Astronomy Without A Telescope – Special Relativity From First Principles

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Einstein’s explanation of special relativity, delivered in his 1905 paper On the Electrodynamics of Moving Bodies focuses on demolishing the idea of ‘absolute rest’, exemplified by the theoretical luminiferous aether. He achieved this very successfully, but many hearing that argument today are left puzzled as to why everything seems to depend upon the speed of light in a vacuum.

Since few people in the 21st century need convincing that the luminiferous aether does not exist, it is possible to come at the concept of special relativity in a different way and just through an exercise of logic deduce that the universe must have an absolute speed – and from there deduce special relativity as a logical consequence.

The argument goes like this:

1) There must be an absolute speed in any universe since speed is a measure of distance moved over time. Increasing your speed means you reduce your travel time between a distance A to B. A kilometre walk to the shops might take 25 minutes, but if you run it might take only 15 minutes – and if you take the car, only 2 minutes. At least theoretically you should be able to increase your speed up to the point where that travel time reaches zero – and whatever speed you are at when that happens will represent the universe’s absolute speed.

2) Now consider the principle of relativity. Einstein talked about trains and platforms to describe different inertial frame of references. So for example, you can measure someone throwing a ball forward at 10 km/hr on the platform. But put that someone on the train which is travelling at 60 km/hr and then the ball measurably moves forward at nearly 70 km/hr (relative to the platform).

3) Point 2 is a big problem for a universe that has an absolute speed (see Point 1). For example, if you had an instrument that projected something forward at the absolute speed of the universe and then put that instrument on the train – you would expect to be able to measure something moving at the absolute speed + 60 km/hr.

4) Einstein deduced that when you observe something moving in a different frame of reference to your own, the components of speed (i.e. distance and time), must change in that other frame of reference to ensure that anything that moves can never be measured moving at a speed greater than the absolute speed.

Thus on the train, distances should contract and time should dilate (since time is the denominator of distance over time).

The effect of relative motion. Measurable time dilation is negligible on a train moving past a platform at 60 km/hr, but increases dramatically if that train acquires the capacity to approach the speed of light. Time (and distance) will change to ensure that light speed is always light speed, not light speed + the speed of the train.

And that’s it really. From there one can just look to the universe for examples of something that always moves at the same speed regardless of frame of reference. When you find that something, you will know that it must be moving at the absolute speed.

Einstein offers two examples in the opening paragraphs of On the Electrodynamics of Moving Bodies:

  • the electromagnetic output produced by the relative motion of a magnet and an induction coil is the same whether the magnet is moved or whether the coil is moved (a finding of James Clerk Maxwell‘s electromagnetic theory) and;
  • the failure to demonstrate that the motion of the Earth adds any additional speed to a light beam moving ahead of the Earth’s orbital trajectory (presumably an oblique reference to the 1887 Michelson-Morley experiment).

In other words, electromagnetic radiation (i.e. light) demonstrated the very property that would be expected of something which moved at the absolute speed that it is possible to move in our universe.

The fact that light happens to move at the absolute speed of the universe is useful to know – since we can measure the speed of light and hence we can then assign a numerical value to the universe’s absolute speed (i.e. 300,000 km/sec), rather than just calling it c.

Further reading:
None! That was AWAT #100 – more than enough for anyone. Thanks for reading, even if it was just today. SN.

79 Replies to “Astronomy Without A Telescope – Special Relativity From First Principles”

  1. If only relativity were that simple.

    Without relativity (e.g., “from first principles”), the time it takes to travel from A to B at a speed c is (B-A)/c. This quantity can only go to zero if B-A is zero or c is infinity. Therefore, by the logic in point 1, either the size of the universe is zero, or the maximum speed is infinity.

    1. Agreed, without relativity.

      Classical physics contains a problem that is a bit like Zeno’s paradox. Classically, you can’t have something moving at zero travel time or it would exist at both points A and B at the same time (and at all points in between). Hence it seems as though you can only asymptotically approach zero travel time and you must arbitrarily decide how small a fraction of time can pass before it’s impossible to go faster.

      Spec Rel gets you out of that conundrum, since from a photon’s perspective it moves between two points without any proper time passing – and by space-time equivalence without it crossing any distance either. Hence from a photon’s perspective B-A is always zero.

      1. Yes, but that doesn’t, in and of itself (i.e., from first principles), argue that this classical view is wrong and therefore there must be a speed limit. There’s no reason the maximum speed limit of the universe couldn’t be infinity in the classical view (and indeed, this was generally assumed before Einstein). Your first conclusion simply does not follow from your argument.

      2. The reason that infinity cannot be the maximum speed is that infinity is not a quantity. If speed were infinite, then the speed would actually be zero because both time and distance would effectively be zero. Given that “infinite” is not a possible quantity for any property, then we must assume a limit imposed by the quantifiable properties of matter, energy, space, and time. Thus we can assume a maximum speed in the classical view.

      3. No, there are many quantities in physics that are unbounded (e.g., temperature, mass). Just because infinity is a useful mathematical representation and doesn’t actually exist in nature does not mean that the quantity you describe it with must be bounded.

