Original Warp Article

8/14/2019 Original Warp Article

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arXiv:gr-qc/0107097v23

Nov2001

Reduced total energy requirements for a modified

Alcubierre warp drive spacetime

F. Loup

Royal Sun Alliance

Avenida Duque De Avila 141 2 Andar 1050 Lisbon Portugal

D. Waite

Modern Relativity

E. Halerewicz, Jr.Lincoln Land Community College

5250 Shepherd Road, Springfield, IL 62794 USA

July 29, 2001

Abstract

It can be shown that negative energy requirements within the Alcu-bierre spacetime can be greatly reduced when one introduces a lapse func-tion into the Einstein tensor. Thereby reducing the negative energy re-quirements of the warp drive spacetime arbitrarily as a function ofA(ct,).With this function new quantum inequality restrictions are investigatedin a general form. Finally a pseudo method for controlling a warp bubbleat a velocity greater than that of light is presented.

1 Introduction

In recent years the possibility of interstellar travel within a human lifetime hasbecome a hurtle for theoretical physics to over come. The discussion involvesthe use of radically transforming the geometry of spacetime to act as a globalmeans of providing apparent Faster Than Light (FTL) travel. In 1994 MiguelAlcubierre then of Wales University introduced an arbitrary spacetime func-tion which provided an apparent means of FTL travel within the frame work of

General Relativity (GR) [1]. The drawbacks of this revolutionary new form ofpropulsion results from the production of causally disconnected spacetime re-gions and major violations of the standard energy conditions within GR. Furtherinvestigations into the warp drive spacetime have suggested that any manipu-lation of the spacetime coordinates in regards to FTL travel requires negative

This address is given for mailing purposes only, although I hold a professional position atthe above establishment my work was carried out independently and does not reflect RSA.;email: [email protected]

homepage: http://www.modernrelativity.com; email: [email protected] of above institution only; email: [email protected]

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http://arxiv.org/abs/gr-qc/0107097v2http://arxiv.org/abs/gr-qc/0107097v2http://arxiv.org/abs/gr-qc/0107097v2http://arxiv.org/abs/gr-qc/0107097v2http://arxiv.org/abs/gr-qc/0107097v2http://arxiv.org/abs/gr-qc/0107097v2http://arxiv.org/abs/gr-qc/0107097v2http://arxiv.org/abs/gr-qc/0107097v2http://arxiv.org/abs/gr-qc/0107097v2http://arxiv.org/abs/gr-qc/0107097v2http://arxiv.org/abs/gr-qc/0107097v2http://arxiv.org/abs/gr-qc/0107097v2http://arxiv.org/abs/gr-qc/0107097v2http://arxiv.org/abs/gr-qc/0107097v2http://arxiv.org/abs/gr-qc/0107097v2http://arxiv.org/abs/gr-qc/0107097v2http://arxiv.org/abs/gr-qc/0107097v2http://arxiv.org/abs/gr-qc/0107097v2http://arxiv.org/abs/gr-qc/0107097v2http://arxiv.org/abs/gr-qc/0107097v2http://arxiv.org/abs/gr-qc/0107097v2http://arxiv.org/abs/gr-qc/0107097v2http://arxiv.org/abs/gr-qc/0107097v2http://arxiv.org/abs/gr-qc/0107097v2http://arxiv.org/abs/gr-qc/0107097v2http://arxiv.org/abs/gr-qc/0107097v2http://arxiv.org/abs/gr-qc/0107097v2http://arxiv.org/abs/gr-qc/0107097v2http://arxiv.org/abs/gr-qc/0107097v2http://www.modernrelativity.com/http://www.modernrelativity.com/http://www.modernrelativity.com/http://arxiv.org/abs/gr-qc/0107097v2


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Loup, Waite, Halerewicz Reduced total energy requirements

energy densities [2]. Alcubierres generic model coupled with causally discon-nected spacetime regions and large negative energy densities make it hard to

build a justifiable model of a working warp drive spacetime. However recently ithas been shown that quantum inequalities can allow for the existence of negativeenergy densities [3], and it has also been found that classical scalar fields cangenerate large fluxes of negative energy in comparison to Quantum Inequality(QI) restrictions [4]. Coupled with recent astrophysical and cosmological obser-vations in support of negative energy in the form of Quintessence [5] we feel thatthere is still new life to be brought into the warp drive spacetime. The oncedistant dream of reaching the stars may become a reality with the aid of thewarp drive spacetime using latin we can express our motto as Ex Somnium AdAstra (ESAA) which translates; from a dream to the stars1. The following workwas made possible as a collaboration of the ESAA motto in order to producea physically justifiable model of the warp drive spacetime2. The inspirationbehind this work was to choose a somewhat less arbitrary means of defining theAlcubierre Warp Drive spacetime. Specifically we have set out the task of re-defining the functions of the warp drive spacetime in order to reduce the overallenergy requirements as seen in respect to the Einstein tensor.

