Are Black Holes Forbidden Mathematically?

Posted by Ethan on November 18, 2009
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In the comments on one of my posts, someone pointed me towards Stephen Crothers, who gives the following argument (in a nutshell) as to why black holes cannot possibly exist:

  1. General Relativity is our theory of gravity, which relates the curvature of space to the gravitational acceleration of objects.
  2. This theory only works in certain regimes; it breaks down at the point of singularities.
  3. A black hole, as predicted by Schwarzschild, is a singularity.
  4. Therefore, since singularities are forbidden by General Relativity, there is no reason to think that black holes exist.

(You can watch his video here, or read his full argument here.) Therefore, he argues, astronomers are wasting their time looking for black holes, since their existence isn’t even a physical prediction.

Talk about not seeing a forest for the trees. The “singularity” is not essential for a black hole to exist. Honestly, it isn’t important at all whether there’s a singularity or not. All that matters, in the real world, is that something is both massive and compact enough so that, within a certain radius, light cannot escape from it. That is the astrophysical definition of a black hole.

So, do they exist? Definitely. Where do you look for incontrovertible proof? The center of the galaxy! There are no two ways around it; there is definitely a black hole there.

How am I so sure? The above image shows the center of our galaxy. There are many, many stars orbiting the central point where the arrows are pointing. We have tracked these orbits over more than a decade, thanks to the UCLA Galactic Center Group. Here’s a screenshot of their results.

From the motion of these orbits, we can figure out what the mass of the object they orbit around is. It turns out to be over 2 million times as massive as our Sun. And yet, we don’t see any light coming from that point. We don’t see a white dwarf, we don’t see a neutron star, we don’t see any object at all.

For a mass that large, you will have a black hole if that mass is confined to a sphere of a diameter of about ten million kilometers. That isn’t hard, considering we have many, many stars that we know of where an entire solar mass is confined to a diameter of about ten kilometers. (These are neutron stars.) If you up the mass, the neutrons at the core will eventually collapse under the tremendous pressures, and collapse farther. There’s a well-known upper limit to how massive a neutron star can be, and it’s less than three solar masses, much less two million.

So you can argue about whether singularities violate General Relativity or not until you’re blue in the face. It doesn’t have a damned thing to do with whether any light gets out of your ultra-dense, massive object. And that’s what we call a black hole, and it exists. Don’t believe it? Then tell me what’s going on at the galactic center.

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Comments

  1. #1 Michael Varney
    November 19, 2009

    Even setting aside the black hole issue in GR, the poster is incorrect saying that GR cannot handle point singularities.
    You can coordinate patch away many forms of singularities. And while in GR the space=time metric is not positive definite, and therefore not m-complete, we can show that it is b-complete, which is equivalent to m-complete in a pd metric.

    So, even if black-holes contained a singularity, and their existence depended on such, GR still admits them as solutions.

  2. #2 Bee
    November 19, 2009

    See also our post Evidence for the Black Hole Event Horizon

  3. #3 Petko Fiziev
    November 19, 2009

    There is another problem with black holes too. Defined in this way you cannot actually observe them. During the course of time there should be some matter that has fallen into the black hole presumably in the form of light. Since it is moving with the speed of light it would look to a distant observer as if the time has stopped for this matter at the point when it reaches the event horizon. So we will never actually see the matter disappearing in the black hole. For us it will just stop moving at the event horizon. Therefore we should see the black holes as bundles of entangled matter that stops moving at some point and not as empty void like in the case with the center of the Milky way.

    Besides this I also do not believe that there are singularities in the nature. There may be a singularity in the theory (as there are in many mathematical theories) but that does not mean that it exists in reality. Also there may be solutions to the equations of Einstein that do not have physical meaning at all. Schwarzschild could be one of them.

  4. #4 mark Phillippoussis
    November 19, 2009

    come on!!! Of course Black holes exist!

    How else would Sam out of Quantum Leap be able to time travel otherwise? Ziggy wouldn’t even exist. Obtuse observations indeed.

  5. #5 Anita
    November 19, 2009

    This always bothered me. I first learnt that the definition of a black hole was what you describe, but after that I kept on hearing and reading things that seemed to suggest that black holes have a singularity in the middle. Where does this idea come from? Is there maybe an astronomical definition of singularity that is different from the mathematical one? Or is it because some metrics predict an actual singularity?

