Exact Relativistic Ideal Hydrodynamical Solutions in (1+3)D with Longitudinal and Transverse Flows Page: 3 of 9
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B. Some Known Simple Exact Solutions
We now recall some known simple exact solutions that
are pertinent for our approach.
One of the most famous examples is the so-called Hwa-
Bjorken solution [31, 32] which is essentially the Hubble
expansion in (1+1)D. The pressure and velocity fields of
this solution are given by
uB. -(1, 0, 0, 0) . (10)
It is more transparent to look at the components of v in
flat coordinates, which are simply vz tanh , v
A generalization of the Hwa-Bjorken solution to radial
Hubble flow in (1+3)D is straightforward. The pressure
and velocity fields are
PIHh. s ,
(2_ p2) 2(1-u)
uxI. = ( coshy , 0, -, 0)
1 - (p/r)2 cosh2 y
with the domain of validity being p < T. In flat co-
ordinates the velocity fields are given in simple form:
v= /t = (x/t, y/t, z/t).
Further generalization of the spherically symmetric
Hubble flow to arbitrary (1+d)D has been done in .
An ellipsoidally expanding hydrodynamical solution has
also been discussed (see the first paper in ). There
have also been study of adding radial flow to a longitu-
dinal Bjorken profile, see e.g. .
III. THE NEW REDUCTION METHOD
In this section, we use a new reduction method to
find solutions for (1+3)D RIHD equations. The general
idea is to first embed known solutions in lower dimen-
sions which automatically solve 2 out of the total of 4-
component hydrodynamics equations, and then reduce
the remaining 2 equations into a single equation for the
velocity field only. As usual, one starts with a certain
ansatz for the flow velocity field: in our case we will
use an ansatz with built-in longitudinal and transverse
radial flow, aiming at possible application for RHIC. It
would be even more interesting to include transverse el-
liptic flow which requires a suitable curved coordinates
(like certain hyperbolic coordinates) other than the one
used here. However generally in those cases, more Affine
connections are non-vanishing, which makes the reduc-
tion method discussed below much more involved: we
will leave this for future investigation.
A. Including Longitudinal and Transverse Flow
We first embed the boost-invariant longitudinal flow as
many numerical hydrodynamics calculations do, which is
a suitable approach for RHIC related phenomenology. To
do that, we simply set v2 = z/t = tanh y, i.e. u = 0.
Next we include the transverse radial flow which is
isotropic in the transverse plane. Radial flow is substan-
tial and important at RHIC. To do so, we introduce the
radial flow field vP and set the flat-coordinate transverse
flow fields to be v vp cos p and Wy =W sin p, which
implies for the curved coordinates uP =y%, and uv 0
with the latter meaning an axially symmetric velocity
field. We note that this ansatz goes beyond a simple
change to cylindrical coordinates, since we require that
u= 0 which considerably simplifies the hydrodynamics
To summarize, in order to describe a situation with
both longitudinal flow and transverse radial flow we have
made the following ansatz for the flow fields u1m in the
coordinates (, y, p, #):
um = (1, 0, tp, 0
vt - vK cosh y ,
S-z1/ 1-t .
Note that we need to require iP < 1.
B. The Equation for Transverse Velocity
With the flow fields given in (12), we can now explicitly
express the stress tensor components. The non-vanishing
ones are given below:
T"T = 72( + p) -p =(-
T vv = 72p ( T p = 7p ,
T = - ,-
For the second equalities in each of the first three lines
we have used the E.o.S (3) to substitute E + p by p/v.
With the above expressions and using (8) (9), the hy-
drodynamics equations (1) then become
T T ,;+ +3- p+ P,P +
Tv; - p,v 0 ,
TP;A = TPP p +
TP A 1~ 0.
p + T'P , + T
p ' T
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Liao, Jinfeng & Koch, Volker. Exact Relativistic Ideal Hydrodynamical Solutions in (1+3)D with Longitudinal and Transverse Flows, article, May 20, 2009; Berkeley, California. (digital.library.unt.edu/ark:/67531/metadc932705/m1/3/: accessed January 18, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.