Fernando Loup
email: spacetimeshortcut@yahoo.com

Is the Natario warp drive a valid candidate for an interstellar voyage to the star system KIC8462852??-Part I

In 2009 the Kepler Space Telescope was launched into space to look for extra-solar planets around distant stars. The portion of the sky covered by the Kepler Space Telescope was(and still is) a small region between the constellations of Cygnus and Lyra.When the Kepler analyzed the star KIC8462852 an anomaly was detected in the starlight distribution with an occultation that reduces the flux of light by 20 percent. Such anomaly as far as we know cannot be explained by natural causes. It is believed that the anomaly is due to an artificial alien megastructure that passes in front of the star blocking its light however this needs further confirmation. KIC8462852 lies at a distance of 600 parsecs.( 1956 light-years) from Earth.(Some works refers a distance of 454 parsecs).Since General Relativity allows arbitrary large speeds(superluminal velocities) under the framework of the warp drive theory we choose the Natario warp drive model in this work in an attempt to determine how large can these arbitrary velocities effectively be.A spaceship would need to attain a velocity of 10000 times light speed in order to make a round trip from Earth to KIC8462852 in a reasonable amount of time(in months not in years) however we would get in the expression of the negative energy density the factor 4, 86 × 10 50 !!!A number with 50 zeros!!!.This number is directly proportional to the magnitude of the number that presents the masses of 1 billion galaxies!!!!.In this work we demonstrate how to reduce the negative energy density requirements from 10 50 to 10 -53 Joules meters 3 or 10 -36 Joules meters 3 Very small values of negative energy density able to sustain a Natario warp bubble at 10000 times light speed in a journey from Earth to KIC8462852. We also describes briefly the star system KIC8462862 using the papers of Boyajian,Schaefer and Wright.

Introduction:

The Warp Drive as a solution of the Einstein field equations of General Relativity that allows superluminal travel appeared first in 1994 due to the work of Alcubierre.( [START_REF] Alcubierre | Classical and Quantum Gravity[END_REF]) The warp drive as conceived by Alcubierre worked with an expansion of the spacetime behind an object and contraction of the spacetime in front.The departure point is being moved away from the object and the destination point is being moved closer to the object.The object do not moves at all 1 .It remains at the rest inside the so called warp bubble but an external observer would see the object passing by him at superluminal speeds(pg 8 in [START_REF] Alcubierre | Classical and Quantum Gravity[END_REF])(pg 1 in [START_REF] Natario | Classical and Quantum Gravity[END_REF]).

Later on in 2001 another warp drive appeared due to the work of Natario.( [START_REF] Natario | Classical and Quantum Gravity[END_REF]).This do not expands or contracts spacetime but deals with the spacetime as a "strain" tensor of Fluid Mechanics(pg 5 in [START_REF] Natario | Classical and Quantum Gravity[END_REF]). Imagine the object being a fish inside an aquarium and the aquarium is floating in the surface of a river but carried out by the river stream.The warp bubble in this case is the aquarium whose walls do not expand or contract. An observer in the margin of the river would see the aquarium passing by him at a large speed but inside the aquarium the fish is at the rest with respect to his local neighborhoods. However there are 3 major drawbacks that compromises the warp drive physical integrity as a viable tool for superluminal interstellar travel.

The first drawback is the quest of large negative energy density requirements enough to sustain the warp bubble. In order to travel to a "nearby" star at 20 light-years at superluminal speeds in a reasonable amount of time a ship must attain a speed of about 200 times faster than light.However the negative energy density at such a speed is directly proportional to the factor 10 48 which is 1.000.000.000.000.000.000.000.000 times bigger in magnitude than the mass of the planet Earth!!!(see [START_REF] Loup | [END_REF], [START_REF] Loup | [END_REF] and [START_REF] Loup | [END_REF]).

Another drawback that affects the warp drive is the quest of the interstellar navigation:Interstellar space is not empty and from a real point of view a ship at superluminal speeds would impact asteroids,comets,interstellar space dust and photons.(see [START_REF] Loup | [END_REF], [START_REF] Loup | [END_REF] and [START_REF] Loup | [END_REF]).

The last drawback raised against the warp drive is the fact that inside the warp bubble an astronaut cannot send signals with the speed of the light to control the front of the bubble because an Horizon(causally disconnected portion of spacetime)is established between the astronaut and the warp bubble.(see [START_REF] Loup | [END_REF], [START_REF] Loup | [END_REF] and [START_REF] Loup | [END_REF]).

We can demonstrate that the Natario warp drive can "easily" overcome these obstacles as a valid candidate for superluminal interstellar travel(see [START_REF] Loup | [END_REF], [START_REF] Loup | [END_REF] and [START_REF] Loup | [END_REF]).

In this work we cover only the Natario warp drive and we avoid comparisons between the differences of the models proposed by Alcubierre and Natario since these differences were already deeply covered by the existing available literature.(see [START_REF] Loup | [END_REF], [START_REF] Loup | [END_REF] and [START_REF] Loup | [END_REF])However we use the Alcubierre shape function to define its Natario counterpart.

In 2009 the Kepler Space Telescope was launched into space to look for extra-solar planets around distant stars. The portion of the sky covered by the Kepler Space Telescope was(and still is) a small region between the constellations of Cygnus and Lyra. The method used by Kepler to detect distant planets is the method of transit when a planet pass in front of the star blocking its light.When the Kepler analyzed the star KIC8462852 an anomaly was detected in the starlight distribution with an occultation that reduces the flux of light by 20 percent. Such anomaly as far as we know cannot be explained by natural causes. It is believed that the anomaly is due to an artificial alien megastructure that passes in front of the star blocking its light however this needs further confirmation

The implications of the existence of aliens or extraterrestrial advanced civilizations capable to assemble large megastructures is beyond the scope of this work and we will not discuss this topic further.

The main reference used in our work about KIC8462852 is the paper of Jason Wright(Et Al) published by the Astrophysical Journal(Apj) which is sponsored by the American Astronomical Society(AAS). We use 3 different versions of the paper: [START_REF] Wright | [END_REF] is the arXiv e-print version submitted to ApJ, [14] is the arXiv e-print version accepted by ApJ and [15] is the official manuscript effectively published by ApJ. Each version is different than each other but in the end the conclusions are the same ones.

At pg 9 in each one of these versions it is mentioned that the anomaly requires 8 large external occulters placed in the outer regions of the alien megastructure and at least another 8 internal occulters placed in the inner regions of the alien megastructure. Also in pg 9 it can be seen that KIC8462852 lies at a distance of 600 parsecs. ( 1956 light-years). At pg 20 in [START_REF] Wright | [END_REF],pg 19 in [14] and pg 18 in [15] it is mentioned that KIC8462852 is surrounded by a swarm of artificial objects (see Appendices C and D).

We use also 2 auxiliary works about KIC8462852:the paper of Tabetha Boyajian(Et Al) in [17] where the anomalies in the starlight propagation of KIC8462852 appeared for the first time and the paper of Bradley Schaefer in [16].His work demonstrates definitely that the anomalies in the starlight propagation of KIC8462852 cannot be explained by natural causes. However in [16] pg 2 it is mentioned that KIC8462852 lies at a distance of 454 parsecs and in [17] pg 9 also a mention to the distance of 454 parsecs. This discrepancy between distances must be clarified. In the abs of [16] it is mentioned that the mechanism of the occultation is not due to a catastrophic planetary event but is an ongoing process with continuous effects.Such an event would require 648.000 comets each one with 200 km diameter all of them orchestrated to pass in front of the star with a regular daily trajectory.According to Schaefer this is physically impossible. (see also pg 8 in [16]).

From pg 8 in [START_REF] Alcubierre | Classical and Quantum Gravity[END_REF] pg 1 in [START_REF] Natario | Classical and Quantum Gravity[END_REF] pg 1 and 2 in [START_REF] Lobo | Classical and Quantum Gravity[END_REF] pg 26 and 34 in [START_REF] Lobo | [END_REF] we known that General Relativity allows arbitrary large speeds(superluminal velocities) under the framework of the warp drive theory and we choose the Natario warp drive model in this work in an attempt to determine how large can these arbitrary velocities effectively be.

A spaceship able to attain a superluminal velocity of 200 times faster than light can cover a distance of 20 light-years in a couple of months but would need 5 years to cover the distance between Earth and KIC8462852 assuming the maximum distance of 600 parsecs ( 1956) light-years. Considering our Natario warp drive moving with vs = 10000 which means to say 10000 times light speed in order to make a round trip from Earth to KIC8462852 at 2000 light-years away in a reasonable amount of time(in months not in years) we would get in the expression of the negative energy density the factor 4, 86 × 10 50 !!! a number with 50 zeros!!! The total amount of negative energy density needed to sustain a warp bubble in a round-trip travel from Earth to KIC8462852 at a speed of 10000 times faster than light is directly proportional to a number with 50 zeros!!!

The sun have a mass of 10 30 kilograms and our galaxy have nearly 400 billions of stars (4 × 10 11 ).Assuming an average mass for each star in our galaxy of about the estimative of the mass of the sun 2 we have for the total mass of the galaxy 10 30 × 10 11 = 10 41 kilograms. a number with "only" 41 zeros.

10 50 is a number 10 9 times bigger than 10 41 so this means to say that the total amount of negative energy density needed to sustain a warp bubble at 10000 times light speed in our journey from Earth to KIC8462852 is directly proportional to the magnitude of the number that presents the masses of 1 billion galaxies!!!! In this work we demonstrate how to reduce the negative energy density requirements from 10 50 to 10 -53 Joules meters 3 or 10 -36 Joules meters 3 Very small values of negative energy density able to sustain a Natario warp bubble at 10000 times light speed in our journey from Earth to KIC8462852

The title of this work is a question:Is the Natario warp drive a valid candidate for an interstellar voyage to the star system KIC8462852??

The answer is:Undoubtely yes.Not only the Natario warp drive can cross the distance between Earth and the star system KIC8462852 with an arbitrarily large superluminal velocity of 10000 times light speed but can also be used to cross enormous distances in our galaxy.

In order to fully understand this work acquaintance with the Alcubierre and Natario warp drive theory is required,We recommend non-familiarized readers to read the works in [START_REF] Alcubierre | Classical and Quantum Gravity[END_REF], [START_REF] Lobo | Classical and Quantum Gravity[END_REF] and [START_REF] Lobo | [END_REF] in order to study the Alcubierre model and the works in [START_REF] Natario | Classical and Quantum Gravity[END_REF], [START_REF] Loup | [END_REF], [START_REF] Loup | [END_REF] and [START_REF] Loup | [END_REF] in order to study the Natario model. Readers with advanced knowledge in General Relativity might want to read the works in [START_REF] Loup | [END_REF] and [START_REF] Loup | [END_REF] in order to study the new formalisms available for the Natario warp drive spacetime. This work is organized as follows:section 2 introduces the shape function for the Natario warp drive and illustrates the problem of the negative energy density.Sections 3 to 9 demonstrates how to reduce the negative energy density to low and affordable levels.Finally sections 10 describes briefly the star system KIC8462862 using the papers of Boyajian,Schaefer and Wright. This work must be regarded as a companion work to our work in [START_REF] Loup | [END_REF].

The Natario warp drive continuous shape function

Introducing here f (rs) as the Alcubierre shape function that defines the Alcubierre warp drive spacetime we can construct the Natario shape function N (rs) that defines the Natario warp drive spacetime using its Alcubierre counterpart.Below is presented the equation of the Alcubierre shape function. 3 .

f (rs) = 1 2 [1 -tanh[@(rs -R)] (1)
rs = (x -xs) 2 + y 2 + z 2 (2)
According with Alcubierre any function f (rs) that gives 1 inside the bubble and 0 outside the bubble while being 1 > f (rs) > 0 in the Alcubierre warped region is a valid shape function for the Alcubierre warp drive.(see eqs 6 and 7 pg 4 in [START_REF] Alcubierre | Classical and Quantum Gravity[END_REF] or top of pg 4 in [START_REF] Natario | Classical and Quantum Gravity[END_REF]). In the Alcubierre shape function xs is the center of the warp bubble where the ship resides. R is the radius of the warp bubble and @ is the Alcubierre parameter related to the thickness.According to Alcubierre these can have arbitrary values.We outline here the fact that according to pg 4 in [START_REF] Alcubierre | Classical and Quantum Gravity[END_REF] the parameter @ can have arbitrary values.rs is the path of the so-called Eulerian observer that starts at the center of the bubble xs = rs = 0 and ends up outside the warp bubble rs > R.

