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My Black Holes brainteaser

  1. Jan 21, 2008 #1
    We know that something with a big mass will collapse and will become a blackhole.
    We know also that an "easy" blackhole is the Schwarzschild solution of the Einstein's equations.
    An object free falling into a blackholes will pass the event orizont without problems but for an external observer he will never arrive to the event orizont.
    So for external observers blackholes won't increase its mass!
    Is it possible? Where do I make a mistake?
    Will the Schwarzschild radius never increase?

    Bye
    random3f

    P.S. I'm sorry for my English, I hope you can understand what I meant.
     
  2. jcsd
  3. Jan 22, 2008 #2
    [An object free falling into a blackholes will pass the event orizont without problems but for an external observer he will never arrive to the event orizon].
    You are correct up to that point. But you have to note that when the object is falling into the black hole, it gradually appears to be red and becomes faint and finally we are unable to see the object object. So the object is not under our visibility, then how could you say that the mass does not increase for a black hole ?! Also the increase in the mass of a black hole is felt due to its gravitational force.
     
  4. Jan 22, 2008 #3
    Analyzing the problem in a classical way, we can find a geodesic equation with an affine parameter (like in Susskind, Lindesay) of an object radially free falling in a Schwarzschild BlackHole. You can see that when object is at [tex]r= 2MG [/tex] (so at Schwarzschild radius), the Schwarzschild time is infinity. So for an external observator the object will pass [tex] r= 2MG[/tex] only after an infinite time. I think. If, ab absurdo, the free falling observator could increase signal frequency, the external observator coud see the same signal for an infinite time (I know that from a quantistic point of view this is impossible, but classically?).
     
    Last edited: Jan 22, 2008
  5. Jan 22, 2008 #4
    Hi Random3f,

    You touch on an important point here. This sort of question has come up many times and a related question is "Where is the infalling mass now or where is the infalling mass really"?

    Asking where something is really makes people here uncomfortable. It is like asking if two observers see each other as length contracted then which observer is really shorter than the other one? One answer is that each observer has there own objective reality and neither can claim to have a greater reality just as neither can claim to be really at rest with some absolute reference frame. However in the case you mention of an object falling into a black hole it is not good enough to just say each observer has their own point of view and leave it at that, because the gravitational mass of the infalling object has to act from a point that is either inside the event horizon or outside the event horizon. An object orbiting the combined mass of the black hole and the infalling mass has to be orbiting mass with a definite location.

    At this point I do not have a definite answer for where the infalling mass "really" is in terms of does it arrive at the singularity in minutes or millions of years from a gravitational point of view but I would tend towards the shorter finite time measured in terms of the proper time of the infalling object.

    In an earlier post I suggested a mechanism for how a significant mass (something the size of a neutron star that would be noticed externally gravitationally) can penetrate the event horizon despite the "time lock" at the horizon. The idea is that the mass of the neutron star in the vicinity of the black hole's event horizon distorts the local gravitational time dilation enough that time is no longer "frozen" at the contact point and it enters easily. To take an extreme example, that is easy to visualise, imagine two black holes of the same mass in contact with each other. The gravitational potential at the contact point is neutralised because of the two equal mass on either side so there is no gravitational time dilation at the contact point and the two black holes merge into one.
     
  6. Jan 22, 2008 #5

    Jonathan Scott

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    Gravitational potential adds up (linearly in Newtonian theory, non-linearly in GR). Time dilation adds up too; nothing is neutralised there. The potential gradient and the related acceleration would however cancel at such a point, if it could exist. However, talking about "contact" between black holes does not seem meaningful; this whole model seems to be based on the idea that black holes are just like very dense masses, which isn't very useful.

    Answering the original point in this thread, if a mass is falling into a black hole, then the combined system of mass+hole obviously has an overall increase in mass-energy compared with the hole on its own, changing the inertial and gravitational mass of the whole system, so the question of when or whether the black hole mass has actually increased doesn't have any obvious measurable consequences.

    The Schwarzschild radius is also not directly observable in a limited amount of time; if some of the mass is very near to the Schwarzschild radius but hasn't actually reached it, it will still have essentially the same gravitational effect on space outside that radius as if it were already part of the black hole; there is no sudden change. (Any attempt to observe the falling mass, by light or for example its magnetic field, will be subject to extreme time dilation).
     
  7. Jan 23, 2008 #6
    Hai everyone in the topic. I want to clear some of the boubts about the black holes. I couldn't do this at a single time, so I shall do it in intervals , starting today. If in any word I write is wrong or if you dont understand please post the reply so that I could clear it in my next post.

    What is a Black Hole ? All the stuff about the creation of the black holes is known to you ( I think so). Puting it in a nut shell..............
    A dying star which is having a mass greater so that its gravitational force is greater than the repulsive force of the elementary particles then the star collapses and has a new form ' black hole '.
    When collapsed the elementary particles does not remain the same as we see them in the normal Universe ( electrons, nuetrons, protons etc.,) but changes into other particles which we have discovered so far or we which we haven't yet. There are many theories on what happens to the particles which are converted to the black hole particles. But no theory is complete.
    The reasons why we didn't have a complete theory will be stated in my next post.
     
