Where does new space come from as the universe gets bigger?

[Mordred]

roflmao, I didn't realize there was an actual model for this, learn something new everyday lol.

just had to google it and pulled a few arxiv articles.

In conclusion, we have constructed a “variable gravity universe” whose main characteristic is a time variation of the Planck mass or associated gravitational constant. The masses of atoms or electrons vary proportional to the Planck mass. This can replace the expansion of the universe. A simple model leads to a cosmology with a sequence of inflation, radiation domination, matter domination, dark energy domination which is consistent with present observations. The big bang appears to be free of singularities.

http://arxiv.org/pdf/1303.6878v4.pdf

just goes to show, metrics is capable of any descriptive lol. I would have to agree though Chronos I seriously doubt it would be simpler myself. Personally I don't think I'll waste much time studying this model except as an alternate viewpoint lol

 

[craigi]

"Any system of particles would get smaller". Does that include the universe itself as a whole? Is the whole universe getting smaller? If so, doesn't that completely defeat the purpose of this model trying to describe an expanding universe? How about a galaxy cluster? A galaxy? Where do we draw the line of "any system of particles"?

Current predictions are that our local cluster will become increasingly isolated. The rest of the universe would partiton off in the same way.

You can model our universe using shrinking matter and variable constants of nature, if you wish. Is that a 'simpler' model? I think not. Some, like Wetterich's model, can resolve certain issues - like a primordial singularity - but, at the price of introducing more issues than they resolve. I've always felt the goal of science is to model reality using the fewest possible variables.

Fewest variables, constants and laws? Even then we like our laws to have a degree of elegance to them. Laws that emerge from symmetry, ideally.

 

[Ken G]

It shouldn't be a different model, it should be the same model. Any model that makes all the same predictions is trivially the same model. All that is different is the language, and we should all know that language does not a physics model make. Certainly all the math should be the same. If any of their math is different, or any of their predictions are different, they should say so, and people should look for the differences. I doubt there are any differences, certainly not if they are talking about what I'm talking about, so it is not a different model. And finally, if all the math is the same, it is clearly not any "more complicated" than the usual arbitrarily chosen language.

 

[craigi]

roflmao, I didn't realize there was an actual model for this, learn something new everyday lol.
...
http://arxiv.org/pdf/1303.6878v4.pdf
...

Hah.

I thought we were just talking about some trivia that Ken G had just made up in the middle of this thread.

 

[Ken G]

Fewest variables, constants and laws? Even then we like our laws to have a degree of elegance to them. Laws that emerge from symmetry, ideally.

Indeed, the symmetries should be the centerpiece. The main symmetry in cosmology is translation symmetry, which is easier to support when nothing is going anywhere, i.e., when the matter is shrinking. If the space is expanding, it's of course the same thing, but a little harder to see that there is translation symmetry because the space is not just sitting there! If space is not said to be expanding, then clocks must be said to be speeding up, because it takes light more time to cross between galaxies. That is to say, the ratio of the time for light to cross between galaxies, and the time for an atom to oscillate, is increasing. The dimensionless speed of light stays the same (all dimensionless constants do), by which I mean, the ratio of the number of rulers crossed, to the number of atomic oscillations during the crossing, stays the same. That is all consistent with the rulers shrinking.

 

[Ken G]

I thought we were just talking about some trivia that Ken G had just made up in the middle of this thread.

Yes, I did just make it up, but then, so did whoever said space is expanding. How is that not just made up, can anyone cite evidence that is happening? Of course not, we have no model of space, we have only the predictions of GR. No part of the mathematics of GR says space is expanding, it's pure made-up language, accepted uncritically as if it was really saying something we could ever test, which then prompts people to ask "where does the space come from." That's my point here, the question emerges from a non-model, no part of GR asserts that space is actually expanding. Indeed, it seems to me one of the main points of all of relativity is noticing the difference between observations and coordinate systems!

 

[Mordred]

The problem is Ken is that it doesn't, stop and think about all the other models it influences, variable gravity? wouldn't that also affect Observation affects due to GR and SR.? This in turn implies remodelling QFT, QED, QCD etc after all we also have to include and describe a time component with an influence just to cover why particle sizes wouldn't be consistent. How many models would a varying Planck mass influence?
A quick google search showed me numerous articles he has written with unusual metrics, Cosmon inflation? "are galaxies Cosmon lumps", Cosmon dark matter the list goes on lol . Sounds to me that its not that simple if he has to redescribe everything we know

 

[Ken G]

The problem is Ken is that it doesn't, stop and think about all the other models it influences, variable gravity? wouldn't that also affect Observation affects due to GR and SR.?

No, it wouldn't affect a single observable that GR predicts, that's obvious because all GR predicts is the dynamics of the metric, and all a metric gives you is the number of rulers that could lay end to end between two events, or the number of times some atom oscillates. That's it, that's all you ever measure, and that's all the mathematics of GR ever refers to. If you doubt that, then feel free to tell me something else it refers to.

How many models would a varying Planck mass influence?

None, the Planck mass does not vary, if expressed in dimensionless form (like how many protons is it).

A quick google search showed me numerous articles he has written with unusual metrics, Cosmon inflation? "are galaxies Cosmon lumps", Cosmon dark matter the list goes on lol . Sounds to me that its not that simple if he has to redescribe everything we know

I don't know quite what this "cosmon" metric is, but if it's what I'm talking about, it cannot be the least bit different from the standard metric. Perhaps they are talking about something different, or perhaps they are talking about the same thing and don't realize it. I'm sure that what I'm talking about is the exact same metric, for the reasons I just gave.

