Pedro F. Gonzalez-Dıaz- On the warp drive space-time

8/3/2019 Pedro F. Gonzalez-Daz- On the warp drive space-time

1/7

arXiv:g

r-qc/9907026v210Apr2000

On the warp drive space-time

Pedro F. Gonzalez-DazCentro de Fsica Miguel Catalan, Instituto de Matematicas y Fsica Fundamental,

Consejo Superior de Investigaciones Cientficas, Serrano 121, 28006 Madrid (SPAIN)(june 23, 1999)

In this paper the problem of the quantum stability of the two-dimensional warp drive spacetimemoving with an apparent faster than light velocity is considered. We regard as a maximum extensionbeyond the event horizon of that spacetime its embedding in a three-dimensional Minkowskian spacewith the topology of the corresponding Misner space. It is obtained that the interior of the spaceshipbubble becomes then a multiply connected nonchronal region with closed spacelike curves and thatthe most natural vacuum allows quantum fluctuations which do not induce any divergent behaviourof the re-normalized stress-energy tensor, even on the event (Cauchy) chronology horizon. In such acase, the horizon encloses closed timelike curves only at scales close to the Planck length, so that thewarp drive satisfies the Fords negative energy-time inequality. Also found is a connection betweenthe superluminal two-dimensional warp drive space and two-dimensional gravitational kinks. Thisconnection allows us to generalize the considered Alcubierre metric to a standard, nonstatic metricwhich is only describable on two different coordinate patches.

PACS numbers: 04.20.Gz, 04.20.Cv

I. INTRODUCTION

General relativity admits many rather unexpected so-lutions most of which represent physical situations whichhave been thought to be pathological in a variety ofrespects, as they correspond to momentum-energy ten-sors which violate classical conditions and principles con-sidered as sacrosant by physicists for many years [1].Among these solutions you can find Lorentzian worm-holes [2], ringholes [3], Klein bottleholes [4], the Gott-

Grants double-string [5,6], the Politzer time machine [7],the multiply connected de Sitter space [8,9], a time ma-chine in superfluid 3He [10], etc, all of which allow theexistence of closed timelike curves (CTCs), so as space-times where superluminal, though not into the past trav-els are made possible, such as it happens in the Alcu-bierre warp drive [11]. Most of the solutions that containCTCs are nothing but particular modifications or gener-alizations from Misner space which can thereby be con-sidered as the prototype of the nonchronal pathologies ingeneral relativity [12].

Time machines constructed from the above mentionedspacetimes contain nonchronal regions that are generatedby shortcutting the spacetime and can allow travelinginto the past and future at velocities that may exceedthe speed of light. A solution to Einstein equations thathas also fascinated and excited relativists is the so-calledwarp drive spaceime discovered by Alcubierre [11]. Whatis violated in this case is only the relativistic principlethat a space-going traveller may move with any veloc-ity up to, but not including or overcoming, the speedof light. Alcubierres construct corresponds to a warpdrive of science fiction in that it causes spacetime to con-tract in front of a spaceship bubble and expand behind,so providing the spaceship with a velocity that can be

much greater than the speed of light relative to distantobjects, while the spaceship never locally travels fasterthan light. Thus, the ultimate relativistic reason makingpossible such a warp drive within the context of generalrelativity is that, whereas two observer at the same pointcannot have relative velocities faster than light, whensuch observers are placed at distant locations their rela-tive velocity may be arbitrarily large while the observersare moving in their respective light cones.

The properties of a warp drive and those of spacetimeswith CTCs can actually be reunited in a single space-time. Everett has in fact considered [13] the formation ofCTCs which arise in the Alcubierre warp drive wheneverit is modified so that it can contain two sources for thegravitational disturbance which are allowed to move pastone another on parallel, noncolinear paths. The result-ing time machine can then be regarded to somehow be athree-dimensional analog of the two-dimensional Gottstime machine [5] in which the CTCs encircle pairs of infi-nite, stright, parallel cosmic strings which also move non-colinearly. In the present paper we consider the physicaland geometric bases which allow us to construct a two-dimensional single-source Alcubierre warp drive movingat constant faster than light apparent speed, that at the

same time can be viewed as a time machine for the astro-naut travelling in the spaceship. This can be achieved byconverting the interior of the spaceship bubble in a mul-tiply connected space region satisfying the identificationproperties of the three-dimensional Misner space [14] .We shall also show how in doing so some of the acutestproblems of the Alcubierre warp drive can be solved.

