Cornelius wrote:

In your original post you posed;

“This however cannot mean that all other inertial frames, moving relative to this one, must now have slower clocks and contracted rods”.

Yes it does. ...

I think the statement "must now have slower clocks and contracted rods" referred to the fact that for purely inertial frames in motion relative to each other, there is no way that both can have a slower clock than the other one. The mutual observation of time dilation and length contraction can then be classified as 'illusionary', or just a measurement problem, I guess.

Truly inertial clocks with a relative velocity (and away from gravitational fields) can only be compared once, when they pass each other. After that comparison is theory (and convention) dependent, e.g., the speed of light postulate or the synchronization of clocks convention.

Obviously, as you said, the "twin paradox" do not suffer from this problem, because at least one observer is non-inertial.

SL: Your Aerospace Watchdog

Cornelius R. Morton
Scruffy, this thread has been a treat to explore and a great learning experience as well as a reinforcement of the KISS principal. There are eight to ten various solutions to the twins paradox available on the web of varying complexity. One of the Minkowski diagrams I tried allowed the traveling twin to view some portion of the stationary twins time line, dependent upon the velocity chosen, during the first part of the flight and the last part of the flight and swept through the central portion. This could be viewed by viewing incremental frames during acceleration but using 1 km increments for a velocity of 0.6c would require some 360,000 frames. Rube Goldberg would be proud. Following is what I have found and my take on the twins paradox.

Everyone seems to agree that the away twin ages less than the home twin. This has also been tested at least twice. In 1971 Havele and Keating had five cesium clocks synchronized. Four were sent around the world on commercial airliners, one pair east and the second pair west, upon their return the clocks were compared to the at rest clock. The results were in accord with GR and SR. GR pertaining to the effects of the earth’s gravitational field ( clocks speed up as the gravitational field decreases with altitude) and SR with the clocks slowing down due to relative velocity. 600 mph did not equal 30,000 ft so the traveling clocks ran a bit faster than the at rest clock but the difference was equal to theory within the accuracy of the equipment. This was repeated in 2005 by the National Physics Laboratory in the UK. The clocks were better and the trip was from London to Washington DC and return. This time the results were within 4% of theoretical. It seems that the only argument remaining is what the twins see, experience, during the experiment.

One item that has not been mentioned concerns the reference frames involved. The home observers’ frame is an inertial frame, if he were blindfolded and seated he would have no way of determining if he was in an un accelerated motion or motionless. He would of course know that the traveling observer was moving with respect to him as he watched the ship take off and accelerate away. As acceleration was involved and would be on several more occasions he knows that he cannot treat the traveling observers’ frame as an inertial frame, in consequence he cannot expect symmetry between their observations. As the traveling observer is in motion he can anticipate that the traveling clock will run slower than his. The traveling observer on the other hand has no trouble realizing he is in motion as any acceleration is felt as a change in velocity and or direction, obviously he is not in an inertial frame. As the traveler has direct knowledge that he is in motion he cannot assign his motion to another object. Thus each observer is limited to his clock and immediate surroundings. The stationary observer may be on a park bench with a pretty girl wondering why time is passing so fast while the traveler is bored staring at his clock wondering why it is so slow.

Lewis Carroll Epstein in his book Relativity Visualized presented a view of spacetime that is a bit easier to get ones arms around. In his view space and time are inseparable and there is only one possible velocity through spacetime and that is the velocity of light. An observer in an inertial frame views himself as only moving through time, not space. A second observer that is moving in relation to the first is seen by the first as moving through space and time. If vt is the velocity through time and vs is the velocity through space and c is the speed of light through spacetime then c^2 equals vs^2 plus vt^2. Then when moving through space vt = ( c^2 – vs^2)^1/2. Normalizing so that all velocities are expressed as a percentage of c, with c = 1 then vt = (1 – vs^2)^1/2 which is the Lorentz transform. This view of spacetime makes it relatively easy to understand how two observers in two inertial frames that are in relative motion can perceive each other as the one in motion with the slow clock. On page 88 of Relativity Visualized, Epstein illustrates the solution of the twins problem and notes the absence of relative motion because the traveling twin is fully cognizant of his motion.

In your original post you posed;

“This however cannot mean that all other inertial frames, moving relative to this one, must now have slower clocks and contracted rods”.

Yes it does. If you select some other frame and determine that his clock is running at some factor, Y, slower than your clock and his rod is shorter than yours by the same factor then in his frame he will see your clock slower than his by the same factor and your rod shorter than his by the same factor. This is because each perceives the other moving at the same relative velocity and gamma is dependent only on relative velocity.

In regards to the two moving vehicles, it is perhaps easier to see if an observer is stationed at the light source. As simultaneity has been established in the description the only concern is what the light observer sees. As each vehicle pass him at the same velocity gamma in each case is the same. He sees each clock running slower than his by the exact same amount and the distance each traveled during the time between their perception of the light flashes is the same. Each vehicle stops in relation to the light source and each other, they are in the reference frame of the light observer. Comparing notes the light observer notes their times are shorter than his by the same amounts and agree with each other.

Scruffy, SL,

Thank you for this.

Yes, I will take elsewhere
http://www.scienceblog.com/cms/comments-about-logunov039s-relativistic-t...
any further comments I might have about the "general theory of relativity".

I am sorry I have overloaded your blog with my long comment on the "general theory of relativity".

I started to make posts to this your blog with a direct focus on your question of the "reality" or otherwise of Einstein's time dilation and length contraction, using the current vehicle of the two-clocks scenario, with my neutral observer demonstration for it. I noted unexpectedly that this simple example pointed to thoughts about the "general theory of relativity", and thought it would be ok to point how this was.

I am still hatching thoughts closely focused on your original question of the "reality" or otherwise of Einstein's time dilation and length contraction. If eventually those thoughts come out briefly, and if you reply to this post that you would like me to do so, then I will post them here when they are ready.

Christopher

Christopher, while your discussions with David and Burt are very interesting, I think the length of your posts are starting to kill this old thread!

Maybe we must consider opening a new thread for continuing this discussion. If you do not want to, tell me and I'll dream up a suitable title and start one.

SL: Your Aerospace Watchdog

David:

Thank you for your extensive reply to my post about the twin clock scenario.

I was trying to create a fresh way of grasping the twin clock scenario, of getting an intuitive common sense take on “Einstein’s time dilation”, a fresh angle on its “reality”.

When I had done that, I saw that the twin clock scenario holds unexpected lessons that I had not originally seen in it, and I tried to indicate what they are. In a nutshell, the twin clock scenario has the core of the reason why the theory of gravity must be built on Minkowski geometry.

I chose to use Minkowski diagrams for the demonstration of the twin clock scenario.

I first offered a list of candidate intuitively simple explanations of the difference in duration registered in the twin clock scenario, in terms of clock rate, clock acceleration, and clock speed, and I invented a neutral observer for whom differences in these were all null and therefore not explanatory of difference.

I rely on the principle that an intuitively reasonable explanation should work for all observers, and I produce an observer who finds that none of the above three simple explanations work. By suggesting a higher dimensional picture than my two-dimensional Minkowski diagram, Burt’s comment brings it to our attention that there are countless other observers for whom they don’t work. Therefore none of the simple three, at least alone, is an intuitively reasonable explanation. This is an intuitive motive for looking for more subtle candidates, still with a common sense or intuitive slant.

My candidate explanation goes like this:

The rate of a clock, the speed of movement, and the acceleration are all differential conditions. They are rates. Besides intensive variables, densities, conveniently represented by tensors, physics is about extensive variables, global quantities. A physical explanation needs an integration of differential conditions, to produce definite results. We want some kind of definite integral.

There are various ways of integrating given differential conditions. One can do a line integral, or an area or volume integral, the result being a number. Or one can find a curve, for example an integral curve of a field of vector rates. There are various ways of setting definitive conditions for that. One can say that the curve must pass through certain points on the boundary or in the interior of the field, or that it must have derivatives with certain values in specified places. Or one might produce a surface as the result of the integration. And so on.

I chose the non-technical word adventure to try to suggest the general intuitive meaning of this variety of ways of setting the definitive conditions for integrating.

Of course it will need to be made precise for each scenario. For example, your choice of established terminology, “spacetime paths”, sometimes also called world lines when they are pervaded by an enduring particle. The mathematically fitting notion of spacetime paths provides an elegant and apt explanation in abstracto, but is bloodless, wanting allusion, not enough for a feeling of intuitive understanding. And I am still working here at a more general intuitive level; that is why I chose to speak of adventure.