  6. #6 Kirk
    November 19, 2009

    While the math is a bit beyond me, after poking around on Stephen Crothers web site it’s clear that this guy is, at best, an unnecessarily argumentative person who’s a bit full of himself. Scientist after scientist take the time to point out his clear errors, to which he responds with insults and accusations of quakery. In Mr Crothers world everyone who disagrees with him is a jerk and an idiot.

    It’s a shame really. Mr Crothers clearly has an aptitude for mathematics and science. He also clearly has become blinded by what appears to be a mistaken theory, this is exacerbated by his corrosive personality. If he would simply have entered into honest discussion with an open mind he likely would have his doctorate and would be able to contribute to science. As it is he’s reduced to an annoying gadfly.

    Before reading Ethan’s post I’d never heard of Mr Crothers, I have, however, been fortunate enough to meet several of the theoreticians with whom he corresponded. Their time is valuable, for more so than that of a failed grad student, and the fact that they took considerable time to respond to and teach Mr Crothers should be respected. Mr Crothers showed no respect and deserves none.

  7. #7 healthphysicist
    November 19, 2009

    Could black holes be concentrations of dark matter rather than baryonic matter?

  8. #8 ajkamper
    November 19, 2009

    Petko:

    HOw would we see this “Entangled matter” from a distance? If it was on the horizon, it couldn’t emit light, because that would get drawn right into the singularity. So even if it weren’t empty void, if I’m understanding you correctly, we still wouldn’t _see_ it.

    Incidentally, can anyone tell me the extent to which the radiation from the accretion disk of a black hole should be red-shifted as it gets closer? I simply don’t know the quantitative physics at all.

  9. #9 Eric Lund
    November 19, 2009

    Anita: I think the issue is that once you overcome neutron degeneracy pressure there is nothing that prevents the mass from collapsing to a point. So a black hole would evolve toward a singularity. As seen by somebody riding the stellar surface through the collapse, the singularity comes about in a finite amount of time.* However, an observer who remains outside the Schwartzschild radius never sees this happen, since for him time dilatation goes to infinity at the Schwartzschild radius.

    *In the GR course I took as a grad student we calculated the trajectory of a universe with no dark energy which collapses in a Big Crunch. It turns out that the maximum radius of such a universe is equal to its Schwartzschild radius, and the universe has a finite lifetime.

    Healthphysicist: There is an old saying that “black holes have no hair”. That is, we can measure a black hole’s mass, charge, and angular momentum, and that’s all. We would have no way of knowing what kind of matter the mass is comprised of after allowing enough normal matter to account for its measured charge.

  10. #10 Dave Lehocky
    November 19, 2009

    Excellent description. With that observation you can’t hold up a red (any object) and say my math predicts this must be blue. fascinating universe we live in.

    Dave

  11. #11 Petko Fiziev
    November 19, 2009

    @ajkamper

    No, I think that we should see glowing and slowing down matter at places where there are black holes instead of empty voids. This view should come from the moments before the matter enters the event horizon.

    One more thing that came to my mind is gravitational lensing. Since the black hole is so massive it should bend the light that comes from objects behind it and magnify or diffract the images thus producing strange pictures on the Hubble photos for example. I’m not quite sure that this is what we see in the center of the galaxies.

  12. #12 healthphysicist
    November 19, 2009

    Thanks Eric!

  13. #13 ScentOfViolets
    November 19, 2009

    IIRC, the concept of a black hole originated over 300 years ago with Isaac Newton. The question really devolves to the definition of what a black hole is. If you define it as a gravitating body whose escape velocity is so large that even light cannot escape, well, yes, black holes have been known to exist for a long time.

    The singularity at the middle? That comes from a different theory, or rather, set of theories, altogether. Even today you get stuff like certain loop gravity formalisms that predict a maximum but finite density from first principles. And very modern up to date stuff it is too.

  14. #14 Joe Postula
    November 19, 2009

    I have the same question as Petko, in all the “pictures” that I’ve seen of the Milky Way’s center, I haven’t seen the tell-tale warped light due to gravitational lensing. Why is that? Are we not able to “zoom in” enough?

    BTW, I love the movies of the stars being flung around the galactic center. It’s crazy to think about stars moving that fast around something.

