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Astronomy Without A Telescope – Warp Drive On Paper

Article Updated: 24 Dec , 2015

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The Alcubierre drive is one of the better known warp drive on paper models – where a possible method of warp drive seems to work mathematically as long as you don’t get too hung up on real world physics and some pesky boundary issues.

Recently the Alcubierre drive concept has been tested within mathematically modeled metamaterial – which can provide a rough analogy of space-time. Interestingly, in turns out that under these conditions the Alcubierre drive is unable to break the light barrier – but quite capable of doing 25% of light speed, which is not what you would call slow.

OK, so two conceptual issues to grapple with here. What the heck is an Alcubierre drive – and what the heck is metamaterial?

The Alcubierre drive is a kind of mathematical thought experiment where you imagine your spacecraft has a drive mechanism capable of warping a bubble of space-time such that the component of bubble in front of you contracts bringing points ahead of you closer – while the bubble behind you expands, moving what’s behind you further away.

This warped geometry moves the spacecraft forward, like a surfer on a wave of space-time. Maintaining this warp dynamically and continuously as the ship moves forward could result in faster-than-light velocities from the point of view of an observer outside the bubble – while the ship hardly moves at all relative to the local space-time within the bubble. Indeed throughout the journey the crew experience free fall conditions and are not troubled by G forces.

Standard images used to describe the Alcubierre drive. Left: Want to make the Kessel run in 12 parsecs? No problem - just compress the Kessel run into 12 parsecs. Right: The Alcubierre concept can be thought of as a spaceship surfing on a wave of space-time. Images sourced from daviddarling.info.

Some limitations of the Alcubierre drive model are that although the mathematics can suggest that forward movement of the ship is theoretically possible, how it might start and then later stop at its destination are not clear. The mechanism underlying generation of the bubble also remains to be explained. To warp space-time, you must redistribute mass or energy density in some way. If this involves pushing particles out to the edges of the bubble this risks a situation where particles at the boundary of the bubble would be moving faster than light within the frame of reference of space-time external to the bubble – which would violate a fundamental principle of general relativity.

There are various work-around solutions proposed, involving negative energy, exotic matter and tachyons – although you are well down the rabbit-hole by this stage. Nonetheless, if you can believe six impossible things before breakfast, then why not an Alcubierre drive too.

Now, metamaterials are matrix-like structures with geometric properties that can control and shape electromagnetic waves (as well as acoustic or seismic waves). To date, such materials have not only been theorized, but built – at least with the capacity to manipulate long wavelength radiation. But theoretically, very finely precisioned metamaterials might be able to manipulate optical and shorter wavelengths – creating the potential for invisibility cloaks and spacecraft cloaking devices… at least, theoretically.

Anyhow, metamaterials capable of manipulating most of the electromagnetic spectrum can be mathematically modeled – even if they can’t be built with current technologies. This modeling has been used to create virtual black holes and investigate the likelihood of Hawking radiation – so why not use the same approach to test an Alcubierre warp drive?

It turns out that the material parameters of even so-called ‘perfect’ metamaterial will not allow the Alcubierre drive to break light speed, but will allow it to achieve 25% light speed – being around 75,000 kilometres a second. This gets you to the Alpha Centauri system in about seventeen years, assuming acceleration and deceleration are only small components of the journey.

Whether the limitations imposed by metamaterial in this test are an indication that it cannot adequately emulate the warping of space-time – which the Alcubierre drive needs to break light speed – or whether the Alcubierre drive just can’t do it, remains an open question. What’s surprising and encouraging is that the drive could actually work… a bit.

Further reading: Smolyaninov, I. Metamaterial-based model of the Alcubierre warp drive.

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36 Responses

  1. J. Major says:

    What about the energy requirements to do all this space-time compressing and expanding? I mean, if it were feasible within “normal” power generating that one might use on a spaceship that’s one thing, but if it turns out to be one of those “infinite energy is needed” situations then it kind of renders the practical application moot, no?

  2. tacitus says:

    It turns out that the material parameters of even so-called ‘perfect’ metamaterial will not allow the Alcubierre drive to break light speed, but will allow it to achieve 25% light speed – being around 75,000 kilometres a second. This gets you to the Alpha Centauri system in about a year or the Gliese 581 system in about five years, assuming acceleration and deceleration are only small components of the journey.

    I can’t believe you got this so wrong, Steve! Traveling at 25% of light speed to Alpha Centauri will get you there in 17.46 years not one and a bit years (i.e. 4.36ly / 0.25 not 4.36ly * 0.25).

