A. H. Taub- Relativistic Fluid Mechanics

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    Fluid Mech. 18 10 0Copyrght 978 by Rvw' rghs

    RELATIVISTIC FLUD

    MEHANCS

    aubMathemaics Department, Universiy of Califoria Berkele California 7

    NOON

    1 Inducy Suvy

    6

    P-lavty mcacs ay b caact as a toy tat scbs tsa o a by mas o v ctos o at ost o vaablsT latt vaabls a o two s t vaabs tmg a o a tsoal ca spac a a ot o abg absolt t o soda uvel cloc. v unc te deede vble t ee w d: ee e e e vbe vi 123) eompos o t vloy vco o som cooat sysm atwo of them are thermoynamc arables, for example he pressure p an the densty

    of the u From a knowlege osuh arabes a oher theroynam arables

    uh a mau c nna ng an cc no S a balcla t at oft bg cos may b scb by tfcoal pc of as a fto op a .

    T t vaabls a as soto o v cosvatoqu. I e cde e ee e e ce a

    + (V) = t covsao o omtm

    a the coneaon oenergyEiE = yj+h

    w w av s t ollowg otatos t smao covo fi =

    = = ' ' s t coc o vscosy

    1ij AT

    d e e otvty

    0066489/78/0150301$0.0031

    An

    nu.

    Rev.

    FluidMech.

    1978.1

    0:301

    -332.

    Downloadedfromarjournals.annualreviews.org

    byPrincetonUniversityL

    ibraryon05/05/09.

    Forpersonaluseonly.

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    302 AUB

    T v covato qato gv abov a vaat a taatgo tat i gat by th -aat go of cla oto

    wth h abay a th h o of a othogoa a t taato

    t' = t+Twth abtay a th Gala taoato:

    wi h bi abiay oa a i aviy i ma ai tcl of wtoa atvty

    Ratvtc yac ao a thoy volvg v t ctothat cb t tat of th h of th katc vaab a twohoyac o owv h t vaabl whch a tl fo b o ot f to a ot a thoa ca ac aabolt t bt ta a ot (ca a vt) a fooacot ot a ac-t a ac wth a t tc wth agat ta w al tak to b (- ++ +). I al latvty ac-t ca Mow ac a t chaact by t act that t xt cooat yt tal o whc th tac btw th vt wtcooat (t,x,y,z) a (t+dt,x+dxy+dyz+dz) gv by

    ds2 c2dt2+dr+dy2+d2.

    a wt t ato a

    ds = dx dx",wh / 0, ,2 ,3 ,

    bbbO = l = x = y x3 = z a t ato ovto bg

    h go of taoato tat cay tal ooat yt to talo t taat Poca go cotg of taoato o th o

    w a a cotat a th cotat fo a ot atx ch that

    v covato law chaactg cal atvtc chac avaat th go

    a ga ooat y ag fo a a o by a afoato

    = y z) = (x) / = 0, , 2 3 ,w a v

    Annu.

    Rev.

    FluidMech.

    1978.1

    0:301-332.Downloadedfromarjournals.annualreviews.org

    byPrincetonUniversityLibrary

    on05/05/09.

    Forpersonaluseon

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    w hav

    d = g"v d dewh

    oxU ax'v = u, a" av

    REAIVISC ID MECHANICS 33

    (.

    hs ga cooa sysm h hso symbos a

    rz tgU'(g.+g,l-g,) 0 wh

    ()owv h Rma cva so

    14)

    .. Mows sac s a a a ss s a abso sacm of s co

    Gaavsc mchacs s om ha o sca avy hah vaabs of h cosao qaos o a cv sacm h mtc enso o hs sactm s t as b o h

    resene o a raiaona e hose soure is eerine aious hsias sch as lcomac os o som cass by h sgavag whos moo s b s. o lcs h avaoal l o whos moo s n m o s sa o b a wh h moo o as hs ga lavsc chacs h a wo ys o oblms1) o m h moo o a s a gv a gavaoa a o m h gavaoa o h sc o a sgavag (a oh s) a o m smaosy h !oo o h (a h sa o h oh s ha may b s).

    I h a y o obms o mus m h sa o h a h

    gavaoa s by sovg h soca Es aos. hs ascb bow

    K Cmi Cit hs a h mmay olowg scos w sha scss som gomcalos o sbmaos o a sac-m Mch o o scsso w o o whh o o h sacm s a a o shal assm ha w aag wh a a cooa sysm wh cooa abs x(p

    =023)

    whch h m s gv by

    2.1)

    Annu.

    Rev.

    FluidMech.

    1978.1

    0:301-332.Downloadedfromarjournals.annualreviews.org

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    30 AUB

    In a oonate yem e o o

    x=xl'(w) 22epeen a e wh w a paamete at labe an eent on e e. The angen

    eo to e gen by

    dxv=-.

    dw

    Te ae ote qanty

    (23

    eemne e nae of he e: If V VI' < 0 the e meke f VI =0

    n an f

    > 0 t paeke. Te pae pat of a n moon

    ae ebe by meke e. Lgt ay ae on nll e. Pale patae ao ae wo-ne of pale.o a v e aam may away b o o a

    L=_1.

    a ae w ofen epae by te ete an ale the pop me o

    The mee e epeenng the patle pah of a egon of he eemne a omalze eo el ae the foeoy eo whh afnton of een n paeme. In he oonae yem e aboe we hae

    an te oluo o a o" uds

    ebe he wolne of the "paleWe may wte e oon of aon .5) a

    . _03= x'; s 3weeg i 0

    2.

