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Chat with Jose Natario

September 17th, 2009 byRichard Obousy

 
 

I recently enjoyed a chat with Dr Jose Natario, author of several influential warp drive papers. Arguably one of Dr Natario's most notable contributions to the field is his demonstration that a mutual contraction/expansion of spacetime is not a required feature of a warp drive, as was commonly believed. You can read the highlights of our conversation below.


Could you briefly describe your research interests and also your contributions to the field of warp drives?

 

[Natario] My research field is General Relativity. Besides warp drives, I have done work on the geometry of the space of light rays, various types of 4 and higher dimensional black holes, spinning objects and the quasi-Maxwell formulation of the Einstein equations.

I have written 2 papers on warp drives. In the first one I set up a fairly general model for a warp drive spacetime, which includes the Alcubierre model as a particular case. I showed that it is possible to have models where there is no overall contraction or expansion of space (although space does get severely distorted on the warp bubble wall). My hope was that one would then get less severe violations of the energy conditions, but that turned out not to be the case: one can easily prove that these
models always violate the positivity of energy. I also showed that there are two other (in my view more serious) problems with the warp drive concept, namely the existence of horizons and infinite blueshift regions.


The second paper studied warp drive spacetimes which have a classical Newtonian analogue. It showed that in these Newtonian versions the gravitational field inside the warp bubble is uniform whenever the bubble is accelerating, thus avoiding tidal forces; all nonuniform gravitational fields are confined to the bubble wall.

What got you interested in warp drives in the first place?

[Natario] The first time I heard about the warp drive was reading the daily newspaper back in Portugal, when I was still an undergrad. I thought it was a nice idea, but didn't really think too much about it. A few years later, when I was in Oxford doing my DPhil, a fellow student showed me the Alcubierre paper, and I made a copy to read at home. The idea was clear: one makes space move around in unusual ways, dragging the spaceship with it. What I didn't understand was why was it necessary for space to expand and contract: couldn't one make it move without expansion? That was the beginning of my first warp drive paper.


It appears on arXiv that far more papers are released on wormhole research than warp drive. Why do you think that this is?

 

[Natario] For one thing the wormhole idea is much older (going back to the Einstein-Rosen bridge), and has been pursued by well known physicists. It is also mathematically more interesting, and probably more realistic (as science fiction ideas go). Finally, it could be relevant to quantum gravity.

 

What do you see as the biggest questions/challenges in current warp
drive research?

[Natario] The question of the horizons. The problem is the following: in all currently known warp drive models part of the warp bubble wall (i.e. the place where the negative energy fields are located) is causally disconnected from the interior. This means that it is impossible for someone inside the warp bubble to send a signal to (hence to control) these regions of the wall. Intuitively, you cannot set up a bubble which is moving faster than light from inside the bubble, because your signals, which travel at most at the speed of light, cannot reach the front part of the bubble wall. (Some people call this the "you need one to make one problem"). A related problem is that whenever the bubble starts travelling faster than light an infinite blueshift region (the analogue of a sonic boom) forms behind it.

 

Warp drive... theoretical curiosity or deep future technology?

[Natario] Because of the two problems I mentioned, I think it will remain a theoretical curiosity.

 

What are you currently working on?


[Natario] Higher dimensional black holes.


  Comment by:   malcolm Shore , 19 September 2009

  


  Comment by:   malcolm Shore , 19 September 2009

  

''Intuitively, you cannot set up a bubble which is moving faster than light from inside the bubble, because your signals, which travel at most at the speed of light, cannot reach the front part of the bubble wall. (Some people call this the "you need one to make one problem"). ''


'Intuitively, you cannot set up a bubble which is moving faster than light from inside the bubble, because your signals, which travel at most at the speed of light, cannot reach the front part of the bubble wall. (Some people call this the "you need one to make one problem"). ''

I have never understood the logic of this, it seems to me that if  the warp bubble wall is created by and sustained by some process such as in ''Casimir energy and the possibility of higher dimensional manipulation'' then when that part of the wall becomes disconnected the process  sustaining  the warping effect disappears however the effect again reaches this region ,the warping effect reappears. In short the warp drive would pulse on and off.
 











