Non-Euclidean Geometry

[Ken G]

The only post where I say I prefer one over the other is followed immediately by a post where I admit I was joking, and where I say that I was trying to get to geometrized units where c = 1. If c = 1, the units of distance cancel with the units of time, they are equal.

Yes, those were the two caveats that you later added to the only two points you've made. As they completely undo both the points, one wonders why you bothered to enter them in the first place. I assumed you had some reason, so responded on that assumption. I am actually making a constructive point here-- and you have so far not. I say when one chooses time or length distances, one should choose time. You have now said, it doesn't matter, without any supporting argument meant to be taken seriously. But, still, one does have to choose one or the other, yes?

I am not arguing for a break, you are.

I'm quite aware of what I have been arguing. Apparently you feel you have been arguing against having a break-- but you've done so by twice favoring distance over time! Then you added that neither of those arguments had any substance-- and so when this is all put together, it comes out no argument at all.

A lot of us have taken that step a long time ago.

The very fact that the Wiki site used length as its fundamental unit undercuts your claim. My comments are not focused on whether or not you personally tend to use distance units for time rather than time units for distance, my comments are focused on the standard we find around us-- and what is wrong with it.

[hhEb09'1]

But, still, one does have to choose one or the other, yes?

I can't imagine why you'd have to. Most people never choose to use one or the other exclusively.

Apparently you feel you have been arguing against having a break-- but you've done so by twice favoring distance over time!

Once was a joke (city blocks), what was the other one?

The very fact that the Wiki site used length as its fundamental unit undercuts your claim.

Not at all. Even you said it, and that's good enough for me.

[Ken G]

I can't imagine why you'd have to. Most people never choose to use one or the other exclusively.

Apparently by "most people" you are not including people who have to write relativity textbooks. The latter group must choose a means of expressing the metric. Even if c=1 is chosen to duck the issue of whether one is thinking in terms of time or distance units, one's prejudices are still exposed in the choices of the symbols, the signs, and the structure of the metric equation. I'm sure you see what I mean there.

Once was a joke (city blocks), what was the other one?

This:

As that link sorta makes clear, time (and mass and energy and momentum) are measured in meters, in geometrized units.

You then termed that a "quibble", but if we agree it has zero physical content, why did you bring it up at all?

[hhEb09'1]

Apparently by "most people" you are not including people who have to write relativity textbooks.

Sure am! they are a small minority though, unfortunately.

Even if c=1 is chosen to duck the issue of whether one is thinking in terms of time or distance units, one's prejudices are still exposed in the choices of the symbols, the signs, and the structure of the metric equation. I'm sure you see what I mean there.

I think so.

This:

Ah, well that's others doing the favoring there. Me, I consider them equal, as I've always said.

You then termed that a "quibble", but if we agree it has zero physical content, why did you bring it up at all?

First, I don't agree--I'm not sure of what you mean by "zero physical content". Going back through the posts, it may be that you thought I had agreed to your assertion that something I said had "no basis at all"--instead, I was just saying I still thought it was a quibble.

Because it represented someone else's approach, not necessarily my own, in the context of this thread--even though it was opposed to yours, I still thought it was a quibble, as opposed to a substantive argument.

Second, apodictic notions of personal preferences just seem out of place, especially when there are physical reasons to not have to choose.

[Ken G]

Ah, well that's others doing the favoring there.

But you keep dodging the point I raised-- you brought up the fact that "others are doing the favoring"-- and you've never said why, or what you thought it proved.

Me, I consider them equal, as I've always said.

And as I've always said, my point is that even those who think they consider them equal must choose a form for the metric, exposing a prejudice that either assists insight or obfuscates it. "They are equal" just doesn't fly, they aren't. To say they are suggests that timelike displacements are physically no different than spacelike displacements-- and I know you know that's false. Again, this is central to the point I'm making, and that you've had no substantive comment on: we experience timelike displacements, we measure them, and we build a physics on their backs. None of those can be said for spacelike displacements. But you are welcome to choose to believe there is no difference, I am merely presenting the argument.

Second, apodictic notions of personal preferences just seem out of place, especially when there are physical reasons to not have to choose.

So you keep claiming, but never actually give any. I have given the reasons why one does have to choose, even if one pretends one does not. Indeed, there are only actual arguments coming from one side here-- the other side does nothing but cite personal preferences and mention what the standard is, with no physical descriptions at all.