      4. but the whole idea above is posited from saying “there must be a maximum speed limit” without any logical justification. As saranova says, if you send the maximum speed limit to infinity, you just get an unbounded quantity – doesn’t seem to be an apriori problem with that – it’s just when you go through the reasoning that Einstein did about the invariance of physics between inertial frames that you discover correctly that there is a speed limit for massive objects.

      5. I wonder if we are confusing a first principles argument with intuitive thinking.

        Humans are perhaps hard-wired to believe that time and distance are absolutes – e.g. even Galileo struggled to convince people of the basic principles of relativity and motion. e.g:
        http://en.wikipedia.org/wiki/Galileo%27s_ship

        Obviously we have this discussion today in the privileged hindsight of Einstein’s discoveries. Nonetheless, the fact that you can reduce your travel time by increasing your speed does (by extension) lead to the conclusion that there exists a state of motion at which travel time reaches zero.

      6. “that there exists a state of motion at which travel time reaches zero.”

        For particles with mass, this is not true in the classical or relativistic sense. Classically, you’d say “as the speed approaches infinity, the travel time approaches zero”. Relativistically, you’d say “as the speed approaches c, the travel time approaches zero.” The former most certainly does not hint at any speed limit, intuitively or from first principles.

      7. If I walk 2km/hr it takes me 30 minutes to walk a kilometer, if I run at 5 km/hr it takes 12 minutes. In a car I can do it in 1 minute. And I haven’t even warmed up my starship yet…

        I am not automatically drawn to the conclusion that I have to go at infinite km/hr to do the distance in 0 mins. It seems quite reasonable that there should be an absolute speed when I can make such drastic reductions to travel time with conventional technologies.

      8. This is simple algebra. a/b = 0 only if a=0 or b=infinity. If b is anything less than infinity (even 10^999999), a/b is still not zero.

      9. Not arguing with the math. As a general principle anything that requires you to divide by zero should be considered a problem rather than a solution.

        I return you to the issue of classical physics requiring an object moving with zero travel time to be in multiple locations at the same time. Either it’s impossible or there’s a problem with classical physics. I think we are agreed the latter is the case.

      10. No, a travel time of zero does not exist for particles with mass (e.g., us). In classical physics, you can get arbitrarily close to travel time=0 by going arbitrarily fast, but you’ll never hit exactly zero and therefore you’ll never be at more than one place at once.

      11. I agree with the above comments, you’re trying to prove SR by presuming that it’s correct to begin with. Your Zenos paradox argument doesn’t make any sense. I could quite happily think of a universe which doesn’t have a speed limit but where, as you get faster and faster, the time to go between two points decreases to zero as the speed goes to infinity. Special Relativity was constructed from an invariance principle, stating that physics must be the same in all non-accelerating frames.

      12. I agree with the above comments, you’re trying to prove SR by presuming that it’s correct to begin with. Your Zenos paradox argument doesn’t make any sense. I could quite happily think of a universe which doesn’t have a speed limit but where, as you get faster and faster, the time to go between two points decreases to zero as the speed goes to infinity. Special Relativity was constructed from an invariance principle, stating that physics must be the same in all non-accelerating frames.

      13. Well yes and maybe but maybe not.
        Photons are called massless particles but do have some mass associated with them (electrons wizzing around create magnetic/gravity fields) so although massless in a classical sense there are strange casses where it looks like a duck but quacks differently….

        I don’t want to go too far out of my depth here – LC and others here have explained this distinction to me before and although we need to invoke some quantum principles, it seems compatible with classical physics (at least in terms of two photons being connected/entagled by Zero space/time but still exerting some force of mass (gravity).

      14. This is ridiculous. saranova is absoutely right, and you are absolutely wrong. For point particles in a classical universe (infinitely divisible in space and time) , there is just no way to cover a finite distance in zero time. And thus there is no absolute speed limit.

        In our real, quantum, relativistic world, things are more complicated, but there is no sense in which you can say a classical universe “must” have a finite absolute speed limit, and your “first principle” #1 is just fallacious.

      15. This is my point about the classical view seeming analogous to Zenot’s paradox.

        We are agreed that you asymptotically approach a particular speed. Whether or not you can actually reach that speed, you should still be able to quantify what it is.

        As others have commented here, saying that something is infinite does not quantify it.

      16. No, you’re asymptotically approaching a particular travel time (zero), while your speed approaches infinity. You don’t need a PhD in physics to understand this; this is high school math.

      17. What do you understand by concepts of “zero” and “infinity”? It seems that your entire argument is based on a logical error and a misconception.

        You seem to have gone from “if zero then very small” to “if very small then zero”. Otherwise you wouldn’t equate ‘drastic reductions’ in travel time with a travel time of zero.

        You also seem to think that infinity is a number. The statement “We are agreed that you asymptotically approach a particular speed” assumes that infinite speed is a particular speed, which it is not (it’s just a misnomer). We, presumably, agree that the limit of velocity as time taken goes to zero (for fixed distance) is infinite. Which means that your argument does not put an upper bound on velocity.

      18. Thanks, it’s nice to know there are a few people who know what they’re talking about.

        Nerlich, please take this article down. You’re clearly not qualified to write it.

