Vol. 4 (1973)
REPORTS
CANONICAL
ON MATHEMATICAL
FORMULATION
No. 2
PHYSICS
OF RELATIVISTIC
HYDRODYNAMICS
I. BIALYNICKI-BIRULA
Institute
of Theoretical
Physics,
Department
Warsaw University, Warsaw, Poland,
University of Pittsburgh, USA
of Physics,
Warsaw
University,
and Department
Warsaw,
of Physics,
Poland
(Receioed February 14, 1972)
Canonical formulation of the relativistic theory of perfect fluids is given. As an introduction to this subject the canonical formalism for charged particles interacting with
electromagnetic
field is briefly discussed. Relativistic hydrodynamics
is studied in several
versions including hydrodynamics
of charged fluids coupled to the electromagnetic
field
and ultrarelativistic
hydrodynamics.
Ultrarelativistic
hydrodynamics
is shown to be
conformally invariant and several implications of conformal symmetry are discussed.
1. Introduction
Various
simplified models have played a significant role in our reaching the present
understanding
of relativistic
quantum
theories. Complete relativistic
quantum
theories
are almost exclusively particle theories, in which particles are described in terms of quantized fields. It may be of some interest to study relativistic field theories which may possess
no particle interpretation,
at least not in the usual sense. We expect this to be the case in the
relativistic hydrodynamics.
In this paper we shall study the canonical formulation
of this
theory. As an introduction
to the main topic we consider
in Section
2 the canonical
formu-
lation of the system of N charged point-like particles coupled to the most general nonlinear
electromagnetic
field. We check the Dirac-Schwinger
relations in order to prove the local
invariance of the theory under the Poincare group. In Section 3 the canonical formulation
of the theory of an ideal compressible
fluid, moving freely, is given and Section 4 contains
a formulation
of the theory of charged fluid coupled to the electromagnetic
field. In Section
5 we consider a special version of the relativistic hydrodynamics,
which is invariant under
the conformal
transformations-the
ultrarelativistic
hydrodynamics
of the fluid moving
freely with the velocity of light. In Section 6 we prove the invariance of this theory under
the conformal
group. The Appendix
contains a summary of relevant properties
of the
conformal group and canonical transformations.
11391
140
I. BIALYNICKI-BIRULA
2. Canonical formalism
of classical electrodynamics
and Z. IWIfiSKI
of N charged point-like particles
We shall study a general theory of electromagnetic field interacting with point-like
charges. The Lagrangian density of the electromagnetic field fPy = 8,A, - &A, will be taken
as an arbitrary function Y(S, P) of the two field inyariants:
S= -af~V(x)f~v(~)=~(~z-B2).
(2.la)
and
P= -~jJX)yX)j=E.B,
(2.lb)
where
E=(f,,
B=
,_fo2
-(f23
(2.2a)
,“fd,
9f31
(2.2b)
,_fiZ),
fVP+Pv~PfAp.
One can define also another
fields D and H,
tensor h’“= -~
(2.3)
a8
8-b '
the components
of which are vector
(2.4a)
(2.4b)
We choose D(x) and B(x) as canonical variables of the fields and CA,A = 1, . . . , N, and zA
=m,vJJl
- ui as canonical variables of the particles. Field and particle equations of mo- ’
tion can be derived from the variational principle with the action integral W having the
form:
w= J d4X9(s, Pj- f
mA 5 dz, - J d4xj, (x) A”(x),
A=1
(2.5)
where
Y(x) = C j d5Z e, a4 (x -CA),
(2.6)
A
and z,, is the proper time parameter of the Ath particle. Equations of motion expressed in
terms of the canonical variables are:
-~+VxH(D,B)=~eA
A
lcA
J”rn,+zj
b(r - L) 9
VD=CeAb(x-L),
A
aB
(2.7)
B)=O,
%+VxE(D,
VB=O,
d’bi
-=eA
dt
EGA, t)+
J z:
n_4
xB(L,
t) >
(2.8)
CANONICAL
FORMULATION
The symmetric energy-momentum
Thy(x)=CmA
A
OF RELATIVISTIC
HYDRODYNAMICS
141
tensor is
f d~:~6,(~-~~)+f~~(x)hl*(x)-g~‘P(x).
(2.9)
A
It satisfies the continuity equation:
a YT’“=O .
