INTRODUCTION TO WORMHOLES
TAKASHI OKAMOTO
[email protected]
Contents
1. Einstein-Rosen Bridge
1.1. Neutral Bridge: The Schwarzschild Solution
1.2. Quasicharged Bridge: The Reissner-Nordström Geometry
1.3. General Bridge Construction
2. Causality Problem
2.1. Topology of Einstein-Rosen Bridge
2.2. Dynamics of the Schwarzschild Throat
2.3. Causality Preserved
2.4. Crossing Bridges
3. Traversable Wormholes
3.1. Criteria for Construction
3.2. Morris and Thorne (1988)
3.3. Weak Energy Condition
3.4. Minimize Exotic Material
3.5. Tension, Stability and Assembly
4. Conclusion
References
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1. Einstein-Rosen Bridge
In 1935 [1], Einstein and Rosen investigated the possibility of obtaining an atomistic theory of matter and electricity which would exclude singularities, and use no
other variables but gµν from general relativity and ϕµ from Maxwell theory. Their
calculations led to representing a particle as a “bridge” connecting two identical
sheets. This bridge is know as the Einstein-Rosen Bridge.
1.1. Neutral Bridge: The Schwarzschild Solution. Consider the Schwarzschild
solution:
1
dr2 − r2 (dθ2 + sin2 θdφ2 )
(1.1)
ds2 = (1 − 2m/r)dt2 −
1 − 2m/r
where r > 2m, θ from 0 to π, and φ from 0 to 2π. Since g11 becomes infinite1 at
r = 2m, we introduce a new variable defined as
(1.2)
u2 = r − 2m
1At this time, physical and coordinate singularities were not distinguished clearly by many
physicists. Singularity was a singularity.
1
2
TAKASHI OKAMOTO [email protected]
Replacing r = u2 − 2m into (1.1) we obtain a new expression for ds2
(1.3)
ds2 =
u2
u2
dt2 − 4(u2 + 2m)du2 − (u2 + 2m)2 (dθ2 + sin2 θdφ2 )
+ 2m
where u varies from −∞ to +∞, thus r varies from −∞ to 2m, and again from
2m to +∞; whereby discarding the region of curvature singularity, r ∈ [0, 2m).
This leads us to an interpretation of the four-dimensional space as two identical
“sheets” corresponding to the asymptotically flat regions around u = ±∞ which
are connected by a “bridge” at u = 0. We can determine this spatially finite bridge.
Taking u as a constant, the area is given as A(u) = 4π(2m + u2 )2 . Obviously,
the minimum area occurs at u = 0, and the area of this “throat” is given as
A(0) = 4π(2m)2 . The region near u = 0 is known as the “wormhole”. We also note
that for this bridge construction we must take m > 0, as if we have assumed m < 0,
our bridge construction will fail since we require the existence of a horizon for this
coordinate transformation to work. Einstein and Rosen concluded that this bridge
characterizes an electrically neutral elementary particle (eg. neutron or neutrino),
and says that particles with negative energy cannot be described as a bridge.
1.2. Quasicharged Bridge: The Reissner-Nordström Geometry. Similar
to the neutral bridge, we can construct a quasicharged Einstein-Rosen bridge. We
have the Reissner-Nordström Geometry in Schwarzschild coordinates
(1.4) ds2 = (1 − 2m/r + Q2 /r2 )dt2 −
1
dr2 − r2 (dθ2 + sin2 θdφ2 )
1 − 2m/r + Q2 /r2
Now, in order for bridge construction, Einstein and Rosen needed to force the
electromagnetic stress-energy tensor
Tik = 14 gik ϕαβ ϕαβ − ϕiα ϕkα
(1.5)
to be negative. We shall see the reason why once we consider the case where m = 0.
We now obtain with this modified geometry
(1.6) ds2 = (1 − 2m/r − 2 /r2 )dt2 −
1
dr2 − r2 (dθ2 + sin2 θdφ2 )
1 − 2m/r − 2 /r2
where is the electric charge. We will set2 m = 0. We get
(1.7)
ds2 = (1 − 2 /r2 )dt2 −
1
dr2 − r2 (dθ2 + sin2 θdφ2 )
1 − 2 /r2
Similar to the previous example, we introduce a new variable defined as
u2 = r2 − 2 /2
(1.8)
Substituting (1.8) into (1.7) we get
(1.9)
ds2 =
2u2
dt2 − du2 − (u2 + 2 /2)(dθ2 + sin2 θdφ2 )
2u2 + 2
This bridge represents an elementary particle without mass.
2Observe here that m is not determined by , and that m and are independent constants of
integration.
INTRODUCTION TO WORMHOLES
3
1.3. General Bridge Construction. We can now generalize this bridge construction. Following Visser [2] we start with a general solution3
