Stable phantom energy
traversable wormhole models
Albert Einstein Century International Conference
Palais de l’Unesco, Paris, France
18 - 22 July 2005
Francisco S. N. Lobo
Centro de Astronomia e Astrofı́sica da Universidade de Lisboa,
Campo Grande, Ed. C8 1749-016 Lisboa, Portugal,
e-mail: [email protected]
Abstract
It has been suggested that a possible candidate for the present accelerated expansion of the Universe is “phantom energy”. The latter possesses an equation of state of
the form ω ≡ p/ρ < −1, consequently violating the null energy condition. As this is
the fundamental ingredient to sustain traversable wormholes, this cosmic fluid presents
us with a natural scenario for the existence of these exotic geometries. In this context,
we investigate the physical properties and characteristics of traversable wormholes constructed using the phantom energy equation of state. We analyze specific wormhole
geometries, considering asymptotically flat spacetimes and imposing an isotropic pressure. Using the “volume integral quantifier” we verify that it is theoretically possible
to construct these geometries with vanishing amounts of averaged null energy condition violating phantom energy. The stability analysis of these phantom wormholes
to linearized spherically symmetric perturbations about static equilibrium solutions is
also carried out. A master equation dictating the stability regions is deduced, and by
separating the cases of a positive and a negative surface energy density, it is found
that the respective stable equilibrium configurations may be increased by strategically
varying the wormhole throat radius. These phantom energy traversable wormholes
have far-reaching physical and cosmological implications. For instance, an advanced
civilization may use these geometries to induce closed timelike curves, consequently
violating causality.
1
1
Wormholes comprising of phantom energy
1.1
Spacetime metric
The spherically symmetric and static wormhole spacetime metric is given by:
ds2 = −e2Φ(r) dt2 +
dr2
+ r2 (dθ2 + sin2 θ dφ2 ) ,
1 − b(r)/r
(1)
Φ(r) is denoted as the redshift function, for it is related to the gravitational redshift
b(r) is called the form function, as it determines the shape of the wormhole.
For the wormhole to be traversable, one must demand that there are no horizons present,
which are identified as the surfaces with e2Φ → 0, so that Φ(r) must be finite everywhere.
1.2
Embedding diagrams
The surface equation of the embedding is given by
Ã
!−1/2
r
dz
=±
−1
dr
b(r)
.
(2)
Throat ≡ minimum radius, r = b(r) = r0 ; embedded surface is vertical, i.e., dz/dr → ∞.
Far from the throat consider that space is asymptotically flat, dz/dr → 0 as r → ∞.
To be a solution of a wormhole, one needs to impose the flaring-out condition
d2 r
b − b0 r
=
> 0.
dz 2
2b2
(3)
At the throat we verify that the form function satisfies the condition b0 (r0 ) < 1.
This condition plays a fundamental role in the violation of the energy conditions.
1.3
Field equations
The field equations provide the following relationships
"
Ã
!
#
1
b
b Φ0
1 b0
,
p
(r)
=
−
+
2
1
−
,
ρ(r) =
r
8π r2
8π
r3
r r
Ã
!"
#
1
b
b0 r − b
b0 r − b
Φ0
00
0 2
0
p(r) =
1−
Φ + (Φ ) − 2
Φ − 3
+
,
8π
r
2r (1 − b/r)
2r (1 − b/r)
r
ρ(r) ≡ energy density; pr (r) ≡ radial pressure; pt (r) ≡ transverse pressure.
2
(4)
(5)
1.4
Violation of the energy conditions
A fundamental ingredient of traversable wormholes and phantom energy is the violation of