1.1 The Alcubierre spacetime

The Story of the Warp Drive begins with the Alcubierre paper [1], where wehave the following metric:

ds2 = dt2 (dx vsf(rs)dt)2 dy2 dz2. (1)

We have also modified the metric signature in (1) to correspond to (1,1,1,1),for reasons which will become clear in later sections. The Alcubierre spacetime

is further defined with the following functions

vs(t) =dxs(t)

dt(2)

rs(t) = [(x xs(t))2 + y2 + z2]1/2. (3)

This spacetime model is often referred to a top hat model, by means of a bumpparameter , which is seen through

f(rs) =tanh((rs + R)) tanh((rs R))

2 tanh(R)(4)

This equation then allows for a warp bubble to develop in which a star ship

may ride. The generalities of the Alcubierre top hat function can be describedby the given volume expansion

= vsxsrs

df

drs. (5)

1The authors would like to note that the ESAA motto was created by Simon Jenks whohelped to formalize the early discussions regarding this work.

2The collaboration of this work was made possible through the online discussions forum;Alcubierre Warp Drive currently available at the following URL:http://clubs.yahoo.com/clubs/alcubierrewarpdrive

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http://clubs.yahoo.com/clubs/alcubierrewarpdrivehttp://clubs.yahoo.com/clubs/alcubierrewarpdrivehttp://clubs.yahoo.com/clubs/alcubierrewarpdrivehttp://clubs.yahoo.com/clubs/alcubierrewarpdrive


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This is where we receive our first scientific definition of a warp drive. Thevolume elements expand behind the spaceship, while contracting in front of it.

The draw back of this work occurs with equation 19, where the energy densityTabnanb violates the weak, strong, and dominate energy conditions as seen byan Eulerian observer [1]:

= Tabnanb = a2 T00 =

c2

8GG00 =

c2

8G

v2s 2

4r2s

df

drs

2. (6)

2 The ESAA spacetime

One can choose to describe the function of the warp drive in relationship to thefunction g(), where = rs and g() = 1 f(). We approach this aspect ina manner not to dissimilar to the one chosen by Hiscock [6] where we impose a

pseudo metric transformation through the following functions

g00 = A()2 [vs(r) g()]

2 (7)

g01 = g10 = vs g() (8)

g11 = g22 = g33 = 1 (9)

In order to reduce the energy requirements arbitrarily we consider the followinglapse function A(ct,). Where have set A = 1 both at the location of the ship,and far from it, but allow it to become large in the warped region following thelapse function A(ct,). Such that the above functions (7-9) have the followingmetric signature in cylindrical coordinates:

ds2 = [(A) vs(r) g()]2 + 2vsg()dctdz

dz2 dr2 r2d2. (10)

We further define (10) with the following coordinate transformation:

z = z

ctvsdct (11)

such that the ship velocity has the following definition vs = 1 = c therebyarriving at

ds2 = (A(ct,)2 vs(ct)2g()2)dct2 2vs(ct)g() cos()dctdr

+ 2vs(ct)g() sin() dctd dr2 2d2 2 sin2()d2. (12)

Thus the exact the solution for the Einstein tensor G00 for (12) is given by

Gct ct = 14

vs(ct)2

g()

sin2()/

A(ct,)2 vs(ct)2g()2

+ vs(ct)2g()2 cos2() + vs(ct)2g()2 sin2()

2(13)

where we have imposed the trigonometric identity sin2() + cos2() = 1.We now just briefly discuss how the dramatic energy reductions can occur

within the warp drive spacetime. First beginning with the Alcubierre metric inthe ship frame coordinates which is stated so that the function of interest is g()instead of f(). Alcubierre originally introduced a gravitational time dilationterm that doesnt limit ship speed in the g00 term of his metric that has been

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U = vsgcA1 (19)

[U] =

cA1

vsgcA1

00

(20)[U] = [ cA 0 0 0 ]. (21)

