some astrophysical aspects of the wormhole spacetime valdir b

SOME ASTROPHYSICAL ASPECTS OF THE

WORMHOLE SPACETIME

Valdir B. Bezerra

Universidade Federal da Paraba

Jo~ao Pessoa, Brazil

Nail R. Khusnutdinov

Kazan State University

Kazan, Russia

1

We investigate the self-energy and the self-force for scalar

massive and massless particles at rest in a wormhole space-

time. We develop a general approach to obtain the self-force

and apply it to a specic prole of the wormhole throat. We

showed that the self-force can be attractive as well as repul-

sive.

3

WORMHOLES ARE TOPOLOGICAL

BRIDGES-CONNECTS DIFFERENTS PARTS OF THE

UNIVERSE

THESE KIND OF TUNNELS IN SPACETIME HAVE

APPEARED IN DIFFERENT CONTEXTS OF PHYSICS

1. ANALYSIS OF BLACK HOLE BACKGROUNDS

A. Einstein and N. Rosen, Phys. Rev. 48, 73(1935)

2. TIME MACHINE

M. S. Morris, K.S. Thorne and U. Yurtsever, Phys. Rev. Lett.

61, 1446 (1988) 1

PARTICLES IN CURVED SPACETIMES/SELF-FORCE IN-

TERACTION

ENERGY-MOMENTUM TENSOR(MATTER/FIELDS) - GRAV-

ITATIONAL FIELDS

GREEN FUNCTION FOR EACH CONFIGURATION

SELF-FORCE - GLOBAL STRUCTURE OF THE FIELD

2

PARTICLE IN THE SPACETIME OF A COSMIC STRING.

THERE IS NO LOCAL GRAVITATIONAL INTERACTION

BETWEEN THE PARTICLE AND THE COSMIC STRING

THERE EXISTS A REPULSIVE SELF-INTERACTION WHICH

IS OF NONLOCAL(GLOBAL) ORIGIN.

3

Wormholes Electromagnetic field Scalar field

A little bit about wormhole

DefinitionWormhole space-time ⇐⇒ a closed surface of minimal areaexists The wormholes with trivial topology:

Nail Khusnutdinov Self-things in wormhole space-time 2/31


The wormholes with non-trivial topology:



The metric of static spherically symmetric wormhole

ds2=−dt2 + dρ2 + r(ρ)2dΩ2,

ρ∈(−∞,+∞).

A wormhole’s throat is described by profile function r(ρ).Definition of the radius of the throat a:

The sphere of minimal area ⇐⇒

r(0) = a,r′(0) = 0.

Flatness far from the throat

limρ→±∞

r2(ρ)

ρ2= 1.



Different profiles of the throat.1) Infinitely short throat (NK and Sushkov 02).

r(ρ) = |ρ| + a,

R = −8

aδ(ρ).

2) Drainhole (Ellis 73).

r(ρ) =√

ρ2 + a2,

R = − 2a2

(ρ2 + a2)2.



3) Wormhole with length of throat τ : r(ρ) = ρ coth ρτ

+ a − τ .

Length of throatRadius of throat = τ

a= 0.2, 1, 5.



Self-force. Electromagnetic field. NK and Bakhmatov, 2008.

Particle at rest in the point ρ′, θ′, ϕ′.

The potential: A0=−4πeδ(ρ − ρ′)

r2(ρ)

δ(θ − θ′)δ(ϕ − ϕ′)

sin θ,

A0=4πeG(x;x′).

Self-Potential: U(x) =e

2A0 = 2πe2Gren(x;x).

3D Green function: G(x;x′) = −δ(ρ − ρ′)

r2(ρ)

δ(θ − θ′)δ(ϕ − ϕ′)

sin θ.

Spherical symmetry: G(x;x′) =

∞∑

l=0

l∑

m=−l

Ylm(Ω)Y ∗

lm(Ω′)gl(ρ, ρ′).

Radial Green function: gl +2r′

rgl −

l(l + 1)

r2gl = −δ(ρ − ρ′)

r2(ρ).



gl = θ(ρ − ρ′)Ψ1(ρ)Ψ2(ρ′) + θ(ρ′ − ρ)Ψ1(ρ

′)Ψ2(ρ),

where

Ψ +2r′

rΨ − l(l + 1)

r2Ψ = 0,

satisfy the boundary and Wronskian conditions

limρ→+∞

Ψ1 = 0, limρ→+∞

Ψ2 6= 0,

W (Ψ1,Ψ2) = Ψ1Ψ2 − Ψ1Ψ2 =1

r2(ρ).

