IOW, just because a finite amount of energy couldn't do the job, doesn't mean that an infinite amount of energy will. Even if you take the limit of infinite energy (and ignore the curving of spacetime that marzojr mentioned this would cause) a massive particle can't escape a black hole -- since a massive particle with "infinite energy" moves like a massless particle, and massless particles can't escape a black hole either.
Just to clear things up, my original statement was "you would need an infinite amount of energy to stop a particle from falling to the singularity (once it's inside the event horizon)", not "you would need an infinite amount of energy to escape the black hole" (which is a completely different thing).
However, as you say, and if I understand correctly, applying any kind of energy, finite or not, would only accelerate the fall into the singularity, no matter how you apply the energy. The only thing you can do is to delay the inevitable, and the maximum delay is achieved by applying no energy at all.
If I understand correctly, in the Schwarzschild solution this is because all geodesics inside the event horizon, including all time geodesics, point directly to the singularity, so simply advancing it time makes the particle move closer to the singularity. Basically you would have to stop time to make the particle not move towards the singularity. (Which raises the question: Assuming you could apply an infinite amount of energy to a particle, would it theoretically be possible to "stop time" for it, ie. make it stop moving in the time axis? Is this something allowed by the GR equations?)
If I understand correctly, in the Schwarzschild solution this is because all geodesics inside the event horizon, including all time geodesics, point directly to the singularity, so simply advancing it time makes the particle move closer to the singularity. Basically you would have to stop time to make the particle not move towards the singularity.
Absent any other forces, all massive particles follow timelike geodesics (definition here) -- this implies that in flat spacetime, all massive particles move at sublight speed, because a path through spacetime corresponding to FTL motion is not a timelike geodesic. In the highly curved spacetime near a black hole, all timelike geodesics point into the black hole. Therefore, all possible paths a massive particle can take point into the black hole, and the massive particle simply has no choice but to fall in, no matter what its energy is.
Warp wrote:
(Which raises the question: Assuming you could apply an infinite amount of energy to a particle, would it theoretically be possible to "stop time" for it, ie. make it stop moving in the time axis? Is this something allowed by the GR equations?)
Sadly, I'm outta my depth here. I have heard stuff said by popular science broadcasters (such as the legendary Dr Karl) along the lines that inside a black hole, spacetime is curved to such an extent that the space dimension pointing inwards to the black hole swaps properties with the time dimension: whereas before you could move freely in the 3 space dimensions but were constrained to move ever forwards in time, now you are constrained to move ever inwards in space but free to move back and forth in the remaining 2 space dimensions and time. If this is true, "time" as defined by "the direction a particle must necessarily go in" is no longer a single direction in space-time, but points in a direction which is affected by the curvature of space-time. I would guess that "time", in this interpretation, is defined at a point in space-time as the direction vector of the axis of the light-cone emanating from that point. Near a black hole, all light-cones point into the black hole so severely that no light can escape it.
Unfortunately, I don't know the numbers or the deep concepts for this, only the pop science version :( Head asplode.
To a layman (like me) there is little difference between the concepts of "you would need an infinite amount of energy to" and "no finite amount of energy is enough to", as they sound like two ways of expressing the same thing. As you say, though, there's probably a difference, at least technically speaking.
Well, what is "an infinite amount of energy"? It is not anything. It is not a well-defined concept, because numbers, as we use them to describe quantities, measures, distances, etc., are only finite. So "you need an infinite amount of energy to [blah]" means absolutely nothing, because there is no such thing as an infinite amount of energy.
On the other hand, saying "no finite amount of energy is sufficient to [blah]" means precisely what it means. There is no quantity of energy sufficient to, say, accelerate a massive particle to the speed of light. Quantities, measures, distances, etc. are finite.
This is why I cautioned against conflating "the way things are" with "the way we describe things" earlier. Because we use complicated mathematical concepts that you don't understand to describe the way things are, you assume that other complicated mathematical ideas that you also don't understand (but seem somehow related, to you) have some bearing on the way things are, too. They don't. Infinity is not a thing, it is a mathematical concept that has no physical meaning.
When I wrote "which is what is happening in a singularity" I was implying "according to GR". (In other words, pure GR predicts that inside a black hole it's impossible for a particle to keep out of the singularity and hence the only possibility is for all matter to compress into a zero-sized point. Of course GR doesn't take into account that according to QM it's impossible for a singularity to happen, which is what causes the dilemma.)
