Waves in an Impossible Sea

Chapter 14 — Elementary Fields: A First, Unsettling Look


Note 6: A subtlety with measuring light’s speed
  • Quote:The faster you try to go, the more [light waves] will seem to recede from you, always at exactly the same speed: c , the cosmic speed limit.

  • Endnote: I’m implicitly assuming that the space around you isn’t rapidly stretching or contracting while you make the measurement, or at least that you are measuring the speed of waves immediately before or after they pass you. This fine point is crucially important when understanding the expanding universe as a whole, though it plays no role here.

  • Discussion

Note 7: You can’t escape from light
  • Quote: “And if you try to run away from light that’s approaching you, it’s hopeless; you’ll never escape it.”

  • Endnote: If you maintain a constant acceleration by running a rocket continuously, you may be able to stay just ahead of a light wave, but the moment you take a breather, it will catch you.

  • Discussion (coming soon)

Note 8: Einstein’s logic about light’s speed, with some math
  • Quote: No matter how you move, light waves approach you from all directions at the same speed, as though you’re always stationary relative to light’s medium.

  • Endnote: That these statements (and others in this chapter) are logically consistent was shown by Einstein, using reasoning supported by math. The math is not so complicated, though too long for an endnote; it’s the reasoning that’s tricky. In any case, Einstein’s claims have all been experimentally verified many times over and are woven into modern technologies.

  • Discussion (coming soon)

Note 9: Why light from moving flashlights reveals nothing about the aether
  • Quote: . . . in order to preserve the principle of relativity, the speeds of the light from the two flashlights must always be equal, as Einstein proposed.

  • Endnote: A subtle but important point: this test of relativity would give the same answer even if the flashlights themselves were moving relative to me inside the bubble. Steady motion of a flashlight relative to the bubble affects the frequency of the light waves that it emits, from my perspective, but has no impact on their speed. The shift in the light’s frequency reveals only my motion relative to the flashlight; it reveals nothing about my motion relative to the aether.

  • Discussion (coming soon)

Note 11: Messing with space and time — with a bit of math
  • Quote: As guessed first by Lorentz and put on a firm footing by Einstein, the passage of time is perspective-dependent. How often a clock seems to tick depends on how fast the clock is moving relative to the person observing it. There’s a similar perspective-dependent distortion of distances.

  • Endnote: These distortions are often called time dilation and length contraction.

  • Discussion (coming soon)

Note 14: How light can have a perspective-independent speed
  • Quote: . . . the speed of light waves, from everyone’s perspective, is always the same. Though this seems logically impossible, and common sense may protest at this seemingly absurd claim, the relativity of times and relativity of distances save the day. They precisely compensate for each other whenever we use our own rulers and our own clocks to measure the motion of a light wave.

  • Endnote: Again, the math behind this is not so complex. What is challenging is to understand what the math means and how to use it correctly.

  • Discussion (coming soon)

Note 15: Group velocity and phase velocity of waves
  • Quote: Higgs waves passing by from any direction cannot exceed the cosmic speed limit—and this is true from every observer’s perspective.

  • Endnote: An important subtlety: for traveling waves whose speeds can vary, there is a distinction between the speed of the wave’s front edge and the speed at which the wave crests move; they are not the same, and I am always referring to the former here. In jargon, these are called group velocity and phase velocity.

  • Discussion (coming soon)

Note 17: The radion field of extra dimensions
  • Quote:To those of us too large to detect that extra dimension directly and aware of only our usual three space dimensions, Einstein’s gravitational field in the full four space dimensions would appear as multiple fields: a gravitational field (bending), an electromagnetic field (leaning), and one more field (compression) often called the radion. The latter is a topic for a different book.

  • Endnote: One such book would be my colleague Lisa Randall’s Warped Passages: Unraveling the Mysteries of the Universe’s Hidden Dimensions.

  • Discussion:

    Extra dimensions are a common phenomenon in the imaginary universes of string theory, though they are not limited to that setting. (If you want to learn more about how to think about extra dimensions, which may seem a strange concept at first, several articles on the subject appear on this website.)

    As seen by a three-dimensional observer, a radion field is a field that resembles the Higgs field; it is “switched on”, or more precisely, has a non-zero average value (in the language of Chapter 15 and beyond.) [It also is non-pointing, in the language of Chapter 19.]

    Radion fields and their particles are a general feature of extra dimensions, and so they are a natural target for experimental searches. Unfortunately they may interact with ordinary matter very weakly, much like gravitons, making them hard to produce and detect. When searching for them, there is no easy and general strategy to follow.

    The discovery of a radion would both confirm the existence of at least one extra dimension and reveal some of its properties. First, the radion field’s average value secretly but directly reflects the size of the extra dimension. (This is often called the “radius” of the extra dimension, hence the field’s name — though it’s really the length or perimeter of that dimension! Yet another linguistic misfortune…sigh…) Second, the rest mass of the radion field’s particles reflects the elasticity of the extra dimension. If the size of the extra dimension can be easily changed, making it relatively elastic (like soft rubber), then the radion particle corresponding to the radion field will have a low rest mass. If instead the extra dimension is stiff (more like hard plastic), then the radion particle will have a high rest mass.

    However, proving that a newly discovered particle is a radion may itself not be simple; it may also be necessary to discover and study corresponding “Kaluza-Klein modes” (mentioned later in this chapter; see endnote 19 below.) And discovering a radion is not a simple matter either, because, as mentioned above, collisions of known subatomic particles may not readily make such particles because of overly weak interactions. Assuming the extra dimension is not completely floppy (rendering radion particles’ rest mass zero), full-fledged radion waves, too, wiil be much more difficult to make than gravitational waves, and they may dissipate very quickly. So the prospects for near-term discovery of these particles is not particularly bright.

    Nevertheless, there are some circumstances in which the discovery might be much easier than I’ve suggested, especially if the extra dimensions aren’t ultra-microscopic. Such possibilities are covered in detail in Lisa Randall’s book, which is worth a read; its general discussion is still relevant and useful, even though some aspects are out of date. Specifically, at the time she wrote it, it was still hoped that perhaps the Large Hadron Collider experiments would discover radion particles (or Kaluza-Klein particles), but this has not yet happened after a decade of data. While the case is not yet closed, it is likely that any discovery of a radion particle will have to wait for a machine of the future, though whether this is the near future or the far future can’t be known.

Note 18: How quantum physics might eliminate space
  • Quote: Physicists do study imaginary universes in which the space we think of as real is only an optional crutch for understanding the world, a crutch that, while sometimes convenient and sometimes not, is never absolutely required.

  • Endnote: This is not, by the way, as crude as simply imagining that our world is someone else’s simulation. What quantum physics can do is much more subtle and interesting than that. Sadly, this lies beyond the scope of this book, but the issues are partly covered in my teacher Leonard Susskind’s book The Black Hole War, in George Musser’s Spooky Action at a Distance, and in Graham Farmelo’s The Universe Speaks in Numbers.

  • Discussion (coming soon)

Note 19: Extra dimensions of space and their Kaluza-Klein “modes”
  • Quote: There are other fields (now called Kaluza-Klein modes) that would arise if space had additional microscopic dimensions.
  • Endnote: These appear when fields in the full set of dimensions exhibit standing waves within the microscopic dimensions.
  • Discussion (coming soon)



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A decay of a Higgs boson, as reconstructed by the CMS experiment at the LHC