Your reply contains the maxim:

“Just nice local measurements, and an accumulation thereof (anything with a memory, like clocks).”

This maxim is in agreement with the local aspects of adventures, but it also suggests a celebrated line of thinking, a line that is often taken to lead to the “general theory of relativity”.

Speeds and accelerations and relative clock rates are local, differential, notions. The maxim leaves open the method of accumulation.

The “general theory of relativity” is a statement of local, differential, conditions, with no free parameters, as you have noted, a fact most admirable and illustrious, even egregium. But it leaves open the method of accumulation, in its particular case, the boundary conditions. This is where it is ambiguous. For the setting of boundary conditions, the use of the “general theory of relativity” requires some theoretical supplementation, and, to be valid in practice, that needs a global Minkowski geometry.

The “general theory of relativity” thinks differentially, in terms of countless local tangent spaces each with its own private Minkowski geometry, its own nice local measurements, but it doesn’t think of putting them together into a single global geometry. So it neglects causality, which requires a global Minkowski geometry, as proved by Robb in 1913.

The twin clocks scenario, unexpectedly to me, turned out to contain the bones of Robb’s discovery, which is sometimes nowadays called the Alexandrov-Zeeman theorem.

In using an accelerated reference frame for the twin clocks scenario, one finds flagrant violations of causality. One finds clocks disappearing into nowhere, appearing from nowhere, and appearing in duplicate at different places at the same time. Indeed, no matter how small the non-zero relative speed of the away-clock, there will be such violations of causality in the accelerated frame.

Violations of causality are physical nonsense. The “general theory of relativity” just carries on as if this didn’t matter, and allows accelerated frames without a fuss. Consequently, it needs to get its boundary conditions from an external source, being alone in itself ambiguous about them.

“… all must be invariant to general coordinate transformations…” This statement expresses an admirable and important concept. For the orthodoxy it has been mesmeric. It is a very general statement, and I think it needs some provisoes.

It is also an admirably true and important statement that “the phenomena of nature care absolutely nothing for what coordinates we may assign to the points of space and time.” This is a valuable and powerful principle of physical explanation. I have used just that principle above for the twin clock scenario. For the orthodoxy I also has been mesmeric.

Neither of these important principles, however, is a law of nature. They are just methodological guidelines, advice about how to write laws of nature. It might also be said that nature doesn’t even care if we don’t use any coordinate system at all, or that nature doesn’t care if we don’t study physics. Half echoing a famous man: For a valid empirical science, laws of nature must take precedence over aesthetics of formalism.

The second statement as it stands is about so-called admissible coordinate systems, that is to say, coordinate systems of spacetime that respect the difference between space and time (but perhaps it was, or perhaps it was not, intended to be more general, and to refer also to coordinate systems that mix up space and time coordinates). It does not state that that just any old arbitrary system of equations, that accurately describe the phenomena of nature, reveals the laws of nature, that is to say, is form-invariant under transformation of spacetime coordinates. The accurate description might have been a lucky accident that does not reveal what we might think of as laws of nature such as we like to see. It took Einstein and Hilbert’s special skills and genius to find a very special just-right system of dynamical equations for gravity that would bring about form-invariance under transformation of spacetime coordinates. Form-invariance does reveal laws of nature. We want our dynamical equations to reveal laws of nature.

While nature does not care what coordinate system we use, nature does care very much not to violate causality, and this is a law of nature. It is a real physical principle, not just a methodological guideline. Methodological guidelines can be validly sidestepped in favour of other valid methodological guidelines. But laws of nature cannot be sidestepped.

Again, while nature does not care what coordinate system we use, we care very much that we can physically empirically verify our statements about nature. This is not a mere methodological guideline: it is an absolute mandate of empirical science. We need coordinate systems that we can use empirically.

These two concerns, causality and empirical verifiability, constrain our choices of coordinate systems. We should distinguish between reference frames and coordinate systems. We may allow a coordinate system that seems to show violations of causality provided that coordinate system has a transformation into another that does not. We will distinguish coordinate systems, that do not violate causality, and can be empirically measured, as reference frames, and use them as bases to generate more general coordinate systems. We may solve a particular problem in terms of an aptly chosen general coordinate system, but we will need the reference frame to guide us out of perplexities of causality violation. Also we will need our reference frame to be physically empirically verifiable, abbreviated by the term admissible. We can physically empirically verify things in a coordinate system only if it respects the difference between time and space. That is to say, it must respect the difference between clocks and measuring rods, for those are our facilities for physical empirical verification. Only a restricted class of coordinate systems for spacetime provide for those facilities.

The following paragraph is very closely based on pages 17-18 of Anatoly Logunov’s Lectures in Relativity and Gravitation: A Modern Look, translated from the Russian by Alexander Repyev, Nauka and Pergamon, first English edition, 1990; some sentences are word-for-word.

The “general theory of relativity” is importantly driven by the fact that the quantities that determine gravity are densities that can be expressed as tensors with respect to spacetime. Consequently we have access to a methodological notion of covariance: “An equation [relating unknown functions of the spacetime coordinates] is said to be covariant [under some arbitrary differentiable transformation of the spacetime coordinate system just when] its new unknown functions expressed in terms of the new [spacetime coordinates] satisfy equations of the same form as the old functions in terms of [the] old [spacetime coordinates].” This notion of covariance is a mighty help in the work of finding a just-right system of equations. But it doesn’t do the whole job; more is needed to achieve form-invariance of dynamical equations. “Form invariance for a metric under some transformation [(…)] is a more stringent requirement than the covariance of equations. This requirement is a constraint on the class of frames of reference: they must be such that when transformed into another the functional form of the metric tensor of spacetime would remain unchanged.”

Putting together these constraints on our coordinate systems and reference frames, we will find that Minkowski geometry is irremovably built into any correct and acceptable system of differential equations for the dynamics of gravity. This is denied by the orthodoxy of the “general theory of relativity”, which thereby invalidates itself. It is respected by the Logunov relativistic theory of gravity.

From a more directly empirical viewpoint, one would ask about the above-mentioned maxim about nice local measurements, do the neutral observer’s sightings through his telescope come within the definition of nice local measurements? Are astronomical observations nice local measurements? If not, would the maxim bar them? Why?

I would like to revise the last paragraph of my post that your reply refers to. “Time is not an event, nor a process, and so it cannot be caused, nor can it be a cause. It does not make sense to say that velocity or acceleration affect time, because such a statement seems to imply that time can be subject to, or patient of, causal agency. Time is in a sense part of the abstract theoretical receptacle that encompasses events and processes, which are causes and effects.”

Moving on from Newton’s theory of time as independent of space, we take up Minkowski geometry. For physics, we demote time from a principal fundamental notion, a notion that together with space defines the eternal absolute Newtonian-Platonic receptacle; we demote time to an abstraction that measures concrete adventures. The eternal absolute receptacle is no longer a primary abstraction for us.

What now for physics is conceptually primary and absolute in its place?

Causality, the causal structure of adventures, of causes and effects, of processes, and, abstractly, of point-events, is the most fundamental concept and the most general law of nature.

It is from causality that we construct the notion of spacetime. We find from causality that if we want a geometry for spacetime, it must be the Minkowski geometry. I have some comments on this in another post http://www.scienceblog.com/cms/einsteins-time-dilation-and-length-contra.... Once we have Minkowski geometry we are in a position to set up arbitrary coordinate systems and consider arbitrary coordinate transforms, according to desirable methodological guidelines, and respecting the laws of nature. This gives us a valid basis for a relativistic theory of gravity, with a geometrisation that sees test particles moving along geodesics and so forth.

The orthodoxy of the “general theory of relativity” proposes to take Minkowski geometry into account just as an afterthought, apparently plucked out of nowhere, not as a foundation because it is the most fundamental law of nature. The orthodoxy of the “general theory of relativity” says “Oh, we can take Minkowski geometry adequately into account by making all relevant tangent spaces have it, and tying them together by using a scale factor that makes the speed of light in a vacuum the same for them all.” But this is too late in the development, and it doesn’t provide the foundation of causality. Just the scale factor is not enough to put them all together into one underlying Minkowski geometry, such as is required by the fundamental law of nature that we call causality.

The ‘equivalence’ story of Feynman and Weinberg is not correct. They get a relativistic theory with general coordinate freedoms from a foundation of Minkowski geometry, but they do not show the converse, getting global Minkowski geometry from an orthodox “general theory of relativity”.