It was Alcubierre himself who raised [11] doubts aboutwhether his spacetime is a physically reasonable one. En-ergy density is in fact negative within the spaceship bub-ble. This may in principle be allowed by quantum me-

1
http://arxiv.org/abs/gr-qc/9907026v2http://arxiv.org/abs/gr-qc/9907026v2http://arxiv.org/abs/gr-qc/9907026v2http://arxiv.org/abs/gr-qc/9907026v2http://arxiv.org/abs/gr-qc/9907026v2http://arxiv.org/abs/gr-qc/9907026v2http://arxiv.org/abs/gr-qc/9907026v2http://arxiv.org/abs/gr-qc/9907026v2http://arxiv.org/abs/gr-qc/9907026v2http://arxiv.org/abs/gr-qc/9907026v2http://arxiv.org/abs/gr-qc/9907026v2http://arxiv.org/abs/gr-qc/9907026v2http://arxiv.org/abs/gr-qc/9907026v2http://arxiv.org/abs/gr-qc/9907026v2http://arxiv.org/abs/gr-qc/9907026v2http://arxiv.org/abs/gr-qc/9907026v2http://arxiv.org/abs/gr-qc/9907026v2http://arxiv.org/abs/gr-qc/9907026v2http://arxiv.org/abs/gr-qc/9907026v2http://arxiv.org/abs/gr-qc/9907026v2http://arxiv.org/abs/gr-qc/9907026v2http://arxiv.org/abs/gr-qc/9907026v2http://arxiv.org/abs/gr-qc/9907026v2http://arxiv.org/abs/gr-qc/9907026v2http://arxiv.org/abs/gr-qc/9907026v2http://arxiv.org/abs/gr-qc/9907026v2http://arxiv.org/abs/gr-qc/9907026v2http://arxiv.org/abs/gr-qc/9907026v2http://arxiv.org/abs/gr-qc/9907026v2http://arxiv.org/abs/gr-qc/9907026v2http://arxiv.org/abs/gr-qc/9907026v2


2/7

chanics, provided the amount and time duration of thenegative energy satisfy Ford like inequalities [15]. How-ever, Ford and Pfenning have later shown [16] that evenfor bubble wall thickness of the order of only a few hun-dred Planck lengths, it turns out that the integrated neg-ative energy density is still physically unreasonable for

macroscopic warp drives. This problem has been recentlyalleviated, but not completely solved, by Van Den Broeck[17] who by only slightly modifying the Alcubierre space-time, succeeded in largely reducing the amount of nega-tive energy density of the warp. A true solution of theproblem comes only about if the warp bubble is micro-scopically small [16], a situation which will be obtainedin this paper when CTCs are allowed to exist within thewarp bubble.

On the other hand, Hiscock has computed [18] for thetwo-dimensional Alcubierre warp drive the stress-energytensor of a conformally invariant field and shown thatit diverges on the event horizon appearing when the ap-parent velocity exceeds the speed of light, so rendering

the warp drive instable. A similar calculation performedin this paper for the case where the interior spacetimeof the spaceship bubble is multiply connected yields avanishing stress-energy tensor even on the event horizonwhich becomes then a (Cauchy) chronology horizon, bothfor the self consistent Li-Gott vacuum [19] and when oneuses a modified three-dimensional Misner space [20] asthe embedding of the two-dimensional warp drive space-time. It is worth noticing that if the astronaut inside thespaceship is allowed to travel into the past, then mostof the Alcubierres warp drive problems derived from thefact that an observer at the center of the warp bubble iscausally separated from the bubble exterior [21] can be

circumvented in such a way that the observer might nowcontribute the creation of the warp bubble and controlone once it has been created, by taking advantage of thecausality violation induced by the CTCs.

