# Exact Relativistic Ideal Hydrodynamical Solutions in (1+3)D with Longitudinal and Transverse Flows Page: 3 of 9

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3

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

constant

pBj. T(i),

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

vy 0.

A generalization of the Hwa-Bjorken solution to radial

Hubble flow in (1+3)D is straightforward. The pressure

and velocity fields are

constant

PIHh. s ,

(2_ p2) 2(1-u)

1 p

uxI. = ( coshy , 0, -, 0)

cosh y

1 - (p/r)2 cosh2 y

(11)

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 [23].

An ellipsoidally expanding hydrodynamical solution has

also been discussed (see the first paper in [22]). There

have also been study of adding radial flow to a longitu-

dinal Bjorken profile, see e.g. [33].

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

equations.

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 ,(12)

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:-2

T"T = 72( + p) -p =(-1)p,

TPP=12tpe+p)+p=( v+1)p,

T vv = 72p ( T p = 7p ,

T = - ,-

7"-2-(13)

(14)

(15)

(16)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 becomeT T ,;+ +3- p+ P,P +

Tv; - p,v 0 ,

TT PP

TP;A = TPP p +

' =p

TP A 1~ 0.0,(17)

(18)

p~

p + T'P , + T

p ' T0, (19)

(20)

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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.