Redshift z

[quantumtopology]

Sorry, I wrote space-like where I meant time-like.

This is what ned wright tutorial says: "We have already seen that the zero density case has hyperbolic geometry, since the cosmic time slices in the special relativistic coordinates were hyperboloids in this model."

And that is what I meant, any light cone is a space-time diagram, anytthing with an coordinate for time and another for space is a spacetime diagram in this sense.

What I pointed out is that the figure from ned's tutorial refers to a special relativistic scenario (that he identifies with a zero density cosmology).

What CC presents is a different type of spacetime diagram with radial coordinates that have both space and time or so I see it.

Let`s not obfuscate. :shrug:

[modest]

Sorry, I wrote space-like where I meant time-like.

This is what ned wright tutorial says: "We have already seen that the zero density case has hyperbolic geometry, since the cosmic time slices in the special relativistic coordinates were hyperboloids in this model."

 

I see. That is, indeed, what it means for space to be hyperbolic. It means that a slice of space in spacetime is an hyperbola. A triangle on that surface will have angles less than 180 degrees.

 

And that is what I meant, any light cone is a space-time diagram, anytthing with an coordinate for time and another for space is a spacetime diagram in this sense.

 

Yes, one dimension is space and another is time. This is the foundation of general relativity.

 

What CC presents is a different type of spacetime diagram with radial coordinates that have both space and time or so I see it.

 

I don't mind if the physical meaning of the diagram corresponds to radial coordinates or Cartesian, I would just like to know what that physical meaning is. Does the angle from the apex correspond to an angle in space as viewed from earth, or maybe it corresponds to a distance as measured from earth, or a time measured between events.

 

How would you use CC's diagram to measure the angular distance between galaxies as viewed from earth? Or, the distance between galaxies as measured by a ruler or the time between events as measured by a clock? I don't know these things. I don't know how the diagram is supposed to correspond to reality.

 

It would be very exciting and worthwhile to get that figured out.

 

~modest

[coldcreation]

Perhaps we could clarify things if I ask: the number of galaxies along a radial line between two outer concentric circles is more than, less than, or the same as the number of galaxies along a radial line between two inner concentric rings...

 

Interesting question and certainly a fascinating topic.

 

First, most (if not all) non-expanding models have no big bang 13.7 billion years ago. The universe would have no age, no beginning. So evolution will have transpired very slowly (the process is much slower, to say the least, than allowed by expanding models). Meaning, the galaxy count at great distances should be very similar to the count observed locally. Detecting evolution in the look-back time would surely be an important step in determining which models can be ruled out.

 

Briefly, to answer your question, there would be virtually no difference, or a very modest difference, in the number of galaxies located, say in the outer spherical shell of Figure 4A as compared to the inner shells. Certainly curvature of the manifold would not change the galaxy count, i.e., there would not be less galaxies near the horizon as a result of hyperbolicity. The only effect that geometry would induce would be to make the surface brightness of galaxies decease rapidly with apparent distance; galaxies would appear dimmer than would expected in a flat Euclidean spacetime, with increasing redshift. So galaxies near the horizon would be very faint. The faintness would make them difficult to count, and some would evade detection, thus making the galaxy count unreliable, i.e., there would be systematic uncertainties in deriving the observational galaxy counts. That's the short answer.

 

It would be a great idea to start a new thread on the topic of galaxy counts and evolution backward in time. I do recall one a while back, where I made the prediction that very little if any, evolution would be found in the look-back time. The physical properties, chemical abundance, metallicity, would be practically the same as those observed locally.

 

Here is a great paper of the subject, for those interested: Near-Infrared Faint Galaxies in the Subaru Deep Field: Comparing the Theory with Observations for Galaxy Counts, Colors, and Size Distributions to K 24.5. There are other as well. One of the best studies of galaxy counts can arguably be found in this book: The Deep universe: Saas-Fee Advanced Course 23, lecture notes 1993, by Sandage, Kron, Longair, et al.

 

 

 

If you want the light-travel distance then the question remains: how do you know by looking at a galaxy for how long the light traveled? [snip]

Distance measures (cosmology) From Wiki: Light travel time or lookback time. This is how long ago light left an object of given redshift. And: Light travel distance (LTD). The light travel time times the speed of light. Of course there is no fool proof method. The above would need to be supplemented when possible with angular diameter distance, luminosity distance, comoving distance, cosmological proper distance, and why not, Hubble's law (taking z = H0d/c, with H0 today's Hubble constant, z the redshift of the object, c the speed of light, and d the "distance"), for more accurate distance measurements.