  Comment by:   louis soltis , 16 July 2010

  

Hi everyone,

My humble opinion:

Reading Martin Tajmar's paper and reviewing his results, increasing the

gravitational field further above the 3.3 noise level, would require pulsed circuit

technology (circuits)..

I'd play with it experimentally but currently, I don't have the time and resources.

 

There are other factors to increase the field as well.

If anyone else is working on these results experimentally and there are published papers, please advise me to location..

These are exciting times, but a direct and concerted effort is needed to see theory and technology to fruition.

Thanks,

God Bless,

lou

 

   
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Warpism

September 03th, 2009 byRichard Obousy

 
 

Building a site thoroughly devoted to the concept of a warp drive has been on my mind for some time now. With the creation and completion of the Interstellar Journey documentary, that you can find on the home page, the time seemed right to build this site.


The scientific warp drive community is currently rather small and there's not much serious research going on right now. The likely reasons for this are two-fold.


1. The Giggle Factor


The giggle-factor is a consequence of using a name for a cutting edge propulsion concept that is taken straight from science fiction. In reality the name is a double-edged sword. When one mentions a 'warp drive' it should be immediately obvious (one would hope) that what is being dicussed is a hypothetical propulsion mechanism that utilizes an asymmetric manipulation of the fabric of spacetime to generate an exotic curvature which allows one to circumvent the traditional limitations of Special Relativity and travel at superluminal velocities.


The reality is that one typically pictures the crew of the Enterprise facing crucial issues of interstellar morality that are usually resolved by Captain Kirk wooing an attractive starfleet officer and using sheer cunning to outwit a robot with a supercomputer for a CPU and generally causing it to overheat in the process.


2. The Miracle


The second reason that warp drives are generally not popular research topics are that exotic and unproven forms of energy are usually necessary to generate the 'warp' and the energies required are so large that even a Type III Kardashev civilization would grumble in disbelief.


These two reasons are enough to deter most physicists away from the study of warp drives and lure them into more conventional research arena's. That said, I do believe that the 'Giggle Factor' plays possibly the strongest role in this aversion to the study of warp drive. I say this because there is another faster-than-light loop-hole in the laws of physics called the Einstein-Rosen bridge, otherwise known as a wormhole.


The wormhole is an equally exotic solution to Einstein's field equations also requiring negative energy and prodigous amounts of energy to sustain. Despite these similiarities, the study of wormholes enjoys a somewhat rich and active research community. A quick peek at Spires (the 'Google' of physics research) and a keyword search for 'wormhole' produces 1248 research papers, with 10 papers being released to arXiv in the month of August alone.


Compare this with the rather abysmal and pathetic 33 research papers when one performs a keyword search for 'warp' and 'drive', with an average of 1 to 2 papers being released a year. It's clear that, despite the theoretical similarities in the two FTL schemes, that warp drive suffers from a serious image problem most likely related to the Giggle Factor.


There's one other possibility why wormholes may be ahead of the curve, and that's the simple fact that wormhole research has a number of 'big name' physicists behind it. This cadre includes: Hermann Weyl, John Wheeler, Kip Thorne and Matt Visser to name a few.


It seems to me that warp drive research could experience a serious 'kick-start' if a few more reputable physicists devoted a little time to their study...anyone have Ed Wittens email address handy? 


 


 


  Comment by:   Berkut , 12 September 2009

  Congratulation for the site, blog and documentary.

I have no working knowledge of theoretical physics, I'm an aerospace engineer,
and I'd very much like to get a hand on this, and YES, I'm a sci-fi geek alike...

1) If you were to advise self-teaching people with a minimal math and physics
background, what book/web site etc.. would you recommend to start with ?

