Friday, May 02, 2014

Laws of physics cannot be hacked

Hackers of physics do not beat Nature; they only fool people

Shaun Maguire, a PhD student who blogs together with John Preskill, spent his childhood hacking computers. It is natural for him to do the same thing to Nature:
Hacking nature: loopholes in the laws of physics
A source of the (especially young) people's excitement about physics is their desire to beat the old laws of Nature and to hack into systems around us. To get unlimited moves in the Candy Crush Saga. To make a compromise with a vendor machine: to acquire the chocolate while paying no money. To be able to subscribe to an ObamaCare website. To surpass the speed of light and to beat the uncertainty principle.

Warp drive cannot work, as I will mention again.

It's a part of the human nature to think that the previous limitations can be circumvented. Our ancestors couldn't get to the Moon; we can. So some people think that if our ancestors couldn't surpass the speed of light, then yes, we can. Or at least, our descendants will be able to. In technology, the slogan "yes, we can" captures a large part of the major advances. But the progress in physics doesn't really uniformly march in this "yes, we can" direction.

Quite on the contrary: most of the progress in modern fundamental physics may be summarized by the slogan "no, you really cannot". You cannot do things that were once thought to be possible. You cannot surpass the speed of light, special relativity tells us, even though Newton thought it was perfectly OK. You cannot concentrate some mass (or entropy) to a smaller volume than the corresponding Schwarzschild radius, general relativity claims, although it was thought to be possible before Einstein.

You cannot measure the position and the velocity more accurately than \(\Delta x\cdot \Delta p=\hbar /2 \) although classical physicists would think that you could. You cannot observe things without affecting them, Heisenberg realized. You cannot perform a mathematical operation without producing some amount of entropy, statistical mechanics implies. You cannot probe geometry at the sub-Planckian distances, quantum gravity teaches us. And so on, and so on. You cannot do many things that used to seem doable.

Most of the progress is going in the opposite direction than the practical "yes, we can" problem solvers seem to assume. Every major revolution in physics is actually connected with some new bans and in most contexts, Nature boasts waterproof law enforcement mechanisms. And because progress in science is really about the falsification of previous theories or ideas, theories that would claim "yes, we can", and because the falsification is irreversible, the finding that "no, you really cannot" do certain fundamental things is here with us to stay.

Shaun discusses five of his projects to hack Nature:
  1. Go faster than light: by Scharnhorst effect
  2. Smash the uncertainty principle: by squeezed measurements
  3. Beat Nyquist-Shannon sampling theorem: by compressed sensing
  4. Defeat all the bases: by the wavelet basis
  5. Circumvent the no-cloning theorem: by postselection
These items share the "hacking character" but their technical content is very different. It is a mixed bag, indeed.

Speed of light: the ultimate speed limit

First, Shaun wants to beat the speed-of-light limit. Last summer, I discussed the warp drive. It's supposed to achieve superluminal speeds because it shrinks the space in front of the spaceship and stretches the space behind it. So it's easier to get forward, and so on.

However, this device cannot work because one would need energy densities of both signs. Negative energy cannot exist. If you could create macroscopic regions with a negative energy density, it would also be possible – compatible with the energy conservation law – to produce such regions along with ordinary regions with a positive energy. But that would mean that such pairs emerge spontaneously and the vacuum would be unstable. Indeed, the ban on superluminal motion is closely related to the stability of the vacuum. That's also why tachyons – originally known as superluminal particles – are signs of instabilities (more).

Special relativity offers us a clear reason why superluminal motion is impossible. In a different inertial frame, such superluminal trajectories would look like trajectories that propagate backwards in time. Once the separation between two points in the spacetime (i.e. two events) is spacelike, you can't invariantly say which of them occurred first. It depends on the observer. But if A is the cause of B, it should be so according to all observers. It shouldn't be possible to revert it. If it were possible, then the laws of physics would also allow you to influence the past and to castrate your grandpa before he had sex with your grandma. That would mean that the castration was performed by a creature that may be proven not to exist, not to be ever born – a logical contradiction. So superluminal propagation of "actual material objects" or "actual information" isn't allowed.

As I discussed in the article on the warp drives, this limitation remains fully valid in general relativity if the special relativity is correctly embedded into general relativity. And there are many ways in which it may be embedded. First, special relativity is valid for small enough, and therefore nearly flat regions of the spacetime. So locally, the space has the Minkowskian geometry and the speed of light is the maximum allowed speed. Also, special relativity may be applied to regions with high curvature which are nevertheless tiny regions relatively to the whole spacetime. So two black holes, although their inner curvature is extreme, cannot move past each other by superluminal speeds in the approximation where the positions of the (small enough) black holes are well-defined points.

(A major "negative example" is the relative speed of two galaxies in an expanding Universe. The spacetime in between the galaxies cannot be considered flat in any sense so the rules of special relativity can't be directly applied to this situation. The relative speed depends on the choice of coordinates and conventions and it may end up being higher than the speed of light.)

Shaun discusses another strategy to beat the speed limit, the Scharnhorst effect. In between two metalic, Casimir plates, there is some negative energy (which makes the plates attract) and this negative energy seems to allow you to reduce the index of refraction to \(n\lt 1\). In other words, the photons of many frequencies up to the electron mass may move faster than light, although \(v/c\) is just \(1+10^{-36}\) or so.

If you blindly believe the values of the indices etc., you may think that you have really beaten the relativistic speed limit, at least by a tiny little bit. But it's really an illusion. The actual causal relationships between events – when the spacetime is viewed with a very fine resolution – still respect the causal structure from the original light cones. At most, the photons that you create at the beginning are linked to perturbations that are not confined to the original locus.

So the superluminal speed is a trick. Let me give you an analogy. Can you send laser beams from the U.S. to Russia in less than 0.01 seconds? Washington D.C. and Moscow are more than 3,000 km apart so it should take more than 0.01 seconds. But you may realize that Sarah Palin can see Russia from her house; she lives in Ice Krym, also known as Alaska. So it's possible for her to shine a laser beam to Russia in less than 0.001 seconds. You see that the apparent "minimum time" only applied if you attributed all the action to the capital cities. If you realize that the initial or final impulse is spread over some regions and you measure the minimum distance between the regions, you will realize that the speed limit has never been breached.

(This Alaskan metaphor is meant to convey the technical idea that if you perform a "nonlocal field redefinition" of fields, the dynamics may look nonlocally in terms of the redefined fields. But there still exist local fields in any Lorentz-invariant theory in whose terms the dynamics is local and respects the relativistic speed limit.)