Einstein’s lofty, epochal, original, and brilliant insights into gravity can make physical sense only in the setting of an overall Minkowski geometry. His “special theory of relativity” is an expression of Minkowski geometry, which expresses all the relativity theory that is needed for the study of gravity, and indeed all the relativity theory that there is. Properly speaking, there is no “general theory of relativity”. Whitehead drew Einstein’s attention to this long ago, but Einstein rejected it, and the orthodoxy has followed Einstein. The proposal that Minkowski geometry drops out of the operative differential equations of gravity is mistaken. To come to terms with all this, perhaps a good start may be to read Logunov’s Lectures, cited above.

Christopher

I have moved this comment to:

http://www.scienceblog.com/cms/comments-about-logunov039s-relativistic-t...

Christopher

Hi, Burt.

Thank you for this.

The added complication of the effect of a gravitational field is indeed harder to understand. Your point is very interesting and important. I am still thinking about this one.

As for a neutral observer for the muons. With the vast profits generated by the licensing of my global warming engine, I have set up an orbiting lab that fires muons in the right direction and speed to do the job of the neutral observer. The lab runs muons in a stationary state to act as the home twin. I have to rely on a symmetry argument to pretend that the travelling muons virtually turn around for a return to base.

I think that the faster moving muons record less time on their way can perhaps be understood on the idea that they hardly interact with anything on the way, and so they have less experience per kilometer travelled. I have not yet read an autobiography of a muon, but I am guessing that it will back me up when it arrives. Muons in a chamber have a good chance to interact with the walls of the chamber, and this counts as experience for them. Light pulses are the extreme case, for they record no experience on the way. This is of course just a kind of Aesop's fable, a sort of allegory.

The definite facts in the life of a muon are its creation and its annihilation. (No taxes are yet levied on muons.) These two facts set up a particular reference frame, inertial between them, privileged for this problem. An distant observer inertial in a reference frame moving with respect to the travelling muon has to do quite a lot of data processing to work out what is happening to the local (travelling) muon. This processing takes time. So he observes the travelling muon to have a long life. I am not sure if this makes sense. I will think about it. This is getting towards a sort of interpretation of your first maxim, I think.

Regards,

Christopher

Hi Christopher.

I think your analysis of what I tried to convey is spot-on. Your own 'crutch' of the neutral observer is also valid and if it helps you (and others) to cope with the complexities, it is valuable.

You wrote:

The purpose of my neutral observer demonstration is to show that there is some intuitive common sense content to the longer time recorded by the home-clock: that intuitive content is that the neutral observer sees the home-clock continuing to travel and tick on its first leg long after the away-clock has finished its leg. This is not related to difference in clock rate, nor to difference in speed of movement; the neutral observer was constructed to ensure that. It is due to a longer leg to be run by the home-clock.

It may become a bit counter-intuitive at times, e.g., the clock onboard the low-Earth-orbit (LEO) satellite that records less time per orbit than Earth clocks, despite the fact that it operates in a gravitational environment where static clocks physically runs faster than Earth clocks.

Another one that is hard to visualize by means of a neutral observer is the muon lifetimes that get "stretched" enough so that a significant quantity reaches Earth's surface, despite their short half-life.

My "first maxim", as you called it, simply says that since each muon is present at the two events (its creation and it hitting Earth's surface) it observes a shorter time interval between the two events than what we do. We are not present at both events.

Obviously, we can only observe the half-life time of 'static' muons, confined to our vicinity, but it shows us that something happens to the time of the fast muons. A different history, as you said before...

Regards,

Burt Jordaan (www.Relativity-4-Engineers.com)

Hi, Burt.

Thank you for this.

We are talking about two maxims that you proposed, and that I found baffling.

Let me speak of the local observer (the one who is inertial and present at both events) and the distant observer (the one who is inertial and present at neither event).

The first maxim is about comparing the time intervals between the two events, estimated respectively by the local observer and the distant observer.

Your new post translates your first maxim into the customary mathematical formulation in terms of inertial reference frames. That of course clarifies things, but is not necessarily what I was looking for. I was more directed towards the ordinary language mode of expression as it stood.

I think my bafflement is with the heavy burden carried by the word "observe".

I find it obvious that the local observer will have a simple task to report his findings: just log the two event readings on his own clock. No bafflement there, I think.

But for the distant observer, I am left wondering does observing mean his assembling the data supplied by his own remotely stationed research assistants with their synchronised clocks, that is to say reporting the time assessed in the reference frame of the distant observer; or, as in your post Twin paradox images (edited again), does it mean his using data from the local observer's remotely stationed synchronized clocks, that is to say reporting the time assessed in the reference frame of the local observer; or using calculations based on his own telescopic sightings and perhaps radar soundings to measure the speed of motion of the inertial observer's clock, and if so, whether his calculation will estimate times in terms of the reference system of the local or of the distant observer? Your comment "'Observe' obviously means compensating for light travel times" as I read it clarifies things to the extent that it excludes just measuring the time according to the the non-inertial observer's own clock between his telescopic sightings, and perhaps radar reflections, of the inertial observer's clock.

If the distant observer is merely distant from, but stationary with respect to, the local observer, he will measure the same times as the local observer, not less, I think, but this is a physical inference, not a logical identity; as a physical inference it might be questioned. Anyhow, this is a marginal case.

Your new post tells me about the first maxim; it tells me, I think, that you mean that the distant observer is estimating the time between the two events in his own reference frame, by one of the several methods available to him to get that quantity. That makes sense of it. Is that in fact what you mean? Only you know for certain what you mean.

What really had me puzzled was your second maxim "In the twin's case, the traveling twin is the only one present at the turn-around event and hence will observe a smaller outbound time." It seems to me that you somehow saw that as just obvious. But how? Your maxim uses the word "hence" as if to mean that knowing that the traveling twin is the only one present at the turn-around event makes it obvious that he will observe a shorter time interval.

Now I see what you had in mind here. You were using the first maxim. You reasoned that since the away-twin is inertial in the outbound leg, the set-out event and the turn-around event for him are two events such that he is present at both, and such that he is inertial in the interval between them, so that he qualifies as a local observer for the first maxim. Moreover, the home-twin is inertial but not present at both events, and so he qualifies as a distant observer for the first maxim. It follows from the first maxim that the away-twin's measurement of the time interval between the twp events is less than the interval that the home-twin will estimate in his reference frame. The logic here is that since the travelling twin is the only one present at the turn-around event, it follows that the home-twin is not present at both events; meanwhile, of course, the travelling twin is present at both; now it follows therefore that the first maxim applies. This is the hidden logic that I was asking for; no great mystery, but with my mind clouded by the puzzles of the meaning of the word observe, I did not notice it. I was wondering why being the only observer present at an event was so important in this situation; that wasn't exactly what was important, it turns out.

It seems I have tried to turn your neat aphorisms into a long-winded and tedious text. That isn't quite my aim! I am just looking for ways of making such neat aphorisms as clear as practicable to the newcomer. It takes a long-winded discussion to set out the problems of doing it neatly.

The purpose of my neutral observer demonstration is to show that there is some intuitive comon sense content to the longer time recorded by the home-clock: that intuitive content is that the neutral observer sees the home-clock continuing to travel and tick on its first leg long after the away-clock has finished its leg. This is not related to difference in clock rate, nor to difference in speed of movement; the neutral observer was constructed to ensure that. It is due to a longer leg to be run by the home-clock. This strikes me as intuitively reasonable according to common sense, while stories of clocks changing rates do not. Stories of changing clock rates and more especially of time-dilation require to some extent a suspension of common sense: that is why people find these things puzzling. I aimed to reduce that puzzlement.

Regards,

Christopher

Christopher:

The statement

an inertial observer present at two events will always observe a shorter time interval between the two events than any inertial observer not present a both events. [Where] 'Observe' here obviously means compensating for light travel time.

Can be seen as a restatement of "time dilation". The "tics" of a clock are events, and the clock is certainly present at each of these events. According to "time dilation" "moving" clocks (clocks that change position over time, as seen in a given [inertial] rest frame) "appear" to "run slower". So, an observer that is not present at the clock "tic" events will observe a longer time interval between the "tic" events than the clock that is present at each of these events (the "moving" clock).

Of course the stipulation of what "observe" means is a way of stating that we are talking about coordinate time of, or the time as measured by, the observer not moving with the clock. This can either be as Burt expresses it, by "compensating for light travel time" (presumably observing the events via light signals from the events and then back calculating the observer's time "simultaneous" to the event), or it can be via a collection of "research assistants" distributed through space with "synchronized" clocks. (The method for "compensating for light travel time" is directly related to the procedure for "synchronizing" the "research assistants". However, depending on how the observer obtains "distance" information the "compensating for light travel time" method may be considered to be more dependent on the spacetime theory involved than the "synchronized research assistants" method.*)

So I hope this helps you see the connection with what Burt is stating.

