I've been told that scientists are now accepting that the singularity that caused the big bang was the size of the universe? Is this true? And if so how can they justify it?
It would seem that for the singularity to be the size of the universe, then the universe would have to be a static model.
I would argue that it would not be able to expand as that would breach the limit set by the singularity. It cannot be more than what it was to stay within the parameters of the model.
And this creation of space-time at the moment of the BB I find dubious.
To have creation one must have something to create it in.
I still feel the assumption of an unstable 2dimensional (no depth) model is a fair one.
There are three assumptions to presume though..
1)space-time is infinite and has always been. There is no beginning and end
2)Energy and matter are finite. There is only so much.
3)Space-time is only 3dimensional(ok.. 4 with time) for the sake of this universe.

I visualise space-time populated with 2D universes, and very unstable they are indeed. Yes, a bit presumptious I know, but the fact our universe is not and stable would suggest that it is because it works.
In this vastness where the universes lie - and as this vastness is infinite, size is irrelevant - there are small pockets of energy that whizz around like meteorites.
Being infinite, these small pockets of energy rarely go anywhere near these 2D universes, but very rarely a fizzing ball hits a one.
Now the outcome of such an impact is filled with variables..
How strong is the ball of energy? How did it hit it? How stable or weak is this universe at this particular time?

With the pocket of energy however there are certain required variables..
It has to be very small.. as to break through the boundary would require a tremendous amount of energy. So less surface area is desirable for a successful penetration.
Then it would also have to have a huge amount of mass to back up the lack of size so it doesn't 'bounce off'.

I would theorise that many bounce off or splatter harmlessly across the boundary, but one small enough and dense enough hitting dead on could break through.
Then we have two things that occur..
Firstly, the 2D universe is forced into a 3D universe as the energy ball cannot exist in a 2D universe, and wins the argument blatently on brute force.
This expansion of the universe to allow the energy pocket to exist leads to a huge release of energy and heat, which would be focused at the rupture where the energy pocket entered.
Therefore, a singularity would have the requirements of size and density, while the eruption of matter and energy would be the universe inflating somewhat like a balloon.
Hence - a big bang!

The trouble I have is defining what the space-time continuum that the 2D universes and the energy pockets exist in would be like.
It cannot be 2D as the energy pockets exist and yet to be 3D would cause problems with the existance of 2D
universes that exist in it.
I could surmise that its just an alternate dimension, but that path is lined with thorns and nettles
Open to suggestions and why I'm barking up the wrong tree

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They say the more complicated something is the easier it is to put a spanner in the works - there is nothing more complicated than the human brain...(Infinite Horizons)

Take a balloon and blow it up. No matter how much air you put into it, the balloon is the size of the balloon (even before you blow).

This is not new, so one one not say that are "now" accepting this. It was the Big Bang theory right from the start.That does not follow, static means unmoving, whereas the Big Bang model is in effect moving, but it is moving everywhere in a similar way. That's why it can be happening to the whole universe at once.
Probably your problem is that the "singularity" is not actually a part of the Big Bang model. The Big Bang model can begin at any point along the way, when there is no singularity. The "singularity" is just a way of saying "we don't know what happened before the model became valid"-- no one takes it seriously, except the popular media.

Ok fair enough.. I read what you are saying, as in it all is and not a part of.. thanks

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They say the more complicated something is the easier it is to put a spanner in the works - there is nothing more complicated than the human brain...(Infinite Horizons)

Having thought on what has been said above, then I would take it that at the moment of the big bang the universe was instantaniously at its maximum size - if being infinite is the maximum size than that counts.
More precisely, the space-time continuum is at its maximum size (infinite) and that matter and energy are still filling it up.
If there is nothing beyond the universe, then it can't be expanding into it! That would presume there is an inside and outside.
So we have a sort of galactic osmosis effect where matter is still expanding from dense regions into not-so-dense (empty) regions.
Even allowing for gravity, I would assume that distance eventually cancels its effect in its control over the distribution of matter.
Would this allow for acceleration of galaxies away from each other? ie.. its too salty here!
Just a thought!

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They say the more complicated something is the easier it is to put a spanner in the works - there is nothing more complicated than the human brain...(Infinite Horizons)

Infinite Horizons,
No!
You are being limited by the limitations of human conception. Just because you cannot conceive of inside an without an outside, does not mean that they don't exist.
According to the big bang theory, the universe (the whole universe) started out as a singularity of zero size (no length, no width, no breadth, no time). It expanded. It didn't expand into anything. It just expanded.
The universe is everything. There is not (and need not be) anything to expand into.