At very short distances, the theory describing the vacuum in between the Casimir plates just doesn't care about the plates – a long-distance effect defining the "environment". It is a relativistic theory so the speed limit is unbreakable. Peter Milonni and Karl Svozil gave another argument showing that if you tried to reliably measure the superluminal speed in between the Casimir plates, you would fail. Their proof ultimately depends on \(\alpha\lt 1\), the small value of the fine-structure constant, but I believe that a more careful argument implies that even if the fine-structure constant were much greater than one, the error in the measurement of the would-be superluminal speed would still be too high. That's surely expected from the electromagnetic duality (or S-duality).

I am convinced that any such setup, with or without Casimir plates or something else, may only create the illusion of a faster-than-light propagation. But actual "usable" objects or information will never move superluminally. If a lamp is spinning sufficiently quickly (but slower than the speed of light) at the center of a huge hollow sphere, the illuminated place on the inner surface of the sphere may move faster than light; its speed is \(v=R\omega\) which obeys \(v\gt c\) for a large enough \(R\) and fixed \(\omega\). But it is no real object. You cannot really transfer any information from the place that was illuminated a second ago to the place that is illuminated now. These two traces of light are not consequences of one another. Instead, both of them are products of a third party, the lamp shining at the center.

Another trick to think that something is moving faster than light was mentioned above, using Sarah Palin. If I make you think that she's in D.C., you will be impressed how quickly she was able to contact a city in Russia. But the trick is that she really lives in Alaska, not D.C., so she can see Russia from her house.

The number of subtleties that may enter similar discussions about the "seemingly superluminal propagation of something" is large and whole books could be written about them. However, what I don't really see is the motviation here. If we could actually send some spaceships to other stars, faster than light, it would be great to know the method how to discredit Albert Einstein. But if you're ready to admit that you won't be able to use such a thing in practice, why would you frantically attempt to invalidate a law of physics that seems clearly valid according to all the evidence? It's not just valid; this insight (or, more generally, special relativity) seems to be one of the two or three main pillars of modern physics.

If the relativistic speed limit were a vendor machine and your survival would depend on your ability to hack the vendor machine (and send some actual information faster than light), well, you would die of hunger. You might still be able to convince billions of people that you were resurrected but it would only be other people, and not Nature, who would be fooled!

I will spend much less time with Shaun's other "hacking projects".

The uncertainty principle

If two observables obey \[

xp-px = i\hbar,

\] then it is straightforward to prove that their uncertainties (given by standard deviations) inevitably obey\[

\Delta x \cdot \Delta p \geq \frac{\hbar}{2}.

\] It's just an unquestionable mathematical proof, one that can be written down and verified. So as long as your physical system obeys the basic rules of quantum mechanics and as long as you correctly identified the observables, the inequality will hold. The inequality is saturated – the \(\geq\) sign may be replaced by \(=\) – if the system is found in a "squeezed coherent state", i.e. a Gaussian wave packet shifted to an arbitrary central point in the position space and the momentum space. I mean \(\psi(x)=A\exp(-Bx^2+Cx)\) for any \(A,C\in\CC\) and \(B\in \RR^+\).

Shaun seems confused and says that modern technologies can make \[

\Delta x \lt \sqrt{\frac{\hbar}{2}}.

\] However, this inequality isn't really dimensionally correct. And even if you add some extra coefficients to make it dimensionally correct, it doesn't matter. If you squeeze the wave packet in the \(x\)-space, it will be streched in the \(p\)-space! The Heisenberg inequality will never be violated.

Lots of claims that the uncertainty principle – or another, more or less equivalent postulate of quantum mechanics – has been violated in an experiment have been analyzed on this blog over the 10 years. One of the first ones was the bold claim by Shahriar Afshar who has debunked Bohr's complementarity. Of course that he hasn't. As long as you carefully interpret what he is doing, each particle he detects is either seen as a wave, or a localized particle, or something in between (with errors in both pictures). But you cannot – and he cannot – see a single photon both with clear properties of a localized particle and a perfect wave. Again, one may only fool the people, not beat the laws of Nature.

The Nyquist-Shannon sampling theorem

This theorem says that you may always perfectly recover the signal if you observe an oscillating function \(2f\) times per second if the highest frequency included in the signal is \(f\). Shaun says that it's often enough to measure the oscillating function even less frequently. But it's only enough if he knows something extra about the function – e.g. that it only contains some frequencies. It isn't surprising that a very special, measure-zero subset of the space of possible functions may be parameterized by a much smaller number of measurements. He isn't really violating any law of physics. The original theorem says that \(2f\) measurements per second are enough; they are still enough!

Not to mention that this is a theorem so we should classify this hacking attempt as an attempt to hack mathematics, not physics. And believe me, mathematics is even harder to hack than physics.


Wavelets are great for the JPEG compression of images etc. Nothing against them, they're important for the modern world (of media). However, there has never been any law of physics that would say that JPEG images were impossible. To go a little bit further, let me point out that the fact that physicists prefer to work with the position basis or the momentum basis (they have pretty good reasons for that) doesn't imply that there aren't any other bases (there surely are other bases, including other useful bases – e.g. the energy eigenstate bases). So the incorporation of wavelets among Shaun's hacking projects is a blunder of a sort.

No-cloning theorem

If the initial state is \(\ket\psi\), the final state of the evolution cannot be \(\ket\psi \otimes \ket\psi\) because the tensor product is quadratic (or bilinear) while all evolution operators in quantum physics are linear operators.

An example of a problem. If \(\ket a\) evolves to \(\ket a \otimes \ket a\) and \(\ket b\) evolves to \(\ket b \otimes \ket b\), then \(\ket a + \ket b \) evolves to \[

\ket a + \ket b \rightarrow \ket a \otimes \ket a + \ket b \otimes \ket b

\] by linearity. But if you wanted the "squaring rule" to work for this sum \(\ket a + \ket b\) as well, you would need this initial state vector to evolve to \[

\ket a + \ket b \rightarrow (\ket a+\ket b) \otimes (\ket a +\ket b)=\\
= \ket a \otimes \ket a +
\ket a \otimes \ket b +
\ket b \otimes \ket a +
\ket b \otimes \ket b

\] which also includes the previously absent \(ab\) and \(ba\) cross terms. So it's no good. Shaun says that one can do some "unambiguous state discrimination" and that one may create a cloning machine that at least works perfectly with some probability \(p\lt 1\) as long as we add some extra unidentified state\[

c\ket{\text{I do not know which state} }

\] to the final state. But this doesn't really help with the basic problem that we don't know whether the final state should contain some admixture of the \(ab,ba\) cross states. We get contradictory answers to that question depending on whether or not we tensor-square the initial state directly or via the linearity rule. So at most, the "squaring engine" may only be OK if the initial state belongs to a particular basis. It will not work for generic superpositions – and quantum mechanics is all about generic superpositions of state vectors!

