Lambda-CDM theory - Woo or not?

[Farsight]

No one can accuse you of hiding behind mathematics.

I don't hide behind anything. I gave sol some mathematics, he ignored it.

When did mathematics, backed by citations to standard references, turn into "bible thumping"?

The bible thumping was you saying that I must be wrong because what I was telling you wasn't in agreement with Hawking and Ellis. That is bible thumping Clinger. Recognise it. Don't do it.

Okay. Let's be even more general, and call it a closed path.

It's still a spherical triangle rather than a triangle.

The sphere is a 2-dimensional surface. On that surface, you cannot connect three geodesics to form a closed path with angles summing to 180 degrees.

The universe isn't a 2-dimensional surface. It's perhaps a "ball", which the common man would call a sphere. Mathematically speaking the sphere is the surface rather than the ball.

From your example, I think you're assuming it's okay to leave the sphere itself and form your triangle within the ball you believe to be enclosed by the sphere. If you're hoping to use that as an analogy for spacetime, you'll have to explain why you're postulating a Euclidean space outside of the spacetime manifold.

I'm using the ball as an analogy for the space of the universe. Spacetime is an "all time view" of the expanding ball. Note that there is no motion in spacetime, or through spacetime, because it's an-all time view. It's a block universe.

That isn't true of all expanding universes, and it certainly isn't true of all coordinate systems.

There's only the one universe, and coordinate systems are merely an artefact of measurement. You can't point up to the sky and say "hey look, a coordinate system".

You have a bad habit of arguing as though there's some privileged coordinate system in which your beloved Euclidean space has been carved out of non-Euclidean spacetime, and that your argument automatically generalizes to all other coordinate systems.

There is, that of the rest frame of the CMBR. See CMBR dipole anisotropy. From this you can gauge your motion through the universe. Whilst this isn't technically an "absolute" reference frame, it doesn't get much more absolute than that.

That unpublished arXiv paper supports what I wrote: Curvature is measured by the Riemann curvature tensor and its contracted forms (the Ricci tensor and the scalar curvature)...

Don't be evasive. Look at the last line of the abstract: The interpretation of gravity as a curvature in space-time is an interpretation Einstein did not agree with. Now sit down and think about how Einstein described a gravitational field as inhomogeneous space. And look at http://iopscience.iop.org/0256-307X/25/5/014

That arXiv paper also agrees with me about the modern view of gravity as "a manifestation of the curvature of space-time."

The modern view is not in line with Einstein.

Peter M Brown, the author of that paper, appears to be arguing that Einstein thought about it the other way around, viewing curvature of space-time as a manifestation of gravity. That's a minor historical point, which we can discuss if and when you acquire a better understanding of general relativity.

We've already been discussing it, and you've dismissed it because you adhere to the modern view.

For now, the more important point is that nothing in that paper supports your technical errors:

• You were flat-out wrong to deny that curvature is defined by the Riemann curvature tensor and its contractions (the Ricci tensor and scalar curvature).
• You were flat-out wrong to deny the curvature of spacetime in most FLRW solutions, including all "dusty" solutions.
• You were flat-out wrong to deny the existence of gravity in FLRW solutions.

The technical error is in FLRW which starts with the assumption of homogeneity and isotropy of space and then allows Σ to range over a 3-dimensional space of uniform curvature, that is, elliptical space, Euclidean space, or hyperbolic space. It has to be Euclidean. Space is flat when it's homogeneous. Spacetime is only "curved" in an arcane mathematical sense when space is expanding, because the homogeneity of space changes over time even though space in the universe remains homgeneous. Light doesn't bend round full circle and end up where it started.

No. You, however, are confusing mathematics with subjective opinion.

You are definitely confusing a spacetime manifold with physical space. Space exists, that's what's out there. It's expanding, we have hard scientific evidence that tells us this. A spacetime manifold is an abstraction.

The following statements are mathematical:

• A non-zero Ricci tensor implies curvature.
• A non-zero scalar curvature implies a non-zero Ricci tensor, hence curvature.
• Most FLRW solutions, including all "dusty" solutions, imply curvature of the spacetime manifold.

When you deny those statements, you are flat-out wrong.

I'm making a distinction between mathemmatical abstraction and reality. Go look at the Ricci curvature tensor, it relates to volume change. The universe is expanding. And in that universe space is homogeneous, at all times. And since a gravitational field is homogeneous space, there is no overall gravity in the universe.

The following statement is your subjective opinion:

• Curvature of the spacetime manifold does not imply gravity.

The universe didn't contract when the matter-energy density was high, now did it. That's not my subjective opinion.

You and Mr Peter M Brown are free to believe that, but Mr Brown at least acknowledges he's out of step with modern presentations of general relativity:

The modern presentation of general relativity is out of step with Einstein.

Do Peter M Brown's quotations from standard references make him a bible thumper?

No. He offers them to support a well-crafted argument.

You are elevating your beliefs over the mathematical definition of curvature that appears quite explicitly in Einstein's field equations. By rejecting the mathematics employed by those equations, you are rejecting those equations.

You're elevating your beliefs over scientific evidence and what Einstein actually said. I'm rejecting the modern interpretation because it is not in accord with hard scientific evidence and the interpretation of gravity as a curvature in space-time is an interpretation Einstein did not agree with.

Einstein's static universe, in which space is spherical, is one of the FLRW solutions. Extending the lane indefinitely brings it back to the bowler's feet.

And WMAP has demonstrated that space is flat. So you can discard that solution.

Yet you say that space is Euclidean

No. WMAP says it: http://map.gsfc.nasa.gov/universe/uni_shape.html

True: the spacetime manifold is a mathematical/scientific model of (some aspects of) the universe.

It's a mathematical model of expanding space over all times. Not space. Not the universe as it is right here right now.

Once again, you are assuming a privileged coordinate system. If I weren't so afraid of being called a bible-thumper, I'd quote Einstein's statement of what he called "the fundamental idea of the general principle of relativity".

I'll quote it for you: "All Gaussian co-ordinate systems are essentially equivalent for the formulation of the general laws of nature".

Spinning a yarn is no substitute for doing mathematics.

You've been spun a yarn, Clinger. Now come on, think it through: start with a high-energy-density universe. It expands, reducing the energy density if you adhere to conservation of energy. What happens next? It expands, reducing the energy density. What happens next? It expands, reducing the energy density. At all times the space within the universe is homogeneous on a large scale, meaning there’s no overall gravity in it. The universe didn’t collapse when it was small and dense. It's going to sail right on past the critical value without a care in the world.