__________________
Any day you wake up on "the right side of the dirt" is a good day.

T. Anderson

IH, try looking at it this way. At time zero, the singularity was not infinitely large, the universe was infinitismally small.

Thank you.. great answers.. sorry to be such a nugget about this, but as kaptain k said 'you are being limited by the limitations of human conception'

A wise man once said: "Ask a question and I will give you the best answer I have.
Don't ask...! Then don't come running to me when you get your fingers burnt!

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They say the more complicated something is the easier it is to put a spanner in the works - there is nothing more complicated than the human brain...(Infinite Horizons)

Look at it this way. Imagine you are a "flat lander", a being who can only perceive 2 spatial dimensions. Flat space is just an infinite plane. Now, from our 3D vantage, we can imagine curving a path of 2D space around. Consider the surface of a sphere, such as the surface of the earth. That space is finite. You can go off in any direction and eventually come back to where you started.

There are other curved 2D surfaces with are infinite. Take the surface of a hyperboloid for example.

Now, imagine a sphere that starts out as a point and inflates, with the radius increasing steadily. The surface area increases. That is an expanding 2D space. Consider some flat landers living on the surface who start out some distance apart. What do they see? They see each other flying away as the surface of the sphere is expanding.

Increase the dimensions by one, the curved "surface volume" of a 4 dimensional sphere (hypersphere) and you have a good toy model of the expanding universe.

While the details get a bit complex mathematically, such an expanding hyperphere whose radius is expanding in a particular way (accelerating, actually) is a solution of the equations of General Relativity for "vacuum" with a cosmological constant. It is the space-time of an empty expanding universe, called deSitter space-time.

Our own universe is not deSitter because there's matter in it and so the space-time is actually different, but conceptually this expanding deSitter "hyperballoon" is fine and dandy.


Now, above we have used the notion of a higher dimensional flat space into which we "embed" a curved lower dimensional space. That allows us to see how that lower dimensional space can curve from our Euclidean mindset.

But we don't need that embedding space. Mathematically, one can abandon that notion and described the curve space of itself without having to resort to higher dimensions. That's how General Relativity works. It describes 3 dimensions of space and one dimension of time together and doesn't worry with or need higher dimensions.

However, considering those higher dimensions can let you paint yourself a helpful picture some times. But remember, it's just a picture that doesn't necessarily have to correspond to anything real.

-Richard

Actually I find this piece very interesting. When blowing up a balloon is there any need to consider the direction of the force or in this case where the application is coming from?

The force where? Inside the balloon, the force is the same in every direction, as it comes from the pressure. There is a force through the nozzle that is moving more air into the balloon, but that's the only anistropic force involved.

The pressure and the amount of air both do contribute to the volume of the balloon, and the change in the amount of air is the cause of the expansion, but I don't see what you're implying, Michael. At any time t_x, the volume V_x does depend on P_x and N_x, but V_x is always V_x independent of how it got there.
[Edit: Just so it's clear to everyone, I'm describing the gas in the balloon using the Ideal Gas Law, PV=Nk_BT.]

Being a little more precise - the presently "observable universe" began in a tiny volume - maybe a singularity, but probably not. The BBT by itself does not specify initial conditions or what went "bang" (what caused the expansion). It addresses the expansion once it got going (and after any "inflation"-like period). Any description of the BB beginning in a state of a singularity is extrapolating the BBT "far" beyond what physics knows.

Thanks for the clarification!

__________________
Any day you wake up on "the right side of the dirt" is a good day.

T. Anderson

In general terms, the mechanisms of blowing air into a balloon, the
compression, pressure, temperature, motions, or other properties of
that air, and the motions and physical properties of the balloon
material have no analogue and no relationship to any properties
of the expansion of the Universe.

The balloon analogy is good for showing how points can all move
farther apart from each other or closer together without any
particular center of expansion/contraction; It is good for showing
how a space (in this case a 2-dimensional space) can be finite
yet unbounded; And it can be good for showing how a closed
Universe might be shaped. That's about as far as the analogy
can be stretched before it breaks.