 

It seems SNe Ia would be the best objects for measuring distance, and curvature, in a spacetime continuum of the type described by GR.

 

 

In essence: by what exact process can you look at a galaxy through a telescope, make measurements, and decide where to put it on your diagram? How do you know the look-back time or the light-travel time distance.

 

There is no need to invent a new method of determining distances in a curved spacetime manifold. The current methods seem to be as good as it gets, considering current technology and potential theoretical limitations. I would venture as to say that generally, distances would likely be further away than currently suspected. But that is just a hunch. The point remains, distances (in the look-back time) of an expanding universe should be very similar to distances in one that is non-expanding.

 

 

The relationship between angular diameter and angular diameter distance depends on the shape, or the curvature, of the manifold (spherical, flat, or hyperbolic). So, you cannot look through a telescope and decide how far a galaxy is by measuring its diameter without assuming the curvature of space.

 

The same is true of the brightness and redshift.

 

That is why a combination of methods for distance measurements must be used. I would think that al measurement technique should be compared to a standard condition: that of Euclidean geometry. If the relationship between angular diameter and angular diameter distance deviates from that which would be expected in a flat spacetime, then determinations and conclusions could be arrived at on the geometric structure of spacetime.

 

 

I think this gets at the heart of things. You presented diagrams A, B, and C saying that they represent observation alone (model-independent so to speak). But, I really think we need to decide what exact observation, or method of observation, is meant to end up looking like one of those diagrams when done.

 

Again, the geometries shown in diagrams A, B and C cannot all be correct. Two of those manifolds will have to be discarded. Two of them will end up in disagreement with observations when put to the test. As it turn out two of the models are already in disaccord with observations (if we are to consider only static spacetime manifold, i.e., non-expanding spaces). Manifold B can be discarded right off the bat, since redshift z would not be observed in a static Euclidean universe (without some for of tired light hypothesis operational, but that is not observed). Manifold B, too, can be discarded off the bat, since it is in disagreement with observations. While it is certain that redshift z would be present in a geometrically spherical universe, it would violate observations with respect to redshift and light curves of SNe Ia.

 

It is true that a full fledged static model consistent with general relativity and theoretical expectations is nonexistent at this time (as far as I know), there are certainly avenues that have been explored and models developed. These divers models have made predictions regarding redshift z, the CMB, galaxy formation and evolution, as well as the abundance of light elements. So progress has been made. But there is still much needed to ensure the viability of such static solutions.

 

I am not discounting the possibility that he universe is expanding, or even accelerating, but it would be foolish on my part to discount a static solution that seems to mimic current observations. It can be demonstrated that non-expanding general relativistic spacetimes reproduce identical cosmological observations as the currently favored theories (G.F.R. Ellis 1978, for example, who developed a model with a spherically symmetric static general relativistic cosmological spacetime). The idea that light is redshifted because it is traveling through a globally curved, four dimensional spacetime continuum was also proposed by I. E. Segal (with a four-dimensional globally hyperbolic (curved), pseudo Riemannian temporal evolution of the spacetime manifold) called chronometric cosmology. The chronometric redshift resembled Weyl’s 1921 model, in which time variations in the non-static case are interpreted as spatial variations in the static case.

 

Where these models flaw-free. Absolutely not. Were they tested against observations. Segal's certainly was. At the time, before the SNe Ia survey(s), the model was deemed inconsistent with observations because it predicted a quadratic redshift-distance relation. Does that mean all static solutions based on GR are untenable? Most certainly not.

 

 

The question I'd like to respond to now is how, physically, does a homogenous and isotropic four-dimensional spherically symmetric globally curved geometrically hyperbolic pseudo-Lobachevskian spatiotemporal manifold, mimic observations currently understood as an accelerated expansion in a flat (or virtually Euclidean) space. And if these models (or something similar, such as represented in Figure 4A) does indeed mimic observations, how can define the shape of the manifold based on observations, and how can we differentiate between two (or more) competing models that are, a priori, consistent with observations and indistinguishable one from the other.

 

This will be the subject of forthcoming posts.

 

 

 

Albert Einstein—The General Theory of Relativity—Chapter 22

Where gravitational fields are present the speed of light is not independent of the source.