2) Here's a more philosophical one : studying epistemology, you find that the
greatest discovery were of "serendip" nature as you english speaking people say.
Not that Einstein was not all at his subject, but he was not trying to tweak physics
to invent nuclear powerplants (not before WWII anyway) : the consequences of his
discoveries reached far beyond his dreams and nightmares I guess. Why would
you think that one should "focus" on "engineering" physics to devise principles
that could be viewed as "pieces of machinery" (although only in equations at this
stage), rather that letting one's mind wander about et see what comes out  ?


  Comment by:   Richard Obousy , 13 September 2009

  

Hi Berkut,

Congrats on being the first to post on the forum! I'll do my best to answer your questions:

1. Any of the books by Paul Davies, Brian Greene and Lisa Randall are a spectacular place to start. I'd probably begin with:

i. The Matter Myth (Paul Davies and John Gribbin)

ii. The Elegant Universe (Brian Greene)

iii. The Fabric of the Cosmos (Brian Greene)

iv. Warped Passaged (Lisa Randall)

2. It's an interesting question. Of course, as theoretical physicists we often let our minds wander as you put it, and often very interesting things come out. Some of these ideas, however, may only be of interest to a small handful of other phyicists working on similar problems so it's unlikely they'd make it to a website designed for popular interest.


  Comment by:   Matt , 13 October 2009

  Why would a Type III Kardashev civilization grumble at the energy needed?  Shouldn't a civilization that advanced already have interstellar capability of some kind, Type III being able to harness the power of an entire galaxy?

I had never heard of The Matter Myth before, although I've read some Davies.  I liked Lisa Randall's book, but at the time I had already read Greene, Hawking, Lederman, and Kaku, so I skipped over a lot of the stuff.


  Comment by:   Richard Obousy , 21 October 2009

  

A Type III Civilization could theoretically harness around 10^37W. According to our calculations we'd need approximately the rest mass of Jupiter converted into energy. This corresponds to around 10^44 J. A quick calculation reveals that it would take a Type III civilization 116 days to generate this much energy...assuming they wanted to devote 100% of their energetic resources.

These are very much simplistic approximations but give us an idea of the scales involved. I hope that answers your question as to why they would grumble.

Yes, a Type III would have to have developed interstellar travel of some form, but that's not to say they would be traveling at FTL speeds.


  Comment by:   rookdiez , 11 February 2010

  part satellite debate 20th response aerosols hypothesis ocean [url=http://www.physicsforums.com]decadal satellite[/url] http://scintilla.nature.com


  Comment by:   orvalhilto , 11 February 2010

  activity era conclude trends comments warm


  Comment by:   Antony , 23 February 2010

  Hi Richard, I was wondering if you have looked at Extended Heim Theory? What are your thoughts on this theory? According to this theory gravity is actually 3 separate forces.  I understand there is some experimental evidence to support it now.  A recent series of experiments by Martin Tajmar et al., partly funded by ESA may have produced the first evidence of artificial gravity. By applying their 'gravito-photon' theory to bosons, Dröscher and Häuser were also able to predict the size and direction of the effect.
The Theory also accounts for the anomalous results from the Gravity Probe B experiments.

http://www2.phys.canterbury.ac.nz/~physrin/papers/SuperFrameDragging2007.pdf


  Comment by:   John B , 23 February 2010

  

Hello Richard:

It's great to finally see a web site with some useful and objective information and commentary on a subject that  just a short time ago would have been a career killer for a lot of physicists. Keep it going.

I have three comments.

Note: The following assumes that the teleport technology, like the FTL technology, is a given. Only the theoretical basis is in question.

