Understand Special Relativity and Time paradox

[PAllen]

I object to giving any physical meaning to simultaneous space. Simultaneity is a convention. For inertial observers (or in an inertial frame used to analyze some overall scenario), there is a standard convention any reasonable person would use; how far it makes sense to extend it (for an observer) depends on how long they have been inertial. For non-inertial observers there is no preferred convention except 'locally'. A non-inertial observer is analogous to the GR situation - only local frames (with standard simultaneity convention reasonably preferred sufficiently locally in time and space).

I believe this is how Einstein viewed it, but that is neither here nor there.

[There is also the sense of relatively arbitrarily chosen simultaneity surfaces used to construct coordinates useful for some problem. Obviously, I don't consider coordinates a feature of physical reality.]

 

[PeterDonis]

I'm really getting confused about what your fundamental objections are

I'm not objecting to any of the statements you have made about what events are in which simultaneous spaces. If those statements are all you've been trying to say, they strike me as too obvious to be worth taking all this time over.

aside from the side bars on interpretations and philosophy

The sidebars are only there because you have made claims about the interpretation and philosophy of simultaneous spaces. If you would refrain from making such claims we wouldn't need any sidebars.

If you had said things like "I find that looking at hyperplanes of simultaneity helps me to make sense of what is going on" (which is pretty much what LastOneStanding said right before you entered the thread to support what he was saying), I doubt we would have had any sidebars. But you insist on saying things like "hyperplanes of simultaneity are fundamental to relativity", which implies (incorrectly) that they are necessary to *any* understanding of relativity, and then claiming that Einstein said so too, which is a strained (at best) interpretation of what he said.

 

[bobc2]

The weekend is over. Can you please make the OP's requests a priority especially since you are concerned that the moderators are going to lock this thread?

ghwellsjr, I have searched through most of my Einstein writings and must concede that I'm not able to find the reference that I am recalling. Of course there is the possibility that I am mistaken in my recollection, so I'll just have to retract my reference to Einstein discretizing the turnaround into incremental boosts (incremental inertial frames) as I've been describing. Of course the concept is not original with me. You were right to have challenged that. If I ever do come up with it I'll let you know.

By the way, you did a very excellent job of explaining the doppler approach. I've read a number of accounts of this, most recently Paul Davies's discussion, and yours is as good as any and better than most--particularly with your use of the diagrams.

 

[bobc2]

I object to giving any physical meaning to simultaneous space. Simultaneity is a convention. For inertial observers (or in an inertial frame used to analyze some overall scenario), there is a standard convention any reasonable person would use; how far it makes sense to extend it (for an observer) depends on how long they have been inertial. For non-inertial observers there is no preferred convention except 'locally'. A non-inertial observer is analogous to the GR situation - only local frames (with standard simultaneity convention reasonably preferred sufficiently locally in time and space).

I believe this is how Einstein viewed it, but that is neither here nor there.

[There is also the sense of relatively arbitrarily chosen simultaneity surfaces used to construct coordinates useful for some problem. Obviously, I don't consider coordinates a feature of physical reality.]

I guess I just don't catch on to your thinking about how to describe external objective reality with objects moving about in space and time without the use of coordinates. And particularly when we need to select the particular coordinate transformations of the Lorentz group if we are to be assured of physical processes unfolding in the various observer spaces in a manner consistent with the laws of physics.

 

[bobc2]

I'm not objecting to any of the statements you have made about what events are in which simultaneous spaces. If those statements are all you've been trying to say, they strike me as too obvious to be worth taking all this time over.

That's all I'm trying to say. Forum members can muse over any possible implications about the simultaneous spaces with regard to physical reality if they are so inclined. In any case I thought the way the order of the brown clock readings, as they are presented in the traveling twins's frames, was kind of interesting after Vandam had pointed it out in another thread (where is Vandam--he was so pasionate about this stuff?). Others may find nothing of interest there. I never intended to get side tracked into the philosophy of solipsism when I first posted--I tried to keep up with responses to new comments and questions but was inexorably drawn into the sid bars.