We outline the rest of the paper as follows. In Sec. IIwe biefly review the spacetime geometry of the Alcubierrewarp drive, discussing in particular its two-dimensionalmetric representation and extension when the appar-ent velocity exceeds the speed of light. The visualiza-tion of the two-dimensional Alcubierre spacetime as athree-hyperboloid embedded in E3 and its connectionwith the three-dimensional Misner space is dealt within Sec. III, where we also covert the two-dimensional

warp drive spacetime into a multiply connected space bymaking its coordinates satisfy the identification proper-ties of the three-dimensional Misner space. By replacingthe Kruskal maximal extension of the geodesically in-complete warp drive space with the above embedding westudy in Sec. IV the Euclidean continuation of the mul-tiply connected warp drive, identifying the periods of thevariables for two particular ansatze of interest. We alsoconsider the Hadamard function and the resulting renor-malized stress-energy tensor for each of these ansatze,comparing their results. Finally, we conclude in Sec. V.Throughout the paper units so that G = c = h = 1 are

used.

II. THE ALCUBIERRE WARP DRIVE

SPACETIME

The Alcubierre spacetime having the properties asso-ciated with the warp drive can be described by a metricof the form [11]

ds2 = dt2 + (dx vf(r)dt)2 + dy2 + dz2, (2.1)with v = dxs(t)/dt the apparent velocity of the warpdrive spaceship, xs(t) the trajectory of the spaceshipalong coordinate x, the radial coordinate being definedby

r =

[x xs(t)] + y2 + z2 1

2 , (2.2)

and f(r) an arbitrary function subjected to the bound-

ary conditions that f = 1 at r = 0 (the location of thespaceship), and f = 0 at infinity.

Most of the physics in this spacetime concentrates onthe two-dimensional space resulting from setting y = z =0, defining the axis about which a cylindrically symmet-ric space develops; thus, the two-dimensional Alcubierrespace still contains the entire worldline of the spaceship.If the apparent velocity of the spaceship is taken to beconstant, v = v0, then the metric of the two-dimensionalAlcubierre space becomes [18]:

ds2 = 1 v20f(r)2 dt2 2v0f(r)dtdx + dx2, (2.3)with r now given by r = (x v0t)2, which in thepast of the spaceship (x > v0t) can simply be writtenas r = x v0t. Metric (2.3) can still be represented ina more familiar form when one chooses as coordinates(t, r), instead of (t, x). This can obviously be achievedby the replacement dx = dr + v0dt, with which metric(2.3) transforms into:

ds2 = A(r)

dt v0 (1 f(r))A(r)

dr

2+

dr2

A(r), (2.4)

where the Hiscock function [18]

A(r) = 1

v20 (1

f(r))

2(2.5)

has been introduced. Metric (2.4) can finally be broughtinto a comoving form, in terms of the proper time d =dt v0 (1 f(r)) dr/A(r),

ds2 = A(r)d2 + dr2

A(r). (2.6)

As pointed out by Hiscock [18], this form of the met-ric is manifestly static. The case of most interest corre-sponds to apparent velocities v0 > 1 (superluminal ve-locity) where the metrics (2.4) and (2.6) turn out to be

2


3/7

singular with a coordinate singularity (apparent eventhorizon) at a given value of r = r0 such that A(r0) = 0,i.e. f(r0) = 1 1/v0.

We note that metric (2.4) can be regarded to be thekinked metric (describing a gravitational topological de-fect [22]) that corresponds to the static metric (2.6). To

see this, let us first redefine the coordinates t and r asdt = A1

4 dt and dr = A1

4 dr, so that metric (2.4) canbe re-written along the real intervals 0 r (i.e.0 r r0) and 0 t as

ds2 =

A(r)

(dt)2 (dr)2 + 2v0(1 f(r))dtdr.(2.7)

Now, metric (2.7) can be viewed as the metric describ-ing at least a given part of a two-dimensional one-kink(gravitational topological charge 1) if we take sin(2) =v0(1f(r)), i.e. cos(2) =

A, so that this metric can

be transformed into

ds2 = cos(2) (dt)2 (dr)2 2 sin(2)dtdr, (2.8)where is the tilt angle of the light cones tipping overthe hypersurfaces [22,23], and the choice of sign in thesecond term depends on whether a positive (upper sign)or negative (lower sign) gravitational topological chargeis considered.