    I guess we’re all allowed a brain fart every now and then… 🙂

  3. Bombadil says:

    Maybe he’s already gone as far as to calculate the effects of time dilation? 🙂

  4. gopher65 says:

    You shouldn’t experience time dialation with an Alcubierre warp drive. So that’s not it. I think he just divided when he should have multiplied. It should have been “about 16 years and about 80 years”, rather than 1 and 4 respectively.

    As to the Alcubierre warp drive only being able to do 25% of the speed of light due to practical limitations, that’s almost exactly what I said years ago when I heard about the idea:-). It was my opinion at the time that the warp drive would be absolutely awesome for intra-system travel, but not really any better than more standard sublight drives (fusion rockets, laser powered solar sails) for interstellar travel.

  5. mavrande says:

    I realize that time dilation was a joke – but my understanding of the Alcubierre drive was that time dilation didn’t actually apply since the theoretical spaceship wasn’t moving through space, but was rather moving space. or something like that?

  6. mavrande says:

    Gopher65 beat me to the punch, and I was too busy looking up Krasinov tubes on Wikipedia to realize it. 😉

  7. gopher65 says:

    I’m faster than a speeding Mavrande. And I can leap over tall… children. As long as they’re short. And they have to crouch down. And a springboard would be helpful.

  8. Lawrence B. Crowell says:

    There are a number of thoughts I have about this. I speculate this might represent a departure between these optical analogues of spacetime physics. There are some very serious problems with these ideas of warp drives and wormholes. I will try to illuminate them after one important comment.

    The question is raised whether we can really “grab” spacetime and make this ripple. The Alcubierre warp drive relies upon being able to compress a region of space in the forwards region, thus reducing the effective distance on travels and creating a sort of gravitation, and expanding the spacetime region in the trailing region. One must be able to “grab” spacetime by some means. There is a big problem here which involves coupling constant. General relativity may be written as

    CURVATURE = k*(mass energy source),

    The constant k = 8piG/c^4, where if you look this stuff up in a book you find that G is pretty small, and c is large. The value of this constant is very small, In natural units this is L_p^2 = 10^{-66}cm^2, which is a unit of area of a black hole horizon. From a practical perspective this means you need an enormous amount of mass-energy source in order to get a significant curvature.

    It is worth pointing out that this coupling constant has a natural unit of area. This is in contrast to the coupling constants of the other gauge fields which are dimensionless, alpha = 1/137 for electromagnetism, g_W = 1/60 for the weak interactions and g_n = 1/10 for the nuclear or QCD interactions. These are comparatively large as well compared to the gravity force. The fact that gravity has a coupling constant that has units of area means that renormalization theory does not work. With other gauge theories quantum amplitudes are defined by a vertex with three lines which meet there. An example is an electron changing it energy by emitting a photon e –> e + photon, where there is a line for the incoming electron, a line for the electron under energy transition and a third for the photon. If I were to have another line, say a second photon the scattering amplitude has a photon-photon interaction which has a coupling constant with natural units of an area. These are removed by renormalization, or shoved to energies above a cut off in energy scale where these problematic terms are irrelevant. So gravity has a feature to it that run afoul with perturbative quantum field theory.

    The Alcubierre warp drive requires this negative energy. This is a big problem. The source field is ultimately quantum mechanical. The weak energy condition says that the source field must has E >= 0, which insures there is a minimum quantum energy level. This is similar to the S-shell in the hydrogen atom, which guarantees the system does not fall into an infinitely negative energy and emit an infinite pulse of energy. This is what Bohr solved with the hydrogen atom, and prevented a runaway collapse of the classical hydrogen atom. Negative energy as a source for spacetime violates the Hawking-Penrose energy conditions and the quantum field which composes this exotic matter has a ladder of quantum states that are on an endless ladder going down to infinity. So this matter is unstable and emits an infinite amount of energy.

    I have not read this paper yet, but I suspect there is a subtle departure between the quantum optics and the spacetime physics. The quantum optics generates a negative energy in a relative setting, which emulates a negative energy in spacetime.

    LC

  9. Doh! Sorry folks – arithmetic was never my strong suit. Did a retro fix on that one.

    To try and restore some dignity, I understand time dilation is not particularly significant below 50% light speed, where, ooh say it takes 8.6 years to Alpha C from the perspective on someone on Earth, for the astronaut it is still going to be around 90% of that – or 7.5 years. 25% light speed might save you 3 months in ageing/travel time (or 6 for the return trip).

