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Relativistic Gases Relativistic Gases 1 Relativistic gases in astronomy Optical The inner part of M87 Radio Astrophysical Gas Dynamics: Relativistic Gases 1/73 Evidence for relativistic motion Motion of the knots in the M87 jet indicates apparent velocities in excess of the speed of light Large scale component velocities are of the order of 0.5-1.0 c Credits: Biretta and colleagues Astrophysical Gas Dynamics: Relativistic Gases 2/73 2 Some relevant aspects of relativity 2.1 Four vectors and tensors Special relativity is based on the geometry of space-time de- scribed by the metric = diag (-1,1,1,1) (1) The interval between two points is 2 0 2 1 2 2 2 3 2 ds ==dx dx – dx +++dx dx dx (2) where x0 = ct (3) Astrophysical Gas Dynamics: Relativistic Gases 3/73 Indices Latin go from 1 to 3; Greek from 0 to 3 Timelike and spacelike Displacements are timelike if ds2 0 and spacelike if ds2 0. For timelike displacements, the proper time, , is given by c2d2 = –ds2 (4) Lorentz factor The Lorentz factor is defined by 1 1 ==------------------- ------------------- (5) v2 1 – 2 1 – ----- c2 Astrophysical Gas Dynamics: Relativistic Gases 4/73 Note that we use for the Lorentz factor corresponding to the bulk motion of the fluid. We use for the Lorentz factors of the particles making up the fluid. Since, for a timelike displacement (the displacement of a mate- rial particle) c2d2 ==–ds2 dx0 2 – dx1 2 – dx2 2 – dx3 2 dx1 2 dx2 2 dx3 2 = dx0 2 1 – --------- – --------- – --------- dx0 dx0 dx0 (6) v2 = c2dt2 1 – ----- = –2c2dt2 c2 Astrophysical Gas Dynamics: Relativistic Gases 5/73 then cd= –1cdt (7) Signature of metric Note the signature of the metric –,+,+,+ . Other expositions of special or general relativity adopt +,–,–,– . The main 4–vector that we deal with here is the fluid 4-velocity, dx dx vi u ==--------- --------- = ---- (8) cd cdt c Astrophysical Gas Dynamics: Relativistic Gases 6/73 where dxi vi = ------- (9) dt and xi are the coordinates of a fluid element through space time. 2.2 Raising and lowering indices Indices are raised and lowered by means of the Minkowski ten- sor. Some results to remember are as follows. The inverse of the Minkowski tensor, is defined by: = (10) = –1111 Astrophysical Gas Dynamics: Relativistic Gases 7/73 Indices are raised and lowered by the Minkowski tensor v = v v = v (11) 3 The stress-energy tensor 3.1 Definition The stress–energy tensor is defined by the following compo- nents: 00 0i T = T T (12) Ti0 Tij Astrophysical Gas Dynamics: Relativistic Gases 8/73 where T00 = Energy density (incorporating rest mass energy etc.) cT0i = Flux of energy in the i direction (13) Ti0 = T0i Tij = Flux of the i component of momentum in the j direction We also have 1 i0 ---T = i component of momentum density c (14) 1 = ----- Flux of energy in the i direction c2 Astrophysical Gas Dynamics: Relativistic Gases 9/73 The relationship of T0i to both momentum and energy Why should we have these two different roles for the compo- nents T0i? Remember that in special relativity, mass and energy are related by Emc= 2 (15) Therefore a flux of energy is equivalent to c2 times a flux of mass. But the mass flux is vi and this is just the momentum density. Hence Energy flux = c2 Momentum density (16) Astrophysical Gas Dynamics: Relativistic Gases 10/73 and 1 Momentum density = ----- Energy flux (17) c2 3.2 Stress energy tensor for a perfect fluid The above characterisation of the stress-energy tensor is valid in general. For a perfect fluid in which the pressure is isotropic and normal to any surface we develop an expression for the stress- energy tensor as follows. Consider the rest-frame of the fluid. This is defined as that frame in which u = 1000 (18) Astrophysical Gas Dynamics: Relativistic Gases 11/73 In this frame, Energy density = e Energy flux= 0 (19) Momentum flux density = pij Hence, e 000 ep+ 000 –p 000 T ==0 p 00 0 000 + 0 p 00 (20) 00p 0 0 000 00p 0 000p 0 000 000p This can be covariantly expressed as T = ep+ uu + p (21) Astrophysical Gas Dynamics: Relativistic Gases 12/73 Relativistic enthalpy The quantity wep= + (22) is called the relativistic enthalpy similar to the corresponding non-relativistic expression h = + p (23) Note that the relativistic expression contains the rest-mass ener- gy. Often we expression the relativistic enthalpy in the form: w = c2 ++ p (24) Astrophysical Gas Dynamics: Relativistic Gases 13/73 where c2 is the rest-mass energy density and = e – c2 is the internal energy. Therefore, relativistic enthalpy = rest mass energy + internal energy + pres- sure. Astrophysical Gas Dynamics: Relativistic Gases 14/73 3.3 Expressions for the components of T in an arbi- trary frame The various components of the stress-energy tensor can then be expressed as: T00 ==ep+ u0u0 – p 2ep+ – p vi T0i = ep+ 2---- c (25) vivj Tij = ep+ 2--------- + pij c2 Astrophysical Gas Dynamics: Relativistic Gases 15/73 4 The relativistic fluid equations 4.1 Energy and momentum equations The relativistic fluid