-- Jeff, in Minneapolis

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http://www.FreeMars.org/jeff/

"I find astronomy very interesting, but I wouldn't if I thought we
were just going to sit here and look." -- "Van Rijn"

"The other planets? Well, they just happen to be there, but the
point of rockets is to explore them!" -- Kai Yeves

Depends on what you mean by balloon. And what you mean by analogy.

Start out in (1, N+1) dimensional Minkowski space and specify a hypersphere whose radius expands in hyperbolic, Rindler fashion,

r^2 - (ct)^2 = (c^2/a)^2 = k^2.

The (1, N) dimensional hypersuface (submanifold) of that "balloon" is an exact solution of the EFE in (1, N) dimensions with a positive cosmological constant, Lambda. For N = 3 (3+1 = 4 spatial dimensions in the higher embedding space), we have the relation Lambda = 3/k^2 = 3(a/c^2)^2.

Solving for 'a',

a = sqrt(Lambda/3)/c^2

which is the constant proper acceleration of the expanding hyper-"balloon" as seen by the (1, 4) Minkowski observer.

Switching to the proper time of an observer riding that balloon, which is most elegantly just a Rindler proper clock, and switching to a "comoving" spherical spatial coordinate system confined to that submanifold, one finds the induced metric:

ds^2 = dt^2 - k^2*cosh^2(t/k) *( dSpherical^2)

Where dSpherical is the familiar spherical line element. That should be immediately recongized as a comoving metric very similiar to FLRW, with a scale factor of k*cosh(t/k).

If you think like I like to, you can switch to static coordinates, the actual radial ruler of a co-mover and let him extend that globally, which is is a notion of "distance" more closely related to SR's notion of same,

ds^2 = (1 - (r/k)^2) (c dt)^2 - [ dr^2/(1 - (r/k)^2) + r^2*dSolidAngle^2]

We thus note a "centrifugal" gravitational potential, and a cute event horizon surrounding us at r = k, which is the Cosmological Horizon. If we drop stuff, it flies away from us radially and smacks into that cosmological horizon.

This is called deSitter space-time. And so contrary to being some crude thing, the notion of a Rindler expanding hyperballoon gives us an exact solution of the big bad EFE for a lambdavacuum. It's a very cute, elegant way to see how an expanding space-time works.

Is our own universe deSitter? No, because there is mass energy there. deSitter is an expanding lambavacuum. But deSitter is a darn good approximation out to a couple GLY.

And the Loopy Quantum Gravity types, along with "crackpot" Garrett Lisi and his E-8 theory are of a mind that deSitter is the natural vacuum background, not Minkowksi. LCDM, indeed, emphasis Lambda.

So contrary to breaking down, I'd say the Rindler/deSitter balloon is a darn good one.

-Richard

We just need a 4 dimentional balloon.

Then it will all make sense.

I know I'll never say that the balloon is "just an analogy" again! But what's the comoving-frame spatial curvature in the deSitter model you are describing? My main problem with the balloon was its tendency to be associated with closed universes. Can't deSitter solutions be open?

Depends on what you mean by open! This is something I still haven't got my head completely wrapped around and may not ever do so, but I'll say this. Remember we're talking *space-time* here. Splitting that into space and time is always *arbitrary*, coordinate dependent.

Look at that comoving metric. At constant time, t, the spatial part is flat as a pancake. It is "open"! No spatial curvature at all. But then look at the static form. Space is certainly curved there, and mathematically that is related to a "spherical" notion. But they are describing the same space-time.

The Riemann curvature of deSitter is constant, 4*Lambda = 12/k^2, and both of those metrics will give you that. But they split space and time differently.

The "proper distance" to the cosmological horizon in the static form is infinite (just like to the horizon in Schwarzschild). And that comes about from the coordinatization -- what you mean by a distant point "now".

Note as the balloon expands, as t --> infinity, you've got all the space in the world. The sphere has gone to infinity. In the static form, the local co-mover, at any t is just extending his tangential ruler out to infinity, and his metric is mapping the actual space-time according to that "slicing".

Don't hold me to the details, but note that this Rindler hypersphere could come in from infinity, shrinking down to a point, then bouncing back, the whole infinite Rindler range of something coming it at near c slowing down to a stop at t = 0 and going back off again, all the while maintaining constant proper 'a'. The co-moving form works with that fine for t < 0, but the static form is only good for t > 0. You'd have to flip something around there, but I'm not sure how it would work.