The existence of the complete one-kink is allowedwhenever one lets to monotonously increase from 0 to, starting with (0) = 0, with the support of the kinkbeing the region inside the event horizon. Then metric(2.8) converts into metric (2.6) if we introduce the sub-

stitution sin = 1 A(r) /2 and the change oftime variable = t + g(r), with dg(r)/dr = tan(2).The region inside the warp drive bubble supporting thekink is the only region which can actually be describedby metric (2.8) because sin cannot exceed unity andhence 0 r r0. In order to have a comple descrip-tion of the one-kink and therefore of the warp drive oneneed a second coordinate patch where the other half ofthe interval, /2 , can be described. This canbe achieved by introducing a new time coordinate [23]

t = t + h(r), with dh(r)/dr = [v0(1 f(r)) k] /A34 , inwhich k = 1, the upper sign for the first patch and thelower one for the second patch. This choice is adopted

for the following reason. The zeros of the denomina-tor ofdh/dr = (sin(2) 1) /A correspond to the twoevent horizons where r = r0 and f(r0) = 1 1/r0, oneper patch. For the first patch, the horizon occurs at = /4 and therefore the upper sign is selected so thatboth dh/dr and h remain well defined and the kink ispreserved on this horizon. For the second patch the hori-zon occurs at = 3/4 and therefore the lower sign isselected for it. The two-dimensional metric for a (in thissense) complete warp drive will be then [23]

ds2 = A(r)dt2 2kdtdr, (2.9)

or in terms of the (t, x) coordinates in the four-dimensional manifold,

ds2 = [A(r) 2kv0] dt2 2kdtdx + dy2 + dz2, (2.10)with r as defined by Eq. (2.2).

Having shown the existence of a connection between

warp drive spacetime and topological gravitational kinksand hence extended the warp drive metric to that de-scribed by Eq. (2.10), we now return to analyse metric(2.6) by noting that the geodesic incompleteness of thismetric at r = r0 can, as usual, be avoided by maxi-mally extending it according to the Kruskal technique.For this to be achieved we need to define first the quan-tity r =

drA(r) . Using for f(r) the function suggested

by Alcubierre

f(r) =tanh[(r + R)] tanh[(r R)]

2 tanh(R), (2.11)

where R and are positive arbitrary constants, we obtain

r =(1 + 2v0)r

1 v20

+(3 + 2v0 v20)v20

(1 v20)2

3v20 + 2v0 1

ln

2v0(1 + v0) tanh(r)

3v20 + 2v0 12v0(1 + v0) tanh(r) +

3v20 + 2v0 1

v0(1 + 3v0)4(1 + v0)

2v0(1 + v0)

ln

1 + v0

2v0(1 + v0)tanh(r)

1 + v0 +

2v0(1 + v0)tanh(r)

+v0

2(1 + v0)tanh(r), (2.12)

where we have specialized to the allowed particular case

for which sinh

2

(R) = v0. Introducing then the usualcoordinates V = t + r and W = t r, so that

ds2 =

1 v20 sinh

4(r)cosh2(r) + v0

2

dV dW, (2.13)

and hence the new coordinates

tanh V = exp

V

4sinh1

v0+1v01

3


4/7

tanh W = exp W

4sinh1

v0+1v01

,

we finally obtain the maximally extended, geodesicallycomplete metric

ds2 =

64

2

1 v

20 sinh

4(r)cosh2(r) + v0

2

sinh1

v0+1v01

2dVdW

sin(2V)sin(2W),

(2.14)

where r is implicitly defined by

tan V tan W = exp

r

2 sinh1v0+1v01

. (2.15)

The maximal extension of metric (2.6) is thus obtainedby taking expression (2.14) as the metric of the largestmanifold which metrics given only either in terms of Vor in terms of W can be isometrically embedded. Therewill be then a maximal manifold on which metric (2.14)is C2 [24].

III. WARP DRIVE WITH INTERNAL CTCS

In this section we investigate a property of the two-dimensional Alcubierre spacetime which will allow us to

avoid the complicatedness of Kruskal extension in orderto study its Euclidean continuation and hence its stabil-ity against quantum vacuum fluctuations. Thus, takingadvantage from the similarity between metric (2.6) andthe de Sitter metric in two dimensions, we can visual-ize the dimensionally reduced Alcubierre spacetime as athree-hyperboloid defined by

v2 + w2 + x2 = v20 , (3.1)where v0 > 1. This hyperboloid is embedded in E3 andthe most general expression for the two-dimensional met-ric of Alcubierre space for v0 > 1 is then that which isinduced in this embedding, i.e.:

ds2 = dv2 + dw2 + dx2, (3.2)which has topology RS2 and invariance group SO(2, 1).

