Black holes

[DeiRenDopa]

The second as-yet-unaddressed of Farsight's posts.

It would seem that you have rather idiosyncratic notions of what GR is, perhaps because you don't quite understand how the math relates to material things like clocks, microwave cavities, and rooms? Or perhaps it's a misunderstanding of how objective, independently verifiable experimental and observational results can be - and are - related to theories in physics (like GR)?

Nope. I've read the original GR, and the scientific evidence squares with it. However it doesn't square with what people say GR says.

For my own purposes, I shall call "what Farsight says is the original GR that Farsight has read" Farsight's GR, FGR for short.

Point of clarification: do Wald and/or MTW contain the "what people say GR says"?

For example, you seem to think that two clocks, separated by a foot or so in elevation are (must be?) in the same reference frame; i.e. that they can both measure 'local' time and that the 'local' is the same.

No. They're just two clocks at different elevations. They're in this room. Or if you prefer, they're in space near a planet. The things we call reference frames are "artefacts of measurement" that have no physical existence.

The use of double quotes, as in ""artefacts of measurement"", by convention, means you are giving this term a meaning that is non-standard, one that is different from what a reader would normally infer.

What do you intend this term to mean, in this context?

And those clocks don't actually measure time. There is no time flowing through those clocks. Can you see it whooshing through? No. What they actually do is clock up some form of regular motion, and display a cumulative total that you call the time.

We most certainly need to get to a clear, mutual understanding of this!

I take a hard-headed approach: time is what a clock measures. The unit of time is the second, which is defined in the relevant SI standard. Two clocks are the same - for the purpose of measuring time - if they agree, when co-located. For me, the rest is philosophical fluff that we would all be better off without.

But, if what clocks "actually do is clock up some form of regular motion", according to you, does that mean that you cannot - even in principle - build a clock based on the nuclear decay of an unstable isotope (to take one example)?

If so, then what are the objective, independently verifiable experimental and/or observational results which are inconsistent with GR?

Hawking radiation.

I wasn't aware that there are objective independently verifiable experimental and/or observational results about Hawking radiation.

Can you cite some please?

So far all I've seen you present is "optical clocks at different elevations don't stay synchronized" (that's a shorthand).

You've also alluded to "the GPS clock adjustment and the Shapiro delay"; what else?

Gravitational lensing.

Noted.

But my question was, and still is, what is it about vacuum impedance that "mentioned before now"? How is it relevant?

Impedance is an electrical property of say a cable, but it applies to space too, which electromagnetic waves propagate through.

How does one measure the impedance of space? And is the impedance of space the same as vacuum impedance?

It applies to alternating current rather than direct current, these both being associated with conduction current, which is the motion of charged particles. You can create such charged particles via pair production, and get the electromagnetic waves back again via annihilation. Those electromagnetic waves are displacement current rather than conduction current, and they wave. They're alternating.

I thought it was photons which created charged particles via pair production, and that particle-antiparticle annihilation produces photons.

And to study photons, we need to use QED, rather than Maxwell's equations.

What am I missing?

 

[ben m]

=
Nope. I've read the original GR, and the scientific evidence squares with it. However it doesn't square with what people say GR says.

Scientific evidence, eh? You think the evidence agrees with F-Einstein and disagrees with Wheeler? Be more specific.

a) Perihelion of mercury.
b) Gravitational redshift.
c) Gravitational lensing.
d) Shapiro delay.
e) The Nordtvedt effect in (e.g.) Apollo lunar laser ranging
f) the Lense-Thirring effect, seen by Gravity Probe B and LAGEOS
g) Binary pulsar spindown


Are you saying that the experimental evidence on these points agrees with "Einstein's original method" and disagrees with relativity-as-it-is-taught?

Sorry, Farsight. When you read a Gravity Probe B paper, and you see a plot showing the data points, and a plot showing the GR prediction, that prediction was generated from modern relativity---MTW-style relativity. If you think MTW is doing something "wrong" for some reason, then you're in trouble---because whatever MTW is doing is the thing that's actually out in the field passing experimental tests.

Unlike you, Farsight, I actually know these people, personally. I know experimental-gravity people at UW, MIT, UCSD, Harvard. Do you think they're out there learning crackpot-Farsight-relativity, bowing down before Schwarzschild coordinates, and devising experimental tests for that? No, they're learning mainstream general relativity theory, including the undergrad-level idea that coordinate-transformations are valid, and testing *that*.

 

[DeiRenDopa]

Third of three.

You confirmed pretty much what I suspected; it seems you don't really understand how the cesium fountain clock works, in terms of determining, as a standard, one second, do you?

