Redshift z

[coldcreation]

CC, that paper from Fischer indeed interests me, even if I don't completely understand it. It has similarities both with the Einstein model with lambda integrated in the energy tensor as negative pressure, and with de Sitter's with the redshift due to a factor multiplying the time displacement part of the metric. What would you say about it? It would be negatively curved, right?

 

I'm not yet certain wether the curvature would be positive or negative. But one thing is certain: Both Fischer model, and the one I've presented, would remain static regardless of the sign of curvature. In other words, even in principle, a Euclidean geometry (or Minkowskian topology, or a Newtonian universe) would remain free-of gravitational collapse. Of course, the flat universe has to be discarded as a possible geometry for the universe, since cosmological redshift would not occur (not because it would be unstable).

 

I'm not sure whether the Fischer model is hyperbolic or spherical. As long as the globally homogeneous spacetime curvature is nonzero, the requirement for stability and redshift z are satisfied. So it really shouldn't make too much difference (though the uncertainty in this respect is still bothersome).

 

My leaning more towards a model with negative curvature is more due to the physical notion that all objects find themselves located at saddle points, than to the philosophically attractive idea that the universe is spatiotemporally infinite. Since a spherically curved universe can also be infinite in spatiotemporal extent (in both direction of the arrow of time), though there does exist the strange (almost paradoxical) idea that the curvature of a sphere come back around on itself. I think this problem is irrelevant in four-dimensions, but it still seems less-intuitive than a hyperbolic topology. In either case, as long as the curvature is constant, where all points and all direction are the same, both solutions are viable.

 

 

Let's get back to Fischer's work: It seems that, unlike the cosmological constant, the pressure to which Fischer refers to is simply a tension of the kind Newton hypothesized (which justified the notion of an infinite static universe). Fischer writes:

 

If we consider the universe as a curved volume of space, homogeneously filled with matter, every volume element will attract its neighbouring elements gravitationally. Due to symmetry, the net force will be zero at every point [...]

 

The tension or negative pressure plays the role of potential energy in the Newtonian picture.

 

As far as Redshift z in concerned, Fischer's concept is more straight forward: "Redshift is a direct consequence of curvature, which causes a change of time scale with distance." See equation 21 and 25. He gets the exact same relation as the critical density, or Einstein-de Sitter model. Whereas in standard cosmology this expanding universe is flat, here it is static universe of constant curvature. And the curvature depends only on the energy density.

 

To a first approximation, Equation 22 shows a linear dependence of the time scale on distance. This, in my opinion, would be the case in both hyperbolic and spherical manifolds. The meaning of this is that redshift z (with the associated time dilation factor) is a function of relative distance from the source to the observer. But, of course, in this case the source objects are not moving radially from the observer. And in accord with equation 20, time depends on distance, so the balance can be fulfilled without any pressure. What is "balanced by a negative pressure in Einstein’s static solution" Fischer writes "is simply compensated by the purely geometrical change of time scale. The negative pressure appears as a property related to the curvature of space, not as an intrinsic property of some matter field."

 

The result is that of a general relativistic universe which doesn't change in size with time (there is no cosmic time, so time depend of distance from the observer, i.e., clocks appear to run slower with increasing distance), gravitating structures fluctuate about a global equilibrium, the global topology is of constant curvature consistent with a homogeneous gravitational field, redshift is therefore a curved spacetime phenomenon, and finally, energy is not only conserved locally but globally.

 

 

 

CC

[modest]

Sorry guys, I've been unexpectedly out of town. I'll reply while I read here... :phones: worst day since yesterday :phones:

 

In the case of the FLRW solutions the cosmic time coordinate is dubious.

 

It's just the time comoving clocks keep.

 

The root of the problem lies in the derivation of FLRW equation(s) that utilizes comoving coordinates and synchronous universal cosmic time t.

 

Every model, and every theory, makes assumptions. The predictions of the theory test the assumptions.

 

It cannot be known whether something (or anything) is "universally true" in any physical theory. It's certainly ambitious to postulate such.

 

The postulates of a theory are, by design, universally true of the theory. It is a postulate of GR that freely moving particles follow geodesics in four dimensional spacetime.

 

Suppose we set ourselves a more modest goal (:)).

 

Let's just say it's an assumption that matter and radiation under the influence of gravity alone follow geodesics according to the general theory of relativity globally, exactly as locally. And let's just say that it is a gross assumption that the dynamics of the universe itself (or in its entirety, whatever that means) understands and obeys the same principle.

 

All you have to say is that the global spacetime metric is a solution of GR.

 

The simple reason is that, if as you imply, there is a slope to the globally homogeneous manifold that cause objects to converge or diverge geodesically towards or away (radially) from one another, then the universe cannot be homogeneous.

 

The 'slope' in gravitational potential that I was talking about is a 3D Newtonian concept and the diverging and converging geodesics are a 4D relativistic concept. I wouldn't describe a manifold as 'sloped' (it sounds like a mixture of the two concepts). I would say that in both the Newtonian and the relativistic sense it is possible for a homogeneous medium to cause the kind of global effect we're talking about so long as it is isotropic from any observer's perspective.

 

In a Newtonian sense, you can use Newton's shell theorem to treat any observer considering a point at distance r as an observer at the center of a uniform sphere considering a point at radius r (this is true because the medium is isotropic for any observer). It is then straightforward to show that potential is constant only at the position of the observer (proved here) and the gravitational field is equal to zero only at the position of the observer (here). The gravitational field increases linearly with r.

