e and is identical. The factor counts the number of photons absorbed so that is a Lorentz invariant. Hence l sin l sin The aberration formula gives sin sin cos sin (106) (107) High Energy Astrophysics: Relativistic Effects 80/93

81 and the lengths l and that l l. Hence, l are perpendicular to the motion, so (108) i.e. is a Lorentz invariant. The emission coefficient S j j Lorentz invariant (109) High Energy Astrophysics: Relativistic Effects 81/93

82 Hence, j 2 is a Lorentz invariant. 5.4 Flux density from a moving source The flux from an arbitrary source is given by V D d F I cosd I d j dv D 2 Observer (110) High Energy Astrophysics: Relativistic Effects 82/93 V

83 Now relate this to the emissivity in the rest frame. Since Therefore, j Lorentz invariant j j 2 j (111) 1 F j dv D 2 V (112) The apparent volume of the source is related to the volume in the rest frame, by High Energy Astrophysics: Relativistic Effects 83/ j dv D 2 V

84 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: Effect of spectral index dv 1 F j dv D 2 V For a power-law emissivity (e.g. synchrotron radiation), j dv D 2 V (114) High Energy Astrophysics: Relativistic Effects 84/93

85 Therefore, j j ---- j (115) F j dv D 2 V (116) This gives a factor of increase for a blue-shifted source of radiation, over and above what would be measured in the rest frame at the same frequency. 3 + High Energy Astrophysics: Relativistic Effects 85/93

86 Example: Consider the beaming factor in a 5 jet viewed at the angle which maximises the apparent proper motion. The maximum occurs when cos. app Hence, cos 1 2 (117) High Energy Astrophysics: Relativistic Effects 86/93

87 3 for a spectral index of (118) High Energy Astrophysics: Relativistic Effects 87/93

88 5.5 Plot of Plot of the Doppler factor as a function of viewing angle. High Energy Astrophysics: Relativistic Effects 88/93

89 Appendix A Line element in momentum space in terms of hyperspherical angles We have the hyperspherical angle representation of a point in momentum space: We can write: p 0 mccosh p 1 mcsinhsincos p 2 mcsinhsinsin p 3 mcsinhcos (119) High Energy Astrophysics: Relativistic Effects 89/93

90 dp 0 dp 1 dp 2 dp 3 A dmc d d d (120) where cosh mcsinh 0 0 sinh mccosh mcsinh mcsinh A sincos sinh sincos mccosh coscos mcsinh sincos mcsinh (121) sinsin cossin cossin sincos sinhcos mccoshcos mcsinhsin 0 High Energy Astrophysics: Relativistic Effects 90/93

91 Hence dp dp dmc d d d A A dmc d d d (122) where High Energy Astrophysics: Relativistic Effects 91/93

92 A cosh mcsinh 0 0 sinh sincos mccosh sincos mcsinh coscos mcsinh sincos sinh sinsin mccosh cossin mcsinh cossin mcsinh sincos sinhcos mccoshcos mcsinhsin 0 (123) On matrix multiplication we obtain: dp dp dmc 2 + d 2 + sinh 2 d 2 + sin 2 d 2 (124) High Energy Astrophysics: Relativistic Effects 92/93

93 A A sinh sinh 2 sin 2 High Energy Astrophysics: Relativistic Effects 93/93