If we start with several copies of the same system, i.e. if the initial state is \(\ket\psi \otimes \ket \psi\), then we can indeed squeeze the quantum information from both copies to one object in the final state. But this isn't really cloning. It's just a redistribution of the quantum information.

I am not saying that there is no interesting "quantum computation science" behind similar comments – see e.g. a comment by Peter Shor who advocates "probabilistic cloning". I am just saying that it is misleading – well, wrong – to suggest that one may circumvent the original problems that make quantum cloning impossible. (It is also wrong to say that the state vector may be measured in a single experiment in any way; the state vector is not an observable and the state vector is not observable.) Quantum cloning remains as impossible as it was before. Just some people sell certain operations that "look like" a violation of the theorem – if you don't look carefully enough – as an actual violation because they want to look cool (or because they genuinely misunderstand the physics). They may look cool but they are talking rubbish, too, and I, for one, find the latter thing more important than the former thing.

The ability of physics to discover limitations that were previously unknown is a part of the wonderful adventure we call science. If you suffer whenever a new limitation, a new "no, you really cannot" dictum – one that is sometimes just temporary but sometimes, it is valid forever and even in principle – is found, you were not born to do science. You might still be a great gift for the practical world and the world of applied science because "yes, we can" is what people want to hear over there. But that's something else than science, which is always happy to learn the new truth – and the truth very often says "no, you really cannot".

Nature is often stringent – whether it's needed for life or not. Physicists still love Her the way She is. Maybe physicists need to have some taste for Femdom, after all.

And that's the memo.


  1. Wow, Lubos, that was a masterfully insightful phrasing of some deep physical notions. Your casting - "yes, we can” vs "no, you really cannot” - is truly great and might well spin up some epiphanies out there. Having encountered many more or less competent engineering types over the years who spout “yes, we can” engineer our way around the speed of light has always been maddening to me. It seems that watching way too many Star Trek movies is rather deleterious.

  2. I really dislike comments (on any blog) which do nothing more than applaud or boo, but I'll make an exception this time. Your essay is very good and much needed in the world.

  3. Isn't there a more obvious reason that the warp drive cannot succeed? The ship itself has to move at a subluminal speed, so no matter how much space you crunch up in front of it in order to actually get anywhere your speed is still limited by light. Presumably the ship is still constrained to 4d spacetime, otherwise we could be soaring off into the 5th dimension and avoid this warp altogether. Or is there a detail that the naive versions of the warp drive that I've heard don't disclose?

  4. Thanks, Tom. I must acknowledge that I was pushed towards this clear articulation of the attitudes by a friend of mine who would be telling me about becoming a higher spiritual being that will be able to do many more things, including the things that the current limited science "isn't capable" of doing.

    This is an "optimistic" attitude to problem solving - her spiritual expectations about the future abilities of the humans may be thought of as a technological optimism on steroids. ;-)

  5. Thanks, even though I tend to dislike them, too. There's still some sense in which I dislike the short content-less *negative* comments more than the positive ones, however.

  6. Right. If you just cut the neighborhood of the ship where the curvature is large enough, you're left with the ordinary flat 4D Minkowski space around it. And nothing in this space - including the hole for the ship and the curvature around it - can move superluminally in this 4D Minkowskii space, so the fact that some curvature is a part of the ship's vicinity doesn't help.

    A new dimension doesn't help, either. The reason why one can't surpass the speed of light is the Lorentz transformation allowing you to reorder spacelike separated events "A and then B" as "B as then A". This Lorentz transformation exists even if there are extra dimensions of space. So extra dimensions beyond the 3+1 dimensions we know do not weaken special relativity in any way.

  7. Dear Lubos, maybe there is a bypass concerning FTL travel. By using squeezed states one could engineer (local) negative energy densities ( ), a requirement for having Alcubierre-like engines ;-)

  8. Right, science definitely speaks clearly for itself. I wish someone would write a scientific history on what physics’ big guns (other than Lorentz) were thinking about between Michelson-Morley in 1887 and that patent dude in 1905. Many conscious states where surely swirling after that negative result.

  9. Could someone help me understand the "no FTL signal" restriction? I understand *what* Lubos is saying above, but I don't understand the "why".

    Suppose I watch the end of a baseball game and then could somehow convey the result at some multiple of the speed of light to a planet orbiting Alpha Centauri. I understand that certain inertial observers would therefore learn of the result before observing the end of the game, but I fail to understand why that's a big deal. It's simply a consequence of the finite speed of light; I don't understand why the consequences of this are so dire.


  10. Negative comments are destructive unless accompanied by an explanation that enables a benefit to be gained. Positive ones may be no more than an act of appreciation, a “thank you”. I see nothing wrong with that.

  11. Hmm, too bad that all such papers are wrong, as I tried to explain but I must have failed - at least from your viewpoint.

  12. Photons' achiral vacuum fails for matter. Parity violations, symmetry breakings, chiral anomalies, and Chern-Simons repair of Einstein-Hilbert action validate vacuum chirality toward hadrons (quarks). Chiral vacuum (a left foot) differentially interacts with otherwise identical enantiomorphic atomic mass distributions (opposite shoes).
    Two geometric Eötvös experiments.

    0.113 nm^3/alpha-quartz unit cell. 40 grams net as 8 single crystal test masses compare 6.68×10^22 pairs of opposite shoes (pairs of 9-atom enantiomorphic unit cells, the test mass array cube's opposite vertical sides). One experiment falsifies 40 years of faerie dust. Hack physics!

  13. Could you please point articles in literature where such a bubble is shown to move with v < c despite of it is stretching/compressing space-time? Maybe, if there is none, you could consider explicitly showing this calculations ... in benefit of us (physics lovers, but still) Star Trek fans ;-) Many thanks!

  14. The problem is that by the same mechanism the people or whoever lives on Alpha Centauri could send a signal back to earth still before the end of the game. Imagine this signal to be a laser beam which creates a hole in the basketball court which is big enough to trigger the cancellation of the game before ending and you should see the problem. (An actual laser beam is light so it will travel with the speed of light by definition, But I think you will get the point.)

  15. Dear Mikael, I am trying to wrap my head around what you just wrote. Here is my attempt to create an example, but it's very likely wrong!