Note this is pure vacuum, there's no mass-energy there, so there's no real singularity of of any sort at t = 0. The only funny business is "space is shrunk to a point". But that is only in the sense of splitting space and time anyway. Space-time itself is something else.

I can sort of dimly see these things but I have a hard time putting it into words. And that makes me mad at myself sometimes. But consider the space-time itself from t = 0 to infinity. That is certainly "open" isn't it, because that balloon has expanded to infinty? And when you're considering space-time as a whole thing together, that's what you're looking at.

From the standpoint of higher dimensional Minkowski observer, at any time on his clock, that balloon has a finite surface volume. But that's his notion of space, his notion of a volume at his notion of 'now'. The co-movers riding it, using their own rulers tangent to that thing, have a different notion of space. And indeed, that global notion of space is arbitrary anyway.

And that the important thing to realize. Notions of whether space is curved or not is an entirely coordinate dependent thing! All the stuff about distances, recessional velocities and all that are coordinate dependent. For example, if we use the co-moving form of the metric, we get a very standard cosmological view. But if we switch to the static metric, we've got a very different notion of distances and velocities.

Global space is a completely arbitrary thing. And so whether it is curved or not, even if it is "open" or not, is likewise arbitrary. And that will sort of drive you crazy because you want space to mean something absolute. But it doesn't. The only absolutes, invariants are found in space-time, the merger of the two.

Someone who understands GR space-times and the math with a talent for explaining it might do better, and probably correct my -1s. But the important thing to appreciate is "space is arbitrary" in a global sense. Space-time is what you have.

-Richard

Again, it makes me mad at myself when I can't put these things into words well. But the basic point here is "space is arbitrary", globally speaking. I almost said space is meaningless, but that's not quite right.

That hit home with me reading about the insanely complicated mess of relativistic rotation. In a nutshell, all the conundrums there come from one simple little fact. If a local observer has "vorticity", which (-1, I'm sure) basically means he has proper rotation along with something else (he could be rotating, but the "something else" might cancel out the problem), then he has no globally consistent notion of space at all. Well, no Riemannian notion of space. If you go to non-Riemannian space, you can come up with something, but it is truly weird.

So the question of whether space is curved or not for an observer riding on the edge of a disc are moot, because no consistent definition of space can be found. Lesson: don't use inconsistent rulers, switch to some other coordinates and forget about it. {This is just an observer thing. Don't spin around in your swivel chair while translationally accelerating, or whatever it is that gives you vorticity. But, if you go to Einstein-Cartan, with torsion, then space-time itself can play this fool stunt and not allow you a globally consistent notion of space at all!}

So that clicked the old light bulb about "space". And I think you're pretty much there as well.

And this mess is why I sort of keep my mouth shut about "cosmological stuff". Cosmologists have a language about distances, velocities and stuff involving this "space" thing. Which comes from a particular coordinatization. And which is utterly arbitrary. Locally, space means something. Globally, it doesn't. Well, it doesn't mean what you think or want it to mean.

So, while some may worry if "space" is open or closed, flat or curved, I don't because that depends on what coordinates you're using in the first place. Space-time has an invariant notion of curvature, but that is something different. Really different.

-Richard

Strange shapes would be ruled out completely then by Occam's Razor. That is why the runaway sphere idea is not considered.

I had noticed it when looking at shapes, if one was to take two Klein bottles and cut them at the lower bend they could be attached in a Mobius-Klein loop. That is probably where I started.

The shape is a bit of a mess but it allows for continuous travel along all surfaces in a forward motion. As a Klein bottle has only one side the addition of a second Klein bottle gives it a second side.

So in theory it should be able to be pulled back together to form a simple sphere of an inner and an outer. It is a truly weird shape. My condolences to anyone trying to determine the mathematics of that shape

I need to correct a little -1, or maybe a -2 here, I just realized. The Rindler hypersphere doesn't contract to a point. Note the cosh in the scale factor -- it never goes to zero. In the standard hyperbolic/Rindler family, which I'll write as r(t):

r^2 - (ct)^2 = (c^2/a)^2 = k^2.

The observer is at the origin which is set up as the Rindler Horizon which makes that hyperbolic form the simplest. At t = 0, r = k. r would be zero only in the case of infinite acceleration.