The square derivative of the Alcubierre shape function is given by:

f (rs) 2 = 1 4 [ @ 2 cosh 4 [@(rs -R)] ] (3)
According with Natario(pg 5 in [START_REF] Natario | Classical and Quantum Gravity[END_REF]) any function that gives 0 inside the bubble and 1 2 outside the bubble while being 0 < N (rs) < 1 2 in the Natario warped region is a valid shape function for the Natario warp drive. The Natario warp drive continuous shape function can be defined by:

N (rs) = [ 1 2 ][1 -f (rs) W F ] W F (4)
This shape function gives the result of N (rs) = 0 inside the warp bubble and N (rs) = 1 2 outside the warp bubble while being 0 < N (rs) < 1 2 in the Natario warped region. Note that the Alcubierre shape function is being used to define its Natario shape function counterpart.For the Natario shape function introduced above it is easy to figure out when f (rs) = 1(interior of the Alcubierre bubble) then N (rs) = 0(interior of the Natario bubble) and when f (rs) = 0(exterior of the Alcubierre bubble)then N (rs) = 1 2 (exterior of the Natario bubble).

The derivative square of the Natario shape function is :

N (rs) 2 = [ 1 4 ]W F 4 [1 -f (rs) W F ] 2(W F -1) [f (rs) 2(W F -1) ]f (rs) 2 (5)
The term W F in the Natario shape function is dimensionless too:it is the warp factor.It is important to outline that the warp factor W F >> |R| is much greater than the modulus of the bubble radius.Note that the square derivative of the Alcubierre shape function appears in the expression of the square derivative of the Natario shape function.

• Numerical plot for the Alcubierre and Natario shape functions with @ = 50000 bubble radius R = 100 meters and warp factor with a value W F = 200 rs f (rs) N (rs) f (rs) According with the numerical plot above when @ = 50000 the square derivative of the Alcubierre shape function is zero4 from the center of the bubble until 99, 996 meters.At 99, 997 meters the square derivative of the Alcubierre shape function is 2, 65 × 10 -251 and starts to increase reaching the maximum value of 6, 25×10 8 at 100 meters from the center of the bubble precisely in the bubble radius decreasing again to the minimum value of 2, 65 × 10 -251 at 100, 003 meters from the center of the bubble.At 100.004 meters from the center of the bubble the square derivative of the Alcubierre shape function is again zero.Note that with respect to the distance of 100 meters from the center of the bubble exactly the bubble radius the powers of the square derivative of the Alcubierre shape function are diametrically symmetrically opposed.We have the values of 10 -77 at 99, 999 meters and at 100, 001 meters.We have the value of 10 -164 at 99, 998 meters and at 100, 002 meters.So the thickness of the warped region is limited or defined by the square derivatives of the shape function when these are different than zero.In the case of @ = 50000 the warped region starts at 99, 997 meters and ends up at 100, 003 meters.The thickness of the warped region is then 0, 006 meters.

Note that inside the bubble the Alcubierre shape function possesses the value of 1 and the Natario shape function possesses the value of 0 and outside the bubble the Alcubierre shape function possesses the value of 0 and the Natario shape function possesses the value of 1 2 as requested.

Also while the square derivative of the Alcubierre shape function is not zero inside and outside the bubble however at the neighborhoods of the bubble radius and possesses the maximum value exactly at the bubble radius the square derivative of the Natario shape function is always zero inside and outside the bubble and possesses also a maximum value at the bubble radius however this value is extremely small when compared to its Alcubierre counterpart.

• Numerical plot for the Alcubierre and Natario shape functions with @ = 75000 bubble radius R = 100 meters and warp factor with a value W F = 200 rs f (rs) N (rs) f (rs) According with the numerical plot above when @ = 75000 the square derivative of the Alcubierre shape function is zero from the center of the bubble until 99, 997 meters.At 99, 998 meters the square derivative of the Alcubierre shape function is 5, 96 × 10 -251 and starts to increase reaching the maximum value of 1, 4 × 10 9 at 100 meters from the center of the bubble precisely in the bubble radius decreasing again to the minimum value of 5, 96 × 10 -251 at 100, 002 meters from the center of the bubble.At 100.003 meters from the center of the bubble the square derivative of the Alcubierre shape function is again zero.Note that with respect to the distance of 100 meters from the center of the bubble exactly the bubble radius the powers of the square derivative of the Alcubierre shape function are diametrically symmetrically opposed.We have the values of 10 -120 at 99, 999 meters and at 100, 001 meters.So the thickness of the warped region is limited or defined by the square derivatives of the shape function when these are different than zero.In the case of @ = 75000 the warped region starts at 99, 998 meters and ends up at 100, 002 meters.The thickness of the warped region is then 0, 004 meters.

Note that inside the bubble the Alcubierre shape function possesses the value of 1 and the Natario shape function possesses the value of 0 and outside the bubble the Alcubierre shape function possesses the value of 0 and the Natario shape function possesses the value of 1 2 as requested.

Also while the square derivative of the Alcubierre shape function is not zero inside and outside the bubble however at the neighborhoods of the bubble radius and possesses the maximum value exactly at the bubble radius the square derivative of the Natario shape function is always zero inside and outside the bubble and possesses also a maximum value at the bubble radius however this value is extremely small when compared to its Alcubierre counterpart.

The previous plots demonstrate the important role of the thickness parameter @ in the warp bubble geometry wether in both Alcubierre or Natario warp drive spacetimes.For a bubble of 100 meters radius R = 100 the regions where 1 > f (rs) > 0(Alcubierre warped region) and 0 < N (rs) < 1 2 (Natario warped region) becomes thicker or thinner as @ becomes higher.In the case of @ = 50000 the warped region starts at 99, 997 meters and ends up at 100, 003 meters.The thickness of the warped region is then 0, 006 meters and in the case of @ = 75000 the warped region starts at 99, 998 meters and ends up at 100, 002 meters.The thickness of the warped region is then 0, 004 meters.

Then the geometric position where both Alcubierre and Natario warped regions begins with respect to R the bubble radius is rs = R -< R and the geometric position where both Alcubierre and Natario warped regions ends with respect to R the bubble radius is rs = R + > R.The thickness of the warp bubble is then 2 × .As large as @ becomes as smaller becomes too.

Note from the plots of the previous pages that we really have two warped regions:

• 1)-The geometrized warped region where 1 > f (rs) > 0(Alcubierre warped region) and 0 < N (rs) < 1 2 (Natario warped region).The geometrized warped region lies precisely in the bubble radius. 5 • 2)-The energized warped region where the derivative squares of both Alcubierre and Natario shape functions are not zero.

The parameter @ affects both energized warped regions wether in Alcubierre or Natario cases but is more visible for the Alcubierre shape function because the warp factor W F in the Natario shape functions squeezes the energized warped region in a region of very small thickness centered in the bubble radius.

The negative energy density for the Natario warp drive is given by(see pg 5 in [START_REF] Natario | Classical and Quantum Gravity[END_REF])

ρ = T µν u µ u ν = - 1 16π K ij K ij = - v 2 s 8π 3(N (rs)) 2 cos 2 θ + N (rs) + rs 2 N (rs) 2 sin 2 θ (6)
Converting from the Geometrized System of Units to the International System we should expect for the following expression 6 :

ρ = - c 2 G vs 2 8π 3(N (rs)) 2 cos 2 θ + N (rs) + rs 2 N (rs) 2 sin 2 θ . (7)
Rewriting the Natario negative energy density in cartezian coordinates we should expect for 7 :

ρ = T µν u µ u ν = - c 2 G v 2 s 8π 3(N (rs)) 2 ( x rs ) 2 + N (rs) + rs 2 N (rs) 2 ( y rs ) 2 (8)
Considering as a simplified case the equatorial plane(1 + 1 dimensional spacetime with rs = x -xs ,y = 0 and center of the bubble xs = 0) we have:

ρ = T µν u µ u ν = - c 2 G v 2 s 8π 3(N (rs)) 2 (9)
Note that in the above expressions for the negative energy density the warp drive speed vs appears raised to a power of 2 and it is being multiplied by the square derivative of the shape function. Considering our Natario warp drive moving with vs = 200 which means to say 200 times light speed in order to make a round trip from Earth to a nearby star at 20 light-years away in a reasonable amount of time(in months not in years) we would get in the expression of the negative energy the factor c 2 = (3 × 10 In the bubble radius the presented value for the Natario shape function in the numerical plots of the previous pages is 0, 5 but actually is a value between 0 < N (rs) < This term is 1.000.000.000.000.000.000.000.000 times bigger in magnitude than the mass of the planet Earth!!!or better:The amount of negative energy density needed to sustain a warp bubble at a speed of 200 times faster than light requires the magnitude of the masses of 1.000.000.000.000.000.000.000.000 planet Earths!!! Note that if the negative energy density is proportional to 10 48 this would render the warp drive impossible but fortunately the term 10 48 is being multiplied the square derivative of the shape function and in the Natario case the square derivative of the shape function possesses values of 10 -102 or 10 -103 completely obliterating the factor 10 48 making the warp drive negative energy density more "affordable" because 10 48 × 10 -102 = 10 -54 Joules meters 3 a very low and affordable negative energy density.So in order to get a physically feasible Natario warp drive the square derivative of the Natario shape function must obliterate the factor 10 48 and fortunately this is really happening with our chosen shape function . Now we need to explain how and why the warp factor W F in the Natario shape functions squeezes the energized warped region in a region of very small thickness centered in the bubble radius.

The Alcubierre shape function and its derivative square are given by:

f (rs) = 1 2 [1 -tanh[@(rs -R)] (10)
f (rs) 2 = 1 4 [ @ 2 cosh 4 [@(rs -R)] ] (11)
The Natario shape function and its derivative square are given by:

N (rs) = [ 1 2 ][1 -f (rs) W F ] W F (12)
N (rs

) 2 = [ 1 4 ]W F 4 [1 -f (rs) W F ] 2(W F -1) [f (rs) 2(W F -1) ]f (rs) 2 (13)
N (rs) 2 = [ 1 4 ]W F 4 [1 -f (rs) W F ] 2(W F -1) [f (rs) 2(W F -1) ][ 1 4 [ @ 2 cosh 4 [@(rs -R)] ]] (14)
N (rs) 2 = [ 1 16 ]W F 4 [1 -f (rs) W F ] 2(W F -1) [f (rs) 2(W F -1) ][ @ 2 cosh 4 [@(rs -R)] ] (15)
The negative energy density in the 1 + 1 Natario warp drive spacetime is given by:

ρ = T µν u µ u ν = - c 2 G v 2 s 8π 3(N (rs)) 2 (16)
Examining now the negative energy density in the 1 + 1 spacetime from the Natario shape function with warp factors:

N (rs) 2 = [ 1 16 ]W F 4 [1 -f (rs) W F ] 2(W F -1) [f (rs) 2(W F -1) ][ @ 2 cosh 4 [@(rs -R)] ] (17)
ρ = T µν u µ u ν = - c 2 G v 2 s 8π [ 3 16 ]W F 4 [1 -f (rs) W F ] 2(W F -1) [f (rs) 2(W F -1) ][ @ 2 cosh 4 [@(rs -R)] ] (18)
The dominant term here is the term resulting from the warp factor which is:

[1 -f (rs) W F ] 2(W F -1) [f (rs) 2(W F -1) ] ( 19
)
This term is composed by two expressions that complementary neutralizes each other giving values of zero inside and outside the bubble squeezing the Natario warped region in a thin layer geometrically placed around the neighborhoods of the bubble radius.

The first expression that neutralizes the square derivative of the Natario shape function inside the bubble is:

[1 -f (rs) W F ] 2(W F -1) (20)
And the second expression that neutralizes the square derivative of the Natario shape function outside the bubble is:

[f (rs) 2(W F -1) ] ( 21
)
Inside the bubble f (rs) = 1 and [1 -f (rs) W F ] 2(W F -1) = 0 resulting in a N (rs) 2 = 0.This is the reason why the Natario shape function with warp factors do not have numerical values for the derivatives inside the bubble.

Outside the bubble f (rs) = 0 and [f (rs) 2(W F -1) ] = 0 resulting also in a N (rs) 2 = 0.This is the reason why the Natario shape function with warp factors do not have numerical values for the derivatives outside the bubble.