(2.10)
In terms of the canonical variables, the components of this tensor read:
X(x,
t)=Too(x,
t)=CJm:+7C:83(X-CA)+E(X,
t)*D(x,
t)-P(E(D,
B), B),
A
Tok(r,t)=~~~S,(x--rA)+(D(x,
Tk’(x, t) =c
4
‘& ”
m-i
OxB(x,
6,(x-<,)-@&
-IP(X,
O)",
t)D’(x,
t)B’(x,
(2.11)
t)-
t)+sk’(-H(x,
t)B(x,
t)+_F(x,
t)).
We define the Poisson brackets (P.B.) between the canonical variables in such a way that
the time derivative of any variable is equal to the P.B. of this variable and the Hamiltonian
of the system. In order to achieve that we assume the following form of basic P.B.‘s:
H4
9
ni}=aAB
eA
@kBk({A),
{&, Bk}=O={~~,Dk}=O={~~, Bk},
(7&, Dj(x)}= e, 6”6,(r {B’(X),Do> = ijkak
6,(x
(2.12)
CA),
- y),
{B’(x),Bk(y))=o = {D’(x), Dk(y)} .
The P.B. for any two functionals of field variables and functions of particle variables
F[D, B 1 <A, ZA] and G[D, B [ <A, aA] has the form:
{P,G)=;Jd3xd3y
-$dC-~G)oi(x),B~(y)}+
6D (x) 6B’(y)
GB’(y) i@(x)
aF ac
aF aG
(2.13)
142
I. BIALYNICKI-BIRULA
The P.B.‘s between the components
relativistic covariance of the theory,
and Z. IWI&3KI
of the energy-momentum
are:
{Too@ 9 0, T OO(Y3 t)} = - [TO%,
{TOO(x, t), TOk(y, t)}= -[Tk’(x,
{TOk(X, t), TO’(y, t)}=[TO’(x,
tensor,
expressing
t) + TOk(y, t)] ak 6,(x-y)
t)+PO(y,
t)ak+Pk(y,
,
(2.14)
t)dk]S&-y),
t)a’]d,(x-y).
We can derive them by a direct computation,
using the equations
ing identities:
aE’ aE’
dHk aH”
DxB=ExH,
>
aD’ aDc
aB”
aBk’
of motion
and the follow-
(2.15)
(2.16)
F(x)ak6~(x-~)=F(~)~k6~(~-~)-~akF(x))6~(~-~).
3. Relativistic
hydrodynamics
the local
of the free fluid
Now we shall consider an ideal compressible
fluid without pressures moving freely in
infinite space in accordance with special relativity.
The states of the fluid will be described by the four velocity field U”(X) and the scalar
rest-mass density p(x) .
The equations of motion consist of the continuity
equation
a,(pu*) = 0
and a relativistic
analogue
of the Navier-Stokes
(3.1)
equation
upa, d = 0.
We shall assume the usual normalization
condition
(3.2)
for the four-velocity
u”(x) U,(X) = 1,
which is, of course,
consistent
The energy-momentum
with equation
tensor
(3.3)
(3.2).
of the fluid has the form
T”‘(x) = p (x) U”(X) u’(x).
(3.4)
As a result of the equations of motion, this tensor satisfies the continuity
equation (2.10).
In order to introduce the canonical formalism we start from the Lagrangian
density for
the fluid:
9((x)= -p(x).
(3.5)
We choose the density of the current (~=puO and the velocity
rewriting the Lagrangian
density in the form
Z(rj?,
v)= -q&v?
v as Lagrangian
variables,
(3.6)
CANONZCAL
The Legendre
FORMULATION
transformation
OF RELATIVISTIC
and its inverse
are:
as
x=-_=
av
143
HYDRODYNAMICS
9,~
~)
(3.7a)
~14
and
IT
(3.7b)
U=
.
\I=
The Hamiltonian
density constructed
in the terms of the canonical variables
.z?(x)=~(a,,
a)=no(p,
to the usual
according
o, and IZ,
x)-Lqp,
procedure
of motion
expressed
_-
u(q,
n))=jfp2+a2=p14,2,
coincides with the time-time component
of the energy momentum
components
of this tensor in terms of canonical variables are:
Equations
and
tensor
Ton(x).
(3.8a)
Other
TOk(X) =7?(X),
(3.8b)
n”(x) T?(X)
Tk’(X) = ---_
.
/z
\ v, +?1:
(3.8~)
for the canonical
variables
may be written
in the form:
a,p=-a, nL01,
J p2+¶C2
(3.9)
a,n'=-a,_~?rXI.
