INTRODUCTION TO WORMHOLES: PRESENTATION NOTES … · INTRODUCTION TO WORMHOLES: PRESENTATION NOTES 9 4. Traversable Wormholes From the last section, we saw that nothing can go through

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INTRODUCTION TO WORMHOLES: PRESENTATION NOTES

TAKASHI OKAMOTO

[email protected]

1. Introduction

• The idea of wormholes predate that of black holes.

• First brought to attention by Flamm1 in 1916 shortly after introduction ofGR.

• Also speculated by Weyl2 in 1928.

• First serious calculations were done by Einstein and Rosen3 in 1935.

• This was named the “Einstein-Rosen Bridge”, as the term wormhole wasintroduced by Wheeler4 in the 1950’s.

2. Einstein-Rosen Bridge

• 1935: Albert Einstein and Nathan Rosen investigated the possibility ofobtaining an atomistic theory of matter and electricity.

• They wanted to know how GR would treat individual particles, and hopedsomehow it can account for quantum phenomena.

• Began by their motivating question [3, p73]:

“Is an atomistic theory of matter and electricity conceivable which, whileexcluding singulatities in the field, makes use of no other field variable thanthose of the gravitational field (gµν) and those of the electromagnetic fieldin the sense of Maxwell (vector potentials, ϕµ)?

1Flamm, L. Beitrage zur Einsteinschen Gravitationstheorie. Phys. Z., 17:448-454, 1916.2Weyl, H. Philosophie der Mathematik und Naturwissenschaft. Handbuch der philosophie.

Leibniz Verlag, Munich, 1928.3Einstein, A. and Rosen, N. The particle problem in the theory of general relativity. Phys.

Rev., 48:73-77, 1935.4J. A. Wheeler, Geons. Phys. Rev., 97:511-536, 1955.

1

2 TAKASHI OKAMOTO [email protected]

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

(2.1) ds2 = (1− 2m/r)dt2 − 11− 2m/r

dr2 − r2(dθ2 + sin2 θdφ2)

where

(2.2) r ∈ [2m,∞) θ ∈ [0, π] φ ∈ [0, 2π].

Introduce a new variable defined as

(2.3) u2 = r − 2m,

and obtain a new expression

(2.4) ds2 =u2

u2 + 2mdt2 − 4(u2 + 2m)du2 − (u2 + 2m)2(dθ2 + sin2 θdφ2).

• u ∈ (−∞,+∞) so twice covers r ∈ [2m,∞).

• Notice now the coordinate change discarded r ∈ [0, 2m).

• Interpretation of the four-dimensional space as two identical “sheets” cor-responding to the asymptotically flat regions around u = ±∞ which areconnected by a “bridge” at u = 0 [see figure 1].

• Taking u as a constant, the area is given as A(u) = 4π(2m + u2)2.

• min{A(u)} occurs at u ≡ u0 = 0, and area of this “throat” is given asA(0) = 4π(2m)2. ⇐= Note the finite size.

• This region near u0 is the “wormhole”.

• Note, we must impose m > 0, for the bridge construction.

⇒ this is due to the fact that the bridge construction is dependent on theexistence of a horizon. no horizon ⇔ no bridge.

• Einstein and Rosen concluded that this bridge characterizes an electricallyneutral elementary particle (eg. neutron or neutrino), and says that parti-cles with negative mass cannot be described as a bridge.

• Key Points:⇒ No new field variables are introduced.⇒ Space is described by two sheets.⇒ A spatially finite bridge which connects the two sheets characterizes thepresence of an electrically neutral elementary particle.⇒ Able to understand the atomistic character of matter as well as the factthat there can be no particles of negative mass.

INTRODUCTION TO WORMHOLES: PRESENTATION NOTES 3

2.2. Quasicharged Bridge: The Reissner-Nordstrom Geometry. Startingfrom the Reissner-Nordstrom Geometry in Schwarzschild coordinates,

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

dr2− r2(dθ2 +sin2 θdφ2).

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

(2.6) Tik = 14gikϕαβϕαβ − ϕiαϕ α

k

to be negative5. We now obtain with this modified geometry

(2.7) ds2 = (1− 2m/r − ε2/r2)dt2 − 11− 2m/r − ε2/r2


where ε is the electric charge. Set6 m = 0.

(2.8) ds2 = (1− ε2/r2)dt2 − 11− ε2/r2


Introduce a new variable defined as

(2.9) u2 = r2 − ε2/2

we get

(2.10) ds2 =2u2

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

• This bridge represents an elementary electrical particle without mass.