 

In a relativistic sense it is less straightforward, but has been rigorously and mathematically proved that the FLRW metric is 1) spatially homogeneous 2) has global curvature 3) converging or diverging geodesics 4) an exact solution of GR, so I would definitely not agree with "the universe cannot be homogeneous". Friedmann solved the homogeneous / isotropic GR universe 88 years ago. Robertson and Walker proved that the metric is an exact solution of GR for the uniform mass case.

 

That follows for the assumption mentioned above (that t is cosmic time). Certain observers will be higher or lower in the field at any given cosmic time.

 

It's a relative matter. Relative to A, B is lower than A in gravitational potential. Relative to B, A is lower than B. Curved spacetime shows better what is happening with converging or diverging geodesics.

 

If you remove that notion, it follows from general relativity—and the principles of differential geometry inherent in the theory—that a Riemannian (or pseudo-Riemannian) manifold can be curved negatively or positively (due to the mass-energy content) yet all points are equivalent. That is because the curvature is constant. There are no coordinate systems, in this respect, that can be regarded as privileged.

 

I believe you've mixed up global curvature with general covariance.

 

Thought experiment question: If you sprinkle the surface of a large sphere of constant radius (and constant curvature) with gravitating objects (say galaxies) initially at rest, what happens? Do all galaxies eventually merge into one big mass at some place on the surface?

 

No, not "at some place on the surface". Space is not a stable background on which things happen. Space would shrink. If the sphere is an analogy for 3D space then the sphere would shrink in 3D. FLRW describes it in 4 dimensions.

 

I would hardly think so!

 

Yeah, I think the physically decreasing space would be every bit real for us poor creatures living in the space as it shrinks to a singularity (does not sound fun). But, that is what the math is telling us. The evidence of physical distances changing with the changing space is strong in the past. Nucleosynthesis makes perfect sense in that scenario as does the CMBR. It sure does explain observations (it, in fact, predicted a whole host of them). Very good evidence that FLRW is a good approximation of what is happening physically.

 

I'd like to come back to a the well-known Newtonian thought experiment Qtop and you discussed earlier:

 

Though you are correct that the potential is the same everywhere, your assumption that the field inside the shell is flat, cannot be justified a priori.

 

According to general relativity, the spacetime is flat. If the shell is uniform then the space inside is Minkowski and outside is Schwarzschild. Flat spacetime inside means that all test objects one might put in there would be at rest relative to one another over time. Newtonian gravity comes to the same physical conclusion with his shell theorem.

 

In fact, the space inside the shell can be curved. I would even speculate that it has to be curved.

 

You could do a google search for "shell theorem" + "general relativity", I should think.

 

Though I'm not sure in this case whether the space inside the spherical shell would be curved spherically or pseudo-spherically, the curvature would be constant, just as the gravitational potential inside the sphere is constant.

 

The gravitational potential in a sphere of uniform density is not constant (unless, that is, you mean the is hollow). A uniform density sphere is graphed in potential here:

 

-source

 

Only at the exact center does the change in V approach zero, and only there is the force zero.

 

If you mean a hollow sphere then yes, the potential should be constant from the center to the shell's interior boundary.

 

Constant gravitational potential (or constant "force") does not mean spacetime of a globally homogeneous Riemannian or pseudo-Riemannian manifold of dimension four with constant Gaussian sectional curvature is equal to zero. It may in fact mean quite the opposite.

 

For the uniform sphere of constant density it should mean curved spacetime. For the hollow shell, it should mean flat spacetimel. I'm pretty sure (near-positive) that's right.

 

Similarly, in a homogeneous gravitational field scenario; just because each observer sees redshift z and time dilation that increases with distance (due to spacetime curvature) it doesn't mean that the global field is sloped downwards (or that the magnitude of curvature increases or decreases) in any particular direction proportionally with distance. Each observer can consider herself, and the objects in the manifold, as at rest relative to the background gravitational field.

 

Yeah, I think the Newtonian approximation of a sloped potential field is indeed an approximation of curved spacetime. I think redshift is modeled quite well with 'expanding space' or 'FLRW' if you like. Its cosmological strength comes from a combination of the very simple and physically reasonable assumptions needed for FLRW, its agreement with general relativity, and most of all, its ability to predict and model actual observations.

 

Likewise, in the curved spacetime scenario (without assuming all objects would gravitationally collapse into a point somewhere, or everywhere), each observer will see very distant (high-z) objects as if they were immersed deeply inside a gravitational well. Yet each observer considers herself at a higher elevation in the field (or at a saddle point). This by no means negates the physical possiblility that the homogeneous gravitational field within which they reside is intrinsically curved.

 

You'd have to have a global metric to say what you say there (in order to solve for null geodesics), and when you have that metric you'd no doubt have to explain why matter is not following inertial time-like geodesics, and even if that could be explained (and I don't see how it could be) then you'd have to have a good 70 years of successful predictions in order to be on equal footing with standard cosmology.

 

Sorry... being called away from the computer...

 

My skepticism is, I think, at least, a little, warranted.

[modest]

By the way Modest, what do you think of the homogenous solution of the link CC gave? http://www.springerlink.com/content/k550472m97528233/fulltext.pdf

Actually, I think it is not a "homogenous solution" in the sense we normally refer to: as "spatially only" homogenous solution. However is not inhomogenous as its equations give a contant density not dependent on radius. I'm trying to find the flaw, do you see some obvios flaw?