Relativity doesn't make any predictions about matter in situations that result in a mathematical singularity in its models. Remember that a "singularity" is a feature of a mathematical model. It is a mathematical idea, not a thing. Physicists understand that relativity is an incomplete description of the universe and that it can't be used to describe situations that it can't describe.
You should also note that it has so far been impossible to test many features of relativity at small scales, so it is not known to apply at those small scales.
Warp wrote:
If I understand correctly, in the Schwarzschild solution this is because all geodesics inside the event horizon, including all time geodesics, point directly to the singularity, so simply advancing it time makes the particle move closer to the singularity. Basically you would have to stop time to make the particle not move towards the singularity. (Which raises the question: Assuming you could apply an infinite amount of energy to a particle, would it theoretically be possible to "stop time" for it, ie. make it stop moving in the time axis? Is this something allowed by the GR equations?)
The answer to that question is yes, but not for any deep physical reason. The reason it is yes is because a false premise implies anything and everything. That is, "false implies false" is true. "False implies true" is also true. So "if something false, then anything, anything at all is true" is true.
You started with a false premise, namely, "if you apply an infinite amount of energy to a particle". Is it true that that premise implies "it is possible to stop time for that particle, making it stop moving in the time axis"? Yes. "False implies anything" is true. That's entirely meaningless, though. There is no such thing as an infinite amount of energy.
"But what if you plug infinity into the physics equations related to whatever I'm talking about?" you ask? You can't. Infinity is not a number. You can't plug it into an equation. Infinity is a mathematical concept that has no physical meaning. There are some mathematical paradigms in which the concept of infinity is well-defined and can be plugged into equations, but the mathematics we use to describe the universe is not one of them. Infinity is not a thing, and there is no such thing as an infinite amount of anything.
When I wrote "which is what is happening in a singularity" I was implying "according to GR". (In other words, pure GR predicts that inside a black hole it's impossible for a particle to keep out of the singularity and hence the only possibility is for all matter to compress into a zero-sized point. Of course GR doesn't take into account that according to QM it's impossible for a singularity to happen, which is what causes the dilemma.)
Relativity doesn't make any predictions about matter in situations that result in a mathematical singularity in its models. Remember that a "singularity" is a feature of a mathematical model. It is a mathematical idea, not a thing. Physicists understand that relativity is an incomplete description of the universe and that it can't be used to describe situations that it can't describe.
I'm not sure I agree with that (but, as I said, I'm a complete layman in physics). The singularity is a prediction of the equations of GR. Whether a singularity happens in reality is a different story (if it doesn't, it simply means that the prediction is wrong, which it could very possibly be).
This is because, assuming GR was 100% correct and QM effects don't apply, it would be impossible for a mass to maintain a non-zero size inside the event horizon of a black hole. As said, simply moving in time would make particles move towards the center of gravity, which is a zero-sized point in the middle of a (non-rotating, non-charged) black hole (in rotating black holes it might not be a point, but it's zero-sized nevertheless, as far as I understand). Hence the only possible outcome of this is that the entire mass is compressed into a zero-sized point. Anything else is impossible. Thus it is a prediction of the GR equations. (Just because GR can't describe this infinitely dense point itself doesn't mean it's not predicting its existence.)
You should also note that it has so far been impossible to test many features of relativity at small scales, so it is not known to apply at those small scales.
I know, but that doesn't change whether GR predicts something or not.
Warp wrote:
If I understand correctly, in the Schwarzschild solution this is because all geodesics inside the event horizon, including all time geodesics, point directly to the singularity, so simply advancing it time makes the particle move closer to the singularity. Basically you would have to stop time to make the particle not move towards the singularity. (Which raises the question: Assuming you could apply an infinite amount of energy to a particle, would it theoretically be possible to "stop time" for it, ie. make it stop moving in the time axis? Is this something allowed by the GR equations?)
The answer to that question is yes, but not for any deep physical reason. The reason it is yes is because a false premise implies anything and everything. That is, "false implies false" is true. "False implies true" is also true. So "if something false, then anything, anything at all is true" is true.
I don't think there's need to nitpick here. But if you want to, then I'll rephrase the question:
Assume you could apply an unlimited amount of energy to a particle inside the event horizon of a black hole. Could it be used to slow down the time of the particle so much that it would delay its inevitable fall into the singularity by more than the lifetime of the black hole itself? Would the GR equations support/allow this to happen (at least in theory)?
