Merged Relativity+ / Farsight

[Reality Check]

I'm sorry RealityCheck,

I'm sorry, Farsight, but in the real world this article is correct.
We know empirically that the universe looks "flat".
We do know that the volume of the universe is more than 20 times bigger than the volume of the observable universe because ... we know empirically that the universe looks "flat". I'm sure that edd and Vorpal and others will back me up on this.
I know that Ned Wright backs me up: Is the Universe really infinite or just really big?

We have observations that say that the radius of curvature of the Universe is bigger than 70 billion light years. But the observations allow for either a positive or negative curvature, and this range includes the flat Universe with infinite radius of curvature. The negatively curved space is also infinite in volume even though it is curved. So we know empirically that the volume of the Universe is more than 20 times bigger than volume of the observable Universe. Since we can only look at small piece of an object that has a large radius of curvature, it looks flat. The simplest mathematical model for computing the observed properties of the Universe is then flat Euclidean space. This model is infinite, but what we know about the Universe is that it is really big.

(my emphasis added)

Huh? I don't think the universe is curved in a higher dimension.

Huh - that is what you have been implying with all your talk of the universe and higher dimensions.
The physics is that the universe in GR has 4 dimensions and only 4 dimensions. There are no higher or even lower dimensions .

So the point is that

Originally Posted by Farsight
Au contraire, the evidence supporting big-bang cosmology suggests a finite universe. And whilst I'm on a roll, there is no evidence whatsoever to support the notion that the universe is somehow curved "in a higher dimension". Zip, zero, zilch. And as for PacMan world, pah.

is a total fantasy about some notion that the universe is somehow curved "in a higher dimension".

People have mentioned that the universe may be a torus or some other form of closed geometry (the "PacMan" world). None of these are a universe curved "in a higher dimension". This is all the intrinsic geometry of a 4 dimensional universe.

 

[Reality Check]

Oh no you can't.

Oh yes I can.
I can even read which you seem unable to do Farsight !
Standard cosmology is based on the Friedmann–Lemaître–Robertson–Walker metric

The Friedmann–Lemaître–Robertson–Walker (FLRW) metric is an exact solution of Einstein's field equations of general relativity; it describes a homogeneous, isotropic expanding or contracting universe that may be simply connected or multiply connected.[1][2][3]

which has no restrictions on the values of its coordinates. For example in reduced-circumference polar coordinates r can take any value.

The simple fact is that a metric is basically a measure of distance. An expanding universe is not that the "size" of the universe increases - it is the measure of distance between points within the universe increases. Thus the theory apples to infinite as well as finite universes.

The only people who would think that an infinite universe cannot expand would be those so ignorant that they think that the universe is expanding into something.
What is the Universe expanding into?

This question is based on the ever popular misconception that the Universe is some curved object embedded in a higher dimensional space, and that the Universe is expanding into this space. This misconception is probably fostered by the balloon analogy which shows a 2-D spherical model of the Universe expanding in a 3-D space. While it is possible to think of the Universe this way, it is not necessary, and there is nothing whatsoever that we have measured or can measure that will show us anything about the larger space. Everything that we measure is within the Universe, and we see no edge or boundary or center of expansion. Thus the Universe is not expanding into anything that we can see, and this is not a profitable thing to think about. Just as Dali's Corpus Hypercubicus is just a 2-D picture of a 3-D object that represents the surface of a 4-D cube, remember that the balloon analogy is just a 2-D picture of a 3-D situation that is supposed to help you think about a curved 3-D space, but it does not mean that there is really a 4-D space that the Universe is expanding into. For objects in our ordinary experience, like the rising loaf of raisin bread dough also used as an analogy to the expanding Universe, there are two ways to see that the object is expanding:

1. The distances between objects are all increasing, so the distance between any pair of raisins increases by an amount proportional to the distance.
2. The edge of the loaf pushes out into previously unoccupied space. Note the distance between any pair of points on the edge increases by an amount proportional to the distance.

The first statement involves the internal geometry of the object, which can be measured by an observer sitting in the object. The second statement involves the external geometry of the object, which can only be measured by an observer outside the object. Since we are stuck within our spacetime, we need to study the internal geometry of space-time, and that is what general relativity does. In terms of internal geometry, any object with the first property above is expanding. Furthermore the Universe is homogeneous so it does not have any edge. Thus it can't have the second property above. But it does have the first property so we say the Universe is expanding.

FYI: I disagree slightly with the highlighted bit. I would put it as either "The universe is assumed in the model to be homogeneous and so it has no physical edge" or "The observed universe is measured to be fairly homogeneous and so we can observe no physical edge".