 

Regards

QTop

 

I'm sorry, I don't have time to look at it. My guess, as this sounds somewhat familiar, is that it solves some static case (and no doubt one that's already been solved), it then changes some aspect of the gravitational field, then moves on assuming that it is still static (in other words: after the field changes they don't go back and solve the equations of motion for the new field) then they assume that light acts a certain way because of the new field while it is static (even though it's probably not static).

 

That would be my guess. I guess in the time that I wrote that I could have skimmed over the paper. Sorry.

 

~modest

 

:phones:

...throwing darts in the dark...

:phones:

[quantumtopology]

I'm sorry, I don't have time to look at it. My guess, as this sounds somewhat familiar, is that it solves some static case (and no doubt one that's already been solved), it then changes some aspect of the gravitational field, then moves on assuming that it is still static (in other words: after the field changes they don't go back and solve the equations of motion for the new field) then they assume that light acts a certain way because of the new field while it is static (even though it's probably not static).

 

That would be my guess. I guess in the time that I wrote that I could have skimmed over the paper. Sorry.

 

~modest

 

:phones:

...throwing darts in the dark...

:phones:

Oh, no problem, whenever you have the time, and if you please could addres my post 679 too, I'm interested in your opinion. Thanks

[modest]

Oh, no problem, whenever you have the time

 

I don't understand it. After reading the relevant section, I don't even understand what "time scales with distance" means. I'm positive I'll be of no help in analyzing the paper.

 

, and if you please could addres my post 679 too, I'm interested in your opinion. Thanks

 

Post 679?

 

“Cosmic time” is nothing more or less than that which is measured by a clock positioned isotropically and moving inertially.

What do you mean by positioned isotropically. Isotropy in our universe is considered pretty much empirically confirmed and I thought according to the copernican principle and cosmological principle (no special place principle) that means isotropy does not depend on position. IOW, if we had isotropy in one location and not in others there would be no homogeneity and the place with isotropy should be at the center, in a special or privileged observing point.

 

I agreed with you. I misspoke.

 

In case you mean post 682:

 

This is tricky, if we strictly follow relativity and consider that Lorentz invariance is valid in a curved spacetime...

 

With a flat Minkowski spacetime certain quantities are invariant under Lorentz transformations. With a curved metric, in curved spacetime, they are not. In other words, according to GR, it is only in flat Minkowski spacetime that Lorentz invariance is not violated. A source saying "in relativity Lorentz invariance is valid in a curved spacetime" would be mistaken.

 

Lorentz invariance cannot be a symmetry of the quantum theory of gravity. First of all, it is not a symmetry in classical general relativity or any other relativistic theory of gravity that incorporates the equivalence principle. Lorentz invariance (by which I mean here global Lorentz invariance) is a symmetry of Minkowski spacetime, which is a particular solution of the classical equations of motion of general relativity. But it has no significance beyond this: it does not come into consideration with studying any other solution, nor does it play any role in the formulation or physical interpretation of the equations of general relativity.

 

http://books.google.com/books?id=K_OfC0Pte_8C&pg=PA507&lpg=PA507&dq=%22global+lorentz+invariance%22&source=bl&ots=IAHneEFtQJ&sig=MDGovs69SWtcnUox4aBl15ZZgmY&hl=en&ei=67umTI3xF4OKlwe_68QX&sa=X&oi=book_result&ct=result&resnum=3&ved=0CCQQ6AEwAjgK#v=onepage&q=%22global%20lorentz%20invariance%22&f=false

 

That seems reasonable if we take note of the fact that when we look at spatial distances we are observing also the past or look-back time.

 

Light comes from a previous time and a distant place, but I don't think that makes the previous assertion reasonable.

 

This admittedly is not the case in standard cosmology where a certain coordinate choice is privileged (the one that originates the FRW metric) and assumed to posses special features such as spatial homogeneity in time spacelike hypersurface slices.And therefore a preferred "cosmic time".

 

Time is orthogonal to space. Cosmology assumes that space is homogeneous and isotropic. Unless the space is completely free of matter and radiation, there will only be one coordinate choice where space is homogeneous. That coordinate choice will be 'privileged' only in so far as it is convenient. Cosmic time is then naturally orthogonal to that spatial slice.

 

So, it seems you are either taking issue with the cosmological principle or mistakenly assuming that arbitrary transformations should leave space homogeneous. Taking issue with the cosmological principle is a valid inductive argument, but you would need the assumption to have produced very bad results. On the contrary, assuming the cosmological principle is true has led to very successful predictions and it is philosophically strong. If you mean to imply that any coordinate choice should have homogeneous spatial slices then your argument would be deductively false.

 

According to GR, there is nothing wrong with choosing a coordinates for practical and convenience reasons, as all coordinate choices should give us the same results in physical experiments. But then we must be coherent and we shouldn't decide that the purely spatial isotropy and homogeneity we seem to get with a coordinate choice is physically relevant if it doesnot appear with other coordinate choices as GR general covariance demands.

 

General covariance would certainly *not* demand that any coordinate choice be spatially homogeneous. If GR is anywhere near correct then we would not and could not expect different coordinate choices to all have spatial homogeneity.

 

Besides, there is reasons to suspect that the apparently "only spatial" isotropy and homogeneity of the universe with the FRW metric is a coordinate artifact, apart from the fact that it doesn't show up with other coordinates

 

Again, the fact that it doesn't show up with other coordinate choices is 100% expected and proper. For it to be otherwise would be problematic.

 

, it demands a certain uniform velocity to be comoving with a supposed flow, and in doing this it is giving us a way to make "closed experiments" of spatial isotropy (see: http://www.phys.ncku.edu.tw/mirrors/physicsfaq/Relativity/SR/experiments.html#Tests_of_isotropy_of_space ) that allows us to distinguish different inertial frames, this is forbidden by the SR principle.

