For me, the interesting case is a black hole. I’ve heard it said that time essentially stops at the event horizon, which I gather means space becomes infinitely prolonged, so isn’t the radius of a black hole infinity? I think Asimov suggested that this means the inside of a black hole is as much a vacuum as anyplace else. So I guess a black hole qualifies as a hollow sphere! :smack:
The mass of a black hole can be considered to be present at the centre of the black hole and the inner region of the event horizon can contain infalling matter.
Time ‘stops’ at the event horizon from the pov of a far observer, for an infalling observer may not even notice that they’ve passed the event horizon.
An observer accelrating with constant extrinsic acceleration relative to some inertial obsrever in flat spacetime will observe something called the Rindler horizon. This has simlair properties to the event horizon of a balck hole.
As the existance of the Rindler horizon suggests many of the properties of the event horizon are a result of the choice of particular spacetime coordinates. systems.
How did this get on here? I canceled the whole post and never sent anything!
It sounds like you’re saying the properties of a physical event depend on the math you use to describe it. Does that mean we’re not really talking about a physical event, but a mathematical one, which might or might not correspond well with the physical event it attempts to describe?
I’m not being critical, I’m just afraid my bringing in black holes has taken the topic into a realm where there is no straight dope, only conflicting mathematical models. Have we entered THE CURVED DOPE?
No, the problem is that the mathematics involved don’t correspond to everyday experiences here on earth. That doesn’t make the description invalid. It’s just the the description and the interpretation will vary for different observers in different reference frames, according to Einstein. Each is equally true and equally valid.
Time never stops for anybody anywhere. Some events may appear to take infinitely long times to certain observers - and they actually do in that reference frame. Infinite time is the opposite of stopped time, however. In every reference frame, including that of a theoretical observer falling into a black hole, time passes at exactly one second per second.
The radius of a black hole is never infinity. It can be precisely measured by the size of its event horizon. (The singularity at the center of a black hole, assuming there is one - theories vary - may or may not be point sized, but the usual depiction of the size of a black hole, and the definition you’re using, is that of the event horizon radius.) I’m sure Asimov never said that the inside of a black hole is a vacuum. That’s not true. It can’t be true. The definition of a black hole is of a certain amount of mass in a small enough radius, so a vacuum is the last possible way to describe one.
I’m afraid it’s just your understanding of black holes that is totally faulty, not the math.
I’m not saying the various descriptions are not true or not valid. It just seems awkward to have to keep shifting frames to describe something. If I had devoted my life to math instead of whatever it is I do (I forget), I think I would be trying to find a frame of reference that lets me think about everything all at once. (Where is the “brain exploding” smily?)
The equations of general relativity describe spacetime and spacetime is the same for all obserevers. However people like to think in terms of space and time and to get space and time from spacetime you need to impose some sort of coordinate system on spacetime.
Different coordinate systems will lead to different spaces and different tiems for different observers. We see the effect of this in special reativity in time dilation and length contraction.
The problems in general relativity are greater, because whilst in special rleativity there’s always obvious global coordinate systems (i.e. coordinate systems that describe the whole of spacetime), in general relativity there are only obvious local coordinate systems (i.e. cooridnatesystems that only describe certain regions of spacetime.
In Schwarzchild cooridnates the event horizon is a cooridinate singularity (a point where the coordinate system breaks down), just as in Rindler coordinates the Rindler horizon is a coordinate singualrity.
Even by choosing what is seemingly a very sesnible coordinate system (i.e. Schwarzchild cooridnates) for Schwarzchild spacetime we find it is not quite a general coordinate system as it cannot describe the region of spacetime inside the event horizon.
Schwarzchild coordinates are an extended cooridinate system that essesntially describe Schwarchild spacetime from the pov of a far away (far away from the black hole that is) observer. If we build a coordinate system around an infalling observer we will a) get a different view of space and time b) we can describe the region inside the event horizon.
Too many dopers crowded into the thread at the same time - they collapsed into one very small, incredibly dense doper, which pulled the post irretrievably out of your grasp and fixed it into the thread.
Ooh, ooh, ooh, I think I know the name of that incredibly dense doper. 
Thanks, Pants. What you say makes sense to me, even if only as a dream makes sense until you wake up.
But surely it’s the coordinate systems that break down at their respective singularities, and not spacetime? In other words, as an observer approaches a black hole from far away and then falls on in, if he weren’t vaporized or otherwise distracted by all the other crap falling in, he wouldn’t know when he fell out of one frame of reference and into another, would he?
I gather all maths leak (mashing Sapir into Gödel in a doubtless silly way). But I wonder if there’s an undiscovered general system around the back somewhere that can describe the whole black hole thing all at once. That’s all. Call it wishful thinking.