    (t=2) 3015 SEASON UNDERWAY


    (t=1) AS ABOVE








    (t=1 + 1 sec) FINAL SCORE YANKEES 3, SOX 0


  16. Dear Eugine S.,
    the basic problem with your way of thinking is that just like Newton you secretly assume the existence of a global time for events on Alpha Centauri and on the earth which all observers agree upon. This does not exist. However all observers will agree that if something travels with the speed of light or slower it will arrive later at whereever place it arrives then when it startet. If something travels faster than the speed of light then at least some observers will say that the signal will arrive earlier than when it started and in fact early enough so that there is time to send a signal back which arrives still earlier at the original place than the original signal started. Now you may say it is not a problem because it is only some observers.But how do you define which observer is right? The whole structure of causality meaning that the cause happens before the effect will break down and nobody ever had a good idea to recover from this mess.

  17. Hmmm... in my example, with Earth and A. Centauri forming the base of an equilateral triangle and the observing (timekeeper) planet its apex, the observing planet does not see an FTL signal arriving earlier than when it started. Can you please tell me where an observer seeing the FTL signal arriving before it started is located in space? Also, is my identification of an impossible disagreement between the timestamped lightspeed and hyperspeed broadcasts accurate and hence sufficient as a refutation of FTL signaling?

  18. Come on, folks, the whole warp drive system was explained years ago by Bob Lazar. That's how the aliens in UFOs get around!

  19. Ive never heard of you. Use your equations to prove something wrong more relevant than FTL tech. Your article comes off as desperate, arrogant, but mostly naive. You are a necessary evil, one I wouldn't want anywhere near myself or my colleagues.

  20. Different observers may not only differ with respect to their position but also with respect to their state of motion. See this link, especially the train example:

    I don't understand your refutation of FTL signaling probably because I don't know which rules you are applying to derive your conclusions Apparently they are not the rules of special relativity so I think you cannot prove or refute anything with them. The basic problem of the faster than light signal is that it will allow you to change something about the game which already happened. So it will not only be a problem of the normal broadcast but for every spetator in the stadium and for all the players etc.

  21. Mikael, thanks for your reply. But it illustrates the problem I have with every explanation I've ever read. Everybody *says* things like "could send a signal back … before the end of the game", but nobody *demonstrates* that this is really the case. Even the moving-train example you point to below simply says that the light beam events are not simultaneous in different reference frames. But again, that's just a consequence of the finite speed of light and is completely unsurprising. It gets me no closer to understanding why FTL signaling implies backward time travel. (And let me be clear that I don't dispute that; I just want to *understand* it.)

    Let me try to be more concrete: I have a magic mechanism for sending signals at twice the speed of light. I send the final score of the game to Alpha Centauri, and my friend there immediately retransmits the signal back to me via the same mechanism. Alpha Centauri is 4 light-years away, so the signal will get there in two years (my time); the total round-trip time will be 4 years (again, my time). Your reply says that the retransmitted signal will arrive back to me before the end of the game. I just don't see that ...

  22. Mikael, thanks for your reply. My first attempt to reply to you failed for some reason, so I apologize if two similar messages appear here.

    Let me be more concrete: Alpha Cen is 4 light-years away. I have a magic signaling mechanism that lets me send information at 2x the speed of light. I send the result of the baseball game to my friend on a planet circling Alpha Cen, and he immediately retransmits back to me via the same mechanism. Staying in my reference frame, the message takes 2 years to reach Alpha Cen, and the reply takes an additional 2 years. Yet your original response to me says that the reply will arrive before the end of the game. I don't see at all how that can be the case.

  23. Clearly Lubos is not one to think outside the box.

  24. Lubos, thanks for taking the time to read my blog post and for expanding upon some of the details here. I especially like your point that physics has been pushing us towards additional constraints. I've never thought of that angle and it's elegant. However, I wish you would have taken a slightly less combative tone because I agree with all of the physics you described here.

    The point of my blog post was to teach people to question assumptions. I think we agree that you shouldn't argue with physics, and especially not mathematics. I closed my blog post with the sentence: "remember that you should never argue with your mother, or with mathematics!"

    However, I think finding edge cases and pathologies is a valuable exercise. I agree that the Scharnhorst effect wouldn't lead to faster than light signaling--but I think it sharpens people's understanding of what the true limitations are. I never said that you can beat the Heisenberg Uncertainty Principle. I said that you can beat the standard quantum limit using squeezed states of light. Again, I think this is an important edge case. And obviously when I wrote sqrt of a dimensionful quantity, I meant of the corresponding dimensionless number--which is right at the sensitivity of what LIGO is trying to measure [10^(-18) m].

    Again, thanks for reading my blog post. I've enjoyed reading your posts for years.


  25. Stephen Paul KingMay 3, 2014, 6:38:00 AM

    "the vacuum would be unstable..." such as what we see in inflation?

  26. Oo look, here comes the grovelling apology. It seems our radical wannabee is just another consensus man after after all. Who would have guessed?

  27. Thanks for the feedback Casper.

  28. Sorry, you're completely confused.

    The (zero-citation) paper you linked to doesn't talk about gravity at all, so it is obviously not true that the scenario over there is stretching or compressing spacetime. It talks about negative energies which is formally enough for superluminal motion but this superluminal speed is illusory.

    Why stretching/compressing spacetime cannot lead to superluminal motion was explained in my article about warp drives.

    I don't want to spend another minute with the crackpot paper you linked to, my time is not that cheap. There are thousands of idiots writing similar junk all the time. If I were discussing every single paper like that, I wouldn't be doing anything else.

  29. Dear Casper, Nature has very sharp laws that hold with the unlimited accuracy. In this sense, Nature is squeezing the whole world not to the human-size box that people like you may think about. It is squeezing Nature and everything else into a virtually infinitesimal box and the task for scientists is to find its precise location and live within this infinitesimal box.

    The idea that scientists are supposed to ignore the existing knowledge - which is your interpretation of "thinking outside the box" - was exactly the idea whose wrongness I was explaining in this very blog post.

  30. Dear Shaun, thanks for your inspiration and your visit. However, I don't think we are on the same frequency.

    It's OK to question assumptions but relativity or quantum mechanics - or the Lorentz invariance and the uncertainty principle - are not just assumptions. On the contrary, they are mostly (revolutionary, super-important) *outcomes* of research in science. They are *results*. It is not right to ignore scientific *results*. And indeed, pretending that scientific results are just "assumptions" is a way to deny them.

    Any "standard" quantum limit that doesn't follow from the Heisenberg limit is really completely non-standard and a misnomer, and by "beating it", one is not violating any laws or questioning assumptions or "hacking Nature".