(1.10)
ds2 = e−ϕ(r) [1 − b(r)/r]dt2 −
1
dr2 − r2 (dθ2 + sin2 θdφ2 )
1 − b(r)/r
Now the horizon is defined by b(r = rH ) = rH and we introduce
u 2 = r − rH
(1.11)
Substituting (1.11) into (1.10), we arrive with the general result
ds2 = e−ϕ(rH +u
2
) rH
+ u2 − b(rH + u2 ) 2
rH + u 2
dt
−
4
u2 du2
rH + u 2
rH + u2 − b(rH + u2 )
−(rH + u2 )2 (dθ2 + sin2 θdφ2 )
(1.12)
Near u = 0 is the bridge connecting the asymptotically flat regions u = ±∞. Near
the bridge, one has r ≈ rH and u ≈ 0 and we get
(1.13)
ds2 ≈ e−ϕ(rH )
u2 [1 − b0 (rH )] 2
rH + u 2
dt − 4
du2 − (rH + u2 )2 (dθ2 + sin2 θdφ2 )
rH
1 − b0 (rH )
Introducing constants A and B, we can rewrite this as
(1.14)
ds2 ≈ A2 u2 dt2 − 4B 2 (rH + u2 )du2 − (rH + u2 )2 (dθ2 + sin2 θdφ2 )
and we can see that this is in the similar form as the neutral and quasistatic bridges.
2. Causality Problem
2.1. Topology of Einstein-Rosen Bridge. Let’s go back to our Schwarzschild
wormhole (neutral Einstein-Rosen bridge). If we take t = v = 0 and θ = π/2, the
surface is defined by the paraboloid of revolution
(2.1)
r = 2M + z 2 /8M
as shown here in figure 1.
Since the Einstein field equations are purely local in character, they tell us nothing
about the preferred topology of the space. We could introduce a multiply connected
space which connects two distant regions of the same asymptotically flat universe,
as shown in figure 2.
This multiply connected universe introduces an issue with causality. There are
essentially two paths to get from a to B. One can either take a path going through
the wormhole or not. Suppose a disturbance travels at the speed of light from
a to B. This disturbance can be outpaced by another disturbance that took the
wormhole route, travelling a much shorter path. It seems that causality is violated,
but Fuller and Wheeler[4] have shown that causality is preserved.
3Visser uses (-,+,+,+), but to keep consistent with Einstein and Rosen, I will be using (+,-,-,-)
4
TAKASHI OKAMOTO [email protected]
Figure 1. The Schwarzschild space geometry at t = v = 0 and
θ = π/2, illustrates the Einstein-Rosen Bridge connecting two
asymptotically flat universes (the “inter-universe” wormhole). (Reproduced from Misner, Thorne and Wheeler [3, fig.31.5a].)
Figure 2. Einstein-Rosen Bridge connecting two distant regions
of a single asymptotically flat universe (the “intra-universe” wormhole). This is described by the same solution (equivalently satisfies
Einstein’s field equations) as in figure 1, but is topologically different. (Reproduced from Misner, Thorne and Wheeler [3, fig.31.5b].)
2.2. Dynamics of the Schwarzschild Throat. When we began the construction
of our Schwarzschild wormhole, we started with the Schwarzschild solution which is
static, with a finite throat with circumference of 2πm. This is true in the region far
away from the throat, since the Schwarzschild solution carries no time dependence.
Can we say that it is the same for the regions close to the Schwarzschild throat?
No! It was argued by Fuller and Wheeler that the Schwarzschild throat is dynamic,
that the throat opens and closes like the shutter of a camera. This “pinch off” of
INTRODUCTION TO WORMHOLES
5
the throat as they called it, happens so fast that even a particle travelling at the
speed of light cannot get through the wormhole. The light will be pinched off and
trapped in a region of infinite curvature when the throat closes. This is illustrated
in figure 3.
We usually think the static time translation, t → t+∆t, leaves the Schwarzschild
geometry unchanged. This is true when we deal with a problem in regions I and
III of the Kruskal diagram. This is not true for r < 2m, since in regions II and
IV, t → t + ∆t is a spacelike motion and not a timelike motion. Thus a surface
t = constant connecting region I through u = v = 0 to region III is not static (see
figure 1). This surface will begin to change just as it moves in the +v direction as
it enters region II.
We can see from figure 3 that the system begins at A (region IV in terms of