In domains ρ > 0 and ρ < 0

φ1|ρ→+∞=ρl,

φ2|ρ→+∞=ρ−l−1,

with the Wronskian

W (φ1±, φ2

±) =A±

r2(ρ).



Problem due to nontrivial topology

Ψ1 =

α1+φ1

+ + β1+φ2

+,ρ > 0α1−φ1

− + β1−φ2

−,ρ < 0, Ψ2 =

α2+φ1

+ + β2+φ2

+,ρ > 0α2−φ1

− + β2−φ2

−,ρ < 0.

limρ→+∞

Ψ1 = 0 =⇒ α1+ = 0,

Matching conditions at ρ = 0 =⇒ α1−, β1

−

limρ→+∞

Ψ2 6= 0 =⇒ No conditions

Matching conditions at ρ = 0 =⇒ Indefinite Green function



Problem due to nontrivial topology

In flat Minkowski space-time there is boundary conditionat ρ = 0 – solution should be finite at ρ = 0 =⇒ the Greenfunction is unique defined.

In wormhole space-time – there is no zero distance, there isminimal distans, throat =⇒ solutions are finite at ρ = 0.

Full Green function is linear combination G = αG1 + βG2,where G1 for α2

− = 0 and G2 for β2− = 0.

Manifest calculations show that G2 is divergent =⇒G = G1 =⇒ wormhole’s electrodynamics is simple.



1. ρ > ρ′ > 0 General Solution

g(1)l (ρ, ρ′) = − 1

A+φ2

+(ρ′)φ1+(ρ) +

1

A+

W+(φ1+, φ2

+)

W+(φ2+, φ2

+)

∣

∣

∣

∣

0

φ2+(ρ′)φ2

+(ρ)

2. 0 < ρ < ρ′ g(2)l (ρ, ρ′) = g

(1)l (ρ′, ρ)

3. ρ < ρ′ and ρ′ > 0, ρ < 0

g(3)l (ρ, ρ′) = − 1

A+

W (φ1+, φ2

+)

W+(φ2+, φ2

+)

∣

∣

∣

∣

0

φ2+(ρ′)φ2

+(−ρ)

4. ρ > ρ′ and ρ′ < 0, ρ > 0 g(4)l (ρ, ρ′) = g

(3)l (ρ′, ρ′)

5. ρ′ < ρ < 0 g(5)l (ρ, ρ′) = g

(1)l (−ρ,−ρ′)

6. ρ < ρ′ < 0 g(6)l (ρ, ρ′) = g

(5)l (ρ′, ρ)



Profile r = a + |ρ|

4πG(1)=1

√

r2 − 2rr′ cos γ + r′2− 1

2aln

∣

∣

∣

∣

∣

1 +2t

1 − t +√

t2 − 2t cos γ + 1

∣

∣

∣

∣

∣

4πG(3)=t

a√

t2 − 2t cos γ + 1− 1

2aln

∣

∣

∣

∣

∣

1 +2t

1 − t +√

t2 − 2t cos γ + 1

∣

∣

∣

∣

∣

,

where t = a2

rr′and cos γ = cos θ cos θ′ + sin θ sin θ′ cosϕ.



Equipotential surfaces. e = 1, position x = 2, y = z = 0.

The small sphere is the throat of the wormhole, ρ = 0. Somesurfaces go under the throat to another universe.



The potential far from the throat has ”bad” behavior

ρ > 0: A0(x;x′) =e

ρ

(

1 − a

2(a + ρ′)

)

+ O(ρ−2),

ρ < 0: A0(x;x′) = − ea

2ρ(a + ρ′)+ O(ρ−2).

But the Gauss theorem is satisfied

4πe =

∫∫

S1

EndS +

∫∫

S2

EndS.

S1 : ρ = R and S2 : ρ = −RSome electric field lines have gone through the throat toanother, invisible domain of space-time.



Self-potential U=e2

4aln[1 − a2

(a + ρ)2],

Self-force F ρ=−∂ρU = − ae2

2r(ρ)31

1 − a2

r(ρ)2

.

Compare with self-force in black hole space-time

F ρ = −∂ρU = −ae2

2r3

√

1 − a2

r2.

