Understand Special Relativity and Time paradox

[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