  31. Lubos, I'm not questioning Lorentz invariance or the Heisenberg uncertainty principle. The "standard quantum limit" is widely used terminology in experimental quantum physics, as is the phrase "going beyond the standard quantum limit." Google it. I would absolutely argue that squeezing the vacuum is hacking nature. To the best of our knowledge, these states don't exist out in the wild--they need to be engineered in labs.

    Compressed sensing is a breakthrough result which put heuristic tools on firm mathematical ground. It shows that a huge range of signal analysis can be done more efficiently than what was previously thought mathematically possible. People knew that heuristics sometimes worked, but they didn't know when to trust the heuristics.

    And so on and so on.

    From the Wikipedia page on Hackers (hobbyists): "a hacker is a person who enjoys exploring the limits of what is possible, in a spirit of playful cleverness."

  32. Dear Shaun, I've read the 2004 paper by Giovanetti, Lloyd, and Maccone, which I think that introduced the SQL. It's just a very naive estimate of the accuracy after several measaurements assuming that the position-momentum correlations must be zero - which they are not. That's it.

    You are either implicitly rejecting the Lorentz invariance or not realizing its power and consequences because the Lorentz invariance *implies* that there can't be propagation of information that is faster than light.

    There is absolulely nothing hacker-like about squeezed states. Squeezed states are completely standard and very simple states in the Hilbert space - and the basic postulates of quantum mechanics guarantee that all states in the Hilbert space (i.a. and all their general complex linear superpositions) are equally allowed and equally real.

  33. Dear Lubos, I would love to have a conversation with you in person someday.

  34. Great article, and nice conclusion too ;-)

  35. I would have to check for the numbers. The thing is that the "finnished" part ends up on the ancient small roads, detouring where a tunnel should be cut or a new bridge built, or an archeologically forced detour ( ancient remains of this and that have been unearthed during the works).

    One of the reasons the EU is financing is because it is mainly roads connecting ports where goods are brought in transit in for the EU countries. Egnatia highway ( finished in 2009) connects efficiently the turkish border with the port of Igoumenitsa to Italy.

  36. The arrogance of man…

    He wants to f**k up Nature, making her his bitch :-)

    What a species; you don’t know if you should love it or hate it :-)

    More seriously we need to distinguish what is fundamental and what is environmental. Fundamental laws of Nature cannot be adjusted to align with human ambition even in principle.

    Similarly you can’t create new fundamental laws. If you create a new universe in the lab the creator (human or alien) will not be able to alter the fundamental laws in it. The same fundamentals laws of the mother Universe will hold. The environmental laws can be altered I guess but these are not laws in the genuine sense, they are properties or characteristics.

    So make a list of fundamental laws and you will automatically know the limits of humankind or more generally of any kind human or alien.

    Even Q ( can't alter the fundamental laws:-)

    A guiding rule for such a list could be that any modification of a fundamental law should not lead to genuine paradoxes.

  37. I'll be the contrarian here and note that the history of the European Economic Community, forerunner of the European Union, has been mostly a success story, with milestones such as the Four Freedoms of people, capital, services and goods (1957), and the completion of the Single Market in the late 1980s. In addition, while the (entertaining!) list above focuses exclusively on crazy EU policies and regulations, it does not list crazy national policies and regulations replaced with something more sensible by Brussels.

    As much as I feel misgivings about greater European political integration (and the Euro currency especially has been a terrible disaster), recent events in Ukraine point to the need for unity in confronting a determined autocratic leader of the largest country on earth flexing its muscles. For too long, Europeans have relied on NATO to supply their defense needs, which has meant mooching from U.S. taxpayers while neglecting to spend what it takes on defense. Supporting Europeans is no longer the top priority for American foreign policy.

  38. There could be an interesting connection between SR and QM if one thinks the speed limit as a ratio of the minimum observable space and time intervals... You altready know that I admire physicists firmly believing in the determinism: they're so funny when they try to explain the arrow of time with entropia, entaglement or other memory and calculation limits...
    "Homey if I could rewind the hands of time
    I woulda cut off the PS2 at 12:49":

    when two lines of a suburban lyric make more sense than two pages of a postdoc articles... :-)

  39. My attempt to contribute something to the discussion late last night was predictably useless, so I hope to make up for it by posting a link found in Wikipedia's faster-than-light article (reference 39):

    To me, it still looks like the diagram assumes that which it is supposed to prove, namely that tachyons B and C travel backwards in time. However, I'm sure that's because of my meager powers of comprehension and not because of any flaw in the diagram's logic.

  40. Ok, I will write it more explicitly to let you understand. The assumption to allow warp drives to work is having negative energy densities (locally). I just linked that paper because not only Casimir effect but also Squeezed states are able to produce that. The main difference here is that negative-energy density states can be attained by starting from states with positive energy densities! There is necessarily no vacuum instability here, as you quoted in your article. Again, there is also no violations of Weak Energy Conditions (but "Strong Condictions" are violated), however, those conditions can be locally bypassed hence the total energy density of the system is still positive. I see you have produced a "hands waving" argument in you other post, but again, it is just an argument without any real calculations! In fact, you have assumed an asymptotically flat space where you embed a "perturbation" (the Alcubierre's bubble) ... well, I would like to see a qualified physicist computing the rapidity of such nondispersive bubble, once we know it is more likely described as a soliton (e.g. again, for an optical analogous:, and not by a simple superposition of plane gravitational waves (as you assumed). Thanks by your patience. All the best.

  41. The Neo-Cons are working very hard to make Russia the "enemy" again. They are literally bragging on Right Wing media how they knew the "cold war" had never ended and Putin is enemy number one (putting aside Bush's gazing into Putin's eyes and seeing his "soul"). Everything that has happened and is happening in Ukraine smells fishy to me. I feel sorry for the Ukrainian peoples who are once again mere pawns in the great global chess game of power politics.

  42. Try putting some people who can make decisions in places where they start to see things happen in different places in a different order.

    See : the Fed released new rules
    and Chicago started trading based on them before they should have known.

    What was supposed to happen is that the stock guys would hear about it a few milliseconds before the commodities guys. Your objection to the natural order of things is that at low speeds it doesn't matter as much. So let the guys from Alpha Centauri jump in a spacecraft at a large fraction of light speed.

  43. kashyap vasavadaMay 3, 2014, 4:42:00 PM

    Hi Lubos: Very interesting post. Although Shaun does not talk about worm holes, I would like to know about current status of worm hole theory. I understand non traversable worm holes proposed by Maldacena-Susskind may be fine because they do not need negative energies. If I remember correctly, traversable worm holes need negative energy blanket. Is this firmly established? Then in your opinion are they out ?

