PhD comics and mini black holes

By Phil Plait | November 20, 2007 10:14 am

The second part of the PhD comic strip I mentioned yesterday is out, and lots of astrobloggers are linking to it. But no one seems to have noticed the glaring error in it:

Black holes are at the center of a lot of misconceptions. Basically, they are objects where the escape velocity is greater than the speed of light. If you throw a rock straight up off the surface of the Earth, you must give it a velocity of 11 km/sec for it to be able to get away from the Earth and not fall back down*. That’s because the Earth’s escape velocity is 11kps. It depends on the size of the object and its mass, or, if you prefer just its density.

Black holes are so small and so massive (or just plain dense) that their escape velocity is faster than light. You literally cannot escape them once inside their ravenous maw. The size of a black hole is actually rather simple to calculate if you know its mass:

radius = 2 x G x mass / c2

G is Newton’s Gravitational constant of the Universe, and is just a number. c is the speed of light. It turns out that for a star like the Sun, it would have to be crushed to a diameter of about 6 kilometers to becomes a black hole. But look at the equation! If I double the mass, the size of the black hole doubles. So it’s really easy to scale this equation to different mass black holes.

The one in the center of the Milky Way Galaxy has a mass of roughly 4 million times the Sun’s mass, so it must be 4 million times bigger, or 24 million kilometers across, far less than the distance between the Sun and Mercury (for comparison, the Sun is about 1.4 million kilometers across right now).

Black holes are small.

However, they can get even smaller. There is a hypothesis that just after the Big Bang, fluctuations in the density of matter may have compressed small amounts of material so much they collapsed into black holes. These are called mini black holes. If you plug in the mass of, say, a typical mountain or asteroid into the equation above, you’ll see that a mini black hole is actually far smaller than an atom!

I won’t even go into Hawking radiation, which says that a black hole that small would have a surface temperature of 10 billion Kelvins. That would make them a bit obvious if they were pelting us; they’d be pretty bright.

I will add that from a centimeter away, the gravity of a mini black hole can be hundreds of times that of the Earth, but from a few meters away you’d hardly notice it.

So the author of PhD comics got tripped up a bit by the scale of things (a black hole the size of a pea would have a mass comparable to that of the Earth). But that’s not too surprising; obviously, black holes are weird, and difficult to comprehend. Now, if only someone were writing a book that had a whole chapter (plus part of another chapter) dealing with the potential dangers of black holes including an easy-to-understand description of how they form, what escape velocity is, and what happens when you fall into one, and what would happen if a mini black hole hit the Earth …

Mwuhahahahaha. But you’ll have to wait until next year for that book.

* Actually, it’s a bit more complicated than this. Escape velocity is for an impulse, a forcer applied all at once. If you apply a smaller force, but do it over a long period of time, you can escape the Earth never having gone 11 kps.


Comments (33)

  1. Andy Varga

    What if the diameter of event horizon the mini black hole was smaller than light wave lengths? Would light skip over/off the horizon or still be sucked in?

  2. cvoid


    Check out “Artifact” by Gregory Benford. Good, hard-SF book that the comic reminded me of. Good read, with some interesting “fictional history” as part of the backstory that is interesting, to say the least.

  3. > far smaller than an atom!

    I’m not being pedantic, Doctor, but what kind of atom? Hydrogen, or something bigger?

  4. gopher65

    I noticed this:D. I was like “Pea size… wait… wouldn’t that be about the mass of Earth??” Muwhahahaha. I’m feeling proud of myself right now for some reason :-P.

  5. Simon

    I’ve got a couple of things here that don’t make sense to me:

    “I won’t even go into Hawking radiation, which says that a black hole that small would have a surface temperature of 10 billion Kelvins. That would make them a bit obvious if they were pelting us; they’d be pretty bright.”

    If they’re black holes, wouldn’t its gravity keep this radiation from escaping into space?

    Also, where does that equation for calculating the size of a black hole come from?

    Just trying to better understand the concept here. :)

  6. Stark

    Simon, Hawking radiation does indeed “escape” from a black hole… though escape is not technically correct. It’s kind of (but not really) like the black hole is evaporating.

    Here’s a nice rudmentary explanation – but it really is a very complex subject to get a good grasp on.

    Suffice to say, if Hawking’s theory is correct, a micro blackhole in your vicinity would be very noticeable and not “black” at all.

    As has been said…black holes are weird. They exist within the fringes of the physical rules we typically understand. Warapping your head around the how’s and why’s of black holes is a good recipe for a headache!

  7. DTdNav

    @ cvoid,

    Is that the book where the artifact hates the spring?

  8. > If they’re black holes, wouldn’t its gravity keep this radiation from escaping into space?

