Understand Special Relativity and Time paradox

[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".

 

[Ken G]

Right (nugatory), invariants are more than just the same in all coordinates, they are coordinate-independent-- meaning you never need coordinates of any kind to know what they are. Again it all gets back to the central concept of objective observation-- raw measurements should never require calculations so they should never refer to any coordinates. That's also why the laws use tensors (including vectors and scalars), because tensors are those objects that have meaning even in the absence of any coordinate system.

 

[Nugatory]

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".

Invariant: a value that can be defined without reference to any coordinate system, and therefore is independent of the choice of coordinate system.

We can demonstrate that something is invariant by exhibiting a coordinate-independent definition of that thing; and I was using proper time as an example.

 

[Ken G]

zonde, what problem are you having with invariants? Their definition is clear, their importance in relativity is clear (whether or not Einstein wanted to name his theory after them), and their connection to the bedrock of science, the concept of objectivity, is clear. A reliance on invariants is by no means restricted to Einstein's relativity, it is there in Newton's physics with Galilean relativity too. All that is different is what the invariants are (in Galilean relativity, one is time, as coordinate time is the same thing as proper time. In Einsteinian relativity, one is proper time, which is different from coordinate time, as it affords a place at the table for spatial separation. Note that Galilean relativity affords no objectivity whatsoever to a concept of spatial separation between events, it is only Einsteinian relativity that allows two events to be absolutely spatially separated, and in the process ushers in constraints on causation that Galilean relativity lacks).

What made this shift in invariants possible is the recognition that you cannot pick your invariants, and objectively absolute quantities, as simply the set of all the things that observers generally agree on, if your set of observers effectively all share the same state. Observers who effectively share the same state, i.e., all observers on Earth working with pre-20th-century experimental precision, will tend to get the same answers for things not because those things are objective, but because the observers are in some sense redundant with each other. That's a very weak form of objectivity!

Einstein discovered a much deeper form of objectivity-- that which observers can agree on even if they are in very different states (i.e., in relative motion at speeds large enough to produce measurable consequences). You first have to either crank up the difference in states, or crank up the precision of the measurement, to replace the weak form of objectivity with the stronger form. Distinguishing those forms is the role of invariants in any theory, including Newton's-- the latter just did it without benefit of any input from observers in any effectively different states (beyond more trivial distinctions like translations and rotations).

 

[Dale]

No, you didn't gave general definition of invariant not relaying
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?

I didn't attempt to give a definition of "invariant" not relying on the concept of coordinates, such a definition wouldn't make sense. If there weren't coordinate systems then you would still know what proper time is, but you wouldn't know that it is invariant.

I thought all of this was already clear from post 101, but hopefully this helps clarify further.

Personally, I cannot see what you think is confusing here.

 

[PeterDonis]

Personally, I cannot see what you think is confusing here.

Part of his confusion may be that different people are telling him different and seemingly incompatible things. You're saying that "invariant" makes no sense unless you have coordinates; Nugatory is telling him "invariants" are things that can be defined without any reference to coordinates. To me this is really an issue of terminology, not physics; but it seems to be confusing terminology.

 

[Ken G]

You're saying that "invariant" makes no sense unless you have coordinates; Nugatory is telling him "invariants" are things that can be defined without any reference to coordinates.

That sounds perfectly compatible to me! For example, if all you have are integers in your mathematics, then it makes no sense to call them "integers" instead of just "real numbers", yet all the same they can be defined without reference to any other real numbers. An invariant is still an invariant even if there is nothing that isn't, but you can't appreciate the distinction.

 

[Dale]

Part of his confusion may be that different people are telling him different and seemingly incompatible things. You're saying that "invariant" makes no sense unless you have coordinates; Nugatory is telling him "invariants" are things that can be defined without any reference to coordinates.

I don't know why that is confusing or why it would seem incompatible. Invariants are things that can be defined without any reference to coordinates but even that definition of the word "invariant" requires the concept of coordinates.

 

[PeterDonis]

Ken G, DaleSpam, I'm not so much focusing on the actual definitions but on the terminology used to describe them. Your posts do help to clarify that terminology; we'll see if zonde picks up on them.

 

[Dale]

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).

The 3D chunk of the 4D physical reality you are describing here, the one representing an observer's experience as it moves through time, is the past light cone, not a surface of simultaneity.