 

[Reality Check]

The point is that we don't know that the actual universe is more than 20 times larger than the observable universe.

The point is that we know

• The size of the observable universe.
• The flatness of the observable universe.

The basic assumption that the observable universe is representative of the entire universe leads to the calculation that the universe is at least 20 times bigger than the observable universe.
If we want to indulge in speculations about the non-observable universe then we can make the size of the universe even bigger or smaller.

 

[edd]

I'm sorry, Farsight, but in the real world this article is correct.
We know empirically that the universe looks "flat".
We do know that the volume of the universe is more than 20 times bigger than the volume of the observable universe because ... we know empirically that the universe looks "flat". I'm sure that edd and Vorpal and others will back me up on this.

Well there are the flat finite topologies you mention later, and the assumption that homogeneity still holds on those larger scales, and maybe something else I've not considered. The 20x or more figure sounds very likely but I'm not sure if say we 'know' it.

 

[Reality Check]

No problem with that, Robo. But [Farsight narrows his eyes] I fear he was saying he doesn't hold with conservation of energy.

Actually what we are saying is that you are mistakenly applying classical and special relativity conclusions to general relativity.
In GR, global energy may or may not be conserved: Is Energy Conserved in General Relativity?

On the large scale the universe is homogeneous and isotropic. Let's set aside "globally isotropic" aside for a moment, because it's loaded

Let us not put "globally isotropic" aside for a moment because it is a fundamental assumption of standard cosmology.
Let's not set expansion to one side because the evidence that the universe is expanding is strong.
In a place where space is homogeneous and isotropic, there's can be gravity and there can be space-time curvature when there is a non-zero stress-energy tensor. That is the condition in standard cosmology where the universe is assumed not to be empty.

And I'm saying what curvature?

The curvature of space-time, duh !
WMAP - Shape of the Universe

We now know (as of 2013) that the universe is flat with only a 0.4% margin of error.

That means that the intrinsic curvature of the observable universe is about zero but does not rule out slightly positive or negative.
If the curvature is exactly zero then a homogeneous, isotropic universe is infinite. The error limits put the possible size of the universe as at least 20 times greater then the observable universe.

Parameters of Cosmology: Measuring the Geometry of the Universe

 

[Reality Check]

Well there are the flat finite topologies you mention later, and the assumption that homogeneity still holds on those larger scales, and maybe something else I've not considered. The 20x or more figure sounds very likely but I'm not sure if say we 'know' it.

Well, we certainly do not 'know' it in the sense of actually measuring it but then that is a really silly meaning in this context .
How can we directly measure a size for the universe that is bigger than the size of the observable universe as Farsight seems to be demanding? Of course we cannot. We can 'know' it in the sense of applying observations to scientific models as we do a lot in science.

 

[Toontown]

Well, we certainly do not 'know' it in the sense of actually measuring it but then that is a really silly meaning in this context .
How can we directly measure a size for the universe that is bigger than the size of the observable universe as Farsight seems to be demanding? Of course we cannot. We can 'know' it in the sense of applying observations to scientific models as we do a lot in science.

Don't try to be too precise. Just tell Farsight the size of the universe is greater than the observable part, but no greater than infinity. That should satisfy him. And if it doesn't satisfy him, we'll know exactly what his game is.

 

[edd]

Well, we certainly do not 'know' it in the sense of actually measuring it but then that is a really silly meaning in this context .
How can we directly measure a size for the universe that is bigger than the size of the observable universe as Farsight seems to be demanding? Of course we cannot. We can 'know' it in the sense of applying observations to scientific models as we do a lot in science.

I'm just saying the range of models could be taken to be broader.

 

[Farsight]

Pretty much yes.

OK thanks edd. I'll pick up on this point in replies to later posts.

I'm saying I thought it was fairly clear that on that one point I agreed, and I felt it was clear what I said the first time, let alone the second.

OK noted. Please accept my apologies for suggesting you were unwilling to acknowledge that I was right.

That's a gross distortion of what sol said, and you know it -- as usual.

It was an exaggeration intended to get across how facile some "hypotheses" are.

Clearly, Mr. Duffield is an advocate of debating by defense through strong and belligerent offense. However, his bombastic tone and continuing straw man tactics are of no avail. We can all see through his bluster and fantasy physics. He is fooling only himself -- and maybe when he looks in the mirror -- not even himself.

My ridicule was actually aimed at you, because you started a multiverse thread, and Max Tegmark classified multiverses lending them credibility, and also wrote a paper suggesting the universe is made of mathematics. If I came out with stuff like that, I'd get ripped to shreds.