J &2
We shall postulate
the following
P.B. relations
{ p(n) 7 nk(y)3 = P(Y) a%+
{~kb9,74~))
= (44
between
the canonical
variables:
- y),
ak f nk(y) al) ~x-Y)
(3.10)
,
{V(X), p((Y))=O.
This will guarantee
F[q, n] of canonical
not only the correct
variables,
value
of the time derivative
for any functional
a,~[~,~i=~FC~,ni,H~,
(3.11)
but also the correct P.B. relations (2.14) between the components
of the energy-momentum
tensor. As in the case of the free scalar field, the free vector field, and Maxwell electrodynamics, one can obtain here an additional relation
(TOO(X), P(y))
where the dot denotes
tions.
=[Pyy)al+P(y)ak-P$tq d,(~-~),
the time derivative.
(3.12)
One must make use here also of the field equa-
144
4. Relativistic
I. BIALYNICKI-BIRULA
and Z. IWIfiSKI
fluid coupled to the electromagnetic
field
In this section we shall consider the system composed of the charged fluid and an arbitrary, in general nonlinear, electromagnetic field. We assume that the charge distribution
of the fluid is proportional to its mass distribution p. The current density is:
jP(x) =$
(x) u”(x).
(4.1)
The full set of equations of motion for the field and the fluid in relativistic notation is:
f& y+fpy,LI
+f7a,p= 0 7
h aS,e=J 4 9
(4.2)
(pu”u@),B=f"@& .
The continuity equation for the current density
@u?,.=O
(4.3)
follows from Maxwell’s equations. The symmetric energy-momentum
T”‘(x) = p (x) u”(x) u”(x) +f “‘(x) h,‘(x) - g”“2’ (x) .
tensor is:
(4.4)
It satisfies the continuity equation.
Following the same procedure as that of Sections 2 and 3 we express all quantities in
terms of canonical variables
and B(x), D(X).
The contkuity
equation and equations of motion can be now written in the form
aO
q+ak
J
i?,n+a,
nkY, =o,
nkX=eP,E(B,
J &2m
D)+--
J
V.B=O,
-;+vxH(D,B)=I qn ,
mJq2+x2
(4.6)
CANONICAL
FORMULATION
OF RELATIVISTIC
HYDRODYNAMICS
145
Again, we postulate the P.B’s between the canonical variables:
{9(-+
&Y))=O,
{VW
nk(Y)I=g,WkMX-Y),
(+) , n’(Y)}
b(4,
Pk(4
=(n’(x) a’+n’(Y) 8’) 63 (x-i)
Bkdy))=O={m,
9 NY)}
+;&x)
~kQ)=O={nk(x),
= 0 = (ok(X) , NY)}
83 (X-Y)&i’k&(X)
(4.7)
B’(y)>,
,
{B’(x), Di(y)}=&‘jka,6,(x--y).
By a direct computation using equations (2.4), (2.15), (2.16) and the last equation of (4.6)
we obtain the correct P.B. relations for the energy-momentum tensor.
5. Special case of relativistic hydrodynamics:
ultrarelativistic
hydrodynamics
The conformally invariant hydrodynamics (a proof will be given in Sec. 6) is obtained
when the fluid moves freely with the light velocity, i.e. when the four-velocity vector field
is a null vector:
u”u,=O.
(5.1)
The equations of motion are still the same as (3.1) and (3.2). The canonical variables are:
fjJ=pllO
(5Ja)
and
(5.2b)
The fields p(x), u”(x) and the traceless energy-momentum
expressed in terms of the canonical variables as follows:
,_J?
9
’
tensor
f2’ J”:
Uk=-_
Tooh0=3(x, t)=
Tok(x
, t) = nk(x, t) ,
p=-T’
J,+,
TpY=puV
can be
(5.3)
t),
(5.4)
n’(x) t) 72(x t)
)
Tkz(x
,t) =
JxZ(x,
’
Equations of motion in terms of the canonical variables are:
(5.5)
146
I. BIALYNICKI-BIRULA
and Z. IWIfiSKI
The definitions of the P.B.‘s between the canonical variables, giving the correct equations
of motion, are the same as in the previous case (Sec. 3)
{n”@), r+(Y)) = (r&x) ak + Xk(Y)a’>63 (X-Y) ,
{U,(&nk(Y)l=V(Y)ak&(X-Y),
W),
(5.6)
p(y)Y))=O.
