Redshift z

[coldcreation]

[

If you set a rotating structure next to a different rotating structure then the two structures may not collapse on their own center because of the centrifugal force, but the two structures will collapse on each other.

 

That depends on the motion and velocity of the structures relative to one other. You can't just assume everything collapses gravitationally. Many things, from planets to superclusters, do not collapse gravitationally, simply because of the motion and velocities of objects relative to others. (See a further explanation regarding stability due to curvature below).

 

Note: in a static universe superclusters are gravitationally bound systems.

 

 

As an analogy, a geodesic in relativity is like a great circle on a sphere, but the reality is different. Because the tangent space of the manifold is Lorentzian in GR, the distance between events along a null geodesic is zero. This is not the case on a sphere. I really think the trouble you're getting is from trying to envision spacetime as a sphere.

 

It is straightforward the extension of a reduced dimension Gaussian sphere, or Riemannian sphere, to a four-dimensional spacetime manifold of constant positive curvature. I have no problem with that. Spacetime is not the surface of a sphere. The surface of a sphere is a reduced dimension representation, where a great circle arc is the path of a photon. That representation, when extended to 3-space obviously changes something. The great circle arc, for example, is no longer seen as a curved trajectory by an observer. The path is seen as a straight line. The path is still a geodesic consistent with a great circle, and distance measurements can be made in the same way. See The Deep Universe link above.

 

Every point in a static homogeneous and isotropic four-dimensional spacetime manifold are equivalent (just as they are on a Riemannian sphere of constant positive curvature).

 

 

If the average radial velocity between superclusters does not increase with distance then...

 

Two superclusters will collapse unless they orbit a common center of gravity.

 

Unless the two superclusters in the previous sentence are orbiting a common center with two other superclusters they will collapse.

 

Unless the 4 superclusters in the previous sentence are orbiting a common center with 4 other superclusters, they will collapse.

 

You could continue this line of reasoning until you reach the size of the visible universe at which point you'll need to say that the visible universe is rotating around a center.[...]

 

False. The visible universe need not rotate. The only requirement is that neighboring superclusters are in motion relative to one another, and, that the global curvature is positive.

 

I could keep saying this over and over again. If you do not look at the equations in the Bruce link, or the Fischer link, I don't think the issue can be resolved.

 

The problem with your reasoning (in the cosmological sense) is that it relies on Newton's theory with absolute space and time, and/or results of the FLRW interpretation of the Einstein field equations, with its cosmic time component, and now, with its pressure parameter (dark energy).

 

When time depends on distance, the balance is fulfilled without any pressure. The term 2/a^2 (of equation 20 here) balanced by negative pressure in the Einstein static solution, is compensated by the geometric change in time scale. What was before attributed to negative pressure appears as a property attached to the curvature of space (Fischer, 2009). Notice the geodesic description that follows from page 72 to 73 of the same link. "Due to the change of time scale with distance moving particles decelerate during their path." Geodesic motion does not, as in Euclidean geometry, mean that the velocity and kinetic energy is constant. "In curved space every motion must be regarded as an accelerated motion. The change is time scale has the same consequence as a retarding force."

 

 

What we have is a universe that is globally stable, with an observed cosmological redshift that increases with distance in a spacetime manifold of constant positive curvature.

 

 

I do not exclude galaxies in superclusters in my statement. The point is that a galaxy (in a supercluster) must be moving 100 million m/s relative to a galaxy (in a supercluster) that is 5 billion lightyears away. If their relative velocity is less than that then they must collapse toward each other. We need only Newton's laws to know this. It doesn't matter if the supercluster 5 billion lightyears away is in a stable orbit with the supercluster right next to it. Unless it is moving 100 million m/s relative to us, it is not stable with us. The larger the distance, the larger the velocity needed for stability.

 

See explanation below...

 

 

As far as there being no center, Newton's shell theorem allows us to make an arbitrary center and get the correct results. The visible universe where we are at the center is just like a nebula which has a center. The fact that the universe may be infinite does not matter because the mass outside the shell can be ignored. A person at the center of a nebula considering what the nebula is going to do is like us at the center of the visible universe considering what it is going to do. The nebula's finite size and the universe's perhaps infinite size is not an impediment because of the shell theorem.

 

 

Fischer, 2009, writes:

It should be mentioned here that a negative pressure appears also in a quasi-Newtonian model of gravity. If we consider the universe as a curved volume of space, homogeneously filled with matter, every volume element will attract its neighbouring elements gravitationally. Due to symmetry, the net force will be zero at every point, but there will be a tension, a negative pressure, analog to the surface tension in the rubber sheet of an air balloon. Work has to be done to ‘blow up’ space to larger size against this gravitational tension. The tension or negative pressure plays the role of potential energy in the Newtonian picture.

 

From the analogy with Newtonian gravity, we see that the tension should decrease with curvature radius as 1/a^2. Taking the limit a → ∞ as the zero-point of the potential, by integration we find that the potential should vary as 1/a. A virtual change of a leads to a decrease of the matter density as 1/a^3 . The potential energy thus changes as 1/a^4 . This behaviour is completely different from the ‘cosmological constant’ introduced by Einstein. Since the change of potential energy is steeper than that of the matter density, the equilibrium is stable against global changes of a, if we regard a as a dynamical property. This is in contrast to Einstein’s static solution, where the balance by a cosmological constant is unstable against variation of a.

(Source: Fischer 2009)

 

Furthermore:

 

It should be noted here that the dependence 1/a^4 is the same as that of relativistic matter or radiation, fields with an equation of state p = q/3. This behaviour must, of course, be expected, as according to Newtonian gravity the sum of potential energy and kinetic energy is conserved. There exists a global equilibrium. Changes of the potential energy by formation of local structures are always accompanied by the generation of kinetic energy. But while the kinetic energy remains concentrated locally, the change of the potential is transmitted into space at the velocity of light. Thus, local perturbations will grow, until kinetic energy or positive pres- sure are in balance again with potential energy. Though the universe is stable as a whole, it is locally unstable, giving rise to the formation of structures, as we observe them in form of stars or galaxies.

(Source: Fischer 2009)

 

 

CC

[coldcreation]

I read from the first paragraph on wiki's 'supercluster' page...

 

"Superclusters are large [...] They are so large that they are not gravitationally bound [...]"

 

I have no reason to doubt that.

 

 

That sentence from Wiki is not entirely accurate, even in the context of an expanding universe. Superclusters represent irregularities in the expansion. The local gravitational effects influence the otherwise smooth velocity field. This disturbing influence is due to the fact that superclusters are bounded gravitationally, at least to some extent. Studying how local motion of clusters and superclusters departs from the smooth expansion rate (of say, the Einstein-de Sitter model, or the critical model) can lead to insight about the topology of the universe (whether it is open, closed or flat). As it turns out, the CMB shows that the Galaxy (along with the Local Group) is moving relative to a smooth Hubble flow. Part of this motion is attributed to the gravitational effect of the Virgo supercluster. But there also seems to be another component influencing the motion of the Local Group that remains unknown at this time (perhaps another nearby cluster that remains hidden by dust). The Local Group appears to be moving away from the Virgo supercluster slower that would otherwise be expected in a smoothly expanding regime. Source

 

Surprisingly, the Shaley Supercluster, "the largest concentration of galaxies in our nearby Universe that forms a gravitationally interacting unit," is in more or less the same direction of the suspected cluster thought to influence the motion of the Local Group relative to the Virgo supercluster. Shapley (after whom the supercluster is named) estimated the distance of this group of galaxies to be 14 time further than the Virgo cluster

 

See too the Great Attractor: "a gravity anomaly in intergalactic space within the range of the Centaurus Supercluster that reveals the existence of a localised concentration of mass equivalent to tens of thousands of Milky Ways, observable by its effect on the motion of galaxies and their associated clusters over a region hundreds of millions of light years across."

 

If indeed these quantitative measures of clustering (and others such as Pisces-Perseus) are exceptional gravitationally bounded regions similar to the average clustering, it could indicate that these exceptional regions are not after all fundamentally exceptional.

 

See, for example: THE PISCES-PERSEUS SUPERCLUSTER AND GRAVITATIONAL QUASI-EQUILIBRIUM CLUSTERING

 

 

Of course, as mentioned earlier, the Wiki sentence in your quote above is entirely inaccurate in the context of a universe where expansion is not operational, i.e., in a static universe superclusters are indeed gravitationally bounded systems.

 

 

 

CC

[modest]

That depends on the motion and velocity of the structures relative to one other. You can't just assume everything collapses gravitationally. Many things, from planets to superclusters, do not collapse gravitationally, simply because of the motion and velocities of objects relative to others. (See a further explanation regarding stability due to curvature below).

 

Note: in a static universe superclusters are gravitationally bound systems.

 

You purposefully miss my point.

 

Also, in a static universe everything is gravitationally bound.

