contents · the wormhole or not. suppose a disturbance travels at the speed of light from a to b....

INTRODUCTION TO WORMHOLES

TAKASHI [email protected]

Contents

1. Einstein-Rosen Bridge 11.1. Neutral Bridge: The Schwarzschild Solution 11.2. Quasicharged Bridge: The Reissner-Nordstrom Geometry 21.3. General Bridge Construction 32. Causality Problem 32.1. Topology of Einstein-Rosen Bridge 32.2. Dynamics of the Schwarzschild Throat 42.3. Causality Preserved 52.4. Crossing Bridges 63. Traversable Wormholes 63.1. Criteria for Construction 73.2. Morris and Thorne (1988) 83.3. Weak Energy Condition 123.4. Minimize Exotic Material 133.5. Tension, Stability and Assembly 144. Conclusion 14References 14

1. Einstein-Rosen Bridge

In 1935 [1], Einstein and Rosen investigated the possibility of obtaining an atom-istic theory of matter and electricity which would exclude singularities, and use noother variables but gµν from general relativity and ϕµ from Maxwell theory. Theircalculations led to representing a particle as a “bridge” connecting two identicalsheets. This bridge is know as the Einstein-Rosen Bridge.

1.1. Neutral Bridge: The Schwarzschild Solution. Consider the Schwarzschildsolution:

(1.1) ds2 = (1− 2m/r)dt2 − 11− 2m/r

dr2 − r2(dθ2 + sin2 θdφ2)

where r > 2m, θ from 0 to π, and φ from 0 to 2π. Since g11 becomes infinite1 atr = 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 + 2mdt2 − 4(u2 + 2m)du2 − (u2 + 2m)2(dθ2 + sin2 θdφ2)

where u varies from −∞ to +∞, thus r varies from −∞ to 2m, and again from2m 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 = ±∞ whichare 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 asA(0) = 4π(2m)2. The region near u = 0 is known as the “wormhole”. We also notethat 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 thiscoordinate transformation to work. Einstein and Rosen concluded that this bridgecharacterizes 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-Nordstrom Geometry. Similarto the neutral bridge, we can construct a quasicharged Einstein-Rosen bridge. Wehave the Reissner-Nordstrom Geometry in Schwarzschild coordinates

(1.4) ds2 = (1− 2m/r + Q2/r2)dt2 − 11− 2m/r + Q2/r2


Now, in order for bridge construction, Einstein and Rosen needed to force theelectromagnetic stress-energy tensor

(1.5) Tik = 14gikϕαβϕαβ − ϕiαϕ α

k

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 − ε2/r2)dt2 − 11− 2m/r − ε2/r2


where ε is the electric charge. We will set2 m = 0. We get

(1.7) ds2 = (1− ε2/r2)dt2 − 11− ε2/r2


Similar to the previous example, we introduce a new variable defined as

(1.8) u2 = r2 − ε2/2

Substituting (1.8) into (1.7) we get

(1.9) ds2 =2u2

2u2 + ε2dt2 − du2 − (u2 + ε2/2)(dθ2 + sin2 θdφ2)

This bridge represents an elementary particle without mass.

2Observe here that m is not determined by ε, and that m and ε are independent constants of

integration.

INTRODUCTION TO WORMHOLES 3

1.3. General Bridge Construction. We can now generalize this bridge construc-tion. Following Visser [2] we start with a general solution3

(1.10) ds2 = e−ϕ(r)[1− b(r)/r]dt2 − 11− b(r)/r


Now the horizon is defined by b(r = rH) = rH and we introduce

(1.11) u2 = r − rH

Substituting (1.11) into (1.10), we arrive with the general result

ds2 = e−ϕ(rH+u2) rH + u2 − b(rH + u2)rH + u2

dt2 − 4rH + u2

rH + u2 − b(rH + u2)u2du2

−(rH + u2)2(dθ2 + sin2 θdφ2)(1.12)

Near u = 0 is the bridge connecting the asymptotically flat regions u = ±∞. Nearthe bridge, one has r ≈ rH and u ≈ 0 and we get

ds2 ≈ e−ϕ(rH) u2[1− b′(rH)]

rHdt2 − 4

rH + u2

1− b′(rH)du2 − (rH + u2)2(dθ2 + sin2 θdφ2)

(1.13)

Introducing constants A and B, we can rewrite this as

(1.14) ds2 ≈ A2u2dt2 − 4B2(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 Schwarzschildwormhole (neutral Einstein-Rosen bridge). If we take t = v = 0 and θ = π/2, thesurface is defined by the paraboloid of revolution

(2.1) r = 2M + z2/8M

as shown here in figure 1.