I'm not sure I agree with that (but, as I said, I'm a complete layman in physics). The singularity is a prediction of the equations of GR. Whether a singularity happens in reality is a different story (if it doesn't, it simply means that the prediction is wrong, which it could very possibly be).
Relativity does not make predictions about the nature of matter, space, and time inside an event horizon. No scientific theory does. A theory is an idea or group of ideas that make testable predictions about the behavior of the universe under certain conditions. There is no such thing as a testable prediction concerning the interior of a black hole. Therefore no theories make predictions about the interior of a black hole.
People enamored with science fiction (many scientists included) like to talk about "what if" scenarios, e.g. "what if these particular theories held in this domain we cannot possibly know anything about, what could that mean?" But those are just flights of fancy. They are for fun, no one takes them seriously, and those that do are engaging in religion, not science.
This is because, assuming GR was 100% correct and QM effects don't apply ...
Stop right there. Your premise is false. False implies anything is true.
... it would be impossible for a mass to maintain a non-zero size inside the event horizon of a black hole. As said, simply moving in time would make particles move towards the center of gravity, which is a zero-sized point in the middle of a (non-rotating, non-charged) black hole (in rotating black holes it might not be a point, but it's zero-sized nevertheless, as far as I understand). Hence the only possible outcome of this is that the entire mass is compressed into a zero-sized point. Anything else is impossible. Thus it is a prediction of the GR equations. (Just because GR can't describe this infinitely dense point itself doesn't mean it's not predicting its existence.)
You should also note that it has so far been impossible to test many features of relativity at small scales, so it is not known to apply at those small scales.
I know, but that doesn't change whether GR predicts something or not.
I mentioned this because I hoped you would understand I was saying, "Even if relativity could make predictions about the nature of space, time, and matter inside the event horizon of a black hole (it can't), it is not known to apply on the scales that result in mathematical singularities in its mathematical models."
Remember, theories are ideas or groups of ideas that make testable predictions about the behavior of the universe under certain conditions. Can you use the equations describing the flow of shear-thickening fluids to make predictions about shear-thinning fluids? No. You can certainly plug in any numbers you want (perhaps numbers corresponding to some shear-thinning fluid) into the equations describing the flow of shear-thickening fluids, but that doesn't change the fact that the scientific theory applies to shear-thickening fluids, and not shear-thinning fluids.
Theories do not make predictions outside of the domains in which they make predictions. Relativity does not make predictions about the nature of space, time, and matter inside the event horizon of a black hole.
Warp wrote:
I don't think there's need to nitpick here. But if you want to, then I'll rephrase the question:
Assume you could apply an unlimited amount of energy to a particle inside the event horizon of a black hole. Could it be used to slow down the time of the particle so much that it would delay its inevitable fall into the singularity by more than the lifetime of the black hole itself? Would the GR equations support/allow this to happen (at least in theory)?
We are talking about precise, well-defined concepts, so I am not nitpicking, I am merely pointing out flaws in your representation of those precise, well-defined concepts. Your rephrased question is as meaningless as the original. There is no such thing as an unlimited amount of energy. Your premise is false. False implies anything is true.
If you want to ask a meaningful question, you have to have a meaningful premise. "Assuming this well-defined concept is true inside the event horizon of a black hole," (which is unknowable, but not intrinsically false), "and you perform this well-defined action, what do you suppose might happen?" Ask a question like that, and you might get an answer.
Relativity does not make predictions about the nature of matter, space, and time inside an event horizon. No scientific theory does. A theory is an idea or group of ideas that make testable predictions about the behavior of the universe under certain conditions. There is no such thing as a testable prediction concerning the interior of a black hole. Therefore no theories make predictions about the interior of a black hole.
Under my interpretation of the meanings of these words: GR makes predictions about the interior of a black hole, but it does not make testable predictions, assuming that information cannot escape a black hole.
Furthermore, any scientific theory which postulates that information can escape a black hole can make testable predictions about the interior of a black hole. If it turns out that information in fact cannot escape a black hole, as seems to be the case, those predictions will be falsified; but a scientific theory does not have to be true, it only has to be testable.
This is because, assuming GR was 100% correct and QM effects don't apply ...
Stop right there. Your premise is false. False implies anything is true.
This is not how logic works. You are confusing necessary truths (logical absolutes) with contingent truths (facts about the world). The laws of physics are not logical absolutes; assuming the negation of a physical law is not a logical contradiction; therefore, assuming the negation of a physical law does not imply arbitrary statements.
