Matter traveling through a wormhole Karina Calhoun Brendan Fay and Ben Kain Department of Physics College of the Holy Cross Worcester Massachusetts 01610 USA

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Matter traveling through a wormhole

Karina Calhoun, Brendan Fay, and Ben Kain

Department of Physics, College of the Holy Cross, Worcester, Massachusetts 01610, USA

We revisit the numerical evolution of Ellis-Bronnikov-Morris-Thorne wormholes, which are con-

structed with a massless real ghost scalar ﬁeld. For our simulations, we have developed a new code

based on the standard 3 + 1 foliation of spacetime. We conﬁrm that, for the massless symmetric

wormhole, a pulse of regular scalar ﬁeld causes the wormhole throat to collapse and form an ap-

parent horizon, while a pulse of ghost scalar ﬁeld can cause the wormhole throat to expand. As a

new result, we show that it is possible for a pulse of regular matter to travel through the wormhole

and then to send a light signal back before the wormhole collapses. We also evolve pulses of matter

traveling through massive asymmetric wormholes, which has not previously been simulated.

I. INTRODUCTION

Considerable attention has been paid to the study

of traversable wormholes ever since Morris and Thorne

showed that a wormhole geometry in general relativity is

possible with matter that violates the null energy condi-

tion [1–3]. A Morris-Thorne wormhole may be realized

with a massless real ghost scalar ﬁeld. By “ghost,” we

mean that the kinetic energy has the opposite sign com-

pared to a “regular” scalar ﬁeld. A static wormhole solu-

tion in this system was ﬁrst discovered by Ellis [4]. The

solution has zero mass and is symmetric across the worm-

hole throat. The solution was generalized by Bronnikov

[5] to a family of static solutions which are generically

massive and asymmetric, with the Ellis solution being a

special case. Recent work on such wormholes includes

Refs. [6–18].

As far as we are aware, all numerical simulations of

wormholes [19–25] have focused on the Ellis-Bronnikov-

Morris-Thorne solutions. Shinkai and Hayward sim-

ulated pulses of regular and ghost scalar ﬁelds trav-

eling through the massless symmetric wormhole [19].

Their code was based upon a dual-null formalism [26].

Gonz´alez et al. studied both the massless symmetric and

massive asymmetric wormholes [20]. They did not simu-

late a ﬁeld traveling through the wormhole, but instead

considered a perturbation applied directly to one of the

metric ﬁelds. The numerical formalism of Gonz´alez et al.

bears similarities to the 3 + 1 formalism we use here,

but they did not incorporate the extrinsic curvature.

Doroshkevich et al. [21] extended the work of [19] and,

like [19], focused on the massless symmetric wormhole

and used a dual-null formalism, as did subsequent stud-

ies [24,25]. Shinkai and Torii used the dual-null formal-

ism to study higher-dimensional versions of the massless

symmetric wormhole in the context of Einstein-Gauss-

Bonnet gravity [22,23].

In this work, we numerically simulate regular and ghost

scalar ﬁelds traveling through both the massless symmet-

ric and massive asymmetric wormholes. For our simula-

tions, we have developed a new code based on the stan-

dard 3+1 foliation of spacetime [27,28]. Using our code,

we conﬁrm results found in [19]. In particular, we con-

ﬁrm that a pulse of regular scalar ﬁeld traveling through

the massless symmetric wormhole causes the throat to

collapse and form an apparent horizon, as does a pulse of

ghost scalar ﬁeld with negative amplitude, while a pulse

of ghost scalar ﬁeld with positive amplitude causes the

throat to expand. As a new result, we show that it is

possible for a pulse of regular ﬁeld to travel through the

wormhole and then to send a light signal back before

the wormhole collapses. We also study matter traveling

through massive asymmetric wormholes, which has not

previously been simulated. Such wormholes are not sta-

tionary, but move. We ﬁnd again that a pulse of regular

scalar ﬁeld causes the wormhole throat to collapse and

form an apparent horizon and a pulse of ghost scalar ﬁeld

can cause the wormhole throat to expand.

In the next section, we brieﬂy describe the 3+1 formal-

ism and present the equations we will be using. These

include equations for both the metric and matter ﬁelds.

In Sec. III, we consider the static limit of these equations

and review the Ellis-Bronnikov-Morris-Thorne solutions

for static wormholes. We then describe adding a pulse

of regular or ghost scalar ﬁeld and how we use these so-

lutions as initial data for our simulations. In Sec. IV,

we brieﬂy describe aspects of our numerical code. We

present our results for the massless symmetric wormhole

in Sec. Vand for massive asymmetric wormholes in Sec.