    Hawking radiation emits from just outside the event horizon (due to the black hole only capturing half of a quantum pair that comes out of the foam), and ends up having escape velocity.

    > Also, where does that equation for calculating the size of a black hole come from?

    Taking the orbital velocity equation, setting orbital velocity equal to the speed of light (defining the event horizon), and solving for radius.

  9. Christine Pulliam

    What, no comment on the whole “zooming all around us and could annihilate you instantly” thing? Seems like bad astronomy to me. :-)

  10. Simon: The radiation actually comes from vacuum fluctuations just above the event horizon. Above the event horizon, the escape velocity is lower than light speed, so light goes fast enough not to be pulled back.

  11. ShavenYak

    It’s worth pointing our that 2Gm/c^2 is the radius of the event horizon, not the collapsed object. I suppose it’s reasonable to say that’s the size of the “hole”, but I think it gives some folks a (probably) mistaken idea that the object itself magically stops collapsing on itself at that size. More likely it continues shrinking, although the knowledge of that fact can never reach us outside the hole.

    Another nitpick: mini black holes the size of mountains wouldn’t have lasted until now if they formed during the Big Bang, at least according to Hawking.

  12. Stark

    To further The Centipedes answer about the equation – it’s the equation for the Schwarzschild radius. Named after Karl Shawarzschild who solved it in 1916. It calculates the radius for a given mass where nothing could stop that mass from collapsing into a gravitational singularity.

    Also, odds are, if micro black holes exist they evaporate (through Hawking radiation) very very quickly. It’s theorized that you could creat a micro black hole in a partical collider like the Large Hadron Collider but such a black hole would wink out of existance nearly instantly. Of course if you could produce a black hole in the LHC then it is very likely that they are being produced (and fairly often) out at the edge of our atmosphere by the natural cosmic ray bombardment of Earth that goes on all the time and is far more energetic than anything humakind can produce a lab. Seeing as how the planet has yet to be destroyed I’d say we’re either wrong about the existance micro black holes or they pose no real risk!

  13. Ken B


    While nothing within the event horizon can escape, things release energy as they approach the event horizon. I’m sure a quick search on “Hawking radiation” would explain more. (Actually, a quick search shows that this particular part is a lot more complex than I thought.)

    As for the equation, as I understand it… (Crossing fingers, hoping not to blow it.)

    The amount of force a given body exerts on another is proportional to the inverse square of the distance. Given no other forces, this force (aka “gravity”) will accelerate the other body towards itself. (And, technically, they both pull each other. But, when talking of things like a black hole versus a person, the person’s mass is so negligible as to be practically irrelevant.) To remain at the same distance, you would need to apply a force which would accelerate it at the same rate in the opposite direction. As you get closer to the main body, the force/acceleration increases. Then the acceleration reaches c, you have reached the event horizon.

    Given “a = Gm/r^2”, and “a = c”, you get “c = Gm/r^2”, which to me gives “r = sqrt(Gm/c)”, which means one of us is wrong. :-)

    A quick search on “Schwarzschild radius” agrees with BA’s formula.

    Well, I’m keeping this post as-is in the hopes that someone can tell me where I went wrong. (Be gentle, please.)

  14. > Then the acceleration reaches c, you have reached the event horizon.

    That’s your error. c is a velocity (m/s), not an acceleration (m/s^2).

    Always check your units! 😉

  15. I disagree on one point. The size of a black hole is *not* a simple calculation. You can use Newtonian physics, and you’d end up with the right answer, but for the wrong reasons. The Newtonian derivation would rely on the fact that kinetic energy = mv^2/2, but in Relativity, the kinetic energy goes to infinity as the speed approaches c. I can’t say I understand why it should come out the same way when we use General Relativity.

    Basically, this is not something an undergrad physicist like me could do.

  16. Jose Natario

    One cannot really understand what a black hole is without General Relativity. Although the Newtonian argument gives the correct Schwarzschild radius, it does so by the wrong reasons. For instance, as Phil said, the escape velocity is just the velocity you have to give an object for it to escape the gravitational field if it coasts along the rest of the way. If you allow for acceleration, an object can escape the field without ever reaching the escape velocity. So a rocket *can* escape a “Newtonian black hole”. In General Relativity, however, you find that *nothing* can escape the black hole – irrespective of its acceleration.