Yes I do.

I'll guess that you have no problem with "maximizes their [the cesium atoms'] fluorescence", but you do seem to have difficulty with "a microwave frequency is found", as in "the microwave signal in the cavity is tuned to different frequencies".

I don't have any difficulty with it at all. I explained how it's like you twiddling the knob on your radio.

Sorta.

So let's start with this: what do you think a "microwave cavity" is? And how do you think you "tune" the microwave signal in such a cavity to different frequencies? To quote you, "this one is hugely important".

No it isn't. Everybody knows what a microwave cavity is, you've got one in your kitchen. Don't waste my time with evasion, go and look at what the NIST clock actually does, and go and think through the issue of finding a frequency when you're defining the second. It's a counting exercise, not some clever circular magic.

No, not really.

I'm going to table this disagreement for now, and focus on your proposal concerning how to measure the speed of light, using a distant pulsar.

Um, you do realize, don't you, that the ideas you express, in your posts in the threads in this part of JREF (at least the ones I'm actively participating in, on GR) are at considerable variance to standard, textbook GR?

Yes. Those textbooks are wrong in some respects. Have a read of the Golden age of general relativity and note the "paradigm shifts". I'm shifting them back to Einstein's original. Hacking through the thicket to the sleeping beauty and all that.

Noted.

That being so, if you wish others to understand you - and that's why, or one important reason why, you post here, right?

I tend to post here to correct some of the pseudoscience bandied about. There's a degree of "Emperors New Clothes" to it, wherein there's no supporting evidence, and if anybody probes deeper they essentially get told you don't understand the maths dear boy. I started posting on this thread because sol was trotting out the waterfall analogy, which paints a picture of a black hole as an object surrounding by inward moving space, flowing like some aether. As it happens Max Tegmark appeared in a Horizon program in front of a waterfall promoting this analogy.

Noted.

then those who seek to understand you need to be confident your understanding of key terms you use is the same as their own, right?

No. You need to be confident that the scientific evidence I present is correct, and then you should be confident in your own ability to think for yourself and follow the reasoning I offer.

That's strange.

You start with "No", but end up agreeing with me!

I can't "follow the reasoning [you] offer" if I am unsure of the meaning of key terms you use in that reasoning.

There are several examples of this already, in this thread alone, for just me. For example, your notions of what clocks are, and how the microwave cavity works in the NIST cesium fountain clock. Others have pointed out that your use of 'speed of light' is, at times, inconsistent (and that's just one example). You apply classical electromagnetism to quantum phenomena (e.g. pair production and annihilation). And so on.

If you find an issue we'll discuss it,

OK.

From our exchanges so far I've learned that you have some very different understandings of key terms and concepts than what I find in standard textbooks. To be sure I do not misunderstand you, I will - often - ask you to define key terms, in your own words. If you reply by pointing to definitions that are standard (or nearly so as never mind), I can be confident of at least that commonality in our mutual understanding.

I don't mind putting some time into this, but if it turns into evasion and distraction on your part, forget it.

Fair enough.

May I take it that the reverse is also acceptable to you? If I find our exchanges turning into evasion and distraction on your part, forget it?

Now that I know - with some degree of certainty - what you mean by "the parallel-mirror light clock", I can proceed to try to understand other things you've posted.

You shouldn't have needed any explanation from me about the paralel-mirror light clock.

Last time I checked, I was not you.

So what I need, and don't need, is not really within your power to determine, is it?

And it's even less in your power to determine what I should need (or should not need)!

But if the word pictures all produce the same results - in terms of quantitative matches between theory and experiment - they're equivalent, right?

No. Have a read of The Other Meaning of Special Relativity by Robert Close. The maths is the same and the quantitative predictions are the same. But the different interpretation delivers understanding that directs endeavour and facilitates scientific progress.

I've not read it, but I realize that we're talking at cross-purposes.

For some people, the word pictures (etc) are extremely important.

For example, they matter a great deal to teachers.

 

[tensordyne]

You stated "Incorrect, and I already did. u = x and v = (-1)^(1/4) y with x and y Cartesian." That's wrong, and I showed you why.

u = x
v = (-1)^(1/4) y

du = dx
dv = (-1)^(1/4) dy

Metric:

u² du² + v² dv²
Upon substitution of u and v one gets

(x)² (dx)² + ((-1)^(1/4) y)² ((-1)^(1/4) dy)²
which equals when noticing that ((-1)^(1/4))^4 = -1,

x² dx² - y² dy²,

as originally advertised. I believe sol invictus that you have made a calculo-algebraic math error.