It is conceivable that there is a universe in which GR is 100% correct and QM effects do not apply, and we can perfectly consistently apply logic to describe that universe.
Warp's question is perfectly reasonable, logical, and meaningful. A paraphrased version could be: "Supposing we programmed a computer to simulate a universe in which GR applies but QM doesn't. Then would X be possible in that universe?" The fact that this universe isn't the same as the physical universe is no logical problem, in the same way that it's ok to reason about SM64 physics despite their violent disagreement with physical reality.
I'm not sure I agree with that (but, as I said, I'm a complete layman in physics). The singularity is a prediction of the equations of GR. Whether a singularity happens in reality is a different story (if it doesn't, it simply means that the prediction is wrong, which it could very possibly be).
Relativity does not make predictions about the nature of matter, space, and time inside an event horizon. No scientific theory does. A theory is an idea or group of ideas that make testable predictions about the behavior of the universe under certain conditions. There is no such thing as a testable prediction concerning the interior of a black hole. Therefore no theories make predictions about the interior of a black hole.
I'm getting the feeling that you are deliberately confusing things instead of acknowleding your understanding of what I'm saying.
What you are effectively saying above is that the Schwarzschild solution of the GR equations for a non-rotating black hole is wrong, and pure science fiction (although I'm sure you'll come up with some explanation of why it isn't, regardless of what you wrote above).
Of course that's obviously not so. The GR equations do describe the geometry of the interior or the black hole, and it's the very reason why we are talking about singularities in the first place and why we say things like "all timespace geodesics point towards the center of the black hole, and applying any energy to a particle inside the event horizon would only accelerate it towards the center". In other words, the GR equations predict what happens inside the event horizon. It's only the singularity itself where the division by zero happens, not the space between the singularity and the event horizon.
You are somehow (maybe deliberately) confusing these GR equations with the concept of "testable prediction" and claiming that since we cannot make any tests about the interior of the black hole (how do you know this?), GR does not predict anything about the interior of the black hole.
This is because, assuming GR was 100% correct and QM effects don't apply ...
Stop right there. Your premise is false. False implies anything is true.
Frankly, I'm getting tired of your nitpicking. You are avoiding the question.
Besides, exactly how do you know that my premise is false? You yourself said that we cannot make any tests about the inside of the event horizon of a black hole. How exactly, thus, can you say that the hypothesis that QM effects don't apply there is a false premise? Can you prove it somehow?
Nevertheless, I was asking purely from the point of view of GR. You can describe what GR says about the situation (even if you "know" somehow that the description does not correspond to reality). Refusing to do so is as silly as refusing to calculate anything using Newtonian physics because you know that they are not completely correct. If someone asks a question about Newtonian physics, would you refuse to answer because you know that the answer would not correspond 100% with reality?
I mentioned this because I hoped you would understand I was saying, "Even if relativity could make predictions about the nature of space, time, and matter inside the event horizon of a black hole (it can't), it is not known to apply on the scales that result in mathematical singularities in its mathematical models."
I was not asking what happens in the singularity. I was asking if it was possible to delay the falling of a particle into the singularity. You are extending the "division by zero" problem of the singularity to cover the entirety of space inside the black hole.
Your rephrased question is as meaningless as the original. There is no such thing as an unlimited amount of energy. Your premise is false. False implies anything is true.
Ok, it's now rather clear that you are simply avoiding the question rather than answering honestly "I don't know".
Do you also refuse to answer any questions about Newtonian physics because you know that they don't correspond to reality? Or do you make an exception in this case if you happen to know the answer?
Assume you could apply an unlimited amount of energy to a particle inside the event horizon of a black hole. Could it be used to slow down the time of the particle so much that it would delay its inevitable fall into the singularity by more than the lifetime of the black hole itself? Would the GR equations support/allow this to happen (at least in theory)?
No, for several reasons.
First: In pure GR, the lifetime of a black hole is infinite. The particle's time would have to stop for it to happen, which won't happen for any massive particle.
Second: inside the event horizon, all timelike and null geodesics end* at the singularity. For this reason, the singularity is called 'timelike': it is in the future of all geodesics, and is unavoidable.