No one believes that a single FLRW solution describes all of spacetime. The "dusty" FLRW solutions are intended only to model spacetime after radiation and other pressure becomes negligible. If the cosmological constant Λ is zero, then the rate of expansion decreases with time. If the matter density is larger than the critical density, then the rate of expansion eventually goes negative, and the universe begins its collapse into a Big Crunch.

It was, and it didn't. Like I said, you've been spun a yarn. Come on Clinger, think it through. Think for yourself. Don't let Hawking do your thinking for you.

If the matter density is less than or equal to the critical density, then the rate of expansion decreases forever but never reaches zero.

Pity the expansion of the universe is speeding up: http://en.wikipedia.org/wiki/Accelerating_universe.

In the paragraph above, I am discussing the FLRW models of the universe. The things I said in that paragraph are theorems of mathematics. The real universe may behave differently because the universe is more complex than the FLRW models and may be wildly different from the FLRW models in ways we have not even begun to suspect.

Which brings us back to where we came in. There are issues with FLRW.

NB: can you try to make your next post a little more succinct please.

 

[Farsight]

Farsight asserted that homogeneous and isotropic spacetimes aren't curved.

I referred to homogeneous and isotropic space. This is the Einstein quote I gave:

”According to this theory the metrical qualities of the continuum of space-time differ in the environment of different points of space-time, and are partly conditioned by the matter existing outside of the territory under consideration. This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that ‘empty space’ in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitation potentials gμν)...

Space, not spacetime!

There's no interpretation, no cause and effect issue:

Oh yes there is. See above.

Einstein's equations say that if there is any stress-energy (and/or a cosmological constant) then there is curvature (and vice versa, so it doesn't matter what causes what), regardless of whether or not there is a lot of symmetry.

Oh really? When Einstein said a gravitational field was inhomogeneous space. Wherein die Ausbreitungs-geschwindigkeit des Lichtes mit dem Orte variiert. And thus the motion of light is curvilinear? And the interpretation of gravity as a curvature in space-time is an interpretation Einstein did not agree with? There's definitely an interpretation issue here.

 

[Reality Check]

He gave the equations of motion rather than the equations of curved spacetime,

He gave the equations relating curved spacetime to mass and energy (the Einstein field equations). These are the equations that give the motion of objects in a gravitational field.
The 'equations of curved spacetime' were already well known. Riemannian's work on geometry was published in 1858 and pseudo- Riemannian manifolds are the mathematical basis of GR.

and took the view that the speed of light varied with gravitational potential, qualifying his SR postulate. This isn't the modern view. See this article and look at the section on General Relativity.

You got something right .
A nice bit of history is that Einstein made a comment in a book that suggests that he thought that the speed of light could vary in GR.
That was physically valid but is not the modern interpretation.

 

[Reality Check]

Of course I've seen them. The issue is in the interpretation, what the individual terms actually represent, what curved spacetime really is, and seeing the underlying simplicity.

Wrong: Curved spacetime has a precise mathematical definition.
The individual terms have exact definitions.
The left hand side is the geometry of spacetime.
The right hand side is mass & energy.

The 00 compononet of the stress-energy tensor

is the density of relativistic mass.

An isotropic, homogeneous distribution of mass means that this density is non-zero. This means that the left side of the EFE is non-zero. This means that spacetime is curved.

Let me try to get this across with a trivial example.
...
Light moves uniformly, in straight lines: your spacetime curvature has gone.

Well duh !
Of course you can cancel the effects of the curvature of spacetime with the correct selection of a mass distribution, e.g. between two bodies with equal mass there is a line where the curvature is essentially zero. A light ray following that line will not be deflected.

 

[sol invictus]

I referred to homogeneous and isotropic space.

Locally and globally homogeneous and isotropic spaces in three dimensions include:

1) 3-spheres, which are positively curved
2) real projective 3-space, which is positively curved
3) hyperbolic 3-space, which is negatively curved
4) Euclidean 3-space, which isn't curved.

Locally and globally homogeneous and isotropic spacetimes in three space dimensions and one time include

5) all metric consisting of any of the above plus time, with a time-dependent scale factor that's usually called a(t).

All such spacetimes are curved, apart from two special cases:

6) a(t) = constant and space 4)
7) a(t)=t and space 3).

This is the Einstein quote I gave:

”According to this theory the metrical qualities of the continuum of space-time differ in the environment of different points of space-time, and are partly conditioned by the matter existing outside of the territory under consideration. This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that ‘empty space’ in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitation potentials gμν)...

Einstein is talking about spacetime, Farsight. You know how I know? Because he refers to ten functions, and if he were talking about space, there would only be six. So either that's a bad translation, or - more likely - he was simply speaking loosely. Regardless, it's totally irrelevant what Einstein said. What matters is what his theory says.

Oh yes there is. See above.

No, there isn't. See above.

Oh really? When Einstein said a gravitational field was inhomogeneous space. Wherein die Ausbreitungs-geschwindigkeit des Lichtes mit dem Orte variiert. And thus the motion of light is curvilinear? And the interpretation of gravity as a curvature in space-time is an interpretation Einstein did not agree with? There's definitely an interpretation issue here.

There's a comprehension issue here.

 

[Vorpal]

Ooh, I was dumb. At first I had a similar idea, deriving the time-related components of Riemann curvature as proportional to a", initially thinking: aha, to vanish this, take a∝t. But then I dismissed it because I incorrectly fumbled the rest to be proportional to k + a'²/a² instead of (k + a'²)/a² as they should have been, and didn't even notice.

 

[W.D.Clinger]

NB: can you try to make your next post a little more succinct please.

I apologize to everyone for the length of this post. It looks to me as though Farsight has been sloppy in his terminology, speaking of "spacetime" when he really meant to say "space", or of FLRW solutions in general when he apparently meant to speak only of FLRW solutions in which 3-dimensional space is Euclidean. Fixing those misstatements wouldn't remove all of his errors, but it would help quite a bit, so I'm going to start by going through a severely abridged history of his recent posts to highlight uses of "spacetime", "FLRW", and other key terms he may wish to walk back.

[size=+1]Review[/size]

"The Friedmann–Lemaître–Robertson–Walker (FLRW) metric is an exact solution of Einstein's field equations of general relativity; it describes a homogeneous, isotropic expanding or contracting universe..."

The first is from Einstein’s 1920 Leyden Address, where he describes a gravitational field as inhomogeneous space. The second is from the wiki FLRW page, which describes a homogeneous isotropic universe. There’s no gravity in that universe. Gravitational fields have been thrown out with the bathwater.

