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.
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.
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.
rhebus and marzojr gave great explanations. I'll make a minor addition by pointing out that it's a bit naive to say things like "you would need an infinite amount of energy to make two particles be at the same place at the same time (which is what is happening in a singularity.)"
First of all, no scientist, when being precise, would ever talk about "an infinite amount of energy" or some such. For example, you've probably heard laypeople say things like "if you had an infinite amount of energy you could accelerate a spaceship to the speed of light." That's not a meaningful statement because there is no such thing as an infinite amount of energy, and it certainly doesn't follow from relativity. What relativity tells us is that no finite amount of energy is sufficient to accelerate a massive object to the speed of light. That certainly doesn't imply there is such a thing as "infinite energy."
Second, it's dangerous to conflate to such a degree "the way things are" with "the way we describe things." As rhebus pointed out, GTR and QM are two ways we have of describing the universe, though they are both neither complete nor even comparable in many meaningful senses. We use them to describe different portions of the universe, and they aren't often useful (or even meaningful) outside of their limited domains.
When physicists talk of singularities and such, it doesn't actually mean they believe there are precise physical manifestations corresponding to the ways mathematical models fail. That is, a "singularity" hidden by the event horizon of a black hole isn't a "thing" as far as anyone can know. It's just a situation where scientists admit our best methods of describing the universe are not sufficient. So to say "[something] is what is happening in a singularity" is wrong simply because it's meaningless. Nothing is known to happen in a singularity because it's not a thing, and our current methods of describing the universe are not sufficient to describe situations that, when they arise, result in mathematical singularities in our models.
It's going on right now and the organizers have been kind enough to put up a really nice quality live stream for our enjoyment: http://konkurs.chopin.pl/en/edition/xvi/online/broadcasting
Click to listen to great piano music from some of the world's best young talent!
On the cereal tip, I watched the entire movie. I'm pretty aggressive about just shutting off movies I don't enjoy and then refraining from voting (I never vote no) ... so I guess I must have liked this movie? I wasn't sure, but I voted yes anyway.
No matter how many people you dispatch with one quick headshot from a sniper rifle a mile away, it won't ever compare beating a man to death with your bare hands.
So I've heard.
I like how all the player controlled characters are like ...
*In a super hurry, running at ~15mph*
*Ponderously juuuuuuuuuuuuuump at ~2mph*
Then again, it was a long, long time ago. I guess hurdling technique wasn't that advanced back then. I bet they'd shit their pants if they saw a Fosbury flop.
Oh, no, sorry, each image is generated from an average of every screenshot from a system. So it's a bunch of games. Not just one. I meant guess the system, not guess the game.
Averaging all the frames from one game would be pretty cool and a fun little quiz. Some games (like Zelda 1) would work better if you could somehow leave off the ubiquitous menu display at the top.
How would one go about doing such a thing if they wanted to?
I can only imagine the snob moue on thundrio's face whenever he sees someone using an emulator in a FORBIDDEN way ... such as to have a little fun.
FUN FACT: "snob moue" is an anagram for "emunoobs"!!
What is it with this shiny ponyta? Is it like the infinity+1 sword of the polemon franchise or something? Does catching it entitle to you to a 1-million dollar cash price? Or is it just for the glory and babes?
The hooves of Ponyta are ten times harder than diamond. I trust this ends the matter.
Other than the obvious choices of amassing wealth and undoing major mistakes, I bet a lot of people would do really bad things just for the experience, and then rewind so they never happened except in your memory.
The real question is, is that immoral?
"So what do we do now"
"You have to finish."
"What?!"
I have known so many of these spoiled, entitled chicks that think the world owes them something. I can only imagine what would happen in her fantasy land ... "Stop the show! She took a watermelon to the face! Her team wins the grand prize, sorry everyone else!"
I can't help but laugh at her attention-seeking comment "I have the worst headache ever," which is almost certainly an outright lie. Her face was probably a bit numb on the side she got hit, but headaches brought on by trauma to the head don't develop that quickly. (They usually take hours to days to develop.) Also, considering the watermelon didn't even break her nose (granted, it hit the side of her face), it seems pretty unlikely that it was any worse than running your face into an airbag at 40+ mph. Surprising, perhaps, and painful to the dermis, but hardly head trauma.
Perhaps a better explanation is: every employee needs a boss. A boss can handle at most 10 people below him/her. Every boss needs another boss, unless s/he is the head of the organization. A boss can only handle bosses below him/her of the same rank.
You mean to say, "A group of 2 or more employees of level n requires an employee of level n+1 to manage them, denoted their boss. A boss may manage at most 10 employees."
Anyway, a slight variation of the equation I posted above suffices for your corrected problem. I leave that as an exercise for the reader :p
2) Assume that a company has X working employees, excluding management. For every 10 employees, a boss is needed to supervise that group. For every 10 bosses on that level, another boss is needed.
For example, for X=100, 11 bosses are needed. For X=101, 14 bosses are needed.
What is a good function for calculating how many bosses are needed for any X number of employees?
What? You should show your derivation for that. One of us is confused because f(101)=f(100) unless f(1)=1 (which I don't think is the case, though your statement of the problem is confusing), but even if f(1)=1 then f(101)=12, not 14.
Anyway, I will first state my understanding of your question in a rigorous fashion so you at least know what question I am answering, even if I don't know what question you are asking:
A company has x level 0 employees. Small numbers of employees are able to manage themselves, however, for each 10 employees at level n, an employee of level n+1 is required to manage them. For convenience, denote any employee of level 1 or greater a boss. What is an equation for the number of bosses a company of x level 0 employees requires?
Ok. Assuming f(1)=0, ignoring your claim that f(101)=14, the following will do:
It may not look pretty but it's quite painless to calculate even for huge numbers as the number of summands grows verrrrry slowwwwwwly. Randil's solution is close but not quite right ... maybe ... depending on what you are actually asking.
It may not look pretty but it's quite painless to calculate even for huge numbers as the number of summands grows verrrrry slowwwwwwly. Randil's solution is close but not quite right ... maybe ... depending on what you are actually asking.