Kruskal diagram) in a pinched off state and as you move up the v coordinate, the
throat opens and reaches a maximum point at D. Finally, the process is reversed
and at G (region II in terms of Kruskal diagram) another pinch off results. For
regions near the throat (u ≈ 0), we have r ≈ 2m.
Figure 3. The dynamical evolution of the Schwarzschild wormhole. For each spacelike slice from the left diagram, corresponding paraboloid is shown on the right. (Reproduced from Misner,
Thorne and Wheeler [3].)
2.3. Causality Preserved. We should now investigate whether a photon (since
it is the fastest particle) will be able to go through the Schwarzschild wormhole
before it pinches off. Fuller and Wheeler [4] provides a full quantitative argument,
6
TAKASHI OKAMOTO [email protected]
but one can be easily convinced that the photon will not be able to pass through a
Schwarzschild wormhole with a qualitative argument aided by the Kruskal diagram.
Figure 4 shows null cones for a particle in region I, II and IV. Timelike particles
are constrained to follow a straight line within 45◦ to the vertical. So it is easily
seen that a particle in region I or III can never crossover to the other side. So
a particle in region I will never be able to crossover to region IV, since it would
require speeds faster than that of light. Also, as soon as it crosses over to region
II, the particle is trapped forever and approaches the singularity.
v
r=0
II
III
u
I
IV
r=0
Figure 4. It is impossible for a timelike particle in region I to
ever cross over to region III. A particle in region II will at some
point hit the singularity.
2.4. Crossing Bridges. When you think about what it means to cross an EinsteinRosen bridge, your ultimate fate is easily described by Visser [2, p.47]
If you discover an Einstein-Rosen bridge, do not attempt to cross it,
you will die. You will die just as surely as by jumping into a black
hole. You will die because you are jumping into a black hole. The
Einstein-Rosen coordinate u is a bad coordinate at the horizon.
Attempting to cross the horizon, say from u = + to u = −,
will force one off the u coordinate patch and into the curvature
singularity.
So stay away from Einstein-Rosen bridges.
3. Traversable Wormholes
From the last section, we saw that nothing can go through the Einstein-Rosen
bridge. They are not traversable since
INTRODUCTION TO WORMHOLES
7
(1) Tidal gravitational forces at the throat are great. Traveller is killed unless
wormhole’s mass exceeds 104 M so the throat circumference will exceed
105 km.
(2) Schwarzschild wormhole is not static but dynamic. As time pass, the throat
starts from zero circumference to a maximum circumference and back again
to zero. This happens so fast that even light will be trapped.
So they are not much fun. We can ask ourselves whether or not traversable4 wormholes exist.
3.1. Criteria for Construction. We should first begin by discussing the criteria
for construction of traversable wormholes (listed in Box 1).
Box 1. Traversable Wormhole Construction Criteria
(1)
Metric should be both spherically symmetric
and static. This is just to keep everything simple.
(2)
Solution must everywhere obey the Einstein
field equations. This assumes correctness of GR.
(3)
Solution must have a throat that connects two
asymptotically flat regions of spacetime.
(4) No horizon, since a horizon will prevent
two-way travel through the wormhole.
(5)
Tidal gravitational forces experienced by a
traveler must be bearably small.
(6)
Traveler must be able to cross through the
wormhole in a finite and reasonably small proper
time.
(7)
Physically reasonable stress-energy tensor
generated by the matter and fields.
(8)
Solution must be stable under small perturbation.
(9)
Should be possible to assemble the wormhole.
ie. assembly should require both much less than the
total mass of the universe and much less than the
age of the universe.
4We use traversable wormhole to mean that a human (or some similar alien) in their spaceship
