Classical and quantum wormholes in a flat -decaying cosmology

F. DarabiDepartment of Physics, Azarbaijan University, Iran .

Abstract

We study the classical and quantum Euclidean wormholes for a flat FRW universe with an ordinary matter density plus a density playing the role of a decaying cosmological term. It is shown that the quantum version of the classical Euclidean wormhole is consistent with the Hawking and Page conjecture for quantum wormholes as solutions of the Wheeler-DeWitt equation.

Introduction Wormholes are usually considered

as Euclidean metrics that consist of two regions connected by a narrow throat.

They have been mainly studied as instantons, namely solutions of the classical Euclidean field equations.

In general, Euclidean wormholes can represent quantum tunneling between different topologies.

Most known solutions of general relativity which allow for wormholes require the existence of exotic matter , a theoretical substance which has negative energy density.

However , it has not been proven mathematically this is an absolute requirement for wormholes.

It is well - known that wormhole like solutions occur only for certain special kinds of matter that allow the Ricci tensor to have negative eigenvalues.

Non - existence of instantons for general matter sources , motivated Hawking and page to advocate a different approach.

They regarded wormholes typically as solutions of quantum mechanical

Wheeler-DeWitt equation .

These wave functions have to obey certain boundary conditions in order that they represent wormholes.

The main boundary conditions are : 1) the wave function is exponentially

damped for large tree geometries,

2) the wave function is regular in some suitable way when the tree-geometry collapses to zero.

An open and interesting problem is whether classical and quantum wormholes can occur for fairly general matter sources.

Classical and quantum wormholes

with standard perfect fluids and scalar fields have already been studied.

The study of - decaying cosmology in this framework has not received serious attention.

In the present work , we shall consider such a cosmology and study

its classical and quantum wormhole solutions.

Classical wormholes

We consider a (FRW) universe

It evolves according to Einstein equation

Where

22

2

2222

1)( dr

kr

drtadtds

ababab g 2

1

),,,( pppdiagab

Einstein equations reads

There is also a conservation equation

pa

k

a

a

a

a

a

k

a

a

22

2

22

2

2

3

0)(3 pa

a

By analytic continuation, we obtain

In FRW models, wormholes are described by a constraint equation of the form

An asymptotically Euclidean wormhole requires that for large , So

itt

322

2

a

k

a

a

na

Const

aa

a

22

2 1

02 a a .2n

We assume the total density as

Substitution for and leads to

By defining

20

30

aavm

,00 .00

0k

30

20

2

2

3

1

aaa

a

aR0

3

We obtain

This equation has the form of the constraint describing an Euclidean wormhole with the correspondence

Therefore, classical Euclidean wormholes are possible for the combined source with any

30

22

2 1

RRR

R

2/3

0

00

3

.3 n

.3

2

By substitution for in the conservation equation we obtain the equation of state

It is easy to show that leads the strong energy condition to hold for the total pressure and total density .

vmvm ppp 3

11

3

2

p

Quantum Wormholes

Quantum mechanical version of the classical equation for is given by

We set q = 0, and study the occurrence of Euclidean domain at large by considering the sign of the potential

R

043602

22

RRR

dR

dqR

dR

dR

R

2340)( RRRU ,0)()(

2

2

RRU

dR

d

For oscillating solutions occur which represent Lotentzian metrics.

For wormhole solutions can occur.

For and the potential is negative. So, quantum wormholes occurs when SEC is valid.

Asymptotically Euclidean property of the wave function is not sufficient to make it a

wormhole.

0)( RU

0)( RU

3

2 00

It also requires regularity for small . we ignore term as when > 2/3. The Wheeler-DeWitt equation (for )

simplifies to a Bessel differential equation with solution

R4R 0R

2

2/)1(30

22/)1(30

12/1

23

2

23

2)(

RYcRJcRR vvq

Where . The wormhole boundary condition at small is satisfied for Bessel function of the kind. In the case of = 4/3 which represents

radiation ( or a conformally coupled scalar field ) dominated FRW universe, Wheeler –

DeWitt equation for q = 0 is written as

23/1 qR

J

0)(202

2

RR

dR

d

which is a parabolic equation with solution in terms of confluent hypergeometric functions

For > 1 and we obtain a regular oscillation at small , and an Euclidean regime for large . Therefore, we have a quantum wormhole

}].;2/3;34/1.[

];2/3;14/1.{[2/exp)(2

0114

20113

2

RRFc

RFcRR

0 04 cR

R

Conclusion The classical and quantum Euclidean wormhole

solutions have been studied for a flat Friedmann- Robertson - Walker metric coupled with a perfect

fluid having a density combined of an ordinary matter source and a source playing the role of a

decaying cosmological term. We have shown that the classical and quantum Euclidean wormholes exist for this combined matter

source which satisfies strong energy condition.