  17. Bob le Moche

    There is one thing I’ve always wondered about black holes: How can they possibly form in the first place? It is my understanding that as space gets more and more bent around a massive object, time gets more and more dilated (Light coming from the object gets more and more red-shifted). At the point where the curvature is such that it prevents light from escaping the red shift would be at its maximum and time at the surface of the object would essentially stop from the point of view of an external observer. How then can any bit of matter reach this point in finite time? It seems to me that a huge mass collapsing on itself would certainly converge towards forming a “black hole”, but never actually do it? I understand that this is all from the point of view of an observer outside of the gravitational field but I don’t see how that has any effect since as far as I know gravitational “information” does not travel faster than the speed of light?
    This might not be the best place to ask about this but could anyone shed some light on this matter?

  18. HvP

    Bob le Moche,

    That’s a very good question for us as observers. In fact, our Bad Astronomer did bring up this point in a past post here:

    Hopefully, a resolution between relativity and quantum theory on these scales might be achieved in the future to help, but right now… it’s hard to get one’s head around.

  19. Alex Besogonov

    Black holes ARE NOT dense!

    Actually, supermassive black holes have density compared to the density of water or even air. That’s because the radius of the black hole’s event horizon grows faster than its mass.

  20. Buzz Parsec

    little nit, Alex. Actually, the radius of a black hole does grow as its mass, but its volume grows as the cube of its radius, hence as the cube of its mass. Density is inversely proportional to volume, so double the mass, and density is 2m/(2r)^3 = 1/4 the original density.

  21. miller: I meant that once you have the equation, getting the radius is easy.

    Everyone: I am sloppy sometimes and say “black hole” when I mean “event horizon”. They are different, but in the public’s mind they are the same thing.

    Alex: right, supermassive black holes are not dense. I remember talking to Roger Blandford (!) one day, and mentioning that; we were walking across Stanford campus at a meeting, and while we were chatting I did a quick mental calculation and realized that an SMBH would have the density of air. IIRC Roger was surprised too! But little ones are dense, and that’s what I meant.

  22. autumn

    Speed of light being equal to all observers, one observing from the surface of an object collapsing to form a SMBH (must be really big to eliminate tidal effects) would observe the hole shrink past the Schwartzschild radius, while an observer outside the event horizon would presumably observe an asymptotic approach to the event horizon, the object getting smaller ever more slowly as it approached the Schwartzschild radius.
    I may be completely wrong here, but it is as I (in an admittedly limited capacity) understand the relativity physics of this particular situation.

  23. David Vanderschel

    Regarding the BA’s footnote clarifying “escape velocity”: Another way of looking at it is that your escape velocity depends on the altitude (above the ‘surface’ (an average) of the earth) from which you are escaping. (Adding a nominal value for the earth’s radius, this is the “r” or “radius” in the formula. It does not work if your start is below the surface.) The higher you are when you start the less velocity you need to escape. Now, if, in the process of acquiring your velocity, you have climbed to a higher altitude, then the escape velocity you need to achieve is less than it was at the altitude where you started.

    An impulse, as described by Phil, is something that physical objects cannot stand. What would be relevant is a situation in which a (large) force is applied over such a short amount of time that there has been little change in altitude when the force stops. (If you were already in orbit before starting the acceleration, then a slower acceleration could initially be applied in a direction roughly parallel to the nearby surface of the earth to achieve escape velocity without first gaining altitude.)

    Actually, “escape velocity” is not about force at all. It is about velocity, period – independently of how that velocity is acquired. But it does depend on that r parameter, the value of which at the point in time when escape velocity is actually achieved is not necessarily the radius of the earth.

  24. Hi Phil,

    you got wrong: 11 km/s is not the speed that you need to reach orbit. It is the second cosmic speed that you need to leave Earth behind entirely.

    To get to orbit, you need 7.9 km/s (first cosmic speed, actually: the orbital speed at sea level). The speed in orbit is actually a bit lower (7.6 or so for the ISS).

    To round things up, the third cosmic speed for leaving the solar system is 42.1 km/s.

    However, you are able to get some speed from Earth’s rotation (depending on launch site and direction) or speed around the sun, respectively.



  25. Martin Moran

    I have been reading about the LHC recently, I found this really interesting and led me on a journey. There was something about the possibility of mini black holes. Is this true?

    I have managed to understand that black holes with a mass that is extremely small are unstable and the ones the LHC would produce would only last a few tenth of a trillionth of a trillionth of a second. There was also the possibility that an exotic remnant might survive?

    But wait, what if in that trillionth of a second the black hole was ‘dropped’ and fell into the center of the Earth!
    Bring the boys on board the ISS home!

  26. NHR

    Some nitpicking! I wanna, too…

    “G is Newton’s Gravitational constant of the Universe, and is just a number.”

    It’s NOT “just a number”, if that is to say it’s dimensionless. It has dimensions of volume per mass per time squared, for example.