the null energy condition (NEC), which is defined as Tµν k µ k ν ≥ 0, where k µ is any null
vector and the Tµν the stress-energy tensor.
1
Tµ̂ν̂ k k = pr (r) + ρ(r) =
8π
µ̂ ν̂
"
Ã
b0 r − b
b
+2 1−
3
r
r
!
#
Φ0
.
r
(6)
Flaring out condition at r0 , and the finite character of Φ(r), we have Tµ̂ν̂ k µ̂ k ν̂ < 0.
Matter that violates the NEC is denoted as exotic matter.
One may prove that the averaged null energy condition is also violated:
s
Z +∞
1
b
1
I = Tµ̂ν̂ k µ̂ k ν̂ dλ = −
e−φ 1 − dr < 0.
2
4π r0 r
r
Γ
Z
1.5
(7)
Phantom energy
[Lobo, PRD 71 084011 (2005); PRD 71 124022 (2005)].
Phantom energy, governed by the equation of state ω = p/ρ with ω < −1, a candidate
for the present accelerated expansion of the Universe, the NEC is violated, i.e., p + ρ < 0.
Despite the fact that the notion of phantom energy is that of a homogeneously distributed
fluid in the Universe, it can be extended to inhomogeneous spherically symmetric spacetimes
by regarding that the pressure in the equation of state p = ωρ is now a negative radial
pressure, and noting that the transverse pressure pt may be determined from the Einstein
field equations.
Now using pr = ωρ with ω < −1, and eqs. (4), we have the following condition
Φ0 (r) =
b + ωrb0
.
2r2 (1 − b/r)
(8)
We now have four equations, namely the field equations, i.e., eqs. (4)-(5), and Eq. (8), with
five unknown functions of r, i.e., ρ(r), pr (r), pt (r), b(r) and Φ(r). To construct solutions
with the properties and characteristics of wormholes, we consider restricted choices for b(r)
and/or Φ(r).
In cosmology the energy density related to the phantom energy is considered positive,
ρ > 0, so we shall maintain this condition. This implies that b0 (r) > 0.
The conditions b0 (r0 ) < 1 and 1 − b/r > 0 are imposed to have wormhole solutions.
3
2
Specific phantom wormhole models
2.1
Asymptotically flat spacetimes
Consider :
Φ(r) = const
b(r) = r0 (r/r0 )−1/ω ,
⇒
(9)
(1+ω)/ω
so that b(r)/r = (r0 /r)
→ 0 for r → ∞, i.e., asymptotically flat spacetime.
The stress-energy tensor components are given by
1
pr (r) = ωρ(r) = −
8πr02
2.1.1
µ
¶3+ 1
r0
r
ω
1
pt (r) =
16πr02
,
µ
1+ω
ω
¶µ
r0
r
¶3+ 1
ω
(10)
Infinitesimal amounts of phantom energy support traversable wormholes
Using the “volume integral quantifier” one may quantify the “total amount” ofRenergy condition Rviolating matter. This notion amounts to calculating the definite integrals Tµν U µ U ν dV
and Tµν k µ k ν dV , and the amount of violation is defined as the extent to which these integrals become negative.
The “total amount” of ANEC violating matter in the spacetime is given by
Z
IV =
"
Ã
e2Φ
[ρ(r) + pr (r)] dV = (r − b) ln
1 − b/r
! #∞
−
Z ∞
r0
r0
"
Ã
e2Φ
(1 − b ) ln
1 − b/r
!#
0
dr .(11)
Wormhole field deviating from the Schwarzschild solution from r0 out to a radius a.
Considering Eqs. (9), for the particular case of ω = −2, Eq. (11) takes the form
Ã
s
IV = r0 1 −
a
r0
!
r
µ
+a 1−
r0
a
¶ ·
r
µ
ln 1 −
R
r0
a
¶¸
.
(12)
Taking the limit a → r0 , one verifies that IV = (ρ + pr ) dV → 0.
Traversable wormhole with infinitesimal quantities of ANEC violating phantom energy.
2.2
Isotropic pressure, pr = pt = p
µ
b(r) = r0 (r/r0 )−1/ω
Consider :
⇒
Φ(r) =
3ω + 1
1+ω
¶
µ
ln
r
r0
¶
,
(13)
The stress-energy tensor components are provided by
µ
¶
1
1
r0 3+ ω
p(r) = ωρ(r) = −
.
(14)
8πr02 r
The spacetime is not asymptotically flat.
Using the “volume integral quantifier”, Eq. (11), with a cut-off of the stress-energy at a,
and considering ω = −2, the volume integral takes the following value
µ
IV = a 1 −
r
r0
a
¶

