Traversable Wormholes
Anthony Carapetis PHS3142
28 October 2011

Abstract

The wormhole is a hypothetical feature of spacetime that connects distant regions of the universe (or even distinct universes) together. While the concept has been around since the earlyyears of general relativity theory, a thorough analysis of their feasibility for human traversal was
not attempted until the mid 1980s. It turns out that traversable wormholes are possible under thelaws of GR if one allows so-called exotic matter with negative energy, which is usually consideredunrealistic but not necessarily impossible. One common objection is that wormholes can link regions that are distant in time as well as space, which has the potential to introduce paradoxes.
This essay gives a mathematical description of some a hypothetical wormhole and reviews theliterature regarding the aforementioned issues.

Natural units with

1c, G = 1 are used throughout.

Introduction

The idea of a wormhole was rst considered by Ludwig Flamm [1] less than a year after Einstein
published his theory of General Relativity (GR). Einstein's eld equation

1Rij − Rk k gij = 8πTij
2(1)places no restriction on the topology of spacetime, it simply relates the geometry with energy andmomentum.

Flamm realised that it therefore seems just as valid to base spacetimes on arbitrary4

dierentiable 4-manifolds as it is to use the usual R , which allows for the insertion of topologicalhandle joining two distant parts of the universe.

By adjusting the metric on the handle, one canthen make the distances measured via the two routes dierent, giving us what has come to be known
as a wormhole.

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Traversable Wormholes

Figure 1: A wormhole between two distant locales of the same universe. In the vicinity of the wormhole,
the fact that there is a long way around is irrelevant; the geometry is the same as in a wormhole
between two dierent universes. Note that the folding of the surface is an extrinsic property of thisparticular embedding that allows us to depict the situation conveniently; the actual geometry is atin that region. Image from [4].

2The Einstein-Rosen Bridge

An explicit example of a wormhole metric was investigated by Einstein and Rosen in their 1935 paper[2] and came to be known as the Einstein-Rosen bridge.

The Einstein-Rosen bridge connects two

distinct universes; but for most purposes, we can easily adapt an inter-universe wormhole metric byjoining up the two universes and smoothing out the metric in between (see Figure 1).
Interestingly, the metric used was exactly that of the Schwartzchild solution, which describes
the spacetime outside a spherically symmetric mass
Schwartzchild metricg = − (1 − 2m/r) dt2 +m.

In the usual spherical coordinates, the1dr2 + r2 dΩ2
1 − 2m/r(2)× S 2 . (In the casewhere the radius of the generating mass is smaller than r = 2m, we have an event horizon at r = 2m√and r < 2m represents the interior of a black hole.) By changing the radial coordinate to u =r − 2mand reecting the metric to negative u, one ndsis singular at r = 2m, and is often only considered on (t, r, θ, ϕ) | r > 2m ' R

2u2
g=− 2
dt2 + 4 u2 + 2m du2 + u2 + 2m dΩ2 .
u + 2m2(3)This is now well-dened and smooth for all values of (t, u, θ, ϕ), so the singularity at u = 0 in the oldcoordinates is simply a property of those coordinates. (There is a genuine singularity at r = 0, but
our new coordinates do not cover this region.) We can now interpret the metric as a wormhole (seeFigure 2 for a visualisation). Noting that dr = 2udu, we nd that in the limit u → ±∞ where we take√u = ± r − 2m the metric tends towards the at metricη = −dt2 + dr2 + r2 dΩ2 ;(4)i.e. we have two universes that asymptotically look like Minkowski space, joined together at the throatr = 2m.

Because we actually have two separate coordinate charts for positive and negative u, wecannot use these coordinates to draw conclusions about geodesics crossing u= 0. (The fact thatdet g = 0 at u = 0 is also a hint that things are not as regular there as they seem.) A complete
analysis [4, Ch 31] using the Kruskal-Szekeres coordinates (which cover the entire extended geometry,
including both universes and both interior regions) shows that there are in fact no geodesics that passfrom one universe to the other. Any geodesic that crosses the horizon ends up in the r < 2m portionof the geometry and ultimately terminates at the singularity r = 0; so we say that this wormhole isAnthony Carapetis

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Traversable Wormholes

Figure 2: Embedding diagram of Einstein-Rosen bridge. We take a slice of the spacetime t = 0, θ = π/2

2u2 + 2m du2 + u2 + 2m dϕ2 , which is then isometrically√
32embedded into R with cylindrical coordinates via the map (u, ϕ) 7→ u + 2m, ϕ, 8mu . This gureuses m = 1, u ∈ (−8, 8), ϕ ∈ [0, 2π). Angles and lengths measured on the surface are the same as
which gives a 2-manifold with metric 4

those in the full spacetime.

not traversable. Two travelers from dierent universes could briey meet inside the event horizon, butthey could never send a message back and would be promptly crushed.