      19. You have some great discussion points saranova. However, precisely the reason I enjoy reading Steve’s articles is that they speak to the person on the street. Like me, who didn’t finish his BA liberal arts and spend most of high school in detention dreaming of Einstein and Schrodingers concepts and trying to understand how they led to such revolutions in science and understanding.

      20. That’s precisely the problem, Wezley, and why this article should be taken down. You think you’re learning, but you’re actually being fed misinformation. Indeed, it’s extremely useful when someone can break down complex ideas and explain them to a non-expert. However, in order to explain ideas, one needs to understand them first, and Nerlich demonstrably doesn’t. This in and of itself isn’t a problem. Not many people understand relativity, and that’s fine. The danger is that Nerlich doesn’t know he doesn’t understand it,
        “agreeing to disagree” with mathematical axioms, and he’s making it worse for those with a genuine interest.

  2. The fact that light happens to move at the absolute speed of the universe is useful to know – since we can measure the speed of light and hence we can then assign a numerical value to the universe’s absolute speed (i.e. 300,000 km/sec), rather than just calling it c.

    The Judge’s view of Einstein.

    1. I had never heard of Zeno’s paradox before, but it looks like it was resolved by calculus. (that is, you can have an infinite series with a finite sum). However, this is not really applicable here…

      1. No fancy mathematics are needed to go beyond the apparent paradox, but mere common sense. Any allusions to Zeno’s inaccurately called “paradoxes” have to take into account the fact that modern philosophy has shown them to be nonparadoxes based on a fallacy.

        Zeno claimed to have proven that motion is ultimately impossible since any moving object must travel over an infinite number of segments of space, which would mean that an infinite amount of time would be needed to get anywhere. His logical error was to mistake the physical realm for the mental realm. In our minds we can divide any stretch of space into as many parts as we wish, to infinity, whereas in the physical world space is CONTINUOUS, not infinitely divided, so that there is no problem at all involved in moving one inch forward, or a thousand kilometers. Moving away from the quandary was that simple, but they were unable to do so in those days.

        The Greek thinkers made a distinction between the sensible world (that we percieve through the senses) and the intelligible world (the world of ideas or the intellect), which we now call the the physical and the mental worlds, so what somebody should’ve pointed out to Zeno was that he was applying to the former the procedures that are valid only in the latter, which was why he arrived at false conclusions. He thought he was being clever, and everybody else thought so, too, and this allowed him to confound them, and in our time people keep falling into his trap, but it was actually the opposite: he was being foolish.

  3. Hey Steve and readers,

    Happy 100th AWAT!! And to celebrate, an article on my hero, Dr. Einstein!! Thanks Steve.

    Growing up my role model used to be evil knevil, but after a few skateboard accidents and puberty, that changed to Einstein.. Too much information? sorry about that.

    Happy Xmas!!

  4. There is also nothing in special relativity that says that nothing can go faster than the speed of light, just that you cannot accelerate something faster than that speed. There are well behaved solutions which go faster than light, but they happen to give you causality problems later on – you cannot use them to state your first posit.

    1. quoting//
      From this he derived that it was actually an upper limit for the velocity to which you can accelerate a massive object. It has nothing to do (as stated in the above article) with a universe having to have an upper speed limit. //

      I am not sure I understand the distinction you are making?

      My understanding is that according to SR, if an object were to exceed the speed of light its mass would become infinite.

      Put another way to get to 99.99999% the speed of light would take nearly all of the energy in the universe to power your vehicle.. To acually reach 100% (not yet exceeding c), your starship would neeed all of the energy in the known universe.. At the same time your starship would be as heavy (massive) as the known universe.

      So if Einstein’s SR is correct, concerning the universe having an upper speed limit, yes it does if we are talking about mass/energy (and not some other imaginary quanta).

      Wezley

      1. Sorry if my statement was confusing. This post is about trying to hand-wave that the universe has an upper speed limit. In fact it is true that there is an upper limit to the speed with which you can accelerate a massive object, but the reason for this isn’t what is quoted in the article – in fact the only reason given seems to be just because it seems like there should be. Also, within special relativity, things can, in principle move faster than light, it’s just that you can’t accelerate a massive object to these speeds.

      2. Jonathan – I have no objection to you considering me wrong, but an accusation of hand-waving implies that I am genuinely attempting to mislead the reader. Ab hominem argument does not help your case.

        Wezley’s point about infinite mass gain and absolute speed is not a helpful way to think of the problem. This is just a misunderstanding of the distinction between proper and relativistic measurement.

        While, to a distant observer, a spaceship may not appear to be able to exceed light speed and hence be interpreted as gaining mass – the pilot does not experience any such mass gain. It is not a proper mass gain, it is a relativistic mass gain.

        I also wrote about this here:
        http://www.universetoday.com/91132/astronomy-without-a-telescope-mass-is-energy/

      3. Hello Steve,

        My example of the starship obtaining the mass of the universe does amplify the relatvisic concept to almost absurdity and assumes an indepenent observer as you so aptly pointed out. However my example was more to the point of addressing the problem we face that to accelerate an object with mass to the speed of light requires the kinetic energy of the known universe to accelerate to 100% of c.