The relations (2.14) between the components of the energy-momentum tensor hold also
in this case.
In all theories mentioned in this paper ten conserved quantities P” and WV
P” = j da, T’“(x),
(r
(5.7)
M”” = 1 do, xCpTv’“(x)
d
are independent of the surface o and they represent the energy, the momentum, the angular
momentum and the moment of energy of the system.
As it follows from the equations (2.14) by a direct integration, the P.B. relations for
these quantities are the same as comutation relations for generators of the PoincarC
group. These quantities generate the Poincare transformations of the fields.
6. Conformal invariance of the ultrarelativistic hydrodynamics
Among theories mentioned in this paper, only free Maxwell electrodynamics and free
ultrarelativistic hydrodynamics are conformally invariant. In both cases (for Maxwell
electrodynamics this is well known) generators of the conformal transformations are of the
form:
translation
P,=
j TgOd3x,
homogeneous Lorentz
transformation
dilatation
acceleration
M,w = j (xp Tv, - xv Tp,) d 3x ,
(6.1)
D = j x’TpO d3x,
Kp = 1(2x, x2 - x’g,,)
T; d3x
(all relevant properties of conformal group and canonical transformations are given in the
Appendix).
The form of the generators of the Poincare group is known. We must only show that
the generators of conformal transformations D and Kp generate the appropriate transformations of the field, i.e. the transformations induced by a dilatation and a special conformal
transformation. The transformation laws of canonical variables can be obtained from the
fact that the energy-momentum tensor and the current vector transform as a tensor density
1
3
and a vector density of weights z and 2, respectively.
CANONICAL
The transformation
FORMULATION
OF RELATIVISTIC
HYDRODYNAMICS
147
laws for canonical variables are:
(6.2)
‘nk(x)= ‘p (x)‘u~(x)‘u~(x)=
where P=f”(i,
y) is the most general transformation of the conformal group.
The transformations of the vector j”(x) and the tensor P’(X) are of the form
(6.3a)
p(2)
22
(i)up
(2)
.
(6.3b)
From the equation
we obtain that
In the case of a dilatation we have
xc= eafp,
A(x)=ew2'
whereas in the case of a special conformal transformation
we have
,
XP,
X” -p-y
1-2cR+~2c2
’
A(x)=
l
1+2cx+c~x~=75
1
*
We shall show now that the fields transformed by one parameter subgroups generated by D
and K= c'K, , respectively, satisfy the same differential equations as fields obtained from
transformations (6.2) in the special case of a dilatation with parameter A and a special
conformal transformation with parameters AC,,. Since for A= 0 these fields satisfy the same
initial conditions, D and K generate appropriate transformations. The differential equations
for ‘v, and ‘ak in the case of a dilatation are:
d ‘P(4
___
dA
= - xfla, ’ P (x) - 3 ‘u,(x),
d’Ak
dL
= -
xpa,
'dyX)
- 4 auk
.
(6.4)
I. BIALYNICKI-BIRULA
148
and Z. IWIfiSKl
On the other hand, fields (n)p,and (‘%? obtained from transformations
tor D satisfy the same differential equations
(A.8) with the gener-
(6.5)
We used here the properties of the canonical transformations (A.8). From these equations
it follows that “‘~=‘~, (‘)zk= ‘zk and for I=a these are the fields transformed by the
dilatation.
The canonical variables transformed by the transformation (6.2) in the case of a special
conformal transformation with parameters ,k, satisfy the differential equations
d’v
-=
dA
irk
- cP(2xPx, - gPVx2) d’ ‘Q,- 6/x,
‘9 - (2x”ck - 2xkc0) G
,
Jn
d ‘nk
= -c~(2XpXY-gpYXZ)aY’~k-8CP~p ‘irkd/l
- (2c’xk - 2ckx9 ‘,rl- (2c’x, - 2c, x*) r2
JZ
+ (2c, xk - 2ckx,) JQ
.
(6.6b)
Canonical variables transformed by a one parameter group of the canonical transformations
.generated by the c“K,, satisfy the same equations:
d (‘)y,
-=R.H.S.
dl
d
of (6.6a) with (‘)g, and %rk instead of ‘9 and ‘2,
(6.7)
.
.