[hhEb09'1]

But you keep dodging the point I raised-- you brought up the fact that "others are doing the favoring"-- and you've never said why, or what you thought it proved.

I brought up geometrizied units to make a different point, unrelated to your crusade to convert to timeism. From my point of view, the disagreement between two such systems is just a disagreement over convention, nothing more. I understand that you feel stongly towards one convention, but I don't.

Again, this is central to the point I'm making, and that you've had no substantive comment on: we experience timelike displacements, we measure them, and we build a physics on their backs. None of those can be said for spacelike displacements.

That remark brought me up short. I'd like to check that. Do we experience spacelike displacements? (yes) Do we measure them? (yes) Do we build a physics on their backs (I think so, if by that you mean they are an integral part of physics). Do you disagree with all of my answers? Which ones?

So you keep claiming, but never actually give any. I have given the reasons why one does have to choose, even if one pretends one does not. Indeed, there are only actual arguments coming from one side here-- the other side does nothing but cite personal preferences and mention what the standard is, with no physical descriptions at all.

Even in "doing relativity", one can maintain the distinction between time and space. One doesn't have to choose one to the exclusion of the other.

Wait, isn't that the point that you are making with the timelike and spacelike comments?

[Cougar]

Whether a geometry is Euclidean or not is dependent on only one thing: the status of Euclid's fifth postulate.

Exactly right. The first four euclidean postulates are retained in non-euclidean geometries, at least the early ones. The first 28 propositions of Euclid's Elements hold in Lobachevsky's early non-euclidean geometry, which he called "imaginary geometry" and published a paper on in 1829.

If they meet, you have Elliptic geometry (first described by Riemannian).... Spherical Geometry, used in measuring distances on a sphere (such as distances on the Earth) is also a non-Euclidean Geometry.

Well, here the history gets more complicated... and interesting. A well-detailed rendition can be found in Robert Osserman's 1995 book Poetry of the Universe, A mathematical exploration of the cosmos. For a chapter or so, Osserman follows the life of Carl Friedrich Gauss, who really opened up the application and further development of non-euclidean geometries with his research into geodesy, where he devised assigning a curvature value for every point on the earth (or any other) surface. This was in the early 1800s.

Then came Lobachevsky and Bolyai, then Ferdinand Minding; (recalling now that Johann Lambert came close to being the founder of non-euclidean geometry fifty years earlier); later Beltrami showed the above non-euclidean geometries were equivalent and proved they were just as viable as euclidean geometry. Then David Hilbert and Henri Poincare "played a role in settling the last few questions and doubts regarding Lobachevsky's geometry." M.C. Escher is clearly the Artist Laureate of this whole "movement"

Then finally,

"The question remains: What is the value of non-euclidean geometry in the "real world"? The answer arrived in the wake of a series of developments that begain with a fundamental rethinking of the entire subject of geometry by one of the great visionaries of mathematics, Bernhard Riemann."

[Ken G]

Do we experience spacelike displacements? (yes)

No, that is not correct. I am using the term "spacelike" in its physically defined sense-- two events are "spacelike" if no observer could ever see them as occuring in the same place, which means that they are not causally connected. Since no observer, not even a hypothetical one, could ever experience both events, we certainly do not experience them (you may be overlooking the fact that all experience is local). What we do, instead, is to experience things locally, and make inferences about what is happening nonlocally that gave rise to our experience. That is not experience, it is mental construction based on experience. That's what I have said all along-- space is a mental construction, time can actually be observed.

Do we measure them? (yes)

No, again all measurements are local. You are confusing a measurement with a mental construction.

Do we build a physics on their backs (I think so, if by that you mean they are an integral part of physics). Do you disagree with all of my answers? Which ones?

All of them, yes. We build physics on the backs of observations, which are local and timelike separated. The physics that we build is what you are talking about-- mistaking the mental constructions we call physics for the observations we use to build that physics.

Even in "doing relativity", one can maintain the distinction between time and space. One doesn't have to choose one to the exclusion of the other.

I am talking about where insight comes from, and I'm saying that space and time are not identical (as is often erroneously argued). There are very clear differences between them, built right into physics (causality being the main one), and when it is recognized that time is the more fundamentally observable, being local, then it is clear that the metric should be viewed as a measure of a proper time interval, not a spatial interval.

[hhEb09'1]

No, that is not correct. I am using the term "spacelike" in its physically defined sense-- two events are "spacelike" if no observer could ever see them as occuring in the same place, which means that they are not causally connected. Since no observer, not even a hypothetical one, could ever experience both events, we certainly do not experience them (you may be overlooking the fact that all experience is local).