1)I may be wrong on this but from what I've read about Quantum Teleportation lately although the actual teleportation occurs in a non-local and instantainious manner the transfer of information regarding set up of the teleportation is still limited by the speed of light. For example if you wish to teleport to say a star 10 light years away you would have to communicate the necessary information regarding what you wish to transport by radio, which would take 10 years. An alien race that receives the info could then ser up their equipment accordingly. You could then teleport the object in question instantly after the information had arrived and allowed them to set up the teleporter. Although this makes true faster then light communication impossible it cuts down on fuel costs. The actual material you wish to transport is aquired at the destination. You need only send information to utilise it.

Assuming I am not to far off the beam in what I just said could a very small and "cheap" wormhole be used for the purpose of sending only information at FTL. The wormhole would then be used in conjunction with quantum teleportation to make FTL transport possible.  

2)With regard to Type III grumblers.

Although it is more a question of politics and psychology then physics the point you made about Type III grumblers has a flip side. The Apollo program cost the US about $24,000,000,000 dollars in the 1960's. For puposes of comparison the total US federal budget in those days averaged around $100,000,000,000. This means the US spent about 90 days tax revenues for a trip to the moon. Of course it was spread out over about 10 years. But the future Type III's could do the same thing, store (tax) the energy over a period of time then use it for a FTL event to cross the galaxy. A civilization functioning at that level of complexity would find some advantage in transmitting information if not material from one end to the other in far less then the 100,000 years otherwise required. Like travel to the moon today (or SST travel over the Atlantic a while back) it would be a luxury, but not necessarily a real budget buster. Of course there is no one to one correspondence between tax revenue and energy production but both are connected to the idea of overall capability of a civilization. There would be a lot of uninhabited Jupiter size planets for fuel in a typical galaxy. Hmmm I wonder if people would eventually end up waiting in line to refuel their warp drives when the supply of Jupiters starts to run low? Nothing truely lasts forever.

3) With regard to comment by Antony:

I certainly am not in a position to say one way or the other what may be the true case in regard to the theories you mentioned but the pdf linked at the bottom of the comment seems to refrute rather then support the Super Frame Dragging hypothesis. In the original articles by Tajmer et al a linear accelerometer was used. The Laser Ring Gyro accelerometers used in the linked paper are, according to that papers authors, more sensitive. As I read their paper they seem to claim that the effect, if it exists, is less then 5% of the value predicted.

John B

 


  Comment by:   Frank , 23 March 2010

  

Hey Dr. Obousy,

Awesome site!  I have always had an interest in this subject and it’s nice to see a scientifically accurate website on the subject.  I just had a few questions.

1. I recently graduated with a B.S. in physics and I am going on for a Ph.D. in aerospace engineering.  Now I know that I won’t be taking the necessary quantum mechanics/string theory courses that I would need, but are there other resources that I could use to stay up to date on developments/become active within the field?

2.  In light of the recent objection to warp drive raised by Edelstein about incoming hydrogen atoms, I was wondering if any of the previous papers on warp drive offer a valid solution.  Many of them claim to solve such problems as the formation of horizons, incoming matter and the high energy requirements, but are their solutions valid?  Also does your warp drive model still apply in the RS model?

3.  In your warp drive paper you stated that the energy requirements could be significantly reduced if it was assumed that it was a thin shell of spacetime instead of a bubble encompassing the volume of the ship.  Is this still a possibility or has it been ruled out?

4.    Are you currently still researching warp drive/the above mentioned topics?

Thanks for your time!

Frank


  Comment by:   Richard Obousy , 25 April 2010

  

Hi Anthony,

Thanks for your post. I've read a little about Heims work and by the looks of things there's some interesting results. It certainly appears as though nature is pushing us toward the realization of extra dimensions. It may be that Heims work turns out to be a low energy approximation to some aspect of string theory, or may provide new ways of thinking about certain aspects of nature. I also think it's important to remain pragmatic and not to 'fall in love' with a theory.