Inside the bubble f (rs) = 1 and [f (rs) 2(W F -1) ] = 1 however [1 -f (rs) W F ] 2(W F -1) = 0 and hence the warp factor product [1 -f (rs

) W F ] 2(W F -1) [f (rs) 2(W F -1) ] = 0.
Outside the bubble f (rs) = 0 and [f (rs)

2(W F -1) ] = 0 however [1 -f (rs) W F ] 2(W F -1) = 1 and hence the warp factor product is also [1 -f (rs) W F ] 2(W F -1) [f (rs) 2(W F -1) ] = 0.
Note that from the statements pointed above when one of the expressions have the value of 1 the other have the value of 0 and vice-versa.This explains how and why each expression complementary neutralizes each other in the regions inside and outside the bubble

In the Alcubierre warped region 9 1 > f (rs) > 0.In this region the derivatives of the Natario shape function do not vanish because if f (rs) < 1 then f (rs) W F << 1 resulting in an [1 -f (rs) W F ] 2(W F -1) << 1 but greater than zero.Consider for example a warp factor W F = 200 and an Alcubierre shape function f (rs) = 1 2 then f (rs) W F = f (rs) 200 = 1 2 200 .Since 2 200 = 1, 6069380442590E + 060 then 1 2 200 = 6, 2230152778612E -061 and 6, 2 × 10 -61 is very small when compared to 1 2 .

.Also if f (rs) < 1 then [f (rs) 2(W F -1) ] << 1 too and using the numbers given above then f (rs) 398 = 1 2 398 .Since 2 398 = 6, 4556246952173E + 119 then 1 2 398 = 1, 5490367659397E -120 and 1, 5 × 10 -120 is also very small when compared to 1 2 .

Note that if

[1 -f (rs) W F ] 2(W F -1) << 1 and [f (rs) 2(W F -1) ] << 1 then their product [1 -f (rs) W F ] 2(W F -1) [f (rs) 2(W F -1)
] <<<< 1 resulting in a very low derivative square for the Natario shape function in the Alcubierre warped region and hence in the Natario warped region with both centered geometrically over the bubble radius.

Note also that inside the Alcubierre warped region 1 > f (rs) > 0 when f (rs) approaches 1 N (rs) 2 approaches 0 due to the factor [1 -f (rs 1) and when f (rs) approaches 0 N (rs) 2 approaches 0 again due to the factor [f (rs) 2(W F -1) ] .Both expressions complementary neutralizes each other giving a very small product and hence a very small square derivative for the Natario shape function.

) W F ] 2(W F -
We will examine the above statement of the expressions that complementary neutralizes each other in the Natario warp drive 1 + 1 spacetime with details in the section 3.

Now we must analyze the more sophisticated case of the Natario warp drive in a real 3 + 1 spacetime where the negative energy density in this case is given by the following expressions(pg 5 in [2]) 10 :

• 1)-3 + 1 spacetime expression for the negative energy density with trigonometric terms:

ρ = - c 2 G vs 2 8π 3(N (rs)) 2 cos 2 θ + N (rs) + rs 2 N (rs) 2 sin 2 θ . (22)
• 2)-3 + 1 spacetime expression for the negative energy density with cartezian coordinates 11 :

ρ = T µν u µ u ν = - c 2 G v 2 s 8π 3(N (rs)) 2 ( x rs ) 2 + N (rs) + rs 2 N (rs) 2 ( y rs ) 2 (23)
9 Remember that the Natario warped region is defined in function of its Alcubierre counterpart 10 see Appendix A 11 see Appendix B

Working with the expanded trigonometric 3 + 1 spacetime expression for the negative energy density we have:

ρ 3+1 = - c 2 G vs 2 8π 3(N (rs)) 2 cos 2 θ - c 2 G vs 2 8π N (rs) + rs 2 N (rs) 2 sin 2 θ (24) ρ 3+1 = ρ 1 + ρ 2 (25) ρ 1 = - c 2 G vs 2 8π 3(N (rs)) 2 cos 2 θ (26) ρ 2 = - c 2 G vs 2 8π N (rs) + rs 2 N (rs) 2 sin 2 θ (27)
Comparing the above expressions with the negative energy density in the 1 + 1 spacetime :

ρ 1+1 = - c 2 G v 2 s 8π 3(N (rs)) 2 (28)
We can see that the term in ρ 1 almost matches the term in the 1 + 1 spacetime except for the trigonometric term in cos 2 θ and this term produces a very low derivative square for the Natario shape function of about 10 -103 and this will be seen in section 3.So the term ρ 2 is the term that really accounts for the negative energy density in the 3 + 1 spacetime.

The dominant expression in ρ 2 is:

N (rs) + rs 2 N (rs) 2 (29)
The expansion of the square in the binomial expression gives:

N (rs) + rs 2 N (rs) 2 = N (rs) 2 + 2N (rs) rs 2 N (rs) + N (rs) 2 (30)
N (rs) + rs 2 N (rs) 2 = N (rs) 2 + N (rs)rsN (rs) + N (rs) 2 (31)
N (rs) + rs 2 N (rs) 2 = N (rs) 2 + rsN (rs)N (rs) + N (rs) 2 (32)
Since the derivative of second order of the Natario shape function N (rs) is a lengthly expression with many algebraic terms then its square N (rs) 2 results in an even more complicated expression with even more algebraic terms.And the product of both the first and second order derivatives N (rs)N (rs) also results in a lengthly expression.Then in order to avoid algebraic complications we must work numerically with the new dominant term which is:

N (rs) + rs 2 N (rs) (33)
Raising to the square only the final numerically evaluated result.

The new dominant term in the expression for the negative energy density of the Natario warp drive in a 3 + 1 spacetime is:

N (rs) + rs 2 N (rs) (34)
The first order derivative of the Natario shape function is given by:

N (rs) = -[ 1 2 ]W F 2 [1 -f (rs) W F ] (W F -1) [f (rs) (W F -1) ]f (rs) (35)
With f (rs) being the first order derivative of the Alcubierre shape function

f (rs) = - 1 2 [ @ cosh 2 [@(rs -R)] ] (36)
The second order derivative of the Natario shape function is given by the lengthly expression:

N (rs) = [ 1 2 ]W F 3 (W F -1)[1 -f (rs) W F ] (W F -2) [f (rs) 2(W F -1) ]f (rs) 2 (37) -[ 1 2 ]W F 2 [1 -f (rs) W F ] (W F -1) (W F -1)[f (rs) (W F -2) ]f (rs) 2 (38) -[ 1 2 ]W F 2 [1 -f (rs) W F ] (W F -1) [f (rs) (W F -1) ]f (rs) (39)
With f (rs) being the second order derivative of the Alcubierre shape function

f (rs) = [ (@ 2 )sinh[@(rs -R) cosh 3 [@(rs -R)] ] (40)
From above we can see that the square of the second order derivative of the Natario shape function N (rs) 2 would result in a very algebraic complicated expression.In order to simplify our study we decompose the second order derivative of the Natario shape function in separated algebraic expressions as shown below

N (rs) = A + B + C (41)
With the expressions for A B and C given respectively by:

A = [ 1 2 ]W F 3 (W F -1)[1 -f (rs) W F ] (W F -2) [f (rs) 2(W F -1) ]f (rs) 2 (42) B = -[ 1 2 ]W F 2 [1 -f (rs) W F ] (W F -1) (W F -1)[f (rs) (W F -2) ]f (rs) 2 (43) C = -[ 1 2 ]W F 2 [1 -f (rs) W F ] (W F -1) [f (rs) (W F -1) ]f (rs) (44)
Then the expressions that really accounts for a numerical evaluation of the new dominant term in the equation for the negative energy density of the Natario warp drive in a 3 + 1 spacetime which is:

N (rs) + rs 2 N (rs) = N (rs) + rs 2 (A + B + C) (45)
Are the following ones:

N (rs) = -[ 1 2 ]W F 2 [1 -f (rs) W F ] (W F -1) [f (rs) (W F -1) ]f (rs) (46) A = [ 1 2 ]W F 3 (W F -1)[1 -f (rs) W F ] (W F -2) [f (rs) 2(W F -1) ]f (rs) 2 (47) B = -[ 1 2 ]W F 2 [1 -f (rs) W F ] (W F -1) (W F -1)[f (rs) (W F -2) ]f (rs) 2 (48) C = -[ 1 2 ]W F 2 [1 -f (rs) W F ] (W F -1) [f (rs) (W F -1) ]f (rs) (49)
Since

N (rs) = A + B + C (50)
With f (rs) being the first order derivative of the Alcubierre shape function

f (rs) = - 1 2 [ @ cosh 2 [@(rs -R)] ] (51)
And f (rs) being the second order derivative of the Alcubierre shape function

f (rs) = [ (@ 2 )sinh[@(rs -R) cosh 3 [@(rs -R)] ] (52)
Note that the most meaningful term f (rs) that occurs in all these expressions raised to powers of the warp factor W F is the Alcubierre shape function.

f (rs) = 1 2 [1 -tanh[@(rs -R)] (53)
And we recall that the Natario shape function defined in function of its Alcubierre counterpart is given by:

N (rs) = [ 1 2 ][1 -f (rs) W F ] W F (54)
Evaluating the first order derivative of Natario shape function which is given by:

N (rs) = -[ 1 2 ]W F 2 [1 -f (rs) W F ] (W F -1) [f (rs) (W F -1) ]f (rs) (55)
With f (rs) being the first order derivative of the Alcubierre shape function

f (rs) = - 1 2 [ @ cosh 2 [@(rs -R)] ] (56)
The dominant term here is the term resulting from the warp factor which is:

[1 -f (rs) W F ] (W F -1) [f (rs) (W F -1) ] ( 57
)
This term is composed by two expressions that complementary neutralizes each other giving values of zero inside and outside the bubble squeezing the Natario warped region in a thin layer geometrically placed around the neighborhoods of the bubble radius.

The first expression that neutralizes the derivative of the Natario shape function inside the bubble is:

[1 -f (rs) W F ] (W F -1) (58)
And the second expression that neutralizes the derivative of the Natario shape function outside the bubble is:

[f (rs) (W F -1) ] (59)
Inside the bubble f (rs) = 1 and [1 -f (rs) W F ] (W F -1) = 0 resulting in a N (rs) = 0.This is the reason why the Natario shape function with warp factors do not have numerical values for the derivatives inside the bubble.

Outside the bubble f (rs) = 0 and [f (rs) (W F -1) ] = 0 resulting also in a N (rs) = 0.This is the reason why the Natario shape function with warp factors do not have numerical values for the derivatives outside the bubble.

Inside the bubble f (rs) = 1 and [f (rs

) (W F -1) ] = 1 however [1 -f (rs) W F ] (W F -1) = 0 and hence the warp factor product [1 -f (rs) W F ] (W F -1) [f (rs) (W F -1) ] = 0.
Outside the bubble f (rs) = 0 and [f (rs

) (W F -1) ] = 0 however [1 -f (rs) W F ] (W F -1) = 1 and hence the warp factor product is also [1 -f (rs) W F ] (W F -1) [f (rs) (W F -1) ] = 0.
Note that from the statements pointed above when one of the expressions have the value of 1 the other have the value of 0 and vice-versa.This explains how and why each expression complementary neutralizes each other in the regions inside and outside the bubble

Note that if [1 -f (rs) W F ] (W F -1) << 1 and [f (rs) (W F -1) ] << 1 then their product [1-f (rs) W F ] (W F -1) [f (rs) (W F -1)
] <<<< 1 resulting in a very low derivative for the Natario shape function in the Alcubierre warped region and hence in the Natario warped region with both centered geometrically over the bubble radius.

We will examine the above statement of the expressions that complementary neutralizes each other in the first order derivative of Natario shape function with details in the section 4.

Evaluating now the term A in the second order derivative of the Natario shape function given by:

A = [ 1 2 ]W F 3 (W F -1)[1 -f (rs) W F ] (W F -2) [f (rs) 2(W F -1) ]f (rs) 2 (60)
With f (rs) 2 being the derivative square of the Alcubierre shape function which is :

f (rs) 2 = 1 4 [ @ 2 cosh 4 [@(rs -R)] ] (61)
The dominant term here is the term resulting from the warp factor which is:

[1 -f (rs) W F ] (W F -2) [f (rs) 2(W F -1) ] ( 62
)
This term is composed by two expressions that complementary neutralizes each other giving values of zero inside and outside the bubble squeezing the Natario warped region in a thin layer geometrically placed around the neighborhoods of the bubble radius.

The first expression that neutralizes the term A in the second order derivative of the Natario shape function inside the bubble is:

[1 -f (rs) W F ] (W F -2) (63)
And the second expression that neutralizes the term A in the second order derivative of the Natario shape function outside the bubble is:

[f (rs) 2(W F -1) ] (64)
Inside the bubble f (rs) = 1 and [1 -f (rs) W F ] (W F -2) = 0 resulting in a A = 0.

Outside the bubble f (rs) = 0 and [f (rs) 2(W F -1) ] = 0 resulting also in a A = 0.