        I would be very happy were this not so and will look foward to our species deveoping FTL Travel, if we figured out how to cirumvent this limitation imposed on baryonic matter being accelerated by e.

        Thanks for your time and explanations from Steve, Torbjorn, LC and others- i honestly admit to having a high school understanding when it comes to applying SR with math/physics.. Happy to learn

        Merry Xmas..

        Wezley

      4. Thanks Wez – the way around all this is the realisation that time dilation allows you to cross vast stretches of the universe within your lifetime – although to an external observer it seems to take you millions of years. So we can potentially go Star Trek as long as we don’t want to go back home afterwards.

        The energy cost may be exhorbitant, but who knows what physics may yet offer up to run a star-drive effectively.

      5. Thanks Steve and most of the posters in this thread..

        When I am contrarian about math/physics subjects an seem to disagree with some smart folks here, in truth it is a layperson mining for understanding of what seem to be simple arguments but can end up getting pretty dern complex, esp concerning possible FTL travel..

        In the case of many of these posts, I will be bookmarking a returing to read the opinions and cross-compare these with the papers available through internet searching to get informed analysis and insights.

        I am grateful to those of you in these forums who entertain us less versed in math/physics/quantum physics and astrophysics phenomena.

        I tilt my hat to the fact that other smart people have designed though experiments such as a spacecraft that could create a gravity well (negative in front or positive gradient at rear of craft) and simply surf the inertia until speed c…
        Assuming we can generate gravity fields, this approach looks promising.. Post-Zephram Cochrane era here we come 😉

        Sometimes these discussions can get heated but I still feel like a kid in awe of it all sometimes…

        To me these videos sum-up why I am addicted to science and astronomy (and why I enjoy being mentally challenged and humbled by all the incredible beauty that is our cosmos:

        The Detailed Universe: This will Blow Your Mind.

        The 4D Universe

        Best,

        Wezley

      6. Thanks Wez – the way around all this is the realisation that time dilation allows you to cross vast stretches of the universe within your lifetime – although to an external observer it seems to take you millions of years. So we can potentially go Star Trek as long as we don’t want to go back home afterwards.

        The energy cost may be exhorbitant, but who knows what physics may yet offer up to run a star-drive effectively.

      7. My apologies if it comes across as ad hominem, I genuinely don’t mean this and I’m sure that you’re not trying to mislead the reader. However, I do have a genuine problem with your argument.

        You are trying to get special relativity from first principles, so what we have to do first is to show that a universe without relativity has some problems (as indeed it does). You are saying that there must be a universal speed limit, but in a classical universe (until you start to ask about the invariance of physics in different rest frames) there isn’t a problem with being able to go arbitrarily fast. The faster you go, the shorter the time to get from A to B but it never actually gets to zero time. I really don’t understand why you claim that this has to happen at some finite velocity.

        Indeed, it does happen at a finite velocity, but only as a consequence of special relativity – you cannot use this property of relativity to get relativity from first principles. For this time to get smaller and smaller as you approach the speed of light, you have to include time dilation in your equations, in which case this is not a first principles argument.

      8. No worries Jonathan – I think this has come down trying to guess what would seem more ‘logical’ in the absence of knowledge about Einstein’s theory.

        I am happy to agree that we disagree.

      9. Hi Steve,

        more to Jon’s argument: the first assumption is fallacious. From that argument, one may infer that the ‘absolute velocity’ be equal to (size of the Universe)/(age of the Universe), roughly 7 times the speed of light, since we are observing lights from the very edge of the Universe emitted that long ago.

      10. “Agree to disagree” might provoke a frustrated grumble amongst those, like me, who’ve scanned this comment section in disbelief. It’s simple maths: zero travel time and finite distance imply infinite speed. This is not an absolute speed limit in any meaningful sense. If the derivation of SR were as straightforward as you have made it sound, it would have been discovered a long time before Einstein.

  5. since i heard from my physics teacher years ago that nothing could go faster than 300,000 mps (approx) i didn’t believe it… why isn’t the constant the speed of time not the speed of light?
    please someone?
    could the speed of time can change???
    Torbjörn Larsson is the smartest person I’ve read on here.. maybe he could set me straight/

    1. This value is a particular number which is found in the equations of electromagnetism (along with many many other fundamental equations of physics). It describes the speed at which light travels, and is related to how charged objects feel a force (two electrons sat some distance apart will feel a force which takes some time to travel in between them, the force essentially travels at the speed of light). It was by looking at these equations and noting that they should hold in all frames (that is, independent of whether or not you are moving) that Einstein realised that the speed of light should be the same whether you are moving or not. From this he derived that it was actually an upper limit for the velocity to which you can accelerate a massive object. It has nothing to do (as stated in the above article) with a universe having to have an upper speed limit.

      1. Cheers Jonathan.
        I just still think that something with a mass could go as quicker than 300,000 mps.. call me dumb.. sorry..