Richard


  Comment by:   Malcolm Shore , 12 May 2010

  

Here is a new warp drive paper that may be found interesting.

http://vixra.org/abs/1005.0024


  Comment by:   M A Shore , 16 June 2010

  

A new paper on warp  drives by Fernando Loup 

http://vixra.org/abs/1006.0028








  Comment by:   M A Shore , 28 February 2011

  

 
 

Horizons(causally Disconnected Regions of Spacetime) and Infinite Doppler Blueshifts in Both Alcubierre and Natario Warp Drive Spacetimes



  Comment by:   M A Shore , 09 October 2012

  


Assunto: warp drive-Open Letter to ResearchGate

 

 

Open Letter to ResearchGate

 

warp drive-is the scientific process of peer review fair and just ??or have failures???

 

My name is Fernando Loup and i am interested in the so-called warp drive spacetimes.

 

The warp drive is an exotic solution of the Einstein Field equations of General Relativity

that theoretically allows faster than light motion.

 

I have some complaints about the so called peer review system.This system is not fair.

For those in ResearchGate unfamiliar with the peer review system here is an introduction:there  are two ways to publish a scientific document:

 

-----------------------------------------------------------------------------

One is to store in an E-print server where the document is available on-line to the general  public.There are in the world 3 E-print servers:arXiv(US),HAL(France) and viXra(UK).

I use the  HAL(the French version of arXiv) as my main E-print server.Both arXiv and HAL are moderated

while viXra is unmoderated.By moderation this means to say that a paper sent to arXiv or HAL is  checked out by people who determines if the paper is in the proper scientific format according  to the E-print standards of quality although moderators do not check the correctness of  equations or scientific statements.

 

Two is to send it to a peer review conventional scientific journal.Conventional journals were  the main way to publish a document until 1991 when arXiv was created.These journals are not

available to the public and only a subscriber can access the content of scientific documents.

The subscription of these journals is so expensive that only universities or laboratories can

subscribe them.But conventional journals are the preferred way to publish documents.

In a conventional journal the editor assigns a referee normally a scientist with a doctorate or  a post-doctorate in the field of subject of the document that will produce the so called

 "referee report" that will tell to the editor if the document can be published..or not...

some times the referee hits the score...sometimes the referee fails..in my case the referee  failed because he seems to do not know basic trigonometry...

how can someone that do not know  basic trigonometry act as a referee...????

-----------------------------------------------------------------------------

 

In order to understand the reason of my complaint about the failures of the peer review system

i will give a brief introduction to one problem that affects the warp drive as a valid  scientific solution of the Einstein Field equations of General Relativity:

 

The so called Quantum Inequality(QI).

 

In order to get acquaitance with the warp drive theory for those interested but still unfamiliar  with this theory i would recommend this paper for an excellent introduction:

 

http://arxiv.org/abs/0710.4474

 

So here we go:the warp drive was discovered in 1994 by the Mexican mathematician Miguel  Alcubierre.Later on in 2001 a second warp drive was discovered by the Portuguese mathematician

Jose Natario proving in fact that the warp drive is a valid family of solutions of the Einstein

Field equations of General Relativity like black holes or wormholes and at the present we know

only two solutions for the warp drive family although perhaps many more members may be  discovered in the future.

 

However the warp drive have a serious drawback:it requires negative energy to be created and

requires lots of it.Classical physics do not allow the existence of negative energy:the  so-called energy conditions but Quantum Field Theory allows the existence of negative energy:

the so-called exotic matter.However the same quantum theory that allows the existence of exotic  matter have a serious drawback:

As larger is the amount of exotic matter created as smallest is  the time of its existence.This is known as a Quantum Inequality(QI).

This means to say that the amount of exotic matter needed to create a warp drive can only exists  for a period of nanoseconds or less making the warp drive impossible for interstellar travel.

Consider a warp drive at 200 times light speed travelling to reach the star Gliese 581 at 20  light-years away.The journey would be completed in two months

but if the exotic matter can  exists only for nanoseconds then the warp drive is physically impossible and a theoretical curiousity.