Inside the bubble f (rs) = 1 and [f (rs)

2(W F -1) ] = 1 however [1 -f (rs) W F ] (W F -2) = 0 and hence the warp factor product [1 -f (rs) W F ] (W F -2) [f (rs) 2(W F -1) ] = 0.
Outside the bubble f (rs) = 0 and [f (rs

) 2W F -1) ] = 0 however [1 -f (rs) W F ] (W F -2) = 1 and hence the warp factor product is also [1 -f (rs) W F ] (W F -2) [f (rs) 2(W F -1) ] = 0.
Note that from the statements pointed above when one of the expressions have the value of 1 the other have the value of 0 and vice-versa.This explains how and why each expression complementary neutralizes each other in the regions inside and outside the bubble

Note that if [1 -f (rs) W F ] (W F -2) << 1 and [f (rs) 2(W F -1) ] << 1 then their product [1 -f (rs) W F ] (W F -2) [f (rs) 2(W F -1)
] <<<< 1 resulting in a very low value for the term A in the Alcubierre warped region and hence in the Natario warped region with both centered geometrically over the bubble radius.

We will examine the above statement of the expressions that complementary neutralizes each other in the term A of the second order derivative of Natario shape function with details in the section 5.

Evaluating now the term B in the second order derivative of the Natario shape function given by:

B = -[ 1 2 ]W F 2 [1 -f (rs) W F ] (W F -1) (W F -1)[f (rs) (W F -2) ]f (rs) 2 (65)
With f (rs) 2 being the derivative square of the Alcubierre shape function which is :

f (rs) 2 = 1 4 [ @ 2 cosh 4 [@(rs -R)] ] (66)
The dominant term here is the term resulting from the warp factor which is:

[1 -f (rs) W F ] (W F -1) [f (rs) (W F -2) ] ( 67
)
This term is composed by two expressions that complementary neutralizes each other giving values of zero inside and outside the bubble squeezing the Natario warped region in a thin layer geometrically placed around the neighborhoods of the bubble radius.

The first expression that neutralizes the term B in the second order derivative of the Natario shape function inside the bubble is:

[1 -f (rs) W F ] (W F -1) (68)
And the second expression that neutralizes the term B in the second order derivative of the Natario shape function outside the bubble is:

[f (rs) (W F -2) ] (69)
Inside the bubble f (rs) = 1 and [1 -f (rs) W F ] (W F -1) = 0 resulting in a B = 0.

Outside the bubble f (rs) = 0 and [f (rs) (W F -2) ] = 0 resulting also in a B = 0.

Inside the bubble f (rs) = 1 and [f (rs

) (W F -2) ] = 1 however [1 -f (rs) W F ] (W F -1) = 0 and hence the warp factor product [1 -f (rs) W F ] (W F -1) [f (rs) (W F -2) ] = 0.
Outside the bubble f (rs) = 0 and [f (rs

) W F -2) ] = 0 however [1 -f (rs) W F ] (W F -1) = 1 and hence the warp factor product is also [1 -f (rs) W F ] (W F -1) [f (rs) (W F -2) ] = 0.
Note that from the statements pointed above when one of the expressions have the value of 1 the other have the value of 0 and vice-versa.This explains how and why each expression complementary neutralizes each other in the regions inside and outside the bubble

20 Note that if [1 -f (rs) W F ] (W F -1) << 1 and [f (rs) (W F -2) ] << 1 then their product [1 -f (rs) W F ] (W F -1) [f (rs) (W F -2)
] <<<< 1 resulting in a very low value for the term B in the Alcubierre warped region and hence in the Natario warped region with both centered geometrically over the bubble radius.

We will examine the above statement of the expressions that complementary neutralizes each other in the term B of the second order derivative of Natario shape function with details in the section 6.

Evaluating now the term C in the second order derivative of the Natario shape function given by:

C = -[ 1 2 ]W F 2 [1 -f (rs) W F ] (W F -1) [f (rs) (W F -1) ]f (rs) (70)
With f (rs) being the second order derivative of the Alcubierre shape function

f (rs) = [ (@ 2 )sinh[@(rs -R) cosh 3 [@(rs -R)] ] (71)
The dominant term here is the term resulting from the warp factor which is:

[1 -f (rs) W F ] (W F -1) [f (rs) (W F -1) ] ( 72
)
This term is composed by two expressions that complementary neutralizes each other giving values of zero inside and outside the bubble squeezing the Natario warped region in a thin layer geometrically placed around the neighborhoods of the bubble radius.

The first expression that neutralizes the term C in the second order derivative of the Natario shape function inside the bubble is:

[1 -f (rs) W F ] (W F -1) (73)
And the second expression that neutralizes the term C in the second order derivative of the Natario shape function outside the bubble is:

[f (rs) (W F -1) ] (74)
Inside the bubble f (rs) = 1 and [1 -f (rs) W F ] (W F -1) = 0 resulting in a C = 0.

Outside the bubble f (rs) = 0 and [f (rs) (W F -1) ] = 0 resulting also in a C = 0.

Inside the bubble f (rs) = 1 and [f (rs) (W F -1) ] = 1 however [1 -f (rs) W F ] (W F -1) = 0 and hence the warp factor product [1 -f (rs

) W F ] (W F -1) [f (rs) (W F -1) ] = 0.
Outside the bubble f (rs) = 0 and [f (rs

) (W F -1) ] = 0 however [1 -f (rs) W F ] (W F -1) = 1 and hence the warp factor product is also [1 -f (rs) W F ] (W F -1) [f (rs) (W F -1) ] = 0.
Note that from the statements pointed above when one of the expressions have the value of 1 the other have the value of 0 and vice-versa.This explains how and why each expression complementary neutralizes each other in the regions inside and outside the bubble

Note that if [1 -f (rs) W F ] (W F -1) << 1 and [f (rs) (W F -1) ] << 1 then their product [1 -f (rs) W F ] (W F -1) [f (rs) (W F -1)
] <<<< 1 resulting in a very low value for the term C in the Alcubierre warped region and hence in the Natario warped region with both centered geometrically over the bubble radius.

We will examine the above statement of the expressions that complementary neutralizes each other in the term C of the second order derivative of Natario shape function with details in the section 7.

3 The expressions that complementary neutralizes each other in a Natario warp drive 1 + 1 spacetime

The Alcubierre shape function is given by:

f (rs) = 1 2 [1 -tanh[@(rs -R)] (75)
And the Natario shape function is given by:

N (rs) = [ 1 2 ][1 -f (rs) W F ] W F (76)
• Numerical plot for the Alcubierre and Natario shape functions with @ = 50000 bubble radius R = 100 meters and warp factor with a value W F = 200 rs f (rs) N (rs) 9, 999940000000E + 01 1, 000000000000E + 00 0, 000000000000E + 00 9, 999950000000E + 01 1, 000000000000E + 00 0, 000000000000E + 00 9, 999960000000E + 01 1, 000000000000E + 00 0, 000000000000E + 00 9, 999970000000E + 01 9, 999999999999E -01 0, 000000000000E + 00 9, 999980000000E + 01 9, 999999979388E -01 0, 000000000000E + 00 9, 999990000000E + 01 9, 999546021312E -01 0, 000000000000E + 00 1, 000000000000E + 02 5, 000000000000E -01 5, 000000000000E -01 1, 000001000000E + 02 4, 539786855834E -05 5, 000000000000E -01 1, 000002000000E + 02 2, 061153636657E -09 5, 000000000000E -01 1, 000003000000E + 02 9, 359180097590E -14 5, 000000000000E -01 1, 000004000000E + 02 0, 000000000000E + 00 5, 000000000000E -01 1, 000005000000E + 02 0, 000000000000E + 00 5, 000000000000E -01 1, 000006000000E + 02 0, 000000000000E + 00 5, 000000000000E -01 According with the numerical plot above when @ = 50000 then from 0 to 99, 996 meters from the center of the bubble we have the region inside the bubble where the Alcubierre shape function f (rs) = 1 and the Natario shape function N (rs) = 0. At 99, 997 meters from the center of the bubble the Alcubierre shape function starts to decrease and we enter in the Alcubierre geometrized warped region 1 > f (rs) > 0.The Alcubierre warped region ends at 100, 003 meters from the center of the bubble.At 100, 004 meters from the center of the bubble we reaches the region outside the bubble where the Alcubierre shape function f (rs) = 0 and the Natario shape function N (rs) = 1 2 . The thickness or the width of the Alcubierre geometrized warped region is then 0, 006 meters.

The geometrized Natario warped region 0 < N (rs) < 1 2 is centered over the radius of the bubble where the Natario shape function N (rs) possesses a value too much close from 1 2 although smaller than 1 2 .Then we can see that N (rs) 1 2 .12 .So we can say that the Natario geometrized warped region starts after 99, 999 meters and ends up before 100, 001 meters.The region inside the bubble for the Natario warp drive goes from 0 to 99, 999 meters and the region outside the bubble starts at 100, 001 meters. The thickness or the width of the Natario geometrized warped region is then smaller than 0, 002 meters.

The derivative square of the Natario shape function is :

N (rs) 2 = [ 1 4 ]W F 4 [1 -f (rs) W F ] 2(W F -1) [f (rs) 2(W F -1) ]f (rs) 2 (77)
And the derivative square of the Alcubierre shape function is :

f (rs) 2 = 1 4 [ @ 2 cosh 4 [@(rs -R)] ] (78)
• Numerical plot for the square derivative of the Alcubierre shape function and the complementary expressions [1 -f (rs) W F ] [2(W F -1)] and f (rs) [2(W F -1)] that neutralizes each other with @ = 50000 bubble radius R = 100 meters and warp factor with a value W F = 200

rs f (rs) 2 [1 -f (rs) W F ] [2(W F -1)] f (rs) [2(W F -1
)] 9, 999940000000E + 01 7, 667648086763E -043 0, 0000000000000E + 000 1, 0000000000000E + 9, 999950000000E + 01 3, 720075984818E -034 0, 0000000000000E + 000 1, 0000000000000E + 9, 999960000000E + 01 1, 804851393312E -025 0, 0000000000000E + 000 1, 0000000000000E + 9, 999970000000E + 01 8, 756510795027E -017 0, 0000000000000E + 000 9, 9999999996275E -9, 999980000000E + 01 4, 248354238773E -008 0, 0000000000000E + 000 9, 9999917966117E -9, 999990000000E + 01 2, 060779370345E + 001 0, 0000000000000E + 000 9, 8209349938841E -1, 000000000000E + 02 6, 250000000000E + 008 1, 0000000000000E + 000 1, 5490367659397E -1, 000001000000E + 02 2, 060779346918E + 001 1, 0000000000000E + 000 0, 0000000000000E + 1, 000002000000E + 02 4, 248354190475E -008 1, 0000000000000E + 000 0, 0000000000000E + 1, 000003000000E + 02 8, 756510695477E -017 1, 0000000000000E + 000 0, 0000000000000E + 1, 000004000000E + 02 1, 804851372793E -025 1, 0000000000000E + 000 0, 0000000000000E + 1, 000005000000E + 02 3, 720075942525E -034 1, 0000000000000E + 000 0, 0000000000000E + 1, 000006000000E + 02 7, 667647999592E -043 1, 0000000000000E + 000 0, 0000000000000E + According with the numerical plot above when @ = 50000 at 99, 994 meters from the center of the bubble the square derivative of the Alcubierre shape function is 7, 66 × 10 -43 and starts to increase reaching the maximum value of 6, 25 × 10 8 at 100 meters from the center of the bubble precisely in the bubble radius decreasing again to the value of 7, 66 × 10 -43 at 100, 006 meters from the center of the bubble.Note that with respect to the distance of 100 meters from the center of the bubble exactly the bubble radius the powers of the square derivative of the Alcubierre shape function are diametrically symmetrically opposed.We have the values of 10 1 at 99, 999 meters and at 100, 001 meters.We have the value of 10 -8 at 99, 998 meters and at 100, 002 meters.

The expression [1 -f (rs) W F ] [2(W F -1)] is zero from the center of the bubble to 99, 999 meters and at the radius of the bubble and beyond changes its value to 1.So inside the bubble this expression is 0 and outside the bubble this expression is 1. The expression f (rs) [2(W F -1)] is 1 from the center of the bubble to 99, 996 meters and at 99, 997 meters starts to decrease reaching its minimum value of 1, 54 × 10 -120 precisely in the bubble radius.At 100, 001 meters its value is zero.

Note that when one of these expressions is zero the other is 1 or possesses values very close to 1.Then one expression neutralizes the other except in the bubble radius but here the value of the product is very low:1, 54 × 10 -120 .This of course obliterates the factor 10 48 .