Third: In GR, as well as in SR, the unaccelerated ("inertial") geodesics are those with maximal lapse of proper time. If you happen to fall into a black hole, you will live longer by not accelerating towards or away from it. With an unlimited amount of energy at your disposal, you will simply reach the singularity as fast as you want to. Of course, if anyone from outside the event horizon could see anything, the unaccelerated observer would reach the singularity faster.
The only "way" to escape from the singularity is to shift to a spacelike geodesic; this is as impossible in GR as it is in SR.
* Technically, there are several timelike and null geodesics that apparently "start" at the singularity and move away towards the event horizon, never reaching it but never falling back into the singularity. These geodesics can, however, be consistently interpreted as the analytic continuation of the geodesics of infalling particles from the outside: they "cross" the event horizon at "t = +infinite", "double-back in time" and fall into the singularity. Note the double-quotes, I am being deliberately imprecise here for the sake of clarity.
First: In pure GR, the lifetime of a black hole is infinite.
Yeah, you are right. I was thinking about the Hawking radiation, but that (if it really exists) would require quantum-mechanical effects, and I did put as a premise that we assume no QM.
The particle's time would have to stop for it to happen, which won't happen for any massive particle.
I was just wondering that since time is relative, if GR allows "slowing down" the time of the falling particle in some way so that its fall into the singularity would be delayed as much as possible.
GR allows sometimes rather surprising things, so it wouldn't be unthinkable (from a layman's point of view) that this could perhaps be possible, according to the GR equations.
(One example of a rather surprising thing that GR allows, something which many people are not willing to believe, is that the distance between two points in space, eg. the distance between two particles, can grow faster than c. Of course this doesn't mean that FTL travel is possible, only that the distance between two points can grow faster than c. If this happens, it effectively creates a horizon such that it's impossible to observe the other point from the first point. This is what eg. the so-called cosmological horizon is all about. IIRC GR also predicts the phenomenon to happen near the surface of a rotating black hole due to so-called "frame-dragging".)
I tire of arguing and it's clear Warp has no desire to learn how to speak precisely about precise concepts, and everyone else just encourages him, for example marzojr with his talk of having "an unlimited amount of energy at your disposal," which is a meaningless concept and distinct from having "any amount of energy at your disposal."
I could go on but I won't because I really don't enjoy arguing with people. As always you all may have the last word, as I'm done, have fun.
Third: In GR, as well as in SR, the unaccelerated ("inertial") geodesics are those with maximal lapse of proper time. If you happen to fall into a black hole, you will live longer by not accelerating towards or away from it. With an unlimited amount of energy at your disposal, you will simply reach the singularity as fast as you want to. Of course, if anyone from outside the event horizon could see anything, the unaccelerated observer would reach the singularity faster.
I tire of arguing and it's clear Warp has no desire to learn how to speak precisely about precise concepts, and everyone else just encourages him, for example marzojr with his talk of having "an unlimited amount of energy at your disposal," which is a meaningless concept and distinct from having "any amount of energy at your disposal."
I honestly can't understand your point.
Assume that someone asked the question: "What is the computational complexity of merge sort?" Then someone answered: "That question is meaningless because calculating computational complexities always assumes an unlimited amount of RAM, and there's no such thing as an unlimited amount of RAM", and outright refused to give an answer to the question. What would you think of that?
That's right. That answer is nitpicking and pointless. Even though calculating the computational complexity of an algorithm always assumes an unlimited amount of RAM, that doesn't make the calculation useless. Algorithm computational complexity is a very useful concept even in real-life situations (where there is always a physical limit to the amount of available RAM).
Assume that someone asked a question like: "Let's assume that you drop a small spherical object in vacuum [...]" and then someone answered "let me stop you right there; there's no such thing as a perfect vacuum, nor is there such a thing as a perfect sphere, the question is meaningless, and I refuse to answer it." What would you think of that?
That's right. Hairsplitting and nitpicking. And completely pointless. You can calculate the falling time of an object assuming a perfect vacuum, and the result you get is not meaningless. In fact, it can have real-life practical uses even if we make assumptions which are not 100% correct (such as assuming Newtonian physics and perfect vacuum).
Now, assume that someone asked a question like: "Let's assume that QM effects are not in effect, what do the GR equations say of this situation?" and then someone answered: "That question is meaningless, I refuse to answer."
See the point?
Assume that someone asked the question: "What is the computational complexity of merge sort?" Then someone answered: "That question is meaningless because calculating computational complexities always assumes an unlimited amount of RAM, and there's no such thing as an unlimited amount of RAM" [..]