I will deal with the issue of v being complex valued in the response to the segment below.

In this latest attempt, you've moved the goal posts. Now x and y are not Cartesian. In fact, they're not real. And despite appearances, the metric above in terms of x and y is not pseudo-Euclidean, it's Euclidean, because y is restricted to a particular line in the complex plane such that the distance element is positive definite.

There is faulty logic in the sentences above. I gave the form of a metric in the u,v coordinate system. I did not specify at first in any way what u and v where. I then gave what they were in that they had the relationship to the Cartesian x,y as given. v being from a subset of the complex numbers is also not a problem. Requiring that v be real is an unnecessary restriction.

How do I know this? Well, you earlier failed to specify the range of u and v (despite being told repeatedly that you needed to). With nothing stated to the contrary, the universal convention is that the coordinates in the metric are real (they cannot be simply complex, else the metric itself isn't real). This has nothing to do with how u and v were "originally" (whatever that means) determined. It simply has to do with their range, which as I've told you from the beginning is part of the metric.

The point I was making did not require ranges for p,q or u,v or whatever it was that the coordinate system was under discussion was. If that is what you want it, OK, here is one:

"What is the shape (not topology) of the following region?"

ds² = dp² + p² dq²
Set of region = {(p,q): 1 < p < 2, 1 < q < 2}
Set of space = {(p,q): 0 < p < 3, 0 < q < 3}

I think this meets your requirements, so let me know the shape if you can. The rest is similar complaints. Maybe I will make a post for the more mathematically literate members of the forum on charts, atlases, Differential Geometry and Cartography.

Everyone take it easy.

 

[sol invictus]

u = x
v = (-1)^(1/4) y

du = dx
dv = (-1)^(1/4) dy

Metric:

u² du² + v² dv²
Upon substitution of u and v one gets

(x)² (dx)² + ((-1)^(1/4) y)² ((-1)^(1/4) dy)²
which equals when noticing that ((-1)^(1/4))^4 = -1,

x² dx² - y² dy²,

as originally advertised. I believe sol invictus that you have made a calculo-algebraic math error.

Not in the slightest - that's obvious, and in complete agreement with what I told you.

There is faulty logic in the sentences above. I gave the form of a metric in the u,v coordinate system. I did not specify at first in any way what u and v where. I then gave what they were in that they had the relationship to the Cartesian x,y as given. v being from a subset of the complex numbers is also not a problem. Requiring that v be real is an unnecessary restriction.

For the third time, your x and y coordinates are not Cartesian. If they were, by definition the metric would be dx^2+dy^2. As for "Requiring that v be real is an unnecessary restriction", see my comment above.

The point I was making did not require ranges for p,q or u,v or whatever it was that the coordinate system was under discussion was. If that is what you want it, OK, here is one:

"What is the shape (not topology) of the following region?"

ds² = dp² + p² dq²
Set of region = {(p,q): 1 < p < 2, 1 < q < 2}
Set of space = {(p,q): 0 < p < 3, 0 < q < 3}

I think this meets your requirements, so let me know the shape if you can.

Yet again, as part of the metric you have to specify the range/identifications/boundary conditions for p and q. Assuming q is identified with q+2\pi, the first region is a section of an annulus (two straight, radial sides, and two sides that are arcs of concentric circles) and the second is a slightly less than half a disk (a pie with a very big slice remaining).

If q is not identified and runs from -infinity to infinity, it's more interesting. Among other things, there's an infinite conical singularity at the origin.

 

[Ziggurat]

"What is the shape (not topology) of the following region?"

ds² = dp² + p² dq²
Set of region = {(p,q): 1 < p < 2, 1 < q < 2}
Set of space = {(p,q): 0 < p < 3, 0 < q < 3}

If q wraps around, I believe you'll have something that looks kind of like a trumpet flower, though I haven't tried hard to figure out exactly what the differences in range will do. If it doesn't wrap around... cut open the trumpet flower.

ETA: never mind, I was thinking the wrong power of p in the second term.

 

[W.D.Clinger]

I'm afraid I have to agree with sol invictus and Ziggurat.

since you claim you can tell what the geometry is just given the form of a metric, here is one.

ds² = u² du² + v² dv²
So what is the geometry like in this case?

This is said to be a metric form, so u and v must be the coordinates defined by some particular chart. That means the space we're talking about is 2-dimensional, and both u and v are real.

Incorrect, and I already did. u = x and v = (-1)^(1/4) y with x and y Cartesian.