The derivative square of the Natario shape function is :

N (rs) 2 = [ 1 4 ]W F 4 [1 -f (rs) W F ] 2(W F -1) [f (rs) 2(W F -1) ]f (rs) 2 (79)
And the derivative square of the Alcubierre shape function is :

f (rs) 2 = 1 4 [ @ 2 cosh 4 [@(rs -R)] ] (80)
• Numerical plot for the square derivatives of both the Alcubierre and Natario shape functions with @ = 50000 bubble radius R = 100 meters and warp factor with a value W F = 200 rs f (rs) 2 N (rs) 2 9, 999940000000E + 01 7, 667648086763E -043 0, 0000000000000E + 000 9, 999950000000E + 01 3, 720075984818E -034 0, 0000000000000E + 000 9, 999960000000E + 01 1, 804851393312E -025 0, 0000000000000E + 000 9, 999970000000E + 01 8, 756510795027E -017 0, 0000000000000E + 000 9, 999980000000E + 01 4, 248354238773E -008 0, 0000000000000E + 000 9, 999990000000E + 01 2, 060779370345E + 001 0, 0000000000000E + 000 1, 000000000000E + 02 6, 250000000000E + 008 3, 8725919148493E -103 1, 000001000000E + 02 2, 060779346918E + 001 0, 0000000000000E + 000 1, 000002000000E + 02 4, 248354190475E -008 0, 0000000000000E + 000 1, 000003000000E + 02 8, 756510695477E -017 0, 0000000000000E + 000 1, 000004000000E + 02 1, 804851372793E -025 0, 0000000000000E + 000 1, 000005000000E + 02 3, 720075942525E -034 0, 0000000000000E + 000 1, 000006000000E + 02 7, 667647999592E -043 0, 0000000000000E + 000 From the plots of the previous page we know that the product

[1 -f (rs) W F ] 2(W F -1) [f (rs) 2(W F -1) ] ( 81
)
is always zero except in the bubble radius giving a non-null square derivative of the Natario shape function in the bubble radius but with a very small value.The final value for the square derivative of the Natario shape function is then 3, 8 × 10 -103 layered over the bubble radius and this obliterates the factor 10 48 in the negative energy density of the Natario warp drive in the 1 + 1 spacetime rendering it physically possible.Note also that this value match the value presented in the numerical plot of the previous section for @ = 50000. We recall the negative energy density in the 1 + 1 Natario warp drive spacetime :

ρ = T µν u µ u ν = - c 2 G v 2 s 8π 3(N (rs)) 2 (82) c 2 G v 2 s 8π 10 48 (83)
(N (rs)) 2 3, 8 × 10 -103 (84)

And the product 10 48 × 10 -103 = 10 -55 Joules M eters 3 resulting in a very low negative energy density even for speeds of 200 times faster than light. [START_REF] Lobo | [END_REF] The expressions that complementary neutralizes each other in the first order derivative of the shape function for a Natario warp drive 3 + 1 spacetime metric Its expression is :

N (rs) = -[ 1 2 ]W F 2 [1 -f (rs) W F ] (W F -1) [f (rs) (W F -1) ]f (rs) (85)
With the first order derivative of the Alcubierre shape function being:

f (rs) = - 1 2 [ @ cosh 2 [@(rs -R)] ] (86)
• Numerical plot for the derivative of the Alcubierre shape function and the complementary expressions

[1 -f (rs) W F ] [(W F -1)
] and f (rs) [(W F -1)] that neutralizes each other with @ = 50000 bubble radius R = 100 meters and warp factor with a value W F = 200

rs f (rs) [1 -f (rs) W F ] [(W F -1)]
f (rs) [(W F -1)] 9, 999940000000E + 01 -8, 756510770143E -022 0, 00000000000E + 000 1, 00000000000E + 9, 999950000000E + 01 -1, 928749850244E -017 0, 00000000000E + 000 1, 00000000000E + 9, 999960000000E + 01 -4, 248354261725E -013 0, 00000000000E + 000 1, 00000000000E + 9, 999970000000E + 01 -9, 357622986115E -009 0, 00000000000E + 000 9, 99999999981E -9, 999980000000E + 01 -2, 061153618432E -004 0, 00000000000E + 000 9, 99999589830E -9, 999990000000E + 01 -4, 539580784989E + 000 0, 00000000000E + 000 9, 91006306432E -1, 000000000000E + 02 -2, 500000000000E + 004 1, 00000000000E + 000 1, 24460305557E -1, 000001000000E + 02 -4, 539580759187E + 000 1, 00000000000E + 000 0, 00000000000E + 1, 000002000000E + 02 -2, 061153606715E -004 1, 00000000000E + 000 0, 00000000000E + 1, 000003000000E + 02 -9, 357622932923E -009 1, 00000000000E + 000 0, 00000000000E + 1, 000004000000E + 02 -4, 248354237576E -013 1, 00000000000E + 000 0, 00000000000E + 1, 000005000000E + 02 -1, 928749839281E -017 1, 00000000000E + 000 0, 00000000000E + 1, 000006000000E + 02 -8, 756510720368E -022 1, 00000000000E + 000 0, 00000000000E + According with the numerical plot above when @ = 50000 at 99, 994 meters from the center of the bubble the derivative of the Alcubierre shape function is -8, 75 × 10 -22 and starts to increase reaching the maximum value of -2, 5 × 10 4 at 100 meters from the center of the bubble precisely in the bubble radius decreasing again to the value of -8, 75 × 10 -22 at 100, 006 meters from the center of the bubble.

The expression [1 -f (rs) W F ] [(W F -1)] is zero from the center of the bubble to 99, 999 meters and at the radius of the bubble and beyond changes its value to 1.So inside the bubble this expression is 0 and outside the bubble this expression is 1. The expression f (rs) [(W F -1)] is 1 from the center of the bubble to 99, 996 meters and at 99, 997 meters starts to decrease reaching its minimum value of 1, 24 × 10 -60 precisely in the bubble radius.At 100, 001 meters its value is zero.

Note that when one of these expressions is zero the other is 1 or possesses values very close to 1.Then one expression neutralizes the other except in the bubble radius but here the value of the product is very low:1, 24 × 10 -60 .

The expression for the first order derivative of the Natario shape function is :

N (rs) = -[ 1 2 ]W F 2 [1 -f (rs) W F ] (W F -1) [f (rs) (W F -1) ]f (rs) (87)
With the first order derivative of the Alcubierre shape function being:

f (rs) = - 1 2 [ @ cosh 2 [@(rs -R)] ] (88)
• Numerical plot for the first order derivatives of both the Alcubierre and Natario shape functions with @ = 50000 bubble radius R = 100 meters and warp factor with a value W F = 200 rs f (rs) N (rs) 9, 999940000000E + 01 -8, 756510770143E -022 0, 00000000000E + 9, 999950000000E + 01 -1, 928749850244E -017 0, 00000000000E + 9, 999960000000E + 01 -4, 248354261725E -013 0, 00000000000E + 9, 999970000000E + 01 -9, 357622986115E -009 0, 00000000000E + 9, 999980000000E + 01 -2, 061153618432E -004 0, 00000000000E + 9, 999990000000E + 01 -4, 539580784989E + 000 0, 00000000000E + 1, 000000000000E + 02 -2, 500000000000E + 004 -3, 11150763893E -054 1, 000001000000E + 02 -4, 539580759187E + 000 0, 00000000000E + 1, 000002000000E + 02 -2, 061153606715E -004 0, 00000000000E + 1, 000003000000E + 02 -9, 357622932923E -009 0, 00000000000E + 1, 000004000000E + 02 -4, 248354237576E -013 0, 00000000000E + 1, 000005000000E + 02 -1, 928749839281E -017 0, 00000000000E + 1, 000006000000E + 02 -8, 756510720368E -022 0, 00000000000E + From the plots of the previous page we know that the product

[1 -f (rs) W F ] (W F -1) [f (rs) (W F -1) ] ( 89
)
is always zero except in the bubble radius giving a non-null derivative of the Natario shape function in the bubble radius but with a very small value.The final value for the derivative of the Natario shape function is then -3, 11 × 10 -54 layered over the bubble radius [START_REF] Loup | [END_REF] The expressions that complementary neutralizes each other in the term A of the second order derivative of the shape function for a Natario warp drive 3 + 1 spacetime metric

A = [ 1 2 ]W F 3 (W F -1)[1 -f (rs) W F ] (W F -2) [f (rs) 2(W F -1) ]f (rs) 2 (90)
With f (rs) 2 being the first order derivative square of the Alcubierre shape function which is :

f (rs) 2 = 1 4 [ @ 2 cosh 4 [@(rs -R)] ] (91)
• Numerical plot for the square first order derivative of the Alcubierre shape function and the complementary expressions [1 -f (rs

) W F ] [(W F -2)] and f (rs) [2(W F -1)
] that neutralizes each other with @ = 50000 bubble radius R = 100 meters and warp factor with a value W F = 200

rs f (rs) 2 [1 -f (rs) W F ] [(W F -2)]
f (rs) [2(W F -1)] 9, 999940000000E + 01 7, 667648086763E -043 0, 0000000000000E + 000 1, 0000000000000E + 9, 999950000000E + 01 3, 720075984818E -034 0, 0000000000000E + 000 1, 0000000000000E + 9, 999960000000E + 01 1, 804851393312E -025 0, 0000000000000E + 000 1, 0000000000000E + 9, 999970000000E + 01 8, 756510795027E -017 0, 0000000000000E + 000 9, 9999999996275E -9, 999980000000E + 01 4, 248354238773E -008 0, 0000000000000E + 000 9, 9999917966117E -9, 999990000000E + 01 2, 060779370345E + 001 0, 0000000000000E + 000 9, 8209349938841E -1, 000000000000E + 02 6, 250000000000E + 008 1, 0000000000000E + 000 1, 5490367659397E -1, 000001000000E + 02 2, 060779346918E + 001 1, 0000000000000E + 000 0, 0000000000000E + 1, 000002000000E + 02 4, 248354190475E -008 1, 0000000000000E + 000 0, 0000000000000E + 1, 000003000000E + 02 8, 756510695477E -017 1, 0000000000000E + 000 0, 0000000000000E + 1, 000004000000E + 02 1, 804851372793E -025 1, 0000000000000E + 000 0, 0000000000000E + 1, 000005000000E + 02 3, 720075942525E -034 1, 0000000000000E + 000 0, 0000000000000E + 1, 000006000000E + 02 7, 667647999592E -043 1, 0000000000000E + 000 0, 0000000000000E + According with the numerical plot above when @ = 50000 at 99, 994 meters from the center of the bubble the square derivative of the Alcubierre shape function is 7, 66 × 10 -43 and starts to increase reaching the maximum value of 6, 25 × 10 8 at 100 meters from the center of the bubble precisely in the bubble radius decreasing again to the value of 7, 66 × 10 -43 at 100, 006 meters from the center of the bubble.

The expression [1 -f (rs) W F ] [(W F -2)] is zero from the center of the bubble to 99, 999 meters and at the radkius of the bubble and beyond changes its value to 1.So inside the bubble this expression is 0 and outside the bubble this expression is 1. The expression f (rs) [2(W F -1)] is 1 from the center of the bubble to 99, 996 meters and at 99, 997 meters starts to decrease reaching its minimum value of 1, 54 × 10 -120 precisely in the bubble radius.At 100, 001 meters its value is zero.

Note that when one of these expressions is zero the other is 1 or possesses values very close to 1.Then one expression neutralizes the other except in the bubble radius but here the value of the product is very low:1, 54 × 10 -120 .