• Notice if we had not set the electromagnetic stress-energy tensor to be neg-ative, bridge construction would have failed, again no horizon ⇔ no bridge.

• m not necessarily have the restriction of being positive.

5We shall see the reason why once we consider the case where m = 06Observe here that m is not determined by ε, and that m and ε are independent constants of

integration.


2.3. General Bridge Construction. We can now generalize this bridge construc-tion. Begin with static, spherically symmetric solution

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

dr2 − r2(dθ2 + sin2 θdφ2).

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

(2.12) u2 = r − rH

Substituting, 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).(2.13)

• 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

ds2 ≈ e−ϕ(rH) u2[1− b,r(rH)]

rHdt2 − 4

rH + u2

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

(2.14)

Introducing constants A and B, we can rewrite this as

(2.15) 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 quasistaticbridges.


3. Causality Problem

3.1. Topology of Einstein-Rosen Bridge. With the Schwarzschild wormhole(neutral Einstein-Rosen bridge), take t = v = 0 and θ = π/2, the surface is definedby the paraboloid of revolution

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

as shown here in figure 1.

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.)

• 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 distantregions of the same asymptotically flat universe, as shown in figure 2.


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.)

• This multiply connected universe introduces an issue with causality.

• There are essentially two paths to get from a to B. One can either take apath going through the wormhole or not.

• If you have some disturbance travels at the speed of light from a to B goingthe “long” way, this disturbance can be outpaced by another disturbancethat took the wormhole route, travelling a much shorter path

• It seems that causality is violated, but Fuller and Wheeler (1962) haveshown that causality is preserved.

3.2. 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 withcircumference of 2πm ⇒ true in the region far away from the throat, sincethe Schwarzschild solution carries no time dependence

• Schwarzschild throat is dynamic in the region of the bridge:⇒ Throat opens and closes like the shutter of a camera.⇒ This “pinch off” of the throat as they called it, happens so fast that evena 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 curvaturewhen the throat closes. This is illustrated in figure 3.

⇒ We usually think the static time translation, t → t + ∆t, leaves theSchwarzschild geometry unchanged.⇒ True when we deal with a problem in regions I and III (timelike regions)of the Kruskal diagram.


⇒ This is not true for r < 2m, since in regions II and IV, t → t + ∆t is aspacelike motion and not a timelike motion.⇒ Surface of figure 2 will begin to change just as it moves in the +vdirection as it enters region II.

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.)

We can see from figure 3 that the system begins at A (region IV in terms of Kruskaldiagram) in a pinched off state and as you move up the v coordinate, the throatopens 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 nearthe throat (u ≈ 0), we have r ≈ 2m.


3.3. Causality Preserved. Will a photon be able to go through the Schwarzschildwormhole before it pinches off?

• Fuller and Wheeler (1962) provides a full quantitative argument, but onecan be easily convinced that the photon will not be able to pass through aSchwarzschild wormhole with a qualitative argument.

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.

• 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 thevertical.

• Particle in region I or III can never crossover to the other side.⇒ Particle in region I will never be able to crossover to region IV, since itwould require speeds faster than that of light.⇒ As soon as the particle crosses over to region II, the it is trapped foreverand approaches the singularity.

• So there is no way of getting from region I to region III, for a photon ornormal timelike particle.


4. 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.

4.1. 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.

• Our construction of the wormhole should at least satisfy criteria (1) to (4).Morris and Thorne calls this the “basic wormhole criteria”.

• (5) to (7) are called “usability criteria” since it deals with human physio-logical comfort.

• 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.


4.2. Morris and Thorne (1988).• Simplified their analysis by first assuming the existence of a suitably well-

behaved geometry.

• Associated Riemann tensor components are calculated and Einstein fieldequations are used to determine the distribution of the stress-energy.

• Finally they ask whether or not this disturibution of stress-energy is phys-ically reasonable or not.

4.2.1. The Metric. Assume the traversable wormhole to be time independent, non-rotating, and spherically symmetric bridges between two universes. Thus our man-ifold should be a static spherically symmetric spacetime possessing two asymptoti-cally flat regions.

(4.1) 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

(4.2) liml→±∞

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

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

(4.3) liml→±∞

{Φ(l)} = Φ±

must be finite.• Radius of the wormhole throat defined by

(4.4) 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)

(4.5) ds2 = e2Φ±(r)dt2 − dr2

1− b±(r)/r− r2[dθ2 + sin2 θdφ2]

• b(r) called the “shape” function since it determines the spatial shape of thewormhole.

• Φ(r) called the “redshift” function since it determines the gravitationalredshift.


