Redshift z

[quantumtopology]

The short answer is no. The Mayer redshift is not the same as the cosmological redshift discussed in this thread. The main difference between the two (the Transverse Gravitational Redshift proposed by Mayer and the curved spacetime redshift discussed in this thread) is the transverse aspect. The transverse redshift, perpendicular to the line of sight or propagation of EMR, as I understand, represents a modification of general relativity. One that appears to have been falsified or shown to be nonexistent, even in the Newtonian limit.

 

The redshift discussed in this thread is entirely based on general relativity, with no modification; albeit there is an extension of the concept of local curvature to global curvature, requiring only an interpretation of GR and its astrophysical implications to cosmology. The mechanism involved and the effect (curvature) are identical both locally and globally. In other words, to disprove or falsify the claim would be to disprove or falsify general relativity. Or, it would have to be shown that gravity is a curved spacetime phenomenon locally, but not globally, ie., that mass-energy curves spacetime around massive bodies, but that the mass-energy density has no (or a very small) influence on the curvature of space on the large-scale (globally). The latter is the current view, but since the total mass-energy density of the universe is unknown, it seems premature to discard the idea on that basis.

 

Note here, the above latter argument does not take into consideration the geometric structure of spacetime in the absence of matter (or mass-energy), which according to GR may also have an intrinsic curvature (with a negative sign, in accord with the de Sitter, or anti-de Sitter metric).

 

Why Mayer would propose a transverse redshift, as opposed to one without the apparently needless transverse component, is beyond me. It's a pity too, since besides the transverse z component much of what here writes, e.g., about empirical observations (and his critique of the concordance model, with its inherent big bang) makes sense.

 

 

 

 

CC

Great, we are in total accordance about this.

[quantumtopology]

Would you say expansion of space is coordinate-dependent or coordinate-independent?

[coldcreation]

Would you say expansion of space is coordinate-dependent or coordinate-independent?

 

It seems the answer depends on the physical metric. Coordinate invariance consistent with observations would seem to follow from predictions.

But that, by no means, implies the correctness of the metric. As discussed throughout this thread, there are two possible interpretation that can be derived from observations.

 

The question as to which coordinate system (one expanding and flat, or one static and curved) best describes nature will have to be determined by observations. So far, neither has been ruled out empirically. My hunch is that the concept of expanding space is a direct artifact of the coordinate transformations. That is, it would appear nonphysical, unnatural. The conclusion would be that space doesn't expand.

 

 

Incidentally, throughout the summer I've been preparing the text and schematic diagrams that attempt to elucidate redshift as a curved spacetime phenomenon, and the mechanism involved in the global stability. That work is not finished, but a qualitative draft version is practically ready for posting. I just have to touch up a few things.

 

Stay tuned...

 

 

PS. Sorry for the delay.

CC

[coldcreation]

Qtop, modest, and others,

 

Explanation 2 in online. It can be found here: Curvature of a globally homogeneous isotropic general relativistic spacetime and the stability of the cosmos

 

I've chosen to post it on an external link since the body of text is too long to post here at scienceforums.com. However, I will be glad to discuss any issues here, of course.

 

The subject matter (along with some new schematic diagrams) have been placed in two main sections but can often be found mixed together since the phenomena discussed are intrinsically related: (1) the physical mechanism for cosmological redshift z in a globally curved general relativistic (nonexpanding) spacetime, and (2) the elucidation of the physical mechanism involved in the global stability process (why universal expansion and contraction are not an option). The surprising feature in the latter discussion is that a cosmological constant is not required to explain large-scale equilibrium (exactly as lambda is not required locally to explain the observed structures).

 

Subsequent posts to the link above will include other related topics, such as the origin and evolution of the large-scales structures, galaxy dynamics, origin and evolution of the CMBR, origin of the light (and heavy) elements and their isotopes (including hydrogen).

 

What needs to follow too, is a rigorous quantitative (mathematical) analysis of the contentions and an comparison with the observational data—against which comparisons can be made with the standard concordance model. What we have now is simply a qualitative study, albeit conceptually based on general relativity along with its inherent non-Euclidean geometric foundation (so it could be worse).

 

Another point to make is that I've repeated some of the material covered in this thread so far (parts of Explanation 1). To that material I've added an Abstract, an Introduction and a note on terminology, amongst other things. I would recommend skimming over that section.

 

Finally, I realize the text is a little lengthy. It could have been reduced (and will be eventually) to just a few pages. After all, the concept is simple for both redshift z and stability. There is no new or exotic physics. It's quite a challenge to write briefly. Writing 100 words on a subject like this is more difficult than writing 1000. Which reminds me of something apt written by an author from my home town in CT: “I didn't have time to write a short letter, so I wrote a long one instead.” (Mark Twain).

 

 

PS. Since this preliminary model is incomplete, I will be adding sections (such as the mathematical proofs, which are now only hyper-linked) periodically. I will point out (here at scienceforums.com) where and when those modifications or updates are made.

 

Those of you that would like to see this endeavor move forward (or see potential drawbacks) are more than welcome to participate in the discussion.

