Redshift z

[coldcreation]

Exactly! If you are thinking that a distant galaxy is at rest relative to us then we must expect it to be at rest relative to us in all frames of reference—including ours. [snip] The world lines of two objects at rest relative to one another neither diverge or converge.

 

Agreed, but one must not confuse different types of geodesics.

 

In general relativity the path of a star (for example) orbiting around a galaxy is the projection of a geodesic of the curved four-dimensional spacetime geometry around the galaxy onto three-dimensional space. The path of the star is not the same as the path of a massless particle, say, that of a photon. Photons will travel the shortest path between two points in a curved spacetime. In a spherically curved manifold, that path corresponds to a great arc (a section of a great circle). The path of the photon is very close to a true geodesic. (This is an idealization, of course, since local deviations or inhomogeneities cause a departure from the true geodesic path, e.g., resulting in the deflection of light, gravitational redshifts, etc).

 

Planets, stars, galaxies and other massive objects, possess an appreciable mass that affects the background gravitational field in which they reside. In qualitative terms, the larger the gravitational field produced by an object compared to the gravitational field within which it resides, the further this object's motion diverges from a true geodesic (great circle arcs).

 

In another way, the smaller the gravitational field produced by a particle, compared to the gravitational field within which it resides, the closer this particle's motion will follow a true geodesic. (Source)

 

 

The geodesic deviation equation shows that curved spacetime = deviating geodesics. If spacetime is curved and matter follows geodesics then we must expect the two bodies are not at rest.

 

Here you confusing the local geodesic motion of objects immersed in a gravitational field, with the dynamics of the universe. Material objects such as planets, stars, and galaxies are pretty much all moving relative to one another. The earth, for example is moving relative to the sun, i.e., the earth is not at rest. That is fortunate for us, since as you imply, otherwise the earth would accelerate towards the sun (your 100% sun screen would quickly become useless).

 

Let's take an example now of two neighboring galaxies. If these galaxies are initially at rest relative to one another they will merge. Fortunately, many galaxies manage to 'survive' encounters with other galaxies since they were never initially at rest relative to one another. So far so good.

 

Now let's up the ante. Consider two large galaxy clusters (or even superclusters). If the two cluster are initially at rest relative to one another they will merge, since spacetime is curved and matter follows geodesics. 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.

 

It is more or less straight forward (albeit, not without difficulty) to extrapolate further, to more complex gravitationally bounded n-body systems, and/or to large scales still ("supercluster complexes", "walls" or "sheets", that can span a billion light-years in length). There is no requirement to stop at any particular scale or level of complexity: even though as Newton once wrote in a similar though more local context "It causeth my head to ache."

 

What will be found is that everything is in motion relative to something else. The observed structures are the result of this type of environmental selection process.

 

 

The problem begins when considering (extrapolating) the universe itself as a self-gravitating bounded system [EDIT: with an FLRW metric].

 

One should not confuse the notion of a stationary (or static) universe with the idea that objects do not move or evolve. Objects need not geodesically converge globally, any more than objects do so locally. The idea that a non-expanding universe would collapse gravitationally cannot be accepted, in light of the pertinent empirical evidence [EDIT: and in light of the theoretical considerations linked below].

 

Nor should one automatically conclude that the size of the universe (the scale factor) or its global topology must change with cosmic time, becoming larger or smaller with time. The idea that the universe must be expanding in order not to collapse is preposterous. [EDIT: The only requirements are that objects are intrinsically moving relative to one another, locally, and that the four-dimensional spacetime manifold is globally curved.]

 

 

 

Honestly CC, the words "static" and "dynamic" do not apply to spacetime manifolds in the way you've been using them. [snip]

 

Certainly, the words "static" and "dynamic" applied to spacetime manifolds in the context relevant to this thread are unambiguous. The definition of these terms are standard and usage straightforward. Simply put, "static" refers to a universe where the scale factor (it's size) does not change with time. A "dynamic" universe has a scale factor that changes with time. In both cases, the universe is considered a spacetime continuum [EDIT: the intrinsic curvature of which, in this case, is directly related to the mass-energy content; and the description of which (according to the metric tensor) is valid in any coordinate system].

 

 

The reason light is affected in a curved spacetime manifold, as you say, is because null geodesics converge or diverge. The same exact principle applies to time-like geodesics of matter.

 

As stated above, the geodesic path of a massive gravitating object is not that of a great arc (on a spherical Riemannian/Gaussian manifold). The greater the mass, the greater the deviation from a true geodesic. However, the geodesic path of a photon is as close as you can get to a great arc (on a spherical manifold).

 

 

 

Is there permitted by general relativity a solution to the Einstein field equations that permits a globally homogeneous gravitational field of constant Gaussian (spacetime) curvature (leading to a universe that does not expand of collapse)?

 

[snip] Everything that I understand and everything that I've read on the subject tells me that the part in parentheses is mistaken.

 

It can never be expected of any individual to be familiar with all the relevant available literature regarding any particular field of study. [EDIT: Nor can it be expected that any individual will intuitively understand everything that she reads.]

 

There are however some relevant works that have already been discussed to some extent in this thread' date=' that should tell you the part in parentheses may in fact be correct (the universe does not expand or collapse), not least of which is a paper entitled An Equilibrium Balance of the Universe, by Ernst Fischer. And, by the same author, Homogeneous cosmological solutions of the Einstein equation (2009). Here, again, is the FULL TEXT in PDF format, and the Abstract:

 

Homogeneous solutions in the framework of general relativity form the basis to understand the properties of gravitation on global scale. Presently favoured models describe the evolution of the universe by an expansion of space, governed by a scale function, which depends on a global time parameter. Dropping the restriction that a global time parameter exists, and instead assuming that the time scale depends on spatial distance, leads to static solutions, which exhibit no singularities, need no unobserved dark energy and which can explain the cosmological red shift without expansion. In contrast to the expanding world model energy is globally conserved. Observations of high energy emission and absorption from the intergalactic medium, which can scarcely be understood in the ‘concordance model’, find a natural explanation.

 

 

Another work discussed to a lesser extent above you may want to familiarize yourself with is The Infinite Universe of Einstein and Newton by Barry Bruce (2003).

 

Product Description

After developing his Law of Gravitation, Newton came to believe that the Universe was infinite and homogeneous on a large scale. Einstein's original intuition was similar to Newton's in that he thought our Universe was static, infinite, isotropic and homogeneous. The field equations of Einstein's general relativity are solved for this universe. One of the three solutions found, the "infinite closed universe", traps light within a finite portion of the universe. This infinite closed universe model is shown to fit all the data of the Hubble diagram better than the Big Bang, and it fits the recent supernova data without having to postulate mysterious dark energy. Using general relativity and the physics which evolved from Newton, the author finds the force of gravity between two point particles. Utilizing this force and the infinite closed universe model, the net force of gravity on a point particle, in arbitrary motion, due to the uniform mass distribution of the universe is calculated by an integration. [...] These results show that the cosmological redshift and the physics that we know are likely the result of the uniform mass distribution of our infinite closed universe and gravity alone.

 

What follows is another example which relates to Einstein's original 1917 world model: Einstein's Static Universe: An Idea Whose Time Has Come Back, A Tribute to Irving Ezra Segal (1918–1998), by Aubert Daigneault and Arturo Sangalli.

 

 

See here too: PHYSICS BEFORE AND AFTER EINSTEIN, Standard Cosmology and Other Possible Universes, Aubert Daigneault (Edited by Marco Mamone Capria, University of Perugia, Department of Mathematics and Informatics, Perugia, Italy. IOS Press, 2005, pp 205-315).

 

Note: CC here refers to Chronometric Cosmology, not Coldcreation, though Coldcreation is mentioned at the bottom of page 315.

 

In these works, some of the difficulties for both the standard model and for alternative solutions are highlighted.

 

 

You might want to take another look at this too: Static Solutions of Einstein's Field Equations for Spheres of Fluid (Richard C. Tolman, 1939).

 

In this well-known work Tolman considered the static Einstein universe as unstable (correctly so) even with the incorporation of the cosmological constant. That problem has since been resolved (see Fischer, 2009, above).

 

Note, en passant, that it has often been concluded that the Tolman surface brightness test (a test of whether the universe is expanding or static) is "consistent with the reality of the expansion" (for a flat geometry and uniform expansion over the range of redshifts observed).

 

This test has generally pitted expanding models against simple static models with a flat geometry. The models proposed above by Fischer and Bruce, and the model proposed here: A General Relativistic Stationary Universe (discussed in this thread) all transpire in a curved spacetime manifold. In qualitative terms, the rate at which photons are received is reduced as each photon has to travel a geodesic path (a section of a great circle). There is both a time dilation factor (1 + z) and an energy loss factor (1 + z) associated with the travel time and distance between the source and the observer. The surface brightness of a standard candle would be dependent of the distance as a function of redshift z, not inversely with the square of its distance. (Just as in an expanding universe, but without the associated angular-diameter distance requirement for comoving objects in an expanding frame).

 

 

 

 

CC

[modest]

Exactly! If you are thinking that a distant galaxy is at rest relative to us then we must expect it to be at rest relative to us in all frames of reference—including ours. [snip] The world lines of two objects at rest relative to one another neither diverge or converge.

Agreed, but one must not confuse different types of geodesics.

 

In general relativity the path of a star (for example) orbiting around a galaxy is the projection of a geodesic of the curved four-dimensional spacetime geometry around the galaxy onto three-dimensional space. The path of the star is not the same as the path of a massless particle, say, that of a photon. Photons will travel the shortest path between two points in a curved spacetime. In a spherically curved manifold, that path corresponds to a great arc (a section of a great circle). The path of the photon is very close to a true geodesic. (This is an idealization, of course, since local deviations or inhomogeneities cause a departure from the true geodesic path).

 

Planets, stars, galaxies and other massive objects, possess an appreciable mass that affects the background gravitational field in which they reside. In qualitative terms, the larger the gravitational field produced by an object compared to the gravitational field within which it resides, the further this object's motion diverge from a true geodesic.

 

In another way, the smaller the gravitational field produced by a particle, compared to the gravitational field within which it resides, the closer this particle's motion will follow a true geodesic. (Source)

 

I recognize the summary, but I'm unsure how you mean for it to apply to the part of my post you quoted.

 

Here you confusing the local geodesic motion of objects immersed in a gravitational field, with the dynamics of the universe.

 

In using general relativity, the same mathematical relationship exits between the metric tensor and the trajectory of inertial particles for both local and global dynamics. The geometry of spacetime is determined by the metric tensor. The particles' trajectory is determined by the curvature of spacetime. I honestly wouldn't be bothered if you disagreed and said that globally matter doesn't tell spacetime how to curve or that curved spacetime doesn't tell matter how to move the way it does locally—but in doing so you aren't disagreeing with me, or pointing out some confusion that I have—you're disagreeing with general relativity.

 

Let's take an example now of two neighboring galaxies. If these galaxies are initially at rest relative to one another they will merge.

 

Where lambda is less than or equal to zero I agree—that will always be the case.

 

Now let's up the ante. Consider two large galaxy clusters (or even superclusters). If the two cluster are initially at rest relative to one another they will merge, since spacetime is curved and matter follows geodesics.

 

Agreed

 

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.

 

I agree.

 

If their relative tangent velocity is greater than [math]\sqrt{ \frac{G(m_1+m_2)}{r}} [/math] or their radial velocity is greater than [math]\sqrt{\frac{2Gm_1}{r}}[/math] they will disperse.

 

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

 

Ned Wright does exactly that in his cosmology tutorial:

 

We can compute the dynamics of the Universe by considering an object with distance D(t) = a(t) Do. This distance and the corresponding velocity dD/dt are measured with respect to us at the center of the coordinate system. The gravitational acceleration due to the spherical ball of matter with radius D(t) is g = -G*M/D(t)2 where the mass is M = 4*pi*D(t)3*rho(t)/3. Rho(t) is the density of matter which depends only on the time since the Universe is homogeneous. The mass contained within D(t) is independent of the time since the interior matter has slower expansion velocity while the exterior matter has higher expansion velocity and thus stays outside. The gravitational effect of the external matter vanishes: the gravitational acceleration inside a spherical shell is zero, and all the matter in the Universe with distance from us greater than D(t) can be represented as union of spherical shells. With a constant mass interior to D(t) producing the acceleration of the edge, the problem reduces to the problem of a body moving radially in the gravitational field of a point mass. If the velocity is less than the escape velocity, the expansion will stop and recollapse. If the velocity equals the escape velocity we have the critical case. This gives

 

v = H*D = v(esc) = sqrt(2*G*M/D)

H2*D2 = 2*G*(4*pi/3)*rho*D2 or

 

rho(crit) = 3*H2/(8*pi*G)

 

For rho less than or equal to the critical density rho(crit), the Universe expands forever, while for rho greater than rho(crit), the Universe will eventually stop expanding and recollapse.

 

In a system with a large number of particles there is no tangent velocity unless the whole thing is spinning around some center so that simplifies things to just one velocity. A particle will not, for example, escape Jupiter unless it has a velocity greater than 60 km/s (that one velocity is all you need to know)

 

What will be found is that everything is in motion relative to something else.

 

Ok, but on a smaller scale we might say that a nebula can collapse into a black hole. It can, and certainly does, happen. If the molecules of the nebula are initially not at rest to one another, but it is instead expanding, then we can solve the velocity above which it will not collapse but disperse. The same reasoning applies to a larger scale 'nebula of galaxies'.

 

One should not confuse the notion of a stationary (or static) universe with the idea that objects do not move or evolve.

 

No doubt. I think I've described "static" as a universe where the average distance between things doesn't increase or decrease over time.

 

Objects need not geodesically converge globally, any more than objects do so locally.

 

Well, that would depend on the initial average radial velocity between things. If it is small or zero then they should converge. If it's greater than the escape velocity then they need not collapse.

 

The idea that a non-expanding universe would collapse gravitationally cannot be accepted, in light of the pertinent empirical evidence.

 

If it is initially not expanding then the only non-collapsing futures should be with a cosmological constant and/or no mass.

 

Nor should one automatically conclude that the size of the universe (or the scale factor) or its global topology must change with cosmic time, becoming larger or smaller with time. The idea that the universe must be expanding in order not to collapse is preposterous.

 

I really don't have any gut feelings about it one way or another. Physics can determine if things will collapse gravitationally or not.

 

Certainly, the words "static" and "dynamic" applied to spacetime manifolds in the context relevant to this thread are unambiguous. The definition of these terms are standard and usage straightforward. Simply put, "static" refers to a universe where the scale factor (it's size) does not change with time. A "dynamic" universe has a scale factor that changes with time. In both cases, the universe is considered a spacetime continuum.

 

That's fine, I was just clarifying that a spacetime continuum does not scale or move and things cannot be at rest or moving relative to it. I think you said something about sprinkling objects onto a four dimensional manifold and considering their motion on the surface. Objects are curves on a 4D manifold and they don't move on the surface, so I wanted to clarify my understanding in so far as that goes.

 

The reason light is affected in a curved spacetime manifold, as you say, is because null geodesics converge or diverge. The same exact principle applies to time-like geodesics of matter.

 

As stated above, the geodesic path of a massive gravitating object is not that of a great arc (on a spherical Riemannian/Gaussian manifold). The greater the mass, the greater the deviation from a true geodesic. However, the geodesic path of a photon is as close as you can get to a great arc (on a spherical manifold).

 

I honestly don't see how you're relating that.

 

True motion is badly approximated by geodesic motion when the test particle contributes a significant amount to the gravitational field, but the contribution is ignored (like it would be in the Schwarzschild metric for a binary system—the larger the orbiting body, the worse the approximation). This wouldn't be the case in a fluid or dust model or a homogeneous solution.

 

But, regardless, my point was that global effects of curved spacetime on light (redshift) and global effects of curved spacetime has on matter (expansion) are both caused when spacetime sets out the world-lines for particles to follow. Expansion and redshift are both caused by diverging geodesics. The same metric that is telling photons what path to take is also telling electrons and nucleons what path to take. The metric should work for both because they are part of the same process.

 

It can never be expected of any individual to be familiar with all the relevant available literature regarding any particular field of study.

 

There are however some relevant works that have already been discussed in this thread, that should tell you the part in parentheses may in fact be correct (the universe does not expand or collapse), not least of which is a paper entitled An Equilibrium Balance of the Universe, by Ernst Fischer. And, by the same author, Homogeneous cosmological solutions of the Einstein equation (2009). Here, again, is the FULL TEXT in PDF format, and the Abstract:

 

I looked at the second one. I 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.

 

Tolman's proof of three static GR universes has stood the test of time and has been reproved in different ways—notably by Robertson and Walker.

 

~modest

[quantumtopology]

Let's see if we can all agree on something.

We agree on the Equivalence principle.

We agree that cosmological redshift (under the apropriate coordinates, see link below) in as much as it can be translated to a radial velocity is a Doppler redshift.

We aree that judging by redshift and brightness measures of supernovaeIa we seem to be in a constantly accelerating frame of reference (usually alluded as accelerated expansion).

 

If we put this three facts together, by the equivalnce principle an constantly accelerated frame is Physically indistinguishable from a gravitational field, and it follows that the cosmological redshift can be interpreted both as a gravitational redshift and as doppler redshift in a constantly accelerating universe.

If this considerations hold we all are right in a certain way , we are just looking from different sides of the coin. CC and I look at distant galaxies as if they were in a gravitational well with curvature proportional to distance, and the standard view is to see them as accelerating away from us, but it's just points of view, the physics are the same.

 

I include a link from a completely mainstream cosmologist, a collaborator of Jim Peebles (if someone defines and decides what Mainstream in cosmology is, that is Jim Peebles) named David Hogg, specially in the last two pages he gets to the same point as me via a completely different reasoning, but you'll see, Modest, that is perfectly possible to consider cosmological redshift as a gravitational redshift without even getting out of hard Mainstream cosmology.

 

Whether or not this makes the concept of "expansion" totally devoid of any explanatory power, is debatable, and I'm not gonna try to convince anyone, if someone is fond of the "expansion"picture is fine with me, as long as it is acknowledged that in the gravitational redshift interpretation of cosmological redshift, that picture is unnecesary.

 

 

http://arxiv.org/abs/0808.1081

[modest]

If we put this three facts together, by the equivalnce principle an constantly accelerated frame is Physically indistinguishable from a gravitational field, and it follows that the cosmological redshift can be interpreted both as a gravitational redshift and as doppler redshift in a constantly accelerating universe.

 

I've made exactly that point in this thread recently. An observer in a de Sitter universe, for example, is like a person sitting at an L1 lagrange point. In Newtonian terms, in each direction there is a decrease in gravitational potential with distance. In each direction there is a gravitational redshift increasing with distance. This is exactly how a de Sitter universe in static coordinates should be interpreted. In Robertson-Walker coordinates it should be interpreted differently, but the two interpretations are physically equivalent and amount to different, but equally valid, coordinate choices.

 

In fact, I just posted this link yesterday: What Causes the Hubble Redshift? Are the light waves "stretched" as the universe expands, or is the light doppler-shifted because distant galaxies are moving away from us?

 

If this considerations hold we all are right in a certain way , we are just looking from different sides of the coin. CC and I look at distant galaxies as if they were in a gravitational well with curvature proportional to distance, and the standard view is to see them as accelerating away from us, but it's just points of view, the physics are the same.

 

Certainly other interpretations are permitted, but hers would 'appear' to be consistent with observations, even though we know the conclusion above (that we are deep in a well, or visa versa) is not the case in the real world, i.e., it is a false conclusion. The effect is a relative one.

 

Expanding space, Doppler shift, and gravitational redshift are all fine interpretations. I've said many times in this thread that these are valid interpretations which represent different valid coordinate choices.

 

The problem I have is that you and CC are concluding that things are static. That is not a logical conclusion based on the 'gravitational well' or 'gravitational redshift' interpretation—or the other interpretations—they are physically equivalent after all.

 

In all three interpretations the redshift is accompanied by a physical change in distance between bodies over time. Think of the person at the earth / sun L1 point. There is a clock one kilometer from him in the direction of the sun and another that is one kilometer from him in the direction of the earth. Everything is initially static. The clocks will be redshifted from his perspective because the light is climbing out of a gravitational well. Time is dilated slower where the clocks are. And, most important to our discussion, the clocks will fall away from him toward the sun and the earth. The distances physically increase.

 

The same thing happens in a de Sitter universe with static coordinates (with gravitational redshift). In the same way that the clocks would fall away from the L1 point and crash into the sun and earth, clocks would fall away from an observer and 'crash' into that observer's cosmic horizon. The cosmic horizon of a de Sitter universe in static coordinates is at a fixed distance. The clocks are redshifted and dilated more and more the closer they get to the horizon not because of expansion or Doppler shift, but because of gravitational redshift. But, the fact remains that the clocks do accelerate toward the horizon and away from the observer. The change in distance is real and physical and physically equivalent to the FLRW expanding space description of a de Sitter universe.

 

~modest

[quantumtopology]

In all three interpretations the redshift is accompanied by a physical change in distance between bodies over time. Think of the person at the earth / sun L1 point. There is a clock one kilometer from him in the direction of the sun and another that is one kilometer from him in the direction of the earth. Everything is initially static. The clocks will be redshifted from his perspective because the light is climbing out of a gravitational well. Time is dilated slower where the clocks are. And, most important to our discussion, the clocks will fall away from him toward the sun and the earth. The distances physically increase.

 

No, they don't with the choice of coordinates that allows the interpretation as gravitational redshift, that is explained in the paper, from page 7:

 

 

"To construct the gravitational family of observers, we require that each member be at rest relative to her neighbor

at the moment the photon passes by, so that there are no Doppler shifts. Initially, it might seem impossible in general

to satisfy this condition simultaneously with the condition that the first and last observers be at rest relative to the

emitter and absorber, but it is always possible to do so. One way to see that it is possible is to draw a small world

tube around the photon path as in Sec. III. Within this tube, spacetime is arbitrarily close to flat. We can construct

“Rindler elevator coordinates,” the special-relativistic generalization of a frame moving with uniform acceleration,

within this tube, such that the velocities at the two ends match up correctly.30 The members of the gravitational

family are at rest in these coordinates. Because they are not in free fall, the members of the gravitational family all

feel like they are in local gravitational fields. Because each has zero velocity relative to her neighbor when the photon

goes by, each observer interprets the shift in the photon’s frequency relative to her neighbor as a gravitational shift.

Because the two families exist for any photon path, we can always describe any frequency shift as either Doppler or

gravitational."

 

And see also section V: Why interpretation matters

 

So the increase of distances you talk about is purely coordinate dependent. In other words it is a property of a congruence, the one defined by the canonical observers in a FRW metric. It's not a property of spacetime.

 

Another more general set of fundamental observers, independent of FRW symmetries, is defined to be at rest in normal coordinates. Expansion vanishes if you use these observers.

 

The same thing happens in a de Sitter universe with static coordinates (with gravitational redshift). In the same way that the clocks would fall away from the L1 point and crash into the sun and earth, clocks would fall away from an observer and 'crash' into that observer's cosmic horizon. The cosmic horizon of a de Sitter universe in static coordinates is at a fixed distance. The clocks are redshifted and dilated more and more the closer they get to the horizon not because of expansion or Doppler shift, but because of gravitational redshift. But, the fact remains that the clocks do accelerate toward the horizon and away from the observer. The change in distance is real and physical and physically equivalent to the FLRW expanding space description of a de Sitter universe.

What happens in a de Sitter universe could have superficial resemblances with what we are discussing here but we are not dealing with such universe here so it could confuse rather than clarify, our universe is not empty.

[HydrogenBond]

One thing that has always bothered me is indirectly connected to the concept of space-time. I like the concept of space-time, but since time is treated as a philosophical abstraction and not a real thing that has potential qualities, aren't we building our castle in the air, using a space-time abstract, instead of a tangible foundation? Once you enter the castle and forget about it floating on the abstract air of time, the entire thing is built extremely well. But what about this floating problem?

 

I often try to define time as a thing of substance to help firm the foundation. But most models seem to prefer that they float on abstract time. Is it possible, that if they sat on a solid or tangible time foundation, it would cause the castle to crack under it own weight?

 

At zero gravity, we don't have the same building needs as on the earth, since we don't need the same structural elements to support the weight of mass. At zero gravity, we can stack a castle on the head of a pin. That design can't work on earth.

[coldcreation]

In using general relativity, the same mathematical relationship exits between the metric tensor and the trajectory of inertial particles for both local and global dynamics. The geometry of spacetime is determined by the metric tensor. The particles' trajectory is determined by the curvature of spacetime. [...]

 

Again, photons will propagate a geodesic path of a great circle arc (in the case of positive curvature of the manifold). Massive objects will travel along geodesic paths defined by the local geometry of spacetime, as the earth around the sun. Notice that the geodesic path of the earth is not at all a great circle arc. Nor is the path of any other massive object. The world line of photons and the world lines of massive objects are two distinct types of geodesic motion.

 

So the idea that all objects will converge geodesically (or diverge in the case of k = -1) is unfounded, since it would imply that all objects freely fall towards (or away from) one another. This is untenable because it means that objects travel the exact same path as photons, but in reverse (in the case of spherical topology). Photons propagate outwards from all sources and in all directions. While objects would converge 'inwards' freely-falling towards every other object in the universe. There is no physical reason why such should be the case.

 

 

Let's take an example now of two neighboring galaxies. If these galaxies are initially at rest relative to one another they will merge.

 

Where lambda is less than or equal to zero I agree—that will always be the case.

 

The merging (or diverging) of two neighboring galaxies has nothing to do with the value of lambda. Lambda has nothing to do with it at all.

 

The point is that neighboring galaxies (cluster or superclusters) will NOT merge when they are NOT initially at rest with each other' date=' depending on the velocities and direction of motion. And since clusters are practically never at rest relative to one another merging is a very unlikely scenario, unless of course velocities are sufficiently small.

 

The key point is that superclusters do not necessarily merge gravitationally (or geodesically) in a static universe (some will and some will not).

 

 

Agreed [...]

 

I agree.

 

 

It's nice to agree once in a while. :)

 

 

 

What will be found is that everything is in motion relative to something else.

 

Ok' date=' but on a smaller scale we might say that a nebula can collapse into a black hole. It can, and certainly does, happen. If the molecules of the nebula are initially not at rest to one another, but it is instead expanding, then we can solve the velocity above which it will not collapse but disperse. The same reasoning applies to a larger scale 'nebula of galaxies'.[/quote']

 

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?). Since all objects are in motion relative to other objects, equilibrium finds a natural explanation. Certainly there are situations where objects merge or disperse. This will continue to transpire. But the idea that all objects in the universe would either converge or diverge is nonsensical, since it excludes the possibility (and inevitability) that some interactions will form quasi-stable self-gravitating equilibrium configurations.

 

The fact that these configurations exist on scales compatible with planetary systems, stellar systems, galactic systems and galactic clusters is a good sign that the same dynamics exists on scale compatible with superclusters. Today, the latter species of objects are erroneously considered to be so large that they are not gravitationally bound. Indeed superclusters are large, and indeed they are bounded gravitationally.

 

 

No doubt. I think I've described "static" as a universe where the average distance between things doesn't increase or decrease over time.

 

More importantly, "static" means that the universe does not shrink or grow with time, i.e., it does not expand or contract, there is no change in scale factor to the metric with cosmic time, proper time, or any other kind of time. In fact, there is no cosmic time at all. In a static universe observers will see the universe with similar features at any given time, and from any coordinate system. All times are essentially equivalent. So observers need not coordinate their clocks. The origin can be considered at every point of the spacetime manifold. The universe is homogeneous and isotropic at every point, and for every observer, meaning that every point in spacetime is equivalent. As such, the physical properties and dynamics of a static universe are governed entirely by the homogeneous mass-energy distribution and the associated gravitational field of constant Gaussian curvature alone.

 

 

Well, that would depend on the initial average radial velocity between things. If it is small or zero then they should converge. If it's greater than the escape velocity then they need not collapse.

 

Yes, but do you see the difference between dynamics of the type described above and the dynamics of the whole universe? Since all objects have differing velocities and directions of motion relative to other objects, there is no global convergence or divergence. As far as geodesics are concerned in a globally curved spacetime there is no force exerted on objects in the manifold that would cause them to all converge. From a purely geometric point of view, there do exist geodesic paths, e.g., great circle arcs, and other local geodesic trajectories (such as that of the earth around the sun). Galaxy cluster A will follow local geodesics that coincide with the local deviations from linearity induced by neighboring galaxy clusters B, C etc. as the fields of A, B, C, etc. interact with each other. Whether merging occurs or not depends on these paths, the velocities of the objects, and the direction of motion. Stability is maintained globally because these and other galaxy clusters (and superclusters) move along local geodesics, NOT great circle arcs. The latter paths are strictly reserved for massless particles such as photons.

 

That is why, in a globally curved spacetime continuum, objects do not all coalesce geodesically, or gravitationally, into one great massive fireball, along with the entire celestial sphere.

 

 

If it is initially not expanding then the only non-collapsing futures should be with a cosmological constant and/or no mass.

 

 

No, not at all.

 

 

The cosmological constant is never necessary. Recall that the field equations required a repulsive force (lambda) that increased with distance in order to counter the attractive gravitational force (in the absence of motion). The same was true of Newtonian cosmology (in the absence of motion).

 

If a universe is initially static (non-expanding), the only requirements for it to remain so are that objects are in motion relative to one another, locally, and that the geometry of the manifold is curved globally (this would be positive curvature according to the links above: see Bruce (2003) and Fischer (2009). The curvature, in effect, acts as a repulsive force or tension, but is entirely related to, and determined by, the mass-energy content (not to some hypothetical vacuum energy), i.e., the balance is maintained without the need for a negative pressure term. Tension results from the gravitational field itself. The equilibrium is physical and follows naturally from the Einstein field equations. (See Homogeneous cosmological solutions of the Einstein equation, By Fischer, pages 72-73)

 

The untenable notion that a non-expanding universe would collapse gravitationally is not physically acceptable.

 

 

[...] Physics can determine if things will collapse gravitationally or not.

 

Agreed.

 

 

[snip] The same metric that is telling photons what path to take is also telling electrons and nucleons what path to take. The metric should work for both because they are part of the same process.

 

 

False.

 

Photons propagate in all directions for the source. In a Riemannian (or semi-Riemannian) spacetime manifold of constant positive Gaussian curvature photons will propagate from point A (any source) to point B (any observer) along geodesic paths (great circle arcs).

 

Electrons, nucleons, planets, stars, galaxies, and so on, will not move along great circle arc paths. And, again, the greater the mass, the greater the deviation from a true geodesic.

 

Clearly, the geometric structure of a globally homogeneous spacetime manifold is not the same as the geometric structure of a locally inhomogeneous gravitational field.

 

 

I looked at the second one. I 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.

 

I think that you will find, if you study the paper more carefully, that it does make sense. The derivation of Einstein's static model is an important one, not to be removed as superfluous, since it is shown how the terms should be physically interpreted (particularly the time parameter and the vacuum energy term). The solution leads to an exact solution of the Einstein field equations consistent with a static Einstein universe. Note: the Bruce paper leads to the same conclusion, albeit by different means.

 

For many decades, a so-called general relativistic universe was synonymous with an unstable universe (an expanding dynamically evolving universe). Interestingly enough, a general relativistic universe is now synonymous with a stable universe (a non-expanding dynamically evolving universe). But the beauty now is that Einstein's theory of gravity is interpreted in a purely geometrical way, exactly the way it was invented and exactly the way it was intended. Indeed there is no reason to use a theory other than GR to describe the large-scale structure and dynamics of the cosmos.

 

 

Remarkable wouldn't you say?

 

 

 

__________________

 

 

 

 

Finally, a note on your answer to Qtop above:

 

 

You still have a problem with the curved spacetime scenario of Qtop, Bruce, Fischer and I, in that you believe a physical change in distance between bodies over time must occur (meaning that instability is unavoidable).

 

Let me assure you that your worries are unjustified, and give you another example as to why your claim is untenable.

 

First, I'd point out that the example you gave above (of a stationary observer sitting at the inner Lagrangian libration point, L1) is a local analogy; meaning that the gravitational field under consideration is not of constant Gaussian curvature of the type we are discussing. The situation between the fields of the earth and sun is inhomogeneous. As such, clocks placed at a distance from the observer along the earth-sun line will indeed accelerate away from L1 (towards the earth and towards the sun, respectively). This is nothing more than a stationary solution of the circular restricted three-body problem. This local analogy cannot be extrapolated to the general case of global dynamics. Why?

 

Globally, what we have is a homogeneous gravitational field (one that results from all undifferentiated matter and energy present in the manifold). The distinction between a globally homogeneous gravitational field and locally inhomogeneous gravitational fields is important, because it means that massive objects will behave differently. In the former, all points are the same, the magnitude of curvature is everywhere identical, like on the surface of a sphere (in the case of positive curvature). In the case of negative curvature (which BTW has been excluded by both Bruce and Fischer) all points are saddle points. Clearly, in the example you gave, all points are not saddle points.

 

Acceleration, of the type experienced locally (by the clocks immersed in the wells of the earth and sun on either side of L1) does not occur globally because acceleration is equal in all directions (there is no preferred direction towards which objects will be displaced gravitationally, yet there is a preferred direction in your analogy). As concerns the universe itself, there is no preferred spatial direction, and there is no preferred temporal coordinate system. All galaxy clusters and superclusters do not gravitate towards one another, not just because they are all in motion relative to one another (which would be a condition sufficient to avoid wholesale collapse, provided intrinsic velocities were adequate on average), but because the universe is homogeneous and isotropic on scales compatible with galaxy clusters.

 

In a homogeneous and isotropic universe all points are the same. In other words, every galaxy cluster (or supercluster) can be considered (and is essentially) at rest relative to the global general relativistic manifold in a locally special relativistic Minkowski regime with a flat Lorentzian metric. No matter where such a system resides, or moves to, the situation remains the same (local deviations in spacetime due to local gravity field curvature do not alter the global result). In relation the the global field of constant curvature, a cluster (or supercluster) will always be in a locally flat space (just as an object on the surface of the earth resides in a neighborhood where curvature is negligible, i.e., where curvature vanishes). In yet another way, the curvature of space at any given location, and at any given time, is always locally zero (relative to the global topology). (Notice that no fine-tuning or flatness problem arises in this situation).

 

This latter condition is not at all incompatible with the field equations of general relativity. Quite the contrary. It is, however, incompatible with the FLRW interpretation of the field equations. So too is this model incompatible with the other analogy you gave (the de Sitter universe), since as Qtop correctly pointed out, the physical universe is not empty. So what may (or may not) happen in a de Sitter spacetime is irrelevant (though not uninteresting as a special-case solution to the field equations).

 

The key point is to work out the general dependence of the effect on gravitating systems entirely based on the concept of constant curvature within the framework of the Einstein field equations (EFE), as opposed to a dependence based strictly on kinematics where the scale factor indubitably changes with cosmic epoch. The kinematic argument can no longer be considered the sole viable solution to the EFE. Thus, what has been interpreted as a recession velocity can also be interpreted as due to a curved spacetime phenomenon (in the absence of instability).

 

To continue the pedagogic exposition, the cosmological pressure term (lambda) used by Einstein as a repulsive effect of the vacuum, is replaced by what can be thought of as a tension (as opposed to pressure) which is associated with the global curvature, and entirely related to the mass-energy content, rather than a fictitious force or dubious stress-energy tensor. The beauty here two-fold in that there is no longer a need for a dark energy component, and lambda (interpreted as a vacuum energy) is no longer required to mediate equilibrium since its 'tension' is directly embodied in the geometric component of the field equations (i.e., it is associated with the mass-energy content, as opposed to a fictitious vacuum pressure). There is no longer an unphysical balance between the attraction of gravity and the repulsion of the Λ-term.

 

Thus, rather than growing exponentially with time under the influence of a negative energy equation of state, the radius of curvature of the geometry of the universe remains constant over time (i.e., curvature is not epoch dependent since there is no change in the scale factor to the metric over cosmic time).

 

 

These are intriguing ideas and appear to offer a plausible explanation not just for the raison d'être of cosmological redshift z but for two of the most fundamental problems of modern cosmology: the flatness problem and that of accounting for the global homogeneity and isotropy of the universe (observed in the CMBR). That's not all. The same stone also kills another related lame duck (or three): Inflation is no longer required to explain the origin of the fluctuations from which galaxies form. Energy conservation is maintained in the context of general relativity, and the laws of physics never break down at some time in the past (since the universe never went through a hot, dense, singular creation phase).

 

All in all, this model removes the chimera associated with the nonconventional precept of spacetime inherent in the FLRW interpretation of general relativity, while restoring the original physical stage upon which general relativity was built: the differential geometry of Gauss (which was embodied in the theorema egregium) and Riemann. This is the theme upon which all observational cosmology concerning not just measurements of spacetime curvature rests—indeed the non-Euclidean nature of the metric properties internal to the manifold can be calculated from data obtained by measurements—but upon which the global dynamics of the universe rests.

 

 

Quite remarkable, is it not?

 

 

 

 

PS. Re- the topic of Tolman's surface brightness test (considered by Sandage to be "by far the most powerful as a test between the expanding and static models"): According to the Robertson equation (1938), a non-expanding universe would have only one factor of (1 + z), from the "energy" effect. So the surface brightness of a standard candle would then decrease only as (1 + z) with distance.

 

Of course, this assumption is erroneous since it assumes a flat manifold. The problem was simply hidden, not dealt with. Had the test been applied to a static curved manifold (of the type postulated by Coldcreation and likely Fischer and Bruce) the result would be consistent with observations, since at least two factors of (1 + z) are required by the curved spacetime interpretation: one factor for the "energy" effect and the other factor for time dilation.

 

 

 

To be continued...

 

 

 

CC

[modest]

I'm very familiar with this subject. Please believe I'm trying to explain and not trying to debate or mislead you.

 

No, they don't with the choice of coordinates that allows the interpretation as gravitational redshift, that is explained in the paper, from page 7:

 

"To construct the gravitational family of observers, we require that each member be at rest relative to her neighbor

at the moment the photon passes by, so that there are no Doppler shifts. Initially, it might seem impossible in general

to satisfy this condition simultaneously with the condition that the first and last observers be at rest relative to the

emitter and absorber, but it is always possible to do so. One way to see that it is possible is to draw a small world

tube around the photon path as in Sec. III. Within this tube, spacetime is arbitrarily close to flat. We can construct

“Rindler elevator coordinates,” the special-relativistic generalization of a frame moving with uniform acceleration,

within this tube, such that the velocities at the two ends match up correctly.30 The members of the gravitational

family are at rest in these coordinates. Because they are not in free fall, the members of the gravitational family all

feel like they are in local gravitational fields. Because each has zero velocity relative to her neighbor when the photon

goes by, each observer interprets the shift in the photon’s frequency relative to her neighbor as a gravitational shift.

Because the two families exist for any photon path, we can always describe any frequency shift as either Doppler or

gravitational."

 

Regardless of the method of finding redshift—be it expansion, Doppler, or gravitational—the situation has only one physical outcome.

 

Consider an elevator accelerating in deep space. There is a clock on the floor and another on the ceiling. The one on the floor will be redshifted and time dilated relative to the one one the ceiling. You can calculate the amount of redshift by Doppler shift or with a gravitational field (ie gravitational redshift). The former is done by assuming that the ceiling always has a velocity relative to the floor (hence, Doppler shift) and is solved here under 'principle of equivalence'. The latter is accomplished by assuming that the two clocks have the same velocity using Rindler coordinates and is solved on wiki's gravitational redshift page under the history section (it calls Rindler coordinates 'accelerated coordinates').

 

By the equivalence principle, these are both valid methods of finding redshift for that situation. The equivalence principle requires that both solutions give the same answer. This does not, however, imply that we have an ambiguous physical situation. GR predicts the future dynamics of the system deterministically and unambiguously. A tape measure stretched between the clocks will measure no change in distance between them. Neither clock is moving inertially, and If one were to detach the clock from the ceiling it would fall and crash into the floor. No argument could be made that the clock on the ceiling wants to stay on the ceiling. It wants to fall. It stays there because we have applied an external non-gravitational force to it.

 

That is what Rindler coordinates do. They apply a non-gravitational force to each coordinate keeping them static on an accelerating reference frame. Anything static in Rindler coordinates will feel an acceleration. The "gravitational family of observers" in the quote above all feel an acceleration. This is not the case in our universe. Galaxies are *not* held static against a gravitational field. Their path is inertial.

 

So, consider the example given in your paper. The first is found right after the quote you gave,

 

When we discuss the Pound-Rebka experiment,31 which measured the redshift of photons moving upward in Earth’s gravitational field, we generally choose 8 to regard observers fixed relative to the Earth (the gravitational family) as more natural than free-fall observers, and hence we interpret the measurement as a gravitational redshift.

 

This is absolutely true. You can use either static coordinates (relative to the tower) or free-falling coordinates. With the former you can solve the expected amount of redshift gravitationally and with the latter you can solve the expected amount via doppler shift. You can not, however, conclude from this that a clock thrown off the top of the tower will remain static relative to the bottom of the tower. The clock will fall to earth's surface—to the bottom of the tower. A static coordinate choice is valid, but it doesn't imply that things under the influence of gravity alone will remain static. That is not the correct deduction (and it's certainly not a deduction that the paper makes).

 

Finally, if you tried to beat a speeding ticket by claiming that the radar results were due to a gravitational redshift, you would in effect be considering a “gravitational family” of prodigiously accelerating observers, with one at rest relative to the radar gun and one at rest relative to the driver. Needless to say, the police officer who gives you a ticket regards this family as extremely unnatural.

 

If the car is approaching the police car then it will crash into it. Using Rindler coordinates (or, "accelerating observers" as it calls them) certainly does *not* imply that the car will never crash into the police car. That would be a bad deduction.

 

In my Lagrange example you certainly could use the method quoted from the paper to determine the gravitational redshift. But, that in no way implies "hey, maybe the clocks really aren't falling away from the L1 point. Maybe they don't crash into the sun and earth". The observer at the L1 point could attach strings to the clocks and hold them against the force of gravity. The tension on the strings and the force required to hold the clocks against their tendency to scatter is real. It can't be transformed away by solving for redshift differently.

 

In a de Sitter universe objects at a distance r from an observer really are drawn toward that observer's cosmic horizon. The observer could tie a string to those objects and hold them against their propensity to free-fall away from the observer, but the objects would notice that. They would feel the force of acceleration just like the clocks in the L1 example would feel a force keeping them from falling away from L1, just like the clocks in the Pound-Rebka experiment feel the force required to keep them static relative to the tower, and just like the person in the car would feel the force needed to decelerate it before it crashes into the police car.

 

Galaxies in our universe are in free-fall—in inertial motion.

 

The clocks will be redshifted from his perspective because the light is climbing out of a gravitational well. Time is dilated slower where the clocks are. And, most important to our discussion, the clocks will fall away from him toward the sun and the earth. The distances physically increase.

No, they don't with the choice of coordinates that allows the interpretation as gravitational redshift

 

I agree, you can interpret redshift gravitationally. I mentioned this months ago. Gravitational redshift does not, however, imply an alternative physical situation. The clocks do crash into the sun and earth. The de Sitter clocks do 'crash' into the cosmic horizon. Solving redshift gravitationally doesn't change that.

 

So the increase of distances you talk about is purely coordinate dependent. In other words it is a property of a congruence, the one defined by the canonical observers in a FRW metric. It's not a property of spacetime.

 

Think about the consequences of what you're saying. Consider a nebula collapsing into a black hole. Observers in the collapsing nebula will observe a blueshift along with the collapse. The observers need not interpret the blueshift as a doppler shift—or with FLRW—they could solve it as a gravitational blueshift. You could not conclude from that option that the nebula will not collapse into a black hole. If the nebula's density is greater than [math]\frac{3 (v/r)^2}{8 \pi G}[/math] it will collapse. If its mass is great enough to overcome its degeneracy pressure then it will form a black hole. The singularity exists on the manifold. It doesn't depend on how we solve or interpret redshift.

 

GR is deterministic. There is only one predicted future regardless of coordinate choice. Deciding that Neil Armstrong was static wouldn't have kept him from reaching the moon. Solving redshift differently doesn't change the dynamics of a system.

 

What happens in a de Sitter universe could have superficial resemblances with what we are discussing here but we are not dealing with such universe here so it could confuse rather than clarify, our universe is not empty.

 

A de Sitter universe in static coordinates is an exact solution of the process approximated in the paper you linked. Using local Rindler coordinates along the path of the photon is an approximation. It's not the gravitational redshift interpretation's fault, but you will not get an exact redshift using that method. The de Sitter metric and the Robertson Walker metric are exact solutions of the two interpretations, but is, as you say, only valid where ρ=0.

 

~modest

[modest]

Again, photons will propagate a geodesic path of a great circle arc (in the case of positive curvature of the manifold). Massive objects will travel along geodesic paths defined by the local geometry of spacetime, as the earth around the sun. Notice that the geodesic path of the earth is not at all a great circle arc. Nor is the path of any other massive object. The world line of photons and the world lines of massive objects are two distinct types of geodesic motion.

 

The earth orbiting the sun is following its geodesic path. Light bent by the gravity of the sun is following its geodesic path. If you disagree with the fundamentals of geodesic motion then you disagree with GR.

 

So the idea that all objects will converge geodesically (or diverge in the case of k = -1) is unfounded, since it would imply that all objects freely fall towards (or away from) one another. This is untenable because it means that objects travel the exact same path as photons, but in reverse (in the case of spherical topology).

 

Your description of geodesics seems very odd to me.

 

Light cones are tilted with curved time so that the photon's path is always 45 degrees off the local time axis. The world line of a massive particle can't be tilted that much. "objects travel the exact same path as photons, but in reverse" doesn't make sense to me.

 

The merging (or diverging) of two neighboring galaxies has nothing to do with the value of lambda. Lambda has nothing to do with it at all.

 

In Newton's theory of gravity the force between two galaxies is given by,

 

[math]{\Phi}\vec{\nabla}=-{\frac{GM}{r^2}}{\vec{r}}[/math]

 

It is always attractive (the right hand side is always negative) no matter what the value of mass or distance. In the general theory of relativity Poisson's force equation becomes,

 

[math]{\Phi}\vec{\nabla}=-{\frac{GM}{r^2}}{\vec{r}}+\frac{c^2{\Lambda}r}{3}{\vec{r}}[/math]

 

The mass term provides an attractive force which decreases with the square of the distance, but there is an additional term with Lambda and this term increases linearly with distance. This now means that it is possible to have a repulsive gravitational force between two masses. Because the first term decreases quadratically and the second increases linearly the right hand side of the equation will be negative at small r and would become positive at a larger r (assuming Lambda is positive). In other words, gravity will be attractive at small distances and repulsive at larger distances.

 

The distance at which gravity becomes repulsive depends on the value of Lambda and the amount of mass. With a large Lambda and a small mass it is permitted by GR to have two neighboring galaxies disperse. To be exact in my agreement with you I dismissed this possibility by saying that galaxies will always be gravitationally attracted if Lambda is zero.

 

The point is that neighboring galaxies (cluster or superclusters) will NOT merge when they are NOT initially at rest with each other, depending on the velocities and direction of motion. And since clusters are practically never at rest relative to one another merging is a very unlikely scenario, unless of course velocities are sufficiently small.

 

Yes, I gave the formula for radial and tangent velocities needed to overcome the force of gravity in my last post.

 

A cluster of galaxies is relatively small compared to the visible universe. The velocity needed to escape one is not much.

 

The key point is that superclusters do not necessarily merge gravitationally (or geodesically) in a static universe (some will and some will not).

 

I agree. The tangent velocity of galaxies in a supercluster are usually sufficient that they orbit the cluster.

 

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?).

 

That property doesn't scale. The peculiar velocity of a galaxy in the Virgo cluster may be sufficient for it to orbit the cluster, but it is nowhere near sufficient for it to orbit the visible universe. The larger the scale, the larger the velocities would need to be. But, the peculiar velocity of galaxies are only so large (the velocity necessary scales, but the velocity itself doesn't scale).

 

The fact that these configurations exist on scales compatible with planetary systems, stellar systems, galactic systems and galactic clusters is a good sign that the same dynamics exists on scale compatible with superclusters. Today, the latter species of objects are erroneously considered to be so large that they are not gravitationally bound. Indeed superclusters are large, and indeed they are bounded gravitationally.

 

It's easy to solve. Use the equation I gave for tangent velocity in my last post. Put in the mass of the visible universe as M1 and the mass of a galaxy as M2 and see what velocity it gives. That will be the tangent velocity necessary to keep a dynamic balance between the galaxy and the mass of the visible universe. It will be much larger than the tangent velocity any galaxy actually has (~1,000 km/s).

 

As far as geodesics are concerned in a globally curved spacetime there is no force exerted on objects in the manifold that would cause them to all converge.

 

Yeah, that's what you keep saying.

 

I'll just say one more time, you're in disagreement with 80 years of theoretical research into GR and physical research in astrophysics. Any GR text book should explain, globally flat spacetime means parallel world-lines and globally curved spacetime means converging or diverging world-lines (just draw the spacetime—it can't not be true). The geodesic deviation equation solves the situation without any ambiguity. To disagree would require a rather compelling case.

 

~modest

[quantumtopology]

I'm very familiar with this subject. Please believe I'm trying to explain and not trying to debate or mislead you.

I don't doubt this, never crossed my mind that you might wanna mislead anybody. I'm just trying to reach a minimum common ground, and judging by your post we are very close to it. In fact most of wat you explain in this post I agree with except something that is central to the discussion and that migh be the origin of misunderstandings

 

Regardless of the method of finding redshift—be it expansion, Doppler, or gravitational—the situation has only one physical outcome.

Sure.

 

 

That is what Rindler coordinates do. They apply a non-gravitational force to each coordinate keeping them static on an accelerating reference frame. Anything static in Rindler coordinates will feel an acceleration. The "gravitational family of observers" in the quote above all feel an acceleration. This is not the case in our universe. Galaxies are *not* held static against a gravitational field. Their path is inertial.

This specific part of the paper is needed to fit reality to the standard model, and obviously does not make physical sense.

 

 

 

This is absolutely true. You can use either static coordinates (relative to the tower) or free-falling coordinates. With the former you can solve the expected amount of redshift gravitationally and with the latter you can solve the expected amount via doppler shift. You can not, however, conclude from this that a clock thrown off the top of the tower will remain static relative to the bottom of the tower. The clock will fall to earth's surface—to the bottom of the tower. A static coordinate choice is valid, but it doesn't imply that things under the influence of gravity alone will remain static. That is not the correct deduction (and it's certainly not a deduction that the paper makes).

 

If the car is approaching the police car then it will crash into it. Using Rindler coordinates (or, "accelerating observers" as it calls them) certainly does *not* imply that the car will never crash into the police car. That would be a bad deduction.

 

I agree, you can interpret redshift gravitationally. I mentioned this months ago. Gravitational redshift does not, however, imply an alternative physical situation. The clocks do crash into the sun and earth. The de Sitter clocks do 'crash' into the cosmic horizon. Solving redshift gravitationally doesn't change that.

 

GR is deterministic. There is only one predicted future regardless of coordinate choice. Deciding that Neil Armstrong was static wouldn't have kept him from reaching the moon. Solving redshift differently doesn't change the dynamics of a system.

 

Now you are confusingly using the concept and the word static, I think you know that when we(CC and I) talk about static spacetime or static metric that has a very specific meaning in differential geometri and general relativity, that has nothing to do with considering all objects to be frozen and unable o move, otherwise you would calling us stupids, I'm sure you are aware that I never implied that a car never crash or that clocks get suspended in the air if I use a different coordinate that gives me its redshift as gravitational, that is why puzzles me that you keep making this comments, unless you use them as strawman arguments.

 

Galaxies in our universe are in free-fall—in inertial motion.

Sure.

 

So let's talk about not having prejudices about situation with only one physical outcome, I repeat again that using a static metric doesn't mean cars don't crash or clocks don't fall under gravity influence, or galaxies are not in inertial motion, if you really believe I think that,no wonder we disagree.

All I say is that is equally erroneous to think that using a determinate coordinatesand metric, the FRW coordinates means that galaxies are accelerating away from us, you can't say that and that they are in inertial motion a t the same time, if they are accelerating away certainly something else than gravity is acting, they call it dark energy or positive lambda to salvage the expanding model but if something like that is really acting on galaxies they should "feel" accelerated and would not be in free fall.

[modest]

I'm sure you are aware that I never implied that a car never crash or that clocks get suspended in the air if I use a different coordinate that gives me its redshift as gravitational, that is why puzzles me that you keep making this comments, unless you use them as strawman arguments.

 

I apologize if I misunderstood. It looked to me like that's what you were saying:

 

In all three interpretations the redshift is accompanied by a physical change in distance between bodies over time. Think of the person at the earth / sun L1 point. There is a clock one kilometer from him in the direction of the sun and another that is one kilometer from him in the direction of the earth. Everything is initially static. The clocks will be redshifted from his perspective because the light is climbing out of a gravitational well. Time is dilated slower where the clocks are. And, most important to our discussion, the clocks will fall away from him toward the sun and the earth. The distances physically increase.

 

No, they don't with the choice of coordinates that allows the interpretation as gravitational redshift, that is explained in the paper, from page 7:

 

It looks—or, it looked—like you were saying that clocks wouldn't fall away from L1 to crash into the sun and earth—that the distance between the clock and L1 could remain constant.

 

All I say is that is equally erroneous to think that using a determinate coordinatesand metric, the FRW coordinates means that galaxies are accelerating away from us, you can't say that and that they are in inertial motion a t the same time, if they are accelerating away certainly something else than gravity is acting, they call it dark energy or positive lambda to salvage the expanding model but if something like that is really acting on galaxies they should "feel" accelerated and would not be in free fall.

 

Even without dark energy or a cosmological constant galaxies would be accelerated. In standard cosmology without a cosmological constant the velocity of a galaxy from our perspective is Hubble's constant times proper distance. [math]v=HD[/math] (Hubble's law) As the distance increases the velocity increases. A galaxy's proper distance increases over time, so its velocity also increases over time (a velocity increasing over time is acceleration). In other words, the further a galaxy gets from us, the faster it's going relative to us.

 

The accelerated expansion that comes with the cosmological constant means that the velocity at a constant proper distance increases over time—essentially, H increases over time.

 

~modest

[coldcreation]

The earth orbiting the sun is following its geodesic path. Light bent by the gravity of the sun is following its geodesic path. If you disagree with the fundamentals of geodesic motion then you disagree with GR.

 

Obviously. But you don't see light traveling around the sun on the same geodesic as the earth, do you. Nor do you see photons traveling around the earth on the same geodesic as the moon, do you. Nor for that matter do you see galaxies, clusters, or superclusters propagating along the same geodesics as photons (e.g., great circle arcs in a spherical spacetime manifold), do you?

 

No. That is because there are different types of geodesic motion. Massless photons are not confined or restricted to the geodesic paths of massive objects (though the fields of these objects will cause a deviation from the true geodesic path of the photon, e.g., causing deflection, gravitational lensing, etc.). And massive objects do not travel on the same geodesic as photons. You see, you can agree with the fundamentals of geodesic motion and agree with GR at the same time.

 

[EDIT} Certainly, all objects reside in the globally curved spacetime continuum (whether it is curved as K = 1, or K = -1), but as mentioned earlier, massive objects (including superclusters) will not freely-fall or be accelerated by the global curvature since locally spacetime is flat, i.e., the origin of the metric is always locally of Minkowski form, just as an observer located anywhere on the surface of the earth measures curvature to be flat in the local vicinity. There is no force acting upon massive bodies resulting from a globally curved spacetime topology. All points are the same. The universe does not, therefor, collapse gravitationally, as you might otherwise suspect.

 

 

Your description of geodesics seems very odd to me.

 

I had a feeling that would be the case. That is why I attempted to explain the situation in several different ways, and with several different examples. I'll try again below.

 

 

Light cones are tilted with curved time so that the photon's path is always 45 degrees off the local time axis. The world line of a massive particle can't be tilted that much. "objects travel the exact same path as photons, but in reverse" doesn't make sense to me.

 

In this sentence, contrarily to the one above, you seem to understand that the geodesic paths of photons and massive particles (or massive objects) are not the same, since in your example a photons path is 45° and massive particles cannot be 45 ° (they "can't be tilted that much"). This is a good beginning, but where not quite there yet. Already this conclusion bring hope, since the first modest quote above seemed to imply that light and massive objects traveled the exact same path.

 

With respect to cosmology, you're saying (correct me if I misunderstand you) that a static universe would collapse, i.e., a static universe has one future (it cannot remain stable). All objects (galaxies, clusters, supercluster etc.) would all converge gravitationally (or diverge; but forget lambda for the moment).

 

In the case that all objects would converge geodesically, this means that all objects would follow the same trajectory as photons, only inwards (not outwards from a source, as photons), the world lines are identical, but towards collapse. Light is propagated outwards in all direction from the source. In other words, your claim is untenable because it means that objects travel the exact same path as photons (in the case where curvature of the manifold is globally spherical), only not outwards from the source, but inwards towards one another in a free-fall trajectory (the universe would shrink). But this contradicts what you write above. 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.

 

It is understandable how you would arrive at such a conclusion (if indeed you do), since according to the standard model, one can reverse the velocity vectors to contraction (rather than expansion) leading to the convergence of all objects toward all other objects in the velocity field. All objects in such a linear velocity field arrive together at the same cosmic time. Indeed, this scenario has massless photons and massive objects confined to the same geodesic. I have argued the untenability of such a regime, and will continue to do so.

 

 

I agree. The tangent velocity of galaxies in a supercluster are usually sufficient that they orbit the cluster.

 

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. (See here, for example: Static, Infinite, Eternal and Self-Sustainable Universe, and Missing Mass In Galaxies Using Regression Analysis In Dynamic Universe Model Of Cosmology)

 

 

The peculiar velocity of a galaxy in the Virgo cluster may be sufficient for it to orbit the cluster, but it is nowhere near sufficient for it to orbit the visible universe. The larger the scale, the larger the velocities would need to be. But, the peculiar velocity of galaxies are only so large (the velocity necessary scales, but the velocity itself doesn't scale).

 

Nothing needs to orbit the visible universe. As long as the peculiar velocity of a supercluster is sufficient, relative to a neighboring supercluster, or superclusters, the objects will not coalesce, merge, or collapse onto each other. Again, orbiting the visible universe is not required at all.

 

 

It's easy to solve. Use the equation I gave for tangent velocity in my last post. Put in the mass of the visible universe as M1 and the mass of a galaxy as M2 and see what velocity it gives. That will be the tangent velocity necessary to keep a dynamic balance between the galaxy and the mass of the visible universe. It will be much larger than the tangent velocity any galaxy actually has (~1,000 km/s).

 

There is no reason to quantify such a thing, since the situation is irrelevant. We're talking about neighboring objects. Superclusters are a fine example of objects that are bounded gravitationally, just like all other objects that reside in any given vicinity. One simply needs to determine the tangent velocities of superclusters relative to one another (not relative to the mass of the visible universe).

 

 

 

As far as geodesics are concerned in a globally curved spacetime there is no force exerted on objects in the manifold that would cause them to all converge.

 

Yeah' date=' that's what you keep saying.[/quote']

 

Here, again, I will refer you to The Infinite Universe of Einstein and Newton by Barry Bruce (2003), Homogeneous cosmological solutions of the Einstein equation by Ernst Fischer (2009) and A General Relativistic Stationary Universe (1996-2010) by yours truly.

 

 

Until you read and understand these works I don't see how the situation can be better explained to you.

 

[EDIT} In one sentence: When the universe (the spacetime continuum with diffuse evenly distributed mass-energy content) is considered homogeneous, isotropic, and when the Gaussian curvature is nonzero and continuous, every point is the same, in space and in time, so there is no preferred direction or 'slope' (there is no preferred geodesic path) in which massive objects will move, be accelerated, gravitate, or freely fall towards one another.

 

 

I'll just say one more time, you're in disagreement with 80 years of theoretical research into GR and physical research in astrophysics. Any GR text book should explain, globally flat spacetime means parallel world-lines and globally curved spacetime means converging or diverging world-lines (just draw the spacetime—it can't not be true). The geodesic deviation equation solves the situation without any ambiguity. To disagree would require a rather compelling case.

 

That's like saying: If it's been around for 80 years, it must be true. If it hasn't been around for 80 years, then it is not true. Here's the problem: It need not be around for 80 years in order for it to be true.

 

It would be a fallacy to claim that because an interpretation of general relativity is popular (e.g., FLRW), or has survived 80 years (e.g., FLRW), it must be valid.

 

Your 80 year statement reminds me of a rhetorical technique that tries to persuade by overwhelming those considering an argument with such a volume of material, and the lengthy time-scale required to attain such, that the argument sounds superficially plausible, appears to be well-researched, and so laborious to untangle and check supporting facts on cosmological scales that the argument might be allowed to slide by unchallenged. (Source)

 

To claim that some evidence, or equation, solves any problem dealing with cosmology without any ambiguity is quite a leap of faith.

 

There is no unambiguous guarantee for the validity of the FLRW interpretation of the Einstein field equations. Indeed, the FLRW interpretation of GR is not defined in such a way as to render all alternatives impossible. There are certainly viable alternatives that do not coincide with the FLRW interpretation, but that do satisfy GR. The problem is that the FLRW models are held to be the only possible options, when in reality there are other exact solutions to the Einstein field equations that happen to be static solutions (see both Fischer, 2009, and Bruce, 2003, linked above).

 

It should never be assumed that because something could happen, it inevitably will happen (e.g., that a static universe would collapse). It would be an inappropriate inference the idea that because redshift z is interpreted as a relativistic Doppler effect, the universe ought to be unstable over cosmic time.

 

Nor should one assume that if an argument for some conclusion appears fallacious (e.g., your interpretation of Qtps gravitational redshift explanation above), then the conclusion is false.

 

Likewise, it would be a fallacy to conclude that a solution to problem of global stability cannot be right because the translation from say german to english is not perfect (recall your sentence: "The key claims of the [Fischer] paper are "explained" in one short and ambiguous paragraph which I don't understand"), especially when the key claims are proved in the language of mathematics and/or differential geometry (exact static solutions of the Einstein field equations).

 

That would be known as the continuum fallacy: The fallacy causes one to erroneously reject a 'vague claim' simply because it is not as precise as one would like it to be. Vagueness alone does not necessarily imply invalidity. In everyday speech, vagueness is an inevitable effect of the usage of language (especially when the language is not your mother tongue). In the case of Fischer, the exact solutions provided and the physical meaning of the terms are far from vague.

 

It would be fallacious, also, to assume that a model is untenable because it has not been proven true after 80 years. For example: "Einstein's 1917 model has failed to wit, therefore all models related to it are wrong." And therefore "It could all be removed as superfluous."

 

 

One of the problems inherent is the standard model that shouldn't be handled lightly is that it's an explanation without actually explaining the real nature of a function or process. For example the big bang (t = 0) is simply brushed under the carpet (physics has no explanation, since all the laws of physics break down). And in way, so too is the accelerated expansion. The concordance model explains the concept in terms of the concept itself, without first defining or explaining the original concept, i.e., dark energy and dark matter have no physical meaning or counterpart in the real world.

 

 

And coming back to your Lagrange point false analogy, with two clock accelerating away from L1: This example was offered as inductive proof for a universal proposition. ("This apple is red, therefore all apples are red."). In another way, local dynamics in an inhomogeneous restricted three-body system is unstable, therefore global dynamics in a homogeneous n-body system must be unstable. The false analogy of an empty de Sitter universe suffers from the same dilemma. The erroneous assumption is that material particles will scatter in a universe filled with a homogeneous distribution of matter, causing clocks to "crash into the cosmic horizon." So the analogies that "clocks do crash into the sun and earth" and that "the de Sitter clocks do 'crash' into the cosmic horizon" are really just a dicto simpliciter ad dictum secundum, otherwise known as fallacy of accident that argues from a special case to a general rule: clocks crash locally, so it must be true that all clocks crash globally.

 

Similarly, it is commonly observed the argument that the cosmological redshift and smoothness of the CMBR (for example) imply the existent of an early hot dense phases followed by an exponential expansion (inflation). But the premises do not establish the truth of the conclusion.

 

These are actually very common quantificational fallacies that occur when (in the former) extrapolating dynamics consistent with local physics to the dynamics of the entire universe.

 

 

 

 

The compelling case for a homogeneous and isotropic, stable, dynamically evolving universe providing a unified description of gravity as a geometric property spacetime entirely consistent with Einstein's general theory of relativity are on the table.

 

 

 

CC

[modest]

Obviously. But you don't see light traveling around the sun on the same geodesic as the earth, do you. Nor do you see photons traveling around the earth on the same geodesic as the moon, do you. Nor for that matter do you see galaxies, clusters, or superclusters propagating along the same geodesics as photons (e.g., great circle arcs in a spherical spacetime manifold), do you?

 

No. That is because there are different types of geodesic motion.

 

They are different geodesics, not different types of geodesic motion. A photon orbits a black hole like the earth orbits the sun.

 

With respect to cosmology, you're saying (correct me if I misunderstand you) that a static universe would collapse, i.e., a static universe has one future (it cannot remain stable). All objects (galaxies, clusters, supercluster etc.) would all converge gravitationally (or diverge; but forget lambda for the moment).

 

Yes.

 

In the case that all objects would converge geodesically, this means that all objects would follow the same trajectory as photons, only inwards (not outwards from a source, as photons), the world lines are identical, but towards collapse. Light is propagated outwards in all direction from the source.

 

Light falling inwards on a source is called a past light cone. The converging geodesics of matter are not identical to a past light cone nor would they need to be in order to be consistent with anything I've said.

 

If matter and light both explode from a point in space then the matter and the light follow the same path through space, but they follow different paths in spacetime. On a spacetime diagram draw the past and future light cone of an event and matter converging and diverging on the same event. The world lines are not (and cannot be) the same.

 

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 distance between events in spacetime joined by a null geodesic is not only "shortest", but zero. For this reason, a great circle is not the best analogy (and certainly not correct mathematically) for a null geodesic in spacetime. More generally, a hypersphere is not a correct description of spacetime. The tangent space needs to be Lorentzian, not Euclidean.

 

While I don't really understand your objection about null geodesics being equivalent to time-like geodesics, it seems possible that thinking of spacetime in terms of a Euclidean sphere might be the source of the problem.

 

...the standard model... Indeed, this scenario has massless photons and massive objects confined to the same geodesic.

 

Certainly not. The standard model in Robertson-Walker coordinates would be something like ds = -dt^2 + d(xyz)^2 where d(xyz) is a spatial hypersurface. Light always follows ds=0 and massive particles follow ds < 0. The geodesic of one cannot be the geodesic of the other.

 

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.

 

That is simply incorrect. The small scale result does not demonstrate the large scale result. Velocity increases with scale. If you weren't trying to debate me then you could quickly come to the truth of this.

 

The speed necessary for one molecule of hydrogen to escape the gravity of another molecule of hydrogen is very small. The escape speed between two people is larger. Larger still is the escape speed between two stars. Larger again between two local galaxies and larger between two groups and larger between clusters and larger again between two superclusters.

 

The larger the scale, the larger the velocity necessary to orbit these larger-scaled things. The velocity under which things are gravitationally bound and above which they are not increases with scale. Because there is a distance / escape velocity relationship, it's easy enough to solve the exact relationship and find some real numbers.

 

The escape velocity for a mass, M, at distance, r, is:

 

[math]E = \sqrt{ \frac{2GM}{r}} [/math]

 

The mass of a spherical collection of galaxies is volume times density:

 

[math]M = \rho \frac{4}{3} \pi r^3 [/math]

 

Substitute the second equation into the first and get the escape velocity as a function of r:

 

[math]E_{®} = \sqrt{ 2 G \rho \frac{4}{3} \pi r^2} [/math]

 

The average density of the universe is [math]10^{-26} kg/m^3[/math] or [math]1.68 \times 10^{-23}[/math] solar masses per au^3. Plugging that into rho and 6.28 for G (in astronomical units) and you get:

 

[math]E_{®} = \sqrt{8.84 \times 10^{-22} r^2}[/math] au/year. Or,

[math]E_{®} = \sqrt{5.6 \times 10^{-36} r^2}[/math] m/s.

 

At 5 billion lightyears (you can easily plug that into the equation directly above) the velocity above which objects are not gravitationally bound in our universe (either tangential or radial velocity—escape velocity is the same either way) is about 100 million meters per second. Of course, galaxies have nowhere near a peculiar velocity that large.

 

If you assume that large-scale distances don't increase over time on average then you clearly cannot conclude that an area of space is somehow immune to gravity wanting to collapse the things in it. A peculiar velocity of a couple hundred km/s may be enough for a galaxy to orbit its local group, but that is not enough for it to orbit a supercluster, or several thousand superclusters. This post proves that such a velocity is insufficient.

 

If you're not following that then consider...

 

If you imagine the universe as a nebula then you should quickly see why your assumption is mistaken. A molecule in a nebula may have a velocity sufficient to escape the gravity of another molecule in the nebula, but this does not imply that the nebula itself will not collapse into a star.

 

I need to hit the sack. I'll read the rest of your post as soon as I'm able.

 

~modest

[Qfwfq]

They are different geodesics, not different types of geodesic motion.

Careful, it depends on what you mean. Photons travel null geodesics, particles travel timelike ones, so they are different types of geodesics. I think this is more useful for sorting out CC's confusion.

 

A photon orbits a black hole like the earth orbits the sun.

That's a bit of a stretch. Perhaps you could say that about a photon at exactly exactly exactly exactly exactly [imath]R_{\rm S}[/imath] but I'm not sure exactly how much sense it makes to say it.

[modest]

...they are different types of geodesics. I think this is more useful for sorting out CC's confusion.

 

Indeed, I meant to say "different types of geodesics, not different types of geodesic motion".

 

Photons travel null geodesics, particles travel timelike ones

 

Reading my post, or the recent discussion in the thread, would indeed show null and time-like geodesics and which particles follow them are very well established.

 

Perhaps you could say that about a photon at exactly exactly exactly exactly exactly [imath]R_{\rm S}[/imath] but I'm not sure exactly how much sense it makes to say it.

No, I don't think you could say that about a photon at [math]r_s[/math]. The photon sphere is 1.5 [math]r_s[/math]. The sphere has zero thickness and it's not stable, but that has no bearing on my point.

 

~modest

[coldcreation]

[

[...] A photon orbits a black hole like the earth orbits the sun.

 

I'd love to see that. :)

 

 

...a great circle is not the best analogy (and certainly not correct mathematically) for a null geodesic in spacetime. More generally, a hypersphere is not a correct description of spacetime. The tangent space needs to be Lorentzian, not Euclidean.

 

While I don't really understand your objection about null geodesics being equivalent to time-like geodesics, it seems possible that thinking of spacetime in terms of a Euclidean sphere might be the source of the problem.

 

It is a fundamental result of the incorporation of differential geometry with in the framework of general relativity that the metric tensor contains all the information concerning the intrinsic geometric structure of space. Gauss was the first to demonstrate how curvature can be determined from the metric tensor (see Weinberg 1972, Gravitation and Cosmology). This is the theme upon which all observational cosmology rests concerning measurements of spacetime curvature. It is straightforward the extension of a homogeneous and isotropic Gaussian two-space to a homogeneous and isotropic three-space continuum (see Longair, 1993, The Deep Universe, pages 362-363).

 

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.

 

Here is where the problems resides: the difficulty is to visualize a great circle path in a four-dimensional spacetime. As it turns out, this path, from the point of view of any observer receiving the signal, is a straight line from the source to the observer, along the line of sight (minus local humps or bumps). The only observational evidence that this path is a geodesic (equal to a great circle arc in 2-space, rather than a straight Euclidean line) is that light is redshifted and clocks appear to run slower in the frame of the source relative to that of the observer (i.e., there is a time dilation factor in addition to redshift z).

 

 

...The small scale result does not demonstrate the large scale result. Velocity increases with scale.

 

 

I'm not sure if you saw my edit or not. I added yesterday two links that explain the situation relative to galaxy clusters. Here is one of them, again, just in case. (I'll quote the relevant parts, but be sure to check the equations). The result is that clusters and superclusters remain in stable equilibrium.

 

In this work: Static Universe: Infinite, Eternal and Self-Sustainable (2008) (PDF format), the author, E. Lopez Sandoval, describes how the distribution of matter organized in clusters at different levels of hierarchy (e.g., clusters, superclusters) leads to a static universe.

 

Also we consider a smoothed potential of this kind of universe and study the effect of gravity in the radiation of the stars: applying the equivalence principle we obtain a mathematical expression for the Hubble’s law and a formula for its redshift that could explain this phenomenon like a gravitational effect. Also we obtain an approximated calculation of the Cosmic Background Radiation (CBR), taking like hypothesis that this radiation is the light of all stars in the Universe that arrive until us with an extreme gravitational redshift. In summary, we present here an alternative explanation for the redshift and CBR, how an alternative to the presented by the Big Bang theory (BBT), or Steady State theory (SST), postulating in consequence a new theory about the structure of the Universe: static, infinite, eternal and self-sustainable.

 

It's interesting to find that there are a growing number of similar solution to the problem of cosmological redshift z in a static Einsteinian spacetime manifold.

 

And, on the topic of cluster and supercluster stability:

 

We know that distribution of matter in Universe (although in local level it seems non homogenous, or until with certain degree of randomness) follows a structure with a distribution function inversely proportional to the nth power of the distance from its rotation center (to level of galaxy, and clusters of galaxies), and diminishes with exponential factor of 1.8 [12, 13, 14]. By extrapolation it is possible to suppose that in a larger scale it decreases with a tendency to a distribution following the inverse square law. Maybe the Universe has a kind of fractal structure that follows a power law (although with variance of scale). We propose this class of distribution, because with this model, to certain scale, the matter reaches the homogeneity. We explain this next: the stars are grouped in galaxies, and galaxies in clusters that are grouped also in a set of clusters (named superclusters). In each level all these groups are rotating around their own center (center of mass) similarly like the galaxies do around its own center (where its density of matter could be infinite due to the presence of a super massive black hole [14, 15]. According with our hypothesis, hierarchic levels finish here, and we think that this grouped matter reach its maximum level and has a common rotation center, nominated for us like maximum gravitational rotation center (MGRC). Similarly, for others supercluster we suppose that they follow a similar distribution (this structure could be considered as the bricks of the Universe, i. e., its maximum unity), and that could to exist infinites similar structures in the infinite Universe. Its interaction among them is only a translational force (because not exists another center of rotation at higher level) and exist a local dynamic but static in average at global level, i. e., they are in a dynamical equilibrium. Therefore, in the following scale the matter distribution no longer follows a radial distribution, and although still this seems random, in average its distribution to this scale goes towards homogeneity. Thus, the density moving away radially of this MGRC is decreasing, since on each scale the separation increase: between stars in the galaxies is of the order of parsecs, and between galaxies in the clusters is in the order of megaparsecs [16], and so successively. Then, the uniformity is reached when to some distance R of any MGRC, the density of the proposed distribution function reaches the value of the average density of the Universe (1.67x10-27 kg/m3). In this scale, the mater inside of each shell of radius r3R is constant. Our model try to agree with the astronomers’ observation about the structure of universe, where to this scale reach the homogeneity, as it is presented in the Atlas of the Universe site [see diagram: The Universe within 14 billion Light Years, below].

 

Summarizing, this infinite system remains in stable equilibrium, since there is not a dominant MGRC, as is proposed in the BBT, and that observationally never was observed. However we propose that exist very much MGRC (we are supposing infinite), and that they are in dynamic equilibrium, between them and with those of the rest of an infinite Universe. This can be deduced directly of the observation, since all centers of gravity imply matter rotating around itself, and some MGRC that dominates the others like a rotation center have never seen. Each MGRC is in dynamic equilibrium with the others, and although is possible that gravitational attraction between two centers dominates to the attraction of the other centers causing a translation, the time that would take a collision is so long due to the great distant among them, that the collision is not common. Besides, the stability of this type of Universe is due that the matter of each gravitational center is rotating or “falling” to its own center, and its time scale is minimally of the order of thousands years. Therefore, although a collision between galaxies is probable, this would takes a long time. For example, a predicted collision Andromeda-Milky way due to take place in three billions light years approximately [18]. In addition, anywhere part of the Universe is compensated with the movement of other centers, in a dynamic re-balancing of forces, allowing a homogeneous distribution of matter on a large scale. [...]

 

Also, how we did in a previous work [15], we will use a “stellar dynamics” model, where each galaxy contributes to the overall gravitational field and we don’t need to know the precise location of each one. In order to obtain an excellent estimation is necessary only to replace this distribution of individual galaxies by a smoothed continuum density. We know that gravitation is a cumulative force, and then we must use Gauss´ law for to obtain its continuum gravitational field. Therefore, each galaxy follows a ‘collisionless dynamic’ around a MGRC, like in a stationary system, influenced by the global gravitational effect, and with weak influence by the local gravitational effects of nearest stars.

 

 

Consider these diagrams.

Source:

 

 

 

The Nearest Superclusters.

 

 

 

The Universe within 1 billion Light Years - The Neighbouring Superclusters.

 

The distribution of galaxies in the universe is far from regular. They tend to clump together into huge supercluster formations. This map shows many of the superclusters within 1 billion light years of us.

 

 

The Universe within 14 billion Light Years - The Visible Universe.

 

Although our knowledge of the large scale structure of the universe is incomplete, many large and small scale features are visible right out to the very edge of the visible universe. The entire universe is fairly uniform, as this map shows. (Again: Source).

 

 

 

The speed necessary for one molecule of hydrogen to escape the gravity of another molecule of hydrogen is very small. The escape speed between two people is larger. Larger still is the escape speed between two stars. Larger again between two local galaxies and larger between two groups and larger between clusters and larger again between two superclusters.

 

Indeed, these velocities a relative. As long as the internal velocity dispersions are sufficient superclusters (and groups of superclusters) can remain gravitationally bound systems. There is no requirement for any significant departure from a smooth random distribution. The larger the scale, the smoother the distribution. These objects need not catastrophically collide or disperse.

 

 

At 5 billion lightyears (you can easily plug that into the equation directly above) the velocity above which objects are not gravitationally bound in our universe (either tangential or radial velocity—escape velocity is the same either way) is about 100 million meters per second. Of course, galaxies have nowhere near a peculiar velocity that large.

 

Individual galaxies need not have peculiar velocities that large. What counts is the velocity of the clusters, or superclusters, themselves, relative to neighboring clusters or superclusters. The results in the link above clearly demonstrate that superclusters are bounded gravitationally, and can (and do) remain is stable equilibrium.

 

 

If you assume that large-scale distances don't increase over time on average then you clearly cannot conclude that an area of space is somehow immune to gravity wanting to collapse the things in it. A peculiar velocity of a couple hundred km/s may be enough for a galaxy to orbit its local group, but that is not enough for it to orbit a supercluster, or several thousand superclusters. This post proves that such a velocity is insufficient.

 

This is a false dilemma. Your example of a galaxy not having sufficient velocity to travel around a supercluster involves a situation in which only one option is considered (a single galaxy relative to the supercluster), when there are other options: notably the fact that a supercluster can have a sufficient velocity relative to another supercluster (or superclusters) in order to remain in quasi-equilibrium configurations. That was the point

 

 

If you imagine the universe as a nebula then you should quickly see why your assumption is mistaken. A molecule in a nebula may have a velocity sufficient to escape the gravity of another molecule in the nebula, but this does not imply that the nebula itself will not collapse into a star.

 

This too is a false dilemma. This time, your analogy involves a situation in which only one 'local' option is considered (particles relative to a nebula), when in fact there are other options: notably the fact that there are many nebula, each with its own particles. And, not only that, there are many clusters of nebulae, and many superclusters of nebulae, each with their own particles. There is no center (as in the case of your nebula) to which all particles (or objects) will accelerate.

 

 

Here, once again, I will refer you to The Infinite Universe of Einstein and Newton by Barry Bruce (2003), Homogeneous cosmological solutions of the Einstein equation (Ernst Fischer, 2009) and A General Relativistic Stationary Universe by yours truly.

 

Until you read these works I don't see how the situation, in the context of a static Einstein universe, can be better explained to you. Both of these physicists have produced solutions to the Einstein field equations that are static solutions. These are spherically symmetric globally curved general relativistic spacetimes. Both authors conclude that light may be in fact redshifted because it is traveling through a four dimensional spacetime continuum of constant positive curvature. Stability is maintained against gravitational collapse. Any local region in a globally homogeneous spherically curved spacetime manifold is flat (Minkowskian), i.e., there is no force exerted on objects attributable to the global field regardless of their location in the universe that would cause all objects or undifferentiated matter to geodesically gravitate towards on another.

 

 

 

Indeed, I meant to say "different types of geodesics, not different types of geodesic motion".

 

See Figure 3 here: Different types of geodesic motion.

 

And here Topological Solitions: "...different types of geodesic motion have been observed on the moduli space."

 

And New Trends in Statistical Physics: an "exhaustive discussion of all types of geodesic motion"

 

Or, Certain Types of Geodesic Motion of a Surface of Negative Curvature.

 

And, "We classified possible types of geodesic motion"

 

Or, The four regular types of geodesic motion

 

 

The list goes on. It is a word game to say that there are "different types of geodesics, not different types of geodesic motion"

 

There are in fact different types of geodesics, and they result in different types of geodesic motion. I hope we can at least agree on that.

 

 

 

CC

[modest]

We know that distribution of matter in Universe (although in local level it seems non homogenous, or until with certain degree of randomness) follows a structure with a distribution function inversely proportional to the nth power of the distance from its rotation center (to level of galaxy, and clusters of galaxies), and diminishes with exponential factor of 1.8 [12, 13, 14]. By extrapolation it is possible to suppose that in a larger scale it decreases with a tendency to a distribution following the inverse square law. Maybe the Universe has a kind of fractal structure that follows a power law (although with variance of scale). We propose this class of distribution, because with this model, to certain scale, the matter reaches the homogeneity. We explain this next: the stars are grouped in galaxies, and galaxies in clusters that are grouped also in a set of clusters (named superclusters). In each level all these groups are rotating around their own center (center of mass) similarly like the galaxies do around its own center (where its density of matter could be infinite due to the presence of a super massive black hole [14, 15]. According with our hypothesis, hierarchic levels finish here, and we think that this grouped matter reach its maximum level and has a common rotation center, nominated for us like maximum gravitational rotation center (MGRC).

 

This electrical engineer clearly has no idea what he is talking about. 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. So long as Newton's or Einstein's law/theory of gravity are correct, this idea is nonsense—it does not provide global stability. It also does not provide isotropy so it is ruled out by observation. If parts of the sky were approaching us at enormous speed and other parts were receding at enormous speed then we could entertain the idea that extremely large scale structures are rotating, but this is not the case.

 

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.

 

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.

 

Indeed, these velocities a relative. As long as the internal velocity dispersions are sufficient superclusters (and groups of superclusters) can remain gravitationally bound systems. There is no requirement for any significant departure from a smooth random distribution. The larger the scale, the smoother the distribution. These objects need not catastrophically collide or disperse.

 

This is getting tedious.

 

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.

 

Unless the 8...

 

Unless the 16...

 

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. Unless we are that center, observations would not be isotropic.

 

The idea just doesn't work.

 

Individual galaxies need not have peculiar velocities that large. What counts is the velocity of the clusters, or superclusters, themselves, relative to neighboring clusters or superclusters.

 

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.

 

 

The results in the link above clearly demonstrate that superclusters are bounded gravitationally, and can (and do) remain is stable equilibrium.

 

If you mean the link that I read, it doesn't prove anything. I read from the first paragraph on wiki's 'supercluster' page...

 

Superclusters are large groups of smaller galaxy groups and clusters and are among the largest structures of the cosmos. They are so large that they are not gravitationally bound and, consequently, partake in the Hubble expansion.

 

I have no reason to doubt that.

 

 

This too is a false dilemma. This time, your analogy involves a situation in which only one 'local' option is considered (particles relative to a nebula), when in fact there are other options: notably the fact that there are many nebula, each with its own particles. And, not only that, there are many clusters of nebulae, and many superclusters of nebulae, each with their own particles. There is no center (as in the case of your nebula) to which all particles (or objects) will accelerate.

 

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.

 

Here, once again, I will refer you to The Infinite Universe of Einstein and Newton by Barry Bruce (2003), Homogeneous cosmological solutions of the Einstein equation (Ernst Fischer, 2009) and A General Relativistic Stationary Universe by yours truly.

 

I'll get back to you.

 

~modest