    There’s a relativity calculator here (though only gives you a dilation factor) http://www.1728.com/reltivty.htm?b0=150

  10. Lawrence B. Crowell says:

    One feature of the Alcubierre warp drive is that special relativity calculations do not apply.

    LC

  11. Torbjorn Larsson OM says:

    The main problem with Alcubierre bubbles is AFAIU that they travel at the speed they are created. Since no one knows how to travel faster than light, no one knows how to make an Alcubierre bubble go faster than light. (Well, duh!)

    Seems to me these models confirm this.

    So this is no physical mechanism that gets you FTL drives. It is merely pointing out how doubtful that is, since it would imply a completely different physics than we know. See Scott Aaronson’s papers on the problems with time travel. (Which FTL effectively is for some light rays.)

  12. Johnson says:

    Perhaps a phase-shifted electrodynamic thruster could be capable of doing much more than 25% of light speed just by applying correctly nowadays technologies.

  13. Lawrence B. Crowell says:

    That is another problem of course. The horizon structure of the warp drive prohibits the device (ship) at the center from being able to control it, which includes creating it.

    The thing that is interesting about the Alcubierre warp drive and related structures is they permit us to theoretically examine the causal structure of physics and its relationship to the nature of quantum mechanics and its relationship to the structure of quantum mechanics.

    LC

  14. DrFlimmer says:

    I remember a public talk of a physics professor about “the physics of Star Trek” (he also made talks about the physics of James Bond or football (soccer for the American people here) — the talks are VERY entertaining!). He also talked about the warp drive and said that “it would be theoretically possible. However, to compress 30 solar masses of energy, which are required for it to work, into the Enterprise is the problem of the engineers!” (Metin Tolan, TU Dortmund, Germany)

    He also calculated that the Enterprise including all personnel weighs roughly 160kg !

    😀

  15. Torbjorn Larsson OM says:

    The thing that is interesting about the Alcubierre warp drive and related structures is they permit us to theoretically examine the causal structure of physics and its relationship to the nature of quantum mechanics and its relationship to the structure of quantum mechanics.

    That is exactly right, I think. Given the energy problems you point out it looks like a gedanken experiment, IIRC it takes more energy than in the visible universe mass content to make one bubble with small enough walls (as given by the energy density required).

    But the models mentioned in the article makes this more accessible to actual verification and elucidation (making and testing more predictions), which points to a healthy physics area in itself. (As the table top “virtual” black hole models do, give or take their actual connection to gravitational black holes.) And of course the actual potentialities in the realization of these “virtual” drives, table top play things or actual drives, implies the same health. (“Play things”, akin to optical tweezers which turned out as good for non-contact transport of individual cells during study.)

    In short, after we remove the excessive and shortsighted initial excitement, there seems to remain something tangible. I’m pleasantly surprised.

    Btw, IIRC the excessive energy requirements was once said to point to a “no naked singularity” type of rule, forbidding actual Alcubierre bubbles. The naked singularity rule is now known to broken (?) as in that it is applicable in the normal case but can be broken in extreme cases, I believe. Similarly with this perhaps.

    Was that the case and what happened with the problem? The exotic matter practical construct difficulties you describe may or may not be tied into it by way of the energy conditions.

  16. ikepod says:

    Hi, carbon copies;-)
    Re: the speed of light barrier(!?)
    Let us just be logical here… it is obvious, that either we apply known physical properties, or go beyond this barrier with new rules… The the Alcubierre warp drive thought model, used here, clearly lacks logic.
    Einstein, the greatest carbon copy of them all, is right, a Lorentzian wormhole is possible. The only question is: HOW do we create, or find wormholes AND what kind of force do we apply to ride them?

  17. Duncan Ivry says:

    Steve Nerlich, your article is a good assessment of Alcubierre’s warp drive, especially the skeptical part — “on paper” etc.

    Because I just wanted to know, some time ago I started going rather intensively through pertinent articles, including that of Smolyaninov.

    It’s remarkable, how some authors bury supposed evidence in favor of warp drive being possible in piles of quotes of previous articles of their own and of others, where, if you look into, you will find the same, again piles of quotes of articles quoting again etc. The way down is accompanied by plain seems-promising-like statements, assessments assessing nothing, deceptive vocabulary (e.g. “engineering” when nothing can be engineered), and a small number of pseudo-proposals of physical experiments (e.g. a nuclear explosion could create a worm hole, just stating this without giving any reason why).