      2. It’s 300,000,000 mps (or 300,000 km/s), but your concern is still the same.

        You’re not in bad company; it took one of the most brilliant minds of the 20th century to argue against an infinite speed limit. It is an extraordinarily unintuitive thing to think about, since all of your intuition comes from interactions at tiny fractions of the speed of light. If the speed limit of the universe were instead 5 m/s, time dilation, length contraction, and the idea of a maximum speed limit would be second nature.

        Unfortunately, this is not the case, so we have to develop sophisticated experiments to test the idea. So far (with the controversial [and generally dismissed] exception of recent CERN results), nothing with mass has been observed to go faster than the speed of light.

      3. That’s because special relativity isn’t obvious, and the whole premise of this post is bogus. It was a clever leap of insight that gave us relativity. If you want to believe something just because it feels right, then go ahead, but that’s not the way science works.

    2. In TL’s absence….

      Imagine you fire a laser pulse at the Moon and watch it travel there. We know it should take 1.5 seconds (from experiment). We also know that the image of the Moon that you see is 1.5 seconds old (since it is 1.5 light seconds in distance).

      If you could observe the laser pulse strike the Moon is less than 1.5 seconds, it would have to be travelling back in time. Not sure that answers your question directly, but should indicate there is are causality problems with believing in infinite speed.

      I also did a story on light speed here if it helps:
      http://www.universetoday.com/89667/astronomy-without-a-telescope-light-speed/

      You ask could the speed of time can change – relativistically yes, that is what SR demonstrates. Your GPS wouldn’t work if the theory was wrong :-).

      1. Thanks, Steve! I am not a Physicist or Mathematician, but your thought:

        Imagine you fire a laser pulse at the Moon and watch it travel there. We know it should take 1.5 seconds (from experiment). We also know that the image of the Moon that you see is 1.5 seconds old (since it is 1.5 light seconds in distance),

        is what I have been thinking for the past 50 years, when I started to become interested in Astronomy and Astrophysics (on an amateurish basis, haha)
        Thanks for your thoughts/questions…

        While I am sure LC and TL are very smart, sometimes thinking “out of the box” and free of mathematics and physics, may even help to develop new ideas to prove wrong 😉

      2. Thank you – I am also not a Physicist or Mathematician. I think it’s a worthy endeavour trying to plain English methods to describe – what LC and/or TL have kindly described as my shaky attempts at heuristics.

        I am not expecting to make any serious advances in science here – my interest is education.

      3. But, the “speed of time” doesn’t change. Length contraction and time dilation are results of transforming an observations of something in the observed frame into observer frame through the Lorenz Transformations …

  6. For the context, these arguments are ancient as a street hooker and a dime the dozen.

    Meanwhile no effort to produce special relativity from mathematics have been successful in peer review, what I know of. The reason we know this is because special relativity is a large part of crankdom.*

    So while I salute those that try to show where Nerlich is wrong, it is hardly worth the time.

    But yes, obviously the first axiom here is the problem, and hand waving that Zeno’s paradox is valid is not the way to persuade mathematicians that this is the way to proceed. Analysis laughs at Zeno’s paradox.

    One way analysis solves Zeno’s paradox is by _not_ dividing with zero, but using a delta-epsilon limit procedure.

    ——————
    * That physics is not mathematics and never can be is a deep insight.

    – It ties into testing and why we need it just not for establishing valididty of observations but also for validity of theory.

    – It also ties into the observation from computer science (CS) that axiomatic math is but a small part of the universe of algorithmic structure, and that physics must inhabit the latter because CS does. There are ties between computability, Turing machines and physics, since computability hinges on computational resources over space and time (energy and relativity bounds).

    1. quoting
      //The reason we know this is because special relativity is a large part of crankdom.* //

      * That physics is not mathematics and never can be is a deep insight.

      Wow pretty bold statements. Yes physics are not mathematics. I fail to see how this makes SR invalid and part of some realm of crankdom??

      Perhaps the physics of SR are on a higher stratum than math & information? My hunch says you are conflating two issues and elevating mathematics above physics.. tsssk..tsssk 😉

      Wezley

    2. Agreed. ‘First principles’ is only useful to formulate theory. Science is about going out and testing the theory.

    3. Thanks for your insights Torbjorn.. Sometimes we argue over degrees of distinction but I always find it an illuminating discussion… In particular quantum physics can be both illuminating, frustrating, and inspirational when new vistas of understanding are reached (to me the modality to get there is less important than the journey to the destination of understanding our emergent cosmos and its rules and constructs as we see them)

      Best,

      Wezley

    1. The web page did not load up and the main page does not make it clear where this is. Your description sounds like the BFSS (Bank, Fischler, Shenker, Susskind) holographic M-theory. This is an interesting development and something I have deeply studied.