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

(4.6) l(r) = ±∫ r

r0

dr′√1− b±(r′)/r′

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

(4.7) limr→∞

{b±(r)} = b±

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

we require both limits

(4.8) 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

(4.9)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.


4.2.2. Tensor Calculations. There are 24 nonzero components. Quoting resultsfrom [?, p.400]

(4.10)

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,

(4.11)

{et = e−Φet, er = (1− b/r)1/2er,

eθ = r−1eθ, eφ = (r sin θ)−1eφ.

This makes the metric Minkowski,

(4.12) 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,

(4.13)

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.


Our nonzero Einstein tensor components are

(4.14)

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:

(4.15)

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,

(4.16) Gαβ = 8πGTαβ

and get,

b,r = 8πGr2ρ,(4.17)

Φ,r = (−8πGτr3 + b)/[2r(r − b)],(4.18)τ,r = (ρ− τ)Φ,r − 2(p + τ)/r.(4.19)

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],(4.20)

τ = [b/r − 2(r − b)Φ,r]/[8πGr2],(4.21)p = (r/2)[(ρ− τ)Φ,r − τ,r]− τ.(4.22)

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.

4.2.3. Stress-Energy at the Throat. We have the condition, r = b = b0 at the throat.This also implies (r − b)Φ,r → 0 at the throat. With (4.21)

(4.23) 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. We have from previous

(4.24)dr

dl= ±

√1− b

r,

and since

(4.25)d2r

dl2=

dr

dl

d

dr

(dr

dl

)=

12

d

dr

(dr

dl

)2

,


we have

(4.26)d2r

dl2=

12r

(b

r− b,r

).

Now, at the throatd2r

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

(4.27)d2r

dl2

∣∣∣∣r0

=1

2r0[1− b,r(r0)] ⇒ b,r(r0) < 1.

Using this and (4.20) at the throat,

(4.28) ρ(r0) ≡ ρ0 <1

8πGr20

and from (4.21)

(4.29) τ(r0) ≡ τ0 =1

8πGr20

combining (4.28) and (4.29) implies

(4.30) ρ0 < τ0.

So this is where we run into trouble.• ρ0 < τ0 ⇒ 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 throughthe throat with radial veolcity ∼ c see their stress-energy tensor (in basisvector eo′ = γet ∓ γ(v/c)er)

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

= γ2[ρ0 − (v/c)2τ0] = γ2(ρ0 − τ0) + τ0(4.31)

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


4.3. Weak Energy Condition.

• Negative density of mass-energy is a direct violation of the weak energycondition (WEC).

• The weak energy condition states that for any timelike vector

(4.32) WEC ⇐⇒ TµνV µV ν ≥ 0.

• Physically, this implies that the weak energy condition forces the localenergy density to be positive measured by any timelike observer.⇒ In terms of principal pressures,

(4.33) WEC ⇐⇒ ρ ≥ 0 and ∀j, ρ + ρj ≥ 0.

• So clearly this condition is violated by the result we obtained previously(τ > ρ).

• Can this violation be physically possible? We do see this violation occur,an example is the Casimir effect.

4.3.1. Casimir Effect. 7 With two parallel conducting plates separated by a smalldistance a, the wave vector is constrained by

(4.34) 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

(4.35) TµνCasimir ≡

~a4

[f1(z/a)ηµν + f2(z/a)zµzν ].

The electromagnetic field is conformally invariant, ie.

(4.36) T ≡ TµνCasimirη

µν = 0.

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

(4.37) T ≡ TµνCasimir =

π2

720~a4

(ηµν − 4zµzν).

We observe that our energy density is thus negative: ρ = −(π2~)/(720a4), violat-ing our energy condition. Similar violations can be seen with Topological CasimirEffect, Squeezed Vacuum and Particle Creation.

7For a more indepth description, please consult other texts.


4.4. 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,(4.38)τ(r) = b0/(8πGr3),(4.39)p(r) = b0/(16πGr3),(4.40)

ζ = ∞.(4.41)

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,(4.42)ρ = τ = p = 0 for r > Rs.(4.43)

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,(4.44)ζ ≤ 0 for |l| ≥ lc.(4.45)

4.5. 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. 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.

(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). At Plank-Wheeler length lp−w ∼ 1.6 × 10−33cm,quantum effects can give rise to foam like multiply connected spacetime.

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INTRODUCTION TO WORMHOLES: PRESENTATION NOTES … · INTRODUCTION TO WORMHOLES: PRESENTATION NOTES 9 4. Traversable Wormholes From the last section, we saw that nothing can go through