    In the end — I would call it a blind alley — there is nothing more on balance favoring the possibility of warp drive than theoretical, and merely theoretical, considerations. Many authors continue chewing on not much more than the formulae describing Einstein’s gravitation theory plus several theoretical solutions like Alcubierre’s warp space metric. Real experimental data is completely lacking. After years and years the proponents of these theories should seriously ask themselves whether this is good science and whether there is no real work to do.

  18. @ Duncan Ivry
    Thanks for that. I don’t know that you can ever get prescriptive about what ‘good science’ is. 90% of it ends up in blind alleys – but you don’t find that out unless you take a walk down them. Alcubierre put an interesting idea out for public scrutiny and I think other theoreticians responded with some polite debunking – but all concerned probably found it a useful debate. If there’s a problem here, perhaps it’s credulous journalism over-interpreting scientific baby steps.

  19. Salacious says:

    Perhaps the EU/PC might know something about this, as in the Star Trek universe, it is driven by circulating warp plasma. No doubt these jackasses also do believe this is ‘true’ science too!

  20. Salacious says:

    The following link is the original reference to this story that appears in the Journal
    “Classical and Quantum Gravity” I.e. Alcubierre, Miguel, “The warp drive: hyper-fast travel within general relativity,” Class. and Quantum Grav., vol. 11, p. L73 (1994),

    “It is easy to convince oneself that, if we define this relative speed as the rate of change of proper spatial distance over proper time, we will obtain a value that is much larger than the speed of light. This doesn’t mean that our observers will be travelling faster than light: they always move inside their local light-cones.”

    This is why;

    “The resulting distortion is reminiscent of the ‘warp drive’ of science fiction.”

  21. Salacious says:

    Steve; The useful arithmetic calculation method of time elapsed appears on pg.7 of the linked article above, (I.e. Equ.15 &16).

  22. Torbjorn Larsson OM says:

    Einstein, the greatest carbon copy of them all, is right, a Lorentzian wormhole is possible.

    I don’t think so, for two reasons:

    1) There is, besides the quantum physics, a classical paradox problem with GR wormholes that in principle permits instant “time” travel.

    2) Besides that we know in general that time travel would break physics severely so that everything would be equally algorithmically easy, same complexity of all systems. But that isn’t what we observe.

    Speaking of logically, I will nitpick your joke and point out that technically we aren’t “carbon copies” but “ribosome-made replicas” since the genome isn’t a blueprint for construction but a recipe for development and control. Just my logical 2 $ (from keyboard inflation – swedish keyboard :-D), because the confused and confusing “blueprint” strawman is so often seen.

  23. cactusmush says:

    So wait… The other day everyone was freaking out because the 1928 cell-phone post “did not belong on this web-site”….

    But here, an article which admits to the following – I quote>>> ….”.as long as you don’t get too hung up on real world physics and some pesky boundary issues”…. is perfectly ok.

    In other words > lame oft-repeated baseless sci-fi pseudo science accompanied by lame power-point quallity visual = ok
    but
    >lame time-travel psuedo science based on a misinterpreted film clip = “it does not belong on this website”.

    Just putting things in context. Nonsense dressed up as science fantasy = nonsense dressed up as paranormal fantasy.

    Personally, I accept speculation posted in articles for the perpose of discourse. It adds value to this excellent website. I am amazed by the fact that otherwise sceptical individuals are aroused by fictional speculative science – if said speculation is accompanied by cheap powerpoint visuals – but not 1928 film footage..

    I look forward to your continued efforts to produce a “warp-drive”. Please make sure you translate your scientifically validated specs into Klingon………… For this – I am sure you think – will add legitimacy to your venture.

  24. Duncan Ivry says:

    Steve Nerlich, thank you for your answer.

    “I don’t know that you can ever get prescriptive about what ‘good science’ is.”

    This should be a matter of course, but, sigh, okay … There is a rather broad consensus in the community of physicists about what good physical science is. The consensus is practiced in physics as a academic discipline and outside of academics, and you will realize it — more or less –, e.g. when you want to get a job as a physicist, when you want to publish in a physics journal, and when you want to acquire a contract. As a good physicist you use a methodical approach to compare the implications of a theory with conclusions drawn from experiments; these have to be documented well; theories are open for falsification; etc. This should suffice.