 

Coldcreation

[quantumtopology]

Qtop, modest, and others,

 

Explanation 2 in online. It can be found here: Curvature of a globally homogeneous isotropic general relativistic spacetime and the stability of the cosmos

 

I've chosen to post it on an external link since the body of text is too long to post here at scienceforums.com. However, I will be glad to discuss any issues here, of course.

 

The subject matter (along with some new schematic diagrams) have been placed in two main sections but can often be found mixed together since the phenomena discussed are intrinsically related: (1) the physical mechanism for cosmological redshift z in a globally curved general relativistic (nonexpanding) spacetime, and (2) the elucidation of the physical mechanism involved in the global stability process (why universal expansion and contraction are not an option). The surprising feature in the latter discussion is that a cosmological constant is not required to explain large-scale equilibrium (exactly as lambda is not required locally to explain the observed structures).

 

Subsequent posts to the link above will include other related topics, such as the origin and evolution of the large-scales structures, galaxy dynamics, origin and evolution of the CMBR, origin of the light (and heavy) elements and their isotopes (including hydrogen).

 

What needs to follow too, is a rigorous quantitative (mathematical) analysis of the contentions and an comparison with the observational data—against which comparisons can be made with the standard concordance model. What we have now is simply a qualitative study, albeit conceptually based on general relativity along with its inherent non-Euclidean geometric foundation (so it could be worse).

 

Another point to make is that I've repeated some of the material covered in this thread so far (parts of Explanation 1). To that material I've added an Abstract, an Introduction and a note on terminology, amongst other things. I would recommend skimming over that section.

 

Finally, I realize the text is a little lengthy. It could have been reduced (and will be eventually) to just a few pages. After all, the concept is simple for both redshift z and stability. There is no new or exotic physics. It's quite a challenge to write briefly. Writing 100 words on a subject like this is more difficult than writing 1000. Which reminds me of something apt written by an author from my home town in CT: “I didn't have time to write a short letter, so I wrote a long one instead.” (Mark Twain).

 

 

PS. Since this preliminary model is incomplete, I will be adding sections (such as the mathematical proofs, which are now only hyper-linked) periodically. I will point out (here at scienceforums.com) where and when those modifications or updates are made.

 

Those of you that would like to see this endeavor move forward (or see potential drawbacks) are more than welcome to participate in the discussion.

 

Coldcreation

 

Hi, CC. First of all Congratulations for such an effort, nice presentation indeed.

I have just finished reading it, and i sincerely think it is on the right track,and does a good job summarizing 5 years of "Redshift z" thread and further extensions on the theme, a few comments:

Good points: sticks to the philosophy of no new fancy or strange physics, discards once and for all the lambda term, the insistence thru all the essay on the obvious (at least to you and me) truth derived from GR that redshift is a curvature effect, it really is puzzling how cosmologist can say the expanding universe is based on GR and not use it to explain the single most important cosmological observation ever, it is an example of selective blindness worth being examined by psychologists., more good points: your concern about stability which I guess you foresee it you are gonna be very questioned about, I agree that it is solved by making the manifold infinite, so there is no center matter can collapse into.

 

Points to improve: I really think you must discard the spherical option and go hyperbolic, I have talked a lot about this in the thread so I won't stress it now. I think I understand now your conceptual distinction of matter and light over their susceptibility to being affected by curvature, but the way you explain it still seems to contradict the E.P. Perhaps changing the wording a little bit would benefit when it comes to explain it to relativists (lke curvature effects are too small at non relativistic speeds ,,etc...).

 

Finally, as you say the next step is sinthesizing to a few pages and present it with the concepts translated to math language, this could be where the real difficulties arise, or not, who knows.

 

Regards

QTop

[coldcreation]

Hi, CC. First of all Congratulations for such an effort, nice presentation indeed.

I have just finished reading it, and i sincerely think it is on the right track,and does a good job summarizing 5 years of "Redshift z" thread and further extensions on the theme

 

Actually, only the Abstract is a summary of the thread (the opening post basically). Otherwise, everything else is a further extension. Part of the point is that both spherical and hyperbolic curvature induce redshift, whereas most of the thread deals with hyperbolic geometry only (as a de Sitter-like effect). The case now is more general. It is purely a general relativistic effect. The Gaussian curvature (spherical of pseudo-spherical, i.e., hyperbolic) removes the necessity of having an "empty" universe, and/or one with a cosmological constant, the latter of which I assumed was operational in the field throughout the thread).

 

Question: do you mean "right track" as opposed to spot on? If so, what do you feel is missing conceptually or qualitatively?

 

 

Good points: sticks to the philosophy of no new fancy or strange physics, discards once and for all the lambda term, the insistence thru all the essay on the obvious (at least to you and me) truth derived from GR that redshift is a curvature effect, it really is puzzling how cosmologist can say the expanding universe is based on GR and not use it to explain the single most important cosmological observation ever, it is an example of selective blindness worth being examined by psychologists., ...