If x and y are Cartesian, then they're real. If y is real, there is no open set of x-y coordinates on which v = (-1)^(1/4) y is real. Hence, by the definition of charts and atlases, v = (-1)^(1/4) y cannot correspond to any overlap map (aka coordinate transformation). tensordyne's coordinate transformation is therefore inadmissible.

I gave the form of a metric in the u,v coordinate system. I did not specify at first in any way what u and v where. I then gave what they were in that they had the relationship to the Cartesian x,y as given. v being from a subset of the complex numbers is also not a problem. Requiring that v be real is an unnecessary restriction.

Sorry, but that restriction is part of the definition of a chart (aka coordinate system). (At the very least, it's part of the definition we use when we're talking about spacetime manifolds.)

Maybe I will make a post for the more mathematically literate members of the forum on charts, atlases, Differential Geometry and Cartography.

Let me propose some common ground on which sol invictus and tensordyne might agree.

A chart determines the open subset of the manifold on which the chart is defined. It also determines the coordinates of all points in that open set.

That's why a chart is often referred to as a coordinate patch or coordinate system.

A full atlas of charts determines the topology of the manifold.

So the charts already tell you everything there is to know about the topology. If you're interested only in the topology, the metric is useless/redundant.

The metric determines the geometry of the manifold.

The topology was already determined by the atlas, but the metric determines what counts as a rigid transformation, establishes a connection on the tangent bundle, allows definitions of parallel transport, geodesics, curvature, etc.

When a metric form refers to coordinates, the metric form is specific to the particular chart (coordinate patch) that defines those coordinates.

So we can have two different metric forms defined on some open subset of the manifold, but those metric forms have to agree when the coordinates of one are transformed into the coordinates of the other via the appropriate overlap map (which is itself defined by the composition of a chart with the inverse of some other chart).

The metric itself is independent of coordinates.

 

[sol invictus]

OK, here is one:

"What is the shape (not topology) of the following region?"

ds² = dp² + p² dq²
Set of region = {(p,q): 1 < p < 2, 1 < q < 2}
Set of space = {(p,q): 0 < p < 3, 0 < q < 3}

Looking back, I see that I might have misread this - by "set of space" do you mean those are the ranges of your coordinates? If so, the region is just as I said above: "a section of an annulus (two straight, radial sides, and two sides that are arcs of concentric circles)".

A more picturesque description would be the following: take a pineapple ring, and cut it twice with two radial cuts separated by a bit less than 60 degrees. The section you cut out is your region.

 

[Brian-M]

When you get to 9,192,631,770 you say that's a second. Then you say that the frequency of those waves was 9,192,631,770 Hertz.

Um, no. You've got it backwards. First we find waves that we say are oscillating at exactly 9,192,631,770 Hertz, and then we count 9,192,631,770 cycles to find out how long second is.

If the light is going at speed X the second has a value Y. If light is going at half X the second has value 2Y. And so on.

Wait, why?

We're not basing the definition of the second on the speed of light, we're basing the definition of the second on a particular frequency of light.

If the speed of light was to slow down by half, the photons used for determining the second would need to have half the wavelength in order to have the right energy level to react with the caesium-133. This means that light would only be able to travel half the distance before the count reaches 9,192,631,770 cycles, or one second.

Regardless of how fast the light is moving, you use it and the second you defined using it, to define the metre. So the metre is always the same.

Okay, you've got me there. The speed of light is used to define the metre... nowadays.

But the speed of light wasn't used to define the metre until 1983, long after the speed of light in a vacuum was determined to be invariant. (In fact, that's why the speed of light was chosen.)

And it wouldn't affect any direct attempt at measuring light in different conditions, since you're not altering the physical dimensions of the measuring device between measurements (which would defeat the point). The device would be built in units of the metre determined by the speed of light through a vacuum in earth-gravity.

So any actual variation in the speed of light would be detected by the device.

You can see light moving. But you can't see time passing.

You can't see light moving any more than you can see time passing. When you "see" a photon, it's no longer moving. It's been absorbed by an atom or molecule in your eye or detection device.

Why this obsession about not being able to see time passing? It's obvious that this assertion holds some special meaning for you, but until you explain your underlying reasoning, this repeated declaration is little more than a non-sequitur for the rest of us.

 

[Brian-M]

This time we have some friends along, each of whom has their own clock, and each clock is of a different kind; one has a grandfather clock, another a quartz crystal clock, a third an optical clock, a ... Oh, and our usual retinue of observers.

Presumably the clocks are kept in a controlled environment, as different timepieces would be affected differently by environmental conditions.