When I read that, I had no way of knowing that Farsight was using the word "gravity" to mean "inhomogeneous space" or a Newtonian 3-dimensional gravitational field instead of the more modern notion (curved spacetime) that was introduced by Einstein, developed by Einstein's contemporaries such as Eddington and Weyl, and expressed nowadays by modern coordinate-free tensor notations instead of the coordinate-specific notations (such as affine connections) that were employed by Einstein and his contemporaries.

Furthermore, the FLRW solutions to which Farsight explicitly referred include entire families of solutions in which space is definitely not Euclidean, and cannot be regarded as flat by any sane definition. So I wrote:

False. Although FLRW solutions (note the plural) assume the universe is homogeneous and isotropic, they allow for matter and gravity.

Farsight reiterated his error:

I reiterate: if you assume that the universe is homogeneous and isotropic, there's absolutely no gravity in it, spacetime is flat.

That's flat-out wrong, even if we substitute "space" for "spacetime". There are entire families of FLRW (hence homogeneous and isotropic) solutions in which neither space nor spacetime is flat. Einstein invented one of those solutions himself: the Einstein static universe.

A concentration of energy results in curved spacetime. This energy has a mass equivalence. Matter exhibits the property of mass but mass doesn't cause spacetime curvature, so much as the energy content of matter. Now go back to that Einstein quote where he referred to a gravitational field as inhomogeneous space, understand that if the energy distribution is absolutely uniform, space is homogeneous, there's no gravitational field, spacetime is flat, and light travels in straight lines.

Farsight is repeating his claim that homogeneous space implies flat spacetime. Even if we substitute "space" for "spacetime", Farsight's claim is plainly incorrect, with Einstein's static universe serving as one of many FLRW counterexamples.

Of course I've seen them. The issue is in the interpretation, what the individual terms actually represent, what curved spacetime really is, and seeing the underlying simplicity.

Although we'd like to rescue Farsight here by substituting "space" for "spacetime", he was responding to Reality Check's observation that the curvature of spacetime appears explicitly in Einstein's field equations. In that context, we can't substitute "space" for "spacetime" without rejecting Einstein's equations, which Farsight has said he does not wish to reject.

As if to prove he does not understand Einstein's field equations, Farsight went on to write:

Let me try to get this across with a trivial example. You have a massive body in space. The energy present in this body "conditions" the surrounding space, the effect diminishing with distance, such that a light beam curves as it transits this space. Now add another massive body near to the first one, and shoot your light beam through the gap between them. It doesn't curve. However you can detect a Shapiro delay, so you can assert that spacetime is still curved. Now chop up your massive bodies into infinitesimal parts and distribute them evenly throughout the universe. Shoot a light beam through it, and that light beam doesn't curve. And you can't detect a Shapiro delay either. Forget about the expansion of the universe for a minute, and think about it. Light moves uniformly, in straight lines: your spacetime curvature has gone.

That example would make sense if we change "spacetime" to "space", but Farsight gave this example in the context of Einstein's field equations, which involve the curvature of spacetime rather than space. Furthermore, as sol invictus eventually pointed out, the example's Shapiro delay proves that curvature (of spacetime) can be detected even when light travels in Newtonian straight lines. Finally, the "forget about the expansion of the universe for a minute" is critical, because curvature would remain detectible if we were allowed to look at anything beyond a particular spacelike slice through spacetime.

I'm not misrepresenting them, I'm pointing out a clear issue. Here's the Einstein quote again followed by the quote from wiki:

”According to this theory the metrical qualities of the continuum of space-time differ in the environment of different points of space-time, and are partly conditioned by the matter existing outside of the territory under consideration. This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that ‘empty space’ in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitation potentials gμν)...

"The Friedmann–Lemaître–Robertson–Walker (FLRW) metric is an exact solution of Einstein's field equations of general relativity; it describes a homogeneous, isotropic expanding or contracting universe..."

...snip...

So you've got light moving uniformly in straight lines. Where's your curved spacetime gone? Space is homogeneous. There's no gravitational field. There is no gravity to make this universe collapse, even when it was much smaller, with a much higher energy density. And let's face it, if it had done, we wouldn't be here.

Although we can change Farsight's use of "spacetime" to say "space" instead, neither he nor we are allowed to make that change when we're quoting others. In the example above, Farsight quotes Einstein saying "space-time" and mentioning the 10 components of a 4-dimensional (spacetime, not space) tensor, but turns right around and scolds me for considering the curvature of spacetime instead of the curvature of a certain 3-dimensional slice of it. To top it off, Farsight himself wrote "spacetime".

Don't throw Humpty Dumpty logic at me, sol. If light travels in straight lines spacetime isn't curved, and that's it.

Once again: It's a fact that spacetime can be curved even if all light travels in straight lines. Farsight evidently meant to say "space" instead of "spacetime".

Unfortunately, he went on (in that same post, just one paragraph later) to scold sol invictus for speaking of something happening in "spacetime" (which happens to be an unambiguous and standard usage, unlike Farsight's):

Nothing happens in spacetime sol, because spacetime is an "all-time-view", a block universe where we draw a world line to represent the motion of a body through space. Things happen in space. It's space that's expanding, not spacetime. Again we see an issue over the distinction between curved space and curved spacetime. Have a read of the FLRW article on wiki and look out for it.

This curvature relies on the expansion of space, which at any one moment is flat and homogeneous and free of any overall gravity.

In that last sentence, Farsight appears to admit that spacetime can be curved even when a particular 3-dimensional slice through spacetime isn't. If Farsight had been willing to admit that most of his own statements about curved spacetime were rendered incorrect by his own sloppiness, there would be less to discuss and this would have been a shorter post.

Your definition of curvature seems to be elevating a manifold into something real rather than mathematical artefact. I think you’re confusing a spacetime manifold with physical space.

Is Farsight accusing me of confusing a 4-dimensional spacetime manifold with 3-dimensional physical space? Or is he accusing me of confusing a 3-dimensional slice of a 4-dimensional manifold with 3-dimensional physical space? Or is he accusing me of confusing a 4-dimensional manifold with a 4-dimensional physical spacetime?

Farsight's accusation was wrong in any case, because I have not confused any of those things, but I answered him as though he were accusing me of the last of those confusions.

[size=+1]Digression[/size]

Thanks. I think the issue with boards like this tends to be intellectual arrogance and dishonesty. Some people who think they understand relativity and cosmology cannot bear a hard evidential argument backed by cool clear logic. When their belief and expertise are challenged, they sometimes start getting nasty. I don't. My name (John Duffield) is public knowledge, and I have my reputation to think about.