By inserting TUU = A2T00c2 into the inequality (15) we have

0

+

A2T00

2 + 20d

h

c33

32240(22)

and

0 +

2

A2

2 dg

d2

d

2

+ 2

0

Gh

c7

3

r2

4

0

(23)

where we can make the replacement

2 = r2 + z2 (24)

with r = const.Pfenning then makes an approximation of the geodesic motion of the Eule-

rian observer. Equation eq. 5.11 (note further references to Pfenning in thissection refer to [7]). Since we are using the ship frame with A(ct,) inserted,we must include its effect here as well

z vg(r)A1(r) (25)

2 = r2 + v2g2A22 (26)

such that

0

+

v2

A2r2 + v2g22

dg

d

2d

2 + 20

Gh

c53

r240. (27)

We make a definition corresponding to Pfennings equation 5.15

=r

[vg(r)](28)

And according to Pefenning 5.16 the integral should be approximately

A(0 + A)

Gh

c5

32

v2

2

4

0

(29)

So the inequality becomes

3

Gh

c52[A(r)]2

v240

v0

rA(r)g(r) + 1

. (30)

This result is the same as Pfenning eq 5.17 with the exception that A(r) is notnecessarily 1. Pfenning then asserts that this result must hold for sample timesthat are small compared to the square root inverse of the largest magnitude ofthe Riemann tensor components as calculated for the local observers frame.

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If for simplicity A is kept approximately constant through the negative en-ergy region, then the largest Riemann tensor component for this spacetime isR00 2(dg/d)2 (v/)2 (31)So in this case, the sampling time must be restricted to

0 = /v (32)

where 0 < < 1.Inserting this into the inequality and looking at the case of large A approx-

imately constant A0 through the negative energy region leads to

(3/)1/2(Gh/2c3)1/2(v/c)A0/2 (33)

Choosing = 0.10, this approximates to

102(v/c)lP lA0 (34)

Now we see that letting A become arbitrarily large also arbitrarily thickens theminimum warp shell thickness. Therefore the -0.068 solar mass calculation [7]had a reachable shell thickness. All that remains then is to divide the Pfen-ning -0.068 solar mass result by an A40 which would allow the thickness chosen,and which will lower the energy magnitude even farther by several orders ofmagnitude.

4 Total Energy Calculations

The energy requirements for the warp drive spacetime can be calculated from

E =

T00dV (35)

For T00 we have

T00 = vsc

4

32G

dg

d

2r2

r2 + z21

A4(36)

Where we assume A = const to be large, one can also see that this is identicalto eq. (14). We now choose

dV = 2 sin()ddd

so that we can write

E = v

2

s

c4

12G0

2dgd2 1

A4

d (37)

which reduces to

EESAA =

0

vs12

c4

G

dg

d

21

A4

2d. (38)

Thus from section 3 we can show similarly that the energy can be given from:

E =

0

v2s c4

12G

1

21

A4

2d. (39)

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5 Post relativistic warp drives?

One of the requirements for a warp drive spacetime which has a velocity greaterthan that of light is the existence of negative energy [2]. However this negativeenergy, exotic matter, quintessence, or however you chose to define it can bemade to pseudoly appear from negative brane tension [8]. More specifically forour case, an Anti de Sitter (AdS) spacetime would b e a logical choice for suchan exploration. The reason for this choice is the fact that acts as a scalarfield to exploit negative energy densities [4]. In reference to [9] we can write anarbitrary metric for a single warped extra dimension that has 3-D rotationalinvariance in AdS space by

ds2 = a2(r, t)dt2 + b2(r, t)dx2 + c2(r, t)dr2 (40)

This equation at a first approximation is identical to eq. (10). Now considering

eq. (12) in reference to eq. (40) a five-dimensional AdS warp drive spacetimecan be given by

ds2 = bg()2dt2 + A(ct, )d2 + (A(ct, ) vs(ct, )