Far from the wormhole throat and from the black hole we havethe same results but with opposite signs

F ρwh=−ae2

2ρ3attractive

F ρbh=+

ae2

2ρ3repulsive

Damour and Solodukhin 07: All effects must be the same.(?)Nail Khusnutdinov Self-things in wormhole space-time 15/31


Profile r =√

a2 + ρ2

Self-potential U=− e2

2π

a

ρ2 + a2,

Self-force F ρ=−∂ρU = −e2

π

aρ

(ρ2 + a2)2.

1 2 3 4 5

Ρa

-0.15

-0.125

-0.1

-0.075

-0.05

-0.025

U ae2

,FΡ a2e2

Asymptotic ρ → ∞

F ρ ≈ −e2

π

a

ρ3.



General form of the profile r(ρ)

Main idea: WKB approach for radial equation. WKBparameter ν = n + 1

2

φ +2r

rφ − l(l + 1)

r2φ = 0.

φ=eS , S =

∞∑

n=−1

ν−nSn.

For l = 0 there is exact solution for arbitrary r(ρ)

ϕ1+ = 1, ϕ2

+ =

∫

∞

ρ

a

r2dρ.



Results

U(ρ) =e2

2

[

−1

r+

1

r

∞∑

k=1

ζH(2k,3

2)j2k(ρ) +

1

aϕ2

+(ρ) − 1

2aϕ2

+(ρ)ϕ2

+(ρ)

ϕ2+(0)

]

,

j2 = −−1 + r2 + 2rr

8=

3

8a1r

2,

j4 =1

128[3+3r4 −12rr−4r2r2−2r2(3+2rr)−32r2rr(3) −8r3r(4)],

j6=1

1024[5 − 5r6 − 30rr − 20r2r2 + 3r4(5 + 6rr) − 192r2r3r(3)

−8r3(r3 + 5r(4)) − r2(15 + 20rr + 12r2r2 + 456r3r(4))

−8r4(11r(3)2 + 20rr(4)) − 16r2r(10(1 + 3rr)r(3) + 11r2r(5)) − 16r5r(6)].



Far from the throat ρ → ∞

U=− e2

4ρ2

a

ϕ2+(0)

+ O(ρ−3)

=− e2

4ρ2

(∫

∞

0

dρ

r2

)−1

+ O(ρ−3).

All information about the specific throat profile is encoded inthe factor

∫

∞

0

dρ

r2(ρ).



Self-force. Scalar field. V. Bezerra and NK, 2009.

Action

S = − 1

8π

∫

(φ,µφ,µ + ξRφ2 + m2φ2)√−gd4x +

∫

jφ√−gd4x.

Energy-momentum tensor

Tµν=jφgµν +1

4π

(

φ;µφ;ν − 1

2gµνφ;σφ;σ − 1

2m2gµνφ2

)

+ξ

4π

(

Gµνφ2 + gµνφ2 − φ2;µν

)

,

Field equation ( − ξR − m2)φ = −4πj,

Scalar Current j(x) = e

∫

δ(4)(x − x(τ))dτ√−g

.



We consider the particle at rest only.

(− m2 − ξR)φ(x) = −4πj = − 4πe√−gδ(3)(x − x′).

The Energy

E = −∫

TµνξµdΣν = −∫

T00√−gd3x.

Integrating by part and taking into account the equation ofmotion we get

E=1

2

∫

jφ√−gd3x

=1

2

∫ ∫

j(x)D(x, x′)j(x′)√

−g(x)√

−g(x′)d3xd3x′

=e2

2Dreg(x, x).

In the above expression we made the renormalization and throwaway the Minkowskian contribution.



Main features of the scalar case.

Ψ′′ +2r′

rΨ′ −

(

m2 +l(l + 1)

r2+ ξR

)

Ψ = 0

For new function Ψ = Φ/r we have scattering problem

Φ′′ +

(

−m2 − l(l + 1)

r2− ξR − r′′

r

)

Φ = 0,

with potential

U = ξR +r′′

r




1. First class of wormholes – without parameter of throat’slength

r = ρ +

∞∑

k=n

bkρ−k,

U = ξR +r′′

r=

4n(ξc − ξ)bn

ρn+3+ · · ·

Critical value ξc = n+14n

. At critical point the potential changesits sign. =⇒ Boundary states ⇐⇒ scattering states.Example – drainhole r =

√

ρ2 + a2: n = 1, b1 = a2

2 .Critical point ξc = 1

2 .




2. Second class of wormholes – with parameter of throat’slength

r(ρ) = ρ + a + cnρne−ρ

τ ,

U = ξR +r′′

r= cn

1 − 4ξ

τ2ρn−1e−

ρ

τ + · · ·

Critical value ξc = 14 . At critical point the potential changes its

sign.