  15. #15 Anita
    November 19, 2009

    neutron degeneracy pressure
    I see. I think I remember something about this now. Thanks.

  16. #16 kasule francis
    November 19, 2009

    That beats my understanding, how is it that in this modern age we are lost in Translation, on issues our basic science is based on.Too many things contradict themselves,the problem seems to originate from the field of theoretical physics, Its time to switch over to physical physics and start afresh ,the universe seems to have thrown us in to disarray.

  17. #17 Thomas Neil Neubert
    November 19, 2009

    Since outside the black hole event horizon is the only spacetime where astronomers can make observations; Ethan’s screenshot image of observations and careful discussion of the calculation of the mass at the center of our galaxy is excellent. Thank you for that snapshot and explanation.

    I can think of no reason to doubt the suggested calculations and the interpretation as a gigantic black hole at the center of our galaxy; since Ethan confines himself to observed phenomenon (i.e. motion of stars outside the event horizon)and the interpretation of those observations. And by the way, I don’t see any extraordinary assumptions or subspecialty bias here. Very nice summary, very nice avoidance of the singularity quagmire, and very nice simplification of the definition of a black hole.

  18. #18 Mu
    November 19, 2009

    Guess the big visible accredition disk only happens in bad Disney movies.
    What happens with the momentum if mass enters the black hole? Stuff circling the black hole can lose a lot of energy via radiation, but what happens if you shoot a mass straight at it for a direct hit?

  19. #19 Warren
    November 19, 2009

    Then tell me what’s going on at the galactic center.

    Vogons, man. It’s the Vogons, working on that hyperspace bypass.

  20. #20 Ted ray
    November 19, 2009

    What’s going on? The black hole is sucking matter out of our un iverse into another one.

  21. #21 Eric Lund
    November 19, 2009

    Guess the big visible accredition disk only happens in bad Disney movies.

    The accretion disk will be there if the black hole has a companion star. Those particles do indeed radiate as they fall toward the black hole, and the reason we know of the stellar mass black holes that we know of is because we detect this radiation.

    Petko is correct that we cannot observe the black hole directly, but he is mistaken on his other points. First, he assumes that black holes always have accretion disks–many but not all do. He is also correct that time as observed outside the event horizon comes to a standstill at the event horizon; however, it does so gradually, not abruptly as he seems to assume. He is also neglecting gravitational redshift–stuff that falls in may radiate in X rays, but as it approaches the event horizon we see those photons at much longer wavelengths.

    One other point to remember is that circular orbits do not exist all the way into the Schwarzschild radius. At a distance of 1.5 Schwarzschild radii a particle would have to move at the speed of light to maintain a circular orbit, so any particle that falls this far in must fall into the black hole (though we can still detect it at this point, because photons radiated away from the black hole can still escape). Circular orbits inside 3 Schwarzschild radii are unstable, so particles inside this range are overwhelmingly likely to fall in (although the particles can still escape if they have or can acquire enough angular momentum).

    As for gravitational lensing: Yes, it could happen. However, there would need to be a background object in the right location pumping out enough photons that we could detect them, and the galactic plane contains a large amount of dust that would tend to reduce the number of detectable photons.

  22. #22 Michael Varney
    November 19, 2009

    And by poster I meant Crothers, not Ethan.

  23. #23 Robert
    November 19, 2009

    The black hole at the center of our galaxy is not currently “feeding.” That’s why the only evidence comes from the orbits of nearby stars.

  24. #24 DaveH
    November 19, 2009

    There was a BBC program on black holes last week, where one of the guys (I forget his name) who collected that data on the orbits of stars at the centre of our galaxy was interviewed. He said “A very massive object, a small space for it to be in. A schoolchild would come to the same conclusion. There’s a black hole there.”

    The fact that it’s dark doesn’t count against it being a black hole.

    As for what happens at the centre of a black hole, all the physicists had the same answer – “Umm…” :-)

  25. #25 DaveH
    November 19, 2009

    Someone has posted that program online (shh):

    Who’s Afraid Of A Big Black Hole?

  26. #26 Nathan Myers
    November 19, 2009

    Um, UCLA Galactic says four million solar masses. True, that’s more than two million. Did you have a reason to low-ball it, such as that we might find 2M more believable than 4M? (I know I do. :-)

    Speaking entirely hypothetically, and presuming an implausibly hard vacuum in the region, what electric charges would the central body and affected stars need in order to experience the accelerations observed? There are many ways to express it, of course, some more accessible than others.