David

P.S. Once again I find something I don't like about the way blockquotes are handled with the new Science Blog formatting: It doesn't allow for non-emphasis text! (At least, it doesn't appear to allow for it using some form of the <em> tags.)

* General Relativity's use of general coordinates allows for any of a number of observational methods. The observational approach determines the numbers used, and hence the coordinates. The determination of the metric (and other tensors of spacetime) can then be used to determine "what happened" by helping one remap the coordinates. (Unfortunately, I doubt that many GR texts explicitly express this nature of general coordinates, though I think most of them have at least one paragraph in which they express how one may use an observation based system [such as telescopes trained on pulsars] to obtain coordinates that are not of the ordinary xyzt variety.)

Hi Christopher, yep, I think I was a bit cryptic in that reply - sorry!

The underpinnings of "an inertial observer present at two events will always observe a shorter time interval between the two events than any inertial observer not present a both events" come from the Lorentz transform, but the simplest way to put is as follows.

The invariant spacetime interval is given by ds^2 = dt^2 - dx^2 for timelike intervals between two events. The observed (squared) time by any observer is hence: dt^2 = ds^2 + dx^2.

For an observer present at both events, dx = 0, so it is obvious that it gives the minimum value for dt (since ds is the same for all observers).

Hope this clears some of it, at least ;)

Regards,

Burt Jordaan (www.Relativity-4-Engineers.com)

Hi, Burt.

Thank you for this and for your invaluable help. The exercise of changing from a class of transform that allows changes scale (a causal transform) to a class of transform that preserves scale (a Lorentz transform) was very useful to me. The scale change doesn't matter for my argument, but keeping the scale constant is more customary. I learnt a bit having to do a Euclid-style straight-edge-and-compass construction for the Lorentz transform. It is to do with the geometry of the hyperbola, just a bit more complicated than the geometry of a circle.

You write:

"an inertial observer present at two events will always observe a shorter time interval between the two events than any inertial observer not present a both events. 'Observe' here obviously means compensating for light travel time.

"In the twin's case, the traveling twin is the only one present at the turn-around event and hence will observe a smaller outbound time."

These matters are obviously so familiar to you that what seems immediately obvious to you is far from obvious to me who is less familiar with them. Your two maxims here are so compressed into few words that your familiarity gives you an advantage. As I am reading the two above maxims of yours, they seem quite baffling. I am not saying they are not right, just that I have the impression that you find them obvious, while I don't. You seem to have access to some logic that isn't apparent to me. Can you expand a little on your two maxims, to show why they seem obvious to you, that is to say, reveal your logic that underpins them?

Christopher

Hi Christopher, sorry for the late reply - been away from computers (and relativity!) for a while.

Your diagrams are now correct, as far as I could establish and so is your conclusion:

Here we see that the simple claim that clocks are slowed by traveling fast is sheer nonsense, neither true nor false, just physically and conceptually meaningless.

But it is true that identically constructed and identically working clocks can disagree on the time lapse between two events (at which respectively they are present to register together) if they have different adventures on the way between.

I've said something similar on various occasions before: an inertial observer present at two events will always observe a shorter time interval between the two events than any inertial observer not present a both events. 'Observe' here obviously means compensating for light travel time.

In the twin's case, the traveling twin is the only one present at the turn-around event and hence will observe a smaller outbound time. One can ignore the "acceleration event" as insignificant and argue the same for the inbound trip.

This way I keep my sanity in relativity's "paradoxes"!

Regards,

Burt Jordaan (www.Relativity-4-Engineers.com)

David:

Thank you for this.

As you note, there is a wide variety of ways of thinking about such matters.

Christopher

sg.hu/listazas.php3?id=1112296594

Christopher:

Unfortunately it appears that you are using the terms "sensation" and "perception" in essentially the opposite way to the way I tend to think of them. So I looked them up.

Since we are talking about light and color, here, I chose the "Lighting Design Knowledgebase". Sensation: (Term of physiology) The immediate result of the stimulation of the sense organs; as distinguished from perception which involves the combination of incoming sensations with contextual information and past experience so that the objects or events from which the stimuli arise are recognised and assigned meaning.

Perception: (Term of psychology) A meaningful impression obtained through the senses and apprehended by the mind. Perception goes beyond plain sensation in that it includes the results of further processing of the sensed stimuli, either conceously or inconceously.

These are actually more closely aligned with my concepts that what I understand your definitions to be, judging by context within your post. Of course I may simply be misunderstanding what you were saying.

To me there are at least three levels, involving increasing amounts of processing within the brain, with increasing degrees of consciousness, or the extent to which the conscious mind is engaged: Sensual stimulus, to sensation, and on to perception.

Just so we may understand each-other's use of terms.

David

P.S. I looked up sensation and perception in other dictionaries as well, and found nothing to contradict what I have expressed, though most didn't do it as directly as what I quoted above. However, I certainly found definitions that muddy the waters, even to the point of stating that these two terms are synonyms of each-other.

David: Thank you for this. Christopher

Christopher:

Congratulations on recognizing that the "color space" solids* are three dimensional manifolds, apparently without anyone telling you such. :-)

If you are saying that, conceptually, at least, the color perception space, at the perception level (as opposed to our assigning coordinate like labels to the individual colors, for our own convenience/thinking) is not some "numerically coordinated set", I may be inclined to agree. However, there is the apparent "fact" that the firing of neurons is equivalent to "numbers", at least at some level. Really, how are we to distinguish them?

However, the physical space in which we dwell is certainly not any kind of "numerically coordinated set", until we humans assign such labels to points within it. This is one of the features of the general coordinate system approach used by General Relativity that strongly appeals to me.

The feature being that the phenomena of nature care absolutely nothing for what coordinates we may assign to the points of space and time. They will "merrily" go about "doing their thing", completely independent of our choice for any such labeling.

Therefore, until the coordinate independent nature of natural phenomena is somehow shown to disagree with nature I consider that any and all theories of natural phenomena must be independent of the assignment of coordinates. In other words, all must be invariant to general coordinate transformations, without having anything other than (unchanging) constants and "dynamical" quantities in their equations.

Incidentally, on the subject of the manifold nature of human color perception, I ran into an article, a while back, that purported to have found that the human perception of shades of gray, in terms of light/dark, was not one dimensional, as one would expect (since we generally consider light/dark, shades of gray, to be able to be characterized by a single parameter). Instead they purported to have found evidence that it is two dimensional. (This makes me wonder whether human perceived color space may be six dimensional, or maybe even more.)

Certainly it is known that our perception of colors does depend upon what other colors are around.

Of course if all we have are "distances" we can get into trouble by assuming a metric relation, when it is possible there is only a norm relation. I'm not sure what the "angle" or "inner product" relations would be, let alone mean, for colors. However, we do know that linear combinations of colors do create other perceived colors for all non-negative linear combinations (meaning linear combinations where the coefficients are all non-negative).

Of course we know that physically realized colors actually lie in a function space (the color spectrum of the pigment/filter/light/whatever), however, our color sense is not so sophisticated as to be able to distinguish this space. What we see is some kind of projection into a space of smaller dimension (almost certainly of finite dimension, but I'm not completely certain that even this can be taken as a given, though I find it highly likely).

Recognizing the possibility that things may not be as simple as we may believe is important to keep in mind as one investigates a given phenomena. However, the nature of science is the continual cycle of: Theory/hypothesis/supposition/idea -> experiment/observation -> analysis/fitting/pondering -> back to theory/hypothesis/supposition/idea. And, as humans, our thinking is usually facilitated by starting with simpler models first. (Besides, Newton's first "Rule of Reasoning in Philosophy" and Occam's razor are designed to favor the simpler models.) So, as evidence accumulates, and as investigators actually look outside the present models to see if things are as simple as we expect/assume, then the science advances.

David

* They are solids only because they are three dimensional (in a "filled figure" sense), not because they are solid objects within our physical space, except when someone makes a physical representation of one.

David:

The idea just came to me as I was writing. I have no idea where it came from. I do not recall reading it, but that is perhaps just testimony to my poor memory. I think I can safely say I did not get it from Spivak, nor from Riemann, because I have read them only recently. When I looked it up for you, I seem to recall a Wikipedia article that mentions the difference between Euclidean and Riemannian geometry and may perhaps lead you to a useful reference, but looking again just now I did not find it again.