 

You're right. Why add artifice to something that has a natural explanation in line with GR. I'm not sure what the truth is though, since both redshift effects (due to a change in the scale factor with time, or as a curved spacetime phenomenon) are virtually indistinguishable.

 

The former requires an initial condition (one that remains outside of known physical laws, i.e., the laws of physic break-down at t = 0), in addition to a period of exponential expansion to solve to causality problem along with a host of other problems (horizon, etc), while the latter mechanism for redshift requires no initial condition.

 

 

...more good points: your concern about stability which I guess you foresee it you are gonna be very questioned about, I agree that it is solved by making the manifold infinite, so there is no center matter can collapse into.

 

The stability, or rather, the instability of the field equations was the single most important problem that lead to big bang cosmology, before 1929. That was why Einstein added the cosmological term, recall. The interpretation of redshift as a Doppler effect followed, in a sense, from the apparent instability. The two factors combined to form the foundation of modern cosmology. So pointing out how stability is maintained in a general relativistic universe is just as important and the origin of redshift z.

 

One more note: I added the sentence about an infinite universe with no center only because it is an obvious conclusion (one that Newton thought of 300 years ago). There is no preferred reference frame. But that is not how stability is maintained. The important point is that even in a homogenous finite universe, or an infinite universe where matter only occupies a finite region of space (with or without a center of mass), stability is maintained. That was exemplified in the text with the argument that the global Gaussian curvature has no gradient locally. All points in the manifold are indistinguishable.

 

As a generalization, the intensity of the global gravitational field curvature is of constant magnitude at every point in the continuum. The source of the global field is the uniformly distributed mass-energy content of the universe, in addition to the gravitational field itself. The concept of a global homogeneous gravitational field of constant Gaussian curvature is conserved in the general theory of relativity, since the magnitude of the Riemann-Christoffel curvature tensor is identically zero at all points. Specifically, a static homogeneous spacetime is one in which the Riemann tensor vanishes (is null). There is no acceleration acting on objects, generated by the global field, since space-time metric is essentially Minkowskian locally for any object in the universe (relative to the underlying local inertial system of reference), while the effective Riemannian curvature becomes significant when the considered distances from the origin O are large. So stability is an inherent property of the Gaussian manifold.

 

But that will not prevent electromagnetic radiation from traveling geodesics.

 

In another way, massive objects are confined to move about on the Gaussian manifold at velocities inferior to that of massless photons, and thus relative to the global field are permanently embedded in a region whereby the intrinsic curvature of the universe remains flat. Every object (including large galaxy superclusters)—just as every observer is confined to view the world from a locally flat rest-frame—is located in a position consistent with flatness relative to the global geometric structure of the Gaussian field. Only curvature of the local submanifold affects intrinsic motion. The photon is unrestricted by such a (local)coordinate system. 

 

 

Points to improve: I really think you must discard the spherical option and go hyperbolic, I have talked a lot about this in the thread so I won't stress it now. I think I understand now your conceptual distinction of matter and light over their susceptibility to being affected by curvature, but the way you explain it still seems to contradict the E.P. Perhaps changing the wording a little bit would benefit when it comes to explain it to relativists (lke curvature effects are too small at non relativistic speeds ,,etc...).

 

Yes, this is a problem that bothers me too: I am unable at this point to unequivocally decide, based on empirical evidence which geometry is operational. The point is that both spherical and hyperbolic spacetimes induce a change in wavelength in EMR towards the red end of the spectrum that increases with distance. To differentiate between the two observationally, requires accurate distance measurements based on a full analytical world-model. As it stands now, I don't see how one geometry could be abandoned in favor of the other. What makes it difficult is that the universe appear locally flat from the rest-frame of an observer in either spacetime. And curvature increases with distance (as seen by redshift and time dilation). In other words, regardless of the sign of Gaussian curvature there is a redshift factor (and none in the case of K = 0, obviously).

 

In a sense, the SNE Ia data could be regarded as consistent with either geometry. How did you rule out the spherical model at this early stage of development?

 

It would appear that according to GR there is a connection between the spatial curvature of the universe and the energy of particles in it: positive total energy implies negative curvature and negative total energy implies positive curvature (Source). In this sense, when lambda is considered nonexistent, the curvature of the universe should be hyperbolic, not spherical, since mass-energy in nonzero and nonnegative. But I have yet to confirm this aspect of the field equations in the context under review here. As it stands now, the left side of the equations appear as geometry and the right side natural philosophy (perhaps without the "natural"). I just found this which looks interesting, on the topic of the field equations: Criticism of the Einstein Field Equation

 

How do you think this concept contradicts the EEP? It is largely based on an extension of the EP to cosmology, i.e, there is an equivalence, observationally, between a coordinate system that is radially expanding and one that is stationary embedded in a curved spacetime, as viewed from the observers inertial frame of reference.

 

 

Finally, as you say the next step is sinthesizing to a few pages and present it with the concepts translated to math language, this could be where the real difficulties arise, or not, who knows.

 

I've been working on this. But it's not easy. I'm pretty sure most of the equations required by the model exist already. It's just a question of placing them in the right context and weeding out the equations (or at least the terms) that are irrelevant.

