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Understanding relativity

I think I finally understood relativity, just now, thanks to the first half hour of Prof. Nima Arkani-Hamed’s Future of Fundamental Physics lecture series. Space-time too. Quantum mechanics is up next, but I’m not holding my breath for that one. Let’s just say I’m not a full-fledged physicist!

I’m still curious about one thing. Arkani-Hamed begins his explanation of relativity at 21:20 with an assumption: if you believe there’s a limit on how far objects can affect each other, at least within a given amount of time, ie how fast they can send a signal or travel…then lots of implications, culminating in relativity.

I understand the intuition behind this, it’s definitely something we’d naturally want to believe, but plenty (most?) of modern physics is counterintuitive. Why did they take this one intuition for granted?

I’m sure I’m just missing something simple, and it’s still a great talk series so far. Looking forward to the rest!


11 thoughts on “Understanding relativity

  1. i’m going to take a few notes here, just as my own memory aids.

    first, relativity. as i mentioned in the post, the key axiom is that there’s a universal speed limit, which happens to be the speed of light. nothing can go faster than it, even communication signals, which includes observing anything else going at the speed of light. that’s why when two objects are moving toward each other, and each is moving at the speed of light, they’re still only getting closer at 1x the speed of light, not 2x.

    time dilation and space-time follow from this. basically, if you bound all speeds by the speed of light, then when you’re going nearly that fast, the difference in speed between you and other moving thing starts to hit that boundary, and something other than speed has to give. that thing is time: at relativistic speeds, your relative time slows down in proportion to how close you are to the speed of light. the analogy is, if space and time are two axes on a plane, and travelling a given distance over a given amount of time is a line, then the speed and relative time are the two sides of any right triangle you can make with that line as the hypotenuse. there are lots of such right triangles, but the relationship is always the same.

  2. quantum mechanics is next.

    we know the core of it well, heisenberg’s uncertainty principle and all: we can’t perfectly predict both a particle’s position and its momentum. the more we pin down one, the less we can be sure of the other. specifically, the lower bound of the product of the two probabilities is planck’s constant, h. in practice, we have to work with probabilities for both.

    (energy and time have a similar relationship, e.g. how long it takes for a radioactive particle to decay.)

    technically, physicists specify quantum positions and velocities as complex vectors, ie vectors where each element is a complex number (termed psi). square the vector, and its length is the probability of that state.

    interestingly, this is why electrons have stable orbits and don’t collapse onto the atomic nucleus. if they did, we’d know both their position (at the nucleus) and their momentum (zero) exactly, which is impossible.

    this allows for crazy unpredictable events like tunneling at the atomic level, but not at the macroscopic level. at our size, tons of other molecules, like the air and the ground, are bumping into us all the time and measuring (observing) our component molecules’ position and momentum, at least to a degree, which makes those effects very unlikely.

  3. next up, quantum field theory.

    relativity is deterministic, quantum mechanics is probabilistic. more importantly, quantum mechanics depends on time as a fixed constant, since it describes the probability of events happening within a give period of time, but relativity mixes up time with space and says they’re kinda equivalent. how do we reconcile them?

    evidently it was hard, and took a lot of work. the answer turned out to be quantum field theory. if quantum mechanics says something could travel faster than the speed of light, e.g by tunneling, quantum field theory describes that not as the movement of matter but the opposite movement of antimatter. (so antimatter isn’t subject to the laws of relativity? not sure.)

    quantum field theory also describes how you can create particle/antiparticle pairs by concentrating enough energy on a given spot in a vacuum. this is one of the things that large particle accelerators do.

  4. next, subatomic particles and unifying the basic forces.

    quantum mechanics combines spatial position with momentum and energy with time, and relativity combines time with space and energy, so using both, we can translate units in any of them to any other. this lets us relate every type of particle directly. that helps us determine that at small enough distances, the four basic forces – gravity, electromagnetism, and strong and weak nuclear – are really derived from the same single force. this is a big deal!

    changing subjects: a particle’s mass depends on, among other things, its speed. photons go the speed of light, so they have no mass. most other particles go slower, though, so they have mass. where do they get it? say there’s a “condensate” throughout the universe. massful particles bump into it at regular intervals, specifically 10^-17cm, which gives them mass. also, the Higgs particle is basically the “ripples” from those condensate bumps.

    finally, cosmology. right after the big bang, mass and energy was pretty uniformly distributed. however, there was a brief period of “inflation” when everything expanded so fast that random quantum mechanical effects weren’t concentrated enough to even out. this resulted in small differences in mass and energy, at most .001%, or so, which eventually resulted in stars, planets, and us. :P

    also: the fact that the universe is expanding is a necessary consequence of general relativity somehow. i don’t fully understand this one.