 

The Quantum Inequalities(QI) for the warp drive were calculated in 1997 by Lawrence Ford and Michael John Pfenning in this paper:

 

http://arxiv.org/abs/gr-qc/9702026

 

Note that although this is an E-print it was published in peer review in 1997 but with a

typographical error in equation 3...the referee dit not spotted the error...not relevant

for the conclusions of the document and it was corrected in 2001..i know the one that sent

an email to the authors of the paper to correct the equation...this is one of the failures

of the peer review system...while in an E-print we must read all the documents carefully

and check by ourselves the scientific content the peer review process is supposed to check every  equation

 because peer review publications are printed in paper and paper costs a lot...

in this case they spent paper to print the wrong equation but fortunately this equation is not  relevant..

.this paper illustrates without shadows of doubt the failures in the peer review  system.

 

Ford and Pfenning calculated that in order to maintain a warp drive the total amount of exotic

matter is of about 10 times the mass of the Universe and this amount would last for a period or  less than nanoseconds..this makes the warp drive impossible.

 

However when Natario discovered the second warp drive no one recalculated the QI for the Natario

warp drive.

 

Well i calculated the QI for the Natario warp drive.Here are the papers:

 

 

http://hal.archives-ouvertes.fr/INSMI/hal-00734603/fr/

http://hal.archives-ouvertes.fr/hal-00734603/en/

 

http://hal.archives-ouvertes.fr/INSMI/hal-00732757/fr/

http://hal.archives-ouvertes.fr/hal-00732757/en/

 

 

The papers shows that for the Natario warp drive ther amount of exotic matter is more reasonable

and this exotic matter can last for more than nanoseconds.It can last time enough to sustain

an insterstellar travel at 200 times light speed to Gliese 581 at 20 light years away.

 

This is due to the distribution of the negative energy in the Natario warp drive very different  than in its Alcubierre counterpart

that makes Natario warp drive survive to the QI.

 

The QI is an infinite integral (integral with the limits of integration in the infinity.one  limit of integration is +infinity and the other is -infinity)

 

Solving the integral by trigonometric substitution we will arrive at two quantities

 

arctan(+infinity) and arctan(-infinity) 

 

now a litte bit of trigonometry:

 

sin(90) = 1 and cos(90)=0  tan(90)= 1/0 = + infinity

 

so arctan(+infinity)=90

 

sin(270) = -1 and cos(270) = 0 tan(270) = -1/0 = -infinity

 

so arctan(-infinity) = 270

 

Browse the picture of trigonometric cycle from Wikipedia..one can see clearly that the  tan(270)=-infinity while tan(90)=+infinity

 

http://en.wikipedia.org/w/index.php?title=File:Unit_circle_angles_color.svg&page=1

 

More of trigonometric functions also from Wiki

 

http://en.wikipedia.org/wiki/Trigonometric_functions

 

The referee of the journal i sent the documents say that tan(270) is not - infinity 

He says that tan(270) is undeterminate so my calculations are not correct.

 

I think this referee or forgot the basics of trigonometry or he never knew it.

In my opinion he was not qualified to refer scientific documents.

In my opinion he was incompetent and what is most unbelievable the journal have one of the  biggest impact factors in the scientific community.

Of course i will not speak here the name of  the journal since i do not want to bring troubles to ResearchGate but those interested to know

the name of the journal can contact me in private and i will have the will

(and the pleasure..yes and i underline the pleasure) to speak the name of the journal.

 

A funny thing:if the division by zero is undeterminated then how about the variation of mass

according to Einstein Special Relativity when v=c??

 

http://en.wikipedia.org/wiki/Mass-energy_equivalence

 

m = m_0 /sqrt[1-(v^2/c^2)]

 

If v=c then sqrt[1-(v^2/c^2)] = 0 and m_0 / 0 = +infinity

 

Science says that a non-zero mass particla cannot travel at light speed or surpass it

because an infinite amount of mass is required to travel at light speed according

to Special Relativity.But it is an infinite mass or an undeterminated mass????