 

 

 

CC

[quantumtopology]

Actually, only the Abstract is a summary of the thread (the opening post basically). Otherwise, everything else is a further extension.

Sure, I just meant that all the points had been touched on the thread on your explanation 1 posts.

 

Question: do you mean "right track" as opposed to spot on? If so, what do you feel is missing conceptually or qualitatively?

I would repeat here what I said in "points to improve". To be spot on you have to choose a specific geometric model and justify it with math, but I understand you are not ready yet.

In a sense, the SNE Ia data could be regarded as consistent with either geometry. How did you rule out the spherical model at this early stage of development?

I have to admit in this case I go by my intuition, if I had a way to rigorously prove it I'd try to publish it. :lol:

 

How do you think this concept contradicts the EEP? It is largely based on an extension of the EP to cosmology, i.e, there is an equivalence, observationally, between a coordinate system that is radially expanding and one that is stationary embedded in a curved spacetime, as viewed from the observers inertial frame of reference.

I agree, it is just that the way you explain the part about matter not being affected by the intrinsic global curvature could be misleading if one doesn't previously understand the concept. Maybe it's just my perception.

 

 

I've been working on this. But it's not easy. I'm pretty sure most of the equations required by the model exist already. It's just a question of placing them in the right context and weeding out the equations (or at least the terms) that are irrelevant.

I agree that probably the hard mathematical work is already done, and the tough part is to locate it and fit it with the concepts. In a way that is what Einstein did with his ideas and the differential geometry of Riemann, Levi-Civita, Ricci .. etc and the superb help of Grossmann.

 

Regards

QTop

[coldcreation]

Gentlemen, I just added some text which I hope will clarify further the issue of Stability in a curved spacetime regime. You'll have to scroll down, since I haven't set up hyperlinks to each section yet.

 

Another couple of paragraphs were added. See Such as collapse...

 

This latter paragraph reveal something quite strange about the Big Crunch scenario, which reflects a strange light on the concept of expanding space in general.

 

I'll post any further updates as they transpire.

 

 

 

 

I would repeat here what I said in "points to improve". To be spot on you have to choose a specific geometric model and justify it with math, but I understand you are not ready yet.

 

The way I see it; both spherical and hyperbolic curvature induce redshift z and time dilation and are consistent with gravitational stability globally. The choice of one geometry over the other (if any at all) should be based on observations (particularly distance measurements). How else, intuition aside, could one Gaussian curvature be favored over another.

 

I've always favored (as you do) the hyperbolic topology over others, but I'm thinking spherical geometry seems to work as well. So I have difficulty excluding it on philosophical grounds.

 

 

 

I agree that probably the hard mathematical work is already done, and the tough part is to locate it and fit it with the concepts. In a way that is what Einstein did with his ideas and the differential geometry of Riemann, Levi-Civita, Ricci .. etc and the superb help of Grossmann.

 

Good point.

 

 

Let me know if you have any further questions or remarks.

 

 

 

CC

[modest]

Hi guys

 

I wasn't going to reply until I had time to read your article, but I just opened it up tonight and unfortunately I don't know when I'll have time. It is quite extensive.

 

I can respond to the first claim in the abstract which I'd guess is the main issue:

 

There are two possible interpretations for cosmological redshift z that show wavelength independence over 19 octaves of the spectrum: (1) A change in the scale factor to the metric, implying the expansion of space and the recession of objects in it (i.e., the radius of the universe or scale-factor changes with time t. (2) The general relativistic curved spacetime interpretation (implying a static metric in a stationary universe).

 

I'd point out that curved spacetime in GR does not imply a static, or stationary, universe. It rather states the opposite.

 

The most basic description of GR, given by John Archibald Wheeler and often quoted, is: "spacetime tells matter how to move—matter tells spacetime how to curve." You are talking, in your quotes above, about two things. 1) how matter moves, and 2) the curvature of spacetime.

 

The relationship between those factors is answered explicitly in general relativity, so there is no need to debate the content of the theory. Material particles under the influence of gravity follow time-like geodesics. The relative path of those particles are related to the curvature of spacetime via the geodesic deviation equation. Put concisely,

 

A number of considerations are worth making at this point. Firstly, as in Newtonian gravity, equation (32) expresses the fact that the separation between two adjacent geodesics will vary if they move in a spacetime with nonzero curvature (i.e. [math] K^{\alpha}_{ \beta } \neq 0[/math]). Secondly, since [math]D^2 {\xi}^{\alpha }/D\tau^2=0[/math] if and only if [math]K^{\alpha }_{ \beta }=0[/math], equation (32) underlines that only in a flat spacetime two geodesics will remain parallel (i.e. with constant separation).

 

Geodesic deviation in General Relativity

 

Standard cosmology is a confirmed result of general relativity by an analysis of the GDE: Deviation of geodesics in FLRW spacetime geometries

 

Parallel woldlines (ie particles at rest with respect to one another in space) stay parallel in flat spacetime. In curved spacetime they either diverge or converge. So, it should be clear that the opposite of #2 is correct—curved spacetime means that the spatial distance between inertial particles changes over time.