  44. Hi Rick,
    the thing is that you are arguing from the point of view of a particular observer, yourself. And from this point of view you are right that there is no movement back in time. But the point about special relatity is that all inertial observers are equally legitimate to describe the situation. And it is relatively easy to see from the Minkowski metric that if a signal moves faster than the speed of light from A to B then there is one observer for which A and B happen at equal time and there is infinitely many observers for which B happens before A.

    It is probably best explained in a picture. If the blue arrow is the signal and the little blue dots are A and B then every point on the red curve is a possible point B for some inertial observer.

  45. Your second to last paragraph, Lubos, is just beautifully written. I continue to be amazed at your ability in English composition.
    One tiny nit is that a comma should follow the word “science” in the last sentence but the paragraph is still a extraordinary piece of good writing.
    I must ask if you devoted a lot of time polishing it.

  46. Thanks for your compliment and loyalty, Gene. The comma has been added, of course. ;-) I was proofreading the wording once and I have found almost nothing to fix. However, before I wrote the text, I was sort of thinking about it for a day. Not the whole day, but for some moments during the day...

  47. Hi Lubos, long time reader. Great post.

    I read the QF post when it came out and found it interesting, and your post is quite illuminating. I don't know too much about the math in relativity etc, I work in ExQOptics so I am much more familiar with squeezed states and LIGO etc and completely agree with your points.

    Although if I may defend Shaun for just a tiny bit, as you yourself rightly pointed out, the ideas he mentioned can be considered hacking nature, but only from the perspective of applied physics or experimental techniques where problems like sub-shot-noise phase resolution, that once looked impossible/difficult, were solved by using some ingenuity. They do not however defeat or challenge or hack the theoretical foundations and understanding of physics in anyway.

    I suppose for every post like Shaun's one would need a post like yours, just to make sure that no laymen/scientists in other fields/students are misled or confused. :P

  48. I think any attempt to exceed the speed of light with an actual signal is excluded by the fact that in quantum field theory any operators commute outside the light cone.

  49. "By the way, people doing QFT on curved spacetimes are working with the Bogoliubov transformations etc. all the time - so general squeezed states routinely appear."

    Right, the amplification of vacuum fluctuations into macroscopic pertubations that occurs during inflation is an example of that.

  50. It is a truth universally acknowledged, that those who favour the EU must be in want of the type of leadership that would make the trains run on time.


    STEADY, BOYS, STEADY ... take your time — make sure you get the lot.

  51. You're correct, I wasn't fully thinking through the extra dimensions comment. To utilize an extra dimension to get anywhere would only end up taking longer, haha.

  52. reminds me of a favorite quote: "physics and chemistry have laws; biology has lawyers looking for loopholes"

  53. The laws of nature cannot be hacked, but Special Relativity is not nature, so SR can be hacked. The speed of light has no upper bound (limit) because of the simple reason that light is a quantum non-local phenomenon, not a classical one, and light (EM waves) do not travel through space. Information (energy and momentum) are transferred non-locally between two distant (distant in space and time) material systems by means of a scattering process. So, there are not free photons flying classically through space, therefore light does not propagate through any space but it is transferred non-locally, and that means we are unable to measure the one way speed of light in any experiment at all. In order to measure the supposed one way speed of light, we would have to synchronise two distant clocks using a convention, but conventions are human artifacts, nature does not need conventions at all in order to work. And that means the one way speed of light is simply not defined in nature, because of the non existence of free photon. If speed of light is simply not defined in nature, how is it supposed you can hack a non defined object?

    So, if you want to hack SR, you only must propose a better theory that can model nature in a more accurately manner than SR's model

  54. Sorry but special relativity surely is a part of Nature.

  55. I will not discuss whether Special Relativity is a discovery or an invention. Politely, Special Relativity is a theory, I believe theories are man-made inventions. A theory may make very acccurate predictions of natural phenomena, but that doesn't mean that theory must be seen as part of nature. If Shaun Maguire want to hack nature and to go faster than light, he only has to define a convention for synchronising two distant clocks that allowed FTL phenomema without causality violations. Surely, that new convention for clocks synchronisation would be part of a theory other than SR, and might do predictions about nature that have to be experimentally tested and not falsified. Hacking is hard ;)

  56. Kashyap, yes exotic matter needed but it is not rally the problem; the real problem is causality violation. Even if exotic matter is allowed by Nature in principle, traversable wormholes lead to closed time-like curves and thus violate causality leading to logical paradoxes (travel in the past).

    This is bad (unless you are a Hollywood writer) so I guess Nature will do something to prevent their formation (unless she likes Hollywood too).

  57. Mikael, thanks for your patience with my questions; I appreciate it.

    I understand the diagram you've shown, but it gets me no closer to understanding because, as Eugene says, it appears to be a diagram that assumes the conclusion.

    What I'd like to see is a concrete example of a defined inertial observer (starting from a given location and traveling in a given direction with a given speed) who would actually see the FTL laser beam from Alpha Centauri (in your example) blow a hole in the basketball court with, say, 12 minutes remaining in the game (the fact that I somehow failed to notice that all of the players were dead would be, of course, the contradiction that would neatly demonstrate the value of causality).

    I'm not requesting that you generate the example, of course. I'm just saying that all of the explanations I've seen put up a diagram such as yours and then call the job done. But it's just an abstraction that is not immediately convincing in the way that an actual worked-out example would be.

  58. Eugene, thanks for the link. The diagram confuses me, too, because John sends tachyon A and it travels forward in time. But simultaneously, he sends tachyon B which travels *backward* in time. All the diagram appears to be saying is that if a tachyon can travel backward in time, then a tachyon signal can travel backward in time ...

  59. Dear Luboš,

    I'd like to ask you how to 'build' the theory of special relativity.

    To be clear, I'm not talking about how one would first learn SR but rather how, having seen the standard approaches in textbooks, one could come to see the construction in its starkest ['minimalisticest' :) ] form.

    Bearing in mind that I'm no physicist, is the following OK?

    I have seen, for example, David Mermin's Relativity Without Light—though that was a long time ago and I have forgotten the details now—and have since skim-read a couple or so of other, later, derivations much on the same 'philosophical' lines.

    If I recall correctly, based on minimal assumptions, such as the isotropy of space, one can derive a general—and necessarily linear—transformation between inertial frames in relative motion which results in a law for the addition of velocities in the form w = (u+v)/(1+K²uv), where K is some constant.