Taking Farsight at his word, I expect he will acknowledge

• his many uses of the word "spacetime" when he apparently meant to say "space",
• his misrepresentations of the FLRW solutions and their diversity,
• the fact that Einstein's field equations are expressed using spacetime curvature, and do not mention the curvature of any 3-dimensional space, and
• the fact that the spacetime curvature that appears in Einstein's field equations is mathematically well-defined.

If Farsight will acknowledge those facts, along with the fact that his personal definition of gravity differs from the notion of Einsteinian gravity in common usage today, then we may be able to move on to a discussion of whether Farsight's personal definition of gravity coincides with Einstein's. In connection with that, I want to quote Vorpal:

Einstein said that the gravitational field is the connection, since he explicitly treated the connection coefficients as the components of the gravitational field. There's a good conceptual reason for this, since it explicitly dissolves any difference between gravity and inertia. But it doesn't affect Clinger's point here, since we're talking about curvature.

I'm really not seeing the difficulty here. Spacetime being curved is by definition having non-zero curvature. Since curvature is given by a tensor, you can't make it go away. If it's nonzero in some frame, it's nonzero in all frames. (Unlike the connection coefficients, which do not form a tensor, and can be transformed away [edit: locally]... which goes right to Einstein's point about things in freefall locally acting just like STR.)

Because the connection coefficients are not tensorial, identifying the connection with gravity leads to a coordinate-dependent notion of gravity, which goes against the spirit of general relativity. As Einstein wrote for a non-technical audience in the book that Farsight has been quoting (Relativity: The Special and the General Theory, italics as in the English translation by Robert W Lawson):

We refer the four-dimensional space-time continuum in an arbitrary manner to Gauss coordinates. We assign to every point in the continuum (event) four numbers, x₁, x₂, x₃, x₄ (coordinates), which have not the least direct physical significance, but only serve the purpose of numbering the points of the continuum in a definite but arbitrary manner. This arrangement does not even need to be of such a kind that we must regard x₁, x₂, x₃, as "space" coordinates and x₄ as a "time" coordinate.

...snip...

The following statement corresponds to the fundamental idea of the general principle of relativity: "All Gaussian coordinate systems are essentially equivalent for the formulation of the general laws of nature."

From Einstein's own words, we can infer that identifying gravity with the coordinate-dependent connection is less than ideal. It seems to me that Einstein probably would have preferred to use the more modern, coordinate-independent tensorial notion had the modern tensor notation been available at the time. Einstein used explicit connections because that was the notation of his time. This may be a case of notion following notation.

[size=+1]Summary of Farsight's errors[/size]

• You were flat-out wrong to deny that curvature is defined by the Riemann curvature tensor and its contractions (the Ricci tensor and scalar curvature).
• You were flat-out wrong to deny the curvature of spacetime in most FLRW solutions, including all "dusty" solutions.
• You were flat-out wrong to deny the existence of gravity in FLRW solutions.

...snip...

The following statements are mathematical:

• A non-zero Ricci tensor implies curvature.
• A non-zero scalar curvature implies a non-zero Ricci tensor, hence curvature.
• Most FLRW solutions, including all "dusty" solutions, imply curvature of the spacetime manifold.

When you deny those statements, you are flat-out wrong.

[size=+1]Farsight's response to my criticisms[/size]

I will now respond to Farsight's response to my criticisms.

When did mathematics, backed by citations to standard references, turn into "bible thumping"?

The bible thumping was you saying that I must be wrong because what I was telling you wasn't in agreement with Hawking and Ellis. That is bible thumping Clinger. Recognise it. Don't do it.

I was saying you were wrong because Einstein's field equations and their mathematical consequences (such as the FLRW solutions) don't agree with you. Rather than write out the derivation of those consequences (which would have made my post even longer!), I referred to standard references in which those consequences are worked out in painstaking detail.

As I mentioned in post #4914, I have found two minor errors in the equations presented by Hawking & Ellis. I suspect there are more, because I still haven't read the entire book and I could have missed some errors even within the equations I studied with some care. If you believe you have found one or more technical errors in Hawking & Ellis that could eliminate the obvious contradictions between your claims and the mathematical consequences of Einstein's field equations, then I urge you to report those errors here.

Until you do so, I will interpret your references to "bible thumping" as a quaint way of admitting you have no substantive criticisms of the mathematics that contradicts you.

The universe isn't a 2-dimensional surface. It's perhaps a "ball", which the common man would call a sphere. Mathematically speaking the sphere is the surface rather than the ball.

Let's speak mathematically. In the FLRW solutions, neither spacetime nor space can be a closed ball. For the solutions in which space is Euclidean, space is homeomorphic but not diffeomorphic to an open ball. For the solutions in which space is a 3-sphere, space is a 3-sphere, and a 3-sphere is not even homeomorphic to a ball.

Lots of people have trouble visualizing a 3-sphere, so I used a 2-sphere for my example. You can criticize my example for being 2-dimensional instead of 3-dimensional, but your criticism of its sphericality is incoherent and suggests (once again) that you are unfamiliar with the FLRW solutions you introduced into this discussion.

You have a bad habit of arguing as though there's some privileged coordinate system in which your beloved Euclidean space has been carved out of non-Euclidean spacetime, and that your argument automatically generalizes to all other coordinate systems.

There is, that of the rest frame of the CMBR. See CMBR dipole anisotropy. From this you can gauge your motion through the universe. Whilst this isn't technically an "absolute" reference frame, it doesn't get much more absolute than that.

Okay, I'll accept that. From now on, we can discuss the FLRW solutions using the standard decomposition into space and time that makes space homogeneous and isotropic. In some of the FLRW solutions, the canonical decomposition yields a non-Euclidean space. You have been arguing as though the canonical decomposition always yields a Euclidean space, which is flat-out wrong.

That unpublished arXiv paper supports what I wrote: Curvature is measured by the Riemann curvature tensor and its contracted forms (the Ricci tensor and the scalar curvature)...

Don't be evasive. Look at the last line of the abstract: The interpretation of gravity as a curvature in space-time is an interpretation Einstein did not agree with.

Yet Einstein chose to express his field equations in terms of the Ricci tensor and scalar curvature. Einstein must have thought the mathematical definition of curvature was more useful than the notion of gravity that Farsight would attribute to Einstein almost a century later.