3The Morris-Thorne Wormhole

The term wormhole was coined by Misner and Wheeler in 1957 [5], and the concept quickly became apopular method of travel in science ction - they are a convenient device to allow rapid interstellar travel
without neglecting the universal speed limit c. Despite this, it was not until 1985 that real scienticinterest arose in the possibility of traversable wormholes. While writing his novel Contact, Carl Sagan
consulted physicists Kip Thorne and Michael Morris for help depicting a believable wormhole, andthey found a remarkably simple example of a traversable wormhole consistent with almost all of GR[3]. They started with the assumption of spherical symmetry and two asymptotic at regions (bothpossessed by the Einstein-Rosen bridge discussed above), and constructed a metric with geodesics
passing from universe to the other in reasonable timescales.

2×S . The coordinate l will measure proper radial distance from the throat (with l < 0 in one universe),
but it is more convenient to construct the metric using the coordinate r where 2πr is the circumferenceof a circle ϕ 7→ (t0 , r, ϕ, π/2) . Assuming spherical and time-translation symmetry, we write down theSimilarly to the Einstein-Rosen bridge, the Morris-Thorne metric uses coordinates t, l, ϕ, θ on R

2general Morris-Thorne metricg = −e2Φ dt2 +1
dr2 + r2 dΩ2
1 − b/r(5)

where Φ, b are adjustable functions of r only. To achieve asymptotic atness, we need this to converge
to η as r → ∞; so we have constraints

lim Φ (r) = 0r→∞

limr→∞b (r)
= 0.
r

(6)

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Traversable Wormholes

Embedding diagram for Morris-Thorne wormhole with Φ (r) = 0, b (r) = 1/r .
Slicet=0,θ
=π/2. Cylindrical embedding map is (r, ϕ) 7→ (r = cosh z, ϕ, z) where z (r) =√
ln r + r2 − 1 . Yellow line shows a geodesic passing from one universe to another.

Figure 3:
taken is= 2m, Φ (r) = 12 ln (1 − 2m/r). We will22
2
2
investigate the simple but fruitful example Φ (r) = 0, b (r) = b0 /r , which yields l = r − b0 andtherefore in coordinates (t, l, θ, ϕ) the metric can be writteng = −dt2 + dl2 + l2 + b20 dΩ2 .
(7)

We can recover the Schwartzchild metric by setting b (r)Figure 3 shows that the geometry appears qualitatively similar to that of the Einstein-Rosen bridge. In
this case, however, we have a valid global coordinate chart with no degeneracies, so what we see is thegenuine spacetime. This means that the trajectories τ 7→ x (τ ) = (γτ, l0 − γvτ, 0, 0) , γ =1 − v2

−1/2

are geodesics that start in one universe at radial position l0 and travel through the wormhole to theother universe. To see this, note that gtt , gll have no dependence on t, l, so all the relevant Christoelsymbols vanish and therefore the acceleration is simply −γ (∂t v) ∂l = 0. Assuming no forces will act
on a particle following this trajectory, we can therefore send matter through the wormhole. For it totruly be deserving of the name traversable we would also want a human to be able to comfortably
travel through it in a reasonable amount of time; so we have a few more requirements. Firstly, thegravitational and tidal accelerations felt by the traveler should be small so that the traveler is not
crushed or spaghettied. For a traveler freefalling on a geodesic, the gravitational acceleration is zero,so there is no issue here. The relative acceleration between x and a geodesic separated from it by asmall vector δx is approximately [4, Ch 11]

Rẋ (δx) = R (δx, ẋ) ẋ = Ri jkl ẋj δxk ẋl .
In this case, there are only a few non-zero curvature components.