 

[Mordred]

lol you should google the model name that Chronos mentioned then look at some of the arxiv articles listed. I nearly laughed my head off.

 

[Chronos]

Christof Wetterich is the leading proponent. He has a number of arxiv papers on the subject. The 'cosmon' is his terminology. It's unclear how his way of reformulating the cosmos actually accomplishes anything.

 

[Ken G]

The first question to ask is, if any of the predictions are different. If it does that, it's a different model. The second question is, if any of the math is different, that leads to those predictions (like Lagrangian physics starts with different math, and is better at different things, but is ultimately equivalent). If it does that, it's the same model, but useful in different ways. What would make cosmons laughable is if Wetterich can't tell the difference. I don't know.

What I am talking about is clearly neither of those, it is exactly the same predictions, using exactly the same math, but a different language for talking about what the math means. So when you say "look, space just expanded there", I say "no, a metric just changed, and you like to imagine that space did something." Pehaps I prefer to imagine that the matter shrank. I think we can all agree it is important to know the difference between a testable prediction, a different mathematical route to a testable prediction, and a simple pedagogical convention for talking about a testable prediction. It would also be laughable if we could not tell that difference!

 

[abitslow]

All models which predict the same things are NOT equivalent. Of course, that is by MY definition of equivalence.
If they were, all we need do is teach every grade school child about tensor calculus, and be done with it.
What is the tax on a bunch of bananas at 0.54$/lb, 3.27 lbs and 6.5% tax? Wait, I need to adjust for the local inflaton field, I'll get back to you once I get some supercomputer time. As simple as possible, but no simpler.

 

[Ken G]

All models which predict the same things are NOT equivalent. Of course, that is by MY definition of equivalence.

I would say they are equivalent, but can be distinguished. It's just semantics, if you'd prefer some other word for models that make all the same predictions, that's fine by me. It's actually not relevant to the picture I'm describing, as the steps for doing the math are precisely the same as for any GR solution. There is no difference at all until the mathematics, which leads to the predictions, is translated into words that say why the predictions come out the way they do. That's the part that isn't really physics at all, which is what I'm saying-- "space is expanding" is not physics, its social convention, much like a coordinate system. It can affect our cognitive resonance, but is ultimately subjective-- when we do an inclined plane problem, some people like to align their coordinate axes with gravity, others with the plane, but the physics is not different, and we do not say one picture is right and the other is wrong unless it yields an answer that does not predict the observations.

 

[guywithdoubts]

I'm still confused, it doesn't look like CosmicVoyager was having a problem with semantics. By space being a "thing," I'm sure he didn't mean a material object, as in made of particles or waves, but rather an entity of a different nature.
If space is not to be thought of as an entity (of whichever nature,) then why would two parallel lines eventually meet (given that the geometry of space allowed for it, I'm by no means trying to discuss this subject) if there is nothing but movement going on? I can understand space as simply being 'distances' between objects, and that gravity affects matter and energy directly, and that no 'actual' warping of a non-existing spatial entity occurs, and that all of this is an aid to help us visualize it. But what about two parallel laser beams? Why would they eventually meet, if all there is between them is distance, and no existing, warped space (as an actual entity)?
I though, precisely, that accepting the existence of an entity (distinct to matter and energy) would provide a mechanism (I'd say medium if it weren't for that darned aether) for this sort of things to happen.

 

[Mordred]

the volume of space is filled with energy-mass, the density influence of that energy-mass is what affects the path of light. In some ways like light passing through a prism or water.
The topography of the universe is determined by a comparison of the actual density (total density) as compared to the critical density. The critical density is represented by the following formula and is a calculated value to have a perfectly flat and static universe.

[itex]\rho_{crit} = \frac{3c^2H^2}{8\pi G}[/itex]

P=pressure (change this to density
c=speed of light
G= gravitational constant.

density is represented by the Greek letter Omega [itex]\Omega[/itex] so critical density is [itex]\Omega crit[/itex]
total density is

[itex]\Omega[/itex]total=[itex]\Omega[/itex]_{dark matter}+[itex]\Omega[/itex]_baryonic+[itex]\Omega[/itex]_radiation+[itex]\Omega[/itex]_{relativistic radiation}+[itex]{\Omega_ \Lambda}[/itex]

this is a copy and paste from the Universe geometry article I posted earlier in this thread, from that you can see the density relations. Density has a pressure relation defined by the equations of state( cosmology) http://en.wikipedia.org/wiki/Equation_of_state_(cosmology)

so an easier way to think of it is the observations of light is influenced by the density influences of energy-mass, much like light flowing through an intergalactic medium. However space itself is best thought of as a change in distance or volume, that is simply filled with the contents of the universe

edit lol I just noticed a mistake in that article, I'm amazed I and other readers never caught it lol I had pressure instead of density for [itex]\rho[/itex] should be density not pressure oops lol

 

[Mordred]

Universe geometry

The origins of the universe is unknown in cosmology. The hot big bang model only covers the history of the universe from 10^-43 seconds forward. Prior to that is described as a singularity. However its important to note that the singularity is not a black hole style. Instead singularity in this case simply means a point in time where our mathematics can no longer accurately describe it. Numerous youtube videos and pop media articles would have you believe our universe exploded from some super particle. This was never predicted by the hot big bang model.