 

Why add artifice to something that has a natural explanation in line with GR. I'm not sure what the truth is though, since both redshift effects (due to a change in the scale factor with time, or as a curved spacetime phenomenon) are virtually indistinguishable.

 

A scale factor is an easy way to describe curved spacetime. The spacetime manifold and the metric with the scale factor are describing the same physical situation.

 

~modest

[coldcreation]

I can respond to the first claim in the abstract which I'd guess is the main issue:

 

I'd point out that curved spacetime in GR does not imply a static, or stationary, universe. It rather states the opposite.

 

Hi modest. Good to have you back on board. Hope you had a pleasant summer.

 

I would argue that curved spacetime does not automatically imply an unstable universe. That point is elaborated upon extensively in the Coldcreation blog. Simply put, that assumption would be supported by the fact that intrinsic Gaussian curvature has the same magnitude or intensity of curvature at all points in the manifold. There is thus no acceleration imparted upon objects in the field attributable to the global curvature. Only local inhomogeneities in the field (the submanifold) induce motion.

 

 

The most basic description of GR, given by John Archibald Wheeler and often quoted, is: "spacetime tells matter how to move—matter tells spacetime how to curve." You are talking, in your quotes above, about two things. 1) how matter moves, and 2) the curvature of spacetime.

 

That theme is actually covered in the text. Here is an introduction to it:

 

The disambiguation between the two types of curvature (local and global) will be made below in order to show that local curvature induces massive bodies to move is certain ways (geodesically)' date=' while global curvature does not. There is a problem with the standard interpretation of general relativity (GR) related to the Einstein equivalence principle, in the absence of a gravitational force (source). It will be shown that the standard concept expressed by Misner et al (1973, p. 5): "Space tells matter how to move. Matter tells space how to curve" is justifiable locally but untenable when global curvature is considered. It can be interpreted from cosmological considerations of the equivalence principle two general ideas (that are incompatible with each the other): there is an equivalence observationally between radial velocity and dilation of the metric. Just as there is an equality of gravitational and inertial masses (in the Newtonian sense), there is an equivalence of acceleration and space curvature (in the Einsteinian sense). Coldcreation blog.

 

See the rest of the story...

 

 

The relationship between those factors is answered explicitly in general relativity, so there is no need to debate the content of the theory. Material particles under the influence of gravity follow time-like geodesics. The relative path of those particles are related to the curvature of spacetime via the geodesic deviation equation.

 

No one is debating the contents of GR. What is debatable, however, is the interpretation of those factors and relationships. The equivalence principle is a prime example of how phenomena can be interpreted differently within the framework of GR (or even SR). Extrapolating the EP to cosmology is even more perilous.

 

GR is not a cosmology, and so interpretations must be made. The difficulty there is to disentangle radial motion from intrinsic curvature, especially cosmologically, since testing a hypothesis (as opposed to a local testing of GR) may reveal itself as highly inconclusive.

 

What you refer too above is, again, related to local gravity fields, not necessarily applicable to one that would be considered global. That remains subject to an interpretation of the field equations. For a more unambiguous result, one must tackle the fundamentals of non-Euclidean geometry, particularly that of Gauss and Riemann (but not to the exclusion of Lobachevsky or Bolayi) and the way in which GR incorporates such.

 

That should become clearer as you review the blog.

 

 

Parallel woldlines (ie particles at rest with respect to one another in space) stay parallel in flat spacetime. In curved spacetime they either diverge or converge. So, it should be clear that the opposite of #2 is correct—curved spacetime means that the spatial distance between inertial particles changes over time.

 

Either spacial distances between inertial particles changes over time (causing redshift and time dilation viewed from O), or spacetime is curved globally (causing redshift and time dilation viewed from O) in a static manifold.

 

Questions as to whether a globally curved spacetime induces motion locally (or even on large-scales) can be resolved by closely examining the properties of non-Euclidean geometry in the framework of a homogeneous and isotropic universe. We are talking about the properties of an Einstein manifold. See too homogeneous Riemannian manifold.

 

 

A scale factor is an easy way to describe curved spacetime. The spacetime manifold and the metric with the scale factor are describing the same physical situation.

 

It's an easy way, but it's not the only way.

 

Either the scale factor changes with time, or it does not.

 

In the former, curvature (K = 1, K = 0 or K = -1) ultimately depends on the rate of expansion being greater than, less than, or equal to the critical value.

 

If the scale factor does not change with time, the Gaussian curvature is nonzero. K = 0 is not an option since there would be no redshift or time dilation. A stationary (non-expanding) universe with nonzero Gaussian curvature would be, a priori, indistinguishable from one with a changing scale factor to the metric, since redshift and time dilation are both present, and increase with further distance from the observer.

 

The question then remains: which is the solution that best describes the physical universe in which we live?

 

The answer to that question, I believe, can be (and will be) determined empirically.

 

 

CC

[coldcreation]

Qtop, modest and others(?),

 

I just read a work that might interest you: On the equivalence of fields of acceleration and gravitation

 

It appears from this work (unless I've misinterpreted it) that hyperbolicity (not sphericity) would be consistent with both general relativity, and the shift of spectral lines towards the red observed astronomically.