 

It is true that general relativity can be interpreted or extrapolated to arbitrarily queer coordinate systems (i.e., they are allowed), that light follows geodesic paths and the constancy of the velocity of light with respect to these peculiar coordinates need not be interpreted as c. However, general relativity also asserts the existence of locally inertial frames, and the speed of light is a universal constant in those frames. The case under scrutiny here related to the latter, since we are talking about a homogenous and isotropic large scale nonzero gravitationally and hyperbolically curves spacetime as view from the locally inertial rest-frame of any observer.

 

So while your remark, and those you quoted by Herr Einstein, are correct in a sense, they are completely irrelevant for the purposes of measuring distances of astronomical objects such as galaxies, Cepheids or SNe Ia is a curved spacetime manifold (though they may be relevant for objects such as quasars and black holes).

 

 

You can measure the brightness and redshift (for example) of a galaxy and that does not directly tell you the curvature. There are multiple models with different curvatures which would give the same redshift / brightness ratio for a single galaxy.

 

If the distance between concentric circles in your diagram is the difference in look-back time then it remains unclear how exactly you want to measure that value in order to plot it without first assuming a particular model or assuming curvature (which is what you're trying to determine and would therefore be circular).

 

I really think, before we discuss the implications of these deviations from linearity we need to answer this first. How does one use a telescope or other instruments to plot data for your diagram—described well enough that we can do it with some real data?

 

This will be the subject of my next post.

 

 

I'd really like to first nail down what physical observations are represented in your diagram.

 

I don't think we're going to get anywhere until you can label a diagram and explain what physical measurements it corresponds to.

 

This too will be the subject of my next post.

 

 

 

 

CC

[coldcreation]

...

 

 

 

Hopefully Figure 6A will clear up a few misunderstandings about the representation of a general relativistic hyperbolic spacetime continuum.

 

This schematic diagram is a cross section of the physical universe. (This it is not a cross section at cosmic time t). All points located outside the origin are in the past relative to the observer (due to the limited velocity c). The present time t is only at the origin O (the observer). So the representation is one that could be observed through a telescope, exactly the way we see the universe.

 

 

 

 

Figure 6A

General Relativistic Hyperbolic Spacetime Manifold.

A slice through the visible universe.

(Design Coldcreation)

 

 

 

Some characteristic features of Figure 6A:

 

 

• This is a cross section of the universe as viewed by any observer (all observers are at the origin O, relative to their own inertial system). Here I've added perspective, but it is a polar coordinate view of the spacetime manifold.
   
   
  
• The 'concentric' circles extending outwards from O (centered on the observer) represent 'spherical' shells of about one billion light years each (in the look-back time). So the outer shell is about 10 billion light years away (I've stopped the illustration there for convenience. The actual horizon is a slightly further).
   
   
  
• This is a homogenous and isotropic universe (assumed on philosophical grounds in accord with the cosmological principle).
   
   
  
• The outer edge of the illustration represents the visible horizon (not infinity).
   
   
  
• Lines A simply indicate a direction (a line of sight) as we, or any observer, looks out into the universe. There is a 360 degree view capability.
   
   
  
• Lines B are photons arriving from sources located anywhere in the coordinate system (all photons that we detect come from somewhere on the manifold). I've made these lines extend from the outer spherical shell, but they would come from light emitting objects located anywhere in the visible universe.
   
   
  
• The slice of sky in the upper left portion of the polar coordinate system is an example of the large scale structures as seen from O (taken from the Sloan Digital Sky Survey, SDSS Galaxy Map). These types of structures would be spread out over the entire manifold, in accord with observations. Here there is only a sample slice.
   
   
  
• Redshift is plotted more or less in accord with contemporary cosmology, though with the perspective plane the plot seems distorted. But there actually may be a nonlinear relation (I'll come back to this in a subsequent post). For now, lets assume the plot to be similar to the one reproduced here below:
  

 

 

 

Now let's take a look at this view of an increasingly distant universe:

 

 

Schematic diagram of the distribution of known galaxy clusters in space. As Earth-bound observers look out from the bottom point toward the top of the cone, they view an increasingly distant and early Universe. Distance (redshift) is marked on the right axis and the corresponding cosmic look-back time is indicated on the left axis.

nasa.gov

 

 

Notice the schematic above, similarly to Figure 6A, plots the look-back time nonlinearly in billions of years, whereas, redshift z is plotted linearly. If one were to change the look-back time plot to a linear function, the corresponding redshift would appear to be nonlinear (I'm not saying that we should do that, just that it is interesting to note). The point is that what appears to be expansion can be interpreted as curvature of the manifold, as in Figure 6A, where there is general relativistic time dilation in the look-back time.