[nitpick]
no, it doesn't assume unlimited ram. Since each computational step can only write to O(1) memory locations, memory complexity is always bounded by computational complexity.
On the other hand it's trivial to find an algorithm that uses no more than O(1) memory, but does not have a computational bound, even with 100% chance of termination.
[/nitpick]
no, it doesn't assume unlimited ram. Since each computational step can only write to O(1) memory locations, memory complexity is always bounded by computational complexity.
Computational complexities always assume an unlimited amount of input and an unlimited amount of available RAM. This is because if you set an upper limit to either one, your (terminating) algorithm becomes O(1): There's a fixed amount of steps that the algorithm will take at most, in other words, it's an O(1) algorithm. The algorithm becomes larger than O(1) only if there's no assumed upper limit.
Of course saying "all algorithms are O(1) in a physical computer because the available RAM is limited" is not a very useful statement. The computational complexity which you get when you assume unlimited RAM is useful information even in a limited setting.
On the other hand it's trivial to find an algorithm that uses no more than O(1) memory, but does not have a computational bound, even with 100% chance of termination.
If the amount of memory is fixed, there's a finite amount of states that the program can be in, and hence there's a finite, fixed amount of computations it can perform at most. Hence there's an upper limit to the amount of steps it can take, making it O(1) (basically, "any input will cause the algorithm to perfom at most n steps, regardless of what that input is" where 'n' is a constant, which is kind of the definition of O(1).) If the algorithm performs more steps than this, it will inevitably be a non-terminating algorithm.
can we talk about physics again please?
to people who have studied university-level physics: what is the hardest part of physics to get your head around? I have heard GR and particle physics as being the hardest; did you find this or was there something you personally found much more difficult?
I did 1 year of physics as part of my compsci degree -- personally, out of special relativity, thermodynamics, quantum (schrodinger equation), and oscillations (normal modes etc), I found thermodynamics by far the hardest to get my head around. The Boltzmann distribution for finite or infinite states, and the concept of entropy, were just too much for me. Special relativity was easy by comparison.
There's one thing that it's not at all clear to me about the Big Bang theory, and I can't find an answer to it.
There was a time in the beginning of the Universe when all the energy in the Universe was inside its own Schwarzschild radius. This would mean that the energy should have been incapable of escaping that radius, and instead collapsed back to a singularity (or whatever is happening inside a black hole). However, it expanded beyond this radius nevertheless. I don't understand how.
I understand that General Relativy allows the distance between two points in space (and consequently the distance between two particles) to grow faster than c, and hence it perfectly allows for superluminal expansion of the Universe (which is ostensibly still happening today, as the observable Universe is, as far as we know, smaller than the entire Universe), which is the currently established hypothesis of the first moments of the Universe (ie. the Universe expanded at an exponential rate at the very beginning).
I don't know, however, if or how superluminal expansion explains the energy of the universe expanding beyond its own Schwarzschild radius at the beginning. Is this the reason, or is it something else?
There's one thing that it's not at all clear to me about the Big Bang theory, and I can't find an answer to it.
There was a time in the beginning of the Universe when all the energy in the Universe was inside its own Schwarzschild radius. This would mean that the energy should have been incapable of escaping that radius, and instead collapsed back to a singularity (or whatever is happening inside a black hole). However, it expanded beyond this radius nevertheless. I don't understand how.
That is an interesting question, one I hadn't thought of. I will try to give a better answer than just "it comes out of solving the Einstein equation", but my physical intuition for this case isn't very good, so take it with a grain of salt.
The problem is that you are considering the universe from the outside, as a spectator of an expanding ball of matter, with vacuum outside. If that were the situation, the universe would indeed be inside its own event horizon. But the actual Big Bang model is about a very homogeneous gas filling everything, so there is no outside point from which to observe this.
Imagine that you are some observer in the very dense gas during the early phases of the big bang. You consider some spherical region somewhere, calculate its mass compared with its radius, and conclude that it would be inside a deep gravitational well. And if you let the radius be big enough, you might conclude that this region should be a black hole.
But this is wrong. You could make the same argument for any spherical region, *including the one centered on you*. I.e. you and every other point is equally deep into the gravitational field as the region you first considered.
Time moves slower the the deeper in a gravitational well you are, and near the event horizon, it moves infinitely slow relative to some point further away. But since you are just as far down in the field as every other point, your time is just as slow as theirs, and everything seems normal to you.