The observable universe which is the portion we can see is a finite, sphere with a radius of 46 Gly, which is equal to 46 billion light years. The 46 Gly particle horizon refers to the today's distance of objects, whose radiation emitted in the past we receive today. The overall size of the universe is not known, it could be infinite or finite. If its infinite now then it would be infinite in the past, a finite value can never become infinite. So why is geometry so important to cosmology if we know the size of the observable universe? The answer to that question lies in how geometry affects the following aspects, Light paths, rate of expansion or collapse and overall shape.

In regards to light paths and geometry a closed universe described as a sphere will have two beams of light emitted at different angles eventually converge. An open hyperbolic universe such as a saddlebag will have those same two light beams diverge. A flat universe will have parallel light paths (provided the beams at emission were parallel to begin with)
You will notice on each image there is a triangle, this triangle represents how the geometry affects our measurements. In a flat curvature the three angles of a equilateral triangle will add up to 180⁰. A positive curvature will add up to greater than 180⁰, a negative curvature will add up to less than 180⁰

Image from http://universeadventure.org

The topography of the universe is determined by a comparison of the actual density (total density) as compared to the critical density. The critical density is represented by the following formula

[itex]\rho_{crit} = \frac{3c^2H^2}{8\pi G}[/itex]

[itex]\rho[/itex]=energy/mass density
c=speed of light
G= gravitational constant.

density is represented by the Greek letter Omega [itex]\Omega[/itex] so critical density is [itex]\Omega crit[/itex]
total density is

[itex]\Omega[/itex]total=[itex]\Omega[/itex]_{dark matter}+[itex]\Omega[/itex]_baryonic+[itex]\Omega[/itex]_radiation+[itex]\Omega[/itex]_{relativistic radiation}+[itex]{\Omega_ \Lambda}[/itex]

[itex]\Lambda[/itex] or Lambda is the value of the cosmological constant often referred to as "dark energy" more accurately it is the vacuum pressure that attributes to expansion.
the subscript "₀"for [itex]\Omega[/itex] shown in the image above denotes time in the present.

Energy-density is the amount of energy stored per unit volume of space or region. Energy per unit volume has the same physical units as pressure, the energy or mass density to pressure relations are defined by the equations of state (Cosmology). see
http://en.wikipedia.org/wiki/Equation_of_state_(cosmology)

[itex]\Omega=\frac{P_{total}}{P_{crit}}[/itex]
or alternately
[itex]\Omega=\frac{\Omega_{total}}{\Omega_{crit}}[/itex]

Geometry in 2D
In developing a theory of space-time, where curvature is related to the mass-energy density, Scientists needed a way of mathematically describing curvature. Since picturing the curvature of a four-dimensional space-time is difficult to visualize. We will start by considering ways of describing the curvature of two-dimensional spaces and progress to 4 dimensional spaces.
The simplest of two-dimensional spaces is a plane, on which Euclidean geometry holds.
This is the geometry that we learned in high school: parallel lines will go off to infinity
without ever crossing; triangles have interior angles that add up to 180. Pythagoras’
theorem which relates the lengths of the sides of a right triangle also holds:
c² = a² + b²
where c is the length of the hypotenuse of the right triangle, and a and b are the
lengths of the other two sides. One can generalize the Pythagorean theorem to three dimensions as well:
c²= a² + b² + c²
see image 2.0 below On a plane, a "geodesic" is a straight line(shortest distance between two points). If a triangle is constructed on a flat 2 dimensional plane by connecting three points with geodesics. The curvature can be represented in 2D, if you establish each angle of a equilateral triangle with
[itex]\alpha[/itex],[itex]\beta[/itex],[itex]\gamma[/itex] for a flat geometry this follows the relation

[itex]\alpha[/itex]+[itex]\beta[/itex]+[itex]\gamma[/itex]=[itex]\pi[/itex].

image 1.0

On a plane, (shown above) we can set up a cartesian coordinate system, and assign to every point a coordinate (x; y). On a plane, the distance ds between points (dx and dy) is given by the relation
[itex]d{s^2}=d{x^2}+d{y^2}[/itex]

If a triangle is constructed on the surface of the sphere by connecting the angles will obey the relation

[itex]\alpha[/itex]+[itex]\beta[/itex]+[itex]\gamma[/itex]=[itex]\pi+{AR^2}[/itex]

image 1.1
where A is the area of the triangle, and R is the radius of the sphere. All spaces in which
[itex]\alpha[/itex]+[itex]\beta[/itex]+[itex]\gamma[/itex]>[itex]\pi[/itex] are called positively curved" spaces. It is a space where the curvature is homogeneous and isotropic; no matter where you draw a triangle on the surface of a sphere, or how you orient it, it must always satisfy the above equation.
"On the surface of a sphere, we can set up polar coordinates "north pole" and "south pole" and by picking a geodesic from the north to south pole to be the "prime meridian". If r is the distance from the north pole, and [itex]\theta[/itex] is the azimuthal. angle measured relative to the prime meridian,"⁽¹⁾ then the distance ds between a point (r; [itex]\theta[/itex]) and another nearby point (r+dr+[itex]\theta[/itex]+d[itex]\theta[/itex]) is given by the relation

[itex]{ds^2} = {dr^2} + {R^2} {sin^2}(r/R)d\theta^2[/itex]

"An example of a negatively curved two-dimensional space is the hyperboloid, or saddle-shape. A surface of constant negative curvature. The saddle-shape has constant curvature only in the central region, near the "seat" of the saddle."⁽¹⁾. David Hilbert proved that a constant negative curvature cannot be constructed in a Euclidean 3D space. Consider a two-dimensional surface of constant negative curvature, with radius of curvature R. If a triangle is constructed on this surface by connecting three points with geodesics, the angles at its vertices [itex]\alpha[/itex]
[itex]\beta[/itex],[itex]\gamma[/itex] obey the relation [itex]\alpha[/itex]+[itex]\beta[/itex]+[itex]\gamma[/itex]=[itex]\pi-{AR^2}[/itex].