Yes, I find it helpful as an example of a manifold that conceptually does not start out as a numerically coordinatised set, but only becomes one as a consequence of its "distance", considered as a function of point pairs. I would be interested to hear of other examples.

Christopher

Christopher:

I'm not so interested in "who first thought of the colour solid as a manifold." What I was asking was whether the ideas was your own, or if you read it somewhere? If it was your own, then I congratulate you, even if others have thought of it first. If you read it somewhere, then I simply would like a reference. (If I really care to know "who first thought" of it, I'll try and pursue it through the reference you provide.)

Simply read what I originally asked

Was the idea yours, or did you read about this idea in one of Spivak's volumes? Was the idea Spivak's, or Riemann's, or someone-else's?

And my secondary question

However, if this idea wasn't your own, did you find this idea helpful in understanding manifolds?

These are the questions I'm seeking answers to.

David

David:

A little grammatical carelessness there, sorry. I meant that the distance in just-noticeable-differences is a metric, within experimental approximation.

I have no idea who first thought of the colour solid as a manifold.

Christopher

All:

Did anyone besides Christopher know that when he was talking about a "colour solid" in previous posts (Re: Re: A different theory of Gravity; considerably edited now, Re3: A different theory of Gravity; considerably edited now, and first introduced in Re: A different theory of Gravity; considerably edited now) he was actually referring to the visual perception "color space" "solid" given in the referenced post Colour space (namely the links http://en.wikipedia.org/wiki/Color_solid, and
http://en.wikipedia.org/wiki/Munsell_color_system)?

If any of you (besides Christopher, of course) recognized this before the referenced Colour space post, I'll feel exceedingly sheepish at not having recognized this. :-) I was picturing something more like a solid with six, or so, colored sides, like maybe a rounded colored cube (maybe like one of the frames of the Windows "3D FlowerBox" screen saver).

I don't know, maybe I just, somehow, got on the wrong track, but I don't see how I was supposed to guess/know that he was referring to something altogether different.

Talk about miscommunication. :-)

David

Christopher:

This is not at all what I meant when I asked

Your the first person (besides myself) that I have experienced that has had such an idea. Was the idea yours, or did you read about this idea in one of Spivak's volumes? Was the idea Spivak's, or Riemann's, or someone-else's?

I was referring to the idea of using a "color space" manifold as an example of a manifold. I was most certainly not asking about whose idea it was to map perceived colors into a "color solid". (I know essentially all that you have expressed in your latest post.)

Incidentally, it would make absolutely no sense to talk about "distance" between colors using some human physical length scale: The units are not compatible (unless someone finds a "fundamental conversion constant" of some kind). Now, while this "color space" manifold is not related to physical space (like spacetime, or some such) that doesn't preclude it from being some "natural space" (like some "perception space", or some part of some "cognitive space"). And just because it doesn't have to do with some physical space in which objects we can touch dwell doesn't make it "metaphorical". (Though its mapping into a solid object within our three dimensional [approximately] Euclidean space most certainly is metaphorical [baring some "fundamental conversion constant" of some kind].)

By the way, did you simply stumble over terminology, or do you not quite recognize the distinction between a manifold (even one with a metric) and a metric itself? You say (emphasis added)

But I think it comes, within the range of experimental approximation, as you say, to be a metric in the sense that Riemann meant.

It certainly is a manifold (there is an essentially continuous neighborhood of points, and the set is open). It can, possibly, be considered to be a manifold with a norm, and possibly even a metric (one would have to determine whether there is an "angle" or "inner product" measure, even though there certainly does appear to be a "distance" measure). However, it most certainly is not a metric, itself, by any stretch of my imagination, at least. :-)

Of course, if "This [the 'metaphorical' nature of a "color space" manifold] is the basis of my story that some geometries are naturally geometrical and others are only metaphorically so", then why have I been getting the impression that you lump all non-uniformly curved manifolds, including "physical space" type manifolds into this "metaphorical" (as opposed to "geometrical") classification? I have no problem considering some manifolds as non-physical, even "metaphorical". I can even handle presenting (or even believing, or allowing others that option) that the four dimensional spacetime is simply a construct (call it "metaphorical" if you want, but I would not present it with this bias, at least not to students), it helps avoid the push-back from those that simply must believe there is three-space that changes over some separate thing called "time". (It appears, at least, to help smooth the transition.)

However, I do believe one should keep in mind that whatever constructs we as humans may employ, they may have little to do with how nature actually works. The best we can ever do is find no conflict between the workings of nature and our constructs. :-) (Again, this is in spite of the fact that I am, at heart, an ontologist: I want to know more than just "the appearances"; I want to know "what's really going on under the hood".)

This is why I always return to the need for experimental/observational correspondence with nature.* Anything that is more or less than this is not science, in my opinion.

David

* Any construct that succeeds at the correspondence test is perfectly viable. Furthermore, if more than one such theory exist that never differ in their predictions of observations or the results of experiments, then the only difference is "interpretation". In which case the choice is simply a matter of preference, though it is usually informed by both Newton's first "Rule of Reasoning in Philosophy" and Occam's razor.

David:

I am no expert on the colour solid. The Wikipedia tells me that it was invented by http://en.wikipedia.org/wiki/Philipp_Otto_Runge, who died in 1810, after long correspondence with Goethe. Riemann was born in 1826.

Colour is not as simple as one might assume, not as simple as I have an idea that Newton assumed? At the risk of being mistaken, and emphasising that I am no expert, I recall that the sensorium continuously constructs a temporary "reference white" as a kind of average of the whole visual field, and then takes ratios of patches of colour to the reference. A sort of projective geometry? This was partly understood by Goethe, a quite remarkable fact of history, I seem to recall. Even today it is not quite readily known perhaps.

The sensation of colour comes from light stimulation of patches of the retina, which has cells which respectively contain just one of four genetically determined visual pigments, white, red, blue, and green. It seems that various linear operations are performed by the network of nerve cells of the retina to produce colour signals for propagation to the brain. The normal four pigment responses are reduced to a three dimensional manifold by the ratio business, I think. Colour-blind people have defects of pigment production, and so they have radically abnormal colour solids, not just minor variations.

The sensations of colour exemplify the subjective qualia of experience in the mode of presentational immediacy. For example, one usually would not say that the note A above middle C is at a higher or lower pitch than yellow. "Secondary quality" was recognised in this sense by Galileo, and discussed further by Locke. According to one view, that I am rather inclined to accept, perception is a thing different from sensation, which is always consciously experienced. Perception is about behavioural response, is in a sense quite objective, is not necessarily immediate, and can be consciously or unconsciously experienced. But the sensation of colour is purely subjective and immediately conscious. (This is just one of the senses of the word conscious, that is relevant to the present context.) In general, one cannot talk or think oneself into seeing something as having a certain colour; it is just immediately present to one. Nevertheless one can make behavioural responses when presented with comparisons, and many repetitions lead to fairly good accuracy. The results have many practical industrial applications. I would not slight animal perception without some careful evidence; different species have different powers of colour perception; of course we do not know what sensations non-human animals have. I think the industrial usefulness of the colour solid depends on its being rather well reproducible and the nearly the same for all colour-perception-normal people.

That there are four pigments places very strict constraints on what can be sensed and what can be perceived, of course. The colours are subjective, and can only be made objective by way of repeated comparisons and so forth. Of course one can easily assign number coordinates to mixtures of monochromatic light beams; by themselves light beams do not have colour in the strict sense of the word. But there is no proper way to assign number coordinates to subjective colours apart from various methods of comparison, which in effect have to presented as a kind of metric.

It seems to me that the notion of distance between colours is therefore not naturally geometrical in the sense of being a measurement of a natural space. For example, I don't know what it might mean to say that red is four inches to the left of green. The notion of distance between colours is metaphorically geometrical in the sense of being a measurement of a metaphorical space. But I think it comes, within the range of experimental approximation, as you say, to be a metric in the sense that Riemann meant.

This is the basis of my story that some geometries are naturally geometrical and others are only metaphorically so. I think it is a distinction worth making.

Christopher

Christopher:

Ah! Why didn't you say so when I first mentioned that I wasn't sure what you were referring to?!?

Yes, I have often thought that one should be able to determine an approximation to a "color metric" based upon "just-noticeable-differences" in color within such color spaces! So, yes, this is, indeed, an example of a manifold, though trying to get down to anything like an "exact" "differential" "color metric" form is probably nigh unto impossible, due to the crudity of human perception. (Actually, really good attempts at such fine level distinctions will probably tell us more about variations between humans, than any "intrinsic" "color manifold".)