According to Vorpal, who (like several other participants in this thread) knows more about this stuff than you and me combined, Einstein identified the gravitational field with the affine connection. As I explained above, Einstein should not have been comfortable with that identification, and several of his contemporaries had already adopted a more modern point of view by 1920. See for example

• Sir Arthur Eddington. Space, Time, and Gravitation. Cambridge University Press, 1920. (Chapter 5 uses the Riemann tensor, Ricci tensor, and scalar curvature to classify the curvature of spacetime.)
• Hermann Weyl. Raum, Zeit, Materie. 1921. (The title of section 15 is "Curvature", and Weyl uses affine connections and the Riemann tensor throughout the rest of the book.)

All of that's just a historical curiosity, however. The central fact is that the only curvature that appears in Einstein's field equations is the mathematical notion of curvature you have been rejecting. It is also a fact that the mathematical notion of curvature is used to classify the spatial component of an FLRW solution, and you have been rejecting that fact also by pretending that all FLRW solutions have flat space.

The technical error is in FLRW which starts with the assumption of homogeneity and isotropy of space and then allows Σ to range over a 3-dimensional space of uniform curvature, that is, elliptical space, Euclidean space, or hyperbolic space. It has to be Euclidean. Space is flat when it's homogeneous.

Unsupported. Semi-wrong. Flat-out wrong. Flat-out wrong.

Let me unpack that for you:

• "The technical error is in FLRW" part is unsupported because you have been unable to point to any technical errors in the FLRW solutions. Aside from your own bare assertion, we have no evidence of any technical error. When some random guy on the Internet is railing against a mathematical theory that was developed, reviewed, studied, and taught by many experts, and has been understood by generations of mathematicians and physicists, it would be a mistake to bet on the random Internet guy.
• "elliptical space" is semi-wrong because the only elliptical space that's homogeneous and isotropic is a 3-sphere. (Other 3-dimensional elliptical spaces are homeomorphic but not diffeomorphic to a 3-sphere.) You should have said "spherical space".
• "It has to be Euclidean" is flat-out wrong. Spherical space is not Euclidean, yet it yields a family of FLRW solutions. Einstein's steady state universe is an FLRW solution with spherical space, and it's just the most famous example of its type.
• "Space is flat when it's homogeneous" is flat-out wrong on several counts. Spherical space is homogeneous and isotropic but not flat. Elliptical space isn't isotropic, but it's homogeneous without being flat. There are many other examples, including some isotropic examples you forgot to include in your enumeration of FLRW spaces.

Spacetime is only "curved" in an arcane mathematical sense when space is expanding, because the homogeneity of space changes over time even though space in the universe remains homgeneous. Light doesn't bend round full circle and end up where it started.

As demonstrated by FLRW solutions with spherical space, including Einstein's steady state cosmology, space can be curved in the familiar spherical sense (albeit 3-dimensional instead of 2-dimensional) even when it is homogeneous and isotropic. In FLRW solutions with spherical space, including Einstein's steady state cosmology, a photon of light (travelling along a geodesic) does indeed end up where it started (speaking spatially, of course).

No. You, however, are confusing mathematics with subjective opinion.

You are definitely confusing a spacetime manifold with physical space. Space exists, that's what's out there. It's expanding, we have hard scientific evidence that tells us this. A spacetime manifold is an abstraction.

Yes, spacetime manifolds are an abstraction. I said so myself in a previous post.

I am not, however, confusing a spacetime manifold with physical space. As I explained in that previous post, the FLRW solutions are models (abstractions) of spacetime. The facts I have stated concerning those models are theorems of mathematics. You have been denying those mathematical facts, apparently because you are elevating your subjective opinions above the mathematical facts.

I'm making a distinction between mathemmatical abstraction and reality.

As am I, except I'm getting that distinction right and you aren't.

Go look at the Ricci curvature tensor, it relates to volume change. The universe is expanding. And in that universe space is homogeneous, at all times. And since a gravitational field is homogeneous space, there is no overall gravity in the universe.

The Ricci tensor is defined on a mathematical abstraction. When you say the universe is expanding, I don't know whether you mean the mathematical abstraction of space is expanding (as in some but not all FLRW models) or that the physical universe is expanding. When you say space is homogeneous in that universe, at all times, I have to assume you're talking about a mathematical abstraction, because the physical universe is definitely not completely homogeneous.

You then insert your subjective opinion ("since a gravitational field is homogeneous space", which probably wasn't even the opinion you intended to express), and conclude that there is no overall gravity in the universe. Since the physical universe in which I live has something that even you might accept as gravity, I have to conclude that your conclusion there was supposed to be about some mathematical model of the universe, not the physical universe. Since you are using the word "gravity" in some subjective sense that you attribute to Einstein, you appear to be making a subjective statement about a mathematical model, which seems pointless.

Let me put it this way: If you want to define "gravity" as a synonym for "homogeneous space" (which is what you wrote above) or "inhomogeneous space" (which is more likely to resemble what you intended to write), then you're free to do so. (Having done so, however, you shouldn't blame anyone but yourself for the confusion that's certain to follow from your redefinition of a technical word that most people (even non-experts!) interpret as something rather different from what you mean by it.) Once you've redefined "gravity" to mean "homogeneous" (or "inhomogeneous") space, then your argument basically boils down to this: All FLRW models have a homogeneous space, so they have "gravity" (in your personal sense above) or don't have "gravity" (in the personal sense you may have meant).

You could have saved yourself and everyone else a bunch of grief by saying that clearly and unambiguously at the outset.

The following statement is your subjective opinion:

• Curvature of the spacetime manifold does not imply gravity.

The universe didn't contract when the matter-energy density was high, now did it. That's not my subjective opinion.

Are you suggesting that a baseball thrown upward becomes subject to gravity only after its vertical velocity changes sign?

Einstein's static universe, in which space is spherical, is one of the FLRW solutions. Extending the lane indefinitely brings it back to the bowler's feet.

And WMAP has demonstrated that space is flat. So you can discard that solution.

WMAP has not demonstrated that all of the FLRW solutions have flat space. Experimental results cannot disprove mathematical theorems; they can, however, show that some of the mathematical models are not realistic.

You, however, have been denying and continue to deny the mathematical fact that some of the FLRW solutions have spherical (or some other non-flat) space. That's yet another example of your failure to distinguish mathematical models from physical reality.

Spinning a yarn is no substitute for doing mathematics.

You've been spun a yarn, Clinger. Now come on, think it through: start with a high-energy-density universe. It expands, reducing the energy density if you adhere to conservation of energy. What happens next? It expands, reducing the energy density. What happens next? It expands, reducing the energy density.

Spinning a yarn is no substitute for doing mathematics.

The change in energy density over time is given by Hawking & Ellis equation (5.10). As I have explained, the rate of expansion decreases with time. Whether that rate of expansion will ever drop below zero depends upon the initial conditions; the equations that determine whether that will happen are well known, and can be found within standard references.