Do you mean, the distance between two events is spacelike?

Fortunately, humans are not local. Is the distance between the ends of a meter bar spacelike or timelike? When I hold my hands on the ends of the meter bar, is the distance between my hands spacelike or timelike? It could be either.

You are confusing a measurement with a mental construction.

At that level, all measurements are mental constructions. I have no problem with mental constructions.

[Ken G]

Do you mean, the distance between two events is spacelike?

No, I do not mean that, I mean the events are spacelike separated. That is an invariant-- the "distance" between them can be zero.

Fortunately, humans are not local. Is the distance between the ends of a meter bar spacelike or timelike? When I hold my hands on the ends of the meter bar, is the distance between my hands spacelike or timelike?

You are casting the laws of relativity into the realm of how we construct our own self-image. You may be assured that nothing that happens in the human body violates causality. All you are saying is that the amalgam of "events" we call a human being includes events that are spacelike separated. I refer to the idealized concept of "observer" used in physics-- it is an important weakness of physics that we have no theory to describe an actual observer, we do all we can to leave the machinations of a real observer out of our physics. Since we are discussing relativity, one can say that the sense of "time" and "distance" to which I refer are the relativistically idealized versions. "Real life" ignores relativity all the time-- if I said "measure out a distance of 5 meters" and you lay a meter stick end to end 5 times, you have ignored the length change in the meter stick due to the varying speed of the Earth's orbit, for example.

Also, if I said "measure the distance from your home to your work", and you used your car's odometer, you are assuming that there wasn't an earthquake fault active during your measurement. We make all kinds of idealized assumptions when we measure distance that we do not make when we measure proper time. This is the point, all distance measurements are inferences of some kind, but a clock just reads a proper time.

At that level, all measurements are mental constructions. I have no problem with mental constructions.

Of course, I did not say you had a "problem" with mental constructions, I said you were confusing the differences between mental constructs and more direct measurements. In physics, it is often quite important to distinguish what you can actually measure from what you have to constuct as part of some picture or theory (this is the amazing recognition at the core of both relativity and quantum mechanics, i.e., of modern physics). It is not as black and white as a "direct" vs. an "indirect" measurement, one must recognize all levels of directness and indirectness. I merely point out those differences, for those who would care to notice them.

[hhEb09'1]

No, I do not mean that, I mean the events are spacelike separated.

That is what I meant--you left that out.

You are casting the laws of relativity into the realm of how we construct our own self-image. You may be assured that nothing that happens in the human body violates causality. All you are saying is that the amalgam of "events" we call a human being includes events that are spacelike separated. I refer to the idealized concept of "observer" used in physics--

You did say "we"

"Real life" ignores relativity all the time-- if I said "measure out a distance of 5 meters" and you lay a meter stick end to end 5 times, you have ignored the length change in the meter stick due to the varying speed of the Earth's orbit, for example.

If I measure out five meters along the wall of my spaceship, before and after liftoff, the marks are in the same place, right?

Also, if I said "measure the distance from your home to your work", and you used your car's odometer, you are assuming that there wasn't an earthquake fault active during your measurement. We make all kinds of idealized assumptions when we measure distance that we do not make when we measure proper time. This is the point, all distance measurements are inferences of some kind, but a clock just reads a proper time.

It measures its proper time, that may not be ours. Who knows what has happened to that clock. Just like a meter stick measures its proper length.

It is not as black and white as a "direct" vs. an "indirect" measurement, one must recognize all levels of directness and indirectness. I merely point out those differences, for those who would care to notice them.

I'll try to do the same

[Ken G]

That is what I meant--you left that out.

And if you knew that, then there was no need to inquire, unless you were just trying to be pedantic. So that's why I pointed out your mistake as well-- one good pedantic argument deserves another, no?

If I measure out five meters along the wall of my spaceship, before and after liftoff, the marks are in the same place, right?

I have no idea at all what you mean by "the same place". If you try to clarify that more precisely I think you will see a lot of what I'm saying.

It measures its proper time, that may not be ours.