So for the full range of t, the sphere shrinks in from infinity, reaches a minumum radius at t = 0, then expands back out again. Since a increases with Lamba, a small lamba means a larger minimum radius.

The insight here is Lamdba ~ 1/k^2. Lamda is thus the inverse of the square of the minimum radius of the hypersphere at t = 0. It's sort of a "radius of curvature", but in space-time sense, the two together.

In the static metric, k, that radius, is just the *distance to the cosmological horizon*. Our Shwarzschild like simple "magic metric factor" is just 1 - (r/k)^2. At r = 0, we're at the top of a potential hill, and everything wants to roll away.

And note every observer has that static metric. Say you and I are embedded in this deSitter space-time some distance apart. I say you fly away from me, with your clock dilating and your light redshifting. You say I fly away from you, and my clock is the one that dilates and my light redshifts. You hit my horizon, and I hit yours. And note the static metric picture is completely "gravitational". You roll off deep into that centrifugal gravity well, and I roll off deep in yours.

From the POV of the higher dimensional Minkowski observer, you and I just start at different spots on the hypersurface and the surface just stretches out as it expands. The "reciprocity" between co-movers is very reminiscent of SR. And that shouldn't be a surprise because from the higher dimensional space, it is SR. We're just blinding those co-movers to the spatial dimension in which they are actually accelerating, and forcing them to use coordinates confined to the hypersurface, tangent to that "hidden" spatial direction. The only "space" they know is that confined to and defined by the surface.

-Richard

Another little -1. Above I said the static form wasn't good before t = 0 and something would need to flip. It needs no such thing and that static form is good all the way to -infinity. Static space-time means just that.

Note what you see on the t < 0 side of a Rindler dude. He comes flying in with negative velocity, *slowing down* all the while, comes to a stop at
t = 0, then flies off again, all the while maintaining constant proper acceleration (and when he stops is a relative thing anyway).

Well, that's exactly what a co-mover would see of the others. At large negative t, he's cold and lonely, sitting in empty space surrouned by a big black horizon. The at some point, he sees stuff become unfrozen on that horizon and come flying in toward him, but deccelerating, braking. At t = 0, everybody stops, waves, then goes flying off toward the horizon again. Ad he's back to being cold and lonely at large positive t. The stopping point is when the hypersphere is at minimum radius. But all the while, everything has a coordinate acceleration *away* from him, things just started out with initial velocity toward him, slowing down.

Cute ain't it? This space-time has constant positive curvature. Everybody feels the same tides pulling them apart. And, if all this rambling of mine has made any half-way sense, you can see that space-time curvature, curvature of this crazy thing with a non-positive definite metric, where we subract some of the coordinate squares rather than adding them all, means something very different that pure spatial curvature.

Now, the opposite, a space-time with constant negative curvature exists. That's called Anti deSitter space. That is very weird...... But it is "collapsing" rather than expanding. But that collapse means something different than we might think because as we saw above, deSitter itself can have a "shrinking sphere" phase, it's just slowing down as it deflates, then inflates again. But always consant positive curvature.

But that shows something interesting. Start out with a (1, N+1) Minkowski space-time. We can write some expression in those N+2 variables and define ourselves a subspace, a submanifold. Anything will do (that doesn't blow up or make a mathematical mess of course). But of all those possible sub manifolds, not all are going to be a proper (1, N) space-time and a possible EFE solution.

And going the other way, not all proper (1, N) space-times can even be embedded in a
(1, N + 1) higher flat space-time. We might might need 2 (or more) time like dimensions to do it.

And that's sort of the difference between Riemannian (curved spaces with positive definite metrics, "space" IOW plus some other conditions), vs psuedo-Riemannian, where you're allowed to have non-positive definite metrics. Which is what space-time has to be to work right.

The embedding space for Anti deSitter requires two *timelike* dimensions. I get off the train there, because the embedding space is even weirder than the subspace.

Put all that in your pipe and smoke it. But not too fast. That's some really wacky tobaccky there.

-Richard

Oh please just a quick question.

Just a curiosity, if a pair of stars one positively charged and the other negative had a stable orbit to cancel attraction could they pass over the t=0 in Anti deSitter space such that neither star actually is at t=0 even though the mid point between the stars would need to pass through that point?

Michael,

I'm not exactly sure what you're asking there. By charge, if you mean electric charge, and a situation where electric repulsion is cancelling gravitational attraction exactly, they would just sit there. Adding local gravity to a deSitter or Anti deSitter background is beyond the scope of what we're talking about here.