 

Oh my...

 

~modest

[quantumtopology]

So, it seems you are either taking issue with the cosmological principle or mistakenly assuming that arbitrary transformations should leave space homogeneous. Taking issue with the cosmological principle is a valid inductive argument, but you would need the assumption to have produced very bad results. On the contrary, assuming the cosmological principle is true has led to very successful predictions and it is philosophically strong. If you mean to imply that any coordinate choice should have homogeneous spatial slices then your argument would be deductively false.

 

You are close but in fact both assumptions follow from each other, If one drops the cosmological principle it seems the most natural thing to assume Local Lorentz invariance that applied to every point which should leave space statistically homogenous, that is, spatially homogenous in any coordinate(which equals spacetime isotropy), and viceversa if one assumes spacetime isotropy then obviously one must drop the cosmological principle.

In any case it all comes dow to change the assumption of the cosmological principle that demands a privileged coordinate spatial homogeneity, and this should depend on purely empirical observations like large scale galaxy surveys: SDSS, 2MASS etc, and they seem to be starting to give us hints that coordinate preferred spatial homogeneity is not the case.

 

 

 

General covariance would certainly *not* demand that any coordinate choice be spatially homogeneous. If GR is anywhere near correct then we would not and could not expect different coordinate choices to all have spatial homogeneity.

I moreless agre with you here , with a precission, General covariance would not demand it in general, unless you considered the matter distribution of the universe in terms of isotropy and homogeneity as a fundamental physical law of our universe. In wich case it would violate general covariance if a differentiable transformation of coordinates changed the form of the law. (from the genral covariance definition of wikipedia). The problem is that I'm not at all sure the distribution of matter in the universe can be considered a physical law, most likely not, it is a purely observational issue.

 

Oh my...

What? Oh, my... here means ??

[coldcreation]

I don't understand it. After reading the relevant section, I don't even understand what "time scales with distance" means. I'm positive I'll be of no help in analyzing the paper.

 

The current standard model (or concordance model) describes the evolution of the universe which depends on a global time parameter, cosmic time, with which the scale factor (or size of the universe) changes. But such a restriction on the field equations is not necessary. Fischer writes: "Instead of assuming that the global distance scale depends on time, we can as well discuss the possibility that the time scale depends on distance and that there exists no global time scale at all."

 

Simply put, a time scale that depends on distance is a concept entirely consistent with empirical evidence (as opposed to a hypothetical "now" or "then"). This is what is observed. The observer measures events that have taken place some time in the past, and at certain distance, relative to her rest-frame. But that's not all, in a curved spacetime not only are spatial distances distorted, but so too are time-like increments. So the actual "time scale may change with distance" in a nonlinear fashion. Redshift and time dilation are manifestations of such a globally curved spacetime. Time depends on distance, since space and time are not isolated from on another. This is a "purely geometrical change of time scale." Fischer continues, "Redshift is a direct consequence of curvature, which causes a change of time scale with distance."

 

For an illustration of such simple general relativistic phenomena, recall the Coldcreation schematic diagrams where the observer in centered at the origin, while spherical shells (measured in light years) are shown to either increase or decrease in distance in the look-back time, depending on the sign of curvature. Time depends on distance, and distance depends on time, in a curved spacetime manifold. (See for example the equatorially sliced diagrams, the cross sections of the visible universe diagrams).

 

The Fischer model and the coldcreation model are almost exactly the same representation of the universe: static, global spacetime curvature and evolving dynamically with time according the laws of physics (locally and globally).

 

 

Interesting wouldn't you say?

 

 

PS. Qtop, did you get my PM?

 

 

CC

[quantumtopology]

Hi, Modest, CC and everyone:

 

This interesting article seems to address some of the points we are debating about assumptions of homogeneity, it's fro 2007 but I think we are basically at the same point. Take a look:

 

http://magickriver.blogspot.com/2007/10/is-universe-fractal-by-amanda-gefter.html

 

Regards

QTop

[coldcreation]

It's just the time comoving clocks keep.

 

Cosmic time is more than that. It is defined for an expanding universe that has the same density everywhere at each moment in time and in conformity with the Hubble flow; where the big bang is the origin of the time coordinate. The fact that this is possible by no means implies that this method of time keeping is the only type that should be used in the context of the field equations. It seems much more natural simply to measure time relative to local clocks, just as it is natural to measure spatial distances relative to local distances. In other words, measurements (both spatial distances and temporal increments) should be determined relative to the frame of reference of the observer. As it turns out, time appears to run slower in the vicinity of distant objects, than it does on the frame of the observer. And, distances appear to be distorted the further the source object. There is no cosmic time in this sense. There is no NOW everywhere in the universe.

 

 

Every model, and every theory, makes assumptions. The predictions of the theory test the assumptions.

 

The problem occurs when assumptions keep piling up, one on top of the other; leading to still more assumptions. This is exactly what has happened to modern cosmology. Every time an assumption has been tested, another more dubious assumption had to be added to the mix. Now we are left with 96% of the universe composed of stuff with no palpable physical counterpart locally.

 

 

The postulates of a theory are, by design, universally true of the theory. It is a postulate of GR that freely moving particles follow geodesics in four dimensional spacetime.

 

On the subject of geodesic motion and homogeneity, I've added a section (two actually) that should clear up the problem (along with a schematic diagram).

