Having done so, I can no longer imagine a 'place' where there is no space.
Regardless of it's shape does the word 'surface' not denote a boundary?
No, not in the mathematical sense of the word 'surface'. When I refer to a surface, the surface is all that exists - there is no above or below. So the surface of the Earth is a 2-dimensional world that is finite but has no boundary. We call it a 2-sphere. If you are a 2d creature constrained to live in this 2-sphere, you can move forwards/backwards and left/right. If you go far enough you will arrive back where you started from without any deviation from your straight path.
The solid Earth is a called 3-ball - it is finite and bounded, and it is bounded by the 2-sphere surface.
A 3-sphere is a 3-dimensional space that is finite and unbounded. It is very difficult to visualise as we cannot build one in space. But if you lived in one, you could move in all three dimensions, and again, if you go far enough you will arrive back where you started from without any deviation from your straight path.
Just to add, a 1-sphere is a 1-dimensional finite unbounded space that similarly loops back on itself. We call them circles. The interior of the circle is the finite bounded 2-ball, or 2-disc. And again, the boundary of the 2-ball is the 1-sphere (i.e. the boundary of a disc is a circle)
So we exist in a universe that can have a finite space surrounded by an infinite border and infinite space surrounded by a finite border.
No! The Universe can be either finite or infinite, and it is without boundary. It could be infinite in all three spatial dimensions, in which case there is obviously no boundary, or it could be finite in all three spatial dimensions but wrapped into a 3-sphere - again, no boundary. It could even be wrapped up into a 3-torus, or something even more exotic, but in each case it is finite with *no* boundary.
Now, there could be boundaries where the Universe as we know it merges into something else, and we consider this in highly speculative theoretical physics... but that's for another day.
Regardless of it's shape does the word 'surface' not denote a boundary? On the surface or not on the surface. Above or below the surface.
No, not in the mathematical sense of the word 'surface'. When I refer to a surface, the surface is all that exists - there is no above or below.
Yes, the surface of the sphere represents a boundary between the inside and the outside of the sphere
So to clear up the controvercy, Percy is quite correct in that the surface of the sphere (2-sphere) *is* a boundary to the inside and outside of the sphere (3-ball).
BUT we can have the 2-sphere quite independent of any notion of there being a 3-ball to which it would be a boundary. In our everyday experience, this is not possible. Every 2-sphere has an "inside" - but mathematically, and when we are talking about the geometry/topology of space-time, this need not be the case.
AND you were actually asking the more general question of whether a general surface is a boundary between that "above" and that "below". In this case, we can have a finite unbounded 2-dimensional surface that does not form a boundary between an inside and an outside of a 3-dimensional space! This is the famous Klein Bottle, whose inside and outside are joined, yet there is no break in the bottle, no opening... imagine a surreally twisted sphere that somehow manages to join its inside to its outside without any break in the sphere! This is another situation where we cannot actually construct a real Klein Bottle in the boring flat 3d space in which we live.
Well, that's not a bad start. But you're in danger of thinking that the Universe has a bounding edge. And you need to start thinking more 4-dimensionally, seeing time as just another physical dimension. Do that, and many of your questions in the other thread will be answered or rendered obsolete.
It's better admit that the terms the Universe and expansion are not compatible. Your might as well talk about the colour of mass. Ex means out, the universe has no outside so it cannot expand for the same reason that mass cannot have colour.
And just to point out that what Alfred says is contradicted by every professional phsyicist and cosmologist working today, every cosmology textbook, and every cosmology paper being published. But Alfred think's he's right - so we just pat him on the head, nod encouragingly, and then wait for him to nod back to sleep again... shouldn't take long.
Sorry, Crankdriver, what I say is contradicted only by the majority of experts belonging to the same club you do and the textbooks written by the club members.
Yes. This club is usually known as "all the physics and cosmology departments in the world".
Those who contradict what I say cannot define the Universe or expansion in any consistent fashion for all the tea in China
We can. We do. We have done. A billion times since Einstein. It is just you who has not gained the ability to understand.