    Some notes:
    – Since Thomas Kuhn’s “The Structure of Scientific Revolutions” we should be aware of the above being “normal” science in contrast to when a paradigmatic shift happens, e.g. from “Newton” to “Einstein”.
    – There are people with an education in physics, and there are certain institutions, journals and peer reviewers, who or which, respectively, are not as professional as they should be.

    The situation regarding warp space is like this (annoyingly repeating myself):
    – Many years of theoretical musings without experiments have gone by.
    – The theory in question is *not* implied by Einstein’s gravity theory.
    – Einstein’s theory is supported by a great amount of data from lab experiments and from observations outside.
    – Warp space theory is *not* supported by experimental data or other observations.
    – Rare proposals for experiments testing warp space give no reason why the experiments could or should work.

    “… you don’t find that out unless you take a walk down them”

    Okay, sometimes you really have to go down more and more alleys. But this is not the case here. As I found in the pertinent, original, scientific publications, the authors go down the same or nearly the same alleys again and again, they “continue chewing …”

    “If there’s a problem here, perhaps it’s credulous journalism over-interpreting scientific baby steps.”

    I told you, that I have gone through the pertinent articles, including that of Smolyaninov, didn’t I? Crystal clear: original, *scientific* articles of the proponents of those theories themselves, no journalism involved. And I found the deficiencies in *scientific* publications.

  25. Lawrence B. Crowell says:

    The Alcubierre warp drive is implied by Einstein’s general relativity. It is a solution which is permitted if you relax the Hawking-Penrose conditions on energy. The solution space to the Einstein field equations are not all physically real. A similar thing actually happens with the Maxwell gauge field theory of electromagnetism in classical wave theory. Quantum electrodynamics cleared some of that up.

    LC

  26. Torbjorn Larsson OM says:

    @ Cactusmuch:

    [What a stingy handle, btw.]

    I am amazed by the fact that otherwise sceptical individuals are aroused by fictional speculative science – if said speculation is accompanied by cheap powerpoint visuals – but not 1928 film footage.

    I don’t think anyone was aroused by the speculative science, which is a downer, but by that the concept has a (putative) actual material counterpart, in analogy with “cloak technology” say. And as “cloak technology” far from actual or perhaps even practical use. Then again, those optical tweezers … who knew!?

    Any 1928 cell phones from clips are anecdote from pareidolia or “photoshop”, you would have to provide actual evidence. They belong to the class of “UFO sightings” and not science.

    @ Duncan Ivry:

    a paradigmatic shift happens

    That is a funny reference in this discussion, since Kuhn was a philosopher that promoted an untested (I believe) theory.

    I’m not conversant with him or his work, but I’ve seen an armchair critique that amounted to something like “a) most scientists detest Kuhn b) he proposed ~ 20 definitions for paradigm, none of them testable; take your pick”. Looked like a fun topic to me. 😀

    @ Lawrence B Crowell:

    It is a solution which is permitted if you relax the Hawking-Penrose conditions on energy.

    In any case GR doesn’t have well and uniquely defined global energy IIRC & AFAIU (having not studied GR, so here’s some more armchair waving from the public stands), so I’m sure you can shoehorn funny and/or unphysical solutions in.

    For example, when you want to know the energy of the universe presumably you have to use its Lagrangian and study the dynamic system behavior instead of GR directly. (I have waved that particular reference around so many times I’ll provide it on request this time around.)

  27. @Duncan Ivry

    Thanks again. I sense the frustration and share a bit of it. I have had a go at passing a skeptical eye over the Pioneer anomaly; the universe in a wormhole; and variable speed of light cosmologies in previous AWATs. All these ideas are (probably) fundamentally flawed, but it’s useful working up an argument as to why they are, even if the outcome is just to confirm that current theory is still the best thing around. Anyhow, I’m no scientist – just a hack science journalist.

  28. Lawrence B. Crowell says:

    Torbjorn Larsson OM: Physical gravity probably does obey the Hawking-Penrose energy condition. General relativity by itself does not. The HP energy condition is, or really are for there are several cases of them, something you have to include. So for somebody who is interested in the foundations of physics and given to mathematical relationships this is interesting to work on. Somehow quantum physics enters into the picture,

    The Lagrangian is defined on 1/2 the phase space variables, where it is defined on configuration variables. The Lagrangian defines the action

    dS = Ldt = pdq – Hdt,

    and the Hamiltionian, which contains energy content, is a Lagrange multiplier that gives the contact manifold of integration. For general relativity the lapse variable N (sort of like a time) and the Hamiltonian obey NH = 0. General relativity is really funny with respect to energy, and in fact there is no general theorem about general relativity which gives conservation of energy. Strange stuff goes on here.