The term A of the second order derivative of the shape function for a Natario warp drive 3+1 spacetime metric is given by:

A = [ 1 2 ]W F 3 (W F -1)[1 -f (rs) W F ] (W F -2) [f (rs) 2(W F -1) ]f (rs) 2 (92)
With f (rs) 2 being the first order derivative square of the Alcubierre shape function which is :

f (rs) 2 = 1 4 [ @ 2 cosh 4 [@(rs -R)] ] (93)
• Numerical plot for the square first order derivative of the Alcubierre shape function and the term A of the second order derivative of the Natario shape function with @ = 50000 bubble radius R = 100 meters and warp factor with a value W F = 200 rs f (rs) 2 A 9, 999940000000E + 01 7, 667648086763E -043 0, 00000000000E + 9, 999950000000E + 01 3, 720075984818E -034 0, 00000000000E + 9, 999960000000E + 01 1, 804851393312E -025 0, 00000000000E + 9, 999970000000E + 01 8, 756510795027E -017 0, 00000000000E + 9, 999980000000E + 01 4, 248354238773E -008 0, 00000000000E + 9, 999990000000E + 01 2, 060779370345E + 001 0, 00000000000E + 1, 000000000000E + 02 6, 250000000000E + 008 7, 70645791055E -1, 000001000000E + 02 2, 060779346918E + 001 0, 00000000000E + 1, 000002000000E + 02 4, 248354190475E -008 0, 00000000000E + 1, 000003000000E + 02 8, 756510695477E -017 0, 00000000000E + 1, 000004000000E + 02 1, 804851372793E -025 0, 00000000000E + 1, 000005000000E + 02 3, 720075942525E -034 0, 00000000000E + 1, 000006000000E + 02 7, 667647999592E -043 0, 00000000000E + From the plots of the previous page we know that the product

[1 -f (rs) W F ] (W F -2) [f (rs) 2(W F -1) ] ( 94
)
is always zero except in the bubble radius giving a non-null value for the term A of the second order derivative of the Natario shape function in the bubble radius but with a very small value.The final value for the term A is then 7, 701 × 10 -103 layered over the bubble radius [START_REF] Loup | [END_REF] The expressions that complementary neutralizes each other in the term B of the second order derivative of the shape function for a Natario warp drive 3 + 1 spacetime metric

B = -[ 1 2 ]W F 2 [1 -f (rs) W F ] (W F -1) (W F -1)[f (rs) (W F -2) ]f (rs) 2 (95)
With f (rs) 2 being the first order derivative square of the Alcubierre shape function which is :

f (rs) 2 = 1 4 [ @ 2 cosh 4 [@(rs -R)] ] (96)
• Numerical plot for the square first order derivative of the Alcubierre shape function and the complementary expressions [1 -f (rs

) W F ] [(W F -1)] and f (rs) [(W F -2)
] that neutralizes each other with @ = 50000 bubble radius R = 100 meters and warp factor with a value W F = 200

rs f (rs) 2 [1 -f (rs) W F ] [(W F -1)] f (rs) [(W F -2)
] 9, 999940000000E + 01 7, 667648086763E -043 0, 0000000000000E + 000 1, 000000000000E + 9, 999950000000E + 01 3, 720075984818E -034 0, 0000000000000E + 000 1, 000000000000E + 9, 999960000000E + 01 1, 804851393312E -025 0, 0000000000000E + 000 1, 000000000000E + 9, 999970000000E + 01 8, 756510795027E -017 0, 0000000000000E + 000 9, 999999999815E -9, 999980000000E + 01 4, 248354238773E -008 0, 0000000000000E + 000 9, 999995918917E -9, 999990000000E + 01 2, 060779370345E + 001 0, 0000000000000E + 000 9, 910512980490E -1, 000000000000E + 02 6, 250000000000E + 008 1, 0000000000000E + 000 2, 489206111144E -1, 000001000000E + 02 2, 060779346918E + 001 1, 0000000000000E + 000 0, 000000000000E + 1, 000002000000E + 02 4, 248354190475E -008 1, 0000000000000E + 000 0, 000000000000E + 1, 000003000000E + 02 8, 756510695477E -017 1, 0000000000000E + 000 0, 000000000000E + 1, 000004000000E + 02 1, 804851372793E -025 1, 0000000000000E + 000 0, 000000000000E + 1, 000005000000E + 02 3, 720075942525E -034 1, 0000000000000E + 000 0, 000000000000E + 1, 000006000000E + 02 7, 667647999592E -043 1, 0000000000000E + 000 0, 000000000000E + According with the numerical plot above when @ = 50000 at 99, 994 meters from the center of the bubble the square derivative of the Alcubierre shape function is 7, 66 × 10 -43 and starts to increase reaching the maximum value of 6, 25 × 10 8 at 100 meters from the center of the bubble precisely in the bubble radius decreasing again to the value of 7, 66 × 10 -43 at 100, 006 meters from the center of the bubble.

The expression [1 -f (rs) W F ] [(W F -1)] is zero from the center of the bubble to 99, 999 meters and at the radius of the bubble and beyond changes its value to 1.So inside the bubble this expression is 0 and outside the bubble this expression is 1. The expression f (rs) [(W F -2)] is 1 from the center of the bubble to 99, 996 meters and at 99, 997 meters starts to decrease reaching its minimum value of 2, 48 × 10 -60 precisely in the bubble radius.At 100, 001 meters its value is zero.

Note that when one of these expressions is zero the other is 1 or possesses values very close to 1.Then one expression neutralizes the other except in the bubble radius but here the value of the product is very low:2, 48 × 10 -60 .

The term B of the second order derivative of the shape function for a Natario warp drive 3+1 spacetime metric is given by:

B = -[ 1 2 ]W F 2 [1 -f (rs) W F ] (W F -1) (W F -1)[f (rs) (W F -2) ]f (rs) 2 (97)
With f (rs) 2 being the first order derivative square of the Alcubierre shape function which is :

f (rs) 2 = 1 4 [ @ 2 cosh 4 [@(rs -R)] ] (98)
• Numerical plot for the square first order derivative of the Alcubierre shape function and the term B of the second order derivative of the Natario shape function with @ = 50000 bubble radius R = 100 meters and warp factor with a value W F = 200 rs f (rs) 2 B 9, 999940000000E + 01 7, 667648086763E -043 0, 000000000000E + 9, 999950000000E + 01 3, 720075984818E -034 0, 000000000000E + 9, 999960000000E + 01 1, 804851393312E -025 0, 000000000000E + 9, 999970000000E + 01 8, 756510795027E -017 0, 000000000000E + 9, 999980000000E + 01 4, 248354238773E -008 0, 000000000000E + 9, 999990000000E + 01 2, 060779370345E + 001 0, 000000000000E + 1, 000000000000E + 02 6, 250000000000E + 008 -6, 191900201472E -045 1, 000001000000E + 02 2, 060779346918E + 001 0, 000000000000E + 1, 000002000000E + 02 4, 248354190475E -008 0, 000000000000E + 1, 000003000000E + 02 8, 756510695477E -017 0, 000000000000E + 1, 000004000000E + 02 1, 804851372793E -025 0, 000000000000E + 1, 000005000000E + 02 3, 720075942525E -034 0, 000000000000E + 1, 000006000000E + 02 7, 667647999592E -043 0, 000000000000E + From the plots of the previous page we know that the product

[1 -f (rs) W F ] (W F -1) [f (rs) (W F -2) ] (99)
is always zero except in the bubble radius giving a non-null value for the term B of the second order derivative of the Natario shape function in the bubble radius but with a very small value.The final value for the term B is then -6, 191 × 10 -45 layered over the bubble radius [START_REF] Loup | [END_REF] The expressions that complementary neutralizes each other in the term C of the second order derivative of the shape function for a Natario warp drive 3 + 1 spacetime metric

C = -[ 1 2 ]W F 2 [1 -f (rs) W F ] (W F -1) [f (rs) (W F -1) ]f (rs) (100)
With f (rs) being the second order derivative of the Alcubierre shape function which is:

f (rs) = [ (@ 2 )sinh[@(rs -R) cosh 3 [@(rs -R)] ] (101)
• Numerical plot for the second order derivative of the Alcubierre shape function and the complemen-

tary expressions [1 -f (rs) W F ] [(W F -1)
] and f (rs) [(W F -1)] that neutralizes each other with @ = 50000 bubble radius R = 100 meters and warp factor with a value W F = 200

rs f (rs) [1 -f (rs) W F ] [(W F -1)]
f (rs) [(W F -1)] 9, 999940000000E + 01 -8, 75651077014E -017 0, 00000000000E + 000 1, 00000000000E + 000 9, 999950000000E + 01 -1, 92874985024E -012 0, 00000000000E + 000 1, 00000000000E + 000 9, 999960000000E + 01 -4, 24835426172E -008 0, 00000000000E + 000 1, 00000000000E + 000 9, 999970000000E + 01 -9, 35762298611E -004 0, 00000000000E + 000 9, 99999999981E -001 9, 999980000000E + 01 -2, 06115360993E + 001 0, 00000000000E + 000 9, 99999589830E -001 9, 999990000000E + 01 -4, 53916861040E + 005 0, 00000000000E + 000 9, 91006306432E -001 1, 000000000000E + 02 0, 00000000000E + 000 1, 00000000000E + 000 1, 24460305557E -060 1, 000001000000E + 02 4, 53916858461E + 005 1, 00000000000E + 000 0, 00000000000E + 000 1, 000002000000E + 02 2, 06115359822E + 001 1, 00000000000E + 000 0, 00000000000E + 000 1, 000003000000E + 02 9, 35762293292E -004 1, 00000000000E + 000 0, 00000000000E + 000 1, 000004000000E + 02 4, 24835423758E -008 1, 00000000000E + 000 0, 00000000000E + 000 1, 000005000000E + 02 1, 92874983928E -012 1, 00000000000E + 000 0, 00000000000E + 000 1, 000006000000E + 02 8, 75651072037E -017 1, 00000000000E + 000 0, 00000000000E + 000 According with the numerical plot above when @ = 50000 at 99, 994 meters from the center of the bubble the second order derivative of the Alcubierre shape function is -8, 75 × 10 -17 however in the bubble radius inverts the signal reaching the value of 8, 75 × 10 -17 at 100, 006 meters from the center of the bubble.In the radius of the bubble the value is 0 due to the term sinh[@(rs -R)] = 0 when rs = R.

The expression

[1 -f (rs) W F ] [(W F -1)
] is zero from the center of the bubble to 99, 999 meters and at the radius of the bubble and beyond changes its value to 1.So inside the bubble this expression is 0 and outside the bubble this expression is 1. The expression f (rs) [(W F -1)] is 1 from the center of the bubble to 99, 996 meters and at 99, 997 meters starts to decrease reaching its minimum value of 1, 24 × 10 -60 precisely in the bubble radius.At 100, 001 meters its value is zero.

Note that when one of these expressions is zero the other is 1 or possesses values very close to 1.Then one expression neutralizes the other except in the bubble radius but here the value of the product is very low:1, 24 × 10 -60 .

The term C of the second order derivative of the shape function for a Natario warp drive 3+1 spacetime metric is given by:

C = -[ 1 2 ]W F 2 [1 -f (rs) W F ] (W F -1) [f (rs) (W F -1) ]f (rs) (102)
With f (rs) being the second order derivative of the Alcubierre shape function which is:

f (rs) = [ (@ 2 )sinh[@(rs -R) cosh 3 [@(rs -R)] ] (103)
• Numerical plot for the second order derivative of the Alcubierre shape function and the term C of the second order derivative of the Natario shape function with @ = 50000 bubble radius R = 100 meters and warp factor with a value W F = 200 rs f (rs) C 9, 999940000000E + 01 -8, 75651077014E -017 0, 00000000000E + 9, 999950000000E + 01 -1, 92874985024E -012 0, 00000000000E + 9, 999960000000E + 01 -4, 24835426172E -008 0, 00000000000E + 9, 999970000000E + 01 -9, 35762298611E -004 0, 00000000000E + 9, 999980000000E + 01 -2, 06115360993E + 001 0, 00000000000E + 9, 999990000000E + 01 -4, 53916861040E + 005 0, 00000000000E + 1, 000000000000E + 02 0, 00000000000E + 000 0, 00000000000E + 1, 000001000000E + 02 4, 53916858461E + 005 0, 00000000000E + 1, 000002000000E + 02 2, 06115359822E + 001 0, 00000000000E + 1, 000003000000E + 02 9, 35762293292E -004 0, 00000000000E + 1, 000004000000E + 02 4, 24835423758E -008 0, 00000000000E + 1, 000005000000E + 02 1, 92874983928E -012 0, 00000000000E + 1, 000006000000E + 02 8, 75651072037E -017 0, 00000000000E + From the plots of the previous page we know that the product

[1 -f (rs) W F ] (W F -1) [f (rs) (W F -1) ] (104)
is always zero except in the bubble radius however due to the term sinh[@(rs -R)] = 0 when rs = R the value of the term C is always 0.