Anti deSitter space, a negative cosmological constant, is something very weird indeed and I don't know enough about it by itself to describe it, much less start worrying about local gravity and other forces inside it.

And t = 0 is a time, not a point in space. Well, every point in the space has a t = 0.

You can certainly have local gravitational systems in a deSitter background, and that has been studied extensively. There are some interesting results. For example, plop a "point mass", a Schwarzschild black hole down in it, and the event horizon occurs at the same radius it does against Minkowski. However, the opposing "expansion field" acts against the local gravity, and there is a balance point where the two exactly cancel.

-Richard

Ken,

If you're still with me after all that mess, I've been thinking about this "closed space" business and why deSitter isn't closed, even though it looks like it from the POV of the higher dimensional Minkowski space. And I've been trying to figure out just what a closed space means anway. There is a coordinate independent notion of a "closed universe", we'll call it, even if coordinate notions of space are arbitrary. But what does it mean exactly?

Let's look more closely at why this deSitter hypersphere isn't closed, at least to the comovers inside it. To keep it simple, let's drop down to 2 spatial dimensions and imagine an expanding circle in the plane. Our subspace would have just one spatial dimension.

To us (we're at the origin in the (1, 2) Minkowski space), every co mover on that circle is a "Rindler dude". And each one has a horizon to us. That horizon for a given point is a just a line that passes through the origin, parallel to the tangent at a that point, which chases off at c. For example, say the circle has radius one at the start, and consider the point initially at
(x, y) = (1, 0). The horizon is just the line x = ct which chases off after it. The point at (x, y) = (0, 1) has a horizon of y = ct, which chases off after it.

Note at the starting point, t = 0, that only half of the total circle is within any observer's horizon. They only see "half" the possible co-movers. They can't see whole 2D "picture" that we can, but they can only see points on the circle not cut off by their horizon line (as we see it). Their own "causal limit" is the piece of the circle still ahead of the horizon line, and the intersection with the circle is the cosmological horizon in their spatial coordinates.

Note that as the sphere expands, and the horizon gets closer and closer (asymptotically), the horizon cuts off more and more of the expanding circle. Any events we see on that circle cut off just don't happen for that co-mover. That's beyond his horizon -- that's the freezing he sees at his cosmological horizon. It's just a RIndler thing, so to speak, except we're forcing him down 1 spatial dimension.

Now, we're restricting things to lie on that line, so a light beam he shoots off would be constrained to follow that circle as it expanded. Even though that circle is a closed path to us at any time, the light pulse could never make it all the way around to get back where it started. The circle expands too fast to allow it.

And that's why the subspace is "open". The light beam can go forever without getting back to where it started, even though it is tracking a closed spatial path as far as we are concerned in the higher space.

And so, a rough notion of a closed universe would be one where a light beam at least could get back to where it started in finite time. We would say spatially closed time-like geodesics exists (and this is not the same thing as a closed time-like curve where something "goes back in time" to come back to the same *event* in space-time but gets back to where it started spatially).

Something could "go off in an inertial path" and get back to where it started. But we've got to be careful there, because an orbit around a local mass is such a closed geodesic, but the whole universe around it doesn't have to be closed. So our defintion isn't yet polished enough, but I think we can sort of see what we're driving at.

It would roughly be "go off in what you think is a constant direction and come back where you started in finite proper time". That might translate to "no such thing as an infinite space-like curve", or something.

So to summarize, we can certainly have closed universes, but deSitter isn't one, even though it looks like it might be from the POV of the embedding space. A closed universe has meaning, but it has to go beyond some coordinate notion of space, which is arbitrary and will trip you up if you try to put too much meaning into it.

-Richard

Right, that's why I came to reject the ds^2 notation in favor of a dtau^2 notation, with dt^2 positive, like you used. Because I feel proper time is the only dynamical variable that is actually observable, and all the rest are another level of abstraction invented to understand why the things that "matter more" to us also require shorter light delays to get the right time ordering. But relativity means that physics as we conceive it is purely local, and so there is not "space", there is a prescription for mapping our physics "here" so that it looks like someone else's physics "there".