 

See: A General Relativistic Stationary Universe and scroll down to a section entitled Geodesics in a homogeneous spacetime of constant curvature. Notice Figure PRMCC and the following section entitled Homogeneous gravitational field. Figure PRMCC represents a homogeneous pseudo-Riemannian manifold of constant positive Gaussian curvature. Placed on the manifold is a galaxy cluster, Cl 0024+17 (ZwCl 0024+1652) from Hubble's Advanced Camera for Surveys.

 

It should be easy to see, if you look at Figure PRMCC, why a light ray would follow a great circle arc towards any observer (the shortest distance between two points), while the galaxy cluster will follow local geodesics (not great circle arcs) that depend on its location relative to other cluster (not shown). Yet, at the same time the cluster is embedded in the Riemannian (or pseudo-Riemannian) manifold.

 

If you disagree with the text or illustration please explain for what reason(s)...

 

 

All you have to say is that the global spacetime metric is a solution of GR.

 

Yes, it is a solution. But there are many possible solutions, one of which can be found here: Homogeneous cosmological solutions of the Einstein equation

 

I highly recommend you read this work.

 

 

The 'slope' in gravitational potential that I was talking about is a 3D Newtonian concept and the diverging and converging geodesics are a 4D relativistic concept. I wouldn't describe a manifold as 'sloped' (it sounds like a mixture of the two concepts). I would say that in both the Newtonian and the relativistic sense it is possible for a homogeneous medium to cause the kind of global effect we're talking about so long as it is isotropic from any observer's perspective.

 

Agreed.

 

But you must remember: once a distance from the reference frame of an observer (from the origin) is specified, the ideas of either rest or motion (or both) naturally follow. If the position vector of the particle relative to a given reference frame changes with time, then the emitting source is said to be in motion with respect to the observer's reference frame. However, if the position vector of the source, relative, again, to the reference frame of the observer, remains same with time, then the source is at rest with respect to the chosen frame.

 

Rest and motion are relative to the reference frame of the observer. It is possible that a source galaxy which appears at rest relative to the observer is in motion, radially. But it is also possible that a source galaxy which appears in motion relative to the observer is actually at rest (in a curved spacetime). Thus, rest and motion aren't absolute terms, rather, they are dependent on the reference frame, and how the observational evidence is interpreted from that frame. In the case here regarding cosmology, there is no distinction observationally between the two, since in both cases proper distances are distorted (either by relative motion, or by relative position in a globally curved manifold) resulting in cosmological redshift z and time dilation that increases with distance.

 

In both cases too, a homogeneous 'medium' (a smooth expanding Hubble flow, or a globally curved gravitational field) causes the kind of global effect which appears isotropic from any observer's perspective. That is why both the expansion and spacetime curvature interpretations are consistent with observations. (Albeit, the latter requires no DE or CDM).

 

Interesting isn't it?

 

 

[...] FLRW metric is 1) spatially homogeneous 2) has global curvature 3) converging or diverging geodesics 4) an exact solution of GR, so I would definitely not agree with "the universe cannot be homogeneous". Friedmann solved the homogeneous / isotropic GR universe 88 years ago. Robertson and Walker proved that the metric is an exact solution of GR for the uniform mass case.

 

I understand your point. But isotropy varies with the cosmic time t coordinate of the FLRW model. Isotropy is an artifact of any solution that uses cosmic time only.

 

The global curvature depends on the deceleration parameter. It is not a true curvature in the Gaussian or Riemannian sense. The geodesics too depend of the deceleration parameter. Without cosmic time and expansion there is no reason why geodesics should converge or diverge globally, i.e., there is no reason why massive gravitating bodies should follow great circle arcs. There is, however, every reason why photons should (and do).

 

 

It's a relative matter. Relative to A, B is lower than A in gravitational potential. Relative to B, A is lower than B. Curved spacetime shows better what is happening with converging or diverging geodesics.

 

Agreed, if we're discussing photons as observed from the observers reference frame, globally (or local self gravitating systems).

 

 

Thought experiment question: If you sprinkle the surface of a large sphere of constant radius (and constant curvature) with gravitating objects (say galaxies) initially at rest' date=' what happens? Do all galaxies eventually merge into one big mass at some place on the surface?[/quote']

 

No, not "at some place on the surface". Space is not a stable background on which things happen. Space would shrink. If the sphere is an analogy for 3D space then the sphere would shrink in 3D. FLRW describes it in 4 dimensions.

 

The idea that space itself can "shrink" or "expand" is also an artifact of the FLRW solutions. There is no guarantee that the universe itself can shrink or expand. The concept of 'curved spacetime' is not synonymous with catastrophic shrinking or wholeslae expansion of space. It is synonymous with a non-Euclidean geometric distortion relative to a given frame of reference (that of any observer). It is a geometric version of the effects of gravity. That is pure general relativity.

 

The fact that space doesn't shrink or expand locally (in a curved spacetime regime, or even a flat Minkowski space-time) is a good reason to believe that space doesn't shrink or expand globally. You don't see, for example, experiments performed in laboratories here on earth that demonstrates, even remotely, that space can contract or expand. The geometry of spacetime is something else. General relativity describes gravity as a geometric property of spacetime. The curvature of spacetime is directly related to the four-momentum (mass-energy and linear momentum) of the matter and radiation present. The relation is specified by the Einstein field equations (a system of partial differential equations). Only when cosmic time is employed does the concept of spacetime itself degenerate to the point of collapse, along with everything in it. Essentially, the idea that space itself can expand or shrink is a consequence of an error in the perception or representation of the time function introduced into the field equations. Certainly expanding and shrinking space is not a natural feature general relativity.