 

[Farsight]

I wanted to find more detail to support my broad statement that energy need not be conserved, since clearly it is very often a powerful principle indeed. Amongst the better explanations by people I am willing to send traffic to is this:
http://www.preposterousuniverse.com/blog/2010/02/22/energy-is-not-conserved/

This is not uncommon knowledge amongst people with some grasp of GR. I find it boggling that someone namechecking Noether and claiming a good understanding of general relativity would find this surprising.

At least I would if I didn't have prior experience of Farsight, anyway.

I'm sorry edd, but Sean is wrong here. Let me try to illustrate by responding to what sol invictus said.

I actually strongly disagree with Sean - or at least, I disagree with the way he explains this. He says people will be confused by referring to gravitational binding energy because it's negative. But Newtonian gravity is not very mysterious, and no one would ever say that energy isn't conserved there - and yet gravitational potential energy is negative for exactly the same reason.

Only there isn't any negative energy at all. There's merely less positive energy in the system that's left, and all the while total energy is conserved. When you drop a brick, some of the mass-energy of that brick is converted into kinetic energy. The brick hits your sandpit and the kinetic energy is dissipated, ultimately as radiation, whereupon the mass-energy of the brick is reduced. We now have a mass-deficit. Strictly speaking one needs to account for the Earth too, but whilst |p|=mv momentum is shared equally, KE=½mv² kinetic energy isn't, as per collisions. The brick has just about all of it, so we discount the Earth. See mass in general relativity.

I mean, if you throw a rock in the air and it comes back down, does anyone shout "energy isn't conserved"? Of course not, because they know about gravitational energy. Cosmology is almost exactly the same.

Good stuff sol. You do work on your rock, you give it kinetic energy, this is converted into gravitational potential energy. At the top of its arc the mass-energy of your rock is higher than what it was when the rock was lying on the ground.

As for defining a local energy density, it's true this can't be done in GR. It's also true that the total energy is identically zero if the universe is closed, which isn't very convenient.

IMHO this is a canard. If all the rocks in the universe fell together, conservation of energy applies. The end product (whatever you want to call it) has the same total energy as the original configuration.

But claiming that energy isn't conserved because of these issues is throwing the baby out with the bathwater. The Einstein action coupled to matter is time-translation invariant, so it conserves energy by Noether's theorem (actually the problems I mentioned arise because it's more than just time-translation invariant, it's invariant under time reparametrizations).

Noted sol.

Edd: you said I think it still helps as a pointer to why some cosmologists aren't concerned about what may look like non-conservation as the universe expands though. I'm afraid some cosmologists are wrong about some things, such as gravitational energy being negative. Another thing I think they're wrong about is the dark-energy density remaining constant, because they aren't accounting for a reducing "strength of space".

 

[Farsight]

Don't try to be too precise. Just tell Farsight the size of the universe is greater than the observable part, but no greater than infinity. That should satisfy him. And if it doesn't satisfy him, we'll know exactly what his game is.

He won't. See how he won't accept what edd said about not knowing the universe is +20 times bigger than the observable universe. He comes out with a barrage of garbage like Let us not put "globally isotropic" aside for a moment because it is a fundamental assumption of standard cosmology. See what Vorpal said, and my response where I pointed out that it's loaded. Globally isotropic means nobody anywhere can see anything unusual thisaway or thataway. It's a "fundamental assumption" which with the WMAP flat universe, leads you by the nose into a infinite expanding universe which has always been infinite. Duh!

 

[Farsight]

Isotropy refers to geometry being independent of direction. Homogeneity does not imply isotropy; a good example of that is a cylinder, which wraps around in one direction but not in another, but is still homogeneous. Thus, by ignoring that condition, you're greatly misinterpreting the statement.

I merely omitted it for brevity. And please note that we have no actual evidence that the universe is in any way cylindrical or toroidal.

BTW, in more common geometric usage unqualified 'isotropy' is the same as 'global isotropy'; I was just a bit more explicit.

Noted.

It isn't, and that's a dangerous attitude to take in a context that's heavily mathematical. Particularly since it's responsible for some of your confusions, including one below.

I'll take more care in future when people refer to spheres, and make sure they make it clear that they're referring to a mathematical sphere rather than the common usage.

Re Please explain. Use a universe that isn't expanding for simplicity, and imagine we shine a light beam in a straight line.