    Logically speaking (again IIRC), so far nothing has been said about the principle that the laws of nature should appear in the same form to all such observers. So up to this point this really is a minimalistic approach — it's all just very basic kinematics at most.

    Now comes the determination of K, somehow. If it turns out that K = 0 then this nicely gives Galileo's take on things, and we can all knock off early and go down the pub. If not, then ... well, that's something different.

    From everyday experience one knows that if K ≠ 0 then it has at least to be very small.

    If one decided to settle this by direct experiment of some sort then it would have to involve very large u and v for there to be a measurable difference. That could be difficult and very expensive. So maybe there's an easier way with something that naturally moves very fast?

    Cut to the chase: light and Maxwell etc. For the laws of electromagnetism to be invariant under a transformation from one inertial frame to another then K² has to equal 1/c². Job done, maybe.

    Assuming I haven't screwed up here, would you reconstruct it in your mind this way? How do you 'see' SR?

  60. kashyap vasavadaMay 4, 2014, 5:22:00 PM

    Thanks Giotis. Problem of causality violation (killing your grandfather!) is an old problem with past time travel.Some people think the solution is the idea of multiverse. What I want to know is,how firm is the requirement of negative energy for traversable worm holes ?Do you know about any review on this stuff? BTW, as you know, the famous theoretical physicist on TV Michael Kaku says he can overcome negative energy problem by Casimir plates and causality problem somehow !! I understand Lubos is against both.

  61. So, theories are man-made and not laws of nature? If SR is not a law of nature then what is? Is QM a law of nature? Please explain.

  62. Gene,

    "If SR is not a law of nature then what is? Is QM a law of nature?"

    To answer deep questions like that I think one needs to think in and out of the box at the same time, all the while maintaining a Zen-like fuzzy entanglement of clarity and uncertainty in one's head so that one can see things clearly for what they really are not.

    I hope that helps. :)

  63. Well, John, you have just set a entirely new standard in ambiguity.
    You’re putting us on, right?

  64. Well, Gene, thank you kindly. You are very complimentary. Great generosity of spirit. Admirable! :)

    It's a funny thing but one never really knows how one's written word will be taken, lacking as it does the exquisite subtleties (or otherwise) that one can convey with a suitably tailored smirk for instance, suitably tailored to one's immediate company that is, one hopes. But all one can do in these circumstances is hope for the best. So I do.

    Be that as it may. In this case your doubt tells me that in the main you probably think I write a lot bollocks, which breaks my heart. :) I'll admit doing that deliberately is sorely tempting on many an occasion but generally I try not to. However, this time I'm afraid the temptation to play the joker was just too much for me so I and went for it, but not quite hell for leather.

    Yes, to be sure, it was a wind-up all right. :)

    Nevertheless, given the above about the lack of body language here and to avoid further doubt, I should add that although it was in reply to you it wasn't directed at you. Nothing like it. In fact it wasn't directed at anyone.

    Think of it more in the way of being the civilised written equivalent of giving stress-free birth to a thundering fart in a crowed lift (or elevator, as you would say) as one jumps out just before the doors shut.

    I hope that helps too. Truly, this time. Straight up! Haha! Brrr-rrup!

    Whoops, there it goes again. :)

    P.S. We are both very aware that we strongly differ on some fundamental social issues but for what it's worth to you I hold no personal animus.

  65. Really, John. Do you know one whit about where I stand on fundamental social issues?
    None of this is personal, of course.

  66. kashyap vasavadaMay 5, 2014, 6:06:00 PM

    Hi Lubos: I reread my guest blog!!! In your reply to Numcraker who asked a question not related to the topic, you said "ST is no local field theory". I can understand it somewhat. Since strings are extended objects they will be non local. But then how do you fix possible violation of Lorentz invariance in case there is one? Can you make some remarks on this?

  67. There is no Lorentz violation in string theory - in the normal, Lorentz-invariant vacua (or maximum AdS/dS symmetry breaking in those vacua). The Lorentz symmetry of a theory - like string theory - does *not* imply that it is a local quantum field theory although, in the usual definitions, the converse holds.

    It's indeed very hard to construct a theory that is Lorentz-invariant but not a local quantum field theory but it still avoids acausal inconsistencies. String theory is the only known theory different from QFT that can do it.

  68. "Do you know one whit about where I stand on fundamental social issues?"

    Well, I certainly wouldn't claim to be an expert on your position, and most definitely not in respect to all of them. Indeed perhaps we see eye to eye on these things.

    Yes, on reflection my 'very aware' in "We are both very aware that we strongly differ on some fundamental social issues" maybe results in too strong a claim, on my part at least.

    I think it's fairly clear where I stand on that relevant 'some' from my posts here but perhaps you haven't read them. To be clear, you've posted on the changing demographic in California and I've mentioned similar concerning the UK.

    Rightly or wrongly, I have gained the strong impression that our respective attitudes couldn't be more different even though, again IIRC, you are far more prudent, let's say, in expressing those attitudes than I am. Indeed, the subtext seems to be that you are very 'relaxed' indeed about the change. But I suppose now I could be very wrong on that score — maybe I should re-read your posts wearing a forensic hat. Of course, what people don't say can be highly significant too. Unfortunately interpreting that is fraught with difficulty and is generally best left out of account.

    Do you know if we "strongly differ" on this relevant 'some', and if so, do we?

  69. Ok,, let me sketch a calculation. But I won't fully work it out. Let's take speed of light c=1. Let's say seen from the earth the signal moves with speed 1.5 (times speed of light) from t1=0, x1=0 to t2=1, x2=1.5.
    According to the Lorentz transformation for an observer A with speed v who sees the signal starting at tA1=0, xA1 = 0, tA2 can be calculated a:
    tA1 = (1-v*1.5)/sqrt(1-v^2)
    If you make v bigger than 2/3, then tA1 becomes negative. Now if you can send a signal back starting in the past and you choose a high enough speed bigger than c then it will still arrive in the past.

  70. Is this actually legal to impose this on people?

  71. kashyap vasavadaMay 6, 2014, 2:44:00 PM

    Thanks Lubos for the clarification. This is a strong point for ST. People who oppose ST should be challenged to produce an alternative theory which would do this.

  72. Reading down the comments on your Delingpole link I see there is a post from "colliemum" thanking a poster for putting up a link to your blog. However, the original post has been deleted!

    WTF's going on here?

    The EU is a huge pressure cooker with the regulator being jammed down tighter and tighter all the time. It will eventually explode unless someone smashes it with a sledgehammer in the meanwhile. That'll still be messy, but less so.