ln 
( ra0 )10
1−
q
r0
a

+
Ã
s
10a + 11r0 − 21r0
a
r0
!
µr
+a
r0
−1
a
¶

( ra0 )21/2
ln  q a
r0
−1

 .(15)
Once again taking the limit a → r0 , one verifies that IV → 0, and as before one may construct
a traversable wormhole with arbitrarily small quantities of ANEC violating phantom energy.
4
3
3.1
Stability analysis
Junction conditions
We shall model specific phantom wormholes by matching an interior traversal wormhole geometry, satisfying the equation of state pr = ωρ with ω < −1, with an exterior Schwarzschild
solution at a junction interface Σ, situated outside the event horizon, a > rb = 2M .
Using the Darmois-Israel formalism, the surface stresses are given by
s
2M
1 
1−
σ = −
+ ȧ2 −
4πa
a

s
b(a)
1−
+ ȧ2  ,
a
³

(16)
´
b
0
2
1  1 − Ma + ȧ2 + aä (1 + aΦ ) 1 − a + ȧ + aä −
q
q
−
P =
8πa
+ ȧ2
1 − 2M
1 − b(a)
+ ȧ2
a
a
ȧ2 (b−b0 a)
2(a−b)

,
(17)
σ and P are the surface energy density and the tangential surface pressure, respectively.
We shall make use of the conservation identity, given by
h
i+
2
σ 0 = − (σ + P) + Ξ
−
a
where Ξ, defined for notational convenience, is given by
i
Sj|i
= Tµν eµ(j) nν
⇒