 

If one were to plot the full 360 degrees (without compression) the schematic would resemble Figures A of this thread, since the billion light year 'spherical' shells appear to diverge hyperbolically, exactly as the Figure A series.

 

 

 

So Figure 6A can be interpreted as consistent with observations. In other words, there are two ways to interpret the structure of this manifold. One way is in accord with the Lambda-CDM model, where space is expanding at an accelerated rate (as seen in the look-back time), with the global structure in quasi-Euclidean form (as determined by e.g., redshift) with a generous dose of DE and CDM.

 

Or, the manifold can be interpreted as non-expanding, with a general relativistic hyperbolic curvature of spacetime. That interpretation would appear, a priori, to be consistent with the current observational data: pending further examination, of course :shrug:.

 

 

Clearly (at least for myself) this type of reduced dimension illustration is more intuitive and thus easier to understand (visualize) than other projections that represent a hyperbolic spacetime. I will be willing to discuss any of them, though it is important that this type of illustration be understood, if one is to ponder how the universe should look to the observer if indeed the universe in which we live is hyperbolically curved.

 

 

So the question posed above remains: How physically, does a homogenous and isotropic four-dimensional spherically symmetric globally curved geometrically hyperbolic pseudo-Lobachevskian spatiotemporal manifold, mimic observations currently understood as an accelerated expansion in a virtually flat (or quasi-Euclidean) space? And if these models (or something similar, such as represented in Figure 4A or 6A) does indeed mimic observations, how can we define the shape of the manifold based on observations, and how can we differentiate between two (or more) competing models that are, a priori, consistent with observations and indistinguishable one from the other?

 

 

I will attempt to answer these questions in the coming posts.

 

 

To be continued...

 

 

 

CC

[quantumtopology]

Does the angle from the apex correspond to an angle in space as viewed from earth, or maybe it corresponds to a distance as measured from earth, or a time measured between events.

I'd say it corresponds to your first option

And points in the same ring would be at the same distance from the observer,

 

the distance between galaxies as measured by a ruler or the time between events as measured by a clock?.

The radial axis measures time (look-back time) according to the clock of the central observer?

Distance between galaxies is both represented by the circular axis(distance along the ring) and the radial axis, I think. (If I'm wrong,CC, please correct me).

[Rade]

Figure 6A

I have a question about Figure 6A. Why is the horizon shown as a smooth line ? It would seem to me that, if we assume a BB, then the horizon should be non-uniform, a type of jagged edge, such as this: .... | | || || | | || ... or this ....^^^^^^^^^^^^^^ ... or some such thing, but not this ... ________________ .... (curved in the hyper geometric model)

 

I mean, consider a seed for a maple tree. When it germinates (a type of BB event for the tree) the geometry of the most outer branches of the mature tree (we can consider this the horizon of the tree in terms of movement of mass and energy over time) is not a smooth line. So, if the geometry of expansion of mass and energy over time in the universe follows some fundamental laws of nature, I would expect that those that study the horizon of the universe could benefit by the study of how a seed becomes a mature tree. Any comments (pro or con) appreciated.

[quantumtopology]

The figure does not assume BB theory, and the horizon is the observable universe.

 

Regards

[Rade]

The figure does not assume BB theory, and the horizon is the observable universe.Regards

OK, thanks, now I see. Point O at the center is located on some observer. But, suppose an observer was present at t = 0 and still at the same location today, would not Figure 6A be what is observed today by that t = 0 observer ? And if so, would not the t = 0 observer then look at Figure A and say, wow, look at what the BB created ?

[coldcreation]

[snip] suppose an observer was present at t = 0 and still at the same location today, would not Figure 6A be what is observed today by that t = 0 observer ? And if so, would not the t = 0 observer then look at Figure A and say, wow, look at what the BB created ?

 

The short answer is no.

 

 

CC

[modest]

If you want the light-travel distance then the question remains: how do you know by looking at a galaxy for how long the light traveled? [snip]

Distance measures (cosmology) From Wiki: Light travel time or lookback time. This is how long ago light left an object of given redshift. And: Light travel distance (LTD). The light travel time times the speed of light.