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).

I am fine with this, but that isn't the assertion you are making. The assertion you are making is that one particular class of 3D chunks is MORE real compared to other 3D chunks. That I disagree with.

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).

I agree that the 3D chunk of the 4D spacetime that is being observed by a particular observer is not arbitrary. It is unique and can be identified without reference to coordinates as the past light cone. Simultaneity is arbitrary so obviously a surface of simultaneity cannot be something which is not arbitrary.

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.

As PeterDonis mentioned, this doesn't work except as an approximation. The laws of physics, written in terms of inertial Lorentz frames, are FALSE except as approximations. It seems like an unjustifiable stretch to identify a known approximation as something so fundamental that it has a unique claim to "reality".

In order to avoid the approximation mentioned by PeterDonis you have to write the laws of physics in a coordinate independent form. Once you have done that, you can no longer appeal to the form of the laws of physics to identify any simultaneity convention as special.

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.

For a non-inertial observer even in flat spacetime this condition does not uniquely define the coordinate system. Both the naive simultaneity convention (where it is valid) and the Dolby and Gull convention have this property.

 

[PAllen]

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?

Over the years, I have run across this statement many times, as a non-controversial claim. However, it seems hard to find a really authoritative source for it on the internet. The best additional link I can give you is the following:

http://hps.elte.hu/~gk/Sokal/Sokal/KLotz.html

with the following at the end of the discussion:

"In actual fact, the theory of relativity is anchored in absolutism-in the concrete of Einstein's two postulates: The velocity of light is a universal constant and the laws of physics are constant. He described these postulates as principles of invariance. An insightful textual analysis of the introductory sections of the 1905 paper would have recognized that the two "postulates" specify unchanging principles that serve as the foundations of the theory. In fact, Einstein called his creation an "Invariententheorie," a theory of invariance. The name "theory of relativity" was coined later in a review by German physicist Max Planck. Einstein resisted that name for years, although he reluctantly bowed to peer pressure. The relativistic features of time and space that led to the term "theory of relativity" are derived from the principles of invariance."

However, this is not quite the sense under discussion. It was really in the process of moving to GR that Einstein stressed contracting the metric with coordinate dependent quantities to construct invariants.

 

[Ken G]

Maybe it would help to understand that invariants are used in many situations outside of physics. For example, if we were interviewing witnesses of a robbery to try and piece together what happened, we might start with asking ourselves, what are the things that all observers agree on. Then we might turn to the things they disagree on, and try to use the things they agree on, the invariants, to understand why they disagreed on the other things. When we have a coherent account of both the things they agree on, and the reasons they disagree on others that is consistent with what they agree on, we can say we understand what happened. That's more or less just what relativity does also.

 

[Dale]

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

Let me try one other approach.

"Invariant" is a property that some quantity may have. The property is not the same as the quantity. The property is defined as that the quantity remains unchanged under a coordinate transform. The definition of the property requires coordinates.

The quantity itself is defined in some other way. Each invariant quantity will have a different definition. However, each invariant quantity, since it remains unchanged under a coordinate transform, must have a coordinate-free definition.

If the concept of coordinates did not exist then all defined quantities would necessarily be invariant, but they would not be called invariant because it would be a meaningless word. It would be a property that every quantity has, so it wouldn't distinguish different quantities.

 

[bobc2]

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.



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...

Of course. That's why my sketch showed the beam in the context of a locally flate region of the 4-D universe. That just means that chunk of the universe is bounded in all four directions. If I cut out a little 3-D chunk of the beam that little chunk is still a real object.

I'm not concerned with tracing simultaneous space completely around the 4-D universe that was depicted in the first pair of pedagogical sketches. When I once asked my physics advisor about that he was annoyed that I was even concerned about it and told me, "O.K., if you are so hung up on that subject, why don't you take some Christoffel symbols and go off and see what you can do about it." I think for now, sufficient unto the day is the evil thereof. Perhaps one would begin with the Ken G post #118 note.

We should be satisfied at this stage if we can say some basic things about the external physical reality that is in our neighborhood of the universe, then later pick up the story with the General theory. I don't think many of our forum members doubt the physical reality of the objects about us--even across our solar system--and perhaps beyond.

...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".