VI. We conclude in Sec. VII. In an appendix, we present

tests of our code.

II. EQUATIONS

The general spherically symmetric metric can be writ-

ten [27,28]

ds2=−α2−Aβr2dt2+ 2Aβrdtdr +dl2

dl2=Adr2+Cdθ2+ sin2θdφ2.(1)

In the 3+1 formalism, we foliate spacetime into a contin-

uum of time slices, where each time slice is a spatial hy-

persurface. dl2is the spatial metric on an individual time

slice. Aand Cparametrize the spatial metric, where A

tells us about the physical distance between coordinates

and Cis the squared areal radius. We deﬁne

R≡√C(2)

arXiv:2210.04905v2 [gr-qc] 29 Nov 2022

as the areal radius, so that the area of a two-sphere is

4πR2.αis the lapse and βris the only nonzero compo-

nent of the shift vector. As we move from one time slice

to the next, αtells us the rate at which proper time in-

creases and βrtells us how the coordinates shift. Both α

and βrare gauge functions, in that once initial data are

loaded onto the initial time slice αand βrcan, in prin-

ciple, be chosen arbitrarily. In addition to these ﬁelds,

there is the intrinsic curvature, Kij, which describes how

a time slice sits in the larger spacetime. The only nonzero

and independent components of the extrinsic curvature

are Krrand KT≡2Kθθ= 2Kφ

φ. All quantities are

functions of tand r.

In a wormhole geometry, ris not restricted to be non-

negative, but instead takes values −∞ < r < ∞. If R

is nonzero at its minimum, then there exists a wormhole

throat which connects the two sides of the minimum. We

deﬁne the wormhole throat radius to be

Rth(t)≡R(t, rmin),(3)

where, on a given time slice, Rhas its minimum at rmin.

We use units such that c=G=~= 1. The Einstein

ﬁeld equations, Gµν= 8πT µ

ν, where Gµνis the Einstein

tensor and Tµ

νis the energy-momentum tensor, in the

spherically symmetric 3 + 1 formalism lead to the evolu-

tion equations

∂tA=−2αAKrr+βr∂rA+ 2A∂rβr

∂tC=−αCKT+βr∂rC

∂tKrr=α(∂rC)2

4AC2−1

C+ (Krr)2−1

4K2

T+ 4π(S+ρ−2Srr)−∂2

rα

A+(∂rα)(∂rA)

2A2+βr∂rKrr

∂tKT=α1

C−(∂rC)2

4AC2+3

4(KT)2+ 8πSrr−(∂rα)(∂rC)

AC +βr∂rC

2C(2Krr−KT)−8πSr

(4)

and the Hamiltonian and momentum constraint equations,

∂2

rC=A+(∂rA)(∂rC)

2A+(∂rC)2

4C+ACKTKrr+1

4KT−8πACρ

∂rKT=∂rC

2C(2Krr−KT)−8πSr,

(5)

where

ρ=−Tt

t+βrTt

Srr=1

ATrr

Sθθ=1

CTθθ

Sr=αT t

(6)

and S=Srr+ 2Sθθdepend on the energy-momentum

tensor and parametrize the matter sector.

Ellis-Bronnikov-Morris-Thorne wormholes are con-

structed with a massless real ghost scalar ﬁeld. As men-

tioned in the Introduction, “ghost” means that the ki-

netic energy has the opposite sign compared to the ki-

netic energy of a “regular” scalar ﬁeld. Since we are in-

terested in sending a regular ﬁeld through the wormhole,

we will include both ghost and regular scalar ﬁelds. The

Lagrangian for our matter sector is then

L=−ηφ

2∂µφ∂µφ−ηχ

2∂µχ∂µχ. (7)

φis the ghost ﬁeld that forms the wormhole and χis the

regular ﬁeld and thus

ηφ=−1, ηχ= +1.(8)

We minimally couple the Lagrangian to gravity through

L → p−det(gµν )L, from which the equations of motion

and energy-momentum tensor follow straightforwardly.

To write the equations of motion, we deﬁne the auxiliary

ﬁelds

Φφ≡∂rφΠφ≡C√A

α(∂tφ−βr∂rφ)

Φχ≡∂rχΠχ≡C√A

α(∂tχ−βr∂rχ)

(9)

and the equations of motion are

∂tφ=α

C√AΠφ+βrΦφ

∂tΦφ=∂rα

C√AΠφ+βrΦφ

∂tΠφ=∂rαC

√AΦφ+βrΠφ

(10)

and

∂tχ=α

C√AΠχ+βrΦχ

∂tΦχ=∂rα

C√AΠχ+βrΦχ

∂tΠχ=∂rαC

√AΦχ+βrΠχ.