could safely travel through the wormhole in a reasonable amount of time and return.
8
TAKASHI OKAMOTO [email protected]
Our construction of the wormhole should at least satisfy criteria (1) to (4). Morris
and Thorne [5, pp.399-400] calls this the “basic wormhole criteria”. (5) to (7) are
called “usability criteria” since it deals with human physiological comfort. Thus we
need to find a solution that will satisfy the basic wormhole criteria, then we tune
the parameters of the usability criteria to suit our needs. We will take the simple
approach of Morris and Thorne [5].
3.2. Morris and Thorne (1988). Morris and Thorne simplified their analysis
by first assuming the existence of a suitably well-behaved geometry. Associated
Riemann tensor components are calculated and Einstein field equations are used to
determine the distribution of the stress-energy. Then they ask whether or not this
disturibution of stress-energy is physically reasonable or not.
3.2.1. The Metric. To keep simple, we will assume the traversable wormhole to
be time independent, nonrotating, and spherically symmetric bridges between two
universes. Thus our manifold should be a static spherically symmetric spacetime
possessing two asymptotically flat regions. We start with
(3.1)
ds2 = e2Φ(l) dt2 − dl2 − r2 (l)[dθ2 + sin2 θdφ2 ]
where l is our proper radial distance. Some key features are listed.
• l ∈ (−∞, +∞)
• Assumed absence of event horizons → Φ(l) must be everywhere finite.
• Asymptotically flat regions at l ≈ ±∞.
• For spatial geometry to tend to an appropriate asymptotically flat limit,
we impose
(3.2)
lim {r(l)/|l|} = 1
l→±∞
or r(l) = |l| + O(1).
• For spacetime geometry to tend to an appropriate asymptotically flat limit
(3.3)
lim {Φ(l)} = Φ±
l→±∞
must be finite.
• Radius of the wormhole throat defined by
(3.4)
r0 = min{r(l)}.
To simplify, we assume there is only one such minimum and it occurs at
l = 0.
• Metric components are at least twice differentiable by l.
We could use this to calculate the Riemann, Ricci and Einstein tensors using this
coordinate system, but it is much easier to use Schwarzschild coordinates. We write
in (t, r, θ, φ)
(3.5)
ds2 = e2Φ± (r) dt2 −
dr2
− r2 [dθ2 + sin2 θdφ2 ]
1 − b± (r)/r
where we introduced b(r) called the “shape” function since it determines the spatial
shape of the wormhole, and Φ(r) called the “redshift” function since it determines
the gravitational redshift. Some key features are
INTRODUCTION TO WORMHOLES
9
• Spatial coordinate r has a geometrical significance. The throat circumference is 2πr and so r is equal to the embedding-space radial coordinate of
figure 1. Also, r decreases from +∞ to some minimum radius r0 as one
moves through the lower universe of figure 1, then increases from r0 to +∞
moving out of the throat and into the upper universe.
• For convenience, demand t coordinate to be continuous across the throat,
so that Φ+ (r0 ) = Φ− (r0 ).
• l is related the r coordinate by
Z
r
l(r) = ±
(3.6)
r0
dr0
p
1 − b± (r0 )/r0
• For spatial geometry to tend to an appropriate asymptotically flat limit,
we require both limits
lim {b± (r)} = b±
(3.7)
r→∞
to be finite.
• For spacetime geometry to tend to an appropriate asymptotically flat limit,
we require both limits
lim {Φ± (r)} = Φ±
(3.8)
r→∞
to be finite.
• Since dr/dl = 0 at the throat (throat is at minimum of r(l)), we have
dl/dr → ∞. Since
dl
1
= ±p
,
dr
1 − b± (r)/r
(3.9)
this implies b± (r) = r0 at the throat.
• Metric components should be at least twice differentiable with r.
• We can simplify things and assume symmetry under interchange of asymptotically flat regions, ± ↔ ∓ or b+ (r) = b− (r) and Φ+ (r) = Φ− (r). This is
not a requirement, just for convenience.
3.2.2. Tensor Calculations. Now, using our standard formulas5, we can compute
the Christoffel symbols and the Riemann curvature tensor. There are 24 nonzero
5Remember
Γα
βγ
=
1
2
g αλ (gλβ,γ + gλγ,β − gβγ,λ )
α
Rβγδ
=
α
α
λ
α
λ
Γα
βδ,γ − Γβγ,δ + Γλγ Γβδ − Γλδ Γβγ
10
TAKASHI OKAMOTO [email protected]
components. Quoting results from [5, p.400]
(3.10)