(8)

Switching to the local reference−1/2∂i ), they are [3]frame of a stationary observer (i.e. the orthonormal frame Ei = |gii |Rθϕθϕ = −Rlθlθ = −Rlϕlϕ =b20(b20 + l2 )2(9)and the other components that can be derived from these via the Riemann tensor's symmetries. Thevelocity of the traveler in this frame isẋ = γ (∂t − v∂l ) = γ (Et − vEl )(10)Anthony Carapetis

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Traversable Wormholes

and therefore the tidal acceleration isRθ lθl ẋl2δxθ Eθ + Rϕ lϕl ẋl2δxϕ Eϕ = −b202γ(b20 + l2 )2 2v

δxθ Eθ + δxϕ Eϕ .

(11)This has a maximum at l = 0, where we nd (for someone 2m tall laying perpendicular to the radial
direction, which maximises the tidal force)atide = 2m γ 2 v 2(12)and therefore we can decrease the tidal acceleration below any given threshold by decreasing thevelocity.
We now have a satisfactory geometry for a wormhole, so the outstanding question is to ask whatmatter distribution generates this geometry. Working in the static orthonormal frame as before andstarting from Equation 9, we nd the non-zero components of the Ricci tensor areRθθ= Rθϕθϕ + Rθlθl = 0Rll= Rlθlθ + Rlϕlϕ = −2Rϕϕ= Rϕθϕθ + Rϕlϕl = 0.
k2

The scalar curvature is therefore R = Rk = −2b0 /b20 + l2

2(13)

b20(b20 + l2 )

2(14)
(15)

and so using Equation 1 we nd we havea diagonal stress-energy tensor:−Ttt = −Tll = Tϕϕ = Tθθ =41
b20.
8π (b20 + l2 )2(16)Exotic Matter

We've now struck our rst real problem with the Morris-Thorne wormhole: the energy density Ttt isnegative. While we have only studied a particular example here, the full analysis [3] shows that for
any choice of b, Φ satisfying (6), there is some observer who sees Ttt < 0, so this is a generic feature ofMorris-Thorne wormholes. Intuitively, the space needs to be negatively curved in order to smoothlytransition from the throat to the asymptotically at universe, so the eld equations necessitate exotic
matter with negative mass to hold it open. While this matter is distributed across all of space (thoughconcentrated near the wormhole) in our simple example, one can nd b, Φ that give safely traversable
wormholes such that T is zero outside of a thin spherical shell [3].
Because Morris and Thorne assumed spherical symmetry in the geometry, we have spherical symmetry in the distribution of exotic matter; so anyone traversing the wormhole will necessarily passthrough it. While the interactions of the exotic matter with other matter is unknown, it is likely that
this would be undesirable. By discarding spherical symmetry, Matt Visser showed [6] the existence ofa wormhole metric where travelers do not come in contact with the exotic matter, and in addition do
not feel any gravitational or tidal forces.
While the eld equations and principle of geodesic motion are perfectly consistent in a universe withnegative energy, it is conventional to impose some restrictions on the energy in order to get realisticresults. Normally one assumes at least the weak energy condition, which states that the energy density
is always measured as non-negative by every observer, i.e. T (V, V ) ≥ 0 for every timelike vector eldV . This is more important than just getting rid of the troublesome concept of negative mass - muchof black hole thermodynamics relies on the weak energy condition [1]; so many would dismiss solutionsof (1) that violate it as non-physical.
All is not lost, however - quantum eld theory (QFT) oers a tantalising possibility in the formof the Casimir eect. If two conducting planes are placed parallel to each other at a distance L, theAnthony Carapetis

6

Traversable Wormholes

boundary conditions suppress electromagnetic eld modes with wavelengths λ > 2L.

As the plates

are brought closer together, these excluded modes increase in energy, and one nds that the vacuum
energy between the plates is lower than that of free space. Because the vacuum energy of free space isconventionally taken as the zero point, this means that there is essentially a negative energy density
between the plates. The Casimir eect has been experimentally validated [7], and is therefore very
good evidence that the weak energy condition does not hold in general. While the Casimir eect itself
is almost certainly not practical for supporting a wormhole, it hints that QFT (or better, a uniedtheory of quantum gravity) can admit negative energy densities. Whether or not usable exotic matter
exists in the universe (or can be manufactured) is another question entirely - there is certainly noempirical evidence for such materials.

5

The Chronology Problem

One common argument against the existence of wormholes is that they introduce time travel paradoxes.
One can easily create a single-universe geometry (as in Figure 1) with a wormhole whose two exits areattached at distant points in time. One can nd closed timelike curves (CTCs; i.e. periodic particle
trajectories) that pass through the wormhole to the past, and then travel back to the future the longway.