Since the Einstein field equations are purely local in character, they tell us nothingabout the preferred topology of the space. We could introduce a multiply connectedspace 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 areessentially two paths to get from a to B. One can either take a path going throughthe wormhole or not. Suppose a disturbance travels at the speed of light froma to B. This disturbance can be outpaced by another disturbance that took thewormhole 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 (+,-,-,-)


Figure 1. The Schwarzschild space geometry at t = v = 0 andθ = π/2, illustrates the Einstein-Rosen Bridge connecting twoasymptotically flat universes (the “inter-universe” wormhole). (Re-produced from Misner, Thorne and Wheeler [3, fig.31.5a].)

Figure 2. Einstein-Rosen Bridge connecting two distant regionsof a single asymptotically flat universe (the “intra-universe” worm-hole). This is described by the same solution (equivalently satisfiesEinstein’s field equations) as in figure 1, but is topologically differ-ent. (Reproduced from Misner, Thorne and Wheeler [3, fig.31.5b].)

2.2. Dynamics of the Schwarzschild Throat. When we began the constructionof our Schwarzschild wormhole, we started with the Schwarzschild solution which isstatic, with a finite throat with circumference of 2πm. This is true in the region faraway 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


the throat as they called it, happens so fast that even a particle travelling at thespeed of light cannot get through the wormhole. The light will be pinched off andtrapped in a region of infinite curvature when the throat closes. This is illustratedin figure 3.

We usually think the static time translation, t → t+∆t, leaves the Schwarzschildgeometry unchanged. This is true when we deal with a problem in regions I andIII of the Kruskal diagram. This is not true for r < 2m, since in regions II andIV, t → t + ∆t is a spacelike motion and not a timelike motion. Thus a surfacet = constant connecting region I through u = v = 0 to region III is not static (seefigure 1). This surface will begin to change just as it moves in the +v direction asit enters region II.

We can see from figure 3 that the system begins at A (region IV in terms ofKruskal diagram) in a pinched off state and as you move up the v coordinate, thethroat opens and reaches a maximum point at D. Finally, the process is reversedand at G (region II in terms of Kruskal diagram) another pinch off results. Forregions near the throat (u ≈ 0), we have r ≈ 2m.

Figure 3. The dynamical evolution of the Schwarzschild worm-hole. For each spacelike slice from the left diagram, correspond-ing paraboloid is shown on the right. (Reproduced from Misner,Thorne and Wheeler [3].)

2.3. Causality Preserved. We should now investigate whether a photon (sinceit is the fastest particle) will be able to go through the Schwarzschild wormholebefore it pinches off. Fuller and Wheeler [4] provides a full quantitative argument,


but one can be easily convinced that the photon will not be able to pass through aSchwarzschild 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 particlesare constrained to follow a straight line within 45◦ to the vertical. So it is easilyseen that a particle in region I or III can never crossover to the other side. Soa particle in region I will never be able to crossover to region IV, since it wouldrequire speeds faster than that of light. Also, as soon as it crosses over to regionII, the particle is trapped forever and approaches the singularity.

v

u

IV

I

II

III

r=0

r=0

Figure 4. It is impossible for a timelike particle in region I toever cross over to region III. A particle in region II will at somepoint hit the singularity.

2.4. Crossing Bridges. When you think about what it means to cross an Einstein-Rosen 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 blackhole. You will die because you are jumping into a black hole. TheEinstein-Rosen coordinate u is a bad coordinate at the horizon.Attempting to cross the horizon, say from u = +ε to u = −ε,will force one off the u coordinate patch and into the curvaturesingularity.

So stay away from Einstein-Rosen bridges.

3. Traversable Wormholes

From the last section, we saw that nothing can go through the Einstein-Rosenbridge. They are not traversable since


(1) Tidal gravitational forces at the throat are great. Traveller is killed unlesswormhole’s mass exceeds 104M� so the throat circumference will exceed105km.

(2) Schwarzschild wormhole is not static but dynamic. As time pass, the throatstarts from zero circumference to a maximum circumference and back againto 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 worm-holes exist.

3.1. Criteria for Construction. We should first begin by discussing the criteriafor construction of traversable wormholes (listed in Box 1).

Box 1. Traversable Wormhole Construction Criteria

(1) Metric should be both spherically symmetricand static. This is just to keep everything simple.

(2) Solution must everywhere obey the Einsteinfield equations. This assumes correctness of GR.