What would an outside observer see? Well, you can't be outside the whole universe, so let us modify the situation a bit: A huge region of the universe is filled by a hot, dense, homogeneous gas like in the Big Bang, but it is surrounded by a large area of vacuum (unlike the Big Bang). In this case, the inside observer would still see mostly the same situation unless he is near the edge. This time, an outside observer is possible, and he would indeed see the part of the universe with all the gas as being inside a black hole - no information from inside this region would be able to reach him, as long as the gas is sufficiently dense and covers a large enough area.
By the way: In the discussion above, I neglected the expansion of the universe. If you solve things properly, and take this into account, you will find that every point in the universe won't be equally far into the gravitational well as yourself after all - every observer will see himself in the center of a gravitational field which points away from him and which grows stronger with distance - it is like he is inside an inverted black hole. No matter which direction he looks in, he sees an event horizon far away. This corresponds to the point where space is expanding away from him at the speed of light. Nothing further away can reach him, as the expansion is carrying it away faster than it can approach.
I hope this helped (and that it was correct).
The problem is that you are considering the universe from the outside, as a spectator of an expanding ball of matter, with vacuum outside. If that were the situation, the universe would indeed be inside its own event horizon. But the actual Big Bang model is about a very homogeneous gas filling everything, so there is no outside point from which to observe this.
You don't need to be "outside" the Universe (if that concept is even possible) for the problem to happen. Just take a portion of the Universe at its initial stages. The majority of the energy in the Universe was certainly within its own Schwarzschild radius even while the Universe itself was already bigger than that (even if by a small margin).
Besides, I don't think you need any external observer for a mass to collapse because of reaching the critical density needed for that to happen. It's not like the external observer somehow triggers the collapse due to the observation.
Imagine that you are some observer in the very dense gas during the early phases of the big bang. You consider some spherical region somewhere, calculate its mass compared with its radius, and conclude that it would be inside a deep gravitational well. And if you let the radius be big enough, you might conclude that this region should be a black hole.
But this is wrong. You could make the same argument for any spherical region, *including the one centered on you*. I.e. you and every other point is equally deep into the gravitational field as the region you first considered.
It's a question of density. The density of my body is not even nearly large enough for it to collapse into a black hole (even though sometimes I can be pretty dense, ha ha!) There's a critical density to every body of mass, that if this critical density is reached, it will collapse. More precisely, the density needs to be such that all the mass is within the Schwarzschild radius of the object (this radius being determined by the mass of the object).
At the beginning of the Universe this density was certainly large enough (because all of the energy in the Universe was compressed into a sphere smaller than the Schwarzschild radius, and consequently the density of the matter was certainly beyond the critical limit).
Time moves slower the the deeper in a gravitational well you are, and near the event horizon, it moves infinitely slow relative to some point further away. But since you are just as far down in the field as every other point, your time is just as slow as theirs, and everything seems normal to you.
I don't understand what this has to do with it. As said, it's just a question of density.
The problem is that you are considering the universe from the outside, as a spectator of an expanding ball of matter, with vacuum outside. If that were the situation, the universe would indeed be inside its own event horizon. But the actual Big Bang model is about a very homogeneous gas filling everything, so there is no outside point from which to observe this.
You don't need to be "outside" the Universe (if that concept is even possible) for the problem to happen. Just take a portion of the Universe at its initial stages. The majority of the energy in the Universe was certainly within its own Schwarzschild radius even while the Universe itself was already bigger than that (even if by a small margin).
Yes, you could easily find regions inside their own Scwharzchild radius in the early universe. The problem is what that implies. It does not imply that each of these regions will quickly collapse into a point.
There are two important effects involved here: The first is gravitational acceleration. This cancels out due to the homogenity - every direction pulls as much as every other direction. The gravitational acceleration has no local effect in this case, but globally it acts to slow down the expansion.
The other effect is gravitational time dialation. The further into a gravitational field, the slower time moves. But again, this is the same for all points, so it has no effect.
Warp wrote:
Besides, I don't think you need any external observer for a mass to collapse because of reaching the critical density needed for that to happen. It's not like the external observer somehow triggers the collapse due to the observation.
Right, the observer does not trigger anything, things do not need to be observed in order to happen. There is one true space-time, but different observers slice it into space and time in different ways, due to the relativity of simultaneity. In the case of strong gravitational fields, this effect is extreme. For example, an observer far away from a black hole will observe that an infalling particle uses an infinite about of time to fall into the black hole, due to the strong time dialation. On the other hand, an infalling observer finds the same fall to take a short time. Even though they are describing the same underlying process.