Metric (3.2) can in fact be conveniently exhibited instatic coordinates (, ) and r (0, r0), definedby

v = v10

A(r) sinh(v0)

w = v10

A(r) cosh(v0) (3.3)

x = F(r),

where

[F(r)]2

=

ddr

2 4v20

4v20(1 + )

, (3.4)

the prime denoting derivative with respect to radial co-ordinate r, with

(r) = v0 (1 f(r))2 (3.5)and

A = 1 + (r). (3.6)

Using coordinates (3.3) in metric (3.2) we re-derive thenmetric (2.6). Thus, the two-dimensional Alcubierre spacecan be embedded in Minkowski space in three dimen-sions. Since Minkowski metric (3.2) plus the identifica-

tions

(v, w, x)

(v cosh(nb) + w sinh(nb), v sinh(nb) + w cosh(nb), x) ,

(3.7)

where b is a dimensionless arbitrary boosting quantityand n is any integer number, make the universal cov-ering [24] of the Misner space in three dimensions, onecan covert the two-dimensional Alcubierre space into amultiply connected space if we add the corresponding

identifications

(, r) ( + nbv0

, r) (3.8)

in the original warp drive coordinates. The boost trans-formation in the (v, w) plane implied by identifications(3.7) will induce therefore the boost transformation (3.8)in the two-dimensional Alcubierre space. Hence, sincethe boost group in Alcubierre space must be a subgroupof the invariance group of the two-dimensional Alcubierreembedding, the static metric (2.6) can also be invariantunder symmetry (3.7). Thus, for coordinates defined byEqs. (3.3) leading to the static metric with an apparent

horizon as metric (2.6), the symmetry (3.7) can be satis-fied in the region covered by such a metric, i.e. the regionw > |v|, where there are CTCs, with the boundaries atw = v, and x2 = v20 being the Cauchy horizons thatlimit the onset of the nonchronal region from the Alcu-bierre causal exterior. Such boundaries are situated at r0,defined by f(r0) = 1 v10 , and become then appropri-ate chronology horizons [12] for the so-obtained multiplyconnected two-dimensional Alcubierre space with v0 > 1.

We have in this way succeeded in coverting a two-dimensional warp drive with constant, faster than lightapparent velocity in a multiply connected warp drive with

4


5/7

CTCs only inside the spaceship, and its event horizon atr0 in a chronology horizon. This is a totally differentway of transforming warp drives into time machines ofthe mechanism envisaged by Everett [13] for generatingcausal loops using two sources of gravitational distur-bance which move past one another. In our case, even

though the astronaut at the center of the warp bubbleis still causally separated from the external space, he (orshe) can always travel into the past to help creating thewarp drive on demand or set up the initial conditions forthe control of one once it has been created.

IV. QUANTUM STABILITY OF MULTIPLY

CONNECTED WARP DRIVE

In this section we shall show that the two-dimensionalmultiply connected warp drive spacetime is perfectly sta-ble to the vacuum quantum fluctuations if either a selfconsistent Rindle vacuum is introduced, or for micro-

scopic warp bubbles. We have already shown that thetwo-dimensional warp drive spacetime can be embeddedin the Minkowskian covering of the three-dimensionalMisner space when the symmetries of this space impliedby identifications (3.7) (that lead to identifications (3.8)in Alcubierre coordinates) are imposed to hold also inthe two-dimensional Alcubierre spacetime with v0 > 1.Since the embedding can be taken to play an analogousrole to that of a maximal Kruskal extension, it followsthat showing stability against vacuum quantum fluctua-tions in three-dimensional Misner space would imply thatthe two-dimensional multiply connected warp drive spacewith v0 > 1 is also stable to the the same fluctuations.