Your informal argument above is similar to an informal argument that a moon rock thrown upward will never fall back to the moon. In reality, whether the rock falls back to the moon depends upon how hard you throw it: Is its initial velocity above, equal to, or less than its escape velocity?

That's a quantitative question. Referring to quantitative mathematics as "bible thumping" isn't going to advance your understanding of physics.

Like I said, you've been spun a yarn. Come on Clinger, think it through. Think for yourself. Don't let Hawking do your thinking for you.

If I were content to let Hawking think for me, would I have found those two minor errors in his equations?

How many errors have you found, Farsight? You have yet to identify a single technical error in any "bible" I've cited.

 

[Farsight]

I wondered what this was about, and based on the abstract, yes, that's basically right (no idea about the content, though). I've posted a quote and translation about Einstein's interpretations of what the 'gravitational field' is in the past.

I read your translation, it seems reasonable, not that I can read German very well. It doesn’t cover much though.

But it doesn't mean what you think it means. The Riemann curvature tensor has a very clear meaning: it's the gravitational tidal forces. That's what geodesic deviation means, and it is a completely objective, coordinate-independent effect. So if we're talking about whether something is curved, the above considerations change nothing.

No problem. Yes, you can't make a real gravitational field go away. There is no such thing as a uniform gravitational field. There's no way you can move when falling into a stellar black hole that will stop tidal forces ripping you apart.

Einstein said that the gravitational field is the connection, since he explicitly treated the connection coefficients as the components of the gravitational field. There's a good conceptual reason for this, since it explicitly dissolves any difference between gravity and inertia.

Do you mean active gravitational mass is the same as inertial mass? If so I’m happy with that.

But it doesn't affect Clinger's point here, since we're talking about curvature.

The real issue at the moment is that people are accustomed to thinking of spacetime as if it's physical space and vice versa.

I'm really not seeing the difficulty here. Spacetime being curved is by definition having non-zero curvature.

There’s a distinction to make. If you fall into a supermassive black hole, the tidal forces are weak so you don’t get spaghettified. The gravitational field is so nearly uniform that locally it looks uniform. Think of the bowling ball analogy. The gradient at your feet is so similar to the gradient at your head, that there’s no curvature between the two gradients. Just like a man falling off a ladder, you don’t feel a tug at your feet, but you still fall in, and passing light still bends. In extremis you’ve still got curved spacetime even without Reimann curvature. I think I read somewhere that the gedanken situation for this is falling towards a massive plane.

Since curvature is given by a tensor, you can't make it go away. If it's nonzero in some frame, it's nonzero in all frames.

Agreed. You can't make a real gravitational field go away.

(Unlike the connection coefficients, which do not form a tensor, and can be transformed away [edit: locally]... which goes right to Einstein's point about things in freefall locally acting just like STR.)

In an infinitesimal region. I never liked this.

If one were to take Einstein's conceptualization seriously, then it follows trivially that every spacetime has gravity.

I don’t follow that. If you are surrounded by homogeneous space you don't fall down. If the space remains homogeneous but is expanding you can say that spacetime is curved, but there's no actual gravity in the space. The gravitational potential is the same everywhere. It varies over time, but it's always the same above you and below you. You don’t fall toward the centre of the universe.

Even the completely flat Minkowski spacetime, because sure enough, there's inertia there and a connection defining the geodesics. So it makes little sense to invoke Einstein to say that some spacetime has no gravity because according to him, every spacetime does.

As above. The moot point is that homogeneous space has no gravitational field that causes objects to move towards the centre. Or indeed space. If space is expanding you can say it's inhomogeneous over time, and that the spacetime is curved. But it's like the whole of space is like the space you go through when you zoom through the gap between two stars. There's gravitational potential in spades. But you don't fall down. Your path is straight. Divide up the energy of those two stars and smear it evenly thoughout a smaller universe. There's gravitational potential in spades. But you don't fall down, and nor do galaxies.

And it doesn't really affect the disagreement here, because what was discussed was curvature anyway.

Noted.

 

[W.D.Clinger]

errata (part 1)

In my long post of yesterday evening, both of my statements about diffeomorphisms were incorrect. I was thinking of isometries, not diffeomorphisms. Before those brain farts can cause even more confusion, here are my corrections (highlighted):

Let's speak mathematically. In the FLRW solutions, neither spacetime nor space can be a closed ball. For the solutions in which space is Euclidean, space is homeomorphic but not and also diffeomorphic (but not isometric) to an open ball. For the solutions in which space is a 3-sphere, space is a 3-sphere, and a 3-sphere is not even homeomorphic to a ball.

"elliptical space" is semi-wrong because the only elliptical space that's homogeneous and isotropic is a 3-sphere. (Other 3-dimensional elliptical spaces are not globally isotropic, although they are homeomorphic but not and also diffeomorphic (but not isometric) to a 3-sphere.) You should have said "spherical space".

I'm aware of several less obvious technical errors in that same post, but I don't think they're worth correcting at this time.

 

[Farsight]

Before Michael Mozina and any other lurkers who've been following along can jump on this, let me point out that the spacetime curvature is zero even though the "SPATIAL curvature" is negative. Because the space is empty, the negative spatial curvature is nothing little more than a mathematical artifact of the Friedmann equations and their conventions for classifying the spatial geometry.

Perhaps a thought expariment might help you with this. Think of space as being the same thing as energy. Imagine it's something like transparent elastic jelly. Start with 100 litres of this elastic jelly under intense pressure in a spherical glass tank. Now add a local concentration of energy by injecting 100cc more jelly with a hypodermic that deftly goes through the glass and repairs it upon withdrawal. You've created a pressure gradient around the injection site. Now send a small mechanical pulse through the jelly. Where the pulse instantaneously is, space is displaced. This is curved space. The derivative of this curvature is a sinusoidal waveform, the electric aspect being the slope, the magnetic aspect the rate of rotation. The pulse travels straight through the jelly until it encounters the pressure gradient, whereupon it starts to veer and travel in a curved path. This is curved spacetime. Another thought experiment is where you don't bother with hypodermics and pulses. You start with 100 litres of elastic jelly under intense pressure. And no glass tank.

 

[W.D.Clinger]

This is EXACTLY why I have never tried to "bark math" or "argue math" with you folks!

Even when someone comes here that's willing to "speak your language" and look into the mathematics, you belittle and berate them every single step of the way.

That's a good way of putting it. Farsight, like you, is merely barking math; he thinks that throwing out a few equations will make him sound like he's "speaking our language". Like you, he is not using math to find out whether his hypotheses are right or wrong.

