Abusing Math for Fun and Profit part 2: From a Certain Point of View

So, you’re as swift as a coursing river, have all the force of a great typhoon, and so on and so forth. But those are mere earthly comparisons. (with the possible exception of the moon thing.) You think you can do better. You want to be the ultimate in physical unstoppability. You want to be a black hole.

Let me check what I’ve got for “black hole.” Good news: You can.

It might be a bit awkward though. There is a thing called the Schwarzschild radius. It’s named after some guy named Schwarzschild, I guess. It has way too many consonants for my taste, but what it does is it tells you how big a black hole of a specific mass would be.

The equation is 2Gm/c^2. G is the universal gravitational constant, your favorite scientist’s favorite number. It’s about 6.67*10^-11 with some units I can never remember. Yup, it’s small. But don’t argue: Any bigger and life as we know it would be impossible. m is the mass of the thing you’re squishing into a black hole. The more stuff there is, the bigger the black hole would be. Pretty straightforward. c is the speed of light in a vacuum, just under 3*10^8 m/s. So c^2 is pretty frickin’ big, and G/c^2 is pretty frickin’ small. If you compressed the entire Earth into its Schwarzschild radius, making it a black hole, it would have a radius of less than nine millimeters. A human (average mass about 62 kg) would be about 10^-24 m across, which is smaller than any object I can compare it to. You aren’t going to get to be a black hole this way.

But you didn’t want to do that anyway. Becoming a black hole by being squished down tight enough? That’s not impressive, it’s not cool, and it would probably ruin your hair. Can you maybe not do that, and do something awesome instead? You bet you can. From a certain point of view. Don’t worry, though; when I say “from a certain point of view,” I don’t mean that the way Obi-Wan does. This isn’t some Jedi code for “I lied.” I mean it literally. There is a point of view, as in an inertial reference frame, in which you can be a black hole.

There is a thing. It is called special relativity. It is pretty well tested and is so right that the guy who came up with it is automatically assumed to be the Smartest Human Ever. Tangent: Go up to a random person and ask who was the Smartest Human Ever and they will probably name the discoverer of relativity. (By last name of course; he’s so famous for being the Smartest Human Ever that his full name is unnecessary.) That is how correct this relativity thing is.

There are two parts of relativity that are relevant here. One: If you’re going really fast (like, really really fast, like significant-fraction-of-the-speed-of-light fast) then you shrink. You look shorter (in the direction you’re moving) to an observer on Earth. This is called Lorentz contraction because in science everything has to have a name. This is so that people reading it know when to be scared.

Yes, I know all motion is relative. I’m going to use “moving” to mean “moving relative to Earth.” You got a problem with that?

Two: If you’re going at the aforementioned really really fast speeds, your mass increases. You get heavier. (As observed by someone who isn’t moving (relative to Earth). From your point of view, you have the same mass and length as before, and it’s everything else that is shorter and heavier. Which point of view is right? They both are. Yes, it’s weird.)

But this is great news! It means that you can be a black hole by moving really fast! That’s much cooler than becoming a black hole by being squished to unimaginably small sizes. You get heavier, which means that the size you have to be squished down to becomes less impossible. And you get shorter, which means you eventually actually reach that size. I promise I’m not just making stuff up to give me the answer I want; this is what is actually true. So if you’re going fast enough, you (as seen from Earth) will be a black hole. And since you’re still not moving relative to yourself, you observe yourself not getting squished and you might even get to survive!

So, when you’re moving fast, you get heavier and shorter. How much heavier and shorter? Conveniently, it’s by the same number. Your mass gets multiplied by a number called gamma, and your length gets divided by the same number. And what is this gamma? As you might have guessed, it depends how fast you’re moving. It’s equal to 1/√(1-v^2/c^2). As I said above, c^2 is really frickin’ big. v is how fast you’re moving. Most of the time, it’s nowhere near c. So v^2/c^2 is basically always close to zero. So 1-v^2/c^2 is only slightly less than 1, the square root of that is even closer to 1, and 1/(that) is very very slightly greater than 1. For you to become a black hole, v is going to be something huge, so v^2/c^2 is almost 1, so you’ll be dividing by something that’s close to zero. Gamma is going to be big.

One way to find how fast you’re moving is to calculate the volume you’ll be taking up when you’re a black hole. Your Schwarzschild radius isn’t going to be the same as when we calculated it above, because you’ll be heavier. So in 2Gm/c^2, m = m0*gamma. (m0 represents your mass when you’re not moving.) 2Gm/c^2 is the radius of a sphere, so you can find the volume from that and set it equal to your current volume divided by gamma. Since you get shorter by a factor of gamma, and your other dimensions don’t change, your volume will be V0/gamma. That gives an equation that you can solve for gamma, and once you know gamma you can solve for velocity. I’ll post the full numbers in a comment, but I used 62 kg for the mass of a human body, 1000 kg/m^3 for its density, and pretended it was a sphere. (There’s an old physics joke about “assume a spherical cow,” but I can’t make that joke here because 1) I don’t want to offend anyone and 2) This isn’t about your mom.) When I plugged the numbers into Wolfram|Alpha, I got that v = c(0.99999999999999999999999999999999999975). c is the speed of light in a vacuum, the absolute universal speed limit. As you can see, you would be moving Very Very Fast, otherwise known as ALL THE SPEED. Even faster than a baseball.