8 The second order derivative of the shape function for a Natario warp drive 3 + 1 spacetime metric

• Numerical plot for the second order derivatives of both the Alcubierre and Natario shape functions with @ = 50000 bubble radius R = 100 meters and warp factor with a value W F = 200 rs f (rs) N (rs) 9, 999940000000E + 01 -8, 75651077014E -017 0, 00000000000E + 000 9, 999950000000E + 01 -1, 92874985024E -012 0, 00000000000E + 000 9, 999960000000E + 01 -4, 24835426172E -008 0, 00000000000E + 000 9, 999970000000E + 01 -9, 35762298611E -004 0, 00000000000E + 000 9, 999980000000E + 01 -2, 06115360993E + 001 0, 00000000000E + 000 9, 999990000000E + 01 -4, 53916861040E + 005 0, 00000000000E + 000 1, 000000000000E + 02 0, 00000000000E + 000 -6, 1919002015E -045 1, 000001000000E + 02 4, 53916858461E + 005 0, 00000000000E + 000 1, 000002000000E + 02 2, 06115359822E + 001 0, 00000000000E + 000 1, 000003000000E + 02 9, 35762293292E -004 0, 00000000000E + 000 1, 000004000000E + 02 4, 24835423758E -008 0, 00000000000E + 000 1, 000005000000E + 02 1, 92874983928E -012 0, 00000000000E + 000 1, 000006000000E + 02 8, 75651072037E -017 0, 00000000000E + 000

In order to avoid the mathematical complexities of a lengthly algebraic expression for the second order derivative of the shape function for a Natario warp drive 3 + 1 spacetime metric N (rs) we decomposed in section 2 the expression for N (rs) in 3 algebraic terms A,B and C and we evaluated numerically and independently each one of the 3 terms that must be added together to provide the final numerical value for N (rs).

From the previous numerical plots in section 5 we know that the term A is always zero except in the bubble radius possessing a non-null value of 7, 70645791055 × 10 -103 and from the previous numerical plots in section 6 we know that the term B is also always zero except in the bubble radius possessing a value of -6, 191900201472 × 10 -45 and from the previous numerical plots in section 7 wee know that the term C is always zero even in the bubble radius .

The final value of N (rs) is also always zero except in the bubble radius.Note that the term B is much larger than the term A so it is the value of B that accounts for the final value of N (rs) in the bubble radius which is -6, 1919002015 × 10 -45

9 The negative energy density for a Natario warp drive in a 3 + 1 spacetime metric

Reviewing the more sophisticated case of the Natario warp drive in a real 3 + 1 spacetime seen in section 2 where the negative energy density in this case is given by the following expression

ρ 3+1 = - c 2 G vs 2 8π 3(N (rs)) 2 cos 2 θ - c 2 G vs 2 8π N (rs) + rs 2 N (rs) 2 sin 2 θ (105)
In section 2 we decomposed the above expression in two terms ρ 1 and ρ 2 given respectively by:

ρ 3+1 = ρ 1 + ρ 2 (106)
ρ 1 = - c 2 G vs 2 8π 3(N (rs)) 2 cos 2 θ ( 107
)
ρ 2 = - c 2 G vs 2 8π N (rs) + rs 2 N (rs) 2 sin 2 θ (108)
Comparing the above expressions with the negative energy density for a Natario warp drive in the the 1 + 1 spacetime given also in section 2:

ρ 1+1 = - c 2 G v 2 s 8π 3(N (rs)) 2 (109)
We can see that the term in ρ 1 almost matches the term in the 1 + 1 spacetime except for the trigonometric term in cos 2 θ and as seen in sections 2 and 3 this term produces a very low derivative square for the Natario shape function of about 10 -103 .So the term ρ 2 is the term that really accounts for the negative energy density in the 3 + 1 spacetime.

The dominant expression in the term ρ 2 is:

N (rs) + rs 2 N (rs) 2 (110)
Since the derivative of second order of the Natario shape function N (rs) is a lengthly expression as shown in section 2 with many algebraic terms then its square N (rs) 2 results in an even more complicated expression with even more algebraic terms.And the product of both the first and second order derivatives N (rs)N (rs) also results in a lengthly expression.Then in order to avoid algebraic complications we also decided in section 2 to work numerically with the new dominant term which is:

N (rs) + rs 2 N (rs) (111)
Raising to the square only the final numerically evaluated result.

• Numerical plot for the terms N (rs) + rs 2 N (rs) and N (rs) + rs 2 N (rs) 2 with @ = 50000 bubble radius R = 100 meters and warp factor with a value W F = 200 rs N (rs) + [(rs/2)N (rs)] (N (rs) + [(rs/2)N (rs)]) 2 9, 999940000000E + 01 0, 0000000000E + 000 0, 00000000000E + 000 9, 999950000000E + 01 0, 0000000000E + 000 0, 00000000000E + 000 9, 999960000000E + 01 0, 0000000000E + 000 0, 00000000000E + 000 9, 999970000000E + 01 0, 0000000000E + 000 0, 00000000000E + 000 9, 999980000000E + 01 0, 0000000000E + 000 0, 00000000000E + 000 9, 999990000000E + 01 0, 0000000000E + 000 0, 00000000000E + 000 1, 000000000000E + 02 -3, 0959501008E -043 9, 5849070264E -086 1, 000001000000E + 02 0, 0000000000E + 000 0, 00000000000E + 000 1, 000002000000E + 02 0, 0000000000E + 000 0, 00000000000E + 000 1, 000003000000E + 02 0, 0000000000E + 000 0, 00000000000E + 000 1, 000004000000E + 02 0, 0000000000E + 000 0, 00000000000E + 000 1, 000005000000E + 02 0, 0000000000E + 000 0, 00000000000E + 000 1, 000006000000E + 02 0, 0000000000E + 000 0, 00000000000E + 000

From the numerical plots in section 4 we know that the derivative of first order of the Natario shape function N (rs) is always zero except in the bubble radius. Its value in the bubble radius is then -3, 11150763893 × 10 -54 . From the numerical plots in section 7 we know that the derivative of second order of the Natario shape function N (rs) is also always zero except in the bubble radius. Its value in the bubble radius is then -6, 1919002015 × 10 -45 .

Then computing the value of the new dominant term in the expression for ρ 2 which is [N (rs)+( rs 2 )N (rs)] we expect to find values only in the bubble radius. Note that the value of N (rs) is much bigger than the value of N (rs) so it N (rs) that accounts for the final value of the new dominant term.

The final value of the new dominant term is then -3, 0959501008 × 10 -43 in the bubble radius. and the final value for the original dominant term in the expression for ρ 2 which is [N (rs) + ( rs 2 )N (rs)] 2 is then 9, 5849070264 × 10 -86 only in the bubble radius. This value also obliterates the factor 10 48 from a speed of 200 times faster than light resulting in a very low negative energy density of 10 48 × 10 -86 = 10 -38 .A very low negative energy density of 10 -38 Joules meters 3 for the term ρ 2 that really accounts for the negative energy density of the Natario warp drive in a 3 + 1 spacetime.

Note that the negative energy density in the term ρ 1 have values of 10 -55 Joules meters 3 according to sections 2 and 3.

The star system KIC8462852

In 2009 the Kepler Space Telescope was launched into space to look for extra-solar planets around distant stars. The portion of the sky covered by the Kepler Space Telescope was(and still is) a small region between the constellations of Cygnus and Lyra. The method used by Kepler to detect distant planets is the method of transit when a planet pass in front of the star blocking its light.When the Kepler analyzed the star KIC8462852 an anomaly was detected in the starlight distribution with an occultation that reduces the flux of light by 20 percent. Such anomaly as far as we know cannot be explained by natural causes. It is believed that the anomaly is due to an artificial alien megastructure that passes in front of the star blocking its light however this needs further confirmation. A complete coverage of all the scientific literature already produced for the star system KIC8462852 is beyond the scope of this work however we selected the 3 most relevant papers which are:

• 1)-paper in [17]:The paper of Tabetha Boyajian(Et Al) where the anomalies in the starlight propagation of KIC8462852 appeared for the first time.

• 2)-paper in [16]:The paper of Bradley Schaefer.This work demonstrates definitely that the anomalies in the starlight propagation of KIC8462852 cannot be explained by natural causes.

• 3)-papers in [START_REF] Wright | [END_REF], [14] and [15].The paper of Jason Wright(Et Al). These papers are different versions of the same paper. [START_REF] Wright | [END_REF] is the arXiv e-print version submitted to ApJ, [14] is the arXiv e-print version accepted by ApJ and [15] is the official manuscript effectively published by ApJ. Each version is different than each other but in the end the conclusions are the same ones. The work of Jason Wright(Et Al) presents the artificial explanation for the anomalies in the starlight propagation of KIC8462852 as being the result of the transit of an alien megastructure passing in front of the star blocking the starlight.Such megastructure is constituted by 8 large external objects in the outer part of the megastructure with more 8 internal objects in the inner part of the megastructure.The purpose of this megastructure is still unknown and the artificiality of the anomalies in the starlight propagation of KIC8462852 still needs confirmation by other independent works. However the paper of Jason Wright(Et Al) was the real motivation that inspired ourselves to write this work.

The implications of the existence of aliens or extraterrestrial advanced civilizations capable to assemble large megastructures is beyond the scope of this work and we will not discuss this topic further but since the work of Jason Wright(Et Al) was accepted and published by the Astrophysical Journal(ApJ) which is sponsored by the American Astronomical Society (AAS) this means to say that at least a scientific team of the editorial board agreed with the artificial explanation for the the anomalies in the starlight propagation of KIC8462852.

We must concentrate ourselves in the basic facts presented in these papers:

• 1)-paper in [16]:In the abs of [16] it is being mentioned that KIC8462852 suffers from unique and inexplicable day-long dips with 20 percent dimming. The dimming is unprecedent for a F-type star and it is also mentioned that KIC8462852 has anything unusual going on.Also still in the abs it is mentioned that the mechanism of the occultation is not due to a catastrophic planetary event but is an ongoing process with continuous effects.Such an event would require 648.000 comets each one with 200 km diameter all of them orchestrated to pass in front of the star with a regular daily trajectory.

In pg 1 there exists another mention to the unique series of dips in the brightness. In pg 2 it is mentioned that KIC8462852 lies at a distance of 454 parsecs.( 1480 light-years) and again the series of dips where the bright of the star faded by 0, 2-20 percent is again mentioned.Also in pg 2 it is mentioned that these unique dips cannot by caused by any defect in the artifact (telescope lenses) such fast variations are inexplicable the comet hypothesis cannot explain the dips in the light curves In pg 6 the unique and inexplicable photometric variations of the light curves is again mentioned. In pg 7 another mention to the starlight dimming by 20 percent. In pg 8 it is mentioned that the occultation mechanism if we try to use comets to explain the anomalies would require 648.000 comets each one with 200 km diameter all of them orchestrated to pass in front of the star with a regular daily trajectory and according to the author this is impossible.

• 2)-papers in [START_REF] Wright | [END_REF], [14] and [15]: In pgs 8 and 9 of [START_REF] Wright | [END_REF], [14] and [15] again there are mentions to the peculiar variations in the observed flux with losses of 20 percent.These events are extraordinary and unlike any other stellar transit of occultation. Also in pg 9 it can be seen that KIC8462852 lies at a distance of 600 parsecs.( 1956 light-years) and again there are mentions to the fact that the dimming is not due to defects in the artifact (telescope lenses) or instrumental reduction.In section 4.2 there are mentioned the difficulties to explain KIC8462852 naturally and all the scenarios failed.In pg 9 it is mentioned that the anomaly requires 8 large external occulters placed in the outer regions of the alien megastructure and at least another 8 internal occulters placed in the inner regions of the alien megastructure. At pg 19 in [START_REF] Wright | [END_REF],pg 17 in [14] and [15] exists a mention to the fact that invoking alien technology to explain anomalous phenomena is a perilous and danger approach .The correct procedure is to try to explain the anomalies using natural phenomena invoking the alien technology only when all the possible natural causes fails to explain the data. At pg 20 in [START_REF] Wright | [END_REF],pg 19 in [14] and pg 18 in [15] it is mentioned that KIC8462852 is surrounded by a swarm of artificial objects (see Appendices C and D)

• 3)-paper in [17]:In the abs of [17] there are mentions to the irregular shaped dips in the stellar flux up to 20 percent in luminosity loss.Also in the abs there exists a mention confirming that the dipping is not caused by defect in the artifacts (telescope lenses).The dipping events last between 5 and 80 days. At pg 2 another mention to the highly unusual dipping source. At pg 9 a mention to the distance of 454 parsecs.( 1480 light-years). At pg 10 a mention to the fact that no other candidate resembles KIC8462852 and again another confirmation that the dipping is not caused by defect in the artifacts (telescope lenses). At pg 15 again mentions to the unexplained dips in the flux up to 20 percent that last for several days and all the scenarios based over natural causes were unsuccessful in the explanation of the observations.

Alien intelligence or extraterrestrial advanced civilizations are delicate topics that must be examined with a strong cautionary note of skepticism and prudence in order to avoid over-enthusiastic misunderstandings or misinterpretations about facts that are not still entirely clear or confirmed.