 

[Nugatory]

I guess I just don't catch on to your thinking about how to describe external objective reality with objects moving about in space and time without the use of coordinates.

It can be done. Any physical situation can be described in terms of proper time along timelike worldlines and the points at which these worldlines intersect lightlike null worldlines.

However, coordinates are a really really convenient calculating tool in many problems... So we use them a lot.

 

[Dale]

I guess I just don't catch on to your thinking about how to describe external objective reality with objects moving about in space and time without the use of coordinates.

Think about different ways you can describe the location of your house. You can give its latitude and longitude. Alternatively you could give some landmarks e.g. 2.3 miles past the post office on Balderdash Rd.

And particularly when we need to select the particular coordinate transformations of the Lorentz group if we are to be assured of physical processes unfolding in the various observer spaces in a manner consistent with the laws of physics.

The thing is that we already know experimentally that physical processes don't in fact transform according to the Poincare group globally, only locally. So we need to write the laws of physics in a manner that is consistent with completely arbitrary coordinate transforms because we know that the Lorentz transforms don't work globally.

Since we need to do that globally anyway, we can also do it locally. We then clearly see that the laws of physics don't care one bit what coordinate systems we use, and the actual laws of physics are expressed entirely in terms of invariant quantities.

 

[PeterDonis]

I never intended to get side tracked into the philosophy of solipsism when I first posted--I tried to keep up with responses to new comments and questions but was inexorably drawn into the sid bars.

I agree that your first post in this thread (#32, unless I missed something) didn't do anything more than draw the simultaneity planes in different frames and comment on them. But your next post (#34) used the word "fundamental":

I certainly have no fuss about doppler. Any special relativity course would not be complete without understanding that. But the real fundamental stuff of special relativity is intimately related to the time dilation, length contraction and hyperplanes of simultaneity as manifest in the Lorentz transformations and the space-time diagrams.

If you had qualified this with "for me", or "in at least one common method of teaching SR", it would have been different. But you made a blanket statement about what's "fundamental", which comes across as being about something more than just what works best when teaching or explaining.

Then, in post #38, you made the statement that I first responded to:

The attempt to replace the direct Lorentz based relativity of simultaneity with the doppler approach is just an argument based on philosophical ideas.

Again, if you had said "I find it much easier to understand and explain SR using relativity of simultaneity, etc., vs. doppler" that would have been different. But you brought in the "philosophical ideas" (that word had only been used once in this thread before your post, and nobody picked up on that one, by LastOneStanding).

 

[zonde]

I disagree completely. Experiment (scientific method) is the foundation of science and what distinguishes it from philosophy.

Discussions about scientific method are philosophy. Improvements in scientific method like falsifiability are philosophy.

The foundation of science is the scientific method. The scientific method requires that a theory make experimental predictions, but doesn't otherwise constrain the method of making those predictions.

Yes

In relativity the experimental predictions of the theory are all invariants.

Can you elaborate on this? First, is invariant defined or is it undefined basic concept?
Because the way it is usually defined i.e. some quantity that does not change under coordinate transformation, is confusing as it is defining invariants using concept of coordinates and consequently coordinate dependant quantities that we are using to construct coordinates. So coordinate dependant quantities are more basic than invariants.

The invariants are also the same for all frames, so I don't know what makes you think that we are ignoring the significance of the first postulate by focusing on them instead of frame-variant quantities.

Invariants are not the same as physical laws. They are certainly two different things.

 

[PAllen]

I guess I just don't catch on to your thinking about how to describe external objective reality with objects moving about in space and time without the use of coordinates. And particularly when we need to select the particular coordinate transformations of the Lorentz group if we are to be assured of physical processes unfolding in the various observer spaces in a manner consistent with the laws of physics.

I don't say you don't use coordinates. But they are analogous to the lines you draw on a globe to label positions. The globe and relief features on it exist independently of what lines I draw. Coordinates are not an aspect of reality. The Lorentz group is simply the group of transforms that preserve the flat space metric in simplest form. The physical principle of relativity is that absolute (inertial) motion cannot be detected. The difference from Galilean relativity is that light speed is included in what is the same for every inertial observer. There is nothing more special about Minkowski coordinates than there is about Cartesian coordinates on a plane (metric is in simplest form). Its geometry is there with no coordinate labels; if I draw polar coordinates, the geometry hasn't changed, only the process of computing things.