Oh, sorry, you answered that. General relativity. So the secret is to grasp spacetime as itself, and not separately as space and time? Can that be done? By a doper of at least middling-incredible density?
He doesn’t. He stays in his same reference frame all the time. Forever. Every second of his life from birth. He can’t help it.
He is also in different reference frames from others not falling into the black hole every second of the way. He can’t help that either.
It’s not wishful thinking. It’s bad thinking. You just have to accept reference frames and go on from there.
I misspoke. Say coordinate system. So yes, he experiences the same reference frame all the way down, but it sounds like Pants is saying that to describe what’s happening to him we have to change coordinate systems somewhere along the way.
What I’m getting at is mathematics have limits, and it’s not always easy (from my frame of reference) to tell if a limit that a mathematician is taking for granted corresponds to a real live physical limit or a mere numerical discontinuity that the universe can casually shrug off.
Things like this…
…set off a little alarm in my head that says “simulation! simulation!,” and I wonder if reality is perhaps a little more complicated (or simple) than I’m being told.
I had heard that mathematics hadn’t yet quite fully described everything to everyone’s satisfaction. If you tell me it has, well, then I guess I can rest assured.
Finally!, my gravitational gradients show up. I’d have to ask the math guys for confirmation, but I don’t think your observer is still observing by the time he gets to the event horizon. Long before the gravitational force becomes so strong that light can’t get out, the structural limitations of the human body would show up. Inverse square says the closer you get, the harder it pulls. The deeper you fall into the gravity well the greater the difference in gravity between any point and another point, lets say 6 feet closer to the source. With 6 feet of difference between his head and his feet your observer will start to feel his spine stretch as his head is subjected to more gravitational force than his feet (he’s an observer, I’m assuming he’s going in head first). Eventually the force will exceed the tensile strength of the human body and what actually enters the event horizon would be a very long, very thin stream of atoms.
I was just reading, I swear! I wasn’t even going to post!
RR
Well, yeah. This is the basis of a zillion science fiction stories. And everybody in this thread has acknowledged this.
And everybody in the thread has also said that the effect when approaching a black hole is a different thing - mathematically and conceptually - than the one you’ve been confusing it with, the effect of gravity at the center of a sphere. We’re all aware of it. Our frustration has been that you keep dragging it in where it doesn’t belong.
It’s also important to understand that when physicists talk of what happens at an event horizon, they usually are conducting thought experiments of idealized bodies that won’t instantly turn to spaghetti. They may use a human for illustration, but they know perfectly well that a human wouldn’t survive the transition.* It doesn’t matter to them. They’re talking about the mathematical principles involved separate from an individual example of messy reality.
*Even this can be misunderstood. A massive enough black hole would have an event horizon large enough so that the gradient might not tear apart a body. A small enough black hole might have an event horizon that would affect only a body cell. People keep talking about black holes as if they were a particular thing rather than a shorthand description for a class of objects. Black holes could theoretically be submicroscopic or the size of the observable universe.
Thatis absolutely correct, only the coordinate system breaks down at a coordinate singularity.
The far observer never falls into the black hole, he always stays far away. So that the cooridnate system we base around him does not say what’s going on in the region bounded by the event horizon is not such a problem.
The infalling obseresver does fall in, so any coordinate system based around him, must be able to describe at least part of this region.
It’s the fact that spacetime can be curved in general relativity stops you from having a general schema to define a general coordinate system in relativty. Unlike special relativity where the spacetime is flat.
With a mass of…about the size of the observable universe :eek:
And special relativity omits gravity, so it can’t be used to describe black holes?
So, to sum up the situation as I see it:
[ul]
[li]Gravity (mass) and acceleration due to gravity are different things.[/li]
[li]The (general) relativistic effect of gravity increases as you approach the center of a mass, regardless of your position relative to the surface (geoid).[/li]
[li]The accelerative effect of gravity decreases (cancels out) as you descend below the surface of a mass. The acceleration cancels out entirely at the center of a solid mass or anywhere inside a hollow sphere (but where did you find one?).[/li]
[li]Black holes are weird.[/li]
[li]Cecil was pretty much right in both the articles to which this thread might refer.[/li][/ul]
I have more questions, but you guys have had enough of me by now!
Going wa-ay back here with a totally minor nitpick.
The gravity potential at the center of the Earth where the potential is smallest, i.e. at a minimum. When you climb up stairs, a mountain, whatever, it takes energy. That energy is going to increasing your potential energy.
Now of course, as long as you are consistent everywhere, you really can define potential positive or negative (like the way heat, Q, is defined with opposite sign in thermodynamics depending on the discipline), but AFAIK, a gravity potential is pretty much always defined as increasing as you go “higher.” (Also when you take the gradient of the potential to get the force field, the signs work out right.)