(3) Solution must have a throat that connects twoasymptotically flat regions of spacetime.

(4) No horizon, since a horizon will preventtwo-way travel through the wormhole.

(5) Tidal gravitational forces experienced by atraveler must be bearably small.

(6) Traveler must be able to cross through thewormhole in a finite and reasonably small propertime.

(7) Physically reasonable stress-energy tensorgenerated 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 thetotal mass of the universe and much less than theage of the universe.

4We use traversable wormhole to mean that a human (or some similar alien) in their spaceshipcould safely travel through the wormhole in a reasonable amount of time and return.


Our construction of the wormhole should at least satisfy criteria (1) to (4). Morrisand Thorne [5, pp.399-400] calls this the “basic wormhole criteria”. (5) to (7) arecalled “usability criteria” since it deals with human physiological comfort. Thus weneed to find a solution that will satisfy the basic wormhole criteria, then we tunethe parameters of the usability criteria to suit our needs. We will take the simpleapproach of Morris and Thorne [5].

3.2. Morris and Thorne (1988). Morris and Thorne simplified their analysisby first assuming the existence of a suitably well-behaved geometry. AssociatedRiemann tensor components are calculated and Einstein field equations are used todetermine the distribution of the stress-energy. Then they ask whether or not thisdisturibution of stress-energy is physically reasonable or not.

3.2.1. The Metric. To keep simple, we will assume the traversable wormhole tobe time independent, nonrotating, and spherically symmetric bridges between twouniverses. Thus our manifold should be a static spherically symmetric spacetimepossessing 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) liml→±∞

{r(l)/|l|} = 1

or r(l) = |l|+O(1).• For spacetime geometry to tend to an appropriate asymptotically flat limit

(3.3) liml→±∞

{Φ(l)} = Φ±

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 atl = 0.

• Metric components are at least twice differentiable by l.We could use this to calculate the Riemann, Ricci and Einstein tensors using thiscoordinate system, but it is much easier to use Schwarzschild coordinates. We writein (t, r, θ, φ)

(3.5) ds2 = e2Φ±(r)dt2 − dr2

1− b±(r)/r− r2[dθ2 + sin2 θdφ2]

where we introduced b(r) called the “shape” function since it determines the spatialshape of the wormhole, and Φ(r) called the “redshift” function since it determinesthe gravitational redshift. Some key features are


• Spatial coordinate r has a geometrical significance. The throat circumfer-ence is 2πr and so r is equal to the embedding-space radial coordinate offigure 1. Also, r decreases from +∞ to some minimum radius r0 as onemoves 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

(3.6) l(r) = ±∫ r

r0

dr′√1− b±(r′)/r′

• For spatial geometry to tend to an appropriate asymptotically flat limit,we require both limits

(3.7) limr→∞

{b±(r)} = b±

to be finite.• For spacetime geometry to tend to an appropriate asymptotically flat limit,

we require both limits

(3.8) limr→∞

{Φ±(r)} = Φ±

to be finite.• Since dr/dl = 0 at the throat (throat is at minimum of r(l)), we have

dl/dr →∞. Since

(3.9)dl

dr= ± 1√

1− b±(r)/r,

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 asymp-

totically flat regions, ± ↔ ∓ or b+(r) = b−(r) and Φ+(r) = Φ−(r). This isnot a requirement, just for convenience.

3.2.2. Tensor Calculations. Now, using our standard formulas5, we can computethe Christoffel symbols and the Riemann curvature tensor. There are 24 nonzero

5Remember

Γαβγ = 1

2gαλ(gλβ,γ + gλγ,β − gβγ,λ)

Rαβγδ = Γα

βδ,γ − Γαβγ,δ + Γα

λγΓλβδ − Γα

λδΓλβγ


components. Quoting results from [5, p.400]

(3.10)

Rtrtr = −Rt

rrt = −(1− b/r)−1e−2ΦRrttr

= (1− b/r)−1e−2ΦRrtrt

= Φ,rr − (b,rr − b)[2r(r − b)]−1Φ,r + (Φ,r)2,Rt

θtθ = Rtθθt = −r2e−2ΦRθ

ttθ = r2e−2ΦRθtθt

= rΦ,r(1− b/r),Rt

φtφ = −Rtφφt = −r2e−2Φ sin2 θRφ

ttφ

= r2e−2Φ sin2 θRφtφt

= rΦ,r(1− b/r) sin2 θ,

Rrθrθ = −Rr

θθr = r2(1− b/r)Rθrrθ

= −r2(1− b/r)Rθrθr

= −(b,rr − b)/2r,

Rrφrφ = −Rr

φφr = r2(1− b/r) sin2 θRφrrφ

= −r2(1− b/r) sin2 θRφrφr

= −(b,rr − b) sin2 θ/2r,

Rθφθφ = −Rθ

φφθ = − sin2 θRφθφθ = sin2 θRφ

θθφ

= −(b/r) sin2 θ,

where basis vectors being used are those (et, er, eθ, eφ). We want to rather be inthe rest frame (ie. r, θ, φ constant) which are related,