We're talking about concepts and principles. One need not be detoured over this kind of minutia. We can never present a completely precise description of external physical objects. We can't even perceive things with precision. Any kind of observation is limited in precision. That does not keep us from conceptualizing the reality. Beyond that, it sounds like you are wanting to restrict reality to the apex of the light cone as I described in an earlier post. I pointed out there the implications of solipsism.

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.

I think my pedagogically designed treatment makes it clear the connection between the epistemology and the ontology. You splitting hairs and introducing red herrings here.

 

[bobc2]

The 3D chunk of the 4D physical reality you are describing here, the one representing an observer's experience as it moves through time, is the past light cone, not a surface of simultaneity.

I thought I had made it clear in my comments that in this pedagogical description we are taking account of time delays in signal transmissions in identifying the simultaneous spaces.

I am fine with this, but that isn't the assertion you are making. The assertion you are making is that one particular class of 3D chunks is MORE real compared to other 3D chunks. That I disagree with.

I have implied nothing of the sort. I've just depicted that the chunk in one observer's simultaneous space is a different chunk than someone else's. It's kind of analogous to a person on one side of the Earth looking at a different piece of the Earth than someone looking at a piece of the Earth on the opposite side of the earth. Both are seeing a piece of external physical reality--one is no more special than the other.

 

[PeterDonis]

That just means that chunk of the universe is bounded in all four directions.

The chunk in which Lorentz 4-D geometry is *approximately* valid, yes. But again, it's only an approximation.

I don't think many of our forum members doubt the physical reality of the objects about us--even across our solar system--and perhaps beyond.

I don't doubt the physical reality of quasars 12 billion light years away (I think that's around the current limit of what we can see). As I made clear in prior posts, that's not what we're disagreeing about.

We're talking about concepts and principles. One need not be detoured over this kind of minutia. We can never present a completely precise description of external physical objects. We can't even perceive things with precision. Any kind of observation is limited in precision. That does not keep us from conceptualizing the reality.

But the strong claims you have made aren't about how we conceptualize reality; they're about how reality *is*. Those are two different things. If all you're saying is that we can use 4-D spacetime to conceptualize reality, of course we can. Nobody is disagreeing with that.

Beyond that, it sounds like you are wanting to restrict reality to the apex of the light cone as I described in an earlier post.

You apparently aren't reading my posts very carefully. Try re-reading my post #88 for a start. I didn't say there that the Sun doesn't exist "now"; I said that the claim that the Sun exists "now" is an extrapolation from the data in a way that the claim that the Sun existed eight minutes ago (when the light we see now was emitted) is not.

I think my pedagogically designed treatment makes it clear the connection between the epistemology and the ontology.

No, it doesn't; it *assumes* connections that aren't necessary. That's the point.

 

[Dale]

I thought I had made it clear in my comments that in this pedagogical description we are taking account of time delays in signal transmissions in identifying the simultaneous spaces.

Then you are not talking about what he experiences but about what he predicts or infers. There is no sense in which a Lorentz hyper surface of simultaneity is "experienced".

I have implied nothing of the sort. I've just depicted that the chunk in one observer's simultaneous space is a different chunk than someone else's. It's kind of analogous to a person on one side of the Earth looking at a different piece of the Earth than someone looking at a piece of the Earth on the opposite side of the earth. Both are seeing a piece of external physical reality--one is no more special than the other.

That is not what I said. I said that you are asserting that a particular class of 3D chunks is more real than others, which you certainly are doing. Furthermore, you have not just implied it, you have stated it explicitly and repeatedly:

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. ...

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 ...

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).

One other thing. Even neglecting the fact that spacetime is curved so the laws of physics cannot be written in terms of Lorentz inertial frames, ie even assuming a perfectly flat universe, you also cannot write the laws of physics in their standard form for a non inertial observer's momentary comoving inertial frame succession of 3 D worlds.

 

[zonde]

"Invariant" is a property that some quantity may have. The property is not the same as the quantity. The property is defined as that the quantity remains unchanged under a coordinate transform. The definition of the property requires coordinates.

The quantity itself is defined in some other way. Each invariant quantity will have a different definition. However, each invariant quantity, since it remains unchanged under a coordinate transform, must have a coordinate-free definition.

If the concept of coordinates did not exist then all defined quantities would necessarily be invariant, but they would not be called invariant because it would be a meaningless word. It would be a property that every quantity has, so it wouldn't distinguish different quantities.