Your the first person (besides myself) that I have experienced that has had such an idea. Was the idea yours, or did you read about this idea in one of Spivak's volumes? Was the idea Spivak's, or Riemann's, or someone-else's?

So, yes, I take it back! :-) The color solids are examples of different coordinatizations of a "color space" manifold.

However, if this idea wasn't your own, did you find this idea helpful in understanding manifolds? Based upon the way you rail against curved spaces as "not being 'spaces'" (as you consider 'spaces') and not being "geometries", either—even though their "inventor", Riemann, considered them so to be—I have to wonder whether you've "got it".

So many questions now that I finally have a picture of what you've been talking about! See how easy it is to have misunderstandings when terms are not properly defined, and how easy it is to clear such misunderstandings up?!?

I only wish you had expressed this long ago, instead of, apparently, assuming that I "knew what you meant"! Now we can talk on the same page, instead of "passed" each-other!

Whew!

David

David:

Thank you for this clarification of which paragraphs you were referring to.

Christopher

David:

When I wrote of the colour solid, I had in mind things such as you will see at http://en.wikipedia.org/wiki/Color_solid, and at
http://en.wikipedia.org/wiki/Munsell_color_system.

Christopher

Christopher:

(In what follows, and from now on, generally, I will simply use the terms metric, inner product, Riemannian, etc. to include both the strictly positive definite forms that mathematicians restrict such to and to the indefinite non-degenerate forms we physicists apply such to. So I hope this will not lead to any confusion. However, since you are so very fond of Minkowski space, I don't think you'll object.)

On the "colour[sic] solid", sou said

I don't see why the colour solid would not pass muster as a manifold with boundary, mapped into an oriented volume element if you like? The oriented volume element is geometrical, I would say. The colour solid in abstracto has, as I understand it, a nice non-geometrical metric, namely the "distance" in just-noticeable-differences; I think that is pretty close to being a metric, at least for smallish "distances"?

Actually, the oriented volume element has nothing to do with a manifold, and is even independent of a metric (though if one has a metric one can use it to "normalize" the volume element). Furthermore, since I guessed (apparently correctly, judging by this paragraph) that you were thinking of the "colour[sic] solid" as an infinitesimal object, I was thinking of the infinitesimal oriented volume element. (A finite oriented volume element is not well defined in any finite sense, since it can be highly deformed without changing its fundamental properties. In addition, if the space is curved, there are even worse ambiguities, and even, potentially, inconsistencies in trying to define a finite oriented volume element.)

In terms of an infinitesimal "space" (your "just-noticeable-differences", though even smaller) the infinitesimal oriented volume element with metric (what you apparently refer to as the "colour[sic] solid") is most closely related to the tangent space associated with a single point of the manifold.* It is a vector space (actually a metric space, when we have a metric), and is an infinite open flat manifold (if you wish). So it is just as much "geometrical" as any other infinite open flat space (especially any such that have a metric of similar character). (Incidentally, you do recognize that a metric provides far more than just "distances", but inner products and thus angles as well?)

In this vector space reside things like velocities, momentum, etc. Hardly just "infinitesimal" quantities. In fact the whole set of tensors, like the curvature tensor, and the mass-energy-momentum-stress tensor reside in this tangent space (or, perhaps more properly, within product spaces of such). Again, not just infinitesimal quantities.

It should be quite obvious that this tangent space is just as "geometrical" as any other vector space (with or without a metric). (Of course there are also the dual spaces of these tangent spaces. These dual spaces are also vector spaces. However, if one has a metric one can map the dual space right back one-to-one and onto the tangent space.)

However, though this tangent space contains arbitrarily large vector and tensor quantities, in general (unless one can identify the manifold with its tangent space[s]) the tangent space can only contain infinitesimal translation ("distance") vectors, since finite translations cary a point within the manifold to a point associated with a different tangent space. (One is permitted to map all the individual tangent spaces into a single vector space of the same character. However, this will involve transforming all the individual metrics into a single metric [which, in general, requires a non-coordinate transformation]. Besides, one really gains nothing, and one must be careful not to fall prey to the temptation to directly compare vectors associated with different points on the manifold, since such can only be compared via "parallel transport" along a path between the points.)

If you have read Riemann's treatise on geometry within Spovak's book, I would hope that he would have been able to help you understand that there is at least as much "geometry" about curved manifolds as there are with (flat) vector spaces. After all, the latter are simply special cases of the former.

On the other hand, if one is insistent that translations be contained within the transformation group of the points (and even other objects) within a manifold, then one will be greatly restricting the possible manifolds. (As the "Relativity without tears" paper points out, constant curvature manifolds are certainly allowed.)

I could go on (and on) about how such a restriction is not called for by nature/observation**, and even how particle physicists are questioning the physicality of the Poincaré group. In fact, some are questioning whether there is any way to incorporate gravity using such. (And even when the Poincaré group is used, now days, it is "gauged", so it is no longer the global symmetry involved in the previous paragraph, but only a local form, quite analogous to the tangent spaces of curved manifolds.)

However, getting back to my comment that lead to our discussion of the appropriateness (or not) of using a "colour[sic] solid" as an illustration of "manifolds". While the "colour[sic] solid" may be illustrative of the tangent space associated with a point on a manifold (though the fact that you apparently thought it was restricted to "just-noticeable-differences" suggests it's even rather poor at that), I would rather use some nice topological surface, or even the surface of an apple (as Misner, Throrne, and Wheeler did—which actually is even more clever when one considers the association between apples and Newton's theory of universal gravitation).

Oh, there's so much more here, but I'm sure this is plenty for now.

David

* Actually, the infinitesimal oriented volume element (with or without a metric) can be said to generate the tangent space.

** After all, why should we conflate the concept of position with that of vectors? How many students get confused at how position "vectors" "must" be anchored at an "origin", while other vectors are free to be unanchored? Or other such issues?

Christopher :

(EDITED: Unfortunately, when I wrote the following reply I had failed to notice that I had used complete in my reference to the quoted paragraphs. I then realized how the inclusion of this modifier in my characterization of "complete 'throwaway' asides" can be interpreted in a negative way that really goes well beyond my intent. I'm ever so sorry. :-{ Please forgive my over characterization of the asides.

Also, Christopher, in your reply to my other post, please hold most of your reply that pertains to these asides for a reply to this post, so we can try to keep separate issues separate. Unless, of course, you really don't see them as separate at all. :-) )

I would say that to at least some extent all four of the following paragraphs are part of a "throwaway" aside. That is not to say that they don't have anything to say. However, they are, to at least some degree, orthogonal to your primary arguments and points. (I did, to only a very slight degree, refer to part of the first of these paragraphs. However, it was a point already made in prior posts.)

In contrast to the foregoing, as Burt has previously pointed out, when it is described in the time system of the neutral observer, the whole adventure will seem paradoxical or unintuitive, or one might even say apparently nonsensical. This is because the neutral observer’s reference frame has a jolt that for him is simultaneous with his turn-around event; at that event, his reference frame suffers an impulsive acceleration. His remotely stationed research assistants instantaneously move respective finite distances.

Let us see what this accelerated reference frame will report for the particular example scenario described in this diagram. The time axes of the two legs of the neutral observer’s reference frame are joined at the event P and its symmetrical partner. Some time after the outset of the travelling clocks from home, a second version of the home-clock appears from nowhere, and the home-clock begins to be present twice simultaneously moving at two different spatial places in the reference system of the neutral observer, one instance moving towards him, the other away from him. This is not at all physically mysterious or strange. It is simply that the reference system is an artefact, with artificial adventures due to the accelerated reference frame. Shortly after that duplicate appearance, the away-clock, at the event T simultaneous with the event L, disappears from this reference system, vanishing into nowhere. Again, this disappearance into nowhere is a mere artefact, not physically strange. Then, after some time, simultaneously with the turn-around event of the neutral observer, the two different records of the moving home-clocks cross each other instantaneously in the same event. The home-clock then continues to be simultaneously twice present moving at two different places in this reference system. After a time the away-clock reappears from nowhere in this reference system. Shortly after that, the home-clock that is moving away from the neutral observer disappears at event M into nowhere. Again these ‘events’ are just artefact. At a time after that reappearance, the three clocks meet in the final event of the scenario.

