# Exact Relativistic Ideal Hydrodynamical Solutions in (1+3)D with Longitudinal and Transverse Flows Page: 4 of 9

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4

The two equations involving derivatives over y and p are

trivially solved by setting p(x) p(r, p) (and the same

for energy density e(T, p) due to the E.o.S) and accord-

ingly tP(x) 'P(r, p). We note, that the simple form

of Eqs.(18,20), are a direct consequence of the vanishing

components uv" 0 in the flow field ansatz, Eq.(12).

Finally we introduce a combined field variable K de-

fined asT + p T2p

p 1T

pm2K-.We then substitute the pressure p in the equations (17)

and (19) and obtain two equations for the fields K and

vP, which can be expressed asDa K,-r+Db'-K,P

D* - K,r + Dc ' K, PD1- K,

D2-K .The coefficients Da, Db, Dc, D1, D2 are given by:

Da (1 - v) + v ,

Di v+ (1 -v)D2

Dl -2v0,0P, ,7 1 - 0/T - 1P , P ,vP ,T

From (22)(23) we obtain

K,1

K

KDcD1 -DbD2

(ln K),T =z2

DaDe- Db

(ln K) = a 2 b1

'P Da - DS[T,p]

Q[T p]first order derivatives a igf and dinf . For In K as a

single function of two variables r, p, the two equations can

be consistent only if the following constraint on second

order derivatives are satisfied a2Ln1K a2Ln1C i.e.

drop dpdra

Op(28)

[OP (r, p)]= 0 .

Thus we only need to solve the above single equation

for the velocity field vP(r, p). Since B, already involve

the first derivatives of vP ,T and iP ,P, the reduced ve-

locity equation 28 is a second-order partial differential

equation for the velocity field. As a minor caveat, the

method applies to the case/region in which ln K is at

least second-order differentiable. This reduction method

can be demonstrated in the more explicit case of (1+1)D

hydrodynamics, see Appendices B and C.

Given the above constraints, we can then solve from

(25) the matter field S directlyK K0 64i'; dr'F[r',p] f9 dp'QLrp']]

(29)

with Ko being the value at arbitrary reference point

To, po.

Finally let us summarize our approach: after including

into the flow field ansatz the physically desired longitu-

dinal and transverse flows, we have reduced the hydro-

dynamic equations into a single equation (28) involving

(24) ONLY the transverse velocity field 'P, and any solution

to this equation automatically leads to the pressure field

which together with the velocity field forms a solution to

the original hydrodynamics equations:

p constant x x e fT dr'Yir', f9 dp9'L' . (30)

-(25)

with the functions , given by

_ ( 1

.F[p(Top)] -x

v(1 - v)(1 -v2)2

[(1- v)D - v]vP , P +

[(1 - 2v2) + 2v(v - 1)t ]ttP ,7

-v[1 - j] [v + (1- vj]/T - iivvJPC. Examination of the Method

We now examine the correctness of the reduced equa-

tion (28) and the solution (30), using the two known sim-

ple analytic solutions (10) and (11) as both of them are

certain special cases of our embedding with longitudinal

and transverse radial flows.

For the 1-D Bjorken expansion, we have vP B-. 0

which leads tovi1

Bj. - ,

v -1 7v ]lpp ,(26)

_ (T 1

v(1 - v)(1 v2

[ + (v - i)]Dp,1 +

[(-1 + 4v - 2v2) + 2v(v - i)tp]t2 ,

+vvjp[ - t ]/T + v[1 -2 j][1 - i) + vD]/p} .(27)

In (25), the function ln K depends (via tP) on two

variables T and p, and we have two equations for the two1

QBj. _ -

P(31)

One can easily verify that the above .j, Q8j. satisfy

the reduced equation (28). Furthermore by inserting

Bj., Q8j. into the solution (30) one finds exactly the

pressure in (10).

For the 3-D Hubble expansion, we have vP Hu. p/T

which leads to

3 v -5/2 2T

T 1 - v T2- p2

1 v -5/2 -2p

QHu. -+ 32

p 1-v T2-p2 (2D2 = 2(1 -v)vPP , P + v(1 - 0)/p

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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/m1/4/: accessed January 20, 2019), University of North Texas Libraries, Digital Library, digital.library.unt.edu; crediting UNT Libraries Government Documents Department.