2g()2)

dct2 + d2 + 2d2 + sin2 d2 (41)

where

d2 =d2

(1 L22)+ 2d22 (42)

with being the curvature scale and L being the length scale of the bulk b.Where ct, = const and d is a unit metric for a 3 brane. From this theAlcubierre warp drive function f() expands and contracts within the (3+1)slice of the AdS spacetime via a scale factor dependent on time ( ct, 0). Since

the five-dimensional AdS spacetime contains slices of the four-dimensional spacethe time coordinates rescale differently at different points in the extra dimensionof the de Sitter space. Thus causing the (ct,) terms to scale differently in the5-D space than it does in the 4-D space, therefore a gravitational wave in 4-D space can appear to exceed the light because the bulk spacetime acquiresdifferent velocities for c = 1 when moving through the extra dimension. Thisspacetime would also act to suppress the need one to make one paradox of[11]. When the 5-D space is static and there is no gauge field one can have amore general solution

ds2 = A(ct,)dt2 + L22d2 + (A(ct,) vs(ct,)2g()2)d2 (43)

with

L2 = 16

25bk (44)

which would essentially describe a warp drive in which v0 < c. What is inter-esting to note about eq. (41) is that it can make the 4-D warp drive functiong(), appear to be generated from a cosmologic term of order g() in AdSspace. Meaning that matter within the warp bubble wouldnt need to becometachyonic to solve the horizon [control] problem of the Alcubierre Warp Driveas c v0 (i.e. the scalar field acts to create a hyper warp drive without largenegative energy requirements), it is however noted that inducing the fifth di-mension is purely a mathematical trick which may have problems of its own. In

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the tradition of Alcubierre the warp drive of eq. (41) just begs to be named afterits familiar counterpart, the hyperdrive of science fiction. We also note that

a similar but more conventional approach was taken by Gonzalez-Diaz [10], inwhich a Alcubierre-Hiscock spacetime is embedded in a three-dimensional Mis-ner space, which fits into the frame work of classical GR as opposed to thisdiscussion.

5.1 a Broeck extended AdS spacetime?

Equation (41), shouldnt look to surprising at first hand, let us look at theBroeck spacetime [12]:

ds2 = dt2 + B2()[dx vs(t)f()dt)2 + dy2 + dz2] (45)

We see that L22 B2(), thus (43) can be interpreted as an AdS Broeck

spacetime, this is because B2() would naturally arise from primed coordinatetransformations [13], eq.(11) is a generic example of this. Thus when comparingequation 5 of [12] to (43), something quite odd occurs to the lapse functionA(ct,) when v0 > c effectively increasing the apparent amount of negativeenergy within the local brane. Moreover a similar scenario for (43) implies acoefficient of order A2bkg()2. The resulting total energy of the bulk spacetimein comparison to (38) when v0 c is thus

Ebk =

0

vsc4

12G

1

A(ct, )22d3 (46)

Notice that the bulk comsomologic term bk acts to flip the negative energyour local brane. This can be expected because a brane with negative tension

can contribute and act positively within a higher dimensional brane [8], andtherefore appear to violate the NEC (Null Energy Condition). This also suggestthat k = K

+ K

, i.e. the center of the Broeck bubble contains agravitational kink.

We now want to discuss a specific boundary condition to illustrate the prop-erties of a hyperdrive. To begin this discussion the lapse function will be givenas

A(ct,) = P W L = fP W L (47)

thus to arrive at specific boundary condition we replace (4) with a function thatresembles eq. 4 of [3] therefore we have the following piecewise function:

fP W L() =

1 < R 2

(1 + [(R

2 )(R +

2 )]/D)

N

R

2 < < R +

21 > R +

2

(48)It is also noted that the R(/2) term could be a boundary equal to or greaterthan one corresponding to some value 1 in standard units) and that thisvalue was chosen arbitrarily, however for this instance we have defined it inmanner which coincides with Pfennings analysis throughout this work. N isthe warp parameter which describes the magnitude of the warp factor:

N()warp = B() = 1 + (49)

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0

1

1

1

0

1

Figure 1: The ESAA spacetime represented one dimensionally in Cartesian

coordinates, with P W L = 1 this spacetime appears hyper-dimensional in com-parison to the Alcubierre spacetime f() = 0. This figure represents the varyingwarp parameter N the minimum value N = B() = 1 gives the Alcubierre tophat f() = 1 (black). The median (gray) and maximum (white) represents anenergy reduction due to a decreased volume as a factor of the warped region. Foran arbitrary boundary condition P W L = (1 + [(R

2)(R +

2)]/D)N,

when R = 100 m the maximum warp factor is N = 9.3, where the lapsefunction becomes singular.

this definition describes how the Alcubierre spacetime differs from our own3.With the boundary condition (48) the energy requirements for this warp

drive spacetime can be given from:

Tnn = 2P W L T

00

= (R + 2

)2N G00

=

(R + 2

)2N

v2s /4

[sin2] [dg()/d]2

(50)

With N = 1 eq. (46) becomes

Ebk =vsc

4

144G

1

(R + 12

)2N

0

4d (51)

orEbkc2

1.7 1038g = 1.2 105 MSun. (52)

The ESAA total energy EESAA is given with N = 1 since v0 = c, this depictsan Alcubierre like spacetime from footnote 3 we see that

N =

1

lim0

+ 1 = 0 + 1 = 1.