      Feynman illustrated how partons (particles which are form factor terms) at different Lorentz boosts can appear and their numbers increase. If an observer is witnessing quantum fields or strings fall onto a black hole this is a form of Lorentz boost, and the partons associated with the field can appear in greater numbers. The mass invariant interval m^2 = E^2 – p^2 for the momentum in the z direction gives the energy

      E = sqrt{p^2 + m^2} = sqrt{p_x^2 + p_y^2 + p_z^2 + m^2}

      and may then be written as

      E = sqrt{p_x^2 + p_y^2 + p_z^2 + m^2} =

      p_z sqrt{1 + (p_x^2 + p_y^2)/p_z^2 + m^2/p_z^2}

      =~ p_z + (p_x^2 + p_y^2)/2p_z + m^2/2p_z

      Now I eliminate the p_z by defining E’ = E – p_z and we get the rather classical nonrelativistic appearing formula

      E’ = (p_x^2 + p_y^2)/2p_z + m^2/2p_z

      The p_z >> p_x or p_y then appears as a mass-like term. For the partons of mass M the p_z then appears as p_z = nM, for n the number of these mass elements, which are partons.

      In this way the system has been boosted to a near light cone condition, and is called the light cone frame or infinite momentum frame. The system then appears time dilated to the point of being slowed to a Newtonian-like system with the mass of the field increased by the increase in the number of partons. Now in a stringy perspective we may think of the string as a type of loaded string (mass beads on a chain studied in classical mechanics courses) and with the increase in the longitudinal boost momentum the number of these mass beads (partons) increases. A string which falls onto a black hole is in the near horizon condition may be considered to be subjected to a succession of Lorentz boosts up to the event horizon. If the observer is witnessing the string fall to a black hole then the string becomes pasted on the stretched horizon as a set of field oscillators with mass M. The sum of these masses then equals the mass of the black hole.

      LC

      1. Reference #7 on this paper is the BFSS paper. The theory reduces things from 10 to 9 dimensions on the infinite momentum frame. The Lorentz contraction along the boost direction reduces the physics by one dimension. The string modes of interest are then transverse (perpendicular) to the boost direction and longitudinal modes then do not contribute. The physics curiously takes on a non-relativistic form as a result, for the transverse physics is time dilated from the perspective of the observer to the point it appears nonrelativistic.

        LC

      2. A theory with 9 spatial dimensions and 1 time dimension obviously differs from a theory with 10 spatial dimensions, the team used 32×32 matrices and in the future intend to use larger ones as they become available, this may endorse the super string theory tidy compound explanation of the thermodynamics of black holes, gravity & electromagnetism and resolve as to why 3 dimensions are enormous yet 6 are diminutive…. assuming the theory is the correct one previous LHC results appear to suggest otherwise.

      3. The 32×32 matrices are irreducible representations of SO(32), which according to the weights is isomorphic to the E_8 exceptional group. This is the case with “32 supersymmetires,” which is the N = 8 maximal supersymmetry.

        LC

  7. The connection between pre-relativistic physics and relativity was with electrodynamics. During the 19th century electromagnetism was developed from the simple Coulomb law for the force between two electric charges, q and q’,

    F = (qq’/4??)1/r^2

    into a theory which combined the magnetic field in a way which described waves of electric and magnetic fields. These waves describe light, radio waves and X-rays discovered at the turn of the 20th century. This connection is what lead to the special theory of relativity.

    The Coulomb law above describes this for according to an electric field F = qE, where the electric field from the other charge is

    E = (q’/4??)1/r^2

    This electric field can be visualized as a set of line or rays which emanate from a central point where the charge q’ resides. This looks a bit like a hedgehog with its spines pointing outwards like little arrows that are tangent to these lines. Now let us suppose we take this charge and move it to another point. Those radially directed lines of force do not instantaneously adjust everywhere in space out to “infinity.” Rather they wiggle into place and this wiggle propagates outwards. This wiggle is a change in the electric field ?E, which is associated with a change in magnetic field ?B that cycle with the changing electric field. This forms a wave that propagates outwards and the speed of this propagation is c = speed of light.

    Einstein as a youth read about this and formed the following thought experiment. Suppose you were watching an electromagnetic wave moving in space. The field equations define this wave propagation according to some varying or oscillating electric and magnetic field. Einstein then pondered what would happen if he were moving on a platform with such a wave. He realized there was a contradiction, for this observer would see this oscillating electric and magnetic field wave stationary in that frame, but where the Maxwell equations for electromagnetic fields described their propagation at the speed of light.

    There ideas at the end of the 19th century about some medium for the propagation of EM waves, called the aether. This meant there was some preferred frame in the universe from which physics could be observed. Curiously, Newton’s laws of motion make no such predictions, but the trends in physics at the time imposed this. Einstein proposed that rather than there being some preferred frame in the universe that in instead the speed of light was invariant, the same everywhere and to all moving observers, and that frames and coordinates including time transformed accordingly. From this was born the special theory of relativity.

    Einstein did formulate things, rather clumsily in a way, from the basis of looking at charges moving in space. The special relativity was in effect buried away in our understanding of electromagnetism since Maxwell derived them in 1865.

    LC

  8. Very interesting article!
    If we assume that some particles have no (measurable) mass, does it mean they are mass-less, or is it matter-less?
    So if at the last big CERN experiment particles arrived at destination at a speed, which suggests FTL velocity, and assuming for a moment that it will be proven without flaw, then what are the ramifications for physics?

    1. The CERN (OPERA) finding is almost certainly measurement error. If neutrinos (which do have slight mass) really moved faster than FTL it would turn physics on its head. It’s an extraordinary claim requiring extraordinary evidence (which we haven’t seen yet).