(3.11)

{et = e−Φet, er = (1− b/r)1/2er,

eθ = r−1eθ, eφ = (r sin θ)−1eφ.

This makes the metric Minkowski,

(3.12) gαβ = eα · eβ = ηαβ ≡

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

,

and the nonzero components of the Riemann tensor are,

(3.13)

Rtrtr

= −Rtrrt

= Rrttr

= −Rrtrt

= (1− b/r){Φ,rr − (b,rr − b)[2r(r − b)]−1Φ,r + (Φ,r)2},Rt

θtθ= −Rt

θθt= Rθ

ttθ= −Rθ

tθt= (1− b/r)Φ,r/r,

Rtφtφ

= −Rtφφt

= Rφ

ttφ= −Rφ

tφt= (1− b/r)Φ,r/r,

Rrθrθ

= −Rrθθr

= Rθrθr

= −Rθrrθ

= −(b,rr − b)/2r3,

Rrφrφ

= −Rrφφr

= Rφ

rφr= −Rφ

rrφ= −(b,rr − b)/2r3,

Rθφθφ

= −Rθφφθ

= Rφ

θφθ= −Rφ

θθφ= −b/r3.


Finally, we contract and find the Ricci tensor, curvature scalar and solve the Ein-stein field equations. Our nonzero Einstein tensor components are

(3.14)

Gtt = b,r/r2,

Grr = −b/r3 + 2(1− b/r)Φ,r/r,

Gθθ =(

1− b

r

)(Φ,rr −

b,rr − b

2r(r − b)Φ,r + (Φ,r)2 +

Φ,r

r− b,rr − b

2r2(r − b)

)= Gφφ.

Non-vanishing stress-energy tensor components should be the same non-vanishingcomponents as the Einstein tensor. We denote the following:

(3.15)

Ttt = ρ(r),Trr = −τ(r),Tθθ = Tφφ = 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) Gαβ = 8πGTαβ

and equating the results of (3.14) and (3.15),

b,r = 8πGr2ρ,(3.17)

Φ,r = (−8πGτr3 + b)/[2r(r − b)],(3.18)τ,r = (ρ− τ)Φ,r − 2(p + τ)/r.(3.19)

What we have here are five unknown functions of r : b, Φ, ρ, τ and p. But if we goback to our original plan, we wanted to be able to “tweak” some parameters so thatwe can get a resonable result for the stress-energy. Since we will be “tweaking” theshape function b(r) and redshift function Φ(r), we rewrite the previous equationsas:

ρ = b,r/[8πGr2],(3.20)

τ = [b/r − 2(r − b)Φ,r]/[8πGr2],(3.21)p = (r/2)[(ρ− τ)Φ,r − τ,r]− τ.(3.22)

In this form, by choosing a suitable b(r) and Φ(r), we will be able to solve for ρand 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

(3.23) τ0 ≡ (tension in the throat) =1

8πGb20

∼ 5× 1011 dyncm2

(1light yr.

b0

)2

,

which is huge. For b0 ∼ 3km, τ0 ∼ 1037dyn/cm2 which is equivalent to the pressureat the center of the most massive neutron star. Taking (3.9) and inverting, we get

(3.24)dr

dl= ±

√1− b

r


and since,

(3.25)d2r

dl2=

dr

dl

d

dr

(dr

dl

)=

12

d

dr

(dr

dl

)2

,

we have

(3.26)d2r

dl2=

12r

(b

r− b,r

).

Now, at the throatd2r

dl2> 0 since r(l) is a minimum at the throat. So

(3.27)d2r

dl2

∣∣∣∣r0

=1

2r0[1− b,r(r0)] ⇒ b,r(r0) < 1.

Using this and (3.20) at the throat,

(3.28) ρ(r0) ≡ ρ0 <1

8πGr20

and from (3.21)

(3.29) τ(r0) ≡ τ0 =1

8πGr20

combining (3.28) and (3.29) implies

(3.30) ρ0 < τ0.