However, we now have v0 > c, thus 1 in standard units, by comparing eq.(38) to eq. (51) we see that = 1, therefore meaning the hyperdrive is defined

3Where B() is derived from eq. 5 of [12] and the warp parameter N we can also definealpha as = N 1.

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through the following parameter

N =1

+ 1 = 1 + 1 = 2.

Thus when applying an arbitrary boundary condition P W L = (b)N for the

warped region, we see that N, would reduce the energy requirements for anarbitrary total energy calculation such as (38). Thus it is seen that under theseconditions the velocity of the warp bubble is to be implicitly interpreted asv0 = 10

2c.In other words using a Pfenning inspired boundary condition it takes just

under one earth mass to achieve a velocity 100 times that of light. From theships frame the apparent velocity is much less from [ 13]:

dx

dt=

dx/dt

1 dx/dt(53)

we see that a fifteen day trip to alpha centauri, would give a local apparentvelocity of v = 0.01002c, thus compounding the aberration effects of [14].

5.2 metric patching

From this it can be seen why the warp drive violates the standard energy con-ditions in GR, assuming a negative brane tension one has [ 8]:

Tkk = Dginducedkk

np(na) = D

npa=1

(na k)2

np(na) (54)

from this the NEC becomes Tkk < 0. When a killing vector K is insertedin (54), for the right hand side we have

Tkk K = T = D

npa=1

(na )2

np(na) (55)

which is exactly what we would expect from (41), with the presence of a gaugefield. In terms of the AdS geometry this is caused by the scale factor (ct, ),such that the time lapse function is given by the relation A2bk = A(ct, )

2, i.e.converting the brane negative energy into positive energy in the bulk frame, assuggested above.

The total energy in the bulk is positive with rs [, a(t)] thus with metricpatching the energy in our three brane can be calculated from:

G = + h + KK+ (56)

Throughout our analysis of the hyperdrive we have considered a metric patchingof (3+1) within an embedding in E4, while in previous works the warp drive hasbeen considered in the form (2+1) embedded in E3 [10]. A similar constructionhas also been considered for transversible wormholes [8] in reference to stringtheory, which conforms to an (n+1) bulk and an ([n-1]+1) brane, again assuggest by Everett the warp drive appears as a special case wormhole (with theexception that this wormhole appears to be a shortcut through hyperspace).

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6 Summary

The goal we set out was to reduce the energy requirements of the warp drivespacetime to more physically realistic amounts. Our analysis has shown thatsimply considering a warp drive spacetime with an arbitrary lapse function candramatically lower the energy requirements within the Einstein tensor. We haveshown that Quantum Inequalities are a poor choice for curved spacetimes, buteven so minor modifications of QIs can also reduce their allowable energy re-strictions. Finally we have derived a mathematical trick which may allow thewarp drive spacetime to survive within the superluminal range. In closing itcan be shown that the warp drive spacetime can not be ruled out because ofunphysical energy conditions alone, by simply choosing an arbitrary functionA(ct,) the energy reductions become rather dramatic. It is however notedthat such an arbitrary function is merely a mathematical devise, within a four-dimensional spacetime there is no mechanism to set N > 1 as far as we know.However, it is somewhat easy to consider N = 2 for a possible hyperdrivespacetime, with our discussion this would correspond to a five-dimensional AdSspacetime (in which a warp bubble could be interpreted to move with a ve-locity that is 100 times that of light), and without experimental verification ofthe fifth coordinate this may also be considered a clever trick of its own. It istherefore noted that the warp drive spacetime appears to have several limita-tions within semi-classical GR, e.g. the violation of the Null Energy Condition(NEC), therefore it is likely that a more plausible warp drive may reside withinpost relativistic corrections of GR, indeed Ex Somnium Ad Astra.