 

It is straightforward the extension of a reduced dimension Gaussian sphere, or Riemannian sphere, to a four-dimensional spacetime manifold of constant positive curvature. I have no problem with that. Spacetime is not the surface of a sphere. The surface of a sphere is a reduced dimension representation, where a great circle arc is the path of a photon. That representation, when extended to 3-space obviously changes something. The great circle arc, for example, is no longer seen as a curved trajectory by an observer. The path is seen as a straight line. The path is still a geodesic consistent with a great circle, and distance measurements can be made in the same way. See The Deep Universe link above.

 

Spacetime is not the surface of any hypersphere (in any number of dimensions). A great circle is not the path of a photon. You missed my point. You didn't respond to it. The length, in spacetime, for a null geodesic is zero. The length, on a hypersphere, for a great circle is non-zero. It is not the same. A space [math]\mathbb{E}^3[/math] will make, but a spacetime in [math]\mathbb{E}^4[/math] is a mistake.

 

False. The visible universe need not rotate. The only requirement is that neighboring superclusters are in motion relative to one another, and, that the global curvature is positive.

 

I could keep saying this over and over again. If you do not look at the equations in the Bruce link, or the Fischer link, I don't think the issue can be resolved.

 

This is good. No longer are you saying that things can be globally static because they have peculiar velocities. You're saying they are in a local equilibrium because they orbit local things (something I've already agreed to) and they are globally static for some other reason. Very good.

 

As far as reading those links—you know from previous discussion that I've read one of them. It didn't make sense, and I don't accept things on faith.

 

 

The problem with your reasoning (in the cosmological sense) is that it relies on Newton's theory with absolute space and time, and/or results of the FLRW interpretation of the Einstein field equations, with its cosmic time component, and now, with its pressure parameter (dark energy).

 

A theory which makes successful predictions before they are confirmed is not a problem. The scientific method is really rather fond of that sort of thing.

 

When time depends on distance, the balance is fulfilled without any pressure. The term 2/a^2 (of equation 20 here) balanced by negative pressure in the Einstein static solution, is compensated by the geometric change in time scale. What was before attributed to negative pressure appears as a property attached to the curvature of space (Fischer, 2009). Notice the geodesic description that follows from page 72 to 73 of the same link. "Due to the change of time scale with distance moving particles decelerate during their path." Geodesic motion does not, as in Euclidean geometry, mean that the velocity and kinetic energy is constant. "In curved space every motion must be regarded as an accelerated motion. The change is time scale has the same consequence as a retarding force."

 

I would rather tend to agree with "The change in time scale has the same consequence as a retarding force". A notable consequence: things get further away from each other over time.

 

Clocks tend to settle where their proper time is most dilated.

 

That sentence from Wiki is not entirely accurate, even in the context of an expanding universe. Superclusters represent irregularities in the expansion. The local gravitational effects influence the otherwise smooth velocity field. This disturbing influence is due to the fact that superclusters are bounded gravitationally, at least to some extent.

 

That is not what "gravitationally bound" means. The poineer and voyager spacecraft which are leaving the solar system are not gravitationally bound to the sun, but the sun's mass does slow them down. To be not gravitationally bound means the velocity is greater than the escape velocity.

 

~modest

[coldcreation]

You purposefully miss my point.

 

Also, in a static universe everything is gravitationally bound.

 

Actually, I didn't miss your point purposely. But I did reread what you wrote, again: "If you set a rotating structure next to a different rotating structure then the two structures may not collapse on their own center because of the centrifugal force, but the two structures will collapse on each other."

 

At first it didn't make sense because essentially you are saying 'they may not collapse, but they will collapse.' But I see now what you mean.

 

Fortunately, things in this universe have a tendency to remain in gravitationally bound quasi-equilibrium configurations without obligatorily collapsing. To argue that everything will eventually collapse in a static universe, even if not on their local centers of mass (due to centrifugal force), seems a rather broad assumption, in light of billions of observed gravitationally bound objects currently enjoying the freedom of motion consistent with equilibrium.

 

Theoretically, the loss of energy via gravitational radiation could force the Earth to fall into the Sun. But, the total energy of the Earth orbiting the Sun (gravitational potential energy plus kinetic energy) is about 1.14 × 10^36 joules of which only 200 joules per second are lost via gravitational radiation. This leads to a decay in the orbit by about 1 × 10^−15 meters per day, which is about the diameter of a proton. At this rate, it would take the Earth about 1 × 10^13 times more than the current age of the Universe to collapse onto the Sun. (Source).

 

"As the orbital speed becomes a significant fraction of the speed of light, this equation becomes inaccurate. It is useful for inspirals until the last few milliseconds before the merger of the objects.

Substituting the values for the mass of the Sun and Earth as well as the orbital radius gives a very large lifetime of 3.44 × 10^30 seconds or 1.09 × 10^23 years (which is approximately 10^15 times larger than the age of the universe)." (Source)

 

That quite a long time. I would even venture as to speculate that the time-scales of such a process of gravitational energy loss in the context of a static curved spacetime continuum tends to lengthen (on average) as scales are increased, and as the number of bodies of a system increases (ad infinitum). The further the distance between objects in a manifold of constant positive Gaussian curvature, the longer it will take for wholesale collapse due to gravitational energy loss.

 

In other words, such a process would be entirely negligible over cosmic time scales consistent with a static universe of the type proposed by Fischer (finite), Bruce (infinite), Qtop or CC (finite or infinite?).

 

 

Spacetime is not the surface of any hypersphere (in any number of dimensions). A great circle is not the path of a photon. [snip] The length, in spacetime, for a null geodesic is zero. The length, on a hypersphere, for a great circle is non-zero. It is not the same. A space [math]\mathbb{E}^3[/math] will make, but a spacetime in [math]\mathbb{E}^4[/math] is a mistake.

 

I think you're having trouble switching from reduced dimension representations to four-dimensional representations.

 

No one has ever claimed that spacetime is the surface of a hypersphere (in any number of dimensions). A great circle (also known as a Riemannian circle) is the path of a photon in a reduced dimension representation such as a Riemann sphere.

 

Note that in the case of an expanding hypersphere (a reduced dimension model of the universe) the path of a photon between the emitting source and that of the observer "must lie uniquely on the expanding "great circle" set of arcs". (Bacinich, Kriz, 1995, see link below).

 

From a cosmological observability perspective, in either a linearly expanding universe, or one in which spacetime is static and curved geometrically, the fundamental nature of the distortion induced by the metrical properties along the 'curved' photon trajectory is equivalent to a great circle arc in reduced dimension. The distortion observed in the expanding regime (as exemplified by redshift z and time dilation) "involves mapping the outward diverging spherical field to an inward converging spherical field." See Figure 6 of Photon Trajectory Attributes of an Expanding Hypersphere, Bacinich, Kriz, 1995.

 

NOTE: This paper examines, using fundamental properties of light cones found in an expanding 4-space, a hyperspherical spacetime, a simplified 2-space model based on well-known great circle geodesic concepts as a means of analyzing photon trajectories. This expanding spacetime is uniquely associated with both past and future light cones corresponding to a geodesic path locus defined by a set of radial null geodesics in expanding hyperspherical space. See too Figure 1 and equation 1. The metric associated with distance between observer and emitting source can be defined in 2-space because the shortest path of photons between observer and source lie exclusively on the expanding great circle arcs. The photon trajectories on great circle arc paths in 2-space correspond to local general relativistic transformations in four dimensional spacetime.

 

The same holds true in a static universe with constant positive Gaussian curvature.

 

The distortion observed in a static nonexpanding geometrically curved spacetime regime (as exemplified by redshift z and time dilation) is attributed to the both the non-Euclidean metric structure of the spatial path between the source and the observer, and the nonlinear function of temporal component that separates two events, i.e., time appears to run slower at the source compared to local clocks at the observers frame. In the case where such a universe is curved spherically (K = +1), the distortion in the path of a photon is identical to that of a great circle arc on the surface of a sphere. The difference when considering this path in a four-dimensional continuum is straightforward. The path appears as a straight line (the line of sight is straight, in the Euclidean sense) but the spatiotemporal intervals and increments along the geodesic path (e.g., in light years) vary with distance and time. See diagram C of Figure 1ABC. See too in the same link Figure 1C and Figure ICb.

 

 

That is the meaning of cosmological redshift z and time dilation in a static universe.

 

 

On a Euclidean plane the path of a photon is a straight line. On a sphere, the geodesic path of a photon is that of a great circle (or an arc section of such). Obviously, if the geometric structure of a static universe (in four-dimensions) is Minkowskian (flat) the path of a photon is not equivalent to that of a great circle arc (in reduced dimension). There is no distortion along the path (no redshift).

 

If, on the other hand, the geometric structure of a four-dimensional manifold is consistent with spherical geometry (K = +1), then the path of a photon is identical to that of the great circle path in reduced dimension. This geodesic path is the shortest distance between two points in a four-dimensional spacetime manifold (of the type in which we live).

 

The derived solution for the distortion along the photon path can be viewed as logically equivalent to one consistent with the general relativistic framework of a globally curved space-time manifold, either caused by the expansion of the hypersphere (in the FLRW sense), or the geometric structure of the spacetime manifold in a static regime.

 

 

This is good. No longer are you saying that things can be globally static because they have peculiar velocities. You're saying they are in a local equilibrium because they orbit local things (something I've already agreed to) and they are globally static for some other reason. Very good.

 

Exactly. This is what I've been saying all along: There is not one physical process alone responsible for the maintenance equilibrium. There are at least two physical processes or properties responsible. One is orbital velocity locally (including scales compatible with clusters and supercluster) and the other is curvature itself.

 

It is the latter upon which we disagree. I've attempted to assure you that your worries are unwarranted but it seems your attachment to the standard unstable model is unshakable. That's fine, as long as you recognize that other possibilities may in fact exist. It would be a pity to allow one's confident belief in the truth or validity of a world-model (e.g., FLRW) to stand in the way of one of the most fundamental aspects of the scientific method: investigating phenomena, acquiring new knowledge, and/or correcting and integrating previous knowledge.

 

Since you appear to have difficulty in visualizing the situation in four-dimensional spacetime, I've given you the reduced dimension analogue. Simply put, locally spacetime is flat. Curvature vanishes locally, just as it does on the surface of the earth locally: meaning that global curvature does not induce gravitational collapse locally (or globally, as we will see below).

 

At scales where curvature does begin to manifest itself (beyond the traditional boundary of the Local Group, where the onset of cosmological redshift is clearly detected), stability is assured on two counts: due to local orbital velocity fields of clusters and superclusters, and due to the curvature of space itself, which, according to Fischer, acts as if it were a "tension" (analogous to surface tension), rather than a repulsive force, effectively canceling out any acceleration tendency. "This tension plays the role of potential energy in the Newtonian picture." (See page 73 of Fischer's Fischer, 2009 for a more analytical explanation, part of which was reproduced in my previous post). The result differs from Einstein's since lambda does not produce stability, whereas the Fischer tension does.

 

My personal view, as explained earlier, is that such a tension is not required, since spacetime is locally flat at all points on the manifold (as on the surface of the earth locally) and so no force is exerted on objects in the field due to the global curvature. But I may have to accept such a tension if there's no other way around the diverging gravitational potential problem associated with both Newtonian and Einsteinian theory.

 

This is one of the main differences with the Fischer model and the one proposed here. Even so, we arrive at the same conclusion: the net force is zero at every point of the manifold, resulting in global equilibrium.

 

 

As far as reading those links—you know from previous discussion that I've read one of them. It didn't make sense, and I don't accept things on faith.

 

This is what you wrote about the Fischer link:

 

It does not make sense to me. Most of the paper is an explanation and derivation of Einstein's static universe. It could all be removed as superfluous. The key claims of the paper are "explained" in one short and ambiguous paragraph which I don't understand.

 

The difference between the Einstein static universe and the Fischer static universe are that, in the latter, the time parameter is no longer cosmic time, and that the cosmological term (lambda) is no longer treated as a repulsive force of the vacuum. There is nothing superfluous that should be eliminated, nor is there anything difficult to understand involved. Everything in the "ambiguous paragraph" (whichever one you refer to) is defined in terms of physical properties in the accompanying Einstein field equations.

 

The key claims of the paper are that the universe is static and curved (as opposed to expanding and flat) and that both global stability and cosmological redshift z are the result of curvature. The derivations of the Einstein field equations show how the author arrived at those conclusions.

 

[EDIT} The dismissal as superfluous of the Einstein field equations on the grounds, a priori, that all solutions must be unstable is unfounded in light of the replacement of cosmic time with a time component that varies with distance. After all, this is exactly what we observe (it's called the look-back time). The time a clock keeps is dependent on distance from an observer. The further away, the slower the clock, relative to that of the observer. The changing of the physical meaning of the negative pressure term (or repulsive force of the vacuum) to a 'tension' is amply justified in the sense that no new physics is required to explain what is operational. The result is No change in the scale factor to the metric over time. The universe neither expands or collapses.{End EDIT]

 

It might help if you specify which "ambiguous paragraph" you refer to, and to clearly state exactly what it is you haven't understood.

 

 

A theory which makes successful predictions before they are confirmed is not a problem. The scientific method is really rather fond of that sort of thing.

 

Which predictions do you refer to? (1) Gamow's 50°K for the CMBR (1961), (2) the accelerated expansion, (3) the value of lambda, (4) the dark energy component, (5) the cold dark matter abundance, (6) the abundance of light elements, (7) the formation of the large-scale structures (e.g., superclusters, the great walls), (8) the average decrease in luminosity of quasars, (9) evolution in the look-back time, (10) the anisotropies of the CMBR spatial power spectrum, (11) the fate of the universe (its topology), (12) the inflationary phase, (13) the age of the oldest stars, (14) the epoch of the so-called dark ages, (15) the local streaming motions of galaxies, (16) the metallicity of high-z objects, (17) the actual density of matter in the universe to the critical density?

 

 

 

I would rather tend to agree with "The change in time scale has the same consequence as a retarding force". A notable consequence: things get further away from each other over time.

 

Another notable consequence is that stability is maintained (that is what the equations in that work prove).

 

 

Clocks tend to settle where their proper time is most dilated.

 

That's relative.

 

Clocks tend to settle in many places, not at some hypothetical mass horizon or big crunch (where their proper time is most dilated).

 

 

 

___________

 

 

Hopefully this post will have served to help make the physical concept of stability in a static universe more palpable. It really is quite simple an approach.

 

It's one thing to reject a static model on the grounds that it is untenable, for whatever reason(s), but it's another thing to reject a model because you don't understand it, and/or because you see the outcome of the FLRW models as the only possible solutions.

 

 

CC

[coldcreation]

I have edited the above post. One of the edits is reproduced here for emphasis, and just in case anyone missed it above:

 

 

NOTE: This paper, Photon Trajectory Attributes of an Expanding Hypersphere examines, using fundamental properties of light cones found in an expanding 4-space, a hyperspherical spacetime, a simplified 2-space model based on well-known great circle geodesic concepts as a means of analyzing photon trajectories. This expanding spacetime is uniquely associated with both past and future light cones corresponding to a geodesic path locus defined by a set of radial null geodesics in expanding hyperspherical space. See too Figure 1 and equation 1. The metric associated with distance between observer and emitting source can be defined in 2-space because the shortest path of photons between observer and source lie exclusively on the expanding great circle arcs. The photon trajectories on great circle arc paths in 2-space correspond to local general relativistic transformations in four dimensional spacetime.

 

The same holds true in a static universe with constant positive Gaussian curvature.

 

 

See other edits above as well.

 

 

CC (from an Apple Store somewhere in Barcelona)

[modest]

I apologize for the delay, CC. I've had some serious unmitigated plight swung my way.

 

Actually, I didn't miss your point purposely. But I did reread what you wrote, again: "If you set a rotating structure next to a different rotating structure then the two structures may not collapse on their own center because of the centrifugal force, but the two structures will collapse on each other."

 

At first it didn't make sense because essentially you are saying 'they may not collapse, but they will collapse.' But I see now what you mean.

 

I should have said "centers" rather than "center".

 

Theoretically, the loss of energy via gravitational radiation could force the Earth to fall into the Sun.

 

Why is that relevant? 99% of the solar system's mass already collapsed into the sun, and not because of gravitational radiation.

 

But, the total energy of the Earth orbiting the Sun (gravitational potential energy plus kinetic energy) is about 1.14 × 10^36 joules of which only 200 joules per second are lost via gravitational radiation. This leads to a decay in the orbit by about 1 × 10^−15 meters per day, which is about the diameter of a proton. At this rate, it would take the Earth about 1 × 10^13 times more than the current age of the Universe to collapse onto the Sun. (Source).

 

That's fine.

 

That quite a long time. I would even venture as to speculate that the time-scales of such a process of gravitational energy loss in the context of a static curved spacetime continuum tends to lengthen (on average) as scales are increased, and as the number of bodies of a system increases (ad infinitum). The further the distance between objects in a manifold of constant positive Gaussian curvature, the longer it will take for wholesale collapse due to gravitational energy loss.

 

In other words, such a process would be entirely negligible over cosmic time scales consistent with a static universe of the type proposed by Fischer (finite), Bruce (infinite), Qtop or CC (finite or infinite?).

 

You've missed the point. My point is not that a stable orbit will degrade over time.

 

Your conclusion also doesn't follow from the premise. Your premise involves the earth which has an orbital velocity relative to the sun and the conclusion involves a static universe in which noting has an orbital velocity relative to anything else over large distances.

 

Spacetime is not the surface of any hypersphere (in any number of dimensions). A great circle is not the path of a photon. [snip] The length, in spacetime, for a null geodesic is zero. The length, on a hypersphere, for a great circle is non-zero. It is not the same. A space [math]\mathbb{E}^3[/math] will make, but a spacetime in [math]\mathbb{E}^4[/math] is a mistake.

 

I think you're having trouble switching from reduced dimension representations to four-dimensional representations.

 

My comment (that a photon's path in spacetime is not a great circle) has nothing really to do with "reduced dimension representations".

 

No one has ever claimed that spacetime is the surface of a hypersphere (in any number of dimensions).

 

Great circles exist on a sphere. When you say that photons make "great circle arcs in a spherical spacetime manifold", you are saying that the spacetime manifold is a sphere.

 

Note that in the case of an expanding hypersphere (a reduced dimension model of the universe) the path of a photon between the emitting source and that of the observer "must lie uniquely on the expanding "great circle" set of arcs". (Bacinich, Kriz, 1995, see link below).

 

That is correct. The path of a photon through space will be a great circle (if the space is spherical). I said this in my last post. Perhaps I should back up and make things clear.

 

Space is three dimensional. If space has constant global positive curvature then it is a hypersphere (a three dimensional sphere). A sphere has intrinsic curvature so you don't have to embed it in a 4th dimension, but it might help to visualize it if you consider it that way. The space we live in, if it is positively curved, is like the surface of a 4 dimensional ball. Hence, any direction you send a photon, it will travel around the ball on a great circle back to its origin.

 

A photon's path through spacetime, however, is not a great circle. Spacetime is four dimensional. It has three dimensions of space and one of time. The link you give, Photon Trajectory Attributes of an Expanding Hypersphere, has a diagram that will allow me to demonstrate,

 

 

The line that I marked red is one dimension of a hypersphere (a three-dimensional manifold). It represents space at some particular time. The red line is a great circle. The blue line is the photon's path through spacetime (a four-dimensional manifold). It is *not* a great circle. If you look at a spacetime metric, like the Robertson Walker metric or de Sitter metric, the distance along the blue line is actually zero. The distance along a great circle is not zero.

 

So, to simplify things, a photon's path through space (given that there is enough mass to close the space) is a great circle. Its path through spacetime is not a great circle. Looking at your comments,

 

The surface of a sphere, extended to three-space (or even to a

four-dimensional manifold

of the type in which we live) has the same radius of curvature at all points in the homogeneous and isotropic spacetime manifold. The path of a photon on such a Gaussian surface (or Riemannian manifold) is best described by a

great circle

arc, and it is correct describe the path mathematically as such.

 

The path of massive objects, in the case where the global topology was spherical, would be equivalent to

great circle arcs (in four-dimensions

now), the shortest path between two points (from wherever they were to the big crunch) in "straight" geodesic lines. That happens to be the same path on which photons propagate.

 

The only reason I corrected you is that you made quite clear you were talking about the four dimensional spacetime manifold—that the photon makes a great circle on that manifold. It really doesn't.

 

NOTE: This paper examines, using fundamental properties of light cones found in an expanding 4-space, a hyperspherical spacetime...

 

The thing that expands in that paper, the hypersphere, the circles in the diagrams, is space. A spacetime manifold that is consistent with GR is not a hypersphere. Sometimes "closed spacetime" or "expanding spacetime" or "hyperspherical spacetime" are used by cosmologists in a way that might be confusing. The thing that is closed, expanding, and a hypersphere, is space—not spacetime.

 

The same holds true in a static universe with constant positive Gaussian curvature.

 

"Static" and "constant positive Gaussian curvature" are mutually exclusive if the universe has non-zero mass content.

 

On a Euclidean plane the path of a photon is a straight line. On a sphere, the geodesic path of a photon is that of a great circle (or an arc section of such). Obviously, if the geometric structure of a static universe (in four-dimensions) is Minkowskian (flat) the path of a photon is not equivalent to that of a great circle arc (in reduced dimension).

 

It is not the "four dimensional" part that makes the path different from a great circle. You can have great circles in four dimensions (e.g. on a 4-sphere). It might be useful to examine the differences between two geometries that can have global constant positive curvature, Riemannian and Lorentzian. Specifically, a Riemannian 4-sphere and de Sitter's metric. They are not the same. The former doesn't make a spacetime in GR and the latter does.

 

 

 

Exactly. This is what I've been saying all along: There is not one physical process alone responsible for the maintenance equilibrium. There are at least two physical processes or properties responsible. One is orbital velocity locally (including scales compatible with clusters and supercluster) and the other is curvature itself.

 

Saying that curvature causes equilibrium is like saying Norwegian winters cause heat stroke. It just makes no sense at all. It's like a Christopher Nolan film. Flat spacetime is static. Curved spacetime is not. All of the reasons, explanations, links, and discussion toward that point and you are casually stating the exact opposite.

 

I'm not entirely sure this discussion is leading anywhere productive.

 

~modest

[coldcreation]

Why is that relevant? 99% of the solar system's mass already collapsed into the sun...

 

The point is that the solar system is stable. None of the planets will collide with each other or be ejected from the system for several billion years (whether due to gravitational collapse, dynamical instability, gravitational scattering, gravitational radiation, or any other process). Once the objects of a given self-gravitating system have acquired most of their material and have settled into nearly circular coplanar orbits the long-term dynamic stability is assured. However much still needs to be understood in this respect.

 

See for example:

 

The long-term dynamical evolution of the solar system

The issue of the long-term stability of the Solar System is of course one of the oldest unsolved problems in Newtonian physics, but recently (largely numerical) work has provided some insight into the problem. In particular, several lines of investigation suggest that the Solar System is subject to the deterministic chaos recently found in many nonlinear Hamiltonian systems. This inherent unpredictability has profound implications for the dynamics of the Solar System, not least of which is the demise of the "clockwork" picture which dominated thinking in the nineteenth century and carried over into much of the twentieth. However, the same numerical simulations that provided evidence for chaos also show that in most cases the timescale for macroscopic changes in the system (e.g. large changes in the eccentricity, crossing of orbits, ejection of bodies, etc) is several orders of magnitude longer that the timescale for unpredictability.

 

See too:

 

Self-Gravitating Systems: Characteristics

 

Long-term dynamical behaviour of natural and artificial n-body systems

 

 

You've missed the point. My point is not that a stable orbit will degrade over time.

 

That was my point too.

 

 

Your conclusion also doesn't follow from the premise. Your premise involves the earth which has an orbital velocity relative to the sun and the conclusion involves a static universe in which noting has an orbital velocity relative to anything else over large distances.

 

The distances we were discussing are compatible with clusters and superclusters. Dynamical stability is acquired at least up to these scales.

 

Beyond that (on a scales compatible with, say, the observable universe), equilibrium is maintained not by orbital velocity, but via the fact that the constant positive curvature in such a static general relativistic universe is homogeneous, i.e., the same at all points in the manifold. Self-gravitating systems are essentially 'freely-floating' relative to the background Gaussian field of constant positive curvature.

 

 

My comment (that a photon's path in spacetime is not a great circle) has nothing really to do with "reduced dimension representations". Great circles exist on a sphere. When you say that photons make "great circle arcs in a spherical spacetime manifold", you are saying that the spacetime manifold is a sphere.

 

A reduced dimension representations is a tool for understanding what happens in a four-dimensional spacetime continuum. In the case under study the four dimensional spacetime manifold is positively curved (the topology is spherical). The path of a photon is consistent (in 4-D) with a great circle arc (in reduced dimension). In a positively curved 4-D manifold the path of a photon is the shortest distance between two points (a geodesic straight line), just as in a reduced dimension illustration. The result is redshift z and time dilation increase with distance. Photons suffer energy-loss and clocks appear to tick slower in the look-back time.

 

The spacetime manifold is not the surface of a sphere. You're confusing a reduced dimension manifold with a 4-D manifold.

 

BTW, what I had written was that you don't see galaxies, clusters, or superclusters propagating along the same geodesics as photons (e.g., great circle arcs in a spherical spacetime manifold), do you? I should have written for clarity; the equivalent of great circle arcs...

 

And the point was that massive objects do not travel the same geodesic path as photons.

 

 

 

That is correct. The path of a photon through space will be a great circle (if the space is spherical). I said this in my last post. Perhaps I should back up and make things clear.

 

That was my point.

 

 

Space is three dimensional. If space has constant global positive curvature then it is a hypersphere (a three dimensional sphere). A sphere has intrinsic curvature so you don't have to embed it in a 4th dimension, but it might help to visualize it if you consider it that way. The space we live in, if it is positively curved, is like the surface of a 4 dimensional ball. Hence, any direction you send a photon, it will travel around the ball on a great circle back to its origin.

 

The idea that a photon will travel around the ball on a great circle back to its origin is an artifact of the reduced dimension representation. In a positively curved four-dimensional spacetime continuum such an artifact is non-existent. A photon does not return to it's point of origin.

 

If by "the space we live in" you refer to the area directly surrounding the earth, then you are correct. If you refer to 4-D spherical universe then you are mistaken. Light does not travel around the "ball." From what you write here it seems definite that you misunderstand what is a 4-dimensional object. If the universe is a 4-dimensional manifold that possess spherical topology then the path of a photon is not at all curved (like the surface of a ball). It is a straight geodesic path. Distances and time-scales will appear distorted from the rest-frame of any observer. If a ray of light is sent out into a 4-D spacetime it does not come back around and hit the originating source.

 

 

A photon's path through spacetime, however, is not a great circle. Spacetime is four dimensional. It has three dimensions of space and one of time...

 

Yes, exactly, and the path is a geodesic straight line. Viewed from the perspective of any observer the line of sight to the emitting source is straight (in the Euclidean sense) but the path is non-Euclidean in that the spatial increments and time intervals along the path vary with distance (i.e., there is distortion proportional to distance).

 

 

The line that I marked red is one dimension of a hypersphere (a three-dimensional manifold). It represents space at some particular time. The red line is a great circle. The blue line is the photon's path through spacetime (a four-dimensional manifold). It is *not* a great circle. If you look at a spacetime metric, like the Robertson Walker metric or de Sitter metric, the distance along the blue line is actually zero. The distance along a great circle is not zero.

 

Here is the problem: you are discussing the path and distance traveled of a photon in a four-dimensional Robertson Walker and de Sitter metric of an expanding universe, while the discussion here involves a reduced dimensional tool for understanding the path in 4-D, consistent with the metric of a static universe of the type postulated by Ernst Fischer's (2009) analytical approach to the Einstein field equations and CC's qualitative description of a Riemannian (or semi-Riemannian) spacetime continuum of constant positive Gaussian curvature.

 

EDIT: Clearly, the illustration you showed is consistent with the idea that in reduced dimensions the path of a photon is a great circle, even in an expanding reduced dimensional universe (that was the point of the link), and that in four dimensions, whether the universe is expanding or not is no longer a great circle, but that identically there is a distortion in the path. Incidentally, I noticed you only reddened one of the great circle arcs in that illustration. Had you reddened all of the great circle arcs you would have seen that the photon path locus centered on the origin of the time cone is precisely along a great arc at any given time t and radial distance r. The curvature of the past light cone locus shows that a significant distortion will manifest itself when an observer peers into the look-back time at distant source objects through a curved spacetime topology. This is so in both expanding and static models. If you look at Figure 4 it will be noticed that this curvature causes the past light cone to converge exponentially in the look-back time of the observer. A analogous distortion (and convergence) occurs in a static spacetime of constant positive Gaussian curvature.

 

My point, in case you missed it, was that the path of a photon in four dimensions is not a curve of great circle arc, but that the path distortion is consistent with a great arc in reduced dimensions.

 

In other words, a great circle arc trajectory projected to a 4-D positively curved static spacetime corresponds to a straight line (a geodesic) between the source to the observer. The path in 4-D is not an arc, a curve, or section of a great circle, it is a straight line, and the distortion along this straight line (a geodesic) manifests itself as redshift z and time dilation. The spatiotemporal increments along the path vary according to distance between the observer and the source. The cause of this cosmological distortion is constant positive Gaussian curvature of the spacetime manifold.

 

 

 

So, to simplify things, a photon's path through space (given that there is enough mass to close the space) is a great circle. Its path through spacetime is not a great circle.

 

Right. The path of photons through spacetime in all directions are straight geodesic lines (excluding local gravitational lensing and deflection). Those paths, in reduced dimension, correspond to great circle arcs. In 4-D the photon path is a straight geodesic line. The fact that the geodesic path (which, again, corresponds to a great arc in less than 4-D) means that light will be redshifted and clocks will appear to run slower with increasing distance from the source.

 

I can repeat that again if you like, in yet another way...

 

 

The only reason I corrected you is that you made quite clear you were talking about the four dimensional spacetime manifold—that the photon makes a great circle on that manifold. It really doesn't.

 

Your confusion here is probably related to switching from a reduced dimension representation to the full 4-D representation. Perhaps I haven't always specified which I was referring to, but it should have been obvious enough.

 

Here goes again: A great circle is by definition a reduced dimension representation in this case. That would be so since a great circle projected in four dimensions would not be a straight line, it would be a curve (or a circular path). Photons obviously do not follow the path of an airplane flying on a great circle arc from JFK to Charles de Gaulle.

 

 

"Static" and "constant positive Gaussian curvature" are mutually exclusive if the universe has non-zero mass content. Saying that curvature causes equilibrium is like saying Norwegian winters cause heat stroke. It just makes no sense at all. It's like a Christopher Nolan film. Flat spacetime is static. Curved spacetime is not...

 

This is a common misconception based on the erroneous idea (myth or belief) that FLRW models are the only possible solutions to the Einstein field equations.

 

Hopefully this discussion (the Redshift z thread in general) will serve as a reference for readers in the coming years to dispel such a myth and to show both qualitatively and quantitatively how two distinct physical processes (for cosmological redshift z and global stability, or instability) are accounted for in both static and expanding models consistent with empirical observations within the theoretical underpinnings of general relativity.

 

As it turns out, the more robust of the two interpretations for redshift z spelled out in the OP is the static curved spacetime approach since no ad hoc dark energy or cold-dark matter (additions of extraneous hypotheses to the theory to save it from being falsified) are required in order to find agreement with observations.

 

 

I'm not entirely sure this discussion is leading anywhere productive.

 

Au contraire. This discussion is everywhere productive. The idea that the universe is inherently unstable and expanding has met with resistance since the inception of the concept. The problem had been a fruitless search for an unknown cause of redshift. But this was a false dilemma since the cause of redshift had been known all along. It was indeed one of the first predictions of general relativity: an effect due to the propagation of light (throughout the entire electromagnetic spectrum) in a curved spacetime. That interpretation has been accounted for in the context of a static universe that displays intrinsic positive Gaussian curvature, in this thread and associated links (particularly those to the works of Ernst Fischer).

 

As stated in the OP there are two viable interpretations for cosmological redshift z. The ruling out of one of these interpretations over the other must be on empirical grounds, if it can't be decided on theoretical grounds.

 

The problem of global stability was a little more difficult to solve. Both Newton and Einstein were hit with the problem of diverging gravitational potentials in a universe that possesses a homogeneous and infinite distribution of mass. But this problem (along with a structurally similar problem of Olbers' paradox) also found resolution (to be discussed again shortly).

 

The solution, surprisingly enough was attributed to the same phenomenon responsible for redshift z: the curvature of spacetime itself.

 

In view of the lingering difficulties that arise with respect to understanding exactly how the problem of diverging potentials (a la Newton) or how stability is maintain in a globally curved spacetime (a la Einstein) can be resolved simply and naturally in a static universe, I will devote the next few posts to an in-depth review of the relevant literature.

 

 

CC

[modest]

Why is that relevant? 99% of the solar system's mass already collapsed into the sun...

 

The point is that the solar system is stable.

 

Like I said in my last post:

 

You've missed the point. My point is not that a stable orbit will degrade over time.

 

When you brought the thought experiment up in post 701 I immediately agreed in post 702 that clusters could have stable orbits. My objection was *not* that stable orbits degrade, so there is no need for you to argue that orbits are stable. My objection is *not* that:

 

Over large distances (billions of lightyears) in a static universe there are stable orbits that will degrade.

My objection is rather:

 

Over large distances there are no stable orbits because there are no tangent velocities sufficient to attain orbit.

 

None of the planets will collide with each other or be ejected from the system for several billion years (whether due to gravitational collapse, dynamical instability, gravitational scattering, gravitational radiation, or any other process).

 

And, that's fine. Here is the premise of your thought experiment from #701 which I agreed with in my first response and we don't need to argue:

 

Then we must expect the two clusters are not at rest. Fortunately, many galaxy clusters manage to 'survive' encounters with other clusters, since they were never initially at rest relative to one another. Those whose relative velocities were inadequate end as mergers. Those whose relative velocities are large and trajectories adequate, will not merge. They may remain in orbit or simply disperse.

 

The conclusion (or, you might say, the implication) that you draw from the premise is:

 

It is straight forward to extrapolate further, to more complex gravitationally bounded n-body systems, and/or to larger scales still.

 

My point is that motion is the key to equilibrium on local scales. No finely tuned velocities or initial conditions are required (nor is lambda). This same process is operational on all scales, up to superclusters (and beyond?).

 

The point is, too, that we can increase the scale with the same qualitative result. As such: The tangent velocity of galaxy clusters in a supercluster are usually sufficient that they orbit the supercluster. And, still larger: the tangent velocity of a supercluster relative to another neighboring supercluster are usually sufficient that they orbit a larger region still (megaclusters?). There's no reason to stop there.

 

I disagree with that conclusion. The velocity needed to orbit a roughly homogeneous collection of matter increases with the size of the collection of matter. A galaxy can easily be stable relative to a cluster with a velocity no more than a couple hundred km/s relative to that cluster, but a galaxy needs a velocity about a third the speed of light relative to a group of galaxies five billion lightyears across to be stable relative to a group that big.

 

In other words, the velocity necessary for one galaxy to orbit another is less than the velocity needed for two clusters to orbit one another, which is again less than the velocity needed for two superclusters to orbit one another, and again less than the velocity needed for two groups of 1,000 superclusters to orbit one another.

 

There are not tangent velocities as large as necessary with scales that large. In a static universe the mechanism that can prevent collapse and provide stability on smaller scales (angular momentum) does not operate on larger scales. If it did then we would see groups of superclusters rotate. Parts of the sky would be doppler shifted very different from other parts of the sky from the velocity needed to establish stability. Over several billion lightyears the velocity would be millions of km/s.

 

So, there's no need for you to respond by proving the premise—that small scaled things do remain stable. That's not the objection. Also, this,

 

False. The visible universe need not rotate. The only requirement is that neighboring superclusters are in motion relative to one another, and, that the global curvature is positive.

 

doesn't address the objection. I didn't object to a statement about local structures being stable for one reason and larger scales being stable for another reason. I was responding to a statement saying that the same mechanism (relative velocity) provides stability at any scale in a static universe.

 

We can rule out that possibility by observation, and even without observations it would be ruled out if one subscribes to Mach's principle.

 

Paragraphs and paragraphs proving that earth's orbit is stable are extremely discouraging. I can't believe that my point is so hard to find.

 

Once the objects of a given self-gravitating system have acquired most of their material and have settled into nearly circular coplanar orbits the long-term dynamic stability is assured. However much still needs to be understood in this respect.

 

See for example:

 

The long-term dynamical evolution of the solar system

The issue of the long-term stability of the Solar System is of course one of the oldest unsolved problems in Newtonian physics, but recently (largely numerical) work has provided some insight into the problem. In particular, several lines of investigation suggest that the Solar System is subject to the deterministic chaos recently found in many nonlinear Hamiltonian systems. This inherent unpredictability has profound implications for the dynamics of the Solar System, not least of which is the demise of the "clockwork" picture which dominated thinking in the nineteenth century and carried over into much of the twentieth. However, the same numerical simulations that provided evidence for chaos also show that in most cases the timescale for macroscopic changes in the system (e.g. large changes in the eccentricity, crossing of orbits, ejection of bodies, etc) is several orders of magnitude longer that the timescale for unpredictability.

 

See too:

 

Self-Gravitating Systems: Characteristics

 

Long-term dynamical behaviour of natural and artificial n-body systems

 

That's fine.

 

The distances we were discussing are compatible with clusters and superclusters. Dynamical stability is acquired at least up to these scales.

 

I wouldn't object to the possibility of superclusters being stable against collapse from the peculiar velocity of their constituent components. Our local group, for example, would need a tangent velocity of only sqrt(GM/r) = sqrt(6.7E-11*1.5E45/5.5E23) = 427 km/s relative to the Virgo supercluster to orbit it. That, conceptually, wouldn't be a problem.

 

All indications are, however, that superclusters are influenced by the Hubble flow meaning the constituent components are mostly not gravitationally bound to the cluster. Using the same example, our local group would need a radial velocity greater than sqrt(2GM/r) = sqrt(2*6.7E-11*1.5E45/5.5E23) = 605 km/s relative to the Virgo supercluster to be unbound to it. Our actual radial velocity is approximately 1300 km/s, so we, like the other groups, are expanding out of our local supercluster.

 

Regardless, the point is that the Virgo supercluster is not orbiting other nearby superclusters nor would a large group of near superclusters be orbiting a large group of further superclusters nor could the scale of this type of equilibrium be increased without increasing the velocity (something which would have drastic observational consequences).

 

The distances we were discussing are compatible with clusters and superclusters... Beyond that (on a scales compatible with, say, the observable universe), equilibrium is maintained not by orbital velocity, but via the fact that the constant positive curvature in such a static general relativistic universe is homogeneous

 

But, you are talking about scales that large...

 

The point is, too, that we can increase the scale with the same qualitative result. As such: The tangent velocity of galaxy clusters in a supercluster are usually sufficient that they orbit the supercluster. And, still larger: the tangent velocity of a supercluster relative to another neighboring supercluster are usually sufficient that they orbit a larger region still (megaclusters?). There's no reason to stop there.

 

...and that's where my objection lies.

 

A reduced dimension representations is a tool for understanding what happens in a four-dimensional spacetime continuum. In the case under study the four dimensional spacetime manifold is positively curved (the topology is spherical). The path of a photon is consistent (in 4-D) with a great circle arc (in reduced dimension).

 

I was objecting to your previous assertion—that photons follow great circles in spacetime (and it is correct to describe the path mathematically as such). I was not objecting to (or asserting anything about) the consistency of 'reduced dimension representations'.

 

In a positively curved 4-D manifold the path of a photon is the shortest distance between two points (a geodesic straight line), just as in a reduced dimension illustration.

 

Ok, my point was that null geodesics are not great circles. I wasn't saying that a photon's path isn't the shortest (either in the metric or in a "reduced dimension illustration".

 

The spacetime manifold is not the surface of a sphere.

 

Right. That's why I said,

 

Spacetime is not the surface of any hypersphere (in any number of dimensions).

 

I don't see any way to reconcile these two statements:

 

 

The spacetime manifold is not the surface of a sphere.

 

The surface of a sphere, extended to three-space (or even to a four-dimensional manifold of the type in which we live) has the same radius of curvature at all points in the homogeneous and isotropic spacetime manifold. The path of a photon on such a Gaussian surface (or Riemannian manifold) is best described by a great circle arc, and it is correct describe the path mathematically as such.

[modest]

...continued

 

 

You're confusing a reduced dimension manifold with a 4-D manifold.

 

There's nothing to get confused about. In GR, space can be a 3-sphere, but spacetime cannot be a 4-sphere. It would be difficult to confuse space with spacetime or to confuse a 3-sphere with a 4-sphere.

 

There's nothing wrong with a sphere. It's a perfectly good manifold—in two, three, four, or however many dimensions. It just doesn't correctly describe spacetime. The distance of a null geodesic needs to be zero which is only the case when the tangent space is Lorentzian. A sphere is Remmanian, meaning the tangent space is Euclidean and the geodesic where dt=dx is a great circle of non-zero length. This would be true in 2 dimensional spacetime, 3-dimensional spacetime, or 4-dimensional spacetime (i.e. reduced dimension illustrations are not relevant to my point).

 

BTW, what I had written was that you don't see galaxies, clusters, or superclusters propagating along the same geodesics as photons (e.g., great circle arcs in a spherical spacetime manifold), do you? I should have written for clarity; the equivalent of great circle arcs...

 

Indeed, the part that I later quoted from that same post does say "equivalent to great circle arcs (in four-dimensions now)".

 

Honestly, if you look at my response in post #713, all I'm saying is that great circles don't make the best analogy and aren't correct mathematically for null geodesics, because null geodesics don't make great circles.

 

And the point was that massive objects do not travel the same geodesic path as photons.

 

No doubt about that

 

 

That is correct. The path of a photon through space will be a great circle (if the space is spherical). I said this in my last post. Perhaps I should back up and make things clear.

 

That was my point.

 

Which is, perhaps, why I said "that is correct". Nonetheless, earlier your point was that photons make great circles on the four dimensional spacetime manifold and that describing them mathematically in that way was correct. It is that earlier point with which I disagree.

 

 

Space is three dimensional. If space has constant global positive curvature then it is a hypersphere (a three dimensional sphere). A sphere has intrinsic curvature so you don't have to embed it in a 4th dimension, but it might help to visualize it if you consider it that way. The space we live in, if it is positively curved, is like the surface of a 4 dimensional ball. Hence, any direction you send a photon, it will travel around the ball on a great circle back to its origin.

 

The idea that a photon will travel around the ball on a great circle back to its origin is an artifact of the reduced dimension representation.

 

No, the ball is an artifact of adding an additional (superfluous) dimension. The 3-sphere correctly describes space with constant global positive curvature. With a 3-sphere manifold, a straight path is a great circle arc. Any straight path will eventually end up where it started. This is true whether or not we add a 4th spatial dimension. Like I said, the 3-sphere has intrinsic curvature so we don't have to put it on a four dimensional ball. We do this only to help visualize.

 

In a positively curved four-dimensional spacetime continuum such an artifact is non-existent. A photon does not return to it's point of origin.

 

It's not exactly clear what you mean. A photon could never return to its event of origin in spacetime (that would be a closed null curve), but a photon surely can return to its spatial position. That would be a closed or "finite and unbound" universe like Einstein's static universe (photons end up where they started).

 

If by "the space we live in" you refer to the area directly surrounding the earth, then you are correct. If you refer to 4-D spherical universe then you are mistaken.

 

You are misunderstanding. "Space" refers to the spatial dimensions of the universe—the X, Y, and Z dimensions. In a closed homogeneous and isotropic universe space is a 3-sphere. It is three dimensional with constant positive curvature.

 

Light does not travel around the "ball."

 

I don't want to be rude. Your reaction indicates that you are maybe a little unfamiliar with manifolds. Light does not, in fact, travel around a ball. Like I said "A sphere has intrinsic curvature so you don't have to embed it in a 4th dimension, but it might help to visualize it if you consider it that way. The space we live in, if it is positively curved, is like the surface of a 4 dimensional ball."

 

You don't have to visualize our three dimensional space as embedded on a 4D ball. The manifold (the 3-sphere) is intrinsically curved, so it exists just fine without making it the surface o fsome higher dimensional thing. It is only a means of visualizing a great circle in three dimensional curved space.

 

I notice you put quotes around the term "ball". Wiki explains the term and the concept here: Wikipedia -- Ball (mathematics)

 

From what you write here it seems definite that you misunderstand what is a 4-dimensional object.

 

I am describing a 3 dimensional manifold—space. I used a 4 dimensional object only to help visualize the curvature of the three dimensional space. You might imagine a 2-sphere. Imagine a two-dimensional creature living on a 2-sphere. He could determine that his space is not flat by making a large triangle and adding the angles. To help him visualize the curvature of his space he could picture it as the surface of a three dimensional ball. He is like a two dimensional creature living on the surface of the earth.

 

In the same way, if our three dimensional space is positively curved then we could visualize our space as being the surface of a four-dimensional ball. This is a common exercise in picturing the curvature of space.

 

If the universe is a 4-dimensional manifold that possess spherical topology then the path of a photon is not at all curved (like the surface of a ball).

 

Nowhere in my explanation that you quoted am I talking about a 4-dimensional manifold. I am describing "space". It is 3 dimensional.

 

It is a straight geodesic path. Distances and time-scales will appear distorted from the rest-frame of any observer. If a ray of light is sent out into a 4-D spacetime it does not come back around and hit the originating source.

 

Correct. A particle that returns to the same event in spacetime is a closed time-like curve. It is paradoxical. It is not what I am describing.

[modest]

...continued

 

 

A photon's path through spacetime, however, is not a great circle. Spacetime is four dimensional. It has three dimensions of space and one of time...

 

Yes, exactly, and the path is a geodesic straight line.

 

So, here I'm talking about spacetime. In the previous paragraph I was talking about space. With that in mind, if you read it again, I think it will be more clear.

 

The line that I marked red is one dimension of a hypersphere (a three-dimensional manifold). It represents space at some particular time. The red line is a great circle. The blue line is the photon's path through spacetime (a four-dimensional manifold). It is *not* a great circle. If you look at a spacetime metric, like the Robertson Walker metric or de Sitter metric, the distance along the blue line is actually zero. The distance along a great circle is not zero.

 

Here is the problem: you are discussing the path and distance traveled of a photon in a four-dimensional Robertson Walker and de Sitter metric of an expanding universe, while the discussion here involves a reduced dimensional tool for understanding the path in 4-D, consistent with the metric of a static universe of the type postulated by Ernst Fischer's (2009)...

 

Comparing the paper to FLRW or an expanding universe is not a problem. The first line of the abstract does the same, and the paper's first two conclusions:

 

1. It provides an intuitively appealing representation of the big bang event as an “expansion of space” process.
   
2. It has a strong analytical basis rooted in global models for black holes and Friedmann-Lemaitre expansion systems.
   

 

EDIT: Clearly, the illustration you showed is consistent with the idea that in reduced dimensions the path of a photon is a great circle, even in an expanding reduced dimensional universe (that was the point of the link)

 

I don't know exactly what you mean by "in reduced dimensions". The diagram is two dimensional and it is meant to depict something that is four dimensional. Two dimensions are omitted. They are both spatial dimensions.

 

and that in four dimensions, whether the universe is expanding or not is no longer a great circle

 

Perhaps I don't follow, but the red line (the spatial path) would be a great circle even if the diagram were 4 dimensional. The blue line (the spacetime path) is not a great circle in either the 2 or 4 dimensional diagram.

 

I noticed you only reddened one of the great circle arcs in that illustration. Had you reddened all of the great circle arcs you would have seen that the photon path locus centered on the origin of the time cone is precisely along a great arc at any given time t and radial distance r. The curvature of the past light cone locus shows that a significant distortion will manifest itself when an observer peers into the look-back time at distant source objects through a curved spacetime topology. This is so in both expanding and static models. If you look at Figure 4 it will be noticed that this curvature causes the past light cone to converge exponentially in the look-back time of the observer. A analogous distortion (and convergence) occurs in a static spacetime of constant positive Gaussian curvature.

 

I would recommend reading part 3 of Ned Wright's tutorial: http://www.astro.ucla.edu/~wright/cosmo_03.htm Notice the spacetime diagram at the bottom is the same as the figure 4 you're talking about. Part 2 sets things up pretty well also.

 

 

My point, in case you missed it, was that the path of a photon in four dimensions is not a curve of great circle arc, but that the path distortion is consistent with a great arc in reduced dimensions.

 

The issue is not "reduced dimensions". A null geodesic in two or three dimensional spacetime is not a great circle.

 

The only reason I corrected you is that you made quite clear you were talking about the four dimensional spacetime manifold—that the photon makes a great circle on that manifold. It really doesn't.

 

Your confusion here is probably related to switching from a reduced dimension representation to the full 4-D representation. Perhaps I haven't always specified which I was referring to, but it should have been obvious enough.

 

You made clear you were talking about the path of a photon in spacetime (you actually explicitly said "the 4D spacetime manifold"). The path of a photon in spacetime is not a great circle regardless of how spacetime is represented.

 

~modest

[coldcreation]

Sorry for the delay in posting a response to the above.

 

In couple of weeks, I will be posting a rather lengthy layout of the situation relative to superclusters as bound gravitating systems is a static universe of constant positive Gaussian curvature. In addition, solutions regarding diverging integrals in a universe with a homogeneous matter distribution.

 

 

The observed distribution and dynamics of matter in space may hold the key to understanding the essence of the physical universe and its evolution in time. The large-scale structures and spatial distribution can place important constraints on formation and evolution as function of time. Thus, accurate predictions for the spatiotemporal distribution of of the bubble-like networks of sheets, filaments, walls, voids, galactic superclusters (along with their component clusters, groups and galaxies) can be used to test the viability of cosmological models, whether static or unstable. Much progress has been made but much has yet to be explored.

 

 

Here is a very brief outline of what is to come:

 

 

Spacetime Curvature

 

Curvature is the central theme of theoretical and observational cosmology. To a large extent today the concept of spacetime curvature relative to the universe itself (the geometry of the manifold) is a profound mystery.

 

 

Equilibrium Solutions

 

Fortunately, methods for computing the N-body problem have grown exponentially in computational efficiency. These methods have been used to simulate the dynamics of systems with as many as 10 billion particles. We will see how these simulations relate to observations and how they fair with the static distribution of the large-scale structures.

 

 

The Local Group

 

The currently favored standard model of the universe (Lambda-CDM) with dark matter and dark energy does not allow for voids that are as large as inferred for the Local Void. It will be shown that such a problem does not arise in the general relativistic static model, since cosmological redshift is not interpreted as resulting from expansion, i.e., a large component of the observed redshift is not due to radial motion away from the local supervoid.

 

 

Superclusters

 

Superclusters are thought to be dynamically unstable systems, not bounded gravitationally. The reason given is usually because the universe is thought to be expanding, i.e., superclusters are moving away from one another, radially, in all directions. This may not be the case, as should emerge.

 

 

The Large-scale Structures

 

The differentiation between models must be made by studying the characteristics that would arise in the case where superclusters are gravitationally bound, in comparison with properties in the case where they are not gravitationally bound.

 

 

Mounting Evidence for Gravitationally Bound Superclusters

 

Direct and indirect evidence with respect to morphology, orientation, flattening, filaments, physical scale (size and mass inferences), richness (multiplicity), nonrandom correlations and redshift (peculiar motions) are analyzed in the context of both expanding models and static models.

 

 

Limits of the Cosmological Principle

 

On the scales of superclusters the cosmological principle does not hold. Deviations from pure Hubble expansion are expected, just as deviations from constant Gaussian curvature would be expected in the static model.

 

 

Redshift Surveys and the Large-scale Structure

 

The measurement of peculiar velocities (and bulk motion) allows the reconstruction of the underlying density field, which can be compared with that derived from the distribution of the visible galaxies. Measurements to date suggest that either (1) there exists a large component of invisible cold dark matter and that superclusters are not gravitationally bound systems (in the expanding case), or that (2) clusters and superclusters are gravitationally bound and that these large-scale structures are interacting and moving relative to one another (in the static case).

 

 

Hierarchical, monolithic collapse vs the top down model in a static universe

 

So far, all observational attempts to distinguish between the two competing big bang models for the formation of galaxies and the large-scale structures have failed. The question of how galaxies form and evolve with time thus remains one of the most important unanswered questions of contemporary astrophysics (Popesso 2006)

 

 

The top-down model in a static universe

 

One of the beauties of the static model of constant Gaussian curvature, contrarily to the standard hierarchical model or the picture of monolithic collapse, is that the formation of the large-scale structures and galaxies themselves occurs from the top-down. In other words, the density fluctuations from which individual and clusters of galaxies are formed cover vast regions of space (perhaps 100 Mpc or more). How does observational evidence support this scenario?

 

 

Voids

 

What is the role played by voids (or supervoids) in the cosmic dance?

 

 

Diverging Integrals in a Homogeneous Universe

 

Structurally similar to Olber's paradox is another problem that had affected the cosmology of both Newton and Einstein: regarding the gravitation potential diverging to infinity in a static homogeneous infinite universe. Problems and solutions are discussed.

 

 

 

 

Stay tuned...

 

 

 

CC

[36grit]

Cosmological Redshift z

 

There are only 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 (often called Doppler effect, (implying the expansion of space and the recession of objects in it, i.e., the radius of the universe changes with time t).

 

(2) The general relativistic curved spacetime interpretation (implying a stationary yet dynamic and evolving universe.

 

If indeed the Doppler interpretation (1) is not correct, a wholesale revision of cosmology is required.

 

Coldcreation

 

I have a bad habbit of sometimes starting from the first question in a forum.

I can see that I'm way out of my league here and I don't want to interrupt anything but, I can't help but wonder if anybody else has considered the possibility that perhaps:

 

 

(3) If the mass creating the light is also creating the space, it would most likely cause a "doppler" like effect. The distance plane imprinted by the light would arrive shortened or lengthened according to to the distance and direction of the given source of the light.

 

 

 

_______________________________________________________________________

[coldcreation]

Hello all,

 

 

After a relatively brief delay :) I've finally uploaded the text mentioned above.

 

 

A General Relativistic Stationary Universe

 

 

The new section, to be found if you scroll down towards the end, is entitled Large-Scale Structures - Superclusters and Supervoids. The topics under review are listed under the headers Spacetime Curvature, Equilibrium Solutions, The Local Group, The Large-scale Structures, Superclusters, Mounting Evidence for Gravitationally Bound Superclusters, Limits of the Cosmological Principle, Redshift Surveys and the Large-scale Structure, Time-scales and the large structures, Top down model of galaxy formation in a static universe, Voids and Supervoids, Re-Redshift z, Converging Integrals in a Homogeneous Universe, Homogeneous (background) gravitational field in general relativity, the problem of diverging integrals, and the inverse-square law of gravity.

 

 

There are several new schematic diagrams. They can be seen at an alternative location as well:

 

Static universe of constant Gaussian curvature

 

Einstein Static Universe of Constant Positive Gaussian Curvature, Figure ESU.

 

 

Some of the other diagrams can also be found here: Our motion with respect to galaxies in the Local Supecluster, Gigaparsec volume N-body simulations, The formation of clusters and large-scale filaments Figure FCLFS (simulation), and so on.

 

For a detailed explanation of these diagrams see A General Relativistic Stationary Universe and the hyperlinks therein.

 

 

Any questions or comments regarding the text and/or diagrams will be discussed here at scienceforums.

 

 

PS. Qtop, you still around?

 

 

Regards

 

Coldcreation, aka DVDjHex

[CraigD]

A General Relativistic Stationary Universe[/center]

Getting a "Request took too long" error page when following this link. :(

 

Can you attach your text to your post, or give a URL that works?

[coldcreation]

Getting a "Request took too long" error page when following this link. :(

 

Can you attach your text to your post, or give a URL that works?

 

 

Hello GraigD,

Yup, there seemed to be a problem (perhaps too many photos).

 

I've separated the posts in two sections.

 

The main page should work now: A General Relativistic Stationary Universe

 

 

And here is the new post:

 

Large-Scale Structures - Superclusters and Supervoids

 

Let me know if there is still a problem with the link.

 

 

Thanks

 

CC

[modest]

Hi, CC.

 

A very well put together document for sure :)

 

The part I bolded:

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). In addition to illuminating how redshift z is caused in a globally curved four-dimensional spacetime manifold, it will be shown how objects (such as galaxy clusters and superclusters) remain stable against gravitational collapse without the requirement of a cosmological constant (or vacuum pressure). It is emphasized that global curvature plays an essential role in cosmology and provides a natural explanation for various empirical observations. Too, it is exemplified this point of view by considering a novel version of Einstein's 1916-1917 world-model, where cosmological redshift z is directly related to the large-scale structure of the universe.

 

still seems backwards to me and I'd love to get Craig's opinion on this, or any fresh opinion.

 

Flat spacetime to me means 'static' (in the sense that we are using the term). The spacetime inside a spherical shell, for example, is flat Minkowski spacetime. Particles in Minkowski spacetime don't want to collapse toward each other or expand away from each other.

 

Curved spacetime to me means non-static (again, in the sense that we are using the term). The spacetime in a uniform density sphere, for example, is curved. It wants to collapse.

 

In flat spacetime there is no gravitational force between objects. The distance between them doesn't change over time. Everything I know wants me to correct the bolded part to say "The general relativistic curved spacetime interpretation (implying a non-static metric in a non-stationary universe)" or "The general relativistic flat spacetime interpretation (implying a static metric in a stationary universe)"

 

~modest

[coldcreation]

Hi, CC.

 

A very well put together document for sure :)

 

Hi modest. Thanks :)

 

 

The part I bolded [...] still seems backwards to me and I'd love to get Craig's opinion on this, or any fresh opinion.

 

I would too.

 

 

Flat spacetime to me means 'static' (in the sense that we are using the term). The spacetime inside a spherical shell, for example, is flat Minkowski spacetime. Particles in Minkowski spacetime don't want to collapse toward each other or expand away from each other.

 

Actually there exist three static solutions of the Einstein field equations for a spacetime of the type we are discussing. One of these spacetimes is effectively Minkownkian, but this type of spacetime is not consistent with observations in the sense that such a universe would be geometrically flat, i.e., there would be no cosmological redshift (unless, of course, the universe were expanding at a finely-tuned or critical rate). The other two spacetime geometries are hyperbolically and spherically curved.

 

The example you give above of an idealized space inside a spherical shell (a la Newton: the shell theorem) is unphysical. Such a spacetime does not exist (at least not in a cosmological context).

 

 

Curved spacetime to me means non-static (again, in the sense that we are using the term). The spacetime in a uniform density sphere, for example, is curved. It wants to collapse.

 

Curved spacetime does not imply non-staticity. Locally, for example, the earth and other planets are embedded in a curved spacetime yet the solar system is not expanding or contracting: it is static.

 

Globally, curved spacetime need not imply instability. Let's take the example of a positively curved spacetime continuum. (1) There is no preferred direction towards which all objects will move and coalesce. (2) Locally, the geometry of a positively curved spacetime manifold approaches flatness, just as the curvature of the earth (in reduced dimensions) tends towards flatness locally (i.e., when the area considered is small, e.g. a few kilometers). Objects are not affected gravitationally by the large-scale geometric structure of spacetime.

 

The example you give above is unphysical (in the cosmological context) as it implies that the "uniform density sphere" is embedded within a larger spacetime manifold, like the collapsing of a cloud of gas is space. Such a cloud has a center (there is a preferred direction) towards which particles can gravitate. All points are not the same. This is not the case for an Einstein universe with positive Gaussian curvature. Such a universe has no center or preferred direction towards which all objects will gravitate. All points are the same.

 

That is the primary difference that arises when considering and comparing a local gravitational field (your analogy) with a globally homogeneous manifold of constant positive Gaussian curvature.

 

EDIT: The other difference arises when considering distance measurements gathered by an observer. In a static Einstein universe of positive Gaussian curvature light emitted from distant objects will be redshifted, and more so with increasing distance, since the magnitude of curvature increases with distance (just as the magnitude of curvature increases with distance on the surface of the earth [in reduced dimensions]). This will not be the case for an observer situated inside a large cloud of gas collapsing under the influence of gravity.

 

 

In flat spacetime there is no gravitational force between objects. The distance between them doesn't change over time. Everything I know wants me to correct the bolded part to say "The general relativistic curved spacetime interpretation (implying a non-static metric in a non-stationary universe)"

 

Just as in an expanding universe the global geometry may be either curved positively, negatively, of flat, in a non-expanding universe the geometry may be spherical, hyperbolic or flat. But again, the flat spacetime is ruled out on empirical grounds (no redshift z).

 

Flat spacetime (if it could exist is a cosmological setting) means not that there is no gravitational force, but that all gravitational forces cancel (a quasi-Euclidean universe). The distances between objects are constantly changing over time, since all objects are in motion relative to others.

 

Note: A general relativistic curved spacetime interpretation for cosmological redshift z implies a static metric in a stationary universe by virtue that redshift z is not a relativistic Doppler effect (the latter of which implies expansion), but an effect due to the passage of light through a curved spacetime continuum. The implication is that the universe may not be expanding.

 

There are other models, without a doubt, where curved spacetime affects the dynamics of a non-stationary universe (the rate of expansion and topology).

 

Whether the general relativistic curved spacetime is stable or unstable (static or expanding) depends on the metrical properties of the manifold. As you know, there exist both static and unstable exact solutions to the Einstein field equations. As it turns out, the stable solutions are both physical and elegant. Indeed, the former is more compelling than the unstable solution(s), since there is no requirement that the universe be dominated by bizarre forms of energy and matter (DE & CDM). The laws of physics need not break-down at some time t in the past. There is no requirement that the observed elements be created primordially, since there's no limit to the time-scale for the synthesis of all the observed elements and their isotopes through natural stellar processes. And finally, the time-scales required for the formation of the observed large-scale structures (e.g., superclusters and supervoids) are not limited to 13.7 Gyr.

 

Noteworthy, is it not?

 

Refer to the text linked above for a more in-depth review of these and other issues.

 

 

CC