Thanks Ken G, PeterDonis and DaleSpam. I suppose I got it.

So let me check it. If I would say that "invariant" has the same meaning as "physical fact" would you (tend to) agree?

 

[Dale]

So let me check it. If I would say that "invariant" has the same meaning as "physical fact" would you (tend to) agree?

Well, "invariant" is well-defined, but I don't think "physical fact" is well defined. However I would tend to agree that under a reasonable definition of "physical fact" that frame invariant facts are more likely to qualify than frame variant ones.

 

[zonde]

zonde, what problem are you having with invariants?

I want to oppose statements like this statement of DaleSpam (suggesting that invariants somehow make physics law more "real" than coordinate dependant quantities). But I wanted to understand what is the motivation behind statements like that.

As PeterDonis mentioned, this doesn't work except as an approximation. The laws of physics, written in terms of inertial Lorentz frames, are FALSE except as approximations. It seems like an unjustifiable stretch to identify a known approximation as something so fundamental that it has a unique claim to "reality".

In order to avoid the approximation mentioned by PeterDonis you have to write the laws of physics in a coordinate independent form. Once you have done that, you can no longer appeal to the form of the laws of physics to identify any simultaneity convention as special.

 

[Ken G]

For me personally, the big "aha" with relativity was the recognition that it was only by past sloppiness that we had gotten away with not distinguishing our actual experiences and measurements (all perfectly local) from the "stories" we tell about our nonlocal environment to make sense of everyone's mutual (local) experiences. Mathematically, a "story" is closely related to the use of a coordinate system, so we might say that the invariants on which an objective description must be based are the common elements of everyone's "stories." We should have always made that distinction, because a story is not the same thing as an experience or a measurement, but we had simply gotten away with not making the distinction because none of the observers were in different enough states to tell different stories, prior to Michelson-Morely.

What is interesting about this is that the role of a story is to help us understand what is happening around us, but the set of all the stories actually includes a lot of extraneous and contradictory accounts that is much more than what actually did happen, objectively. Invariants help us cull down the extraneous embellishments we have built into our stories, to focus on what actually carries true information about nature. So the difference between what is invariant, and what is coordinate dependent, is a lot like the difference between what actually happens, and what is our effort to come to terms with what happens. Our effort to make sense of reality is something more than objective reality, and Einstein is essentially chastising us for our sloppiness in allowing those two things to be treated as if they were the same. That's related to writing the laws of physics in a coordinate-free way: it culls out the stories, and gives "just the facts, ma'am."

 

[zonde]

Well, "invariant" is well-defined, but I don't think "physical fact" is well defined. However I would tend to agree that under a reasonable definition of "physical fact" that frame invariant facts are more likely to qualify than frame variant ones.

If we take "invariant" as basic term it does not have to be well defined. In fact basic terms can't be well defined.

It's like with axiomatic systems and primitive notions (undefined terms).

 

[PeterDonis]

I would tend to agree that under a reasonable definition of "physical fact" that frame invariant facts are more likely to qualify than frame variant ones.

Actually, now that we seem to be clearing up confusion, I feel the need to insert some more.

The sorts of things we have been calling "frame-invariant" actually can include a lot of things that could also be called "frame-variant". For example, consider the energy of an object. It varies from frame to frame; but given any frame, I can easily construct an invariant that expresses the energy of the object as measured in that frame. I just pick an observer at rest in the frame whose worldline crosses that of the object at a chosen event, and take the inner product of the observer's 4-velocity at that event with the object's 4-momentum at that same event:

E = \eta_{\mu \nu} u^{\mu} p^{\nu}

The number E is usually thought of as the "time component" of the object's 4-momentum in the given frame, and hence as a "frame-variant" quantity; but as I've written it above, it should be obvious that E is an invariant; it's the inner product of two 4-vectors, and inner products are preserved by Lorentz transformations.

If I look at this inner product in a different frame, p^{\nu} will have different components, but so will u^{\mu}, because I defined u^{\mu} as the 4-velocity of a particular observer in a particular state of motion. That observer won't be at rest in the new frame, so u^{\mu} in the new frame will have spatial components as well as a time component; and that will compensate for the change in the components of p^{\nu} in just the right way to keep the inner product E constant.