    2.

    (2.

    ae eqe to be e paamet eqaton of a hypefae he nal hypefae on w he pae e loae a ope me = Te fo aabe, s wh we al enoe a , fom a omong oonae ytm n paetme. Tey ae mla o agangan oonae of peeay yoynamb te na yefae nee no be the ypefae 0 een n e ate paetme Mnkowk pae an e oonae ae neal oneUne the anfomaon

    Annu.

    Rev.

    FluidMech.

    1978.1

    0:301-332.Downloadedfromarjournals.annualreviews.org

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    AIVII I MI 30

    X* = t uion o t inti yu s = 0 bs

    X*4

    X*4X*i;0).

    T x* e gene vng nes. Ens 2) e en be nes

    ax uxx*

    ee n e eenn e X e e nsn f en ese vese e ne s seee

    n e gene vng ne sse e ve e nens f e f

    vey ve gven bv

    Hn

    whee g:v omonn o mti tno n t omovn oonytm.

    rn r l ln rA ve e wt omonn v n lon y wt ngen ve s s neg e ns ng y i sses e en

    (3.)

    w w v u mioon o not t ovin iviv wit t to mti g o tim tt i < i n y uion uv iy

    2

    o.Ve es iy t ution

    V = U1 33 to uo m-l not. t i onun o t nitiono# n t t t ovint ivtiv o mi no vi

    Annu.

    Rev.

    FluidMech.

    1978.1

    0:301-332.Downloadedfromarjournals.annualreviews.org

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    6 TA

    Hence the tnent vecto u to cuve uderoe Fermi-Wke trnort etes Eqution (33 t w not undergo pel tnport e tfy Equton ule e cuve eec

    A e f fu vec

    th fy he eu34)

    wh

    t- 28 (35

    o fom n rhnorml er. We m cntuct uch tetd on cuve r b chn ut f he bve eun n even of he cuvewth mtr vu 5 = 5 an wth

    =u( nd thn tot he tetrd to te event wh meter vue Fem rt Such tt contn n thorm t a 2ohol to r Th fm "tl fe of eference f n erv whoew-lne me f efeence e elvc ezn hNew cep f "tg fme

    e e

    rje rbtr vecto c-e n t t f)

    We hve

    (3

    In ddon

    9

    m be eee he mec he heeme ce h hevect x) he even xhe cmone f he fu vec }f. fm 4 mx We h ee

    b Aa the eement f he nvee m ht

    n

    }rA =

    b v w }' u, }O - hu

    d the m e cred ou ove = ,,

    Annu.

    Rev.

    FluidMech.

    1978.1

    0:301-332.Downloadedfromarjournals.annualreviews.org

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    REATIVI I MEI 07

    4 Sc ScmW ha that th lt t ld u d a thpaat al pa thh a hpfa L pat a lt f th

    euaindx = uxd

    Wh th lt a wtt a

    te euan

    = 0

    = 12

    a th paat at fLI w ha a dbd b th at

    i ),

    th th at btad (4 b bttt 42 naey

    x# x()s x(s)

    1

    (42

    (4

    dene a ienina uace n acetie If te cue gien by uatin2 i a ced cue te tdinina uace cad a tue

    Te ec edax

    }I=_ tat t th f paat th fa It a fEuain (4 at

    aA# au

    as a(44

    e eent beled by cnant ue ie n cue n e cnguencegen by uain 1 Tu een n e dine and n te dine + eeen neigng aice a i a diaceent ec gien by Te caa

    2 (4

    wh a th pal t f a thal ttad a b dda te patal coote o the ecto lat t h wldl

    Wh a that th d Wal tapt ay hwby dtat Eat (4. wth pt t ad Eat ( ad

    (4.4 that

    (6

    Annu.

    Rev.

    FluidMech.

    1978.1

    0:301-332.Downloadedfromarjournals.annualreviews.org

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    08 TAB

    wh

    ad u; d h aa da h d

    W a dp u; b g h p pa h a lw

    wh

    2( =(u#v+ Uv#)hdh -%8h"

    u+u+auaui8h"

    (48

    (4.9)

    wh h a d h da da I a hd ur ad ha

    10

    Ea 6 a h a aa da a dgg a da Th a b alg alg

    h wdl abld b ad g Wa ppagad ax w#># ad8 db pl h a ha ad pa h ld ghbg pal ad a db h ala

    h

    11

    wh + h da h ad f' h aaag d h agla h al pa ch a b # mm a h c c W ha

    W#u# = .h cd

    W# 0

    h a ad cd ha

    u# pJ:#

    12

    wh ad fa caa c h a u# ppal h al h hpa

    fx = a

    Annu.

    Rev.

    FluidMech.

    1978.1

    0:301-332.Downloadedfromarjournals.annualreviews.org

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    Wav Vlciy

    an

    XXX 0

    RAVII I MC

    51)

    dn a yr ra n a-ti i.. a trdnna a. atng wit at

    X = cntandm a a a tw-dma a in atim T may gadd a ty t t wd w t y a X vai.

    T ma t i gvn y 4 ad it may wn a

    (5.2

    where Ul is e tangent vector o te world-line of an observer eauated a apoin ofL,

    53

    and

    H

    (5.4)

    T v W dtmi ' ata dit aga