Yes, that is the definition of light-travel time, but I was asking how you would measure it. The purpose in asking was that you would recognize it is simply not possible. Light travel time is not directly observable. One must have a model with which to use other data and calculate the look-back time. Because you indicated that distance on your diagram is expressed in look-back time it is therefore not possible to construct your diagram without a working model.

 

By working model I specifically mean a formula that will take information like redshift and brightness and give a look-back time. I can tell you that such a formula will, as a matter of course, assume curvature.

 

I have to be honest, you don't have such a model or method—no way to label and compare data or make predictions. You're very good at dressing up your posts, but I don't see anything that comes close to supporting your assumptions.

 

It seems SNe Ia would be the best objects for measuring distance, and curvature, in a spacetime continuum of the type described by GR.

 

You can't directly measure curvature with SN-Ia, but you can see what any homogeneous, isotropic GR universe would predict as observables for those supernova. You can do that at this site:

 

http://faraday.uwyo.edu/~chip/misc/Cosmo2/cosmo.cgi

 

But that is just a hunch. The point remains, distances (in the look-back time) of an expanding universe should be very similar to distances in one that is non-expanding.

 

The site above will also disprove that assertion.

 

If the relationship between angular diameter and angular diameter distance deviates from that which would be expected in a flat spacetime...

 

The angular diameter distance is a model-dependent function of angular diameter and therefore can't be independently measured and compared.

 

Again, the geometries shown in diagrams A, B and C cannot all be correct.

 

Sure enough :rolleyes:

 

(G.F.R. Ellis 1978, for example, who developed a model with a spherically symmetric static general relativistic cosmological spacetime). The idea that light is redshifted because it is traveling through a globally curved, four dimensional spacetime continuum was also proposed by I. E. Segal (with a four-dimensional globally hyperbolic (curved), pseudo Riemannian temporal evolution of the spacetime manifold) called chronometric cosmology. The chronometric redshift resembled Weyl’s 1921 model, in which time variations in the non-static case are interpreted as spatial variations in the static case.

 

I'd need to see a peer-reviewed paper making that same claim—a static solution of GR consistent with observation.

 

And if these models (or something similar, such as represented in Figure 4A) does indeed mimic observations, how can define the shape of the manifold based on observations, and how can we differentiate between two (or more) competing models that are, a priori, consistent with observations and indistinguishable one from the other.

 

There are many methods to distinguish between models. To rule out a static, isotropic, homogeneous, general relativistic model one simply needs to note 1) the universe has at least some energy density and 2) the Hubble constant is non-zero.

 

There are 3 GR line elements that are static, isotropic and homogeneous. Einstein's static universe is the only one that satisfies #1 and it does not satisfy #2. There is, therefore, no static, isotropic and homogeneous general relativistic model of the universe that is consistent with observation. This can likewise be proven with the Friedmann equations which are the exact solution of GR for an isotropic and homogeneous universe.

 

However, general relativity also asserts the existence of locally inertial frames, and the speed of light is a universal constant in those frames. The case under scrutiny here related to the latter, since we are talking about a homogenous and isotropic large scale nonzero gravitationally and hyperbolically curves spacetime as view from the locally inertial rest-frame of any observer.

 

So while your remark, and those you quoted by Herr Einstein, are correct in a sense, they are completely irrelevant for the purposes of measuring distances of astronomical objects such as galaxies, Cepheids or SNe Ia is a curved spacetime manifold (though they may be relevant for objects such as quasars and black holes).

 

Curved spacetime = speed of light not globally equal to c. This is well known.

 

Hopefully Figure 6A will clear up a few misunderstandings about the representation of a general relativistic hyperbolic spacetime continuum.

 

This schematic diagram is a cross section of the physical universe. (This it is not a cross section at cosmic time t). All points located outside the origin are in the past relative to the observer (due to the limited velocity c). The present time t is only at the origin O (the observer). So the representation is one that could be observed through a telescope, exactly the way we see the universe.

 

You have the horizon at z = 1.4

 

[*]This is a cross section of the universe as viewed by any observer (all observers are at the origin O, relative to their own inertial system). Here I've added perspective, but it is a polar coordinate view of the spacetime manifold.

 

You've been misusing "polar coordinate". A diagram, like the one you made, can be mathematically described in polar or Cartesian coordinates. The diagram itself is not one or the other.