 

The metric expansion of space is usually qualified as an increase of metric distance between distant objects in the universe with cosmic time. Of course, there is no way to measure the actual increase of metric distances. Even so, the expansion is thought to be an intrinsic expansion. But in reality, it may very well be the byproduct of an extrinsic agent or method used for time keeping that erroneously affects geodetic motion of space itself. The idea is a product of human conception rather than something inherent in nature.

 

Observationally, we could also conclude that the metric distance between objects in the universe, as measured in the look-back time, remains more or less constant in a four-dimensional spacetime manifold of constant intrinsic Gaussian curvature.

 

In the former, distances change with cosmic time. The manifold is one big expanding preferred coordinate system.

 

In the latter, there is no preferred time coordinate. The numerical values at the spatiotemporal coordinates of any given object (compared to local clocks and atomic spectral lines) are uniquely determined and based upon the geometric properties of the spacetime that separates the observer and the source (e.g., a distant SNe Ia, Galaxy or galaxy cluster). The appropriate metric can easily be established mathematically, based on redshift z and time dilation.

 

 

You'd have to have a global metric to say what you say there (in order to solve for null geodesics), and when you have that metric you'd no doubt have to explain why matter is not following inertial time-like geodesics, and even if that could be explained (and I don't see how it could be) then you'd have to have a good 70 years of successful predictions in order to be on equal footing with standard cosmology.

 

My skepticism is, I think, at least, a little, warranted.

 

“The skeptic does not mean him who doubts, but him who investigates or researches, as opposed to him who asserts and thinks that he has found” ~(Miguel de Unamuno, Spanish Author and Philosopher, 1864-1936)

 

 

 

CC

[coldcreation]

Gentlemen,

 

I found this today. It's not dissimilar to what we've been discussing:

 

The Infinite Universe of Einstein and Newton by Barry Bruce (2003).

 

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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. This net force of gravity is found to be equal to the force of inertia. These calculations explain Newton's First Law, Newton's Second Law, and the equivalence of inertial and gravitational mass. In addition, by the extension of Einstein's general relativity to two-body interactions Newton's Third Law is elicited. 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.

 

 

The work can also be viewed here: Google Books

 

 

The global geometry reduces to locally flat (Minkowski spacetime) near the observer (the origin).

 

See for example Section 1.2.2 Nature of the Solutions.

 

In section 1.3.1 the Author writes the frequency shift relation for an infinite open and an infinite closed universe (and shows that there is no frequency shift in the flat case). Only two of these metric solutions allow redshift z (the solutions for positive and negative curvature). The Author shows that the closed universe fit the data of the Hubble diagram better than the big bang theory.

 

 

This model is an infinite static (nonexpanding) homogeneous and isotropic solution consistent with both general relativity and Newtonian cosmology. (See equations 1.37a - 1.38c)

 

These results show that the cosmological redshift z is likely the result of the uniform mass distribution of an infinite closed universe and gravity alone.

 

These are static solutions to the Einstein field equations. Note, the time parameter is not cosmic time. Using Einstein's field equations, this model unites the goal of Einstein, Mach, and Newton all at once (according to the author).

 

 

This model is (a priori *) to some extent in line with the concepts of Fischer, Qtop and coldcreation. Though, from what I understand Bruce does not consider a global Gaussian curvature as the cause of redshift. But the general result will be the same since z is a gravitational effect (Section 1.2, p. 9).

 

 

(* I've not studied the entire theory, since many of the books pages are absent in the preview.)

 

 

 

CC

[coldcreation]

Qtop, check your PM inbox. :unsure:

 

Modest, your king is in check, your move... :rolleyes:

 

 

CC

[HydrogenBond]

If you look at a neutron star, for neutron density to form, the neutrons need to get very close, regardless of reference. If the laws of physics are the same in all references, within any given reference we can create actual movement into nuclear distances, independent of reference.

 

Relativity appear to be talking about the reference effects. The laws of physics are same in all references, since there are secondary effects that are not dependent on SR and GR. If they were dependent, the laws would not be the same in all references, but would be reference dependent as a function of GR and SR.

 

This suggest two sets of theories working at the same time. One possible explanation is due to the propagation of the attraction effect from gravity, at the speed of light. This is not reference dependent, since it moves at C, separating it from the reference dependency of GR.

 

An interesting consideration is graviton red shift as it moves into expanding space-time, but at the speed of light. Just as a red shift of energy, lowers the energy quanta, this would lower the energy quanta of the gravitational force effect, thereby altering which laws of physics which can become active in that GR reference. If we move away from a neutron star, we get to a point where neutron density is not a very likely thing, since the red shift lowers the potential energy effect of C-speed propagation.

 

If we went into a black hole, we get gravity blue shift, where potential energy increases, changes which laws are in effect. One would expect going from neutron density into the next logical phase change.

[coldcreation]

What follows is fundamental to the entire discussion.

 

 

Referring back to Figure PRMCC:

 

Here's a (global) thought experiment (touched on above) in question-form.

 

If you sprinkle the surface of a large four-dimensional spherical Riemannian (or pseudo-Riemannian) manifold of constant Gaussian curvature K = 1, with self-gravitating objects (say galaxies, or clusters of galaxies) initially at rest, (1) what happens?

 

(2) Do all galaxies merge into one big mass at some place on the spherical surface?

 

(3) Does the spherical surface itself collapse or shrink (meaning that the radius of the sphere itself cannot remain constant)?

 

Or, (4) does the manifold remain stationary, static (meaning that the radius of the sphere itself remains constant), i.e., there is no change in the scale factor to the metric?