equations are T T , ------------- = 0 (26) x To see why these are the equations, let us examine the different equations corresponding to = 0 and = i. Astrophysical Gas Dynamics: Relativistic Gases 16/73 Time component 00 0j = 0 T ,0 + T ,j = 0 1T00 T0j --------------- +0----------- = c t xj (27) T00 cT0j ------------ + ------------------- = 0 t xj In words: Energy density --------------------------------------------+0 divergence Energy flux = (28) t Astrophysical Gas Dynamics: Relativistic Gases 17/73 Spatial component i0 ij = i T ,0 + T ,j = 0 1Ti0 Tij -------------- +0--------- = c t xj (29) c–1Ti0 Tij ------------------------ +0--------- = t xj In words: Momentum density ------------------------------------------------------+0 d i v e r g e n c e M o m e n t u m f l u x d e n s i t y = t These are the appropriate conservation laws for energy and mo- mentum. Astrophysical Gas Dynamics: Relativistic Gases 18/73 4.2 The conservation of number density Assuming that our fluid consists of particles that do not trans- form into other particles, then their number is conserved. Let n = Number of particles per unit proper volume (30) i.e. n is the number density of particles in the rest frame of the fluid. We often use the baryon number density for n. The conservation law for number is nu 1n nvi c -----------------= 0 ----------------- + -------------------------- x c t xi (31) n nvi -------------- +0------------------- = t xi Astrophysical Gas Dynamics: Relativistic Gases 19/73 This makes sense since the number density in an arbitrary frame is n. Rest mass density Let m be the average rest mass of particles in the fluid. The rest- mass energy density in the rest frame is nmc2. The rest-mass en- ergy density in an arbitrary frame is c2 = nmc2 (32) 5 Non-relativistic limit The non-relativistic limit is instructive since it • Confirms that we have the right equations • Clarifies the relationship between energy flux and momentum Astrophysical Gas Dynamics: Relativistic Gases 20/73 density 5.1 Nonrelativistic limits of the components of the energy-momentum tensor In proceeding to the non-relativistic limit, we express the energy density in the form: enmc= 2 + (33) where is the internal energy density. The T00 component T00 = 2ep+ – p = 2nmc2 ++ p – p (34) = nmc2 + 2 + p – p Astrophysical Gas Dynamics: Relativistic Gases 21/73 Now v2 –12/ 1v2 = 1 – ----- 1 + -------- c2 2c2 (35) v2 2 = 1 + ----- c2 So 1v2 1 T00 1 + -------- c2 + + p – p = c2 + ---v2 + (36) 2c2 2 So we can see that the energy density in an arbitrary frame, breaks up into the rest mass energy density plus the kinetic ener- gy density plus the internal energy density. Astrophysical Gas Dynamics: Relativistic Gases 22/73 The components T0i vi vi T0i ==ep+ 2---- nmc2 ++ p 2---- c c vi vi = c2---- + 2 + p ---- c c (37) 1v2 vi v2 vi 1 + -------- c2----1+ + ----- + p ---- 2c2 c c2 c vi 1 vi vi = c2---- ++---v2---- + p ---- c 2 c c Astrophysical Gas Dynamics: Relativistic Gases 23/73 1 1 v 2 + p Momentum density ==---T0i vi + ----- vi + ---------------- vi c 2 c c2 (38) 1 Energy flux density ==cT0i c2vi + ---v2vi + + p vi 2 The components Tij vivj vivj Tij ==ep+ 2--------- + pij nmc2 ++ p 2--------- + pij c2 c2 vivj vivj c2--------- ++2 + p --------- pij (39) c2 c2 vivj v2 = vivj + pij +++ p --------- Higher order in ----- c2 c2 Astrophysical Gas Dynamics: Relativistic Gases 24/73 The term nmvi This originates from the continuity equation for the number of particles. nmvi = vi (40) 5.2 Non-relativistic equations Continuity equation The conservation equation for rest-mass nm nmvi ------------------- +0------------------------ = (41) t xi Astrophysical Gas Dynamics: Relativistic Gases 25/73 is, without approximation: vi ------ +0--------------- = (42) t xi Momentum equation The relativistic form c–1Ti0 Tij ------------------------ +0--------- = (43) t xj Astrophysical Gas Dynamics: Relativistic Gases 26/73 becomes (44) 1 v 2 + p vi + ----- vi + ---------------- vi 2 c c2 -------------------------------------------------------------------------------- + (45) t vivj vivj + pij + + p --------- c2 --------------------------------------------------------------------------- = 0 xj Astrophysical Gas Dynamics: Relativistic Gases 27/73 The leading terms are the usual non-relativistic terms, the others are of order vc 2 by comparison. Hence, the momentum equations become: vi vivj + pij --------------- +0------------------------------------- = (46) t xj The energy equation The relativistic form is: T00 cT0j ------------ +0------------------- = (47) t xj Astrophysical Gas Dynamics: Relativistic Gases 28/73 With the non-relativistic expressions: 1 1 c2 + ---v2 + c2vj + ---v2vj + + p vj 2 2 --------------------------------------------------- +0-------------------------------------------------------------------------------- =(48) t xj In this case we need to go beyond the zeroth order expression since this is c2 c2vj ----------------- +0--------------------- = (49) t xj i.e.
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