It's like a stranger walking into a big band rehearsal.

S: "You guys are terrible. I'm your new first chair."
B: "Is that so? Let's hear it. Take the first solo on 'Beguine'. One, two ..."
S: "Wait, wait, I don't have this 'sax solo', I have an interpretive dance." <dances>
B: "That dance is terrible, but if it inspires a good solo then you're in. Let's hear the solo."
S: "But my dance is already the best possible sax solo." <begins dancing. Appears incompetent.>
B: "Baloney. Let's hear something."
S: "I don't bark on command." <dances> "But I'm clearly the new first chair."
B: "This is a band. The first chair needs to play sax solos."
S: "OK, blowing air into metal tubes is your membership totem? Fine. I'll play along. hissssss HONK SPLAT SQUEEEEEEONK BLAAAT. There, see? I can play sax." <dances>
B: "Ha ha ha ha! Go away."
S: "First you tell me to play music, and when I do you're all critical! This is why I don't bother with the music."

Nominated.

 

[sol invictus]

Perhaps a thought expariment might help you with this. Think of space as being the same thing as energy. Imagine it's something like transparent elastic jelly. Start with 100 litres of this elastic jelly under intense pressure in a spherical glass tank. Now add a local concentration of energy by injecting 100cc more jelly with a hypodermic that deftly goes through the glass and repairs it upon withdrawal. You've created a pressure gradient around the injection site. Now send a small mechanical pulse through the jelly. Where the pulse instantaneously is, space is displaced. This is curved space. The derivative of this curvature is a sinusoidal waveform, the electric aspect being the slope, the magnetic aspect the rate of rotation. The pulse travels straight through the jelly until it encounters the pressure gradient, whereupon it starts to veer and travel in a curved path. This is curved spacetime. Another thought experiment is where you don't bother with hypodermics and pulses. You start with 100 litres of elastic jelly under intense pressure. And no glass tank.

That has absolutely nothing to do with what W.D.Clinger was talking about. Literally absolutely nothing. You quite clearly have no idea what that was about.

You know what the correct analogy is? Spherical coordinates in ordinary 3D Euclidean space. In spherical coordinates, the metric is

[latex]$ds^2=dr^2 + r^2 d\Omega_2^2$[/latex],

where d\Omega is the area element on a 2-sphere. Hey look - that's "curved space", since spheres are curved! And indeed, the curvature of the particular 2 dimensional spaces that I've chose to foliate 3-space with is not zero. But obviously this does not mean that the 3D curvature isn't zero. An almost identical choice of coordinates in 4D gives the metric Clinger and I were discussing.

I know that is going to go completely over your head, but it might be of interest to someone else.

 

[sol invictus]

Ya know.....

This is EXACTLY why I have never tried to "bark math" or "argue math" with you folks!

Even when someone comes here that's willing to "speak your language" and look into the mathematics, you belittle and berate them every single step of the way.

In the case of Farsight, he's posted very little math. The few equations he posted weren't wrong, but simply irrelevant.

You automatically ignore the beliefs of the folks that wrote the formulas you're talking about too.

In many cases we understand the formulas better than those that wrote them down originally. That's certainly true of Maxwell, and it's probably true of Einstein as well (for instance, I know a lot about GR that wasn't understood until after Einstein died).

The great thing about physics and math is that beliefs aren't very important, because physics and math exists outside of individual humans. So for instance if Einstein said something wrong about GR it really doesn't matter, because others later can correct his error.

You really don't give one hoot about the math. You simply use it as a "weapon", as a way to belittle and put down individuals. That's exactly what I expected. Thanks for demonstrating exactly how pointless it is to attempt to publicly discuss mathematics with you folks. Wow!

That's total nonsense, MM. GR is a mathematical theory. It's a set of equations. How can you discuss a set of mathematical equations without ever using math?

 

[Michael Mozina]

In the case of Farsight, he's posted very little math. The few equations he posted weren't wrong, but simply irrelevant.

FYI, my criticism was not directed at you or edd. I think you and edd have been the only ones attempting to debate the points "cleanly", in your case through GR proper. I have no problem with that. The band-aid comment was quite entertaining from my perspective.

In many cases we understand the formulas better than those that wrote them down originally. That's certainly true of Maxwell, and it's probably true of Einstein as well (for instance, I know a lot about GR that wasn't understood until after Einstein died).

Well, maybe. In your particular case, as it applies to GR theory and the MATHEMATICAL aspects of GR theory specifically, I'm certain that you're right. In terms of the conceptual aspects, I'm not sure.

In the case of Alfven, I seriously doubt your statements are applicable. AFAIK you're the only one I've met here that seems to have an excellent understanding of subatomic energy exchanges and EM transfers of energy. The rest of this crew seem pretty oblivious to subatomic physics if you ask me.

The great thing about physics and math is that beliefs aren't very important,

Well, yes and no. Do you believe it's 'ok' to put "God" into some constant in a GR formula?

because physics and math exists outside of individual humans.

Well, the "interpretation" can be quite subjective if you ask me.

That's total nonsense, MM. GR is a mathematical theory. It's a set of equations. How can you discuss a set of mathematical equations without ever using math?

Well, for starters I can question the empirical legitimacy of stuffing "magic" into one of those constants, right?

 

[sol invictus]

Well, yes and no. Do you believe it's 'ok' to put "God" into some constant in a GR formula?

Huh?

Well, for starters I can question the empirical legitimacy of stuffing "magic" into one of those constants, right?

I've never understood what you mean by that, Michael. Honestly - I really have no idea. I think that if you understood the theory a little better, you'd realize that what you're saying here doesn't make sense, but I'm not even sure of that.

The cosmological constant is a parameter in Einstein's theory. It's a number that enters into his equations, very similar to Newton's constant G. Both of those numbers have to be fixed by observation, since the theory doesn't tell you what they are. So you go out and do observations, you determine the best-fit values for those two parameters, and that's your theory. Then you can test the theory by doing more observations and seeing if they match the predictions of the theory.

That's really all there is to it - there's no "magic".

 

[Reality Check]

AFAIK you're the only one I've met here that seems to have an excellent understanding of subatomic energy exchanges and EM transfers of energy. The rest of this crew seem pretty oblivious to subatomic physics if you ask me.

Then your opinion is really, really wrong.
ben m for example works with subatomic physics every day.
I certainly have a better knowledge of subatomic physics than you (even if my formal educaiton in it is a couple oif decades old!).