V),k
-=R.H.S.
dl
of (6.6b) with (‘)~JIand %ck instead of ‘p and ‘rck
and the same initial conditions. This ends the proof that PK, are indeed the generators of
special conformal transformations. Any transformation of the conformal group can be
.obtained as a canonical transformation generated by
G=aD+c’K
P+d’M
w +dP P’
generators of the conformal group are constants of motion. For D and K,, this
follows from the vanishing of the trace of the energy-momentum tensor
All the
$=
j T”,,d3x=0,
(6.8)
52J.
xCTYVd3x=0.
CANONICAL
FORMULATION
OF RELATIVISTIC
HYDRODYNAMICS
149
This guarantees that the transformations generated by these generators are symmetry
transformations, so this theory, as well as Maxwell electrodynamics, is conformally invariant.
It is easy to calculate the P.B. relations between the hfteen generators from the most general P.B. relations of the components of the energy-momentum tensor (2.14), using only the
fact that this tensor is traceless. These P.B. relations are the same as commutation relations for generators of the conformal group (A.7).
Appendix.
The conformal
group
The conformal group is a subgroup of general space-time coordinate transformations
x”+‘x”(x) which leave the Minkowsky metric invariant up to a scale factor depending on x:
1
axA a9
gAp=;l(‘x)glrv
-a‘9 a‘XV
where
0
grip= i 8 -i
0
-1
0
0
0
y
8
.
(A.1).
o-1
Fifteen basic transformations of this group are translations, homogeneous Lorentz transformations, scale transformations (dilatations) and special conformal transformations (accelerations)
translation
‘xp=tP(x,
a)=xp+ap,
6x’=6a”,
homogeneous Lorentz
transformation
‘xl’ = l”(X) W) = nq xv,
6x” = h’,
dilatation
6xp = 6ax”,
acceleration
‘xc = d”(x , ol)= e”x” ,
‘xl’=
syx
)
The most general transformation
c) =
xp-Px2
xv,
(A.21
6x” =(2xPx” - QW) 6c,.
l-2cx+x2c2’
of this group will be denoted by
‘xl’=f”(x , Y) ,
(A.3p
where y’, . . . . y15 denote 15 parameters of this group. Realization of this group in the space
of fields #(x) (of some tensorial character) is defined in the usual manner:
V(x)
=A (x 3YYB P,“(f_
‘(x , Y)),
(A-4)
where A(x, r)“B is some finite dimensional matrix depending on the tensorial character
of the field. For example, the transformation laws for the electromagnetic field are:
)f&)= gg
f,r(.p(X,
where
n=f-‘(X,
y).
y)),
64.5%
I. BIALYNICKI-BIRULA
150
and Z. IWIfiSKI
The operators F(x, 7) which induce the change of the argument of the field, F(x, Y)#(x)
= q2A(f-1(x, r)) are given as follows:
translation
T(x, a)=exp[-a,?],
homogeneous Lorentz
transformation
L(x, W)=exp[-+o,,z”“],
dilatation
D(x, cl)=exp[-c&J,
acceleration
S(x, c)=exp[-c,rP],
64.6)
These differential operators satisfy the algebra of the generators of the conformal group
CT,, %I=o 3
(A-7)
cJ$I
94 = -%7,,~+~,J.
For the canonical formulation of the theory it is necessary to find the canonical transformations (and their generators) of the canonical variables corresponding to the conformal transformations of the fields. Elements of one parameter subgroup of canonical
transformations generated by the generator G(t) transform the canonical variables in
the following way
O}+&)r{G(t), P”(x,t>}}+...,
‘“‘$<x, f)=p% t>+~(G(t), v%,
64.8)
%&,
f)=nng(x, t>+A{G(t),
nng(x,
f))+;(W),
{G(t), n&,
t>>>+...
CANONICAL
These transformations
FORMULATlON
are canonical
{‘Qv,, %},,
OF RELATIVISTIC
HYDRODYNAMICS
151
because of the fact that
x = { a,34~)lc = {(I) P , (Aw,,,
wlr *
(A-9)
New variables (‘)#(x, t) and @)rcs(x, t) satisfy in general (when G(t) depends on time)
canonical equations of motion with a new Hamiltonian gA((t), which is a solution of
differential equation
dqp+dT-(G(t),
i&(t)}=0
(A.10)
with the initial condition tillrlzO= H. These transformations are symmetry transformations if the generators are constants of motion. In a conformally invariant theory, generators of conformal transformations of the fields satisfy the algebra of conformal generators (A.7).