t
Rrtr














t

Rθtθ








t

Rφtφ









r

Rθrθ













r

Rφrφ













θ


Rφθφ



t
r
= −Rrrt
= −(1 − b/r)−1 e−2Φ Rttr
r
= (1 − b/r)−1 e−2Φ Rtrt
= Φ,rr − (b,r r − b)[2r(r − b)]−1 Φ,r + (Φ,r )2 ,
t
θ
θ
= Rθθt
= −r2 e−2Φ Rttθ
= r2 e−2Φ Rtθt
= rΦ,r (1 − b/r),
φ
t
= −Rφφt
= −r2 e−2Φ sin2 θRttφ
φ
= r2 e−2Φ sin2 θRtφt
= rΦ,r (1 − b/r) sin2 θ,
r
θ
= −Rθθr
= r2 (1 − b/r)Rrrθ
θ
= −r2 (1 − b/r)Rrθr
= −(b,r r − b)/2r,
φ
r
= −Rφφr
= r2 (1 − b/r) sin2 θRrrφ
φ
2
2
= −r (1 − b/r) sin θRrφr
= −(b,r r − b) sin2 θ/2r,
φ
φ
θ
= sin2 θRθθφ
= −Rφφθ
= − sin2 θRθφθ
= −(b/r) sin2 θ,
where basis vectors being used are those (et , er , eθ , eφ ). We want to rather be in
the rest frame (ie. r, θ, φ constant) which are related,
(
et̂ = e−Φ et , er̂ = (1 − b/r)1/2 er ,
eθ̂ = r−1 eθ , eφ̂ = (r sin θ)−1 eφ .
(3.11)
This makes the metric Minkowski,

1
0
0
0
 0 −1
0
0 
,
≡
 0
0 −1
0 
0
0
0 −1

(3.12)
gα̂β̂ = eα̂ · eβ̂ = ηα̂β̂
and the nonzero components of the Riemann tensor are,
(3.13)
 t̂
Rr̂t̂r̂










Rt̂


 θ̂t̂ t̂θ̂
Rφ̂t̂φ̂


Rr̂

θ̂r̂ θ̂


 r̂

R


 φ̂r̂φ̂

Rθ̂
φ̂θ̂ φ̂
t̂
r̂
= −Rr̂r̂
= Rt̂r̂t̂r̂ = −Rt̂r̂
t̂
t̂
= (1 − b/r){Φ,rr − (b,r r − b)[2r(r − b)]−1 Φ,r + (Φ,r )2 },
= −Rθ̂t̂ θ̂t̂ = Rt̂θ̂t̂θ̂ = −Rt̂θ̂θ̂t̂ = (1 − b/r)Φ,r /r,
= −Rφ̂t̂ φ̂t̂ = Rt̂φ̂t̂φ̂ = −Rt̂φ̂φ̂t̂ = (1 − b/r)Φ,r /r,
θ̂
= −Rθ̂r̂θ̂r̂ = Rr̂θ̂θ̂r̂ = −Rr̂r̂
= −(b,r r − b)/2r3 ,
θ̂
φ̂
= −Rφ̂r̂ φ̂r̂ = Rr̂φ̂φ̂r̂ = −Rr̂r̂
= −(b,r r − b)/2r3 ,
φ̂
= −Rφ̂θ̂ φ̂θ̂ = Rθ̂φ̂φ̂θ̂ = −Rθ̂φ̂θ̂φ̂ = −b/r3 .
INTRODUCTION TO WORMHOLES
11
Finally, we contract and find the Ricci tensor, curvature scalar and solve the Einstein field equations. Our nonzero Einstein tensor components are

2

Gt̂t̂ = b,r /r ,


3

+
2(1 − b/r)Φ,r /r,
Gr̂r̂ = −b/r
(3.14)
b,r r − b
Φ,r
b,r r − b
b
2

Φ
−
Φ
+
(Φ
)
+
−
G
=
1
−
,rr
,r
,r

 θ̂θ̂
r
2r(r − b)
r
2r2 (r − b)



= Gφ̂φ̂ .
Non-vanishing stress-energy tensor components should be the same non-vanishing
components as the Einstein tensor. We denote the following:


Tt̂t̂ = ρ(r),
(3.15)
Tr̂r̂ = −τ (r),


Tθ̂θ̂ = Tφ̂φ̂ = p(r),
where ρ(r) is the total mass-energy density, τ (r) is the radial tension per unit area,
and p(r) is the pressure in the lateral direction. Now we use,
(3.16)
Gα̂β̂ = 8πGTα̂β̂
and equating the results of (3.14) and (3.15),
(3.17)
b,r
=
8πGr2 ρ,
(3.18)
(3.19)
Φ,r
τ,r
=
=
(−8πGτ r3 + b)/[2r(r − b)],
(ρ − τ )Φ,r − 2(p + τ )/r.
What we have here are five unknown functions of r : b, Φ, ρ, τ and p. But if we go
back to our original plan, we wanted to be able to “tweak” some parameters so that
we can get a resonable result for the stress-energy. Since we will be “tweaking” the
shape function b(r) and redshift function Φ(r), we rewrite the previous equations
as:
(3.20)
ρ = b,r /[8πGr2 ],
(3.21)
(3.22)
τ
p
=
=
[b/r − 2(r − b)Φ,r ]/[8πGr2 ],
(r/2)[(ρ − τ )Φ,r − τ,r ] − τ.
In this form, by choosing a suitable b(r) and Φ(r), we will be able to solve for ρ
and τ . Then with that we finally determine p.
3.2.3. Stress-Energy at the Throat. From (3.9) we have the condition, r = b = b0
at the throat. This also implies (r − b)Φ,r → 0 at the throat and thus using (3.21)
we have
2
1
11 dyn 1light yr.
(3.23)
τ0 ≡ (tension in the throat) =
∼ 5 × 10
,
8πGb20
cm2
b0
2
which is huge. For b0 ∼ 3km, τ0 ∼ 1037 dyn/cm which is equivalent to the pressure
at the center of the most massive neutron star. Taking (3.9) and inverting, we get
r
dr
b
(3.24)
=± 1−
dl
r
12
TAKASHI OKAMOTO [email protected]
and since,
d2 r
dr d
=
dl2
dl dr
(3.25)
dr
dl
=
1 d
2 dr
dr
dl
2
,
we have
1
d2 r
=
2
dl
2r
(3.26)
Now, at the throat
(3.27)
b
− b,r .
r
d2 r
> 0 since r(l) is a minimum at the throat. So
dl2
d2 r 1
=
[1 − b,r (r0 )] ⇒ b,r (r0 ) < 1.
dl2 r0
2r0
Using this and (3.20) at the throat,
(3.28)
ρ(r0 ) ≡ ρ0 <
1
8πGr02
τ (r0 ) ≡ τ0 =
1
8πGr02
and from (3.21)
(3.29)
combining (3.28) and (3.29) implies
(3.30)
ρ0 < τ 0 .
So this is where we run into trouble. ρ0 < τ0 says that at the throat, the tension
exceeds the total mass-energy density. Materials with the property τ > ρ > 0
is called, “exotic”. This makes things troublesome because it forces an observer
moving through the throat with radial veolcity ∼ c see their stress-energy tensor
(in basis vector eô0 = γet̂ ∓ γ(v/c)er̂ ) [5, p.405]
Tô0 ô0
(3.31)
= γ 2 Tt̂t̂ ∓ 2γ 2 (v/c)2 Tt̂r̂ + γ 2 (v/c)2 Tr̂r̂
= γ 2 [ρ0 − (v/c)2 τ0 ] = γ 2 (ρ0 − τ0 ) + τ0
for sufficiently large γ, to have negative density of mass-energy.
3.3. Weak Energy Condition. Negative density of mass-energy is a direct violation of the weak energy condition (WEC). The weak energy condition states that
for any timelike vector
(3.32)
WEC ⇐⇒ Tµν V µ V ν ≥ 0.
Physically, this implies that the weak energy condition forces the local energy density to be positive measured by any timelike observer. In terms of principal pressures,
(3.33)
WEC ⇐⇒ ρ ≥ 0 and ∀j, ρ + ρj ≥ 0.
So clearly this condition is violated by the result we obtained previously (τ > ρ).
So we may investigate whether this violation can occur or not. At least we can see
some examples of observing this violation, due to quantum effects. An example of
energy condition violation is the Casimir effect.
INTRODUCTION TO WORMHOLES
13
3.3.1. Casimir Effect. 6 With two parallel conducting plates separated by a small
distance a, the wave vector is constrained by
nπ
(3.34)
kz =
.
a
By symmetry, the stress-energy can depend only on the spacetime metric η µν ,
normal vector ẑ µ and the separation a. So introducing two dimensionless functions
f1 (z/a) and f2 (z/a) we can write by dimensional analysis
~
[f1 (z/a)η µν + f2 (z/a)ẑ µ ẑ ν .
a4
The electromagnetic field is conformally invariant, ie.
(3.35)
(3.36)
µν
TCasimir
≡
µν
T ≡ TCasimir
η µν = 0.
With this we find the relationship between f1 and f2 , and it can be shown that
(3.37)
µν
T ≡ TCasimir
=
π 2 ~ µν
(η − 4ẑ µ ẑ ν ).
720 a4