[itex]{ds^2} = {dr^2} + {R^2} {sinH^2}(r/R)d\theta^2[/itex]

image 1.2

A negative curvature is an open topography

If a two-dimensional space has curvature or flat which is homogeneous and isotropic, its geometry can
be specified by two quantities k, and R. The number k, called the curvature constant, R is the radius

k = 0 for a flat space,
k = +1 for a positively curved space,
k = -1 for a negatively curved space

Geometry in 3D
A two dimensional space can be extended to a three-dimensional space, if its curvature is homogeneous and isotropic, must be flat, or have uniform positive curvature, or have uniform negative curvature.

The 3 possible metrics for homogeneous and isotropic 3D geometries can be represented in the form ds²=dr²=S_k(r)²dΩ²

where
dΩ²=dθ²=sin²d[itex]\phi[/itex]²[tex]
S\kappa(r)=
\begin{cases}
R sin(r/R &(k=+1)\\
r &(k=0)\\
R sin(r/R) &(k=-1)
\end {cases}
[/tex]

If a three-dimensional space is flat (k = 0), it
has the metric

ds² = dx² + dy² + dz² ;

expressed in cartesian coordinates or

[itex]{ds^2} = {dr^2} +{r^2}[d\theta^2 + {sin^2} d\phi^2][/itex]

If a three-dimensional space has uniform positive curvature (k = +1), its
metric is

[itex]{ds^2} = {dr^2} +{R^2}{sin^2}(r/R)[d\theta^2 + {sin^2}\theta d\phi^2][/itex]

A negative curvature in the uniform portion has the metric (k=-1)

[itex]{ds^2} = {dr^2} +{R^2}{sinH^2}(r/R)[d\theta^2 + {sin^2}\theta d\phi^2][/itex]

Geometry in 4D

Thus far we have discussed the 2 and 3 dimensional components. The Friedmann-Lemaitre-Robertson-Walker metric (FLRW) can be used to describe the 4D dimensions with the use of a(t). a(t) is the scale factor. See the redshift and expansion article for more information or the cosmocalc.
http://cosmology101.wikidot.com/redshift-and-expansion
http://cosmocalc.wikidot.com/start

Scale factor in a homogeneous and isotropic universe describes how the universe expands or contracts with time.
The FLRW metric can be written in the form

[itex]d{s^2}=-{c^2}d{t^2}+a({t^2})[d{r^2}+{S,k}{(r)^2}d\Omega^2][/itex]

references
(1)"Introductory to Cosmology" Barbera Ryden"
images 1.0,1.1 and 1.2 (see (1))
(2)"Modern Cosmology" Scott Dodelson
(3)"lecture notes, Introductory to Cosmology" Dr. Ka Chan Lu

 

[Mordred]

there corrections applied

 

[Ken G]

I can understand space as simply being 'distances' between objects, and that gravity affects matter and energy directly, and that no 'actual' warping of a non-existing spatial entity occurs, and that all of this is an aid to help us visualize it. But what about two parallel laser beams? Why would they eventually meet, if all there is between them is distance, and no existing, warped space (as an actual entity)?

It sounds like you are saying you want the spacetime manifold to be a real entity. What I'm saying is, we don't want to attribute it any real properties than are more than we need for it to support GR, since GR is the only tested theory we have here. I'm not a mathematician, and they are the ones who keep careful track of what aspects you need to keep the same on the various possible manifolds and metrics you could have that would induce all the same observable physics. All I'm saying is that there is one example which pretty obviously induces the same physics, which is a universe that globally respects a cosmological principle, and either has space expanding with age, or bound systems contracting with age. The "entity" of the spacetime manifold might sound like it is doing two different things in those cases, but it could just be the same thing seen from a different perspective. The distinctions are superfluous, like looking at a painting from different angles. What do we usually do with superfluous elements? Ask the aether!

 

[phinds]

I'm still confused, it doesn't look like CosmicVoyager was having a problem with semantics. By space being a "thing," I'm sure he didn't mean a material object, as in made of particles or waves, but rather an entity of a different nature.
If space is not to be thought of as an entity (of whichever nature,) then why would two parallel lines eventually meet (given that the geometry of space allowed for it, I'm by no means trying to discuss this subject) if there is nothing but movement going on? I can understand space as simply being 'distances' between objects, and that gravity affects matter and energy directly, and that no 'actual' warping of a non-existing spatial entity occurs, and that all of this is an aid to help us visualize it. But what about two parallel laser beams? Why would they eventually meet, if all there is between them is distance, and no existing, warped space (as an actual entity)?
I though, precisely, that accepting the existence of an entity (distinct to matter and energy) would provide a mechanism (I'd say medium if it weren't for that darned aether) for this sort of things to happen.