  27. #27 Ethan Siegel
    November 19, 2009

    Nathan,

    There have been two different techniques used to measure the mass of the black hole at the galactic center. The technique used by the UCLA galactic group came second, and gave a much larger estimate than the first technique, which yielded 2.7 million solar masses.

    I decided to play it safe and give the lower of the two numbers, but the UCLA group’s work is probably more reliable, and is definitely more straightforward.

  28. #28 bryan242
    November 19, 2009

    As far as I can understand it, GR’s prediction of singularities can be understood as just an indication of its own incompleteness as a theory, on the scale of quantum gravity;

    but also, if you’re interested in astrophysics and not quantum gravity, you can use GR’s prediction of gravitational collapse into an event horizon, and stick to considering only the event horizon and its mass, charge & angular momentum and its interaction with the exterior universe, a regime in which GR works great, and stay agnostic about what’s under the hood of the event horizon, without ever having a problem.

    The world has an inexhaustible supply of quacks like this no-singularity guy who are more interested in arguing than learning and are just a big waste of time when you could be actually learning about the universe.

  29. #29 Douglas Watts
    November 19, 2009

    The most famous grav lensing images are the Hubble pix of the Abell 2218 galaxy cluster and Galaxy Cluster 0024+1654 in Pisces. In the latter the lensing cluster is 5 billion light years from Earth and the lensed galaxy is est. at 10 billion light years. Not surprisingly these are incredibly faint tiny and faint objects which can only be seen because the viewing angle from Earth is not near the light and dust filled center of our galaxy.

    In contrast the area of the image that Ethan shows in the Milky Way galactic center is so thoroughly crowded with dust and stars that the image of a lensed star behind the imputed black hole might simply be invisible simply due to a) dust and b) too much competing light from other stars.

    Another thought. The two classic lensing pix cited above are created by clusters of galaxies, not black holes. A black hole of 2 million solar masses is puny compared to a galaxy of Milky Way size, which is 100 billion solar masses, and really puny compared to a cluster of galaxies, which would be pushing 500-1,000 billion solar masses. And based on the lensing pix I have seen thus far, the phenomenon is pretty uncommon and limited to galaxy clusters as the lenses. And even then, it’s not like every galaxy cluster shows a lensing effect. Most do not.

    As an exercise, grab a hi-res Hubble pic of the Whirlpool or Pinwheel galaxy and you can see a number of tiny galaxies way way farther out behind them. They are not lensed.

    It might be that the minimum mass of the lensing object(s) must be orders of magnitude greater than a 2 million solar mass black hole to make a lens visible from Earth.

  30. #30 Douglas Watts
    November 19, 2009

    A more mundane explanation for the lack of visible grav lensing of stars in the pic above is that viewed from Earth, stars near the galactic center are point sources. Their discs are far too small to be discerned. In contrast galaxies are big enough to discern their shape and dimensions even at enormous distances. The “smearing” of the lensed galaxies behind Abell 2218 et al. is due to the fact that even from Earth, they are not point sources of light, but have discernible shapes, lengths and widths. It is kind of hard for me to imagine how a point source could create a visible lens since there are no dimensions to smear and stretch in the first place.

  31. #31 AndrewN
    November 19, 2009

    Well, this is so disappointing. I came here expecting some new work on the subject that would both expand our knowledge and excite me and all I got was a tedious argument from ignorance :(

    Even with my limited knowledge of astronomy I could tear huge holes in the so called chain of logic. Ho hum, maybe next time.

  32. #32 Random
    November 19, 2009

    AndrewN, let’s hear it. Why next time?

  33. #33 MadScientist
    November 20, 2009

    @Random: I think he means maybe next time he’ll find something interesting rather than “We currently use theory X. My mistaken understanding of Theory Y says that it must have condition Z. My mistaken understanding of Theory X says condition Z cannot exist. Therefore Theory Y must be wrong.” The argument is a montage of logical fallacies; using the same pretentious argument we could say that Einstein’s general theory of relativity is wrong because Newton’s laws don’t predict certain observed phenomena.