 

I believe that the answer(s) to this Gedankenexperiment can be answered (though perhaps not unambiguously) in light of our current knowledge (based on theoretical and empirical evidence collected to date). Certainly, the answers to the questions depend on how the parameters are treated and used (if at all) in the Einstein field equations.

 

 

The answer to questions (2 and 3) raised by this thought experiment may in fact be a resounding NO. Objects may not coalesce into one big massive Globoid. This is the problem of stability or instability of a globally homogeneous Riemannian (or semi-Riemannian) manifold of constant sectional curvature.

 

This is the problem Einstein faced with his 1916-1917 world-model. Newton suffered a similar problem (some 230 prior to Einstein) when gravity was treated as an attractive force operating in a Euclidean universe.

 

 

The Standard Scenario:

 

It is believed according to the current interpretation of general relativity that the actual sphere would shrink. In other words, objects would not all group together somewhere on the sphere. Rather, the objects would freely-fall towards one another, resulting in the shrinking of the sphere. Einstein added the cosmological constant (lambda) to the field equations in order to prevent such a collapse. It's easy to see that any perturbation in the equilibrium induced by lambda in relation to the mass content would result in either expansion of collapse of the sphere.

 

The curious feature here (in reduced dimensions) is that the galaxies would actually move (freely fall) towards the center of the sphere. All objects would eventually come into contact at the center of the sphere. But since the surface of the sphere represents the spacetime manifold itself, there is no center to the sphere. In fact there is no interior or exterior to the sphere. The surface of the sphere is all there is. And yet the free-fall occurs in the direction of the center. Space shrinks in this case (like a deflating balloon). (And depending on the value of lambda, space can expand). Essentially, all objects on the manifold free-fall (accelerate) geodesically along with space.

 

The 'attractive' nature of the gravitational force (a la Newton) forces objects to gravitate towards each other. But there is no center to the surface of a sphere (a la Einstein) towards which to merge (there is no center-point towards which objects accelerate). This leads to a unique and remarkable conclusion: The universe itself must shrink. In other words, the motion of objects in the manifold (which is induced by the gravitational force) causes space to curve, increasingly, while the scale factor (the size of the sphere) becomes smaller. Motion of the objects causes space to curve continually and increasingly.

 

The final result is that the entire universe (the sphere in our thought experiment) becomes compact, small, dense, hot, increasingly so, until the physical laws break down at some unspecified boundary condition. While the mass-energy content remains the same globally, curvature increases unimpeded, until the magnitude of curvature is so high that the sphere tends to become point-like. In the language of an expanding FLRW model, when we reverse the clocks, all spacelike, timelike, and lightlike, or null geodesics, converge to a point. So, what originally presented itself as a large sphere with objects randomly spaced upon it, becomes a minuscule fireball: a Big Crunch, where space, time, matter and energy are united.

 

The above scenario was described by modest as such:

No, not "at some place on the surface". Space is not a stable background on which things happen. Space would shrink. If the sphere is an analogy for 3D space then the sphere would shrink in 3D. FLRW describes it in 4 dimensions.

 

[...] the physically decreasing space would be every bit real for us poor creatures living in the space as it shrinks to a singularity (does not sound fun). But, that is what the math is telling us.

 

I responded to that in Post #389.

 

 

Another question imposes: Is there permitted by general relativity a solution (to the Einstein field equations) that resolves the problem of instability, leading to the stability of a globally homogeneous gravitational field of constant Gaussian curvature (meaning that the universe does not expand of collapse)?

 

 

 

CC

[modest]

Sorry guys, I missed the replies in the control panel and I've been really busy.

 

You are close but in fact both assumptions follow from each other, If one drops the cosmological principle it seems the most natural thing to assume Local Lorentz invariance that applied to every point which should leave space statistically homogenous

 

Local Lorentz invariance doesn't imply spatial homogeneity. The comment I was responding to was,

 

"But then we must be coherent and we shouldn't decide that the purely spatial isotropy and homogeneity we seem to get with a coordinate choice is physically relevant if it doesnot appear with other coordinate choices as GR general covariance demands."

 

you imply that GR somehow wants every coordinate choice to be spatially homogeneous. Is that not what you are implying, or have you changed your mind?

 

, that is, spatially homogenous in any coordinate(which equals spacetime isotropy), and viceversa if one assumes spacetime isotropy then obviously one must drop the cosmological principle.

 

Spacetime is not isotropic in our universe. We can make an isotropic metric, like Minkowski spacetime, but we would have to fault it for not agreeing with observation.

 

~modest

[modest]

Cosmic time is more than that... It seems much more natural simply to measure time relative to local clocks, just as it is natural to measure spatial distances relative to local distances.

 

The duration of a supernova at z=1 is twice that of a local supernova from our perspective. It may be 20 seconds according to our clock and 10 seconds according to a local clock. There is no problem in using either measure of time so long as you don't mix them up.

 

As it turns out, time appears to run slower in the vicinity of distant objects, than it does on the frame of the observer. And, distances appear to be distorted the further the source object. There is no cosmic time in this sense. There is no NOW everywhere in the universe.

 

Just like in SR, the relative nature of time does not make simultaneity nonexistent—it makes it relative.

 

But you must remember: once a distance from the reference frame of an observer (from the origin) is specified, the ideas of either rest or motion (or both) naturally follow. If the position vector of the particle relative to a given reference frame changes with time, then the emitting source is said to be in motion with respect to the observer's reference frame. However, if the position vector of the source, relative, again, to the reference frame of the observer, remains same with time, then the source is at rest with respect to the chosen frame.