 

[*]The outer edge of the illustration represent the visible horizon (not infinity).

 

The visible horizon of a static universe would have z=infinity (otherwise, why is it a horizon?).

 

So Figure 6A can be interpreted as consistent with observations. In other words, there are two ways to interpret the structure of this manifold. One way is in accord with the Lambda-CDM model, where space is expanding at an accelerated rate (as seen in the look-back time), with the global structure in quasi-Euclidean form (as determined by e.g., redshift) with a generous dose of DE and CDM.

 

Or, the manifold can be interpreted as non-expanding, with a general relativistic hyperbolic curvature of spacetime. That interpretation would appear, a priori, to be consistent with the current observational data: pending further examination, of course :phones:.

 

It is non-sequitur. You've given no method of measuring, plotting, or solving any distance or data. No method of solving or interpreting redshift. All you've done is assume that static space will somehow agree with observation. You've essentially dressed up the SDSS map.

 

I'm sorry, but I don't believe we've gotten anywhere. If we are going to explain redshift with statically curved space then the first step should be to derive some relationship between redshift and some other observable. Light travel time would not be a good choice because it isn't an observable.

 

~modest

[modest]

Does the angle from the apex correspond to an angle in space as viewed from earth, or maybe it corresponds to a distance as measured from earth, or a time measured between events.

I'd say it corresponds to your first option And points in the same ring would be at the same distance from the observer,

 

I was referring to this image.

 

~modest

[quantumtopology]

There are many methods to distinguish between models. To rule out a static, isotropic, homogeneous, general relativistic model one simply needs to note 1) the universe has at least some energy density and 2) the Hubble constant is non-zero.

 

the second one rests on a previous (a priori) assumption: if you assume a static model then it's zero (you just assume the scale factor to be constant), if non-static it's non-zero.You would be assuming the Hubble law and assuming the redshift to indicate a velocity, wich is what you try to prove in the first place. Therefore this second way to rule out a static universe doesn't work, I'll provisionally agree with method number one.

 

 

There are 3 GR line elements that are static, isotropic and homogeneous.

 

Ain't you forgetting the negative curvature static solutions Friedmann published in 1924?

[modest]

the second one rests on a previous (a priori) assumption: if you assume a static model then it's zero (you just assume the scale factor to be constant), if non-static it's non-zero.You would be assuming the Hubble law and assuming the redshift to indicate a velocity, wich is what you try to prove in the first place.

 

The Hubble parameter and the scale factor are derived from GR. I stated GR as an assumption.

 

Ain't you forgetting the negative curvature static solutions Friedmann published in 1924?

 

All three of the static universes mentioned are Friedmann universes. Friedmann's equation can solve any isotropic, homogeneous, GR universe.

 

~modest

[quantumtopology]

The Hubble parameter and the scale factor are derived from GR.

 

That assertion is debatable.Sure enough, not directly but making further assumptions.

 

 

I stated GR as an assumption.

Then both your methods could reduce to this assumption

 

 

All three of the static universes mentioned are Friedmann universes. Friedmann's equation can solve any isotropic, homogeneous, GR universe.

 

 

Who said otherwise? I am only reminding you that in his 1924 (so not in the 1922) paper he gives another 4 static solutions based in GR and negative curvature spacetime, 3 of them with vanishing density like de Sittter's.

[modest]

That assertion is debatable.Sure enough, not directly but making further assumptions.

 

Then both your methods could reduce to this assumption

 

This is what I said:

There are many methods to distinguish between models. To rule out a static, isotropic, homogeneous, general relativistic model one simply needs to note 1) the universe has at least some energy density and 2) the Hubble constant is non-zero.

Assuming isotropy (which I stated) and homogeneity (which I stated) the following is an exact solution of GR (which I stated),

[math]H^2 = \left(\frac{\dot{a}}{a}\right)^2 = \frac{8 \pi G}{3} \rho - \frac{kc^2}{a^2} + \frac{\Lambda c^2}{3}[/math]

[math]\dot{H} + H^2 = \frac{\ddot{a}}{a} = -\frac{4 \pi G}{3}\left(\rho+\frac{3p}{c^2}\right) + \frac{\Lambda c^2}{3}[/math]

No solution to the above equations is both static and agrees with observation. None of that is a matter of debate.