We can observe that our energy density is thus negative ρ = −(π 2 ~)/(720a4 ),
violating our energy condition. Similar violations can be seen with Topological
Casimir Effect, Squeezed Vacuum and Particle Creation. [2, pp.125-126].
3.4. Minimize Exotic Material. Since exotic materials are so troublesome, one
may want to minimize the use of it. The amount of exotic material is quantified by
a dimensionless function ζ(r) = (τ − ρ)/ρ. We have the following scenarios.
(1) Use exotic material throughout the wormhole, but make the density of
exotic material fall off rapidly with radius as one moves away from the
throat. An example of this is to take b = const and Φ = 0. This yields
(3.38)
ρ(r) =
0,
(3.39)
τ (r)
(3.40)
(3.41)
p(r) = b0 /(16πGr3 ),
ζ = ∞.
= b0 /(8πGr3 ),
This is unattractive since it has huge ζ but the density drops with r.
(2) Use exotic material as the only source of curvature, but have it cut off
completely at some radius Rs . So
(3.42)
(3.43)
ζ>0
ρ=τ =p=0
for r < Rs ,
for r > Rs .
But there is a more effective way than this.
(3) Confine the exotic material to a tiny region (−lc < l < +lc ) centered at the
throat. Around this region should be surrounded with normal matter. We
then have
(3.44)
(3.45)
ζ>0
ζ≤0
for |l| < lc ,
for |l| ≥ lc .
6For a more indepth description, please consult other texts.
14
TAKASHI OKAMOTO [email protected]
3.5. Tension, Stability and Assembly. Earlier we said that a traversable wormhole should be safe for a traveller to go through. But it seems very uncomfortable
for someone to go through a throat that experiences torque equivalent to that of a
neutron star core (§3.2.3). Two workarounds are suggested.
(1) Build a long vacuum tube (diameter b0 ) through the throat and have
the stresses of the tube wall to hold the exotic matter out. This breaks the
spherical symmetry of our solution, but even before that good luck trying
to find the tube material!
(2) Hope that the exotic material couples very weakly (like neutrinos) to the
traveller. Then even with the high stress and density, the traveller can go
through the throat without noticing much effect.
We cannot talk too much about stability of the wormhole, since this relies heavily
on the behavior of the exotic material. Whether naturally stable or unstable, there
could be ways to stabilize the wormhole, but again without knowing the behavior
of the exotic material, it is hard to analyze.
Finally, the actual assembly relies on topology change. This will probably need to
be addressed after gravity has been properly quantized. This may be understood by
taking a quantum mechanical picture of spacetime, like that of the spacetime foam
introduced by Wheeler (1955) [7]. At Plank-Wheeler length lp−w ∼ 1.6 × 10−33 cm,
quantum effects can give rise to foam like multiply connected spacetime.
4. Conclusion
The idea of a wormhole has come from an attempt to form an atomistic model of
GR to the idea of traversable wormhole that connects two points from two different
universes, or universe on its own. But the reality of such, comes with serious
problems that cannot be proven (or disproven) at the present time. Quantum
gravity seems to be what can attempt to give us real evidence of the (non)reality
of traversable wormholes. Until gravity is quantized, we’ll just have to wait.
References
1. A. Einstein and N. Rosen, Phys. Rev. 48, 73 (1935).
2. M. Visser, Lorentzian Wormholes: From Einstein to Hawking. AIP, New York, 1996.
3. C. W. Misner, K. S. Thorne, and J. A. Wheeler. Gravitation. W. H. Freeman and Company,
San Francisco, 1973.
4. R. W. Fuller and J. A. Wheeler, Phys. Rev. 128, 919 (1962).
5. M. S. Morris and K. S. Thorne, Am. J. Phys. 56, 395 (1988).
6. M. S. Morris, K. S. Thorne, and U. Yurtsever, Phys. Rev. Lett. 61, 1446 (1988).
7. J. A. Wheeler, Phys. Rev. 97, 511 (1955).
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