ETA: You can leave the pendulum-based clocks like grandfather clocks at home, as they'd be useless. Any change in gravity would change the period of the pendulum, meaning they won't agree with the other clocks. The same applies for any gravity-based timepieces such as hourglasses and water-clocks.

We visit all the places we went to on our 'gravitometer tour'.

This time we find something strange and wonderful (or not): at each location, all the clocks tell the same time (within their error bars/uncertainties)!

With me so far? Any questions or comments?

We're all with you so far.

Oh, one more thing we need to agree on: how to measure the speed of light.

Can you please describe how we can do this, using devices/equipment/techniques/etc which incorporate - at whatever critical point necessary - the definitions of the second and the meter?

How about a long evacuated pipe with a couple of mirrors, a narrow-beam laser LED and a photosensor built into it? The laser would be aimed at an angle so that it bounces between the mirrors a few hundred times before striking the photosensor.

If you only need to detect a change in the speed of light, you can forget about units of distance, and use any units of time you want, as long is it's precise. Just measure the difference in the delay it takes for the light to be detected by the photo-sensor with an atomic clock. If the tube is long enough, you might be able to get by with a high precision quartz timer instead.

 

[tensordyne]

Perverse, Pedantic and Prosaic.

(note: R is the set of the reals.)

I'm afraid I have to agree with sol invictus and Ziggurat.

That is unfortunate.

This is said to be a metric form, so u and v must be the coordinates defined by some particular chart. That means the space we're talking about is 2-dimensional, and both u and v are real.

According to Sadri Hassani in his book "Mathematical Physics"{MP}, a differential curve is given by:

26.2.1 Definition: (page 770)
A differential curve in the manifold M is a C^∞ map of an interval of R to M.

In Riemannian Geometry we integrate differential curves using a metric. Let M be defined as follows.

V = {v: v = (-1)^(1/4)y where y belongs to R}
M = {(u,v): u belongs to R and v belongs to V}

M is a manifold. Choose whatever C^∞ map of an interval you may desire. No problems here, but perhaps it is the use of v as a coordinate that is disquieting? On page 764 of {MP} this issue is resolved. I want to note though that I did give a metric, both in form and in terms of measure. There is nothing wrong with the metric for (u,v) because it allows one to calculate all the usual things one might want to calculate.

from page 764

Let U_P denote a neighborhood of P. When we say that this neighborhood looks like an m-dimensional Cartesian space, we mean that there exists a bijective map φ: U_P → R^m from a neighborhood U_P to a neighborhood φ(U_P) of φ(P) in R^m, we can define functions xⁱ: U_P → R such that φ(P) = (x¹(P), ..., x^m(P)). These functions are called coordinate functions of φ. The numbers xⁱ(P) are called coordinates of P. The neighborhood U_P together with its mapping φ form a chart, denoted by (U_P, φ).

Let U_P = M for the M as given before. M fits the bill and it should be pretty obvious what φ should be. Coordinate functions map coordinates in one chart to coordinates in another chart. (u,v) act like coordinates in all ways that matter. If I give you values for u and v, you only get one point. (u,v) are transformable into elements belonging to the set R².

Not allowing v to be complex is merely discrimination against non-real valued coordinate variables. I personally do not like being racist against complex numbers.

If x and y are Cartesian, then they're real. If y is real, there is no open set of x-y coordinates on which v = (-1)^(1/4) y is real. Hence, by the definition of charts and atlases, v = (-1)^(1/4) y cannot correspond to any overlap map (aka coordinate transformation). tensordyne's coordinate transformation is therefore inadmissible.

Let U = R².
Let V be defined as before.
Let φ_V: V → R² (define this globally or locally, doesn't matter)
Let φ_V: R² → R² (define as identity map)

From the book again:

page 764 and continuing where the last book quote left off.

Now let (V_P, μ) be another chart at P with coordinate functions μ(P) = (y¹, ..., y^m(P)). It is assumed that the function μ ∘ φ^-1: φ(U ∩ V) → μ(U ∩ V), which maps a subset of R^m to another subset of R^m, possesses derivatives of all orders. Then we say that the two charts are C^∞-related. Such a relation underlies the concept of smoothness in the definition of a manifold. A collection of charts that cover the manifold and of which each pair is C^∞-related is called a C^∞ atlas.

So...

Sorry, but that restriction is part of the definition of a chart (aka coordinate system). (At the very least, it's part of the definition we use when we're talking about spacetime manifolds.)