This conclusion is in sharp contrast with some recent re-sult obtained by Hiscock who has shown [18] that thestress-energy tensor for a conformally invariant scalarfield propagating in simply connected two-dimensionalAlcubierre space diverges on the event horizon if theapparent velocity of the spaceship exceeds the speed oflight. He obtained an observed energy density near the

horizon proportional to

f (1 v10 )2

, which in factdiverges as r r0.

Metric (3.2) can be transformed into the metric of athree-dimensional Misner space explicitly by using thecoodinate re-definitions

v = cosh x1

w = sinh x1 (4.1)

x = x2.

Then,

ds2 = d2 + 2 dx12 + dx22 , (4.2)which is the three-dimensional Misner metric for coor-dinates 0 < < , 0 x1 2 and 0 x2 .

This metric is singular at = 0 and, such as it happensin its four-dimensional extension, it has CTCs in theregion < 0. Note that one can also obtain the line ele-ment (4.2) directly from the two-dimensional Alcubierremetric with v0 > 1 by introducing the coordinate trans-formations

=i

A(r)

v0(4.3)

x1 = i arcsin[cosh(v0)] (4.4)and

x2 = F(r), (4.5)

with F(r) as defined by Eq. (3.4). Let us now considerthe Euclidean continuation from which metric (4.2) be-comes positive definite. This is accomplished if we rotateboth coordinates and x1 simultaneously, so that

= i, x1 = i, (4.6)

where and can be expressed in terms of the Alcu-bierre coordinates r and by means of transformations(4.3) and (4.4). The covering space of the resulting metricpreserves however the Lorentzian signature. This mightbe an artifact coming from the singular character of met-ric (4.2), so it appears most appropriate to extend firstthis metric beyond = 0 in order to get the Euclideansector of the three-dimensional Misner metric, and henceinvestigate the stability of the superluminal, multiplyconnected two-dimensional warp drive spacetime against

vacuum quantum fluctuations. Extension beyond = 0of metric (4.2) is conventionally made by using new co-ordinates [24] T = 2, V = ln + x1 and x2 = x2, sothat

ds2 = dTdV + T dV2 + dx22 , (4.7)or by re-defining V = Y + Z and T T dV = Y Z,

ds2 = dY2 + dZ2 + dx22 (4.8)and

Y

2

Z2

= V

T T dV ,Y

Z

Y + Z =

t

T dV

V .

(4.9)

Metric (4.8) becomes positive definite when continuingthe new coordinate Y so that Y = i. By using thiscontinuation together with rotations (4.6) in expressions(4.9), we can deduce that the section on which and Zare both real corresponds to the region defined by e1

2 , x1 1/2, and that there exist two possible ansatzeaccording to which the continuation can be performed.One can first set

5


6/7

exp(i) =

i exp

2 Z2

Z

exp

2

+

2

exp(i), (4.10)

where only coordinate turns out to be periodic on

the Euclidean sector, with a period b = 2. Rotatingback to the Lorentzian region, we then have b = 2.Ansatz (4.10) should be associated with the self consis-tent Rindler vacuum considered by Li and Gott for four-dimensional Misner space [19], instead of the Minkowskivacuum with multiple images originally used by Hiscockand Konkowski [25]. Introducing then Rindler coordi-nates defined by v = sinh and w = cosh in thethree-dimensional covering metric (3.2), so that it be-comes

ds2 = 2d2 + d2 + dx2,

we can compute the Hadamard function for a conformallyinvariant scalar field in Rindler vacuum to be

G(1)(X, X) =1

22

sinh [( )2 + 2] , (4.11)

with X = ( , , x), X = (, , x), and

cosh =2 + 2 + (x x)2

2 . (4.12)

Using the method of images [25] and the usual defini-tions of the regularized Hadamard function and hencethe renormalized stress-energy tensor [18], we finally get

for the latter quantity an expression which is proportionalto

1

3

2

b

3 1

,

which, if we take b = 2 according to the above ansatz,obviously vanishes everywhere, even on the (Cauchy)chronology horizon at = 0.