So this is where we run into trouble. ρ0 < τ0 says that at the throat, the tensionexceeds the total mass-energy density. Materials with the property τ > ρ > 0is called, “exotic”. This makes things troublesome because it forces an observermoving through the throat with radial veolcity ∼ c see their stress-energy tensor(in basis vector eo′ = γet ∓ γ(v/c)er) [5, p.405]

To′o′ = γ2Ttt ∓ 2γ2(v/c)2Ttr + γ2(v/c)2Trr

= γ2[ρ0 − (v/c)2τ0] = γ2(ρ0 − τ0) + τ0(3.31)

for sufficiently large γ, to have negative density of mass-energy.

3.3. Weak Energy Condition. Negative density of mass-energy is a direct vio-lation of the weak energy condition (WEC). The weak energy condition states thatfor any timelike vector

(3.32) WEC ⇐⇒ TµνV µV ν ≥ 0.

Physically, this implies that the weak energy condition forces the local energy den-sity to be positive measured by any timelike observer. In terms of principal pres-sures,

(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 seesome examples of observing this violation, due to quantum effects. An example ofenergy condition violation is the Casimir effect.


3.3.1. Casimir Effect. 6 With two parallel conducting plates separated by a smalldistance a, the wave vector is constrained by

(3.34) kz =nπ

a.

By symmetry, the stress-energy can depend only on the spacetime metric ηµν ,normal vector zµ and the separation a. So introducing two dimensionless functionsf1(z/a) and f2(z/a) we can write by dimensional analysis

(3.35) TµνCasimir ≡

~a4

[f1(z/a)ηµν + f2(z/a)zµzν .

The electromagnetic field is conformally invariant, ie.

(3.36) T ≡ TµνCasimirη

µν = 0.

With this we find the relationship between f1 and f2, and it can be shown that

(3.37) T ≡ TµνCasimir =

π2

720~a4

(ηµν − 4zµzν).

We can observe that our energy density is thus negative ρ = −(π2~)/(720a4),violating our energy condition. Similar violations can be seen with TopologicalCasimir Effect, Squeezed Vacuum and Particle Creation. [2, pp.125-126].

3.4. Minimize Exotic Material. Since exotic materials are so troublesome, onemay want to minimize the use of it. The amount of exotic material is quantified bya dimensionless function ζ(r) = (τ − ρ)/ρ. We have the following scenarios.

(1) Use exotic material throughout the wormhole, but make the density ofexotic material fall off rapidly with radius as one moves away from thethroat. An example of this is to take b = const and Φ = 0. This yields

ρ(r) = 0,(3.38)τ(r) = b0/(8πGr3),(3.39)p(r) = b0/(16πGr3),(3.40)

ζ = ∞.(3.41)

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

ζ > 0 for r < Rs,(3.42)ρ = τ = p = 0 for r > Rs.(3.43)

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. Wethen have

ζ > 0 for |l| < lc,(3.44)ζ ≤ 0 for |l| ≥ lc.(3.45)

6For a more indepth description, please consult other texts.


3.5. Tension, Stability and Assembly. Earlier we said that a traversable worm-hole should be safe for a traveller to go through. But it seems very uncomfortablefor someone to go through a throat that experiences torque equivalent to that of aneutron star core (§3.2.3). Two workarounds are suggested.

(1) Build a long vacuum tube (diameter � b0) through the throat and havethe stresses of the tube wall to hold the exotic matter out. This breaks thespherical symmetry of our solution, but even before that good luck tryingto find the tube material!

(2) Hope that the exotic material couples very weakly (like neutrinos) to thetraveller. Then even with the high stress and density, the traveller can gothrough the throat without noticing much effect.

We cannot talk too much about stability of the wormhole, since this relies heavilyon the behavior of the exotic material. Whether naturally stable or unstable, therecould be ways to stabilize the wormhole, but again without knowing the behaviorof the exotic material, it is hard to analyze.

Finally, the actual assembly relies on topology change. This will probably need tobe addressed after gravity has been properly quantized. This may be understood bytaking a quantum mechanical picture of spacetime, like that of the spacetime foamintroduced by Wheeler (1955) [7]. At Plank-Wheeler length lp−w ∼ 1.6× 10−33cm,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 ofGR to the idea of traversable wormhole that connects two points from two differentuniverses, or universe on its own. But the reality of such, comes with seriousproblems that cannot be proven (or disproven) at the present time. Quantumgravity seems to be what can attempt to give us real evidence of the (non)realityof 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.

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