An undeterminated mass of +5kg or + 5 grams can travel at light speed or even surpass it!!!

Ahhhhh for one things the division by zero is infinite but for others is undeterminate???

How sweet!!!how convenient!!!!how disgusting!!!!

This illustrates the bureaucracy of science and the incompetence of this referee.

Those that may ask me for the name of the journal will get a big surprise due to the

so-called "impact factor" and "scientific reputation" of this journal..well in my opinion

this journal cannot be considered a journal but instead..a tabloid...

 

To terminate here is a peer review published paper about the injustices ..of peer review!!!

 

http://vixra.org/abs/0907.0020

 

Abstract

 

After a short overview of arguments pro and contra peer reviews, examples of gross misjudgement  are compiled, followed by an attempt to identify some frequent,

recurrent patterns of  unjustified rejection of scientific manuscripts. A few specific questions are studied in more  detail: the claim for still more precise and comprehensive

definitions, the right way of  handling "parallel theories", and the frequent misuse of the term "pseudoscience". Finally,  practical rules to improve refereeing, and

"basic rights of authors" are proposed, together with  a word of encouragement for future authors.

 

Comments: 10 pages, Journal reference - Journal of Scientific Exploration, Vol. 22, No. 2,  pp.233 – 243, 2008

 


  Comment by:   M A Shore , 11 October 2012

  

 

Malcolm according to the link below Xavier is right    arctan is defined between –pi/2 and + pi/2


http://en.wikipedia.org/wiki/Inverse_trigonometric_functions

 

according to this site below the division of 1 by 0 is undeterminate


http://en.wikipedia.org/wiki/Exact_trigonometric_constants

 

but according to this site below the division of 1 by 0 is infinite  and we have ranges from – infinite to + infinite

 

also according to this site tan=sin/cos      and sin(270) =-1 and cos(270) = 0    so tan(270)=-1/0 = -infinite or should be –infinite  according to the rules “by the book”

 

but unfortunately arctan(-infinite) is considered to be defined between –pi/2 and +pi/2      so perhaps the referee was mentioning the fact that what is undeterminate is the arctan(tan(270)) and not the division of 1 by 0    arctan is not defined for 270  he was not entirely clear

 

http://en.wikipedia.org/wiki/Trigonometric_functions

 

from the link below we can see that sin(90)=1 cos(90) = 0 and sin(270)=-1 and cos(270) = 0 so the definition of tan=sin/cos gives ranges from –infinite to + infinite according to the site above

 

although what is being considered here is perhaps the fact that tan(270) = - tan(90)  and not tan(270)=sin(270)/cos(270) both gives – infinite…but the inverse function don’t heve definition for 270


http://en.wikipedia.org/wiki/File:Unit_circle_angles_color.svg

 

so the second paper without warp factors prove Natario warp drive is so impossible as the Alcubierre one

 

Howevet the first paper with warp factors

 

http://hal.archives-ouvertes.fr/INSMI/hal-00732757/fr/

http://hal.archives-ouvertes.fr/hal-00732757/en/

 

may still be rescued back because a sampling time less than or equal to 20 billion years and not greater than or equal to 20 billion years would still make Natario warp drive a valid candidate for interstellar travel

 

as for me I will take vacations from this topic…lots of work trying to find an integration method to solve the QI for Natario warp drive and after all defeated by a sample forgotten definition of high school trigomometry

 

 


  Comment by:   Fernando Loup , 16 October 2012

  

work updated....Natario warp drive is still valid for insterstellar space travel

the >= sign was corrected to a <=

the maximum time we can observe a Natario warp drive is 20 billion years



  Comment by:   Fernando Loup , 02 November 2012

  

the other work is now corrected too    Natario warp drive is a valid candidate for interstellar travel


  Comment by:   malcolm Shore , 04 December 2012

  

A new warp drive paper can be found at

http://hal.archives-ouvertes.fr/hal-00760456/en/

   
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