    And yeees, I know you astronomers often use some weird units, where some thing or another is _defined_ to be dimensionless, to keep the math as simple as possible. But anyways, G is not _generally_ “just a number”.

    Or maybe I just don’t read that right, wouldn’t be the first time…

  27. Donald Smith

    Hi, Phil!

    I have found that the Newtonian “escape velocity equals c” definition of the event horizon is too misleading. Since escape velocity means the velocity you need to give an object at a radius r from a massive object, if you give that object at r a slightly smaller velocity, it will go a long way out before falling back. This allows you to imagine someone just inside the event horizon throwing a baseball to someone outside — after all, the ball doesn’t *have* to “escape”; it just needs to get to the person at the larger radius.

    The right way to think of it requires explaining light cones, and what it means to be in the “future” of a particular event. Near a massive object, light cones “tilt” in towards the object, and for an event at the event horizon, all the events that constitute that event’s future lie on or within the horizon. It cannot influence any events outside the horizon.

    Of course, it usually takes a few weeks to explain light cones and the schwarzschild metric. :-)

    Secondly, I have a problem with saying SMBH are “as dense as air”. Presumably, most of the volume of space inside the event horizon is empty, with all the mass concentrated at the center. I teach my students that it doesn’t make physical sense to take an average when the quantity you are trying to average is changing, and that certainly is the case here. The bulk of the mass of an atom takes up about one-trillionth of the volume of the atom. It doesn’t make sense to talk about the “average density” of the atom by dividing the mass of the atom by its total volume. That number will not be representative of the density at *any* point within the atom, so why use it? There would be a local mass density at any point around the BH, but once you get inside the innermost stable orbit, you’re not going to get an equilibrium situation (unless you have a steady mass accretion rate, but presumably the local mass/energy density at the EH will be much less than the presumed signularity at the center), and besides which, the space is highly curved, so the volume is going to be greater than 4/3pi r^3. Although if you can do curved space volume integrals in your head while walking across campus, I am very impressed. :-)

    Anyway, I just think it’s tricky and a little misleading to talk about “the density” of a black hole. I mean, yes, you can certainly divide mass by volume, but I don’t think that number is particularly meaningful, and it’s certainly misleading to compare it to air. That’s like saying because saturn has a lower average density that water, it would float. Well, if you brought it to Earth and tried to float it, the Earth would sink in it! What is *meant* is that something small with the same density as Saturn’s average density would float in water. I’ve found that the former formulation leads to bizarre misconceptions in students, and I think the same thing would happen in this black hole situation.

    Don Smith
    Guilford College

  28. bestonnet

    11 kilopicoseconds as Earth’s escape velocity makes no sense.

  29. Anne

    Saying that a mini black hole is really hot doesn’t mean all that much, since it’s so tiny – the power output of a black body (which these should be) depends on the temperature (huge) and the area (tiny). It happens that the product is in fact huge, so they actually are bright, but mentioning how hot they are isn’t that relevant. In fact, the fact that they’re so hot but also so tiny means that they don’t actually emit much visible light as primary radiation.

    I saw a neat paper on the arxiv a few months ago that proposed that the earth already had swallowed some tiny primordial black holes and that they were providing some of the extra heating it took to keep the core liquid. I don’t know how the accretion rate depends on hole size in that kind of environment, but the authors certainly seemed to think that there would be enough of a balance between accretion and evaporation to keep the hole from either devouring the earth or evaporating into a flash of particles over astronomical timescales…

  30. zeb

    Taking the event horizon of a black hole as is radius for the density problem is incorrect. The event horizon isn’t a phyiscal object, just a place in spacetime where there are no longer any exit trajectories. For the density of the object (i.e. the singularity), you would have to put in the radius of the singularity itself (which is really really really^47,000 small). So the density of a black hole is really really large.

  31. Wouldn’t you consider a singularity to be infinitely dense?

  32. Folcrom

    Black Holes and Warped Space-Time
    was release 25 years ago.
    Very good book if you can find a copy.

  33. TC

    Is there anything energetic enough to leave the universe?

    What is the escape velocity of the universe?

    Would an observer outside of the universe be able to ‘see’ the universe?

    If the universe is expanding then what is it expanding into and why cant we get across the possibility of ‘space’ existing outside of this observable universe?

    Black holes are detected due to their effects (gravitational influence, radition of particles being shredded at the EH etc etc)

    What is the ‘smallest’ black hole detected?

    Has a mini black hole ever been detected?

    Has a black hole ever been detected where there was none before (at nova, snova, whatever)?

    Lastly, where did all the matter come from?

    Sweet dreams, enjoy your lives and the best health you can maintain, dont be frightened of dying, its ok, its really ok, you can trust me.


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