Applying the definitions given from the book to the sets and functions defined earlier, it should be clear that everything is OK. Overlap conditions are a check. (Diff/Hom)eomorphisms, Topology and any other nit-noid thing you could probably think up are a check as well. I have a chart, an atlas and everything else. This is about the nicest case you could have.

I get the feeling though maybe that the problem is with going from R² → V and then considering the V elements as "coordinates". V has a lot of added structure. For one, it is isomorphic to R (and a vector space) over the group whose group product operation is defined as normal addition. Again, I really do not see the big deal here. The whole time I am only doing maps from powers of R to powers of R or isomorphic versions thereof.

This line of objection about V to me seems either utterly prosaic or overly pedantic. Perhaps a little of both.

Let me propose some common ground on which sol invictus and tensordyne might agree.

Sounds like an excellent proposition to me.

A chart determines the open subset of the manifold on which the chart is defined. It also determines the coordinates of all points in that open set.

That's why a chart is often referred to as a coordinate patch or coordinate system.

Sounds good.

A full atlas of charts determines the topology of the manifold.

So the charts already tell you everything there is to know about the topology. If you're interested only in the topology, the metric is useless/redundant.

Sounds good.

The metric determines the geometry of the manifold.

The topology was already determined by the atlas, but the metric determines what counts as a rigid transformation, establishes a connection on the tangent bundle, allows definitions of parallel transport, geodesics, curvature, etc.

Define metric please.

When a metric form refers to coordinates, the metric form is specific to the particular chart (coordinate patch) that defines those coordinates.

So we can have two different metric forms defined on some open subset of the manifold, but those metric forms have to agree when the coordinates of one are transformed into the coordinates of the other via the appropriate overlap map (which is itself defined by the composition of a chart with the
inverse of some other chart).

Generally sounds good.

The metric itself is independent of coordinates.

Have to be careful here. The meaning of metric has not been pegged down. Plus, there are number of different meanings customarily associated with metric.

 

[Farsight]

You are using "the speed of light" in two different senses above, and you have commited an equivocation fallacy of sorts.

No I haven't.

The Shapiro delay experiment is a non-local measurement which measures the time taken for light to get from one point to another, distant point and back again. This is not the value which is denoted by the symbol "c" in the formula Z₀ = μ₀c for the vacuum impedance.

It's the Shapiro delay. Even a child can work out that the light is delayed because it goes slower when it skims the sun. Ah, I see some bright spark has removed the Einstein quote from the wiki article.

In any freely-falling laboratory small enough that tidal forces are irrelevant, light always goes at the same speed. It this universal constant which "c" denotes in that formula.

No it doesn't. We know it doesn't because an optical clock in that falling laboratory ticks slower and slower the lower it gets. Our parallel-mirror light clock will tick slower and slower too. All electromagnetic phenomena will be similarly affected, included that within the bodies of the observers within that laboratory.

Of course, even if c did vary from point to point your argument above would be a non-sequitur. The value of c by itself is insufficient to determine Z₀; you must also specify how μ₀ changes from place to place. Obviously it is constant in standard classical physics, but since you have already claimed that other fundamental constants vary, perhaps you should be clear on your position with regard to μ₀.

It obviously isn't constant, and this is even more obvious when one understands the unification of electric and magnetic fields into the electromagnetic field. But that's one for another day. Meanwhile: I don't need to specify how it changes from place to place, all I need to do is demonstrate that it does.

Nevertheless, let's suppose you manage to cook up a vacuum in which the speed of light truly varies; suppose that light is refracted in some manner as it moves through space.

You mean like in gravitational lensing? You might like to read up on that, ct. Here, try this little article. Or take a look at Inhomogeneous Vacuum: An Alternative Interpretation of Curved Spacetime. Here's the abstract:

"The strong similarities between the light propagation in a curved spacetime and that in a medium with graded refractive index are found. It is pointed out that a curved spacetime is equivalent to an inhomogeneous vacuum for light propagation. The corresponding graded refractive index of the vacuum in a static spherically symmetrical gravitational field is derived. This result provides a simple and convenient way to analyse the gravitational lensing in astrophysics".

Theories of gravitation based on a scalar (in your case, c) that varies through space are not mathematically equivalent to GR. Therefore, if there is nothing else to your model, whatever you are discussing here is not GR and you at least have to show that it is equivalent as far as experimental observations go. Look up "scalar theories of gravitation" to get an idea of what problems you will need to overcome.