(11)

From the energy-momentum tensor, we have the matter

functions

ρ=1

2A"ηφ Π2

C2+ Φ2

φ!+ηχ Π2

C2+ Φ2

χ!#

Srr=1

2A"ηφ Π2

C2+ Φ2

φ!+ηχ Π2

C2+ Φ2

χ!#

Sθθ=1

2A"ηφ Π2

C2−Φ2

φ!+ηχ Π2

C2−Φ2

χ!#

Sr=−1

C√A(ηφΠφΦφ+ηχΠχΦχ),

(12)

which follow from Eq. (6).

The solution to the equations listed in this section give

the numerical evolution of the system. Note that the

ﬁelds φand χdecouple and do not have to be determined.

Instead, for matter ﬁelds, we need only determine Φφ,

Πφ, Φχ, and Πχ.

We can use the solution to compute various quantities.

For example, the Misner-Sharp mass function, m(t, r), in

spherically symmetric systems is given by [27,28]

m=1

8√C4C+ (CKT)2−(∂rC)2

A.(13)

The ADM mass, M, is given by the large rlimit of m.

In a wormhole geometry, we can take r→ ±∞, and we

should not expect the two limits to give the same value.

Instead, the two limits give the total mass as viewed from

either side of the wormhole.

We will be interested in whether an apparent horizon

forms, which we will use as an indicator for a black hole.

Apparent horizons satisfy [20,27,28]

C√AKT∓∂rC= 0,(14)

where we use the upper sign for r > rmin, where rmin

marks the minimum of C, and the lower sign for r < rmin.

Finally, we can compute null geodesics, rnull(t), which

obey

drnull

dt =±α

√A−βr,(15)

where we choose the sign depending on whether we want

to compute a right-moving or left-moving geodesic.

III. INITIAL DATA

To numerically evolve the system, we must place ini-

tial data on the initial time slice. Our code can then

evolve the initial data forward in time. Our initial data

will include a wormhole and a pulse of ghost or regular

matter.

We assume our initial data are time-symmetric, which

sets βr=Kij= 0 [28]. Under this assumption, the

Hamiltonian constraint in Eq. (5) and the KTevolution

equation in Eq. (4) become

∂2

rC=A+(∂rA)(∂rC)

2A+(∂rC)2

4C−8πACρ

0 = 1

C−(∂rα)(∂rC)

αAC −(∂rC)2

4AC2+ 8πSrr,(16)

the evolution equations for Krrand KTin Eq. (4) can

be combined to give

∂2

rα=∂rA

2A−∂rC

C∂rα+ 4παA(ρ+S),(17)

and the Misner-Sharp mass function in (13) reduces to

m=√C

21−(∂rC)2

4AC .(18)

A. Static wormholes

We ﬁrst discuss the wormhole before including pulses.

For now, then, we drop the regular ﬁeld by setting χ= 0.

We use a static wormhole solution for our initial data.

By static, we mean that spacetime is time independent,

which can be achieved by assuming φis time indepen-

dent. The ghost ﬁeld equations of motion in (10) reduce

0 = ∂rαC

√AΦφ(19)

and the matter functions in (12) reduce to

ρ=Srr=−Sθθ=ηφ

Φ2

2A,(20)

along with Sr= 0.

Static wormhole solutions can be found analytically

[4,5,29,30] for

α=1

√A, C =A(r2

0+r2),(21)

where r0is a constant. For α= 1/√A, the top equation

in (16) is unchanged, while the bottom equation and Eq.

(17) become

∂2

rA=2(∂rA)2

A−(∂rA)(∂rC)

C−8πA2(ρ+S)

0 = (∂rA)(∂rC)

2A2C+1

C−(∂rC)2

4AC2+ 8πSrr.

(22)

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MattertravelingthroughawormholeKarinaCalhoun,BrendanFay,andBenKainDepartmentofPhysics,CollegeoftheHolyCross,Worcester,Massachusetts01610,USAWerevisitthenumericalevolutionofEllis-Bronnikov-Morris-Thornewormholes,whicharecon-structedwithamasslessrealghostscalareld.Foroursimulations,wehavedevelopedane...

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Matter traveling through a wormhole Karina Calhoun Brendan Fay and Ben Kain Department of Physics College of the Holy Cross Worcester Massachusetts 01610 USA

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