Your claim about some preferred meaning to your chosen 'simultaneity space' is equivalent to insisting that only cartesian coordinates are valid on a plane. Even more, that if we draw some arbitrary curve on a plane, and then want treat it as a coordinate axis, we must use lines perpendicular to its tangent at each point.

 

[Dale]

Discussions about scientific method are philosophy. Improvements in scientific method like falsifiability are philosophy.

I already covered this back in post 83 with my statement: "I cannot think of any philosophical proposition that has any scientific value except for those which are essentially restatements of Bayesian inference."

I was specifically thinking of falsifiability and Occham's razor, both of which can be derived from Bayesian inference, which is the mathematical framework for inductive reasoning. So falsifiability is not a counter-example and I stand by my previous assertion.

Can you elaborate on this? First, is invariant defined or is it undefined basic concept?
Because the way it is usually defined i.e. some quantity that does not change under coordinate transformation, is confusing as it is defining invariants using concept of coordinates and consequently coordinate dependant quantities that we are using to construct coordinates. So coordinate dependant quantities are more basic than invariants.

The term "invariant" itself is indeed defined as a quantity that does not change under coordinate transformations, so that term does presuppose the definition of coordinates etc. However, each invariant physical quantity can be defined without reference to coordinates.

For example, proper time can be defined physically as the time measured by a clock. It can also be defined geometrically as the integral of the spacetime interval along a timelike path. Neither of these definitions require coordinates. Similarly with the other invariant quantities used in physics.

You can define all of your physical theories in terms of these invariant quantities without reference to coordinates. Then, once you add coordinates, you can note that all of the quantities that show up in your physical theories are invariants, and you can refer to them collectively as "invariants" without at all implying that they are less basic than coordinates and coordinate-dependent quantities.

 

[zonde]

However, each invariant physical quantity can be defined without reference to coordinates.

So how do I tell apart invariants from everything else?

 

[Ken G]

So how do I tell apart invariants from everything else?

Perhaps your issue is you have not yet connected the concept of "invariant" with the concept of "objectivity", but that is an important connection to draw because all of science is based on what can be objectively established. The latter means, that which different observers can agree on based on their measurements. Or another way to say it is, physics is about predicting measurements, given that we know some things about the state of the measurer. Relativity adds to that the beautiful idea that the laws we use to establish what those predictions must not depend on those things, only the predictions themselves must. Before relativity, this important distinction was not made-- a measurement established something as true for all observers. With relativity, we found that a measurement only establishes something as true for that measurer, and an observer in a different state might arrive at a different measurement, but can still use the same laws of physics to predict either one of those measurer's results--so long as they account for the measurer's state.

Hence we suddenly had a need for the concept of translating between measurement outcomes, and one way to do that is via invariants. But the invariant is more than just a mathematical trick for doing the translation-- it is the thing that the measurements are referring to, in the sense that it is the thing that is objective. So to me, the main lesson of relativity is that measurements are only "objective" if we keep track of the state of the measurer, whereas the invariants we construct from the measurements are objective in the true sense of being the same for all observers. That's also why a special relativistic invariant is indeed just a kind of mathematical trick, a means to an end, whereas a general relativistic invariant is actually what the laws of physics must refer to (at least, insofar as general relativity is a good theory of physics). That was, after all, Einstein's primary motivation for GR-- he never liked singling out the inertial observers, and I imagine that's because it didn't seem very objective to do so.

I think an analogy can help us see deeper into what objectivity means. Imagine a "chick flick" that is being reviewed. Let's rampantly overgeneralize and say that women like this movie and men find it boring. Now imagine a male reviewer who pans the movie and a female reviewer who says it's oscar-worthy. Are either of those reviews making objective claims about the movie? No, the objective claim, and the best review, are simply the statement that women will like this movie and men will hate it (again ignore the absurdity of such sweeping generalizations about movies). Can we say if the movie is good or not? No, we cannot, there is no objective way to do that-- all that is objective is to account for how each person will experience the movie. And how can we tell how each person will experience the movie? By considering what is invariant about that movie-- what aspects can men and women both agree this movie has? So even though we might thus say that experiencing a movie is something subjective, we can still say that accounting for that experience is objective. It is the latter, not the former, that underpins science, and hence the need for invariants.