More independent works about these 16 occulters are needed in order to confirm(or disproof) the extraterrestrial origin of the anomalies in the starlight propagation of KIC8462852.

However since no known natural phenomena is able to explain these anomalies then the hypothesis of aliens or extraterrestrial civilizations might be the correct one after all.

But one thing is clear:there exists a clear discrepancy in the measures of the distance between KIC8462852 and Earth presented in these works:The star cannot stay at 454 parsecs or at 600 parsecs at the same time!! This discrepancy must be solved.

For the Natario warp drive in an interstellar travel from Earth to the star system KIC8462852 we consider the largest distance of 600 parsecs or 1956 light-years.

11 Conclusion:Is the Natario warp drive a valid candidate for an interstellar voyage to the star system KIC8462852??

From pg 8 in [START_REF] Alcubierre | Classical and Quantum Gravity[END_REF] pg 1 in [START_REF] Natario | Classical and Quantum Gravity[END_REF] pg 1 and 2 in [START_REF] Lobo | Classical and Quantum Gravity[END_REF] pg 26 and 34 in [START_REF] Lobo | [END_REF] we known that General Relativity allows arbitrary large speeds(superluminal velocities) under the framework of the warp drive theory and we choose the Natario warp drive model in this work in an attempt to determine how large can these arbitrary velocities effectively be.In order to do so we must recall the equations of the negative energy density in the Natario warp drive 3 + 1 spacetime presented in section 2:

ρ 3+1 = ρ 1 + ρ 2 (112) ρ 1 = - c 2 G vs 2 8π 3(N (rs)) 2 cos 2 θ (113) ρ 2 = - c 2 G vs 2 8π N (rs) + rs 2 N (rs) 2 sin 2 θ (114)
Note that in the equations above the term

c 2 G vs 2 8π (115)
Appears with the superluminal velocity raised to the square and as fast as the ship goes by then more and more amounts of negative energy density are needed in order to sustain the warp bubble. Considering our Natario warp drive moving with vs = 10000 which means to say 10000 times light speed in order to make a round trip from Earth to KIC8462852 at 2000 light-years away in a reasonable amount of time(in months not in years) we would get in the expression of the negative energy density the factor c 2 = (3 × 10 8 ) 2 = 9 × 10 16 being divided by 6, 67 × 10

The total amount of negative energy density needed to sustain a warp bubble in a round-trip travel from Earth to KIC8462852 at a speed of 10000 times faster than light is directly proportional to a number with 50 zeros!!!

The sun have a mass of 10 30 kilograms and our galaxy have nearly 400 billions of stars (4×10 11 ).Assuming an average mass for each star in our galaxy of about the estimative of the mass of the sun 13 we have for the total mass of the galaxy 10 30 × 10 11 = 10 41 kilograms. a number with "only" 41 zeros. 10 50 is a number 10 9 times bigger than 10 41 so this means to say that the total amount of negative energy density needed to sustain a warp bubble at 10000 times light speed in our journey from Earth to KIC8462852 is directly proportional to the magnitude of the number that presents the masses of 1 billion galaxies!!!! This would turn of course the warp drive impossible but we also know from sections 2,3 and 9 that the values of the square derivatives of the Natario shape function being these derivatives of first and second order possesses extremely low values as shown below:

(N (rs)) 2 Very small values of negative energy density able to sustain a Natario warp bubble at 10000 times light speed in our journey from Earth to KIC8462852 This section started with a question:Is the Natario warp drive a valid candidate for an interstellar voyage to the star system KIC8462852??

The answer is:Undoubtely yes.Not only the Natario warp drive can cross the distance between Earth and the star system KIC8462852 with an arbitrarily large superluminal velocity of 10000 times light speed but can also be used to cross enormous distances in our galaxy.

Unfortunately although we can discuss mathematically how to reduce the negative energy density requirements to sustain a warp drive using warp factors we dont know how to generate the shape function that distorts the spacetime geometry creating the warp drive effect.So unfortunately all the discussions about warp drives are still under the domain of the mathematical conjectures.

However we are confident to affirm that the Natario warp drive will survive the passage of the Century XXI and will arrive to the Future.The Natario warp drive as a valid candidate for faster than light interstellar space travel will arrive to the the Century XXIV on-board the future starships up there in the middle of the stars transforming the scenario depicted in the science fiction novel Star Trek from an impossible dream into a physical reality and helping the human race to give his first steps in the exploration of our Galaxy

G

The Alcubierre expressions for the Negative Energy Density in Geometrized Units c = G = 1 are given by(pg 4 in [START_REF] Natario | Classical and Quantum Gravity[END_REF])(pg 8 in [START_REF] Alcubierre | Classical and Quantum Gravity[END_REF]): In this system all physical quantities are identified with geometrical entities such as lengths,areas or dimensionless factors.Even time is interpreted as the distance travelled by a pulse of light during that time interval,so even time is given in lengths.Energy,Momentum and Mass also have the dimensions of lengths.We can multiply a mass in kilograms by the conversion factor G c 2 to obtain the mass equivalent in meters.On the other hand we can multiply meters by c 2 G to obtain kilograms.The Energy Density( Joules meters 3 )in Geometrized Units have a dimension of G we retrieve again ( Joules meters 3 ). 15 . This is the reason why in Geometrized Units the Einstein Tensor have the same dimension of the Stress Energy Momentum Tensor(in this case the Negative Energy Density)and since the Einstein Tensor is associated to the Curvature of Spacetime both have the dimension of

1 length 2 .
G 00 = 8πT 00 (123)

Passing to normal units and computing the Negative Energy Density we multiply the Einstein Tensor (dimension 1 length 2 ) by the conversion factor c 4 G in order to retrieve the normal unit for the Negative Energy Density ( Joules meters 3 ).

T 00 = c 4 8πG G 00 (124)

Examine now the Alcubierre equations: vs = dxs dt is dimensionless since time is also in lengths. y 2 +z 2 rs 2 is dimensionless since both are given also in lengths. f (rs) is dimensionless but its derivative df (rs) drs is not because rs is in meters. So the dimensional factor in Geometrized Units for the Alcubierre Energy Density comes from the square of the derivative and is also 1 length 2 .Remember that the speed of the Warp Bubble vs is dimensionless in Geometrized Units and when we multiply directly G to obtain the Negative Energy Density in normal units Joules meters 3 the first attempt would be to make the following:

ρ = - c 4 G 1 32π
vs 2 f (rs) But note that in normal units vs is not dimensionless and the equations above do not lead to the correct dimensionality of the Negative Energy Density because the equations above in normal units are being affected by the dimensionality of vs.

In order to make vs dimensionless again,the Negative Energy Density is written as follows: In pg 9 of [START_REF] Wright | [END_REF], [14] and [15] it is mentioned that the anomalies in the light propagation of KIC8462852 requires 8 large external occulters placed in the outer regions of the alien megastructure and at least another 8 internal occulters placed in the inner regions of the alien megastructure.

ρ = - c 4 G 1 32π ( vs
The image above although not entirely correct shows 4 occulters17 in the external region of the alien megastructure and 2 occulters in the internal regions of the alien megastructure. This image is useful to transmit the idea of how the alien megastructure might looks like.

Recall the fact that at pg 19 in [START_REF] Wright | [END_REF],pg 17 in [14] and [15] exists a mention to the fact that invoking alien technology to explain anomalous phenomena is a perilous and danger approach .The correct procedure is to try to explain the anomalies using natural phenomena invoking the alien technology only when all the possible natural causes fails to explain the data. At pg 20 in [START_REF] Wright | [END_REF],pg 19 in [14] and pg 18 in [15] it is mentioned that KIC8462852 is surrounded by a swarm of artificial objects.

The purpose of this alien megastructure is still unknown and the artificiality of these occulters needs confirmation by other independent works. In the image above the star lies in the center of a "sphere" with all the occulters in the "surface".More than 8 occulters are being shown.

Although this image is not also entirely correct like the previous one if we project two concentric "spheres" one internal and another external with occulters placed in both "spheres" 8 in the inner "sphere" and 8 in the outer "sphere" we can get another idea of how the alien megastructure might looks like.

  -11 giving 1, 35 × 10 27 and this is multiplied by (3 × 10 12 ) 2 = 9 × 10 24 coming from the term vs = 10000 giving 1, 35 × 10 27 × 3, 5 × 10 23 = 4, 86 × 10 50 !!! a number with 50 zeros!!!

2 G v 2 s

 22 3, 8 × 10 -103 (117) [N (rs) + ( rs 2 )N (rs)] 2 9, 5 × 10 -86 (118) And these terms appears in the equations of the negative energy density completely obliterating the factor c 8π 10 50 allowing the Natario warp drive to perform a trip from Earth to KIC8462852 at 10000 times light speed. As a matter of fact we have: 10 50 × 10 -103 10 -53 Joules meters 3 (119) 10 50 × 10 -86 10 -36 Joules meters 3 (120)

Live

  Long And Prosper As Captain Jean-Luc Picard would say:Make It So!!! 12 Appendix A:Dimensional Reduction from c 4 G to c 2

1 length 2

 12 and the conversion factor for Energy Density is G c 4 .Again on the other hand by multiplying 1 length 2 by c 4

1 length 2

 12 from the Negative Energy Density in Geometrized Units by c 4

Figure 1 :

 1 Figure 1: Artistic Presentation of the star system KIC8462852 with some of the 8 large occulters . (Source:Internet)

Figure 2 :

 2 Figure 2: Alternative Artistic Presentation of the star system KIC8462852 with some of the 8 large occulters . (Source:Internet)

  8 ) 2 = 9 × 10 16 being divided by 6, 67 × 10 -11 giving 1, 35 × 10 27 and this is multiplied by (6 × 10 10 ) 2 = 36 × 10 20 coming from the term vs = 200 giving 1, 35 × 10 27 × 36 × 10 20 = 1, 35 × 10 27 × 3, 6 × 10 21 = 4, 86 × 10 48 !!! A number with 48 zeros!!!The planet Earth have a mass 8 of about 6 × 10 24 kg

	5

  As already seen.The same results are valid for the Natario Energy Density

											c	) 2 f (rs)	2 [	y 2 + z 2 rs 2 ]	(127)
	ρ = -	c 4 G	1 32π	(	vs c	) 2 df (rs) drs	2	[	y 2 + z 2 rs 2 ]	(128)
	Giving:
	ρ = -	c 2 G	1 32π	vs 2 f (rs)	2 [	y 2 + z 2 rs 2 ]	(129)
	ρ = -	c 2 G	1 32π	vs 2 df (rs) drs	2	[	y 2 + z 2 rs 2 ]	(130)
	Note that from
	ρ = -	c 4 G		1 32π	(	vs c	) 2 f (rs)	2 [	y 2 + z 2 rs 2 ]	(131)
	ρ = -	c 4 G	1 32π	(	vs c	) 2 df (rs) drs	2	[	y 2 + z 2 rs 2 ]	(132)
	Making c = G = 1 we retrieve again
	ρ = -	1 32π	vs 2 f (rs)	2 [	y 2 + z 2 rs 2 ]	(133)
	ρ = -	1 32π	vs 2 df (rs) drs	2	[	y 2 + z 2 rs 2 ]	(134)

do not violates Relativity

remember that the major part of the stars in our galaxy are red dwarfs much lighter than the sun

tanh[@(rs + R)] = 1,tanh(@R) = 1 for very high values of the Alcubierre thickness parameter @ >> |R|

not exactly zero but possesses extremely low values and we are limited by the floating-point precision of our software

Remember that we are limited by the floating-point precision of our software

remember that the major part of the stars in our galaxy are red dwarfs much lighter than the sun

assuming each ring-shaped form surrounding the star as an occulter

13 Appendix B:The Natario warp drive negative energy density in Cartezian coordinates

The negative energy density according to Natario is given by(see pg 5 in [START_REF] Natario | Classical and Quantum Gravity[END_REF]) 16 :

In the bottom of pg 4 in [START_REF] Natario | Classical and Quantum Gravity[END_REF] Natario defined the x-axis as the polar axis.In the top of page 5 we can see that x = rs cos(θ) implying in cos(θ) = x rs and in sin(θ) = y rs Rewriting the Natario negative energy density in cartezian coordinates we should expect for:

Considering motion in the equatorial plane of the Natario warp bubble (x-axis only) then [y 2 + z 2 ] = 0 and rs 2 = [(x -xs) 2 ] and making xs = 0 the center of the bubble as the origin of the coordinate frame for the motion of the Eulerian observer then rs 2 = x 2 because in the equatorial plane y = z = 0.

Rewriting the Natario negative energy density in cartezian coordinates in the equatorial plane we should expect for: