[coldcreation]

.


On spiral structure formation in a galaxy with a bar-like nucleus

A new approach to the problem of the formation of galaxy spiral structures having a rotating bar-like nucleus is offered. The process of disk formation due to matter accretion onto the disk is considered in terms of the solution of the key problem on the motion of the matter element in the equatorial plane of the galaxy in the corotation resonance region. It is shown that in the vicinity of unstable libration points high-density regions are formed, which elongate with time following the separatrix shape and forming thereby spiral arms.

...Let us consider a galaxy with a central bar-like structure rotating in the equatorial ..... related to the saddle-type libration point S 2 in Figures la...


...in a realistic dynamical model, describing motion in a barred galaxy. Expanding the global model in the vicinity of a stable Lagrange point, ...

My bold.


CC Cheers

[coldcreation]

Here is another example: 19. The Bar Instability

Astronomy 626: Spring 1995

Orbits in strong bars

As an example of a strong bar, BT87 use the logarithmic potential

1 2 2 2 2 2
(4) Phi(x,y) = - v_0 ln(R_c + x + y / q ) ,
2
where v_0 is the asymptotic circular speed, R_c is the core radius, and q is the axial ratio of the potential; q < 1 for a bar elongated along the x axis. A contour plot of Phi_eff for this potential (BT87, Fig. 3-13) displays five places where
d d
(5) -- Phi_eff = -- Phi_eff = 0 .
dx dy
These equilibrium positions (in the rotating frame of reference) are called the Lagrange points. One, conventionally known as the L3 point, occupies the origin and occurs even in a non-rotating potential. Points L1 and L2 fall along the x axis, while L4 and L5 lie along the y axis; these are the places where the centrifugal pseudoforce balances gravity. L3 is a minimum of Phi_eff and hence is always stable, while L1 and L2 are saddle points and thus are always unstable. The points L4 and L5, though they mark maxima of Phi_eff, are stable for a logarithmic barred potential.

Motion in the vicinity of a Lagrange point (x_L,y_L) may be studied by expanding the effective potential in powers of x-x_L and y-y_L (BT87, Ch. 3.2.2). The key results are outlined here (see also BT87, Ch. 4.6.3). In the special case of a non-rotating potential with a finite core radius, a star near the L3 point executes independent and generally incommensurate harmonic motions in the x and y directions. For the case of a rotating potential the motion may likewise be decomposed into the sum of two periodic motions: one a retrograde motion about an epicycle, and the other a prograde motion of the guiding center. Because two motions are involved, it follows that orbits near the L3 point must have another integral of motion in addition to E_J. Similar results are obtained at the L4 and L5 points when these are stable.

Numerical integration of Eq. (1) provides a way to study orbits which do not stay close to a Lagrange point (e.g. Contopoulos & Papayannopoulos 1980). Just as in the earlier discussion of orbits in triaxial systems, here too each closed, stable orbit parents an orbit family. Close to the core of a barred potential the only important orbit families are the prograde x1 family, which is aligned with the bar, and the retrograde x4 family, which is nearly circular. For slightly smaller values of -E_J two new types of closed orbits may arise (BT87, Fig. 3-17): the stable x2 orbits and the unstable x3 orbits. Both are elongated perpendicular to the bar, but only the x2 orbits, which are rounder than x3 orbits of the same E_J, can parent an orbit family. At yet-smaller values of -E_J these perpendicular orbits disappear, and finally the x1 orbits likewise vanish when -E_J is small enough for the star to reach the L1 and L2 points. At comparable values of -E_J one may also find closed orbits circling the L4 and L5 points.

Vashé zdorov'ye



CC

[Qfwfq]

Originally Posted by freeztar

The mods here do not give infractions for being wrong.

And his wording does not consitute any unsupported claim either, where he talks about "a working hypothesis" and refers to "having argued that...".

Originally Posted by UncleAl

1) Quantitatively predict Mercury's perihelion precession.
2) Compare vacuum free fall of light to that of massed bodies.
3) Quantitatively predict GPS satellite corrections for relative velocity and position in Earth's gravitational well versus ground observers.
4) Rationalize anomalous contraction of s-orbitals in gold (creating its color) and mercury (lowering its melting point).

He doesn't need to, GR adresses these points and CC is not aiming to replace it. Unk, you let the baby run out with the bath.

Learn to distinguish between GR and cosmology. Also distinguish between the part of GR consequent to SR + the equivalence principle and that which requires the Einstein equation; the latter takes a bit of tweaking and fiddling. These things differ as to how certain of them we can be. There is still a lot which is conjectural in cosmology, including dark matter and dark energy.

Originally Posted by PhysBang

This is one of the least offensive posts in this thread!

Let's take a look at a really offensive sentence:

Uncle Al actually called CC an ass, while there was nothing offensive in what you quoted of CC.

Originally Posted by PhysBang

The poster tells an untruth by saying that, "The point of this thread is to determine (to show) exactly what that mechanism [for organizing gravitating bodies] is and why it is so important, not just for astronomy, but for cosmology and physics in general."

Perhaps 'determine' was a somewhat ill-chosen word (BTW not synomymous with 'show'), but I wouldn't call that a lie. Perhaps even it would be sufficient to say 'aim' in lieu of 'point'.

Originally Posted by PhysBang

If the poster could address UncleAl's pertinent questions, then I will retract this claim. However, this will not happen.

Which pertinent questions?

PhysBang you are being trollish. I have not analysed CC's arguments enough to make critical discussion of them; you and Uncle Al are welcome to do so as long as it is done appropriately. This is one thing these boards are for and it would be more interesting than your conduct so far.

[modest]

Originally Posted by coldcreation

Spacetime curvature in and of itself is not the physical mechanism of gravity.

You’re right and I’m an idiot for that strawman.

I clearly forgot there aren’t any textbooks that tell me why or how mass curves spacetime. Why stuff makes the universe present itself as non-Euclidian. Why or how the postulates of GR are correct or much in the way of conceptualism behind them.

I suppose it’s an often-asked question that is just as often attacked. People would say it’s a philosophical question or that it isn’t part of science. Maybe they’d say our human minds are incapable of understanding or conceptualizing what the math tells us. Or, they may just as well say someone is an ass for asking. Whatever the response, it never seems to be a straight-forward answer to the question. For my part in that, I apologize and offer the best answer I have:

I don’t know or understand the physical mechanism behind gravity. I don’t know if the answer can be found in cosmology so I obviously don’t know if you’re going about it the right way or not.

The next best thing I can offer is something I think you may have already read:
The Ontology and Cosmology of Non-Euclidean Geometry

I will quote the conclusion for anyone who hasn’t:

Just because the math works doesn't mean that we understand what is happening in nature. Every physical theory has a mathematical component and a conceptual component, but these two are often confused. Many speak as though the mathematical component confers understanding, this even after decades of the beautiful mathematics of quantum mechanics obviously conferring little understanding. The mathematics of Newton's theory of gravity were beautiful and successful for two centuries, but it conferred no understanding about what gravity was. Now we actually have two competing ways of understanding gravity, either through Einstein's geometrical method or through the interaction of virtual particles in quantum mechanics.
Nevertheless, there is often still a kind of deliberate know-nothing-ism that the mathematics is the explanation. It isn't. Instead, each theory contains a conceptual interpretation that assigns meaning to its mathematical expressions. In non-Euclidean geometry and its application by Einstein, the most important conceptual question is over the meaning of "curvature" and the ontological status of the dimensions of space, time, or whatever. The most important point is that the ontological status of the dimensions involved with the distinction between intrinsic and extrinsic curvature is a question entirely separate from the mathematics. It is also, to an extent, a question that is separate from science--since a scientific theory may work quite well without out needing to decide what all is going on ontologically. Some realization of this, unfortunately, leads people more easily to the conclusion that science is conventionalistic or a social construction than to the more difficult truth that much remains to be understood about reality and that philosophical questions and perspectives are not always useless or without meaning. Philosophy usually does a poor job of preparing the way for science, but it never hurts to ask questions. The worst thing that can ever happen for philosophy, and for science, is that people are so overawed by the conventional wisdom in areas where they feel inadequate (like math) that they are actually afraid to ask questions that may imply criticism, skepticism, or, heaven help them, ignorance.

-modest

[modest]

Originally Posted by PhysBang

If the poster could address UncleAl's pertinent questions, then I will retract this claim. However, this will not happen.

The two lines of arithmetic UncleAl is talking about are those of relative time dilation of velocity and gravitation (I assume). The former can be derived fully from Newton’s laws of motion plus the consistency of the speed of light for all inertial frames while the second can be gotten with the former plus the equivalence principle of GR.

Both have been tested very accurately as UncleAl pointed out. We can therefore infer with a very, very, very high degree of confidence that space-time shows and has non-Euclidean properties near a mass.

The two relevant postulates are the consistency of c between observers and the equivalence principle. Both come from observation with no theory or foundation that I know of. We always measure light at c and we notice falling objects fall and feel no acceleration while doing so. Why are these the case and what the hell does the latter have to do with mass/energy? The easy answer is that mass curves space-time. But, this is fundamentally unsatisfying. I’m not saying it’s wrong. On the contrary - it is correct enough to imply something is happening (a very real interaction) that we don’t understand - at least, I don’t understand. It is unsatisfying because it’s incomplete. Why and how does mass curve space-time? And, what is this space-time stuff and how does it curve anyway? Does it curve into a higher dimension? Does space curve into time or time into space? As de Sitter originally proved and others have confirmed, there are different cosmological models depending on the answers to these questions. Questions that I'd add don't get answered in GR.

It may be that GR hints at something beyond the way the standard model of particle physics currently does and Newton's law of gravity did. I’m a staunch supporter of GR and standard cosmology and I can accept this.

That said, UncleAl is right. Any theory of gravity needs to get to the equations and predictions he mentions. If it is gravitons or something else entirely - it has a mountain of observational evidence to agree with most of which came from GR. While I haven’t read this thread fully, it seems no one here is denying this. The “physical mechanism of gravity” needs to agree with, support, and lead to GR. As everyone agrees on this it seems ridiculous to throw it around as an attack.

Originally Posted by PhysBang

This is something that is not even wrong.

From what I’ve read - I mostly agree. However, I also see things that can be refuted.

[Moontanman]

Originally Posted by UncleAl

STURGEON WAS AN OPTIMIST

Move this thread to the

http://www.mazepath.com/uncleal/analysis.jpg

forum where it belongs. There is a perceptible difference between a parlour and a water closet. Each has its own properly accommodated audience.

Could you be more specific about Sturgeons optimisim

[Little Bang]

The nuts and bolts of gravity is the warping of time.

Moving down the page is increasing gravity.

____________

Clock A between the lines is distance traveled by a
____________ particle or wave.


Clock B
__________



Clock C
___________


Clock A runs slightly faster than B and B slightly faster C. This means a particle or wave traveling through A toward the center of gravity covers less distance than when it moves through B and less through B when it travels through C. A proof of this would be the red shift of light coming out of a gravity well

[coldcreation]

Originally Posted by modest

You’re right and I’m an idiot for that strawman.

I clearly forgot there aren’t any textbooks that tell me why or how mass curves spacetime. Why stuff makes the universe present itself as non-Euclidian. Why or how the postulates of GR are correct or much in the way of conceptualism behind them.

I suppose it’s an often-asked question that is just as often attacked. People would say it’s a philosophical question or that it isn’t part of science. Maybe they’d say our human minds are incapable of understanding or conceptualizing what the math tells us. Or, they may just as well say someone is an ass for asking. Whatever the response, it never seems to be a straight-forward answer to the question. For my part in that, I apologize and offer the best answer I have:

I don’t know or understand the physical mechanism behind gravity. I don’t know if the answer can be found in cosmology so I obviously don’t know if you’re going about it the right way or not.

The next best thing I can offer is something I think you may have already read:
The Ontology and Cosmology of Non-Euclidean Geometry

I will quote the conclusion for anyone who hasn’t:


-modest

Very good posts modest. Once again, thanks to your intervention, this thread (as others before) is becoming interesting.

I was originally going to post more examples of gravitating systems that exhibit a geometrical strucutre consistent with Lagrangian mechanics (and there are many): to show that there is a definite relation between all bounded systems (regardless of scale or complexity). And, that this relationship leads to a deaper understanding of the concept of spacetime curvature - whereby there is an intrinsic competition between objects (the maxima of potential) and 'empty' space (the minima of potential), via the gravitational field.

I will continue to post these, as the discussion necessitates (particularly those related to barred galaxy structure). For now though, let me get right to the heart of the debate: the elucidation of the physical mechanism behind the gravitational interaction.

To understand what is happening in the field (something which we cannot see) we have to look at (not only) how objects (or a test particle move) within the combined fields of massive bodies, but too, where they eventually find their equilibrium position (where they end up 'at rest'). That was the purpose of presenting examples of systems observed in nature.

If we accept that many gravitating systems display a propensity to collect, group, coalesce, arrange themselves, in a pattern consistent with that which is observed locally (in the solar system), i.e., that both general relativity and Lagrangian dynamics are operational ubiquitously, then a striking conclusion can be drawn: There exists in the nature of gravitating systems (and thus there exists in Nature) a (I'm going to get a cup of coffee...) fundamental property inherent in the fabric of space responsible for the competition between massive bodies and the underlying space within which they are immersed, resulting in the curvature of spacetime: a tension or distortion (in, or of) the manifold. This tension (or curvature), juxtaposed between objects (the maxima) and points of equilibrium, certain Lagrange points in the potential (the minima), is/are distributed in such a way that allows systems to remain stable (for whatever timescale), or at least allows objects to remain in stable orbits (e.g., at L4 or L5 zones), thus forming patterns observed at all scales. This propensity, possibly coupled with a 'natural selection' process during the initial formation of such systems, and the resulting velocities requirments attained for objects to remain in orbit, are the reason why celestial bodies do not end up in one great massive fireball: i.e., tend to “fall down” as Newton wrote, “into the middle of the whole space and there compose one great spherical mass.”


Let's examine how this interplay might work.

Spacetime curvature (gravity) must be treated as a deviation or departure from linearity, i.e., the deviation occurs away from linearity, from a flat, Euclidean, Minkowski spacetime and in accord with GR. All gravity (curvature), whether hyperbolic or spherical in geometric form (whether a compression or a stretching of spacetime), is considered a departure from the standard zero condition. For convenience, let us consider gravity a positive departure from linearity.

What we have is an absolute scale for gravity (spacetime curvature) that begins at absolute zero, flat, Euclidean spacetime, and becomes increasingly curved as the gravitational potential increases. Euclidean spacetime, however ideal that state described as gravity-free, where a test particle introduced into the field will experience no net acceleration, i.e., it will feel no force.

We set thus a lower limit on the gravitational field curvature for the spacetime manifold, and in doing so a reveal a basic property (or two) of spacetime: There exists a fundamental limit inherent in nature that manifests itself as field-free space. Beyond that limit, even in principle, spacetime cannot be curved, i.e., there is no ‘beyond’ that limit (just as there are no negative temperatures on the Kelvin scale). Mathematicians invented the concept of ‘field’ to articulate how a specific quantity might vary from point to point in space. What we are saying here is that there exists a fundamental boundary at the state where the field is reduced to zero, at critical points, when interacting fields cancel out. This value of zero for curvature, though not the absolute zero value of gravitational potential energy at infinity, is a relative potential (where fields cancel to zero, i.e., at the inner Lagrange saddle point), meaning that a zero curvature can exist in a wide variety of potentials and in all systems where two or more interacting fields cancel (which by definition is in all N-body system), resulting in a net force of zero.

Terminolgy: A global minimum value for gravitational potential energy (PE) is also a local minimum value (or point), i.e., there is no value less than absolute zero (the value at infinity). On the other hand, a local minimum value (consistent with the value of PE at certain Lagrange points) is not necessarily equal to the global minimum value. (Respectively, local and global are synonymous with relative and absolute). For the purpose of this discussion, the fact remains, the local value of gravitational potential at L1 is zero, the gravitational gradient vector vanishes.

Here is an example of a saddle point (See Critical Points of Functions of two variables):

The result of this interaction and reduction of potential in the combined fields is that there are smooth peaks (minima potentials) and troughs (or wells) the depth of which depends on the relationship between the mass-density of the object(s) inside the well, and the minima reference frame. So far, so good. The metric properties of spacetime adapt to accommodate mass, creating or inducing a dynamic stress or surface tension on the manifold: always in relation to the local minima (which is always zero).

So gravity is not caused by mass-energy. Spacetime is distorted (compressed, not stretched, as we will see later) in the presence of mass-energy as a result of tension against flat (otherwise 'empty') space. Spacetime curvature (gravity) is a property of the four-dimensional vacuum surface, just as surface tension is a property of the surface of a liquid. In the case of surface tension, the liquid is not being stretched downward (say, when an insect walks or glides on the surface of water), since water tends to ...

The physical behavior of the gravitational field (the deviation from linearity) in the presence of massive objects cannot be understood without taking into consideration the tension or stress created on the original surface (the vacuum manifold itself - like the surface of water before being disturbed). This 4-D surface, along with its associated tension in the presence of mass-energy, governs not only the shape objects can assume when immersed in the vacuum, or the degree of contact a massive body can make with another body, but too (and as a result of both the former and latter), the geometric structure and dynamics of the entire system (the placement or location of objects in relation to others, the ability and propensity to attain orbital velocity required for maintaining equilibrium, and possibly too the threshold or maximum density allowable for a given area of point: more on this later).

Applying general relativity and Lagrangian mechanics to the forces and interactions that arise due to 'manifold tension' in the presence of mass-energy accurately predicts the behavior of the system.

The nature of the vacuum ground state is that there is a tendency to minimize potential energy, to minimize its energy state, to minimize its surface area, to remain as flat and as empty as possible.

As a result of potential energy minimization, the vacuum substrate (the 4-dimensional spacetime manifold, or surface) will assume (or tend to assume) the smoothest and flattest shape allowable. When massive objects populate the manifold the tendency toward flatness, linearity, is still present, thus creating stress or tension in the field. Mathematical proof that smooth shapes or flatness minimize surface area can be found with the use of Euler-Lagrange Equations.

Here is another example of saddle points (See Critical Points of Functions of two variables):

By symmetry, it seems safe to conclude that the local minima (Lagrange points, the local minima of curvature) are at the origin of the gravitational phenomenon, i.e., the physical mechanism of gravity must be associated with this minima, or it must at least be taken as a starting point for the elucidation of the physical mechanism.

It can be shown that gravitating systems display mechanical equilibrium and that this equilibrium is attributable to the interplay between the local maxima and the local minima of potentials: both, in effect canceling each other out. So, going out on a limb, I hypothesize that gravitating systems are in mechanical equilibrium; the sum of the forces, and torque, on each massive body in the system is zero. In another way, GR explains the observed equilibrium without fine-tuning orbital velocities. (Recall, Newtonian mechanics was unable to do so, since gravity was considered an attractive force. The only way to get around this problem would be to consider Lagrangian points repulsive). This means, too, that the effects of GR do not only over-ride Newtonian gravity when velocities are close to c, or when mass-energy density tends to the global maxima. Indeed GR is the theory of choice, as well, when mass and velocities are small.

Here is an example of visible Lagrange points within the context of a galaxy; the infamous ESO 566-24 four-armed barred spiral galaxy, optical image, slightly enhanced by CC (See here the original photo and simulation): Note the similarity with the above contour plots of maxima and minima potentials.

I will continue to pursue the geometric argument in the next few posts, since it is through this pathway, only, that the physical mechanism of gravity can be understood.




CC
*

[Little Bang]

No comment on post #27. I'll rephrase it. Momentum = mv, v = d/t. If t runs slower as you move into a gravity well v is increasing, therefore momentum is increasing. A particle or wave will always move toward increasing energy.

[modest]

Originally Posted by Little Bang

The nuts and bolts of gravity is the warping of time.

Originally Posted by Little Bang

No comment on post #27. I'll rephrase it... A particle or wave will always move toward increasing energy.

I'm not sure what you mean by moving toward increasing energy. However, I agree it is the warping of time that causes a particle near a mass to be accelerated toward the mass in the spatial dimensions. That is certainly the pic GR paints.

-modest