Acknowledgement

The authors would like to express their sincere gratitude towards Miguel Al-cubierre for his helpful discussions and appreciated comments throughout thedevelopments of this work.

A Selected exact contrivariant solutions of the

Einstein tensor

The exact solutions to the ESAA metric (12) were produced in Maple VI withthe aid of GRTensorII and are as follows (and for that reason vs is representedby in the following calculations):

Gct = 14

4A(ct,)2 + 4(ct)2g()2 cos2() 4(ct)2g()2

+ (ct)2g()sin2()

g()

+ 4(ct)2g()sin2()

g()

cos()(ct)

/

A(ct,)2 (ct)2g()2 cos2() + (ct)2g()2 sin2()

2(57)

G = 14

2A(ct,)2

ct (ct)

+ 4A(ct,)2(ct)2g()cos()

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2(ct)

ctA(ct,)

A(ct,) + (ct)4g()2

cos()sin2

()

g()

+ 4(ct)4g()3 cos()3 4(ct)4g()3

cos() + 4(ct)4g()3 cos()sin2()

g()

sin()

2

A(ct,rho)2 (ct)2g()2

+ (ct)2g()2 cos2)() + (ct)2g()2 sin2()2

(58)

B Selected exact covariant solutions of the Ein-

stein tensor

Again these are the solutions to the metric (12), with the vs to modifications:

G =1

4

8A(ct,)

A(ct,)

+ 4(ct)2g()sin2()

g()

2sin2()

8(ct)2g()

g()

A(ct,)2 (ct)2g()2

+ (ct)2g()2 cos2() + (ct)2g()2 sin2()

(59)

Gct ct = 1

4(ct)2

4A(ct,)3

2

2A(ct,)

g()2 sin2()

+ 4(ct)2g()3 sin4()

2

2g()

A(ct,)2

+ 4(ct)2g()3 cos2()sin2() 2

2 g() A(ct,)2 4A(ct,)2(ct)2g()3

2

2g()

sin2()

4(ct)2g()4 sin2()A(ct,)

2

2 A(ct,)

cos2()

+ 4g()sin2()

2

2 g()

A(ct,)4

+ 4(ct)2g()3 cos2()sin2()

2

2g()

A(ct,)2

4A(ct,)(ct)2g()3

g()

A(ct,)

sin2()

+ 3(ct)4g()4 sin2()

g()2

cos2()

+ 4(ct)2g()4 sin2()A(ct,) A(ct,) 4(ct)2g()3 sin2()A(ct,)2

g()

4(ct)2g()4 sin4()A(ct,)

A(ct,)

8g()4 cos2()(ct)2A(ct,)

A(ct,)

8g()4 cos4()(ct)2A(ct,)

A(ct,)

12g()4 cos2()(ct)2A(ct,)

A(ct,)

sin2()

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8g()3 cos2()(ct)2

g()A(ct,)2

3(ct)2 sin4()

g()2 A(ct,)2g()2

3(ct)2 sin2()

g()

2g()2 cos2()A(ct,)2

+ 4(ct)2g()4

A(ct,)

2sin2()

3(ct)4g()4

g()

2sin2()

4(ct)2g()4

A(ct,)

2+ 3(ct)4g()4 sin4()

g()2

+ 6A(ct,)2

(ct)2

g()2

g()2

sin2

()

4g()sin2 A(ct,)3

A(ct,)

g()

+ 4(ct)2g()3 sin2()A(ct,)

A(ct,)

g()

cos2()

+ 4(ct)2g()3 sin4()A(ct,)

A(ct,)

g()

4(ct)2g()4 sin()2

A(ct,)2

cos2()

+ 8(ct)2g()3 cos4()

g()

A(ct,)2

+ 12(ct)2g()3 cos2()

g()

A(ct,)2 sin2()

+ 4(ct)2

g()3

sin4

()

g()

A(ct,)2

8g()2 cos2()A(ct,)3

A(ct,)

4g()2 sin2 A(ct,)3

A(ct,)

+ sin2()

g()

2A(ct,)4

+ 8g()cos2()

g()

A(ct,)4

+ 4g()sin2()

g()

A(ct,)4

/

A(ct,)2 (ct)2g()2 + (ct)2g()2 cos2()

+ (ct)

2

g()

2

sin

2

()2

(60)

References

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[2] Olum K. Superluminal travel requires negative energies. Phys. Rev. Lett.81 (1998) 356770 gr-qc/9805003

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Documents

Original Warp Article