  5. next: reconciling quantum mechanics with gravity.

    here’s a puzzle: Stephen Hawking showed that black holes radiate particles – when particle pairs appear and only one is past the black hole’s event horizon – and they do uniformly, i.e. independent of what went into them. does that mean lost information! ie, entropy decreased? quantum mechanics says that the initial state and final state are deterministic, with a one to one mapping, so this seems like a contradiction.

    this lasted a long time, but in the end, Hawkings calculation was subtly wrong, quantum mechanics survived. the information in black holes isn’t lost after all. furthermore, another calculation showed that a black hole’s entropy (ie number of possible states) varies with its surface area , not its volume. you can fully describe it with just its boundary! which leads to…

    the behavior of n dimensional system (eg space, space time) can be fully described by observable quantum field that lives in the *n-1 dimensional boundary of that space.* for example, the edge of the universe. this is generally called a hologram, since it generates an extra dimension.

    how does this work? position in the extra dimension corresponds to energy at that flattened point on the boundary. how about gravity? draw a line between two particles, intersect it with the boundary. the quantum attraction (ie vacuum polarization, ie the “condensate”) between those two boundary processes describes the gravitational force.

    side note: string theory. motivated by problems with microscopic black holes at around the Planck length. “strings” are one dimensional particles (strings, specifically loops) at around that size. they’re really nice mathematically, but the quantum-mechanics-on-the-boundary idea above eventually collapsed string theory back into traditional particle theory for arbitrary dimensions (membranes etc) and sizes.

  6. next: spacetime is doomed.

    so, we can measure the energy of the vacuum due to quantum fluctuations. the calculation turns out to depend entirely on the Planck constant. oddly, when you use it to measure the energy in a given volume, the energy increases as the volume decreases.

    we can use that to determine how fast the universe should be expanding. the result is that the universe doubles in something like 10^-43 seconds, not the 10^80 or so we observe. that’s a huge error! off by 10^120!

    it turns out there’s a classical physics effect (what is it?!) that almost perfectly negates the 10^120 difference. basically, the quantum fluctuation energy stops increasing below the Planck length, 10^-17cm. still, crazy coincidence, so it’s suspicious. what’s going on at 10^-17cm?

    another high level question: in a universe with violent microscopic quantum fluctuations, how is there macroscopic order?

    the answer comes from cosmology. as stars orbit galaxies, they should slow down as they get farther away, but they don’t, they converge on a constant speed. this is the result of dark matter. (it makes up most of the matter in the universe. same with dark energy.) empirically, the dark matter effect is due to its interactions at 10^-17cm. aha!

  7. …so, we’re still trying to reconcile an ordered macro universe with a violent, unordered microscopic space. supersymmetry to the rescue!

    supersymmetry in a nutshell: all particles have superpartners (selectrons, photinos) that move in perfect symmetry with normal particles. however, they move in quantum dimensions according to quantum numbers where arithmetic isn’t symmetric, e.g. a * b = -b * a! notably, that means a * a = -a * a, so a can only be 0, so you can’t move two steps in the same direction, since you’d just go back to the beginning.

    so, superpartners can’t have violent fluctuations, so the fluctuations must be absent. ok then!

    also, the photon’s superpartner, the photino, behaves in such a way that it’s a good candidate to be dark matter. it’s a good sign when new theories fit into more than one existing framework!

    in the same vein, a unifying theory of physics should show that all four basic forces should collapse into one (ie be equal) at short enough distances. when we try with normal quantum mechanics and gravity, they get close but not equal. however, if we add in super symmetry, they converge on a point perfectly at 10^-30cm! (for the record, this is just math so far, no experimental proof yet…but some evidence was announced just today.)

  8. wrapping it up with…parallel universes!

    string theory has 10^1000 possible solutions, and each has different configurations with slightly different vacuum energy. the total range of energy is a small band around 0. the energy difference between neighbors is roughly .5^1000.

    in a configuration with positive vacuum energy, when a quantum tunneling event happens, it generates a lower energy bubble where the particle disappears. that bubble is effectively a separate universe with the lower vacuum energy. it expands out at the speed of light, taking over everything it touches. this happens again and again, inside the bubbles, and everything converges on a universe with the lowest possible energy. (when the energy goes negative, space contracts instead of expanding, and you get a big crunch.)

    HOWEVER. our universe’s vacuum energy is positive, so our universe is accelerating. bubbles still appear, and bubbles inside them, but our universe is expanding just as fast as them, or faster, so they never combine. this is called eternal inflation. it means that every possible universe (ie vacuum energy) currently exists in some bubble somewhere! it’s an infinite fractal, where each bubble has its own particles, physical constants, etc. all of this is a direct consequence of the fact that positive vacuum energy makes the universe expand.

    for any matter to coalesce, the vacuum energy has to be within an order of magnitude of ours, or less. so our configuration is very special and unusual, in order to support us. this is just more of the anthropic principle, writ large.

    the last big remaining open question: can we see the other universes in their bubbles? is it even theoretically possible? if so, how would we do it?

  9. aha! i finally heard an answer to my initial question the other night: where did we get the very initial assumption that there exists some ultimate speed limit?

    basically, as particles with mass (ie not photons, etc.) speed up, their energy increases. that’s part of relativity. as they approach the speed of light, their energy approaches infinity. because of that, they can’t go faster than the speed of light.

    the speed of light itself isn’t really important; what is important is that we can empirically see that massive particles do gain energy as they speed up. it just also happens that the speed at which their energy hits infinity is the speed of light.


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