The point of all this is that focusing attention on "invariants" does not cost us anything. We can still talk about all the quantities that we would normally think of as "frame-variant", like components of vectors; we just have to define them properly. When we do, we see that they represent perfectly good "physical facts". The number E is not just the "time component" of p^{\nu} in a particular frame; it represents the physical fact that a particular observer, in a particular state of motion, measures a particular object to have a particular energy.

 

[Dale]

I want to oppose statements like this statement of DaleSpam (suggesting that invariants somehow make physics law more "real" than coordinate dependant quantities). But I wanted to understand what is the motivation behind statements like that.

I dislike the word "real" so I wouldn't say that they are more "real" written in a coordinate independent fashion. I would say that they are more accurate, which is true since it avoids the approximation I mentioned above. I could also say that they are more general, which should be obvious I hope.

 

[Ken G]

The number E is not just the "time component" of p^{\nu} in a particular frame; it represents the physical fact that a particular observer, in a particular state of motion, measures a particular object to have a particular energy.

It sounds like you are drawing the distinction between the purely mathematical notion of a "coordinate system" and the purely physical notion of a "frame of reference." It's true that once we choose a convention for creating coordinates, then there is a one-to-one mapping between the two notions, but since we don't have to agree on any such convention, how things depend on coordinates is more general, and less physically important, than how they depend on reference frames (states of the observers).

We should probably dispense with redundancies right away by picking a particular coordinate convention (like Einstein simultaneity in the absence of gravity, or comoving-frame coordinates in cosmology), and noting that the rules of tensors automatically (and trivially) navigate for us the redundancies of other coordinate conventions. There's no physics in that yet, it's just the mathematical requirement that when we tell our stories using coordinates, we will need the mathematical forms of tensors to keep the stories the same, even when all the observers are in the same state. The physics appears when we ask how different states of the observers will affect their observable outcomes, which is where every particular type of relativity (i.e., the appropriate metric in the contractions) distinguishes itself. Indeed I would argue that Einstein's relativity should not be called the theory of relativity, nor the theory of invariants, because neither term distinguishes it from every other set of relativistic invariants. We should call it Einsteinian relativity, or the theory of Einstein invariants.

 

[PeterDonis]

It sounds like you are drawing the distinction between the purely mathematical notion of a "coordinate system" and the purely physical notion of a "frame of reference."

Yes, that's one way of looking at it. The 4-velocity u^{\mu} is the timelike vector of the observer's frame. As such, it's a coordinate-free geometric object; sometimes it's convenient to describe it in a particular coordinate chart, but we can reason about it without doing that.

 

[Dale]

For example, consider the energy of an object. It varies from frame to frame; but given any frame, I can easily construct an invariant that expresses the energy of the object as measured in that frame. I just pick an observer at rest in the frame whose worldline crosses that of the object at a chosen event, and take the inner product of the observer's 4-velocity at that event with the object's 4-momentum at that same event:

E = \eta_{\mu \nu} u^{\mu} p^{\nu}

This should be expected. When anyone performs a measurement the outcome of that measurement is frame invariant. Otherwise different frames would predict that the same experiment would generate different numbers. So there must be some mechanism for converting frame variant components into frame invariant scalars.

However, I think that it is important to note that the invariant quantity you labeled E is NOT the energy, except in the rest frame of the observer. So "energy" is frame variant, but a particular measurement of energy produces an invariant number. Other frames will disagree that the number produced by that measurement represents the energy.

I wish I knew a better way to state that.

The point of all this is that focusing attention on "invariants" does not cost us anything. We can still talk about all the quantities that we would normally think of as "frame-variant", like components of vectors; we just have to define them properly. When we do, we see that they represent perfectly good "physical facts". The number E is not just the "time component" of p^{\nu} in a particular frame; it represents the physical fact that a particular observer, in a particular state of motion, measures a particular object to have a particular energy.

Well said. But again, the number E is not the energy except in one frame.

 

[PeterDonis]

I wish I knew a better way to state that.

I wish I did too. You're quite right, the way I stated it is open to misinterpretation, but so is every other way I can think of.

 

[Thinker007]

I wish I knew a better way to state that.

It was pretty clear to me. :)

 

[Alain2.7183]

hence the "Andromeda paradox" of this thread

What is the "Andromeda paradox"?

 

[Nugatory]

What is the "Andromeda paradox"?

http://en.wikipedia.org/wiki/Rietdijk–Putnam_argument