That's what relativity is trying to tell us, and it was completely new to science at the time, but then quantum mechanics came along and gave us additional reasons to track what the observer is doing. Personally, I'd say the main lesson of physics of the 20th century is that we can never again imagine that physical reality has an existence completely separate from how we perceive it. But then again, Einstein never liked quantum mechanics!

And on the matter of the "reality" of the concept of relativity of simultaneity, I agree completely with DaleSpam. What's more, I'd say the well-known "Andromeda paradox" that bobc2 is talking about makes pretty clear the unreality of the entire concept of global simultaneity. We should have learned from relativity that simultaneity is a strictly local concept whose usefulness gets diluted with larger and larger (invariant) separation between the events. Maybe this lesson will someday prove false, but it's all we have to go on at the present moment (pun intended).

 

[PeterDonis]

So how do I tell apart invariants from everything else?

Mathematically, they are picked out by being either Lorentz scalars, like the invariant mass of an object, or integrals of Lorentz scalars, like the proper time along a curve. Contrast this with, for example, energy, which is a component of a 4-vector.

 

[Ken G]

(nitpick: 4 vectors are also invariants, as are the tensors of GR. Also, the laws built from those objects are themselves invariants, as are the invariant parameters embedded in those laws like e and c. But measurements are always scalars, so the positivist might further restrict the "real" invariants to just the scalars, whereas a more rationalistic philosopher might allow all the classes of tensorial invariants, and the parameters of the theory, to be considered "objectively real." Personally, I hold that no quantity that has units is something real, but here we are just talking about what can be considered an invariant. So perhaps we must allow zonde that even the concept of an "invariant" contains some troubling elements, a suggestion that we have not penetrated the mystery completely!)

 

[PeterDonis]

4 vectors are also invariants, as are the tensors of GR. Also, the laws built from those objects are themselves invariants

I've seen the term "invariant" used both ways, in the strict sense I used it, and in the more permissive sense you used it. I've also seen the term "covariant" rather than "invariant" used to refer to objects like 4-vectors and tensors whose components change when you change frames. I agree this is more an issue of language than physics; the laws certainly use 4-vectors and tensors as well as scalars.

as are the invariant parameters embedded in those laws like e and c.

Yes, those are basically Lorentz scalars whose values are the same at every event.

But measurements are always scalars

Or the integrals of scalars over a curve or region.

 

[zonde]

I think an analogy can help us see deeper into what objectivity means. Imagine a "chick flick" that is being reviewed. Let's rampantly overgeneralize and say that women like this movie and men find it boring. Now imagine a male reviewer who pans the movie and a female reviewer who says it's oscar-worthy. Are either of those reviews making objective claims about the movie? No, the objective claim, and the best review, are simply the statement that women will like this movie and men will hate it (again ignore the absurdity of such sweeping generalizations about movies). Can we say if the movie is good or not? No, we cannot, there is no objective way to do that-- all that is objective is to account for how each person will experience the movie. And how can we tell how each person will experience the movie? By considering what is invariant about that movie-- what aspects can men and women both agree this movie has? So even though we might thus say that experiencing a movie is something subjective, we can still say that accounting for that experience is objective. It is the latter, not the former, that underpins science, and hence the need for invariants.

It's not the same. We can't predict what male reviewer will think about it from the things that woman reviewer have told.

For example:
I say this stuff you are telling me is a crap. And my statement is invariant. Someone else can say: "zonde says this stuff Ken G told him is a crap."
But it's says nothing what others will think about it.

 

[Ken G]

You are right, the point is just that we must admit some ambiguity exists even in the meaning of that which is "invariant." Objectivity is the underpinning concept, yet we can be troubled by these various different forms of "things that are the same for all observers." Does a law have to be the same for all observers in the same way that a proper time along a path does, or the charge of the electron? In relativity, there is no need to distinguish these flavors of invariance, as the theory is built from all of them, but future theories that relax the postulates of relativity might need to navigate those differences. For example, a proper time over an infinitesmal interval has a perfectly good reason to be considered an invariant in relativity, as it results from a metric inner product over that interval, but what justification do we have that the charge of the electron is the same in all reference frames? It's not really part of the structure of the theory, it is simply Occam's razor.

 

[Dale]

So how do I tell apart invariants from everything else?

You already asked this in post 99 and I already answered it in post 101. That is what definitions are for, after all, to allow you to "tell apart" different things.

Did you have a specific question or objection about 101, because if you just ask the same general question that I already answered then you will get the same general answer I already gave.

 

[Mentz114]

... but what justification do we have that the charge of the electron is the same in all reference frames? It's not really part of the structure of the theory, it is simply Occam's razor.

I believe SR requires that the charge is the same in all frames to avoid causal paradoxes. If this was relaxed there would have to be other changes to avoid the paradoxes.

 

[Ken G]

It's not the same. We can't predict what male reviewer will think about it from the things that woman reviewer have told.

Well, it's just an analogy, but I think we actually can do that. The female reviewer describes what it is she likes about the movie, and we can see the invariant elements of the movie underneath her review-- and draw our own conclusions about whether or not we want to see that movie. Indeed, I'd say that's pretty much just how we use movie reviews.

For example:
I say this stuff you are telling me is a crap. And my statement is invariant. Someone else can say: "zonde says this stuff Ken G told him is a crap."

That's why any reasonable movie reviewer, or physics forum member, would never limit their comments to something so useless as a simple value judgement.

 

[Ken G]

I believe SR requires that the charge is the same in all frames to avoid causal paradoxes. If this was relaxed there would have to be other changes to avoid the paradoxes.

You may be right, but even so, that would still make the charge a different kind of invariant than a proper time along a path. After all, we don't really know that causal paradoxes are disallowed, we just find that the other postulates don't allow them (although some strange situations might exist in GR).

 

[bobc2]

I really didn’t need a tutorial on the meaning and application of coordinate systems. However, the comments directed at my posts have helped me understand better the source of some of our misunderstandings. I’m going to give a shot at clarifying the sense in which I’ve characterized the simultaneous Lorentz spaces as significant.

You will have to indulge the use of the block universe model. I am going to use this model of a 4-dimensional universe existing “all at once” as a real physical structure. Don’t get excited over this, I will only use this model in a pedagogical context. It’s just a prop that can hopefully be used to illustrate the ideas I’m trying to express. You can completely disregard the model after its intended use has ended.

First, I will use this fictitious model to illustrate the sense in which Lorentz frames should not be considered unique. The goal is to set up a kind of analogous picture from which we can draw some distinctions. Play like we have a universe that is quite roughly represented by the sketches below. Imagine you are some hyperintelligent hyperdimensional being who has a “birds-eye” view of the entire 4-dimensional universe (we are unconcerned about the impossibility of this). You can apply any of an extremely huge (perhaps infinite) number of charts or coordinate systems to identify locations, events, and extended geometric objects embedded in the universe, etc. Two such arbitrarily selected coordinate systems are depicted here in a universe with various extended 4-dimensional objects present (blue curves).

Notice that the path lengths of the extended objects, in fact the geometry of the objects, are established completely independently of the selection of coordinates. I hope we are all on the same page with this general concept and can recognize the analogous situation with regard to our own universe as described by relativity theory.

Now, perhaps you can understand the sense in which I consider the local Lorentz spaces of special relativity as special and in what sense I regard them as associated with an external 3-D physical reality. I don’t abandon the more fundamental reality as understood when contemplating the 4-dimensional structure in the above sketches. But, there is the 3-dimensional observer’s experience that evolves with time as he (or something involving his consciousness) moves along his worldline at the speed of light (pedagogically speaking). At the outset we should point out that this 3-D external world available to the observer is in fact a 3-D chunk of the 4-D physical reality (we’re still doing pedagogy).