To vastly simplify Mordred's excellent exposition, your problem is that you are thinking of "parallel lines" as being Euclidean (in flat space) but in cosmology they are not. In the real world, things travel on geodesics (the cosmological equivalent of straight lines) and geodesics can diverge and meet in ways that Euclidean parallel lines cannot.

 

[Frank Weyl]

Even then we like our laws to have a degree of elegance to them. Laws that emerge from symmetry, ideally.

Hello craigi,
Could I ask if you were specifically referring to symmetry as: eightfold, global,local, patterns, symmetry break(ing) or symmetry groups per se?

 

[Frank Weyl]

Universe geometry

The origins of the universe is unknown in cosmology. The hot big bang model only covers the history of the universe from 10^-43 seconds forward. Prior to that is described as a singularity. However its important to note that the singularity is not a black hole style. Instead singularity in this case simply means a point in time where our mathematics can no longer accurately describe it. Numerous youtube videos and pop media articles would have you believe our universe exploded from some super particle. This was never predicted by the hot big bang model.

The observable universe which is the portion we can see is a finite, sphere with a radius of 46 Gly, which is equal to 46 billion light years. The 46 Gly particle horizon refers to the today's distance of objects, whose radiation emitted in the past we receive today. The overall size of the universe is not known, it could be infinite or finite. If its infinite now then it would be infinite in the past, a finite value can never become infinite. So why is geometry so important to cosmology if we know the size of the observable universe? The answer to that question lies in how geometry affects the following aspects, Light paths, rate of expansion or collapse and overall shape.

In regards to light paths and geometry a closed universe described as a sphere will have two beams of light emitted at different angles eventually converge. An open hyperbolic universe such as a saddlebag will have those same two light beams diverge. A flat universe will have parallel light paths (provided the beams at emission were parallel to begin with)
You will notice on each image there is a triangle, this triangle represents how the geometry affects our measurements. In a flat curvature the three angles of a equilateral triangle will add up to 180⁰. A positive curvature will add up to greater than 180⁰, a negative curvature will add up to less than 180⁰

Image from http://universeadventure.org

The topography of the universe is determined by a comparison of the actual density (total density) as compared to the critical density. The critical density is represented by the following formula

[itex]\rho_{crit} = \frac{3c^2H^2}{8\pi G}[/itex]

[itex]\rho[/itex]=energy/mass density
c=speed of light
G= gravitational constant.

density is represented by the Greek letter Omega [itex]\Omega[/itex] so critical density is [itex]\Omega crit[/itex]
total density is

[itex]\Omega[/itex]total=[itex]\Omega[/itex]_{dark matter}+[itex]\Omega[/itex]_baryonic+[itex]\Omega[/itex]_radiation+[itex]\Omega[/itex]_{relativistic radiation}+[itex]{\Omega_ \Lambda}[/itex]

[itex]\Lambda[/itex] or Lambda is the value of the cosmological constant often referred to as "dark energy" more accurately it is the vacuum pressure that attributes to expansion.
the subscript "₀"for [itex]\Omega[/itex] shown in the image above denotes time in the present.

Energy-density is the amount of energy stored per unit volume of space or region. Energy per unit volume has the same physical units as pressure, the energy or mass density to pressure relations are defined by the equations of state (Cosmology). see
http://en.wikipedia.org/wiki/Equation_of_state_(cosmology)

[itex]\Omega=\frac{P_{total}}{P_{crit}}[/itex]
or alternately
[itex]\Omega=\frac{\Omega_{total}}{\Omega_{crit}}[/itex]

Geometry in 2D
In developing a theory of space-time, where curvature is related to the mass-energy density, Scientists needed a way of mathematically describing curvature. Since picturing the curvature of a four-dimensional space-time is difficult to visualize. We will start by considering ways of describing the curvature of two-dimensional spaces and progress to 4 dimensional spaces.
The simplest of two-dimensional spaces is a plane, on which Euclidean geometry holds.
This is the geometry that we learned in high school: parallel lines will go off to infinity
without ever crossing; triangles have interior angles that add up to 180. Pythagoras’
theorem which relates the lengths of the sides of a right triangle also holds:
c² = a² + b²
where c is the length of the hypotenuse of the right triangle, and a and b are the
lengths of the other two sides. One can generalize the Pythagorean theorem to three dimensions as well:
c²= a² + b² + c²
see image 2.0 below


On a plane, a "geodesic" is a straight line(shortest distance between two points). If a triangle is constructed on a flat 2 dimensional plane by connecting three points with geodesics. The curvature can be represented in 2D, if you establish each angle of a equilateral triangle with
[itex]\alpha[/itex],[itex]\beta[/itex],[itex]\gamma[/itex] for a flat geometry this follows the relation

[itex]\alpha[/itex]+[itex]\beta[/itex]+[itex]\gamma[/itex]=[itex]\pi[/itex].

image 1.0

On a plane, (shown above) we can set up a cartesian coordinate system, and assign to every point a coordinate (x; y). On a plane, the distance ds between points (dx and dy) is given by the relation
[itex]d{s^2}=d{x^2}+d{y^2}[/itex]

If a triangle is constructed on the surface of the sphere by connecting the angles will obey the relation