Thus I don't think of space as existing for one observer, but rather as a connection between a real observer and a hypothetical observer if we imagine ourselves somewhere else watching some local physics unfold. In other words, I think of relativity as something extra-- physics is local, relativity is how we conceptualize space to connect to somebody else's local physics. In that sense, Newton got the local physics right, but not the action-at-a-distance spatial connections. That's also why I like writing the metric as
ds^2 = f(s,t)*(dt^2 - dtau^2)
because it's supposed to be a construct for how to make distance based on how one observer measures time, relative to an observer who was at "both events" in question. I suppose I'll never succeed in getting it this way in all the textbooks, and indeed the "high priests" might have a reason for not liking my way.
Not utterly-- indeed I would say cosmology is the only place where the concept of space makes sense in a non-arbitrary way. It's all based on the cosmological principle-- that's what selects comoving coordinates as the "special" ones, the ones where you don't need to explicitly treat a global (universal) concept of gravity. Why should we need universal gravity to understand our local physics? We shouldn't-- it should only come up when we connect to someone else's physics. Again, universal gravity isn't physics at all, it's in the connections.

Your comments on this thread have been highly insightful, thank you.

I find it really interesting to read what you write both Ken G and publius. I will have to go over it a few more times I am sure and even then I will only be scratching the surface. Thank you both for an interesting discussion.

I did see a while back a close up picture of one of the strands in the web of density from one of the great observatories. It had the filament looking like a twisted strand of fairly flat pasta. The stars were in lines along both edges of the strand like there was some connection. I have Googled a few time to see if I could find the photo again but no success yet. Do you think anybody here on the forum might know of it?

Cheers

The only thing I would say to that (Who was it who was bad to say "Let me say this about that", Nixon?) is we do have curvature, and that is local. If we define "universal gravity" as the curvature, which is pretty much the high priest definition of gravity, then deSitter (and any cosmological solution with curvature) does have it.

It's just very small, at least for us. Nereid posted this a while back:

Dark energy, General Relativity, etc - a fun paper for many readers (like publius)!

As I rambled about a little above, it's very interesting to plop a local mass down in de Sitter and see what happens. It's very neat for spherical symmetry -- the the magic metric factor is just (1 - R/r - (r/k)^2 ). That would describe a "co moving" black hole. The amazing thing is the event horizon, R, now an "inner horizon", is the same as normal, R.

And, an even more amazing thing happens. Null geodesics remain the same in these coordiantes as the k/Lambda term cancels out completely (but only for null geodesics -- time-like geodesics will be different). That led many to say Lamdba thus has no observable effect on gravitational lensing.

But in that paper and another, ironically Wolfgang Rindler himself showed that was wrong. The trouble was coordinates -- the "far away" observer is not flat, and his local ruler and sense of direction is actually different, affected by Lambda. I thought that was a good lesson -- coordinates will trip you up, and they trip up even the high priests when they're not careful.

Those "Schwarz-Sitter" (be careful when you say that fast ) coordinates don't even correspond to any observer -- they would be the coordinate of an observer at r = 0 if the black hole weren't there.

Anyway Rindler and his co-authors used that to come up with a value of Lamdba in line with other estimates by using lensing data from distant sources.

However, for the solar system, Lamda effects light bending around the sun by only 1 part in 10^20 for us on earth. It should affect orbits by around in 1 part in 10^13.


-Richard

Before anyone asks, I don't have a clue what "Schwarz-Sitter" would look like, how it would modify our embedding space picture. It might mess things up so much that our (1, 4) higher space was no good anymore, and we'd need even more dimensions. I just don't have a clue.

For example, consider a helix. That's a curve, but you need 3 dimensions to embed it in, you can't confine it to a plane. And in 4D, you can imagine a type of curve there that required all four dimensions. That's the motivation for forgetting about embedding spaces -- describe the curvature of the geometry you're dealing with directly without worrying about embedding it something.

And the singularity in the full black hole form would mess things up even more. A curvature singularity means your "hypersuface" does something nasty. It would be some type of "kink" or knot, tear, or something.

-Richard

Let me make one thing perfectly clear: Ronald Reagan was The Man
who said, "Well, I have this to say about that." It resignates with
the American people. Just a horrible, horrible mess.

-- Jeff, in Minneapolis

__________________
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"I find astronomy very interesting, but I wouldn't if I thought we
were just going to sit here and look." -- "Van Rijn"

"The other planets? Well, they just happen to be there, but the
point of rockets is to explore them!" -- Kai Yeves