This introduces the possibility of a given initial condition violating itself when it propagatesaround the loop, as in the classic example of a man killing his own grandfather. This is not, however,particular to wormholes: there are many known spacetimes with CTCs, often with otherwise reasonableconditions (i.e. no violation of energy conditions) [9]Because a full theory of quantum gravity has not yet been developed, the chronology problem has
a number of resolutions [1], from conservative to radical:• Conject that our particular universe is globally hyperbolic, implying no CTCs and therefore thatMorris-Thorne wormholes do not exist. (It can be shown [10] that in the case of classical GR,
any Morris-Thorne wormhole can be manipulated into a conguration that contains a CTC.)

This is often known as the boring physics conjecture.

• Hawking's chronology protection conjecture: assume that quantum mechanics will (via someyet to be determined mechanism) intervene to forbid the existence of macroscopic CTCs. Whilesome have attempted to prove/disprove this conjecture using the theory of semiclassical quantum
gravity, these analyses are possibly outside the regime where such an approximation is valid [11];so a full theory of quantum gravity is required before this can really be investigated. Dependingon the nature of the mechanism, this could possibly preclude the existence of wormholes, but
may admit them as long as they do not contain CTCs.

• The Novikov consistency conjecture: accept time travel, and simply require that it be consistent,i.e.

conjecture that the universe will only accommodate globally consistent solutions.

From

one point of view, this does not modify our laws of physics at all: a non-consistent time loop
would result in a discontinuity of the metric or some eld, meaning it would no longer be asolution to the governing PDEs. However, this may remove the locality of the physical laws. We
can take the state (i.e. the spatial topology, the metric and any auxiliary elds) on one local
spacelike hypersurface {t0 } × U and use it as initial conditions to nd a unique local solution on[t0 , t0 + ) × U . The Novikov conjecture implies that in a spacetime where {t0 + τ } × U connects
back up with {t} × U via geodesics, such a solution may not in fact be correct if it does notextend to a smooth global solution.

• The radical rewrite conjecture: discard altogether the requirement of a single consistent timeline
and allow multiple branching timelines. This requires a fundamental modication of our theory,
even at the macroscopic level. One possible model for this is the extension of general relativity
to a certain class of non-Hausdor manifolds, allowing a single-sheeted universe to split into
two [1].

Anthony Carapetis

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Traversable Wormholes

The last two conjectures would admit wormholes - under the consistency conjecture they would beexactly as described earlier, but after radical rewrite they would take somewhat of a dierent form tot the new model.

6Conclusions

Traversable wormholes certainly exist as solutions to the classical equations of general relativity if
and only if

we allow the violation of the energy conditions. The exotic matter required to do so isunlike anything yet discovered or manufactured, but is potentially possible due to quantum eects.
The relationship between wormholes and time travel depends entirely on which of the four chronology
conjectures is true - quantum gravity may intervene to rule them out entirely, allow them while barring
time travel or place no restriction upon them at all. Ultimately, the feasibility of traversable wormholes
cannot be determined until we either have a validated theory of quantum gravity, or direct empiricalevidence.

References[1] M. Visser, Lorentzian Wormholes, Springer-Verlag (1996)[2] A. Einstein and N. Rosen, Phys. Rev.

48 (1935)[3] M. S. Morris and K. S. Thorne, Am. J. Phys.

56 5 (1988)[4] C. W. Misner, K. S. Thorne and J. A. Wheeler, Gravitation, W. H. Freeman and Company(1973)[5] C. W. Misner and J. A. Wheeler, Ann. Phys.
[6] M. Visser, Phys. Rev. D

2 6 (1957)

39 (1989)[7] G. Bressi, G. Carugno, R. Onofrio and G. Ruoso, Phys. Rev. Lett.
[8] F. J. Tipler, Phys. Rev. Lett

88 (2002)

37 14 (1976)[9] M. Visser, The quantum physics of chronology protection, Proceedings:

The future oftheoretical physics and cosmology, Cambridge, 7-10 January 2002.
[10] J. Friedman, M. Morris, I. Novikov, F. Echeverria, G. Klinkhammer, K. Thorne and U.
Yurtsever, Phys. Rev. D[11] M. Visser, Phys. Rev. D

42 (1990)

55 8 (1997)