  34. #34 Vicki
    November 20, 2009

    If that’s really Crothers’s reasoning, any high school student should be able to spot the error. Step 2 is “General relativity doesn’t know how to handle singularities,” and then step 4 is “since general relativity forbids singularities.” “Does not know how to handle” is very different than “forbids,” and this looks like a restatement of the point that general relativity can’t tell us about conditions inside a black hole.

    (The more usual conclusion here would be “see, we have a black hole, therefore general relativity is wrong!” which is also an error, because “cannot handle” doesn’t mean “is inconsistent with.”)

  35. #35 AndrewN
    November 20, 2009

    AndrewN, let’s hear it. Why next time?

    MadScientist has it right, next time there might be something interesting because there certainly wasn’t this time.

  36. #36 Random
    November 20, 2009

    AndrewN,

    I must apologize. I thought you were being critical of Ethan Siegel arguments rather than Crothers’ logic. I misunderstood and had a kneejerk reaction :(

    But yes, this was a quick summary of what a black hole is and good observational evidence pointing to the existence of this phenomenon.

  37. #37 Random
    November 20, 2009

    Another blogger has gone deeper into Crothers’ work, if I may post a link to it:

    http://dealingwithcreationisminastronomy.blogspot.com/2009/06/some-preliminary-comments-on-crothers.html

    Crothers has posted a counter argument on that thread.

  38. #38 healthphysicist
    November 20, 2009

    Slightly off-topic…back onto the Dark Energy series….if the universe is expanding (volume increasing) and the dark energy density (energy per volutme) is constant, that suggests dark energy is being “made”.

    What is it being made from? A conversion of gravitational potential energy to dark energy?

  39. #39 T_U_T
    November 20, 2009

    black holes do exist, but singularities don’t. Why ? Consider an outside observer of a spherical mass collapsing into a black hole. What will you see ? You will see the collapsing mass redshifted, and also slowing in time as it approaches the schwarzschild radius, the more it is close to it, the slower it will get. And consider the hawking radiation. The collapsing sphere starts slowly radiating away energy, which causes the schwarzschild radius to decrease. Soon the infalling matter will be slowed so much that its velocity, as observed by an external observer will be only as slow as the the shrinking of the schwarzschild radius, so they never actually meet all the way till the black hole is radiated away completely, ending in a burst of gamma rays.
    And now, consider an observer sitting on the surface of the collapsing sphere. He will observe, as he approaches the schwarzschild radius increase of the unruh effect causing both the schwarzschild radius to recede and lifting him up by radiation pressure, all while he sees the outside observer to be blueshifted and extremely rapidly aging, till the last mass below him will be carried away by the last jolt of high energy radiation and the internal and external observer can meet, one just a few seconds old, the other like 10^200 years.

    This is, how I understand it. Of course I could be wrong, so, if you see where I am mistaken, feel free to point it out.

  40. #40 Bob
    November 20, 2009

    Remember, as a star collapses, it passes from the macroscopic world of Classical Physics to the microscopic world of Quantum Mechanics where new rules take over.
    When it passes that Quamtum threshold the rules of GR cease to exist, so ruling out Black Holes on the basis that GR doesn’t allow them is just plain wrong!

  41. #41 DaveH
    November 20, 2009

    T_U_T, you say
    “the internal and external observer can meet”

    but I don’t think they can even communicate.

    Tbh, I don’t see how any of that sheds light on the centre of a black hole. Worse, I keep wanting to spell “centre” with “er”. Damn html, damn the internet, damn America – y’all brainwashed me finally.

  42. #42 T_U_T
    November 21, 2009

    but I don’t think they can even communicate

    They can. After the black hole disappeared by hawking radiation. Long time, sure, but after a finite amount of time the black hole disappears and anything that was inside is kicked out to the universe once again.

  43. #43 Hush
    November 21, 2009

    Ethan,

    Crothers is not alone. I expect Rachel Beans’ work to further undermine the fallacy and myth of the Astrophysical Society’s tenacious, desperate embrace of black holes.

    1.)http://arxiv.org/abs/0909.3853

    2.)http://blogs.discovermagazine.com/cosmicvariance/2009/10/12/a-new-challenge-to-einstein/

    I’m sure with the signatures of new incoming, upcoming future lensing datasets to bolstered Beans’ findings, we will be engaged again in a lively forum discussion ostensibly in favor of Crothers work.