 

I've argued that the Gaussian curvature is constant (in the case of either K = 1, or K = -1) while this paper concludes hyperbolic curvature requires a non-constant curvature, in order to agree with GR (or contrary to a particular interpretation of GR). But our conclusion that redshift is a curved spacetime phenomenon is not rendered invalid on this account, for the simple reason that we (or I) have postulated homogeneity and isotropy. This of course is based on philosophical grounds, but that is not all. It is also based on the idealization that evolution in the lookback time is negligible (or even nonexistent). Evidently, evolution with time is an attribute of any universe (static or expanding). So the idea of varying density, or varying mass, over time is unavoidable. This means that Gaussian curvature in the physical universe can be non-constant, i.e., non-linear with time and distance from O). Such a conclusion does not take away from our hypothesis, on the contrary, it would support it (see, e.g., SNe Ia and the non-linearity the data represents).

 

Anyway, Qtop, in the above link, you will see the level of complexity such deliberations incorporate, exemplified by the equations involved (let alone the intuitive and nonintuitive qualitative issues).

 

In the final outcome, the spherical model may have to be abandoned. But at least its presence as an alternative solution (with an opposite sign) remains theoretically viable, as a spacetime where redshift and time dilation are inescapable phenomena. If your (and my) intuition does not fail us, then the Lobachevsky-Bolyai metric structure of space (or spacetime) may indeed turn out to be the geometry of choice when it comes to representing the large-scale structure of the physical world and its evolution in time.

 

I will continue researching this topic until I can unambiguously arrive at a conclusion (whether the global field should be spherical or hyperspherical) based on GR and non-Euclidean geometry alone. Then, and only then, will observations make complete sense within the framework of a static, stationary, non-expanding, yet dynamic and evolving universe.

 

 

Best regards

 

CC

[quantumtopology]

Qtop, modest and others(?),

 

I just read a work that might interest you: On the equivalence of fields of acceleration and gravitation

 

It appears from this work (unless I've misinterpreted it) that hyperbolicity (not sphericity) would be consistent with both general relativity, and the shift of spectral lines towards the red observed astronomically.

 

I've argued that the Gaussian curvature is constant (in the case of either K = 1, or K = -1) while this paper concludes hyperbolic curvature requires a non-constant curvature, in order to agree with GR (or contrary to a particular interpretation of GR). But our conclusion that redshift is a curved spacetime phenomenon is not rendered invalid on this account, for the simple reason that we (or I) have postulated homogeneity and isotropy. This of course is based on philosophical grounds, but that is not all. It is also based on the idealization that evolution in the lookback time is negligible (or even nonexistent). Evidently, evolution with time is an attribute of any universe (static or expanding). So the idea of varying density, or varying mass, over time is unavoidable. This means that Gaussian curvature in the physical universe can be non-constant, i.e., non-linear with time and distance from O). Such a conclusion does not take away from our hypothesis, on the contrary, it would support it (see, e.g., SNe Ia and the non-linearity the data represents).

 

Anyway, Qtop, in the above link, you will see the level of complexity such deliberations incorporate, exemplified by the equations involved (let alone the intuitive and nonintuitive qualitative issues).

 

Hi, everyone

 

CC, I already knew this author (Lavenda) from another paper http://arxiv.org/pdf/0805.2240 but this one you linked is new to me.

He makes interesting points but I think it is important to remark several warnig signs in his articles before giving him much credit. When in page 16 he introduces the non-constant curvature he does it because that is necesary for his assumption that the Equivalence principle must be wrong, and on the same page he also casts doubts on Einstein's field equations. So this guy denies validity of the General relativity theory based on arguments apparently not very reliable.

On top of that from the other paper I linked(where he also questions validity of EP and GR, it's evident he only considers expanding cosmologies (perhaps the fact that he assumes expansion as self evident influences his need to discard the EP and thus GR).

 

I'm the first to question anything that seems to be taken for granted. But one must not be too radical either at the risk of not having nothing from what to build something sensible.

I think keeping GR as a solid starting place is a good thing that we have maintained and should be very careful or have very good reasons before we reject it.

[modest]

Hi modest. Good to have you back on board. Hope you had a pleasant summer.

 

:)

 

 

I would argue that curved spacetime does not automatically imply an unstable universe. That point is elaborated upon extensively in the Coldcreation blog. Simply put, that assumption would be supported by the fact that intrinsic Gaussian curvature has the same magnitude or intensity of curvature at all points in the manifold. There is thus no acceleration imparted upon objects in the field attributable to the global curvature.

 

Great circles converge on the surface of a sphere. The sphere's curvature is 'global'. With globally curved spacetime geodesics will converge (or diverge, depending on the sign of curvature). That means things get closer together (or further apart) in space over time.

 

If the curvature of spacetime is global then the 'tidal force' between any two particles of distance d will be the same.

 

If the curvature of spacetime is globally flat then the 'tidal force' between any two particles of distance d will be the same and equal to zero.