    In my view, although they are all bad, ACTA and spying in cars are the worst, in that order but they come very close to each other.

    The establishment and the legacy media are one and the same for all practical purposes. They must hate the internet and the resulting lack of control they now have over what information and range of opinion is available to people. So it's all just a matter of time.

    I don't know if any of you remember that thick lying bitch, Margot Wallström (haha!), "First Vice President of the European Kommission for Institutional Relations and Kommunication Strategy [2004-2010]", but she started a blog in about 2004 to 'sell' the EU. Pure propaganda. She got slaughtered in the comments and so did her europhiliac supporters. It was great fun while it lasted. Despite the slaughter (or perhaps even because of it — giving up immediately would have entailed losing mega-acres of 'ace), it ran for a few years but eventually the slime had to face up to the reality that she and her twee lying blog were an utter laughing stock and shut the thing down. It was the first and, to my knowledge, the last such blog, one where the fcukwit EU high and mighty would attempt to connect™ (engage™) with "the citizens™" (a truly repulsive word), whom they must have assumed were even thicker than they are (no doubt a lot are — just look at what people vote for still) but plenty aren't, as the disastrous experiment so clearly showed. Yes, it was schadenfreude on steroids for us EU-haters! :)

    She was another one very keen on civil society™ — a euphemism for EU-funded leftard/greentard activist groups in the community™ 'spreading the message' and dominating the agenda, all cancerously and rabidly pro-EU of course. These people like to talk to themselves, and only themselves (it's bubble-maintenance mode 24/7 for them): "There's no need to involve the great unwashed, all those horrible little people who don't count. We'll let them know what to think and how vote when we're ready. Peasants!"

    So if they can't control it or at least dominate it then it's obvious they are going to attempt to neuter it. And that's what this is really all about.

    Knowing where you are in your car and your regular movements can be very handy too. I bet the NKVD and Gestapo would each have given their right arm to have had such a handy method for locating ... citizens.

    Shhhhh! Listen! Wait for it!

    "Well, if you're doing nothing wrong, what's the problem?"

    There it goes. Sheep to the slaughter. Millions and millions of them.

  73. OT:

    Re your Delingpole link again.

    A poster there put up a link to Andrew Klavan: Multiculturalism Explained.

    I thought it very good, including his nice jibe at us in England — I know what he means: you only have to hear the bliar and the camermoron for two seconds to get the picture. The pair of them are always banging on about the UK having "influence" (principally in and via our connection with the EU and through that onto "the world stage") but Klavan's take on that "influence" is far more accurate, at least in their case. It's understandable why he conveniently extrapolates that the the whole nation. I don't blame him. The mugs here vote for it.

    I'd never heard of Klavan before. I must watch out for him in future. Meanwhile here's another of his little gems:

    Andrew Klavan: How to Behave During an Islamic Massacre

  74. What's the difference between Odessa's House of Trade Unions and Berlaymont?

    Answer: With luck, none — it's just a matter of time. Also it won't be a tragedy.

    Mutiny Of The Lab Rats: Europeans Grow Weary Of EU Experiment

    P.S. & O.T., i.e. something different:

    Is the difference between a difference in degree and a difference in kind a difference in degree or a difference in kind?

    I say yes, indeed it is! :)

  75. "If it were possible, then the laws of physics would also allow you to influence the past and to castrate your grandpa before he had sex with your grandma."

    Could you please give me clear instructions on how to use aucubiere drive to do just that?

  76. Sure. Just find a civilization that is moving by a speed like c/2 relatively to our civilization, and ask it to construct the superluminal vehicle, and enter this spaceship.

    From their viewpoint, the vehicle still moves forward in time. From the Earth's viewpoint, it moves backward in time. So you may just use this vehicle, jump in, and when you jump out, it's the year 1950 and you may castrate your grandpa by the usual tools.

  77. That could get me to the right time but I would be in the middle of nowhere. If I turn the ship around I will return to Earth's future.

  78. There is nothing wrong about traveling to the future. It's called waiting. Special relativity allows you to travel arbitrarily far into the future within your lifetime because time slows down at speeds close to the speed of light.

  79. But such trajectory would go backwards in time in all reference frames. The original assumption was that from the alien point of view we are traveling forward in time. I draw a picture wich show how would such trip tranform into the Earth RT. Because Lorentz transformation transforms superluminar trajectories into superluminar trajectories there could be no causal link between me and my own past.

    Image description:
    A ... strating point
    T ... my destination
    C ... point with the same time coordinate as T
    D ... point of my return to Earth

  80. Jesus Christ, what's so difficult about it?

    You may travel from Earth 2014 to Earth 1954 by a two-way spaceship journey, with the intermediate stop at some distant place in 1984 (Earth time).

    The first part of the path from Earth 2014 to distant place 1984 goes backward in time according to Earth's frame (and other frames), but it is forward in time according to some moving civilization A, so you may use this civilization's warp drive.

    The second part of the path from the distant place 1984 to Earth 1954 is also backwards in time from Earth's viewpoint (and others), but it is forward in time from the viewpoint of another alien civilization B that is moving in the opposite direction relatively to the Earth than A does, so you may use the B's warp drive to realize the second part of the journey.

    If the civilization allowed superluminal traffic of any sort and if the laws of physics were invariant under the Lorentz transformation, you could get to the past on the Earth. Contradiction, bullshit, QED. Please read this comment at least 5 times if you still happen to have fundamental problem with this point - which is nothing else than the basic point "special relativity prohibits superluminal traffic" that we were taught at the high school

  81. Ok, I think I get it now. So I need to accelerate to c/2, warp, change speed to -c/2, warp back and stop. That is a bit more complicated than you originaly suggested.

    But I have a problem with your second argument. I understand we can't have both faster than light travel and causality, but who says we must have causality? Course of actions which would lead to time travel is arguably to deliberate to ever occur naturally. What if causality works not because it's a fundamental law of nature but only because nobody is messing with time travel?

  82. For comments that merely express appreciation, I agree, but I dislike positive comments that cheer on the blogger in a controversy when the commenter shows no sign of knowing enough to have an opinion about it.
    I notice that Gene said something to you about a missing comma. This makes me feel a little freer to say something about your English for the first time in a while. You do very well with the articles, not perfect, but very well. Nothing to really worry about there. But I would like to point out that you over-use the present perfect; you sometimes use it when the simple past would be better. When something was over and done with in the past... - e.g., it would be wrong to say, "My wife has made my lunch yesterday." OTOH, there's nothing wrong with "Galileo has taught us that...," although "Galileo taught us that..." would do just as well.