Although, in spite of the Ford-Pfennings requirement[16], this is allowing the existence of stable superluminal,multiply connected warp drives of any size, this choicefor the vacuum has two further problems. First of all,previous work by Kay, Radzikowski and Wald [26] andby Cassidy [27] casts compelling doubts on the mean-ing of the Cauchy horizon, and hence on the validity ofthe conclusion that the stress-energy tensor is zero alsoon such a horizon. On the other hand, having an Eu-clidean section on which time is not periodic (which im-plies nonexistence of any background thermal radiation)while the spacetime has an event horizon appears to berather contradictory. It is for these reasons that we tendto favour the second possible ansatz implementing theEuclidean continuation of the three-dimensional Misnercovering, which appears to be less problematic. It reads:

exp

i

Z

= exp

2 Z2

2Z2

2

exp

2

+ 2

2

exp

i

+ 2

2

, (4.13)

where both the Euclidean time and the Euclidean co-ordinate are now periodic, with respective dimensionlessperiods = 1/2 and = 22. The physical time pe-riod in two-dimensional Alcubierre space at faster thanlight apparent velocity, Al, can be related to period by means of the expression

v0Al = 4

grrAldr

d

r=r0

.

Using then the Euclidean continuation of Eq. (4.3), onecan obtain Al = 4/A(r0), which corresponds to abackground temperature, TAl = A(r0)

/4, that is the

same as that which is associated with the event horizonof the spaceship and was first derived by Hiscock [18].Rotating back to the Lorentzian section we see that time becomes again no longer periodic, but x1 keeps stilla periodic character with period b = 22. If we ad-here to ansatz (4.13), this would mean that the Misnerspace itself should be modified in such a way that itsspatial volume would vanish as time approaches zero[20]. When calculating then the regularized Hadamardfunction, one should use a method of images which oughtalso to be modified accordingly with the fact that the pe-riod of the closed spatial direction is time dependent. Ifwe use a most general automorphic field and impose con-

stancy for the frequency of the general solution of thewave equation [20], one is led [20] to a time quantizationthat unavoidably implies an also strictly zero value forthe renormalized stress-energy tensor, everywhere in thewhole two-dimensional, superluminal Alcubierre space.The price to be paid [20] for this quantum stability isthat the CTCs developing inside the warp bubble andactually the warp bubble itself should never exceed a sub-microscopic size near the Planck scale, so avoiding not

just the unwanted Hiscock instability, but also any viola-tion of negative energy-time Ford like [15,16] inequalityand hence any unphysical nature of the warp drive. Apossible remaining question is whether the CTCs withinthe bubble might produce new divergences, at least ifHawkings chronology protection conjecture [28] is cor-rect. However, the semiclassical instabilities leading tochronology protection are actually of the kind which areprecisely prevented in our above model.

V. SUMMARY AND CONCLUSIONS

Among the achronal pathologies which are present ingeneral relativity and that can be associated with thesymmetries of the Misner space, we consider in this paper

6


7/7

the two-dimensional warp drive with an apparent velocitywhich is faster than light, and whose spaceship interioris multiply connected and therefore nonchronal. Afterreviewing the geometrical properties of the Alcubierre-Hiscock two-dimensional model, it has been proved thatthere exists a close connection between warp drives and

gravitational kinky topological defects, at least in twodimensions. Generalizing to the standard Finkensteinkinked metric to allow a complete description of the one-kinks, we also generalize the Alcubierre metric in such away that it can no longer be described in just one coor-dinate patch.

The geodesic incompleteness of the Alcubierre-Hiscockspace at the event horizon for superluminal warp bubbleshas been eliminated by first extending the metric be-yond the horizon according to the Kruskal procedure, andthen by an embedding in a three-dimensional Minkowskispace. It has been also shown that, if the latter space isprovided with the topological identifications that corre-spond to the universal covering of the three-dimensional

Misner space, then one can convert the interior of thewarp spaceship into a multiply connected space withclosed timelike curves which is able to behave like a timemachine.

The problem of the quantum stability of the two-dimensional, multiply connected warp drive spacetimehas been finally considered. Using the three-dimensionalMisner embedding as the maximal extension of the two-dimensional warp drive space, it has also been shown thatthe divergence encountered by Hiscock on the event hori-zon for the simply connected case is smoothed out, whilethe apparent horizon becomes a regularized (Cauchy)chronology horizon. We argued as well that the unphysi-

cal violation of the negative energy-time inequalities canat the same time be circumvented because the most con-sistent quantum treatment for dealing with vacuum fluc-tuations in the multiply connected case leads also to theresult that the size of the spaceship bubble and its closedtimelike curves must necessarily be placed at scales closeto the Planck length.