Let's take a particular flat 2-torus for simplicity. Imagine a square of side length 2, with Cartesian coordinates with origin at the center and axes aligned in the obvious way (parallel to sides). But unlike a square, the sides are identified: (x,-1)~(x,+1) and (-1,y)~(+1,y). The coordinates refer to the same point on the manifold. A more convenient definition is ℝ²/ℤ, or if we're talking about topology, a 3-torus is the product of three circles S¹×S¹×S¹, though both of these may be more cryptic for people unused to these areas of mathematics.

That means if you're at the origin and you shine a light beam along the x-axis, it will eventually hit you in the back: it will go from (0,0) to (1,0), which is the same as (-1,0), and proceeds still along the x-axis to (0,0). Similarly for many other directions, but not all: if the slope of the light beam direction is irrational in these coordinates, then it will never reach the origin again, although it may come arbitrarily close. Thus the flat torus fails to be isotropic, as not all directions behave in the same way.

A flat 3-torus is completely analogous. Note in particular there's no mention of any higher-dimensional space. Any embedding to a higher-dimensional manifold would be purely a convenience for some purposes and not at all necessary for any intrinsic properties.

Vorpal, this is a most unsatisfactory explanation. We have flat space, light goes straight, no higher dimensions, and yet light hits me in the back! You have employed mathematical abstraction to invent an non-real science fiction scenario.

Not so. The flat torus is flat and edgeless, but not infinite. In a sense all valid mathematical theorems are tautologous, but your interpretation of it is still incorrect.

Only we have no evidence that the universe is in any way any kind of flat torus, and people do say "the universe must be infinite".

Like I said, in the common geometric usage, unqualified 'isotropic' means 'globally isotropic' already.

Noted. But please note that when Einstein referred to a gravitational field as non-homogeneous non-isotropic space, the word global is not present.

You're making essentially the same mistake as above. A 4-sphere is indeed a hypersphere, but a 3-sphere is also a hypersphere. The usual sphere such as the surface of a common ball or (a somewhat idealized) Earth is a 2-sphere because it is 2-dimensional. The topology of the Einstein static universe is ℝ×S³, one dimension of time and three dimensions of space, the latter of which form a 3-sphere.

OK.

Of course it can. There's nothing wrong with the FLRW family of solutions.

Apart from the fact that two out of three (or three out of four) solutions are definitely wrong. They can't all be right.

Perhaps you're thinking of there being no movement through space whenever the pressure is uniform.

Not at all. What I'm essentially saying is light goes straight and there is no overall gravity when the pressure is uniform.

But in the comoving frame of FRLW solutions where pressure is so 'balanced', the galaxies are stationary (on average) and space is expanding.

And light goes straight and there is no overall gravity. No problem.

"On a poscard", huh? As an idealization, imagine an infinite line of equally spaced galaxies:

    ...-*-*-*-*-*-...
  ...-*--*--*--*--*-...
...-*---*---*---*---*-...

There's nothing whatsoever conceptually difficult about space between galaxies stretching.

There is. The space doesn't stretch, it's under pressure. It expands. It can only do this if the line isn't infinite, and there's a place where the pressure is not counterbalanced.

That's like saying that a Euclidean plane is not a space but a surface embedded in 3-space. An embedding may be practically useful for certain things, but it isn't necessary. Besides, the flat 3-torus we're talking about cannot be embedded in (Euclidean) 3-space. See above for one way to think of a flat 2-torus without any higher-dimensional space; a flat 3-torus is straightforwardly analogous and has been covered in sol invictus's post.

Hopefully Toontown is man enough to say he will not accept hypothetical mathematical abstraction in lieu of convincing explanation backed by hard scientific fact.

 

[edd]

He won't. See how he won't accept what edd said about not knowing the universe is +20 times bigger than the observable universe.

I think it's a little unfair to complain he doesn't accept my word as absolutely true.

You clearly don't, after all, in most other circumstances.

 

[Reality Check]

Only there isn't any negative energy at all.

Wrong, Farsight: Gravitational potential energy is always negative.
Every time that you raise a brick, it gains gravitational potential energy.
Every time that you drop a brick, it loses gravitational potential energy.

IMHO this is a canard. If all the rocks in the universe fell together, conservation of energy applies.

As anyone can tell this is a straw man argument since sol invictus was not talking about the rocks in the universe .
Count the number of rocks in sol invictus's post, Farsight - there is a grand total of 1.

I'm afraid some cosmologists are wrong about some things, such as gravitational energy being negative.

I'm afraid that you are making things up, Farsight.
No cosmologist says that gravitational energy being negative.
What cosmologists say (and anyone who bothers to learn) is that gravitational potential energy is negative.