 

If the distance between A and B increases over time in one frame then it will increase over time in all frames. In other words, if A is in motion relative to B then no change of reference frame will change that fact.

 

... But it is also possible that a source galaxy which appears in motion relative to the observer is actually at rest (in a curved spacetime).

 

Objects cannot be said to have motion or to be at rest in spacetime. Things don't move in sapcetime. Movement is a change in distance over a change in time. Spacetime represents both distance and time so things don't move on it. In otherwords, things can move in space but things cannot move in spacetime.

 

I have a similar problem with your thought experiment,

 

Here's a (global) thought experiment (touched on above) in question-form.

 

If you sprinkle the surface of a large four-dimensional spherical Riemannian (or pseudo-Riemannian) manifold of constant Gaussian curvature K = 1, with self-gravitating objects (say galaxies, or clusters of galaxies) initially at rest, (1) what happens?

 

(2) Do all galaxies merge into one big mass at some place on the spherical surface?

 

(3) Does the spherical surface itself collapse or shrink (meaning that the radius of the sphere itself cannot remain constant)?

 

The 4-dimensional manifold cannot move. A three-dimensional manifold that represents space can shrink, but a spacetime manifold simply exists and can't be said to move.

 

The answer to the thought experiment should be that the geodesics converge or diverge. That is to say, the spatial distance between their world lines increases or decreases over time.

 

~modest

[coldcreation]

If the distance between A and B increases over time in one frame then it will increase over time in all frames. In other words, if A is in motion relative to B then no change of reference frame will change that fact.

 

If the distance between A and B does not increases over time in one frame then it will not increase over time in all frames. In other words, if A is at rest relative to B then no change of reference frame will change that fact.

 

The point is that a source galaxy which appears in motion, moving radially (judging from redshift z) may actually be at rest in a curved spacetime continuum (judging from redshift z).

 

 

Objects cannot be said to have motion or to be at rest in spacetime. Things don't move in sapcetime. Movement is a change in distance over a change in time. Spacetime represents both distance and time so things don't move on it. In otherwords, things can move in space but things cannot move in spacetime.

 

You're missing the point. Either the universe is expanding, or it is not expanding. These two different world-views depend on the interpretation of redshift z. The former requires a Doppler (or Doppler-like) effect, the latter is interpreted as a curved spacetime phenomenon. (Note: that would not be a curved space phenomenon).

 

 

The 4-dimensional manifold cannot move. A three-dimensional manifold that represents space can shrink, but a spacetime manifold simply exists and can't be said to move.

 

You're still missing the point. Either the universe collapses, expands, or stays the same, i.e., there is either a change in the scale factor to the metric, or an intrinsic Gaussian curvature that affects light in a static manifold (a spacetime manifold, of course).

 

 

The answer to the thought experiment should be that the geodesics converge or diverge. That is to say, the spatial distance between their world lines increases or decreases over time.

 

Well, now, that depends on the answer to this question: 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 or collapse)?

 

 

CC

[modest]

If the distance between A and B does not increases over time in one frame then it will not increase over time in all frames. In other words, if A is at rest relative to B then no change of reference frame will change that fact.

 

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.

 

The point is that a source galaxy which appears in motion, moving radially (judging from redshift z) may actually be at rest in a curved spacetime continuum (judging from redshift z).

 

You either mean 'at rest relative to curved spacetime' or 'at rest relative to us in curved spacetime'. The former wouldn't make sense to me because things can't move in 4D spacetime. If you mean 'at rest relative to us in curved spacetime' then the idea is incompatible with general relativity. The world lines of two objects at rest relative to one another neither diverge or converge. 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.

 

You're missing the point. Either the universe is expanding, or it is not expanding. These two different world-views depend on the interpretation of redshift z. The former requires a Doppler (or Doppler-like) effect, the latter is interpreted as a curved spacetime phenomenon. (Note: that would not be a curved space phenomenon).

 

Both with curved spacetime and a special relativistic doppler shift interpretation the increase in physical distance is real. In one view, objects are moving through space away from us and in the other view objects are at rest with space as space expands. Both interpretations have the same physical meaning: the distance between the two things is physically increasing.

 

Are galaxies really moving away from us or is space just expanding?

 

 

This depends on how you measure things, or your choice of coordinates. In one view, the spatial positions of galaxies are changing, and this causes the redshift. In another view, the galaxies are at fixed coordinates, but the distance between fixed points increases with time, and this causes the redshift. General relativity explains how to transform from one view to the other, and the observable effects like the redshift are the same in both views. Part 3 of the tutorial shows space-time diagrams for the Universe drawn in both ways.

Frequently Asked Questions in Cosmology

 

Also: 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?

 

You're still missing the point. Either the universe collapses, expands, or stays the same, i.e., there is either a change in the scale factor to the metric, or an intrinsic Gaussian curvature that affects light in a static manifold (a spacetime manifold, of course).

 

Honestly CC, the words "static" and "dynamic" do not apply to spacetime manifolds in the way you've been using them. There is no higher 5th dimension of time in which a 4D manifold can move (or not move). The concept of 'motion' applies to 3D objects, but not 4D events, world-lines, and manifolds.

 

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.

 

Well, now, that depends on the answer to this question: 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)?

 

That seems to me like asking if chemistry allows one atom of oxygen to combine with two atoms of hydrogen (making the most massive and most deadly molecule known to man). Everything that I understand and everything that I've read on the subject tells me that the part in parentheses is mistaken.

 

~modest