Hasn't anybody else here rumbled this blind-em-with-maths rewriting of history? This is what Einstein actually said:

1911: If we call the velocity of light at the origin of co-ordinates cₒ, then the velocity of light c at a place with the gravitation potential Φ will be given by the relation c = cₒ(1 + Φ/c²)
1912 : On the other hand I am of the view that the principle of the constancy of the velocity of light can be maintained only insofar as one restricts oneself to spatio-temporal regions of constant gravitational potential.
1913: I arrived at the result that the velocity of light is not to be regarded as independent of the gravitational potential. Thus the principle of the constancy of the velocity of light is incompatible with the equivalence hypothesis.
1915: the writer of these lines is of the opinion that the theory of relativity is still in need of generalization, in the sense that the principle of the constancy of the velocity of light is to be abandoned.

Only the word he used was geschwindigkeit. Speed. And c is a speed, not a velocity. The SR principle was the constant speed of light, not the vector-quantity.

The closest thing I've found to a decent attempt at what you're proposing is the "polarisable vacuum" model of Puthoff. His model of gravitation, based on a varying ε₀ and μ₀, matches the predictions of GR in some areas (redshift, light deflection, precession of the perihelion of planetary orbits) but not in others (particularly those relating to gravitational radiation and frame-dragging).

Noted. He's mentioned in the Inhomogeneous Vacuum paper above. Which doubtless some here will say has not been peer-reviewed, or will decry in some other fashion.

I'll quickly mention one more problem with this whole idea. You have placed a lot of emphasis on vacuum impedance, as though it could be an explanation for gravitational time dilation and the like. However, not all phenomena are electromagnetic in nature. Even if you did have a model in which the vacuum impedance varies from point to point, somehow reducing the speed of photons passing through that space, how does that affect weak and strong nuclear interactions? In detail, I mean, rather than just vaguely alluding to electroweak unification (which would not help you with strong interactions, anyway).

I'm not giving you any detail. But I will give you this:

http://outreach.atnf.csiro.au/education/senior/cosmicengine/images/cosmoimg/pantipannihilation2.gif

 

[W.D.Clinger]

Perverse, Pedantic and Prosaic (part 2)

Not allowing v to be complex is merely discrimination against non-real valued coordinate variables. I personally do not like being racist against complex numbers.

As mathematical arguments go, that isn't one.

You quoted Hassani's definition on page 764 as your authority. That definition says the coordinates are real.

I'm sorry that Hassani (and every other reference I've checked) offends your personal sense of the racial characteristics you have chosen to attach to mathematical objects, but your racist language is not a mathematical argument.

This line of objection about V to me seems either utterly prosaic or overly pedantic. Perhaps a little of both.

It's a matter of using accepted definitions. When you ignore accepted definitions while pretending to champion the proper mathematical definitions, it's a little churlish of you to blame us for your miscommunication.

Have to be careful here. The meaning of metric has not been pegged down. Plus, there are number of different meanings customarily associated with metric.

True that.

I am, however, using one of the meanings customarily associated with the term. That's an improvement over your practice of inventing a meaning that's contrary to the universally accepted customary meaning.

 

[Farsight]

You have not cited any "hard scientific evidence" other than the references to the effects predicted by GR .
this post of yours is a list of quotes from Einstein before the publication of GR in 1917. No surprising physics in what he states.

A bit of sceintific woo from you though " My particular position with GR is Einstein's position: that the speed of light varies and space is inhomogeneous":

1. Einstein's position in the quotes is that the speed of light varies
   It is unclear whether he is talking about proper or coordinate speed from your quotations but othe rposters point out that itis coordinate speed.
2. "space is inhomogeneous" is nonsense.

That statement is foolish on a couple of levels.
Firstly who cares about "Einstein's GR"?
What matters is GR. That is the scientific theory. You seem to want people to ignore almost 100 years of progress in science just to obsess on the original presentation of GR by Einstein.

Secondly: GR as taught today is exactly in line with Einstein's GR with the obvious exception of the use of a non-zero cosmological constant. The mathematics is the same aside from notational differences. The predictions from GR are the same.

When I read posts like this, I am reminded of "conversations" I've had with young-earth creationists. I really mean that, I'm not just saying it to snipe or evade. It doesn't matter what evidence your offer, they dismiss it and proclaim that it isn't evidence at all. They absolutely will not shift from what they've had drummed into them, even though there is no evidence to support that position. I'm sorry, I'm not going to reply in detail to this one, apart from saying this: I am reminded of this week's Horizon, and of a saying I picked up somewhere a few years back: it's like the shutters are down and there's nobody home.

 

[Farsight]

Amazing! Not only "all local clocks" but all local physics experiments whatsoever. Why, it's almost as there were some "law of relativity", in which the laws of physics do not care what time-coordinates the observer has chosen to use!