 

[edit] --------->

 

Perhaps I misunderstood what you meant. I'm not sure what "both your methods could reduce to this assumption" means

 

<-------- [/edit]

 

Who said otherwise? I am only reminding you that in his 1924 (so not in the 1922) paper he gives another 4 static solutions based in GR and negative curvature spacetime, 3 of them with vanishing density like de Sittter's.

 

Friedmann's 1924 paper is usually cited as demonstrating expansion. You'll have to show me exactly what you're referring to.

 

~modest

[quantumtopology]

Perhaps I misunderstood what you meant. I'm not sure what "both your methods could reduce to this assumption" means

Ultimately , method number one is implicitly assumed in GR, if it's not correct me; and method number two is not derived from GR but the equations you show are valid within GR if one makes the further assumption that redshift Z rflects velocity and therefore radius of universe increases with time and therefore there exists a non constant a , and so on. Given this assumption added to GR you get exact solutions of those equations.

 

Friedmann's 1924 paper is usually cited as demonstrating expansion. You'll have to show me exactly what you're referring to.

 

The one usually cited is the 1922 paper "On the curvature of space" . I'm refering to the 1924 "On the Possibility of a World wih Constant Negative Curvature of Space" SpringerLink Home - Main

 

I"ll try to find it . I have it somewhere

[modest]

Ultimately , method number one is implicitly assumed in GR

 

By "method 1" I guess you mean this:

1) the universe has at least some energy density

and 2) the Hubble constant is non-zero.

It is something that may or may not be true of our universe. If the universe has mass then #1 is true. If #1 is true then Minkowski space and de Sitter space are ruled out as possibilities because they both require a world without mass.

 

Both Minkowski space and de Sitter space do follow from GR—perhaps that is what you mean.

 

and method number two is not derived from GR

Again, I guess "method #2" means:

1) the universe has at least some energy density and

2) the Hubble constant is non-zero.

This is, again, something that may or may not be true of the universe—it is an observation. If it is true then Einstein's static universe model is ruled out as a possibility.

 

I'm not sure if that is what you are talking about exactly.

 

but the equations you show are valid within GR if one makes the further assumption that redshift Z rflects velocity

 

No, the equations follow from GR assuming isotropy and homogeneity. Redshift and scale factor are not assumptions but conclusions. They follow from 1) GR 2) isotropy 3) homogeneity. If those three things are true then you get redshift and a scale factor (i.e. the two Friedmann equations I gave).

 

and therefore radius of universe increases with time and therefore there exists a non constant a , and so on. Given this assumption added to GR you get exact solutions of those equations.

 

No, that doesn't sound right. The Friedmann equations themselves are exact solutions of general relativity. If you plug radiation and matter content and the other variables into the equations then they will tell you how the universe acts (and will act and has acted). They model the universe assuming the universe is isotropic, homogeneous, and follows the physics of general relativity.

 

The one usually cited is the 1922 paper "On the curvature of space" . I'm refering to the 1924 "On the Possibility of a World wih Constant Negative Curvature of Space" SpringerLink Home - Main

 

I"ll try to find it . I have it somewhere

 

I can't find a translation. The only reference I can find talking about the paper and mentioning a static universe is:

Friedmann shows [in his 1924 paper] that static solutions for a world with negative space curvature are obtained only with matter having zero or negative density; the latter possibility has no physical meaning, while the former corresponds to a non-zero cosmological term, i.e. to vacuum as it is understood today. (As was subsequently shown, such a static world is a part of de Sitter’s world if the vacuum density is positive.)

Alexander A. Friedmann: the man who ... - Google Books

 

That is what I would expect. It is the second of the static GR universes derived here:

The Three Static Line Elements

 

These three possibilities,

[math]v’ = 0[/math]

or [math]\rho_{00} + p_0 = 0[/math]

or both [math]v’ = 0[/math] and [math]\rho_{00} + p_0 = 0[/math]

lead respectively to the Einstein, to the de Sitter, and to the special relativity line elements for the universe as we may now show in detail...

Relativity, thermodynamics, and ... - Google Books

The above 3 are the 'static' (Einstein's is the only non-trivially static) solutions. Friedmann's equations (i.e. FLRW) can model all three.

 

Given the amount of work that has been done with general relativity and observational cosmology we are very much left with the choice of rejecting general relativity or rejecting static cosmology. Einstein and de Sitter realized this some 80 years ago and an incredible amount of supporting evidence has since agreed with them in their choice.

 

~modest