    LC

  29. Aqua says:

    http://science.nasa.gov/media/medialibrary/2010/07/27/bounce_512x288.m1v

    Okay.. so this magnetic reconnection or ‘spacequake’ had the energy of a mag. 5.5 earthquake… That’s a lot of energy!

    Harvesting that energy might be done by creating the largest induction field one could and attaching via pulsed a magnetic field to portions of that E/M wave?

    I like using a sphere of ultra light composite geodesic structure to build a sturdy(!) 10 KM diameter superconducting antenna frame, which would support spiderweb fashion the field generator and vessel – suspended separately. The nuclear powered field generator creates a magnetic bubble around the spacecraft which rides the wave front(s) as they pass to accelerate thereby ‘surfing’ those waves…

  30. Aqua says:

    Then… to get back… you tack.

  31. Aqua says:

    The shape of the magnetic field could be controlled by passing current through any of several thousand antennas mounted on the vertices of the spherical structure.

    How fast would I get ‘fried’ in a device like this at one of the Lagrangian Points? LOL!

    oTay… on the other hand, what would it be like to ‘play around’ in Jupiter or Saturn’s mag. fields for some kicks? If you get where I’m going?

  32. Torbjorn Larsson OM says:

    Physical gravity probably does obey the Hawking-Penrose energy condition.

    Surely that must be a controversial claim!? As mentioned, my knowledge of GR is a null set, so I look at this from an armchair perspective:

    AFAIU there are Penrose–Hawking singularity theorems, that rely on extending energy conditions to high energies, i.e. in effect outside the known quantizable effective theory which breaks down there. (The quantized term series diverge.)

    Also I note that HPSTs, which aren’t a unique theorem, mostly rely on the different energy conditions so again aren’t unique.

    There doesn’t seem to be a well defined sense where physical gravity obeys an energy condition!? The work I linked to [edit in posting: omitted to link to this time around, I see] shows that the universe does so in a standard cosmology setting with gravity and matter, with a resulting energy that corresponds to the average energy density of spacetime and with conservation of energy in the thermodynamical sense.

    I assume the strange stuff happens when GR, an effective theory, is applied to unphysical systems where it strictly doesn’t. Somewhat akin to arguing that singularity theorems, that points to unphysical solutions and so are “effective” in some loose sense, describes the actual physics of singularities.

    Another source of strange stuff could be that local solutions are non-unique, as opposed to the one they (that paper again) found for cosmology. Then again, maybe their technique hasn’t been used elsewhere as of yet.

  33. Aqua says:

    Plasma from a fission reaction would be used to power a MHD electric generator. That power would be distributed via woven superconducting carbon nanotubes to thousands of superconducting electromagnets mounted on the surface of the sphere. Energizing combined magnets will be used to create and shape the desired fields.

  34. Aqua says:

    Anyway.. that’s the best ‘warp drive’/wave rider I could come up with! ~@; )

  35. Lawrence B. Crowell says:

    The Hawking-Penrose singularity theorems are different from the Hawking-Penrose energy conditions: The strong, dominant and weak energy conditions. For those who know what tensors are the momentum-energy tensor T_{ab} is the source of the curvature of spacetime. The indices are a = {0,1,2,3}, for 0 corresponding to time and 1, 2, 3, corresponding to special directions (denoted with middle index I or j), such as x, y, and z in Cartesian coordinates. Here T_{00} = energy, T_{ii} = p_i = momentum and there can be off diagonal terms as well, which I will not discuss. These energy conditions are

    Strong T_{ab}U^aU^b >= 0

    Dominant: For U^a a timelike vector T_{ab}U^a is also timelike

    Weak: T_{00} >= 0.

    The weak condition is the most general, and called weak because it imposes the least restrictions general relativity.

    The weak energy condition for a quantum source is the averaged weak energy condition >= 0, is compatible with quantum mechanics or quantum field theory, for it provides a minimal energy level so quantum eigenvalues of energy have a minimum. If this is violated there is no minimal bound on the eigen-spectrum of the quantum field. This causes all sort of havoc with physics.

    LC

  36. Lawrence B. Crowell says:

    I said above special directions, where I meant spatial directions.

    LC

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