(18)
s
1  b0 a − b
b
³
´ + aΦ0  1 − + ȧ2 .
Ξ=−
2
b
4πa 2a 1 −
a
a
(19)
Using ms = 4πa2 σ, the surface mass of the thin shell, and taking into account the radial
derivative of σ 0 , Eq. (18) can be rearranged to provide the following relationship
µ
ms
2a
¶00
= Υ − 4πσ 0 η ,
(20)
with the parameter η defined as η = P 0 /σ 0 , and Υ given by
Υ≡
4π
(σ + P) + 2πa Ξ0 .
a
(21)
Equation (20) will also play a fundamental role in determining the stability regions.
3.2
Equation of motion
Rearranging Eq. (16), we deduce the thin shell’s equation of motion, i.e., ȧ2 + V (a) = 0,
with the potential given by
µ
V (a) = F (a) −
ms
2a
¶2
µ
−
aG
ms
¶2
.
(22)
where ms = 4πσa2 is the surface mass of the thin shell, and for computational purposes and
notational convenience, we have define the following factors
F (a) = 1 −
b(a)/2 + M
a
and
5
G(a) =
M − b(a)/2
,
a
(23)
Linearizing around a stable solution situated at a0 , we consider a Taylor expansion of
V (a) around a0 to second order, given by
1
V (a) = V (a0 ) + V 0 (a0 )(a − a0 ) + V 00 (a0 )(a − a0 )2 + O[(a − a0 )3 ] .
2
(24)
Evaluated at the static solution, at a = a0 , we verify that V (a0 ) = 0 and V 0 (a0 ) = 0.
From the condition V 0 (a0 ) = 0, one extracts the following useful equilibrium relationship
µ
ms
Γ≡
2a0
¶0
µ
a0
=
ms
¶"
µ
a0 G
F −2
ms
¶µ
0
a0 G
ms
¶0 #
,
(25)
which will be used in determining the master equation, responsible for dictating the stable
equilibrium configurations.
The solution is stable if and only if V (a) has a local minimum at a0 and V 00 (a0 ) > 0 is
verified. The latter condition takes the form
¶µ
¶
µ
ms
ms 00
< Ψ − Γ2 ,
(26)
2a
2a
where Ψ is defined as
F 00
−
Ψ=
2
3.3
"µ
aG
ms
¶0 #2
µ
aG
−
ms
¶µ
aG
ms
¶00
.
(27)
The master equation
We shall use η as a parametrization of the stable equilibrium, so that there is no need to
specify a surface equation of state. Thus, using Eqs. (20) and (26), one deduces the master
equation given by
σ 0 ms η 0 > Θ ,
(28)
where η0 = η(a0 ) and Θ is defined as
´
a0 ³ 2
1
Γ −Ψ +
ms Υ .
(29)
2π
4π
Now, from the master equation we find that the stable equilibrium regions are dictated by
the following inequalities
Θ≡
η0 > Θ,
η0 < Θ,
if
if
σ 0 ms > 0 ,
σ 0 ms < 0 ,
(30)
(31)
with the definition
Θ≡
Θ
.
ms
σ0
(32)
We shall now model the phantom wormhole geometries by choosing the specific form and
redshift functions considered in Section 2, and consequently determine the stability regions
dictated by the inequalities (30)-(31). In the specific cases that follow, the explicit form
of Θ is extremely messy, so that we find it more instructive to show the stability regions
graphically.
6
3.4
Stability regions
3.4.1
Asymptotically flat spacetimes
Consider :
Φ(r) = const ,
b(r) = r0 (r/r0 )−1/ω ,
(33)
To determine the stability regions separate the cases of b(a0 ) < 2M and b(a0 ) > 2M .
(i) For b(a0 ) < 2M , i.e., for σ > 0. a0 lies in the range 2M < a0 < 2M
2.5
ω=−2
ro/M = 1
2
³
2.5
´−(1+ω)
.
ω=−2
ro/M = 0.25
2
1.5
2M
r0
1.5
η
η
1
1
Stability region
0.5
0.5
Stability region
0
2.2 2.4 2.6 2.8
3
α
3.2 3.4 3.6 3.8
4
0
4
6
8
α 10
12
14
16
Figure 1: Plots for σ > 0, i.e., b(a0 ) < 2M . We have defined α = a0 /M .
(ii) For b(a0 ) > 2M , i.e., ms (a0 ) < 0. Separate the cases of r0 < 2M and r0 > 2M .
If r0 < 2M , the range of the junction radius is given by a0 > 2M
10
8
6
8
6
4
´−(1+ω)
ω=−2
1.5
Stable
ro/M =
4
2
2
η0
η0
–2
–2
–4
–4
–6
–6
Stable
–8
–10
2M
r0
10
ω=−2
ro/M = 0.5
Stable
³
Stable
–8
10
12
14
16
α
18
20
22
24
–10
3
4
α5
6
Figure 2: Plots for σ < 0, considering r0 /M < 2. We have defined α = a0 /M .
7
7
3.4.2
Isotropic pressure, pr = pt = p
µ
Consider :
Φ(r) =
3ω + 1
1+ω
¶
ln (r/r0 )
b(r) = r0 (r/r0 )−1/ω ,
and
(34)
Separate the cases of b(a0 ) < 2M and b(a0 ) > 2M .
(i) For b(a0 ) < 2M , we have ms > 0, and the condition r0 < 2M is imposed. Thus the
stability region is given by 2M < a0 < 2M
³
2M
r0
´−(1+ω)
10
10
ω=−2
=1
8
ro/M
6
ω=−2
1.5
ro/M =
8
4
2
6
η0
η
–2
4
–4
–6
Stability region
2
Stability region
–8
–10
2.2 2.4 2.6 2.8
3
α
3.2 3.4 3.6 3.8
4
0
2.3α 2.4
2.2
2.1
2.5
2.6
Figure 3: Plots for an isotropic pressure phantom wormhole. We have defined α = a0 /M
and considered ω = −2 for both cases.
(ii) For b(a0 ) > 2M , then ms (a0 ) < 0. As before, we shall separate the cases of r0 < 2M
and r0 > 2M . For r0 < 2M , the range of a0 is given by a0 > 2M (2M/r0 )−(1+ω) .
30
100
80
20
60
40
20
η0
10
Stability
Stability
region
region
Stability
region
Stability
region
η0
–20
–10
–40
–60
ro/M =
1
ω=−2
–80
–100
5
6
7
α
8
9
ro/M =
1.5
ω=−2
–20
–30
10
3
3.5
4
4.5
α
5
5.5
6
6.5
Figure 4: Plots for an isotropic pressure phantom wormhole, for b(a0 ) > 2M and r0 < 2M .
We have defined α = a0 /M and considered ω = −2 for both cases.
8