  8. Jan 24, 2008 #7
    I stand corrected. You are right that gravitational potential is additive and does not cancel out so my suggestion that time dilation would cancel out is wrong :frown: It would seem this Wikipedia article on gravitational Time Dilation http://en.wikipedia.org/wiki/Gravitational_time_dilation in the section headed Inside a non-rotating sphere stating “The implication is that the gravitational time dilation reaches its maximum at the surface of the non-rotating massive spherically-symmetric object, and that the gravitational time dilation reaches its minimum at the center of the sphere.” is wrong too.

    I now think the correct answer is more along the lines that the event horizon expands to enclose the infalling particle. The gravitational potential at the event horizon of a non rotating black hole is –GMc^2/(2R). If we set G=1 and c=1 then the surface potential is always -1/2 for any black hole, whatever its mass. If we bring two equal mass black holes to within 4 Schwarschild radius ([tex]R_s[/tex]) of each other (the event horizons would be separated by a distance of 2 [tex]R_s[/tex] at this point) the mid point between them would have a combined potential of -1/2. I.e there is already zero time dilation at the midpoint. If the masses of the black holes are completely located within singularities at their centres the two black holes are already technically a single black hole while their singularities are 4 [tex]R_s[/tex] apart. This is because the Schwarzschild radius of a black hole with mass 2M is 4 [tex]R_s[/tex] and our pair meets that criteria. This is nice and neat but there are complications. Points between the mid point and the event horizons will have gravitational potentials that are even lower than the unperturbed potential at the surface of a black hole. (less than -1/2).The time dilation factor in this separation region (outside the nominal event horizons of the two separate black holes) is the imaginary solutions to the square root of a negative number. What that means in real physical terms, I do not know.

    For the conventional interpretation of a black hole with all its mass contained in a singularity of zero volume it would have been better to say midpoint rather than contact point. The conventional interpretation is not without its own problems. The concept of mass within zero volume is incompatible with Quantum Mechanics for example. An alternative model could have a mass distribution within a black hole of radius r, such that 2GM/(r’c^2)=1 within any sub volume of radius r’ and the time dilation anywhere within the black hole would be zero. It would be time dilation that would prevent the mass of the black hole collapsing to a singularity of zero volume. In this interpretation the maximum density anywhere within a black hole of any mass would never exceed the planck mass. What is it, that rules out that interpretation, or is just that singularities with imaginary time and possibilities of matter being ejected to parallel universes or white holes is the more interesting but possibly less plausible interpretation?

    If the infalling mass was significantly large the gravitational lensing of the combined black hole and object would be observably different in the case of the two objects completely merged or as unmerged pair. The orbit of particle orbiting the combined black hole + infalling mass would be detectably different in the two cases also.
     
  9. Jan 24, 2008 #8

    Jonathan Scott

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    I agree that the Wikipedia entry saying that time dilation is minimized at the center of the sphere is wrong. I think someone forgot that even though for most purposes you can model the inside of a hollow sphere as flat space, the time dilation and potential are determined by the surrounding distribution of mass.

    If an object is combining with a black hole, then from the point of view of an observer externally, anything relating to that object will be dilated by the same time dilation factor. For example, if it is orbiting, then the closer it gets, the slower its orbit, until you can no longer see any (quadrupole) wobble in the system, depending on how long you want to wait. There is no point at which you can distinguish between things happening too slowly to observe and the merge having "really" occurred, as from an external point of view that takes an infinite time (which is a well-known paradox of black holes).

    At this point, I should admit that I'm now considering the possibility (after reading papers by Leonard S Abrams and others)that Karl Schwarzschild and Marcel Brillouin were correct and a mass actually has its physical center at Schwarzschild r=2m, and that Hilbert was therefore unjustified in reinterpreting Schwarzschild's radial coordinate, and there are no black holes. Although I agree that the vacuum solution can be mathematically continued in Kruskal-Szekeres coordinates past r=2m, if the point mass is at that radial coordinate then you can't reach a black hole. I don't find the theoretical arguments in either direction totally compelling yet, so I guess we'll have to wait for some experimental evidence, but the fact that many black hole candidates appear to have a strong intrinsic magnetic field does seem to support the idea that they are not really black holes.
     
    Last edited: Jan 24, 2008
  10. Jan 25, 2008 #9
    Hai friends,
    I want to clear you all one important note about the black holes.
    Earlier I had stated that no theory about the black holes is incomplete............
    The reasons are that
    1. Einstein's theory is not valid inside the black hole so, we cant use this theory.
    2. Quantam mechanics, firstly, does not allow the 'point singularity'.

    All the explanations we presently give are all based on these both theories. So any new theory should satisfy these two theories. But that is impossible.
    So we are in need of a new theory........... " The Grand Unification Theory ".
    The concepts of this theory and how it will be delt in my next post.
    [ all the concepts about the mathematical formulation and the concepts of the paradoxes will be delt in a very easy manner, so that every one will understand throughly].
     
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