Another thing I think they're wrong about is the dark-energy density remaining constant, because they aren't accounting for a reducing "strength of space".

Oh goody - writing complete gibberish or total ignorance again, Farsight .

The "strength of space" nonsense though could be the cosmological constant (i.e. the cost of having space-time) which by definition is constant and does not reduce.
Cosmologists are correct about dark-energy density remaining constant when caused by the cosmological constant because it is a simple derivation from GR that the cosmological constant has a constant energy density:
Farsight: The energy density of the cosmological constant is constant!
First mentioned 19 July 2013 and basically ignored for 5 days now.

 

[Reality Check]

He won't. See how he won't accept what edd said about not knowing the universe is +20 times bigger than the observable universe.

That is an fairly dumb statement.
I accept that we do not know the universe is +20 times bigger than the observable universe in the sense that you, Farsight, may mean it as in we have an actual measurement.
I accept that we do know universe is +20 times bigger than the observable universe in the generally accepted meaning of the word know in science, i.e. the evidence supports it.

Globally isotropic in cosmology means that looking in all directions looks the same at large enough scales. It is an assumption that the standard cosmological model uses. It is not physically true except for large scales, thus "globally".

A homogeneous, isotropic universe that is flat implies that universe is infinite. WMAP shows that the universe is very close to flat which implies that the universe is flat. The error limits on the WMAP measurements place a lower limit on the size of the universe which is about 20 times the size of the observable universe.
WMAP - Shape of the Universe
Parameters of Cosmology: Measuring the Geometry of the Universe
Duh !

What actually leads scientists into an infinite universe that expands and remains infinite is nothing to do with the WMAP measurements . It is the actual meaning of the word expansion in cosmology.
It is not a change in the size of the universe because an infinite universe has no size (infinity is not a number) . It is a change in the metric which is basically the distance between points.

 

[Reality Check]

Apart from the fact that two out of three (or three out of four) solutions are definitely wrong. They can't all be right.

Farsight, you are the only person here who is claiming that all of the solutions have to mathematically and physically right.
Everyone else here understands that all of the solutions are definitely mathematically correct.
Everyone else here understands that all of the solutions cannot be physically correct because the universe cannot have multiple values of intrinsic curvature in the model.
Everyone else here understands that observations show that one solution (a flat universe) is probably physically correct.

 

[Reality Check]

Vorpal, this is a most unsatisfactory explanation. We have flat space, light goes straight, no higher dimensions, and yet light hits me in the back!

Farsight, if you do not understand Vorpal's explanation then how much dumbed down do you want it made ?
Seriously though - if you are this ignorant about cosmology then you should learn about cosmology before you make uninformed assertions about cosmology.

Given a flat, finite universe, light goes straight and light does hit you in the back! Vorpal's example of a flat torus shows this behavior for a situation that you should be able to understand.

Perhaps you should reverse the question - what if this light travelled in a straight line and never "hit you in the back"? Where do you think that the light goes in this finite universe?
There are some options:

• It goes outside of the universe (but that "outside " does not exist!).
• It gets absorbed at the "surface" of the universe.
  But that violates the assumption of homogeneity and we are looking at an empty (except you and this ray of light) universe.
• It is reflected and somehow never comes back to you.
  But see above and this places you at a privileged position in the universe.

 

[Toontown]

Hopefully Toontown is man enough to say he will not accept hypothetical mathematical abstraction in lieu of convincing explanation backed by hard scientific fact.

I really don't think Vorpal is selling torus space. I think Vorpal simply pointed out that it is conceivable for a flat universe to be not much larger than the observable part, but not without being strangely cramped.

Anyhow, you slapped an impenetrable edge on it to keep it small, so you're as much as arguing against yourself. "Mathematical abstraction", indeed.

http://www.youtube.com/watch?v=WANNqr-vcx0

 

[Farsight]

It beats me how I can see how the back of my own head in a flat universe Toontown. If you'd care to explain how that Asteroids universe actually works, I'm all ears. If you can't, what other options are there?

Does anyone else get the feeling that words are flowing out like endless rain into a paper cup?

Reality Check is a pain, Frau. He's one of those guys who just doesn't listen.

I think it's a little unfair to complain he doesn't accept my word as absolutely true.

Like I said, the guy just isn't listening.

You clearly don't, after all, in most other circumstances.

If you back up what you say with evidence and/or references and/or sound logic, I'll accept what you say. By the way, I would urge you to pay attention to what I said about conservation of energy and gravity.

 

[trebor]

He's one of those guys who just doesn't listen.

Like I said, the guy just isn't listening.