Again you are bringing in the practical realities of "us". We don't do that in physics, expressly because we haven't the vaguest idea how to do that. All physics involves idealizations, and the idealization we use for clocks is that they are "good" clocks, and the clock "has a reference frame". Formally, there's no such thing as a reference frame, but you can see how we need to get around that little problem to do any physics at all. We do not, however, need to define a concept of two marks "being in the same place". What being in the same place means in relativity is simple-- every observer is always in the same place, from their perspective. That is all that "being in same place" ever means in a physically meaningful way.

You can say the two marks identify specific locations in the spaceship, perhaps counting a certain number of atoms-- but to what extent are they in the same place? Reference frames are local things, so cannot connect two separate places. Einstein used global inertial frames in special relativity-- but later discovered, with general relativity, that there's no such thing physically. Global reference frames are just mathematically spliced together local reference frames, the latter of which being physically meaningful-- the idea that a global frame applies to a single observer is a mental construct that works approximately well in many situations, like the concept of distance itself. I think the reason a lot of people have trouble conceptualizing general relativity is that their minds have been subverted by picturing the global inertial frames of special relativity. Better would be to learn general relativity first, and then see special relativity as an approximation that applies approximately well on local tangent spaces. The way it is done now is kind of like not telling geography students that the Earth is a sphere until they get to graduate school!

Who knows what has happened to that clock.

That's why we have the idealization of a "good clock".

Just like a meter stick measures its proper length.

And if I had a light-year-stick instead, do you think it would measure its own proper length too? That's the whole problem-- rulers only work when they are local enough to only be able to measure distance infinitesmals in the tangent space. But "real" distance is not in the tangent space, it's in the real space-- and cannot be measured by a single ruler. Distances have to be measured by getting a lot of different observers together, each with their own ruler. But that must also come with the instructions of what observers to pick, i.e., you do not have a distance until you have a coordinate system cobbled together from real observers. That's a very different animal, this is my point. Have you never wondered why there is no concept of "proper distance" in cosmology, but there is a "proper age" of the matter in the universe?

[hhEb09'1]

And if you knew that, then there was no need to inquire, unless you were just trying to be pedantic. So that's why I pointed out your mistake as well-- one good pedantic argument deserves another, no?

What mistake?

I have no idea at all what you mean by "the same place". If you try to clarify that more precisely I think you will see a lot of what I'm saying.

You make the marks on the wall, you don't have to remake the marks on the wall, is what I meant.

Again you are bringing in the practical realities of "us". We don't do that in physics, expressly because we haven't the vaguest idea how to do that.

I was just parallelling your discussion of an earthquake changing the distance after we've measured it.

I think the reason a lot of people have trouble conceptualizing general relativity is that their minds have been subverted by picturing the global inertial frames of special relativity.

I have it on good authority that most people's troubles start well before that!

That's why we have the idealization of a "good clock".And if I had a light-year-stick instead, do you think it would measure its own proper length too? That's the whole problem-- rulers only work when they are local enough to only be able to measure distance infinitesmals in the tangent space. But "real" distance is not in the tangent space, it's in the real space-- and cannot be measured by a single ruler. Distances have to be measured by getting a lot of different observers together, each with their own ruler. But that must also come with the instructions of what observers to pick, i.e., you do not have a distance until you have a coordinate system cobbled together from real observers.

I'm not sure where you're going with this.

The same objections that you make for rulers can be made for clocks. They're finite entities, whereas our idealizations are not. That's the whole point of the equivalency--whatever problems occur with space, you can find the analog in time.

That's a very different animal, this is my point. Have you never wondered why there is no concept of "proper distance" in cosmology, but there is a "proper age" of the matter in the universe?

Do you mean, like the radius of the universe?

[Ken G]

What mistake?

The implication that I should have said "the distance between two events is spacelike" rather than simply "spacelike events". The latter is actually less misstated than the former.

You make the marks on the wall, you don't have to remake the marks on the wall, is what I meant.

Yes, time elapses for the marks-- they have a proper time associated with them, that is correct. And if you like, you could choose a convention that the distance between them should be measured at a consistent proper time for all the atoms that connect them. Or, a different convention for the distance between them could be chosen, for example that distance could be length contracted if the rocket accelerates.

I was just parallelling your discussion of an earthquake changing the distance after we've measured it.

Parallel away-- that is what is necessary to show my point. Earthquakes, acceleration, or just a changed convention for associating simultaneity-- all can change the distance between the marks. It is not an invariant property of the marks. Marks don't have an invariant-- events do.

I have it on good authority that most people's troubles start well before that!

True enough-- I mean the accomplished ones who study well what they are told and still can't get it, because it's being told badly.

The same objections that you make for rulers can be made for clocks. They're finite entities, whereas our idealizations are not.

No, the same objection does not apply to clocks-- because clocks don't measure time in a tangent space, they measure actual proper time along the manifold (to within the inherent idealizations required for an object with finite size, treated classically, etc.). That's the whole point of the lack of equivalency.

That's the whole point of the equivalency--whatever problems occur with space, you can find the analog in time.

Well, if that were actually true, then my point would be erroneous, that is clear. But there is no such equivalency, that's the most common mistake that people are actually taught. Time and space are not equivalent in relativity, they can't be, because relativity treats causality as an invariant-- and you could never get that if they were actually equivalent.

Do you mean, like the radius of the universe?

Yes, exactly like that-- that is exactly the flaw that people don't understand. Look up the "radius of the universe" and see all the different definitions that concept admits-- it's pure convention, usually either the distance light can travel in the age of the universe. Now try to define that concept without it really being an age, and you will see my whole point-- the "radius of the universe" is really a time masquerading as a distance. Any effort to make it a real distance, without reference to time, will suffer from nonuniquenss of the convention you choose. There is no such problem with proper time, it is the time a clock registers from the beginning. Every counterexample you pose is exactly the point I am making, and merely serves to prove how widespread the misconception is.

[hhEb09'1]

The implication that I should have said "the distance between two events is spacelike" rather than simply "spacelike events". The latter is actually less misstated than the former.

What are spacelike events?

Yes, time elapses for the marks-- they have a proper time associated with them, that is correct.

I didn't say anything about proper time there. The point is, using the same meter stick will get you the same marks, even though subject to 20% contraction relative to the first reference frame.

Parallel away-- that is what is necessary to show my point. Earthquakes, acceleration, or just a changed convention for associating simultaneity-- all can change the distance between the marks. It is not an invariant property of the marks. Marks don't have an invariant-- events do.

Whether with time or space, somehow we have to translate the event of marking or ticking into a useable measurement, separate from the events.

True enough-- I mean the accomplished ones who study well what they are told and still can't get it, because it's being told badly.

Agreed.

No, the same objection does not apply to clocks-- because clocks don't measure time in a tangent space, they measure actual proper time along the manifold (to within the inherent idealizations required for an object with finite size, treated classically, etc.).

It seems to me that your comments within the parentheses hides a lot of the problem. In theory, we could do the same thing with space. Why not?

Time and space are not equivalent in relativity, they can't be, because relativity treats causality as an invariant-- and you could never get that if they were actually equivalent.

Never get what?

There is no such problem with proper time, it is the time a clock registers from the beginning.

Which clock?

[Ken G]

What are spacelike events?

You know perfectly well, they are events that are acausal. Was there ever any uncertainty in what we are talking about here?

I didn't say anything about proper time there.

I know you didn't, that's the problem.

The point is, using the same meter stick will get you the same marks, even though subject to 20% contraction relative to the first reference frame.

Not necessarily, it depends on the details of how the measurment is done. For example, I could make two marks from outside the ship, than ast the ship is taking off and leaving me behind, I could measure the distance again as it goes by-- and get a different answer. You might say, no fair, now I'm not in the reference frame of the ship-- but that's my whole point, there is no reference frame of the ship in formal terms-- reference frames are local. There is only a reference frame of a tangent space, which the ship approximates.

Whether with time or space, somehow we have to translate the event of marking or ticking into a useable measurement, separate from the events.

No, that's just it-- we don't have to "translate" time at all, proper time is what a clock measures, period. That is as true in the tangent space as it is for the age of the universe.

It seems to me that your comments within the parentheses hides a lot of the problem.

No, there's a difference between a real problem in the way relativity works, fundamentally, and the usual idealizations we have to make when we include a real observer into the physics (i.e., we never do).

In theory, we could do the same thing with space. Why not?

I have already answered that several times: because there is no such thing as a global reference frame, only an arbitrary global coordinate system. Time is inherently local, space is inherently nonlocal. That is the very nature of causality in a nutshell, and the difference between events that are spacelike vs. timelike separated.

Never get what?

The difference between causality and acausality. No such fundamental difference could exist if time and space were equivalent, as you imagine.

Which clock?

Any clock. That's the point.

[hhEb09'1]

You know perfectly well, they are events that are acausal. Was there ever any uncertainty in what we are talking about here?

So what is the problem with "the distance between two events is spacelike" then?