 

The former does not imply the latter. The surface tension in a balloon filled with air may be the same everywhere on the surface of the balloon, but that does not imply that it is everywhere equal to zero.

 

~modest

[coldcreation]

Great circles converge on the surface of a sphere. The sphere's curvature is 'global'. With globally curved spacetime geodesics will converge (or diverge, depending on the sign of curvature). That means things get closer together (or further apart) in space over time.

 

Yes, I understand your point, but one must not forget that we are considering a surface (of 4-dimensions) that has intrinsic curvature. This is not a surface (e.g., a sphere or pseudo-sphere) embedded in a Euclidean space. This means that trajectories of photons will appear distorted (geodesically) as they propagate towards an observer (with the consequential redshift/time dilation factors), making the distance of the source (e.g., galaxy) appear closer or further, depending on the signature of curvature, than would be the case in a flat Euclidean manifold.

 

Objects don't geodesically move closer or further from the observer in space over time. That is because the global field (which is indeed a gravity field) is homogeneous and isotropic. The local neighborhood of any object (e.g., a galaxy cluster) is effectually flat with respect to this field. (Of course local inhomogeneities, intrinsic local gravity fields, will induce interactions and intrinsic motion between objects). There is no directional acceleration imparted on objects, due to the global field.

 

Connecticut has a flat topology, for example, compared to the United States as a whole. Even more dramatic, Cannes, France, is flat compared with a surface the size of Africa (see Figure G3). Likewise, the geometry of spacetime in the vicinity of the Local Group is flat (local curvature aside) compared with the global Gaussian curvature of the universe. Though the Local Group interacts gravitationally with its neighboring clusters, these interactions are a local phenomenon essentially independent of the intrinsic field associated with global Gaussian curvature.

 

An important point to make is that even at distances where the global Gaussian curvature begins to manifest itself, say at distances compatible with the separation between galaxy superclusters, there is no influence or acceleration induced by the Gaussian curvature imparted upon those neighboring superclusters, since the magnitude of curvature is the same in all directions, i.e., there is no spatiotemporal change in the gradient (or intensity) of a globally homogeneous gravitational field governed by the laws of general relativity.

 

 

If the curvature of spacetime is global then the 'tidal force' between any two particles of distance d will be the same.

 

Tidal force primarily results when gravity fields are not uniform (or homogeneous). At any point in the field(s), in such cases (which are local, not global), the tidal force is approximately the gravitational force (at that point) minus the gravitational force at the center of mass. Globally, there is no center of mass, and the filed is uniform. There are no tidal forces in a globally homogeneous and isotropic Gaussian/Riemannian manifold.

 

Clearly, a global Gaussian manifold of constant curvature exists in the absence of any tidal forces. In another way, a uniform gravitational field does not have tidal forces. By definition, there are no differential gradients in a spacetime of constant Gaussian curvature, no matter what the signature (K = 1, K = 0, or K = -1).

 

 

If the curvature of spacetime is globally flat then the 'tidal force' between any two particles of distance d will be the same and equal to zero.

 

We agree. But in the case where spacetime has constant nonzero Gaussian curvature, i.e., not globally flat, there is no tidal force between, say, galaxy clusters due to this field, because the value (or magnitude) of curvature depends only on how distances are measured on the surface—and distances are indeed distorted when measured from a rest-frame sufficiently removed from the emitting source—not on the way it is isometrically embedded in space. This result is the content of Gauss's Theorema egregium. The Gaussian curvature is determined by the inner metric of the surface without any reference to ambient space: it is an intrinsic invariant. (Source).

 

 

The former does not imply the latter. The surface tension in a balloon filled with air may be the same everywhere on the surface of the balloon, but that does not imply that it is everywhere equal to zero.

 

It needs not be equal to zero everywhere, or anywhere. As long as the curvature has the same value or magnitude at every point in the manifold equal to x the result is the same as if the gradient were zero everywhere. It is null. The "surface tension" to use your analogy, is the same everywhere. This is similar but not identical to the situation that arises in the context, again to use your analogy, when the forces of surface curvature tension and pressure are balanced according to the Young-Laplace equation. That's as far as the analogy goes.

 

In the case under review here, vacuum pressure (e.g., of the cosmological constant type) is not required for the maintenance of stability.

 

Remarkable, wouldn't you say?

 

 

CC

[modest]

Yes, I understand your point, but one must not forget that we are considering a surface (of 4-dimensions) that has intrinsic curvature. This is not a surface (e.g., a sphere or pseudo-sphere) embedded in a Euclidean space.

 

10. The sphere has constant positive Gaussian curvature.

 

Gaussian curvature is the product of the two principle curvatures. It is an intrinsic property which can be determined by measuring length and angles and does not depend on the way the surface is embedded in space

 

Eleven properties of the sphere

 

Objects don't actually move closer or further from the observer in space over time.

 

They do according to general relativity.

 

Tidal force is primarily results when gravity fields are not uniform (or homogeneous).

