Supersonic Man

May 17, 2016

let’s try something easier: special relativity

Filed under: Hobbyism and Nerdry — Supersonic Man @ 5:21 pm

This article has been promoted to my website, here.



  1. To generalize from the examples above and give yourself practical formulae, you need to use the dilation factor, commonly called gamma (γ). This is much easier to define and describe if you go by its reciprocal, which is sometimes called alpha. This reciprocal is a number between zero and one — it’s one when the subject in question is stationary and approaches zero as the subject’s velocity approaches lightspeed. The relation is that the speed (as a fraction of c) squared, plus alpha squared, equals one. So if you graph it, you get a quarter circle. If you graph gamma itself, you get a curve which looks kind of circular at first and then shoots up to infinity, with no intuitively obvious shape to it. The formula is γ = 1 ∕ √1 – |v|²∕c², where |v| is the object’s speed (the magnitude of its velocity vector).

    The rate which time progresses in the moving frame of reference is the stationary rate divided by gamma. The foreshortened length along the axis of movement is the stationary length divided by gamma. The mass is the rest mass (m) times gamma.

    If people ever really do fly around in relativistic spaceships, they’ll probably want to make use of a “false velocity”, which is the real velocity times gamma. In these terms acceleration adds up in a linear way: like, if you accelerate at 1/400 c per day for 800 days, your false speed is 2 c, and the distance you cover across the galaxy is, as measured by your local clocks but by the galaxy’s stationary ideas of distance, two light days per day. Framing it this way makes sense in human terms because you’ll be looking at a map of stars which says your destination is some number of light years away, and you’ll be looking at your local calendar and wanting to know how many days of your own time it’ll take to get there.

    This fake speed gives you nice linear predictions of how fast you will cover ground between two landmarks, given how much you accelerated. It correctly states various properties such as your momentum (which is fake velocity times rest mass)… but it doesn’t work for kinetic energy: if you use that fake speed and your rest mass in the old ½mv² formula, you overestimate your kinetic energy, but if you plug in your real speed, you underestimate it. The real formula is just relativistic mass minus rest mass, expressed as energy, or (γ-1)mc². This approximates ½mv² at lower speeds, but there’s no clean way to express it in terms of fake speed.

    Comment by Supersonic Man — May 24, 2016 @ 12:44 pm | Reply

    • Yeah, unlike other equations which still work if you define the terms right, the Newtonian kinetic energy formula ½mv² is actually wrong in its overall shape. It’s close at low speeds (the error is barely one percent at 15% of lightspeed) and you can get even closer by using the relativistic mass, i.e. ½γmv², but that is still different from (γ-1)mc².

      To make it work, the “½” in that second approximate formula would have to be replaced with a special correction factor: at a tenth of lightspeed it’s 0.501, at half of lightspeed it’s 0.536, at three quarters it’s 0.602, at nine tenths it’s 0.696, at 99% it’s 0.876, and at lightspeed (for massless particles) it’s 1. This is why I was never able to see how kinetic energy made sense when I tried to grapple with relativity in the past — for slow particles kinetic energy is half of momentum times speed, but for the fastest ones it’s momentum times speed with no ½. So it wasn’t that I was failing at math: the ½mv² formula really just doesn’t apply.

      Comment by Supersonic Man — May 24, 2016 @ 11:41 pm | Reply

      • For a while I was trying to make the old ½mv² formula work by integrating over a varying m, but it still doesn’t fly. The basic idea was to allow for the fact that the kinetic energy it’s already got adds inertia to any further acceleration. But the way that energy goes to infinity as v approaches c isn’t fully accounted for by that — the cumulative effect of inertia feedback pushes the energy higher, but not towards infinity.

        Comment by Supersonic Man — July 24, 2016 @ 12:26 pm | Reply

  2. You could say that the fundamental formula for special relativity, from which the rest follows, is

    ΔS² = Δx² + Δy² + Δz² – cΔt²

    where x, y, and z are the three coordinates in space, t is the coordinate in time, Δ means the difference between the coordinates of two locations, and ΔS is the resulting “spacetime interval”.

    Alternatively, for two points in spacetime (events) denoted a and b, some people prefer to express each coordinate explicitly as a difference, like instead of Δx say (xb – xa).

    This formula does count distances in time as imaginary relative to distances in space. In essence, the theory of special relativity consists of the assertion that this interval measures a combined spacetime distance which remains unchanged regardless of an observer’s frame of reference.

    But this formula wasn’t written by Einstein. It from Hermann Minkowski, a former teacher of Einstein’s whose response to the special relativity theory was to reframe it in terms of geometry. People trying to solve special relativity problems often do so using Minkowski Diagrams based on this formula. It’s usually considered a better way to understand the theory than Einstein’s original approach was. And yeah, it’s what works for me.

    Apparently general relativity can also be boiled down to a single one-line formula, but only if you take pages of explanation to define the terms in it. Many of the letters stand for complex entities known as tensors. If you expand those terms you get ten partial differential equations.
    the Einstein Field Equation

    Comment by Supersonic Man — June 23, 2016 @ 9:14 am | Reply

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