 

[Farsight]

I was looking through the thread and this caught my eye. I thought it was worth a quick out-of-order response:

You got something right. A nice bit of history is that Einstein made a comment in a book that suggests that he thought that the speed of light could vary in GR. That was physically valid but is not the modern interpretation.

Take a look at this physicsworld report on a super-accurate optical clock. It's so precise that you can see two of these clocks losing synchronisation when they're separated by only a foot of vertical elevation. Now take a look at wiki re time dilation, and note the bit that says consider a simple clock consisting of two mirrors A and B, between which a light pulse is bouncing. When you simplify those precise optical clocks to parallel-mirror light clocks, when they lose synchronisation this is what's happening:

|~~~~~~~~~~~~~|
|~~~~~~~~~~~~~|

The modern interpretation will tell you that these two racehorses finish in a dead heat. Don't swallow it RC, look to the evidence instead. And read A World Without Time. The point is that whether it's a pendulum clock, a quartz watch, or an atomic clock, clocks clock up some kind of regular motion, not "the flow of time", which is merely a figure of speech. So when the motion goes slower, the clock goes slower. This is true for the NIST caesium fountain clock too.

Think it through.

 

[Michael Mozina]

Then your opinion is really, really wrong.
ben m for example works with subatomic physics every day.

You're right. Ben's ok too when he sticks to the physics.

I certainly have a better knowledge of subatomic physics than you (even if my formal educaiton in it is a couple oif decades old!).

Pffft: "Electrical discharges in plasma? What discharges?" Please!

 

[Reality Check]

I was looking through the thread and this caught my eye. I thought it was worth a quick out-of-order response:

Take a look at this physicsworld report on a super-accurate optical clock. It's so precise that you can see two of these clocks losing synchronisation when they're separated by only a foot of vertical elevation.
...wikireference not needed...
The modern interpretation will tell you that these two racehorses finish in a dead heat. Don't swallow it RC, look to the evidence instead. And read A World Without Time. The point is that whether it's a pendulum clock, a quartz watch, or an atomic clock, clocks clock up some kind of regular motion, not "the flow of time", which is merely a figure of speech. So when the motion goes slower, the clock goes slower. This is true for the NIST caesium fountain clock too.

Think it through.

Thought it through and it is obvious that Farsight should never bet on racehorses !
The modern interpretation will tell you that one racehorse finishes in a dead loss.

The point is that we measure time using clocks.
The physical fact is that when the motion goes faster, the clock goes slower. This is basic special relativity.
Don't swallow it Farsight, look to the evidence instead:
Experimental basis of Special Relativity

Tests of Time Dilation and Transverse Doppler Effect
...
Review Article

• G. Gwinner, “Experimental Tests of Time Dilation in Special Relativity”, Mod. Phys. Lett. 1, 20, no. 11 (2005), pg 791.A general review article.

The Ives and Stilwell Experiment

• H.E. Ives and G.R. Stilwell, “An Experimental Study of the Rate of a Moving Atomic Clock”,J. Opt. Soc. Am. 28 pg 215–226 (1938); JOSA 31 pg 369–374 (1941).This classic experiment measured the transverse Doppler effect for moving atoms.
• Otting, Physik. Zeitschr. 40, 681 (1939).-
• Hasselkamp et al., Z. Physik A289 (1989), pg 151.A measurement that is truly at 90° in the lab. Agreement with SR to an accuracy of a few percent.

See also Mandelberg and Witten.
Measurements of Particle Lifetimes

• Rossi and Hoag, Physical Review 57, pg 461 (1940).Rossi and Hall, Physical Review 59, pg 223 (1941).Rasetti, Physical Review 60, pg 198 (1941).Redei, Phys. Rev. 162 no. 5 (1967), pg 1299.Various measurements of the lifetimes of muons.
  See also: Bailey et al.
• Durbin, Loar and Havens, Physical Review 88, pg 179 (1952).-
• D. Frisch and J. Smith, “Measurement of the Relativistic Time Dilation Using Mesons”, Am. J. Phys. 31 (1963) 342.Measurements of the lifetimes of pions. An interpretation was given by: Terell, Nuovo Cimento 16 (1960) pg 457.
• Greenberg et al., Phys. Rev. Lett. 23 no. 21 (1969), pg 1267.More accurate measurement of pion lifetimes.
• Ayres et al., Phys. Rev. D3 no. 5 (1971), pg 1051.Measurements of pion lifetimes, comparison of positive and negative pions, etc.
• Burrowes et al., Phys. Rev. Lett. 2 (1959), pg 117.Measurements of Kaon lifetimes.

Doppler Shift Measurements

• Kaivola et al., Phys. Rev. Lett. 54 no. 4 (1985), pg 255.McGowan et al., Phys. Rev. Lett. 70 no. 3 (1993), pg 251.They compared the frequency of two lasers, one locked to fast-beam neon and one locked to the same transition in thermal neon. Kaivola found agreement with SR's Doppler formula is to within 4×10−5; McGowan within 2.3×10−6.
• Hay et al., Phys. Rev. Lett. 4 (1960), pg 165.A Mössbauer absorber on a rotor.
• Kuendig, Phys. Rev. 129 no. 6 (1963), pg 2371.A Mössbauer absorber on a rotor was used to verify the transverse Doppler effect of SR to 1.1%.
• Olin et al., Phys. Rev. D8 no. 6 (1973), pg 1633.A nuclear measurement at 0.05 c, in very good agreement with the prediction of SR.
• Mandelberg and Witten, Journal Opt. Soc. Amer. 52, pg 529 (1962).Measured the exponent of the quadratic Doppler shift to be 0.498±0.025, in agreement with SR's value of ½.

 

[Michael Mozina]

Huh?

I've never understood what you mean by that, Michael. Honestly - I really have no idea. I think that if you understood the theory a little better, you'd realize that what you're saying here doesn't make sense, but I'm not even sure of that.

The cosmological constant is a parameter in Einstein's theory. It's a number that enters into his equations, very similar to Newton's constant G. Both of those numbers have to be fixed by observation, since the theory doesn't tell you what they are. So you go out and do observations, you determine the best-fit values for those two parameters, and that's your theory. Then you can test the theory by doing more observations and seeing if they match the predictions of the theory.

That's really all there is to it - there's no "magic".

That all sounds quite logical. The problem from my perspective is that you folks aren't actually deriving that number from direct observation, rather that number comes from a subjective "interpretation" of data, typically of redshift data.

http://arxiv.org/PS_cache/arxiv/pdf/1109/1109.6571v1.pdf

There are typically multiple ways to interpret that very same data. What makes a "dark energy" interpretation preferable in your mind, particularly since you can't even tell me where it comes from?