Examples:

r=ρ coth ρτ

+ a − τ,r=ρ tanh ρ

τ+ a.

We have correspondingly n = 1, b = 1/2 and n = 1, b = −1/2and therefore the critical value ξc = 1

4 .Infinitely short throat r = |ρ| + a (NK and Sushkov) belongs tothe second class.



3D section is conformally flat

Let us consider the 3D flat space in spherical coordinatesR, θ, ϕ:

dl2fl = dR2 + R2dΩ2.

New radial coordinate ρ: R = r(ρ)eσ(ρ) with

σ = ±∫ ρ dx

r(x)− ln r(ρ).

Using this coordinate system we obtain

dl2fl = e2σ(dρ2 + r(ρ)2dΩ2) = e2σdl2wh.

Therefore gwhik = e−2σgfl

ik and the section is conformally flat. Forthis reason due to Hobbs1968 the self-force is zero for ξ = 1

8 andm = 0.



The case r = |ρ| + a and m = 0

U=−ae2(1 − 8ξ)

4r2Φ

(

a2

r2, 1, 1 − 4ξ

)

.

Φ

(

a2

r2, 1, 1 − 4ξ

)

=

∞∑

n=0

(1 − 4ξ + n)−1(a

r

)2n

.

The self-force is zero for ξ = 18 and it tends to infinity for

ξ → ξc = 14 .



The case r = |ρ| + a and m 6= 0

For ξ = 18 the self-force is zero and at critical point 1/4 + ma/4

it is infinity:

Ξ = 0

ma = 0, 0.1, 1

0.5 1.0 1.5 2.0

Ρ

a

-0.4

-0.3

-0.2

-0.1

a U

e2

ma = 1

Ξ =1

10,

1

5,

1

4,

1

3

0.5 1.0 1.5 2.0

Ρ

a

0.1

0.2

0.3

a U

e2



General profile and m = 0

U(ρ) =e2

2

[

−1

r+

1

r

∞∑

k=1

ζH(2k,3

2)j2k(ρ) + g

(1)0 (ρ)

]

,

j2=−ζ−1 + r′2 + 2rr′′

8,

j4=3ζ2

128

(

r′2 + 2rr′′ − 1)2 − rζ

16

(

2r′′r′2 + 4rr(3)r′ + r(

2r′′2 + rr(4)))

.

All terms proportional to ζ = 1 − 8ξ.

g(1)0 (ρ, ρ)=− 1

A+ϕ2

+(ρ)ϕ1+(ρ) +

1

2A+

(

ϕ1+

ϕ2+

+ϕ′1

+

ϕ′2+

)

0

ϕ2+(ρ)ϕ2

+(ρ),



There is no solution for zero mode for arbitrary r. For ξ = 18

there is general solution (due to conformal invariance)

ϕ1 =1√re

R ρ dy

r(y) , ϕ2 =1√re−

R ρ dy

r(y) ,

and self-force is zero. Far from the throat ρ → ∞

U ≈ − e2

2ρ2A.

Constant A depends on the global structure of the space-time.



Drainhole r =√

ρ2 + a2

g(1)0 (ρ, ρ) =

cos(2µ arctan ρa) − cos(πµ)

2aµ sin πµ, µ =

√

2ξ.

Ξ = 0,1

10,

1

9,

1

7,

3

10

1 2 3 4 5

Ρ

a

-0.1

0.1

0.2

0.3

a2 F

e2

Asymptotic ρ → ∞

U ≈ − e2

2ρ2

aµ

tan πµ,∀ξ

U ≈ −e2aπ

16ρ(1−8ξ), ξ → 1

8

U ≈ − e2a

2πρ(ξ − 12)

, ξ → 1

2


DISCUSSION AND CONCLLUSION

SELF-INTERACTION/SCALAR PARTICLES AT REST

NONMINIMAL COUPLING WITH CURVATURE

MAIN PARAMETERS

MASS OF THE FIELD

COUPLING CONSTANT 1

ξ = 1/8, THE SELF-FORCE IS ZERO

ξ < 1/8, THE SCALAR PARTICLE IS ATTRACTED

ξ > 1/8, THE SCALAR PARTICLE IS REPELLED

SELF-FORCE IS SINGULAR WHEN ξ GOES TO 1/2

2

WE FOUND GENERAL EXPRESSION FOR THE SELF-

FORCE.

THANK YOU FOR ATTENTION.

3

some astrophysical aspects of the wormhole spacetime valdir b

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