 

Consider de Sitter space. Consider if you were in a de Sitter space. Any particle that you examined some distance from you would have a Newtonian 'force' pushing it away from you (supported here). That 'force' would be proportional to distance so that something at 2d would have twice the force as something at d. If the two objects were tethered together then the line would be under tension. In a Newtonian sense, that is a tidal 'force'.

 

It needs not be equal to zero everywhere, or anywhere. As long as the curvature has the same value or magnitude at every point in the manifold equal to x the result is the same as if the gradient were zero everywhere.

 

It really just seems like you're mixing up the effects of curved spacetime with flat spacetime. 'Constant curvature' is not the same as 'flat'.

 

~modest

[coldcreation]

 

Eleven properties of the sphere

 

I didn't find anything in this link that supports your position. Number 10, which you quote supports the point I am trying to make clear. A sphere has constant positive Gaussian curvature. The key word there is constant.

 

A Gaussian manifold of constant curvature such as a sphere (K = 1) or a pseudosphere (K = -1), has the same magnitude of curvature, the same gravitational potential (in Newtonian terms), at all points on the manifold. The constancy of curvature of a Gaussian manifold has the obvious meaning that it's the same at all points.

 

It need not be zero to have the same degree of curvature everywhere. The curvature changes not from point to point. The space is curved nevertheless. Thus objects are not displaced geodesically due to Gaussian curvature, but the spectral lines of electromagnetic radiation emitting objects are affected as photons travel across spacetime geodesically to the observer.

 

Obviously those are very important points.

 

 

They do according to general relativity.

 

That depends on your interpretation and extrapolation of GR, from local to global considerations.

 

 

Consider de Sitter space. Consider if you were in a de Sitter space. Any particle that you examined some distance from you would have a Newtonian 'force' pushing it away from you. That 'force' would be proportional to distance so that something at 2d would have twice the force as something at d. If the two objects were tethered together then the line would be under tension. In a Newtonian sense, that is a tidal 'force'.

 

De Sitter space is a particular (peculiar even) matter-free vacuum solution of Einstein's field equation with a repulsive (positive) cosmological constant, corresponding to a positive vacuum energy density and negative pressure.

 

This is indeed an interesting solution, but not particularly relevant here. The topic under review is based on a universe that contains matter and energy (i.e., it is not empty) as a source of Gaussian curvature. There is no cosmological constant in this model, no Newtonian 'force' (no tidal force) or vacuum energy pushing particles away from observers.

 

 

 

It really just seems like you're mixing up the effects of curved spacetime with flat spacetime.

 

Not at all. :)

 

 

'Constant curvature' is not the same as 'flat'.

 

True. 'Constant curvature' represents a physical property of spacetime (in this case of 4-dimensions) whereby the surface can be described by a complete Riemannian manifold of constant sectional curvature K. Here we are discussing two types of curvature: with positive and with negative signatures, K = 1 and K = -1 respectively. These are globally homogeneous curved spacetimes (the properties of which are consistent with the laws of general relativity). These manifolds are not globally flat, though they tend towards flatness locally (where special relativity is sufficient for, say, measuring local distances).

 

The interesting feature is that even though the field is curved intrinsically there is no preferred direction towards which objects will gravitate (no center of gravity). There is no difference in the gradient or slope from any arbitrary point A to any arbitrary point B that would otherwise force objects to move geodesically towards or away from one another. There are no lines of force (if you prefer) pointing in any particular direction, no privileged reference frames.

 

Take for example a negatively curved manifold. Every point can be considered a stationary saddle-like point. That is so because all points on a homogeneous Gaussian manifold manifold are equivalent. Objects or particles in a negatively curved Gaussian manifold will not gravitate towards or away from one another (as they would at a Lagrange point L1) for the simple reason that this field (contrary to the vicinity of L1) is a homogeneous manifold of constant curvature. Hence, all points are stable points with respect to neighboring points on the global field.

 

Just as objects immersed in a Gaussian manifold of constant positive curvature remain free-of global gravitational instability, so too do objects in a manifold of constant Gaussian curvature with a negative sign remain free-of gravitational instability.

 

This result is remarkable because it means that the universe itself does not collapse or expand as a function of time, depending on the sign of curvature.

 

 

See Curvature of a globally homogeneous isotropic general relativistic spacetime and the stability of the cosmos for a more in-depth review, along with the pertinent schematic diagrams.

 

 

 

Regards,

 

CC

[quantumtopology]

 

A Gaussian manifold of constant curvature such as a sphere (K = 1) or a pseudosphere (K = -1), has the same magnitude of curvature, the same gravitational potential (in Newtonian terms), at all points on the manifold. The constancy of curvature of a Gaussian manifold has the obvious meaning that it's the same at all points.

Just a clarification that might help: In GR curvature and gravitational potential are different things, the equivalent of the potential in GR would be the metric tensor, and the equivalent of GR's curvature is (in vector calculus terminology) the divergence of the gradient of the potential, that is usually referred to in short as the Laplacian of the potential.

So strictly speaking a constant curvature is not the same as the same gravitational potential at all points on the manifold. Or that is my humble understanding.

 

Regards

QTop