Furthermore, it is possible for a region to be expanding on the inside, but contracting on the outside. Consider a sub-balloon expanding from the side of another balloon, with the sub-balloon itself growing bigger, but the contact point shrinking. Depending on the configuration, this could look like a black hole from the outside, even though it is expanding on the inside. Another example would be a house shrinking on the outside but growing larger on the inside.
The point of this digression was that even though observers don't cause things to happen, you still need to take care which observer you are making predictions for. Specifically, just like "will fall to the left" and "will fall to the right" depend on the motion of the observer, "will proceed quickly" and "will proceed slowly" also depend on the observer's position in the gravitational field, and even things like "will collapse" and "will expand" can be relative to the observer, as in the example above.
Yes, you could easily find regions inside their own Scwharzchild radius in the early universe. The problem is what that implies. It does not imply that each of these regions will quickly collapse into a point.
I thought that's exactly what GR predicts. Because of the spacetime geometry inside the event horizon, particles have no other direction to go than the singularity at the center. (Even moving forward in time makes the particle go towards the singularity.)
There are two important effects involved here: The first is gravitational acceleration. This cancels out due to the homogenity - every direction pulls as much as every other direction.
I don't buy that without further explanation. Total gravity always pulls towards the center of mass of the object/gaseous substance/whatever. For example, any point on or inside the Sun will have a gravitation towards the center of the Sun, because that's where the total gravitation points to (and is the reason why stars collapse inwards when their fusion reaction is too weak to maintain the shape of the star).
Even if at the beginning the entirety of the energy in the universe was totally evenly distributed, the total gravity would nevertheless still point to the center of mass of all this energy. (Of course I'm a complete layman in physics, so there may well be something I'm not understanding here.)
The other effect is gravitational time dialation. The further into a gravitational field, the slower time moves. But again, this is the same for all points, so it has no effect.
I don't quite understand that, but "gravitational time dilation" gave me an idea for the reason why the Universe did not simply collapse right back into a singularity: It expanded faster than gravity waves could move (AFAIK gravity does not propagate instantaneously, but at velocity c or something along the lines), and hence an event horizon did not have time to form, as the energy escaped faster than the gravity well could propagate.
That actually makes sense.
It seems that black holes can be characterized by three (and only three) quantities: its mass, angular momentum and electric charge. For example the Reissner-Nordström metric is a solution to the general relativity equations for a charged, non-rotating black hole.
I don't understand. Electric charge is mediated by photons. Photons cannot escape a black hole. How can they mediate anything in this case? A black hole swallows photons, it doesn't exchange them with anything.
While I had heard of virtual particles numerous times before, I had never actually studied any details, so this was new information to me.
I suppose the next question would be: Why aren't virtual particles bound to the same laws and limitations as regular particles? Why is a virtual photon able to interact from the inside of a black hole's event horizon with the outside, seemingly unhindered by the geometry of spacetime?
To be quite frank, I think that this answer is terrible and should never be used. Ever.
Black holes, such as the Reissner-Nordström, are a feature of general relativity; virtual photons (and virtual particles in general) are a feature of quantum field theory. Trying to explain how general relativity black holes interact by using a different theory which is completely and thoroughly incompatible with general relativity at a fundamental level is not an explanation at all.
Moreover, the very idea of the eletromagnetic field (and other fields) being "mediated by photons" is alien to general relativity -- this idea also has its origins in quantum field theory. While the idea might be useful when doing "semi-quantum" gravity analysis, there is no need to use this when talking about general relativity proper. And who knows, the very notion of particle-mediated fields may turn out be wrong when an actual theory of quantum gravity is found.
The actual answer for the question is conservation of stress-energy-momentum: given the expression for electromagnetic energy in terms of the potential fields, you can deduce Maxwell's equations from the contracted Bianchi identities (see this for some information on these identities) on all but a few hyper-surfaces in which the electric and magnetic fields are orthogonal; and if you have Maxwell's equations, the Lorentz force can be deduced from those very same contracted Bianchi identities, regardless of the geometry of the situation.
This happens because electromagnetic fields warp space-time in very specific ways in general relativity, and the gravitational field itself is responsible for coupling the charges to the electromagnetic fields; this is easier to see in action in the Hilbert-Palatini variational principle formulation of general relativity.