So the speed you’d have to be traveling to be a black hole is kind of a big deal. Like, almost the speed of light. How close? Very. Put it this way: The difference between that speed and light speed is about 6*10^-29 m/s. That probably doesn’t mean much, but you can tell it’s not a lot because of the 10^-29. Here’s a comparison: It’s about a hundred quintillion times slower than continental drift. That’s right: The difference between this speed and light speed is that incomprehensibly small. In fact, you’d be going faster than the speed of light through air. You turn on a light, it takes a bit of time to fill the room. You’d have to be going fast enough to outrun that. (The speed limit, c, is “the speed of light through vacuum.” There’s no rule saying you can’t go faster than “the speed of light through air;” it’s just hard.)

You’re probably not going to be going that fast in the near future. So you’re not going to get to be a black hole from the earth’s point of view. That’s OK, though. You’d have to be a couple hundred million times the weight of the Sun to be a black hole from earth’s reference frame, and you’re not that fat.

But remember when I said that you could be a black hole from a certain point of view? If you were moving Very Very Fast, you’d be a black hole from earth’s point of view. But even if you’re completely stationary on earth, think of a Point Of View moving that fast toward or away from you. If you were there, you’d be a black hole from earth’s point of view. But here’s the thing: If it’s moving at that speed relative to you, then you’re moving at the same speed relative to it. So the point of view from which you are a black hole is:

Any point that is moving (relative to you) at at least 0.99999999999999999999999999999999999975 times the speed of light.

Mission accomplished. Feel free to use this in any boasts or insults that would be improved by describing someone as a black hole. Just don’t mess up the numbers or the Earth ends up the size of a peanut.

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One thought on “Abusing Math for Fun and Profit part 2: From a Certain Point of View

  1. comparativelysuperlative Post author

    The promised numbers. People who don’t like math, feel free to skip this comment. Also, may shame be upon you and all associated with you for your disrespect of the language in which the universe is written. But go ahead and skip it if you want to.

    Since there’s no gamma button on my keyboard, I’ll use “y.” Sorry about that.
    m = y*m0. m is your mass, as measured from Earth, and m0 is your mass when you’re not moving.
    l=l0/y. l is your length as seen from Earth…basically like with mass.
    V=V0/y. Like above, and V is volume, or how much space you take up. If your length gets cut in half, then so does your volume, so this follows from the thing with length.
    Rs = 2Gm/c^2. Rs is the Schwarzschild radius, or how the radius of how big you’ll be when you’re a black hole.
    But Rs is the radius of a sphere. So we can use the formula for the volume of a sphere to find how much space you’ll be taking up. The volume of a sphere is 4/3*pi*r^3. In this case, r=Rs=2Gm/c^2.
    So the volume you’ll take up, call it Vs for Schwarzschild volume, is 4/3*pi*(2Gm/c^2)^3.
    Vs = 4/3*pi*8*G^3*m^3/c^6. You can tell this is tiny because it says “divided by the speed of light to the sixth power.”
    Substituting m0*y for m:
    Vs = 32/3*pi*G^3/c^6*m0^3*y^3.
    Vs = V0/y, so:
    V0/y = 32/3*pi*G^3/c^6*m0^3*y^3.
    Multiply both sides by gamma:
    V0 = 32/3*pi*G^3/c^6*m0^3*y^4.

    Now we’ve got everything in terms of the measurements from your point of view, like what you would get if you measured it now. So now it finally gets simpler.

    The average mass of a human body is about 62 kg, about 136.7 lbs. And a human body has about the same density of water, approximately 1000 kg/m^3. This saves me the trouble of looking up the volume of a human; it’s about .062 cubic meters.

    .062 = 32/3*pi*G^3/c^6*62^3*y^4. Now we just have to do some multiplication and division (and maybe a fourth root) and find the value of gamma.

    y^4 = .001*3*c^6/(32pi*G^3*62^2) This is something we can plug into Google Calculator or Wolfram|Alpha. It knows the values of G and c, and is happy to take care of the powers for you. The answer is that y^4=1.89670098*10^73, and so y=2*10^18. More or less. So, that’s number gamma is, as you can see, big. That’s what your mass gets multiplied by and your volume gets divided by.
    It is also equal to 1/√(1-v^2/c^2) = y.
    So now we’re almost done. Now that we know y, we can find the value of v, and that tells us how fast you have to be moving.
    2*10^18 = 1/√(1-v^2/c^2).
    √(1-v^2/c^2) = 1/y
    1/y^2 = 1-v^2/c^2
    v^2/c^2 = 1-1/y^2
    v^2 = c^2(1-1/y^2)
    v = c√(1-1/y^2)
    v = c√(1-1/(2*10^18)^2)
    That stuff under the square root is something you can plug into Wolfram|Alpha. Apparently it’s equal to 0.99999999999999999999999999999999999975.
    v = c(0.99999999999999999999999999999999999975)

    Reply

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