You talk as if the laws of physics constitute some deity, ben. They aren't. Space is the way that it is, and other things too, such as light. We describe how these things behave, we note the underlying symmetries, and we draw inferences that we label as "the laws of physic". That's all.

But I, Farsight, will designate the correct coordinates anyway! Just for the heck of it! Using a mythical pulsar, at Absolute Rest, in Flat Space at Zero Gravitational Potential. Heck, let's put it at the Geometric Center of the Universe, and we'll have it spinning Right-Side Up.

If you prefer, we'll use the CMBR. Now face up to it ben, look at your own psychological response. I've given you ample evidence and a sound argument, and you can't counter it. So ask yourself why you're being emotional rather than rational. Shall I tell you why? It's because your unconscious mind cannot face up to being wrong. And what you need to do, is get a hold of it.

 

[Farsight]

As far as I know, Fizeau used several different experimental setups, each with light and mirrors.

And we need to be pretty specific, I think, so would you mind spelling out the actual setup you recommend, in more detail?

Yes I would, stop wasting my time. Everybody knows the setup, see English Wikipedia.

OK, but as we will be doing our experiments in many different environments, including in deep space, we need a way to establish what "horizontal" is; how do you recommend we do that?

You note which way things fall down, then you take an orthogonal direction.

We may have a problem here Houston. Or not; can you explain how we "time it [the back-and-forth travel time, in some Fizeau-like set-up] using the distant pulsar"?

Are you for real? Go do some research.

 

[W.D.Clinger]

OK, but as we will be doing our experiments in many different environments, including in deep space, we need a way to establish what "horizontal" is; how do you recommend we do that?

You note which way things fall down, then you take an orthogonal direction.

ETA: Farsight's definition of horizontal is observer-dependent, therefore coordinate-dependent. As the dialogue above continues, DeiRenDopa will explain the significance of that coordinate-dependence, and its implications for signals coming from the distant pulsar.​

 

[Farsight]

In this round, our aim is to develop a model. We want the model to be able to 'explain' (I'll explain in a minute) all our detailed, objective, independently verifiable, quantitative results....

Sure thing.

 

[DeiRenDopa]

This time we have some friends along, each of whom has their own clock, and each clock is of a different kind; one has a grandfather clock, another a quartz crystal clock, a third an optical clock, a ... Oh, and our usual retinue of observers.

Presumably the clocks are kept in a controlled environment, as different timepieces would be affected differently by environmental conditions.

ETA: You can leave the pendulum-based clocks like grandfather clocks at home, as they'd be useless. Any change in gravity would change the period of the pendulum, meaning they won't agree with the other clocks. The same applies for any gravity-based timepieces such as hourglasses and water-clocks.

And interesting side project might be to do research on why some clocks do not 'keep the same time' as the others. With Farsight's response in hand ("What they [clocks] actually do is clock up some form of regular motion, and display a cumulative total that you call the time"), I'd have explicitly added a 'radioisotope clock'.

We're all with you so far.

Good, thanks.

Oh, one more thing we need to agree on: how to measure the speed of light.

Can you please describe how we can do this, using devices/equipment/techniques/etc which incorporate - at whatever critical point necessary - the definitions of the second and the meter?

How about a long evacuated pipe with a couple of mirrors, a narrow-beam laser LED and a photosensor built into it? The laser would be aimed at an angle so that it bounces between the mirrors a few hundred times before striking the photosensor.

If you only need to detect a change in the speed of light, you can forget about units of distance, and use any units of time you want, as long is it's precise. Just measure the difference in the delay it takes for the light to be detected by the photo-sensor with an atomic clock. If the tube is long enough, you might be able to get by with a high precision quartz timer instead.

This may be somewhat similar to the Fizeau-type set-up Farsight mentioned, except for the clocks.

Myself I think the actual set-up will depend heavily on the sorts of hypotheses we're aiming to test. For example, if - per Farsight - they have to do with variations in c over distances as small as ~1 meter, then what you've proposed may not be suitable.

 

[Farsight]

....So, re-writing F's original sentence with these clarifications, and adding back the full context (I've had to do some paraphrasing):

F: The optical clock uses aluminium rather than caesium, and a UV frequency rather than a microwave frequency, but it works along the same lines, and employs electromagnetic phenomena. When electrons move by doing a spin-flip, they emit electromagnetic waves, which move away more slowly from an optical clock at an elevation of a foot (or so) above an otherwise identical optical clock.

Right?

Sigh. Wrong. They move away faster. Remember this: they're slower when they're lower.