[itex]\alpha[/itex]+[itex]\beta[/itex]+[itex]\gamma[/itex]=[itex]\pi+{AR^2}[/itex]

image 1.1
where A is the area of the triangle, and R is the radius of the sphere. All spaces in which
[itex]\alpha[/itex]+[itex]\beta[/itex]+[itex]\gamma[/itex]>[itex]\pi[/itex] are called positively curved" spaces. It is a space where the curvature is homogeneous and isotropic; no matter where you draw a triangle on the surface of a sphere, or how you orient it, it must always satisfy the above equation.
"On the surface of a sphere, we can set up polar coordinates "north pole" and "south pole" and by picking a geodesic from the north to south pole to be the "prime meridian". If r is the distance from the north pole, and [itex]\theta[/itex] is the azimuthal. angle measured relative to the prime meridian,"⁽¹⁾ then the distance ds between a point (r; [itex]\theta[/itex]) and another nearby point (r+dr+[itex]\theta[/itex]+d[itex]\theta[/itex]) is given by the relation

[itex]{ds^2} = {dr^2} + {R^2} {sin^2}(r/R)d\theta^2[/itex]

"An example of a negatively curved two-dimensional space is the hyperboloid, or saddle-shape. A surface of constant negative curvature. The saddle-shape has constant curvature only in the central region, near the "seat" of the saddle."⁽¹⁾. David Hilbert proved that a constant negative curvature cannot be constructed in a Euclidean 3D space. Consider a two-dimensional surface of constant negative curvature, with radius of curvature R. If a triangle is constructed on this surface by connecting three points with geodesics, the angles at its vertices [itex]\alpha[/itex]
[itex]\beta[/itex],[itex]\gamma[/itex] obey the relation [itex]\alpha[/itex]+[itex]\beta[/itex]+[itex]\gamma[/itex]=[itex]\pi-{AR^2}[/itex].

[itex]{ds^2} = {dr^2} + {R^2} {sinH^2}(r/R)d\theta^2[/itex]

image 1.2

A negative curvature is an open topography

If a two-dimensional space has curvature or flat which is homogeneous and isotropic, its geometry can
be specified by two quantities k, and R. The number k, called the curvature constant, R is the radius

k = 0 for a flat space,
k = +1 for a positively curved space,
k = -1 for a negatively curved space

Geometry in 3D
A two dimensional space can be extended to a three-dimensional space, if its curvature is homogeneous and isotropic, must be flat, or have uniform positive curvature, or have uniform negative curvature.

The 3 possible metrics for homogeneous and isotropic 3D geometries can be represented in the form ds²=dr²=S_k(r)²dΩ²

where
dΩ²=dθ²=sin²d[itex]\phi[/itex]²


[tex]
S\kappa(r)=
\begin{cases}
R sin(r/R &(k=+1)\\
r &(k=0)\\
R sin(r/R) &(k=-1)
\end {cases}
[/tex]

If a three-dimensional space is flat (k = 0), it
has the metric

ds² = dx² + dy² + dz² ;

expressed in cartesian coordinates or

[itex]{ds^2} = {dr^2} +{r^2}[d\theta^2 + {sin^2} d\phi^2][/itex]

If a three-dimensional space has uniform positive curvature (k = +1), its
metric is

[itex]{ds^2} = {dr^2} +{R^2}{sin^2}(r/R)[d\theta^2 + {sin^2}\theta d\phi^2][/itex]

A negative curvature in the uniform portion has the metric (k=-1)

[itex]{ds^2} = {dr^2} +{R^2}{sinH^2}(r/R)[d\theta^2 + {sin^2}\theta d\phi^2][/itex]

Geometry in 4D

Thus far we have discussed the 2 and 3 dimensional components. The Friedmann-Lemaitre-Robertson-Walker metric (FLRW) can be used to describe the 4D dimensions with the use of a(t). a(t) is the scale factor. See the redshift and expansion article for more information or the cosmocalc.
http://cosmology101.wikidot.com/redshift-and-expansion
http://cosmocalc.wikidot.com/start

Scale factor in a homogeneous and isotropic universe describes how the universe expands or contracts with time.
The FLRW metric can be written in the form

[itex]d{s^2}=-{c^2}d{t^2}+a({t^2})[d{r^2}+{S,k}{(r)^2}d\Omega^2][/itex]

references
(1)"Introductory to Cosmology" Barbera Ryden"
images 1.0,1.1 and 1.2 (see (1))
(2)"Modern Cosmology" Scott Dodelson
(3)"lecture notes, Introductory to Cosmology" Dr. Ka Chan Lu

thank-you Mordred for your elegant solution,
Question: Would your triangle have the same topological 'argument' placed upon a mobius or Riemann sphere..?(!).

 

[craigi]

Hello craigi,
Could I ask if you were specifically referring to symmetry as: eightfold, global,local, patterns, symmetry break(ing) or symmetry groups per se?

I wasn't referring to any particular instance of symmetry in the laws of physics. Simply that we prefer that laws of nature are fundamentally due to symmetry, to laws that have a more complex form or laws that have unexplained constants.

 

[Frank Weyl]

I wasn't referring to any particular instance of symmetry in the laws of physics. Simply that we prefer that laws of nature are fundamentally due to symmetry, to laws that have a more complex form or laws that have unexplained constants.