So, we emphasize that a 3-D chunk of the universe is every bit as real as the 4-D structure of which it is a part (still doing pedagogy). And as we say this, we are not invoking any coordinate system. However, next, we must point out that an observer’s 3-D experience—the particular 3-D chunk of the 4-D reality that is being experienced—is not just any arbitrary chunk of reality. It is unique in this sense: It is constrained by the organization and patterns of objects that are presented in the observer’s continuous sequence of Lorentz simultaneous 3-D spaces. In short, the sequence of 3-D chunks are selected out for an observer in a way that assures that a photon worldline will always bisect the angle between his X4 and X1 axes (speed of light will then be the same for all observers). Further, that sequence is constrained in a manner that results in experiencing the same laws of physics as are experienced by all other observers within their individual inertial Lorentz frames. So, the observer’s “view” of the world (after accounting for signal transmittal time delays, etc.) is associated with 3-D cross-sections of the 4-D universe defined by the Lorentz simultaneous spaces. The coordinates themselves are not the reality, but they do help us identify a chunk of physical reality (such as that unique 3-D chunk of a 4-dimensional wooden beam).

So, there seem to be two things at work with the observer (noticed in the context of the 4-D universe sketched above): 1) He moves inexorably along his worldline and 2) His view along his X1 axis (suppressing X2 and X3 for simplicity) is constrained to a direction resulting in photon worldlines bisecting the angle between the X4 and X1 axes. The Lorentz coordinates help us understand the context of the 3-D world we live in and its relationship to the larger 4-D structure.

The sketch below is intended to help with this envisioning these ideas (in a pedagogical sense, i.e., dismiss acceptance of any actual physical reality if you are inclined). Imagine an observer at rest in the black frame and another observer moving at relativistic speed with a wooden object moving along with him. The black plane and the blue plane identify the relative cross-section views experienced by the black and blue observers. The bar is a 4-dimensional object (a static object in the static 4-D block universe). But, each of the observers experiences at any instant (as shown by black and blue planes in the sketch) a three-dimensional bar. For the black guy in his world the object is moving, and for the blue guy the bar is at rest. The black and blue coordinate systems assist us in identifying the different 3-D chunks of the real object being “viewed” by each observer.

So, it is in this sense that we say that the Lorentz simultaneous spaces play a necessary role in identifying the 3-D chunk of reality.

 

[bobc2]

Discussions about scientific method are philosophy. Improvements in scientific method like falsifiability are philosophy.


Yes


Can you elaborate on this? First, is invariant defined or is it undefined basic concept?
Because the way it is usually defined i.e. some quantity that does not change under coordinate transformation, is confusing as it is defining invariants using concept of coordinates and consequently coordinate dependant quantities that we are using to construct coordinates. So coordinate dependant quantities are more basic than invariants.


Invariants are not the same as physical laws. They are certainly two different things.

You've provided some really good observations with your posts. I hesitate to compliment you for fear of bringing you some of my disrepute.

 

[PeterDonis]

bobc2, I have no problem with your description of how a "block universe" model would work, and how local Lorentz frames are defined for observers. It's a perfectly valid model, but you are making claims about it that go beyond what is justified.

I consider the local Lorentz spaces of special relativity as special

...

a 3-D chunk of the universe is every bit as real as the 4-D structure of which it is a part

A spatial slice of a local Lorentz space is not a 3-D chunk of the *universe*. It's only a 3-D chunk of a local patch of spacetime. Strictly speaking, in fact, if the spacetime is curved (i.e., if gravity is present), the "local Lorentz space" is really only valid at a single event; extending it beyond that single event is an approximation, not "fundamental reality".

each of the observers experiences at any instant (as shown by black and blue planes in the sketch) a three-dimensional bar.

This is a fact about the observer's experience, not about the bar. And if you really dig into it, you find that it's a fact about the way human cognitive systems are organized, not about the bar. Human cognitive systems are wired to experience a sequence of 3-D worlds evolving in time. That in no way proves that reality "actually is" a sequence of 3-D worlds evolving in time.