Accelerated reference frames do not preserve causality. The previous paragraph describes an example of an accelerated reference frame that, if mistakenly interpreted as physical, would seem to violate causality. The physicist knows from experience that it is the peculiarity of the accelerated reference frame that is the reason for the nonsensical appearance, and that nature does not violate causality. Real physical clocks do not appear from nowhere, nor do they disappear into nowhere, nor do they occur in two separate places simultaneously. That is why the physicist uses Minkowski spacetime and inertial reference frames, which preserve causality. Alfred Arthur Robb in 1913 showed the tight connection between causality and Minkowski spacetime by an argument based on causal ordering.

The foregoing demonstration is in terms of a fixed scale for the diagram, but this is not necessary. The Lorentz transformation is a Minkowski isometry, but any of a more general class of transformation of frames would have served the purpose of the demonstration. The scale of the transformed figure does not matter for the demonstration, and is therefore arbitrary for this purpose. Only the shape, defined by the angles, of the transformed figure matters; any similarity transform would do. All that matters for the present purpose is that the transformation preserve causality, that it be a member of the causality group, which is the Poincaré group composed with the group of dilations, (changes of scale, the same for every dimension). The Lorentz transform requires a little extra geometrical construction work, beyond what would be needed for a causality-preserving transform that does not preserve scale, less work to construct but still enough to demonstrate the point of concern here.

The fact is that the first two or three paragraphs would probably have had a greater impact as a separate post, possible in reference to someone else's post.

I can see where you were caught up in your causality comments (particularly from the third paragraph) and so brought in the additional aside of the fourth paragraph.

Of course I'm certainly not immune to such diversions myself. So it was perhaps quite unfair for me to even mention such. For that, I'm sorry. :-{

However, when I do succumb to such diversions within a single post* I usually try to move them down to a footnote or a "P.S." But I'm sure there are times you may be able to point to where I had somewhat similar "throwaway" asides within some of my longer posts. (These small editing boxes can make it difficult to see when such is happening. I do, however, try to preview what I have written several times, and even edit a number of times after posting, unless I'm heavily constrained by time.)

David

* I usually try to put such diversions/asides in separate posts, at least if they appear to be getting too large.

David:

Thank you for this. It will help me to consider my reply if you will please post me as to just which paragraphs you refer to when you write of

"at least two paragraphs that are complete "throwaway" asides."

Christopher

Christopher:

It appears to me that you multiply many words but say little. (You have at least two paragraphs that are complete "throwaway" asides.)

Your graphic constructs are just fine. However, your conclusions (presuming that the embolden sentences are conclusions, as they read to be, along with your last one or two paragraphs) leave much to be desired.

First, you conclude, based upon the "neutral" observer's outward going reference frame

The difference in total time registered is therefore not due to difference in the test clocks’ rates, nor in their speeds, nor in their accelerations. It is because one is inertial and the other is not; they have different adventures. Since the home-clock’s adventures are all inertial at home, and the comparison is made at home, it registers most time there.

If this were the case, then why do the two "away" clocks not match when they return to "home"? Both are non-inertial, neither has a "home" like "adventure".

This concept of different "adventures", in and of itself, provides no ability to compare clocks, or anything else, between different "adventures" (the terminology of choice is spacetime paths, by the way). An additional structure/construct/characteristic is required for such.

By your reckoning one could say that the reason one side of a triangle is always shorter than the sum of the other two sides is because the other two sides have "different adventures". Yet the comparisons available for Spacial Relativity (SR) are quite comparable to those of Euclidean Geometry: The only difference is in the nature/character (signature) of the metric (Euclidean vs. Minkowskian). In my mind, at least, there can be nothing simpler. No metaphysical waffling about "different adventures", or changes in simultaneity (especially resynchronization of clocks or "remotely stationed research assistants instantaneously move[ing]" in space or time [I know this isn't your doing, Christopher, it's a problem with those that insist upon seeing everything via inertial reference frames without thinking deeply about what that would mean physically]). Just nice local measurements, and an accumulation thereof (anything with a memory, like clocks).

On the matter of so called "time dilation"

Here we see that the simple claim that clocks are slowed by travelling fast is sheer nonsense, neither true nor false, just physically and conceptually meaningless.

In a sense, you are correct (and a somewhat similar thing can be said of so called "length contraction"). In the sense of the metric nature of spacetime, in analogy with the metric nature of Euclidean space, the characteristics of different speeds ("boosts", in terms of the Lorentz transformations) are analogous to characteristics of different rotations in Euclidean space. So nothing has actually changed, in the sense of the things themselves relative to themselves, only their "appearances" or their "projections" back into the "original" (or some other) coordinate system.

This doesn't, however, make the "rotation" (boost, in the Minkowski case) any less "real", or the changes in "appearance" any less "real", since they do have real, measurable consequences. It just means that there is a perspective (in terms of how one thinks about what one is observing) available that renders such issues as "time dilation" and "space contraction" in such a form that it's more analogous with "appearance", in a non-the-less measurable way (just as the "height" of a square changes when one rotates it in Euclidean space).

It's a mater of perspective. (Sorry for the double entendre, but I felt that it fit.)

This, of course, has bearing on your next "conclusion"

But it is true that identically constructed and identically working clocks can disagree on the time lapse between two events (at which respectively they are present to register together) if they have different adventures on the way between.

Or, rather, they "can disagree on the time lapse between two events (at which respectively they are present to register together) if they" take different paths through spacetime. The metric nature of spacetime, just as with Euclidean space, means that different paths may have different path lengths. In Minkowski spacetime, due to the expectation of the orientation of any observer's own time axis (namely along one's own path in spacetime), this path length is proportional to the "proper" time accumulated along the path. (Since the proportionality is constant, for the most "perfect" of clocks [meaning that they are as insensitive as theoretically possible to changes in motion, accelerations, etc.], a suitable scaling can be defined such that they are identical.)

Unfortunately, your paragraph pertaining to "velocity" vs. "acceleration" "affecting" (or not) "the speed of clocks" is flawed. This is primarily due to your reliance on some fictitious need for "zero re-settings of clocks". After all, any clock can always be effectively re-set to whatever time (including zero) one may desire at any time—even particle "clocks". (If nothing else, it can always be accomplished via the logging system.) In fact, another rather interesting feature of particle "clocks" (meaning the "clock" that is related to particle decay) is that they are always continually being re-set to zero. The particle decay rate/probability does not depend on how long the particle has been in existence. The particle has no memory, its "clock" has no memory.

Of course you are correct that "The term ‘synchronisation’[sic] means coordinated re-setting of zeros of several clocks." And, yes, "It usually does not refer to clock rates." However, as I have tried to point out in prior posts (and alluded to above), such "synchronization" is not all that helpful in so many instances, especially in the case of non-inertial "reference frames". (Actually, I consider the term "non-inertial reference frame" to be an oxymoron. But to each his own.)

As per your last paragraph/conclusion

Time is not an event, nor a process, and so it cannot be caused, nor can it be a cause. It does not make sense to say that velocity or acceleration alter time, because such a statement seems to imply that time can be subject to, or patient of, causal agency. Time is in a sense part of the abstract theoretical receptacle that encompasses events and processes, which are causes and effects.

You are correct that time, itself, is not an event, anymore than position is. Of course the "time measured on a clock as it passed a certain position" is an event. Also, all the times expressed by a given clock are events (a sequence of such).

Any single even has a label/tag (or ordered tuple of such) associated with it, within any given observer's system of reckoning (we usually codify this as a coordinate system, and in SR, for inertial observers, we have a prescription for choosing a particular coordinatization). However, the labels/tags, or the individual parts of any such labeling/tagging system (such as the individual slots of tuples), are not, in themselves, events.

However, when one talks of "time" being "altered" by motion (speed, acceleration, whatever) one can still be correct if one is referring to how such may "alter"/"affect" some "natural" choice of "time coordinate". On the other hand, if one is referring to events, one should be more careful in one's choice of terms, and refer to the "passage of time as measured by .." or some such. I agree that we shouldn't refer to time as some thing that exists unto itself. We have labels (or, rather, parts thereof) that we may call "time", and we have devices/things that can be used to log/"mark" "time", or the passage thereof, but there isn't some independent "entity" as "time". (Such terminology stems from a "common language usage" like issue. It stems from our "common" perception of the passage of personal time that we have a tendency to universalize. [We have to retrain ourselves not to think in such terms, just as we had to retrain ourselves to recognize that the people around us do not share our thoughts.])