ACKNOWLEDGMENTS

The author thanks C.L. Siguenza for useful discussions.This work was supported by DGICYT under Research

Project No. PB97-1218.

[1] K.S. Thorne, Black Holes and Time Warps (Norton,New York, USA, 1994); M. Visser, Lorentzian Worm-holes (American Institute of Physics Press, Woodberry,USA, 1996).

[2] M.S. Morris, K.S. Thorne, and U. Yurtsever, Phys. Rev.Lett. 61, 1446 (1988); M.S. Morris and K.S. Thorne, Am.J. Phys. 56, 395 (1988); B. Jensen and H.H. Soleng, Phys.Rev. D45, 3528 (1992).

[3] P.F. Gonzalez-Daz, Phys. Rev. D54, 6122 (1996).[4] P.F. Gonzalez-Daz and L.J. Garay, Phys. Rev. D59,

064020 (1999).

[5] J.R. Gott, Phys. Rev. Lett. 66, 1126 (1991).[6] J.D.E. Grant, Phys. Rev. D47, 2388 (1993).[7] H.D. Politzer, Phys. Rev. D49, 3981 (1994).[8] J.R. Gott and Li-Xin Li, Phys. Rev. D58, 023501 (1988).[9] P.F. Gonzalez-Daz, Phys. Rev. D59, 123513 (1999).

[10] P.F. Gonzalez-Diaz and C.L. Siguenza, Phys. Lett. A248,412 (1998).

[11] M. Alcubierre, Class. Quantum Grav. 11, L73 (1994).[12] K.S. Thorne, in Directions in General Relativity, Proceed-

ings of the 1993 International Symposium, Maryland; inhonor of Charles Misner, edited by B.L. Hu, M.P. Ryan,Jr., and C.V. Vishveshwara (Cambridge University Press,Cambridge, England, 1993), Vol. 1.

[13] A.E. Everett, Phys. Rev. D53, 7365 (1996).[14] C.W. Misner, in Relativity Theory and Astrophysics I.

Relativity and Cosmology, edited by J. Ehlers (AmericanMathematical Society, Providence, RI, USA, 1967).

[15] L.H. Ford, Phys. Rev. D43, 3972 (1991); L.H. Ford andT.A. Roman, Phys. Rev. D55, 2082 (1997).

[16] L.H. Ford and M.J. Pfenning, Class. Quantum Grav. 14,1743 (1997).

[17] C. Van Den Broeck, A warp drive with reason-able total energy requirements, gr-qc/9905084; On the(im)possibility of warp bubbles, gr-qc/9906050.

[18] W.A. Hiscock, Class. Quantum Grav. 14, L183 (1997).[19] Li-Xin Li and J.R. Gott, Phys. Rev. Lett. 80, 2980 (1998).[20] P.F. Gonzalez-Daz, Phys. Rev. D58, 124011 (1998).[21] A.E. Everett and T.A. Roman, Phys. Rev. D56, 2100

(1997).[22] D. Finkelstein and C.W. Misner, Ann. Phys. (N.Y.) 6,

230 (1959).[23] D. Finkelstein and G. McCollum, J. Math. Phys. 16, 2250

(1975).[24] S.W. Hawking and G.F.R. Ellis, The Large Scale Struc-

ture of Space-Time (Cambridge University Press, Cam-bridge, England, 1973).

[25] W.A. Hiscock and D.A. Konkowski, Phys. Rev. D26,1225 (1982).

[26] B.S. Kay, M.J. Radzikowski and R.M. Wald, Commun.Math. Phys. 183, 533 (1997).

[27] M.J. Cassidy, Class. Quantum Grav. 14, 3031 (1997).[28] S.W. Hawking, Phys. Rev. D46, 603 (1992).

7
http://arxiv.org/abs/gr-qc/9905084http://arxiv.org/abs/gr-qc/9906050http://arxiv.org/abs/gr-qc/9906050http://arxiv.org/abs/gr-qc/9905084

Documents

Pedro F. Gonzalez-Dıaz- On the warp drive space-time