    Althought Crothers’ math was admittedly beyond most posters here, that did not prevent them from dismissing Crothers work – a most unfortunate stance.

    Perhaps they will embrace Rachels’ work with less bias.

    Nevertheless, thank you all for your innate, if not passing interest.

    Hush

  44. #44 DaveH
    November 21, 2009

    @41 – I’m not familiar with the idea of black holes as long-term storage, but it’s intriguing.

    Yet I fear the external observer would be waiting in vain. Our intrepid traveller would be torn apart into elementary particles. And then would he not be Hawking-radiated away with the rest of the black hole?

    @42 – What are the differences between Mr Crothers’ objections and Rachel Bean’s paper, in terms of

    (a) methodology (broadly speaking)
    (b) physical scale

    not forgetting

    (c) circularity

    ?

  45. #45 Ethan Siegel
    November 21, 2009

    Hush,

    Rachel Bean’s work has absolutely nothing to do with Stephen Crothers’ work or with the existence of black holes.

    It is a statistical likelihood analysis asking if general relativity is cosmologically the most likely theory of gravity on the largest scales. Which is not evidence of anything, simply a suggestion.

  46. #46 Random
    November 21, 2009

    Hush,

    Did you understand Crothers’ math?

  47. #47 Cusp
    November 27, 2009

    Hi – I work on BHs and have examined Crothers maths and it is very wanting. He is cherry picking a particular coordinate representation of Schwarzschild and doesn’t appear to understand the transformation into the version presented today. It is simple to check that the textbook solution is a valid solution by feeding it through the field equations.

    The almost laughable aspect of Crothers rant is that Schwarzschild is a “vacuum solution” – and he clearly has never used Poisson’s equation with a point mass (which is also a vacuum solution).

    Dragging Rachel Bean into this mis-understands what she is doing – testing alternative gravity does not mean she is rabidly opposed to GR, only that she is being a good scientist and questioning everything.

  48. #48 yoron
    March 1, 2010

    39..

    you say “And consider the hawking radiation. The collapsing sphere starts slowly radiating away energy, which causes the schwarzschild radius to decrease. Soon the infalling matter will be slowed so much that its velocity, as observed by an external observer will be only as slow as the the shrinking of the schwarzschild radius, so they never actually meet all the way till the black hole is radiated away completely, ending in a burst of gamma rays.”

    You draw several false conclusions here. The proposed Hawking radiation will be the last thing to disassemble a black hole, in fact it will be at the end of the universe. It have nothing to do with the frames you describe.

    You use two frames of reference as I see it. The one of the infalling object and one at rest with the black hole observing, a so called ‘far observer’. The far observer will observe what you say, but from inside the frame of the infalling object, when measuring f.ex heartbeats against a time device, he will find that the time is as ‘usual’ for him, not differing from at Earth, what will differ is the tidal forces. He will also find himself falling into that black hole in a, for him, measurable time. You have to be careful with what frames you use.

    Furthermore, both Hawking radiation, and our infalling observer are more or less in the same frame of reference when compared to the ‘far observer’, if you see my drift here :) which means that the Hawking radiation will become almost infinitely redshifted as it gets released from the EV (event horizon) of that immense gravity well, and so take an immense time to reach the far observer.

    As I see it.

  49. #49 yoron
    March 1, 2010

    Hmm, better strike that last one at 48. “and so take an immense time to reach the far observer.” As light always move at ‘c’ from all frames of reference no matter your velocity/ acceleration I won’t swear to that one. Although it will be red shifted it will from the infalling persons view move away from him at ‘c’ as well as reaching the far observer at ‘c’ too, the main point being that it still will be immensely red shifted. But light do have a momentum and at what point will that be so weak and the light so red shifted as to disappear from our far observers view?

  50. #50 robby
    April 13, 2010

    Forgive my ignorance and feel free to flame on(as if i really cared) but how on earth did time and space ever get jumbled up together into space-time? it would seem to me that no matter how much gravity would be condensed into a tiny space..a second is a second. How could time possibly be altered by the force of gravity? Im no physicist, just a bit interested in the subject and therefore have no background in this field so be gentle =]

  51. #51 mm
    May 25, 2010

    you guys are so cool :)

  52. #52 tori
    May 7, 2011

    totally weird dude!!!!!!!!!!!!