So, in conclusion, you're not too far off in most things. There's just a few areas that could use some clarification or the addition of alternate perspectives (in the "ways of thinking" sense). The worst trouble that I saw, however, involved your reliance upon some concept of "arbitrary zero re-settings of clocks" and a false perception that such is impossible for "particles" in order to argue that statements to the effect "that it is velocity, not acceleration, that affects the speed of clocks" is "meaningless". I'm sorry, such doesn't hold together. (However, an argument can be made that the theoretically best possible clocks are unaffected by both velocity and acceleration, including gravitation. But this is within a particular perspective [think proper time vs. path length].)

David

P.S. You cannot make any "decisive comparison" in the "reference system of the neutral observer" any more than you can make any "decisive comparison" in any other reference system. (Even if we disregard the fact that the "reference system of the neutral observer" is only inertial for part of the total problem.)

David:

Thank you for this.

Christopher

Christopher:

On the matter of

One likes to be free to check a theory in diverse ways that one does not at first anticipate, and one (at least those of us who are not perfect) cannot be sure that one's logic is so perfect that one can say "Oh, look, I have found the way to check this theory once and for all!"

First, refer to what I quoted from Feynman about always doubting.

Second, I hope you recognize that there can be no such thing as "Oh, look, I have found the way to check this theory once and for all!" At least not in the affirmative, since there is simply no way to "prove" a theory, unlike proving theorems in mathematics*: One can only "prove" a theory incorrect (inconsistent with reality/nature); one can only invalidate (falsify) a theory.

Third, there's absolutely nothing to keep anyone from testing anything, via observations and/or experiments, concerning any theory any human being has or ever will come up with. Yes, there are some things within General Relativity (GR) for which "the results are assumed by the formalism". However, such does not make any "desirable empirical checks logically meaningless". After all, the experiments are not governed by the theory, only by nature, and the ingenuity of the experimenter/observer.

For instance, GR "assumes" the equivalence of inertial and gravitational mass**, but that hasn't prevented experiments from being run to test this assumption.

Now, as I understand the Whitehead/Logunov concept, at least, they suggest, at least, that there is some underlying Minkowski space (flat spacetime) underlying the "observable" ("effective") "metric-field" that GR want's to call the actual "metric". Fair enough. The only question is whether there is some experiment and/or observation that can distinguish between the predictions of such a theory and those of GR. It's as simple and as complicated as that! :-)

If the underlying Minkowski space is unobservable, as is the case with the self consistent spin-2 (tensor) "particle"/"field" theory of Feynman (and I think Steven Weinberg as well***), then even if one considers such to differ from GR at some "gut"/"fundamental"/conceptual/ontological level, it is completely equivalent in the only area that truly matters: observationally and experimentally. (And I say this as an unabashed ontologist!) Anything else is a matter of interpretation/preference. (Though both Newton's first "Rule of Reasoning in Philosophy" and Occam's razor will argue that if one cannot observe something, or any consequence thereof [such as the presence of some unobservable Minkowski space/metric] then one really shouldn't keep it around.)

As for the charge of "precariousness" you apparently lay upon GR, to which you say you are referring to its "reliance on things that are not easily checked". I say "guilty as charged". (Even more so with regard to how little wiggle room the theory gives itself.) Of course all human theories have had this "issue". We as humans have to start somewhere, after all.

One of the "biggies", in my opinion, is the "assumption" that spacetime is a continuum. How can we check that it's not simply some discreet "space" that's just too finely divided for us to tell the difference?

Ah, but that's another one of those things I said can, at least in principle, be tested by any sufficiently ingenious experimenter/observer! True, at this point we can only say that any lack of continuity has to be below some threshold, that any discreetness must be below some size. This is no different than bounds upon how massive the photon or the "graviton" can be! (Incidentally, last time I checked, the constraint on the graviton's mass was tighter than that of the photon.)

If the theory is "precarious" in that it depends on something that's not absolutely "nailed down solid", then that's an opportunity for an ingenious experimentalist and/or observer! Have at it! (See what Feynman has to say about the potential for excitement physicists can look forward to if any of our great theories are falsified! What fun!)

As for the "ambiguous and so untestable" charge that is apparently lain down by Logunov, I'll have to check into what he is actually accusing GR of before I make a judgement.

David

* Even theorems in mathematics are only "if ... then ..." statements (with the best, in my opinion, being "if and only if ... then ..." statements, which are really only two coupled "if ... then ..." statements). So if their proposition is not satisfied that you have no guarantee that the result will hold. (Of course the "if and only if ... then ..." cases go further to guarantee that the result does not hold, in such case.)

** Actually it only "requires" the constant ratio of what one could measure of such, but that's beside the point.

*** I have found this theory expressed with Misner, Thorne, and Wheeler's Gravitation, with references to a number of Feynman's papers/writings. It only makes sense that two great particle physicists would come up with a particle/field theoretical explanation for gravity. What's really interesting is that when they try to make it completely self consistent they find that the original Minkowski metric of the theory is no longer present in any of the equations, including the equations of motion of the matter. So the Minkowski metric/space is no longer observable! And what they are left with are equations that are exactly equivalent to GR! There is nothing that is observationally different. Hence their statements that it is "the same as GR" (even though, at a conceptual level, at an ontological level, it does differ from GR).

I am new,I hope it is ok to reply.This intense reflexion make me do some introspection and it will not end,now.I was honestly mad about some part and also strongly happy about other.This bring to my rigid undiplomatic hard head,a question?I love when it is good for me.I say love.I hate what could be wrong for my believe.It is to early for anwser my self,and may be I am not directly to the topic it self,surely I grasp the depth.beautiful ,and thank you ,phil

Hi, Burt.

Last time I chose an unsuitable would-be “neutral observer”. I stipulated that he move so as to be halfway in spatial distance between the two test clocks; bad choice. Also my diagram had some glitches which you reported; thank you for that. Let me try again.

Let us look at a Minkowski diagram of the twin clocks scenario for the study of “Einstein’s time dilation”, drawn in the reference frame of the home-clock. The two clocks are identically constructed and simultaneously zeroed at home. The home-clock stays inertial. The away-clock moves out from the home event, O, and back again, travelling inertially at one and the same speed except at the turn-around event, T´, which in the reference frame of the home-clock is simultaneous with an event M´ at the home-clock. We compare the durations of time that these two test clocks register between their separation event, when the away-clock sets out, and their re-meeting event, when the away-clock arrives back. The scenario has two legs, outward and homeward. The total times can be found and compared by putting together the two leg times.

The outward-homeward symmetry about M´T´ means that we need look closely at just one leg, let it be the outward.

For the sake of a demonstration, we introduce a suitably chosen neutral observer. The neutral observer’s clock moves between the two test clocks, so that in his reference frame the two test clocks move at equal and opposite velocities, away from him in opposite senses; this is the desired demonstrative symmetry that makes the two test clocks run at equal rates and move at equal speeds.

In the next Minkowski diagram, the broken lines are world lines (dashed) and lines-of-simultaneity (dotted) for the home-clock (blue), the away-clock (red), and the neutral clock moving at equal and opposite velocities between them (green), plotted in the frame of reference of the home clock. The full lines are for the same, plotted in the frame of reference of the neutral clock. The diagram shows a transformation of reference frames that takes the broken lines into the full lines. Events labelled for the home-clock reference frame are primed, for example M´; for the neutral observer’s reference frame unprimed, for example M. The transformation takes the turning point T´ into T; it takes the "mid-time" event of the home-clock M´ into M; and it takes the "halftime" event of the outward leg of the home clock, H´, simultaneous with T´ as judged in the reference frame of the neutral observer, into H. c and -c mark the light cone of the origin O. x marks the distance and t the time axes.

In the reference frame of the home-clock: The turn-around events, M´ , P´ , and T´, though simultaneous, will occur not when the neutral observer is at half spatial distance between the two test clocks; but rather, when the neutral observer’s clock is spatially nearer to the away-clock than to the home-clock:
spatial distance M´P´ > spatial distance P´T´.

The two reference frames are related by a Lorentz transform. It requires that space be isotropic, that is to say, that if observer A moves inertially at velocity v with respect to observer B, then observer B must be travelling inertially at velocity –v with respect to observer A. The Lorentz transform preserves scale. Also it preserves collinearity. The events M, P and T are collinear in a line of simultaneity in the reference system of the neutral observer, because they are simultaneous for the home-clock’s reference system, at M´, P´, and T´.

The transformation diagram was drawn (with the aid of my textbook from schoolboy days) by elementary straightedge and compass methods known to Euclid, according to simple rules of Minkowski geometry. No numerical calculations here.

We will make our decisive comparison in the reference system of the neutral observer: The turn-around event T of the away-clock is simultaneous with an event H at the home clock that occurs before the turn-around event P of the neutral observer’s clock