      Particles like photons have no rest mass – although they do have momentum. Mass is an intrinsic property of matter – so no, you can’t have massless matter 🙂

      1. I tend to also think the OPERA results are flawed. Unfortunately there has been no announcement of any finding of such an error.

        Particles which move faster than light suffer from a couple of big problems. The invariant mass is the momentum-energy conjugate to the invariant interval or proper time. The rest mass of a particle is given by

        mc^2 = sqrt{E^2 – (pc)^2}

        Here p is momentum, and if p = 0 (particle at rest in your frame) you get E = mc^2. If a particle moves faster than light then pc > E and the mass is imaginary.

        The momentum is replaced by p — > -i?? and E — > -i??/?t in quantization and this equation for the invariant mass becomes the Klein-Gordon equation for the scalar wave ?

        ?^2? – ?^2?/?t^2 = (m^2/?^2)?

        The constant m^2/?^2 acts in a potential term (m^2/?^2)|?|^2, and if the mass is imaginary this potential term is quadratic and negative. This potential term derives this wave equation according to the Euler-Lagrange equation, which is something I will bypass for now. For ? = 0 the potential term is zero, but it becomes more negative as ? increases. This means that the vacuum state is unstable. If tachyons exist it would mean the quantum vacuum is not stable. In effect the vacuum would be producing particles or radiation in an ultraviolet divergence.

        LC

  9. It’s probably worth mentioning that point 1) begs the question by assuming that there is an absolute speed in order to try and prove that there is an absolute speed.

    Interestingly, even though the author assumes that there is said speed limit, his argument (when the mathematical errors are fixed) doesn’t show any such thing. As has been previously stated:
    if t = d/v then
    t = 0 if d = 0 or if v = infinity m/s.

    However v = infinity m/s is nonsense (v is an actual velocity and therefore cannot be set to infinity m/s). This means that if d is not zero then there is no value of v that can make t = 0. (It is true to say that the limit of t when v -> infinity is 0, but the author talks about values, not limits, so we will too)

    The author claims “At least theoretically you should be able to increase your speed up to the point where that travel time declines to zero – and whatever speed you are at when that happens will represent the universe’s absolute speed.”

    As shown, you can’t “increase your speed up to the point where that travel time declines to zero” and so you are never in a position to say ‘Ah ha! I have found the universe’s absolute speed!’.

    In summary, even though the author assumes the truth of the very thing he wants to prove he fails to prove that it is true.

    1. I should have been more precise that “infinity m/s” is not a value, but rather a limit that shows the velocity is not bounded, and I think that caused a bit of confusion. Thanks for making that point clearer.

  10. As a layperson, I can’t help noticing apparent resurrection of two abandoned ideas:
    (1) the “luminiferous aether” and the quantum vacuum with its virtual photons;
    (2) Lord Kelvin’s atomic “vortices” and the inward dimensions of string theory.

    “Everything old is new again.”

    1. To number (1):

      You are talking about two VERY different things. In fact, the “luminiferous aether” has been proven wrong by experiments, while virtual photons can actually be measured (at least their effect).

      Virtual photons are a consequence of quantum mechanics and the uncertainty principle. It is, in fact, possible for a certain amount of time to create energy from nothing, as long as it vanishes again after that time.
      This fundamental principle also explains radioactivity, for example.

  11. This is a disappointing post for several reasons. First, statement 1) is provably false if taken at face value. This point has been made tirelessly in the comments section, and the author has failed to successfully resolve this problem. He has attempted to support this claim — advertised as a “first principles” statement from which special relativity can be derived — by invoking special relativity itself (specifically, by considering time dilation in the photon’s rest frame — a frame that does not even exist). The logic is circular and the physics is wrong. For starters, from which of your starting assumptions are we to derive Lorentz symmetry? The added reference to Zeno’s paradox appears confused and unmotivated; when pressed further about the role of Zeno’s paradox as supporting claim 1), the author dismisses it as “analogy”. Given the clear opposition that readers have been voicing to elements of this article, I think the author owes unequivocal, rigorous responses that don’t rely on vague analogies.

    I would say that the author should be prepared to “agree to disagree” with anyone who understands special relativity. This piece, and especially his confused and evasive responses to the comments, miserably fails to convey the possibility that he should be considered one of them. If the quality and accuracy of this piece are acceptable at this site, and if the author’s resistance to consider the comments of experts is the norm around here, I’m afraid this column is securely headed for the crackpot dustbin.

    1. To Brian and saranova,

      This great free series, AWAT, has been my favourite series on the Interweb for months; bar none.

      If the lively and informed debate below is anything to go by, I see no harm in Steve’s articles and feel Steve is providing great talking points many people from many different fields get involved in.

      Lets let truth become self evident in an economy of ideas.

      Instead of attacking the messenger could we please stick to topics at hand? What’s the danger? We laypersons might learn by making our own informed decisions and choosing which posters discussion points to accept?

      I can only detect genuine interest and encouragement in these articles and have myself ratched-down my own rhetoric at times realising the most productive approach is listening to a range of ideas and deciding which to personally accept.