You will no doubt object that you are talking about physics, not human cognitive systems. But when you use the word "experience", and talk about "experiencing" 3-D worlds, you are, implicitly, talking about human cognitive systems, unless you first establish a great deal of supporting framework that you have not established. You would have to, for example, show that *any* observer, regardless of how it was constructed physically, would have to "experience" a sequence of 3-D worlds evolving in time. You have not done anything to establish that.

So, it is in this sense that we say that the Lorentz simultaneous spaces play a necessary role in identifying the 3-D chunk of reality.

The word "necessary" here is the problem. It's one way of understanding the physics, certainly. But "necessary" implies that it's the *only* way. You haven't shown that.

 

[zonde]

Einstein several times said he wished the word relativity was never used - the theory should be called the theory of invariants.

This statement caught my interest and so I tried to google - Einstein "theory of invariants". I got this link http://www.economist.com/node/3518580. It says:
"Abraham Pais, a physicist who wrote what is generally regarded as the definitive scientific biography of Einstein, said of his subject that there are two things at which he was “better than anyone before or after him; he knew how to invent invariance principles and how to make use of statistical fluctuations.” Invariance principles play a central role in the theory of relativity. Indeed, Einstein had wanted to call relativity the “theory of invariants”."

And that's it. I tried too google - Abraham Pais "theory of invariants". It gave google book about Einstein where the phrase "theory of invariants" is used two times. But it's not even close to the idea that relativity should be called “theory of invariants”.

Do you have some other source for that statement? Or did you mean that Einstein said he wished the word relativity was never used but the part about theory of invariants is your own addition?

 

[zonde]

You already asked this in post 99 and I already answered it in post 101. That is what definitions are for, after all, to allow you to "tell apart" different things.

Did you have a specific question or objection about 101, because if you just ask the same general question that I already answered then you will get the same general answer I already gave.

No, you didn't gave general definition of invariant not relaying on coordinate dependent quantities.

You gave as an example proper time. So how do you know that "proper time" is invariant assuming you accidentally forgot what is coordinate system?

As I understand it we would have to refer to different observers (people). Say something like - if different observers can agree about amount of some quantity then this quantity is "invariant".

 

[Ken G]

Another surprise of modern physics to bear in mind comes from cosmology, where we find the best current theory is that there is a meaningful global concept of simultaneity which appears in a special coordinate, the "comoving frame" coordinates at rest with respect to the cosmic microwave backgroud. What's more, 3D spatial slices in those coordinates do indeed appear to be Euclidean, i.e., flat like a Lorentzian inertial simultaneity plane. This would seem to be a remarkable coincidence, that after all the efforts of GR to destruct the physical meaningfulness of global Lorentzian inertial planes of simultaneity, we come full circle when describing the universe at its largest scales. It seems the concept of simultaneity is meaningful on local scales, and on the grandest possible scales, where it completely breaks down is in between, on galactic scales (hence the "Andromeda paradox" of this thread). Do we have a duality here, that dynamics on the largest scales is dual to dynamics on the smallest scales, but in between we have a mess?

 

[Nugatory]

You gave as an example proper time. So how do you know that "proper time" is invariant assuming you accidentally forgot what is coordinate system?

We can measure proper time without any coordinate system at all - and if we aren't using any coordinate system there's nothing to accidentally forget. Consider a sample of radioactive material moving on a timelike world line between two points A and B in space-time. The fraction of the material that is decayed at point A is something that we know without recourse to any coordinate system; likewise the fraction that is decayed at point B. The difference between the two is a direct and coordinate-free measurement of the proper time on the path between A and B.

 

[zonde]

We can measure proper time without any coordinate system at all - and if we aren't using any coordinate system there's nothing to accidentally forget. Consider a sample of radioactive material moving on a timelike world line between two points A and B in space-time. The fraction of the material that is decayed at point A is something that we know without recourse to any coordinate system; likewise the fraction that is decayed at point B. The difference between the two is a direct and coordinate-free measurement of the proper time on the path between A and B.

This is not answer to my question. Can't you see it? Read again my question and your answer. You are giving definition of "proper time" not definition of "invariant".