Hello again craigi,
AS you have probably realized I am interested in gauge fields, superstrings and symmetry groups. ( gauge fields,as you know, are related to the structure of space-time itself),
I was unhappy with the SU(2) x U(1) which was built out of the symmetry group SU(2) , which describes the weak nuclear force, and U(2) for the electromagnetic field.
Then along came the the new symmetry which was made by combining SU(3)--corresponding to the gluon force--with SU(2) X U(1) for the electroweak force.
SU(5) didn't last very long at the expansion (Big bang) as it separated into two groups very rapidly.
Also The SU(5) predicted new bosons with enormous masses---10^15 times bigger than the mass of the proton. Also the next SU(10) group had implications that the normal left-handed neutrino has a right-handed partner of enormous mass...
Still unhappy!
Along came Schwarz and Green and gave us a single choice of symmetry with elegance..the beautiful SO(32) symmetry.
Very happy!

 

[Mordred]

thank-you Mordred for your elegant solution,
Question: Would your triangle have the same topological 'argument' placed upon a mobius or Riemann sphere..?(!).

yes, Riemann geometry, is used extensively in the 4 dimensional geometry aspects of space-time geometry. The FLRW metrics can be converted to a variety of differential geometry forms. Though the proper uses of each must follow GR and SR rules, the FLRW metric is an exact solution to the Einstein field equations.

this lengthy articles shows the usages and risks involved in the various differential geometry forms. As well as covering the FLRW metric aspects in the later chapters.

http://www.blau.itp.unibe.ch/newlecturesGR.pdf

 

[bapowell]

Along came Schwarz and Green and gave us a single choice of symmetry with elegance..the beautiful SO(32) symmetry.
Very happy!

Except that it fails miserably to reproduce the Standard Model...

 

[Drakkith]

thank-you Mordred for your elegant solution,
Question: Would your triangle have the same topological 'argument' placed upon a mobius or Riemann sphere..?(!).

On a side note, please don't quote the entirety of very long posts. It just clutters up the thread.

 

[Mordred]

On a side note, please don't quote the entirety of very long posts. It just clutters up the thread.

just to add to this you can refer to a specific post by clicking the post number in the top right. of that post, it will open a new internet window then just copy and paste the address
for example using the Geometry article post

https://www.physicsforums.com/showpost.php?p=4720016&postcount=86

not that I mind seeing my articles posted :P

edit one other PF aid. this post covers how to use the Latex commands to type mathematical expressions for this site
https://www.physicsforums.com/showpost.php?p=3977517&postcount=3

 

[Drakkith]

just to add to this you can refer to a specific post by clicking the post number in the top right. of that post, it will open a new internet window then just copy and paste the address
for example using the Geometry article post.

Son of a... I didn't know that Mordred... thanks!

 

[Mordred]

Son of a... I didn't know that Mordred... thanks!

no problem, its useful for large posts such as the Universe geometry and Expansion and redshift article. LOL coincidentally the method was showed to me by one of the moderators. To reduce clutter of my reposting that very same article.

 

[ptalar]

Space is simply volume filled with matter and energy, we have tried explaining that to you numerous times. Space itself is not a material. It is simply volume filled with matter and energy. However space itself does not have a fabric and is not a form of energy or matter, it is simply volume filled with the matter and energy content of the universe.

If this is the case then why does space expand faster than the speed of light? If the Universe is 13.8 billion years old we should only see back in time to 13.8 billion light years. Yet the visible Universe is roughly 46 billion light years. The rebuttal, as I understand it, is that space can expand faster than light because it is nothingness. Nothingness does not follow the laws of physics.

But it was said above that space, essentially, includes matter and energy. So it is not nothingness. Is not this matter and energy subject to the speed of light limitation, or is the universe expanding (creating space) into an existing space of matter and energy? Not clear to me. Or are we alluding to Dark Energy and Dark Matter which is called Dark because we do not understand it yet. But then I am just a lowly mechanical engineer. But I absolutely enjoy reading these threads. I do get a lot of insight from them.

 

[Drakkith]

If this is the case then why does space expand faster than the speed of light?

A few things about this. First, expansion is measured by a RATE, not a speed. By that I mean that it doesn't make sense to talk about the speed of expansion because that speed will change depending on how far away your two points of comparison are. For example, galaxies recede from each other at an increasing velocity of about 70 km/s for every megaparsec (Mpc) they are apart. So two galaxies 10 Mpc's apart will be receding from each other at approximately 700 km/s.

However, the RATE of expansion does not change in this manner. The time it takes for the distance between them to double is the same whether they are 10 Mpc's apart or 10,000 Mpc's apart.

Also, it gets us nowhere by talking about the expansion of "space itself". In reality, expansion is measured by comparing actual objects that exist within spacetime.

If the Universe is 13.8 billion years old we should only see back in time to 13.8 billion light years. Yet the visible Universe is roughly 46 billion light years.

There is no contradiction here. We CANNOT see back further than 13.8 billion years because light has not had time to travel any longer than that. (A little shorter than 13.8 billion years actually. Light wasn't free to travel until after Recombination occurred around 378,000 years after the big bang and the universe became transparent to EM radiation)

However, we have measured the radius of the observable universe and it is indeed approximately 46 billion light years from Earth to the edge, making it about 92 billion light years in diameter. Note that years is a measurement of time, while light years is a measurement of distance.