      I would feel deprived if Steve didn’t contribute his time and energy freely to writing these articles.

      All the best,

      Wezley Jackson

      1. I appreciate the reply Wezley. My concern is that laypeople do not necessarily have the background to navigate through all the technical details that might come up in a discussion like this. I am wary of writers like Steve who appear more interested in defending an incorrect viewpoint than in *learning* and entertaining fruitful discussion. This is not an even-sided argument: indeed,I happen to know that some of the people posting comments here are actual theoretical physicists, people who have studied and thought deeply about relativity theory. The author, as evidenced by the conceptual errors in his post and the imprecision of his responses, has not. This is not intended to be a slight against the author; however, it is *not* OK to peddle nonsense under the illusion of authority, with the attitude that the interested reader will sort it out. Some might. But some might read this article and get a very wrong idea about relativity.

      2. I agree with Brian 100%. Teaching others your willful ignorance under the illusion of authority is how you train global warming deniers and creationists who think their position is backed by science. Just because Steve’s conclusions are correct doesn’t mean the path he took to get there is any less dangerous. Science is a way of thinking, and this article — especially the author’s responses to our comments — is not science. To those who don’t recognize that, it’s very harmful.

        I’m not saying Steve’s heart isn’t in the right place, and I’m not saying Steve should stop writing. He should just stop writing about things he doesn’t know about with an air of authority. I’d be far more understanding of his mistakes if he really tried to understand what he did wrong, but he doesn’t seem interested in that. Agreeing to disagree just because you don’t understand something is no way to learn. That phrase is the epitome of unscientific thinking. Let’s not agree to disagree, let’s ask questions, find answers, and/or do experiments until at least one of us is demonstrably wrong.

  12. AWAT is my favourite series on the site but I have to say I think there are a couple of points here that could perhaps have been clearer.

    The key point is that Einstein’s paper was not based on the premise that there was a maximum speed, though that can be used as another approach to the same conclusion. In fact his postulate was “that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body.” See the second paragraph of his paper here:

    http://www.fourmilab.ch/etexts/einstein/specrel/www/

    The key is that Hertz had showed that the electromagnetic field should support wave and that the speed could be derived from the pemitivity and permeability of the vacuum, it was related to the square root of their product. Both these factors are simple “scalar” numbers like temperature with no intrinsic direction, you could find their values by measuring the charge stored in a Leyden Jar and the magnetic field produced by a solenoid, so as fundamental constants, they must have the same value for everyone who makes the measurement in a sealed lab regardless of their motion relative to the rest of the universe. That means the speed they measure for light emitted from any source must also have the same value, no matter how that source is moving realtive to the experimental apparatus. From that postulate, he demonstrated that simultaneity could not be absolute but was dependent on the coordinate system used which is what broke the connection to Newtonian physics. His genius lay in questioning something so intuitive that nobody else doubted it.

    If that was all he did, it might have been seen as an academic excercise but one of the key problems of the time was the discrepancy between Newtonian mechanics and the motion of recently discovered electrons in “cathode ray tubes”. Showing how the new way of looking at time solved that problem was essential in demonstrating that the theory was grounded in experimental evidence.

    The other minor point is that the term “absolute speed” has unfortunate connections as many cranks use it to mean distance through absolute space divided by absolute time, i.e. using the Newtonian concepts which relativity replaced. It is only semantics but if the first use of the phrase was changed to “… deduce that the universe must have a maximum speed …” and so on, it could avoid that potential confusion.

  13. AWAT is my favourite series on the site but I have to say I think there are a couple of points here that could perhaps have been less misleading.

    The term “absolute speed” has unfortunate connections as many people use it to mean distance through absolute space divided by absolute time, i.e. using the Newtonian concepts which relativity replaced. It is only semantics but if the first use of the phrase was changed to “… deduce that the universe must have a maximum speed …” and so on, it could avoid that potential confusion.

    The other point is more relevant. Einstein’s paper was not based on the premise that there was a maximum speed, though that can be used as another approach. In fact his postulate was “that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body.” See the second paragraph:

    http://www.fourmilab.ch/etexts/einstein/specrel/www/

    The key is that Hertz had showed that the electromagnetic field should support wave and that the speed could be derived from the pemitivity and permeability of the vacuum, it was related to the square root of their product. Both these factors are simple “scalar” numbers like temperature with no intrinsic direction, you could find their values by measuring the charge stored in a Leyden Jar and the magnetic field produced by a solenoid, so as fundamental constants, they must have the same value for everyone who makes the measurement in a sealed lab regardless of their motion relative to the rest of the universe. That means the speed they measure for light emitted from any source must also have the same value, no matter how that source is moving realtive to the experimental apparatus. From that postulate, he demonstrated that simultaneity could not be absolute but was dependent on the coordinate system used which is what broke the connection to Newtonian physics. His genius lay in questioning something so intuitive that nobody else doubted it.

    If that was all he did, it might have been seen as an academic excercise but one of the key problems of the time was the discrepancy between Newtonian mechanics and the motion of recently discovered electrons in “cathode ray tubes”. Showing how the new way of looking at time solved that problem was essential in demonstrating that the theory was grounded in experimental evidence.

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