They are inept at defining their terms and their definitions are contradictory as a result so ultimately it's themselves and not me that they are contradicting.
Yes Alfred, every physicist and mathematician involved in relativity and cosmology is completely inept. All of us. And you are not. Alfred's world sounds like a nice play to be.
Those outside the club are having a good laugh at the whole thing together with my cat.
Your laughing? What, while it us to that takes all the grant money? Us, that get to build and play with the largest machines ever buily by man? Us, that are approached for our scientific and mathematical ability? Us, that after our scientific careers, are employed around the world in some of the highest paying jobs? Keep laughing Alfred... keep laughing
Do you want to say that the only requirements for a black hole is to be a light trapping region surrounded by an event horizon and that in your version of what the entity should be it may not have a singularity at its center?
I don't want to say that - that is simply the way it is. And we don't even need the event horizon, as we could never have the globally required god's-eye view to determine if it exists. An apparent horizon is good enough.
yet as you must know gravity is a relation of at least two bodies.
I certainly must not know, as it is blatently untrue. Depending on context, gravity is the reaction of the metric to itself and the stress-enegy distribution (essentially the mathematics of General Relativity); or it is the reaction of a test body to the metric (essentially the physics of General Relativity.)
Inverse square law is talking about mutual relation between two masses as a function of distance between them.
Yes, if we are talking about Newtonian gravity and mechanics. But as we are discussing black holes, we are actually talking about General Relativity and have left such basic concepts long behind.
Gravity seems to be crossing the horizon in either direction.
Gravity does not "cross" anything. The space-time metric has a particular form around the black hole, and external bodies will move according to that form. There is no gravity "force" at this level of discussion, and gravity cannot be "trapped" inside the horizon - although one could look at propagating fluctuations in the metric (gravitational waves) and these are indeed trapped.
Actually, an almost identical situation occurs with the electromagnetic field, where propagating fluctuations are trapped (i.e. light cannot escape), but static field configurations can straddle the horizon as we see with electrically and magnetically charged (Reissner-Nordstrom) black holes.
When they define event horizon, they talk also about information going in one direction only. Does your theory hold that gravity of a black does not represent any information passed to the galaxy spinning around it?
Information does become trapped until the point where the black hole evaporates away. But we are now moving through semi-classical relativity into quantum gravity, and this certainly does not belong in a discussion that is still clinging to Newtonian concepts and terminology.
So, Crankdriver, you say that gravity is a reaction of metric to itself and to stress-energy tensor. Metric is a map, the tensor is the vectors or markers on that map. Are you telling the cat that gravity is map playing with itself with no territory needed to be present in the relation?
Couldn't put it better myself. Ok, I could, but it's close enough.
Are you aware that Einstein himself held black holes to be a bunch of superstitious nonsense?
That would be rather incredible if true, given that Einstein died in '55 and John (Wheeler) didn't coin the term Black Hole until '67 (although there are reports of the term as early as '64)
In 1939, when Einstein wrote the paper of which you are so fond, it had only recently been appreciated that the Schwarzschild "singularity" (at r=2M in geometric units) was not actually a real singularity, but merely a coordinate singularity (such as r=0 in polar coordinates) and later became known as the Schwarzschild radius. Oppenheimer with his student Snyder started looking at collapse to and through the Schwarzschild radius, but Einstein was dubious as to such an extreme area existing physically.
Einstein attacks the problem by building the mass for his Schwarzschild solution out of orbiting bodies. He discovers that even before the Schwarzschild radius could be reached, his orbiting bodies were being forced to orbit at the speed of light, which of course is impossible. He reasons that this behaviour amongst others will prevent collapse down to the radii in the region of the Schwarzschild radius.
His error is assuming that he can build such a mass out of orbiting bodies in the vicinity of the Schwarzschild radius. It is not possible as there are no stable orbits for massive objects close to the Schwarzschild radius. So his reasoning is incomplete and incorrect.