Understand that to measure the radius of the observable universe requires us to understand how expansion works, otherwise we would get the wrong value. In a non-expanding, static universe, the diameter of the observable universe would only be about 13.8 billion light years, increasing by one light year ever year. However, in an expanding universe, the galaxies we see now have actually receded from us since their light left them, making the observable universe larger than it would appear to be if you only consider the time that light has had to travel.

The argument is that space can expand faster than light because it is nothingness. Nothingness does not follow the laws of physics. But you said space includes matter and energy. Which is it?

That is not the argument. Our primary theory for understanding the universe on its largest scales is General Relativity. Under the rules of GR, the expansion is the result of dynamic geometry, not the result of space expanding as if space were something that was actually moving. Space is not moving. The concept of space being something that can move does not even apply under GR. We can set up frames of reference at different points within spacetime and watch the effects that this dynamic geometry has on them, but we cannot see "spacetime itself" nor can we assign a frame of reference to it since it is the underlying framework upon which everything occurs.

 

[Ken G]

Also note that, once again, just imagining that matter shrinks makes that question go away as well. Poof, no "where does the space come from," no "how can objects move away from each other faster than c." This doesn't mean matter "really does" shrink, any more than space "really expands", it just makes the point that these are all just pictures and cannot be taken seriously enough to worry about questions like these. Questions that persist in all perspectives are the "real" questions-- questions that stem from a particular picture, but go away in a different picture, are seen to stem from the pictures, not the physics.

 

[craigi]

The rebuttal, as I understand it, is that space can expand faster than light because it is nothingness. Nothingness does not follow the laws of physics.

Not quite. The laws of physics are models and we do model nothingness. In fact, it's important to explain the expansion of the universe.

There's a common misconception that claims that there is nothing that can travel faster than the speed of light. In the model of special relativity, information and matter can't travel faster than the speed of light, but that isn't to say that there isn't anything that can. To illustrate this, if you project an image off into the distance and rotate your projector, there is a distance beyond which the projected image is traveling faster than the speed of light. The image is still constructed from the interaction of matter and information is still transmitted, but no matter or information moves faster than the speed of light, nevertheless the projected image can travel at unbounded speeds.

 

[bapowell]

If this is the case then why does space expand faster than the speed of light? If the Universe is 13.8 billion years old we should only see back in time to 13.8 billion light years. Yet the visible Universe is roughly 46 billion light years. The rebuttal, as I understand it, is that space can expand faster than light because it is nothingness. Nothingness does not follow the laws of physics.

No, the observable universe is 46 byl across because it is expanding. Simple as that. As space expands, it brings photons along with it: as they recede from Earth they have a velocity [itex]v_{\rm rec} + c[/itex]: the first term is that due to the Hubble expansion, and the second, [itex]c[/itex], is the local speed of light. Of course, as the universe expands [itex]v_{\rm rec}[/itex] continuously changes. If you integrate this speed over the time the universe has existed (13.6 by), you get something larger than 13.6 bly because photons recede with speeds surpassing that of light the whole time. The universe itself "expanding faster than light speed" has nothing to do with it, and as Drakkith has explained, is not a correct way to think of expansion.

 

[Mordred]

If this is the case then why does space expand faster than the speed of light? If the Universe is 13.8 billion years old we should only see back in time to 13.8 billion light years. Yet the visible Universe is roughly 46 billion light years. The rebuttal, as I understand it, is that space can expand faster than light because it is nothingness. Nothingness does not follow the laws of physics.

But it was said above that space, essentially, includes matter and energy. So it is not nothingness. Is not this matter and energy subject to the speed of light limitation, or is the universe expanding (creating space) into an existing space of matter and energy? Not clear to me. Or are we alluding to Dark Energy and Dark Matter which is called Dark because we do not understand it yet. But then I am just a lowly mechanical engineer. But I absolutely enjoy reading these threads. I do get a lot of insight from them.

The others have already answered the majority of your questions. However I would add the term "dark" in both dark energy and dark matter are kind of stuck to science. Although we do not completely understand them. There is a lot we do. We may not know what dark matter is, however we know how it influences the universe mass density calculations. In fact dark matter is the larger gravitational influence. Baryonic (what stars etc are made of) matter is a small fraction of the gravitational influence. In Dark energy the mystery is more that we do not know what mechanism that keeps the density of dark energy constant. As a negative pressure influence its effects are easily understood in regards to expansion. Dark energy is a contributor to the cosmological constant. Matter influence is positive vacuum, the cosmological constant is a negative influence.

Due to the ratio of dark matter and baryonic matter, dark matter is the largest contributor to positive vacuum. Due to the sheer volume of space the cosmological constant is the largest contributor to the negative vacuum pressure. So yes much of the dynamics of expansion is primarily due to the pressure relations of the cosmological constant and dark matter. More so than other forms of energy and baryonic matter

P.S I'm just a lowly Electronic controls tech, I simply spent several years studying cosmology textbooks, and asking questions on PF. Cosmology is my favorite hobby lol. Although some of the posters on this thread are physicists. I won't say who, that is their privilege to divulge

hint the tools and links to understand basic cosmology have already been posted throughout this one thread. So they don't need to be reposted. However my signature also contains many of those tools on the http://cosmology101.wikidot.com/main link. The http://www.einsteins-theory-of-relativity-4engineers.com/LightCone7/LightCone.html. Provides a handy calculator to understand the history and future expansion history. http://cosmocalc.wikidot.com/start is where to find the information on how to use that calculator