Not necessarily, it depends on the details of how the measurment is done. For example, I could make two marks from outside the ship, than ast the ship is taking off and leaving me behind, I could measure the distance again as it goes by-- and get a different answer. You might say, no fair, now I'm not in the reference frame of the ship-- but that's my whole point, there is no reference frame of the ship in formal terms-- reference frames are local.

And clocks are not local either, they're finite.

No, that's just it-- we don't have to "translate" time at all, proper time is what a clock measures, period. That is as true in the tangent space as it is for the age of the universe.

Any clock. That's the point.

We are talking about your "proper age" of the universe. Which clock are you using?

[Ken G]

So what is the problem with "the distance between two events is spacelike" then?

The spacelikeness is not determined by the distance alone. There's no such thing as "the distance between two spacelike events" in general. What distance can I tell you is between them that you will be able to say they are spacelike separated? You can say the minimum distance is nonzero, but that doesn't answer the question "what is the distance between spacelike events"? For times and timelike events, on the other hand, I can answer that-- it is the proper time between them. The physical difference is simply that timelike separated means an observer can be at both events, and so can measure an invariant (subject to some technicalities) proper time interval between them. But spacelike separated means no observer can be at both events, and the spacelike minimum distance can be defined in many ways, because it cannot be measured directly-- how does one make direct measurements on acausal events? They are not accessible to the same observer, one must use nonlocal proxies and make inferences based on assumptions-- or involve light and clocks!

And clocks are not local either, they're finite.

As I said, that has nothing to do with relativity, it is a limitation of all physics that we never take into consideration because we have no way to do it-- we have no theory for taking into consideration the complex amalgamation of events that we call a measuring apparatus. Instead, events are idealized as points in spacetime. Your objection says nothing about the difference between space and time-- the differences of which I speak are fundamental to the theory of relativity in particular and the way physics is idealized in general. You see, the problem is not that real clocks must exist over a finite space or that real rulers must exist over a finite time, it is that a real clock can measure an arbitrarily large proper time, because its reference frame is local, whereas no ruler can measure an arbitrarily large distance, it can only measure infinitesmal distances in the tangent plane we call the reference frame of the ruler. Again, one could not define "the reference frame" of a large ruler, as it would require that signals travel along the ruler at infinite speed.

We are talking about your "proper age" of the universe. Which clock are you using?

As I said, I'm using any idealized clock, corresponding to a local reference frame that connects any event of interest to the Big Bang singularity. (No part of my argument requires that the "proper age" be the same. Note that in relativity, "proper" does not mean "correct" or "unique", it means "own". Nevertheless, it is a useful property of our universe that the typical matter we deal with has a similar proper age, 13.7 billion years. Universes that do not obey that rule could still obey relativity, that is no part of my argument. It is the mere existence of a proper time of which I speak-- that's the age of the universe for that hypothetical idealized clock.)

[hhEb09'1]

ThYou can say the minimum distance is nonzero, but that doesn't answer the question "what is the distance between spacelike events"? For times and timelike events, on the other hand, I can answer that-- it is the proper time between them. The physical difference is simply that timelike separated means an observer can be at both events, and so can measure an invariant (subject to some technicalities) proper time interval between them.

Ah.

Proper time between timelike-separated events is not an invariant.

PS: wait, you're not talking about proper time, you're talking about an invariant proper time, is that right?

[Ken G]

Proper time is an invariant, in the relativistic sense that it does not change when looked at from another reference frame, expressly because it is always referenced to a particular physically possible (though quite likely hypothetical) observer. One of the most fundamental rules of physics is that any observer, real or hypothetical, follows whatever course that constructively interferes, and does not follow what destructively interferes, and that interference is controlled by the proper time of various contributing hypothetical processes for the observer (which could be a single particle, anything with a reference frame). Thus we tend to say that what happens is an extremum of the proper time, as that's where you get constructive interference. So it's not a coincidence that the proper time of an actual observer is always less than the conceptualized elapsed time for reference frames that don't track the actual process-- for it is that minimum proper time that made the process happen in the first place, that allowed the observer to be something physically possible (like not going faster than c, etc.).

[hhEb09'1]

But different observers, going between the same two events, can have different proper times. The events themselves are not separated by an invariant proper time, right?

[Ken G]

But different observers, going between the same two events, can have different proper times. The events themselves are not separated by an invariant proper time, right?

In relativity, the word "invariant" never applies to two different objects, like two twins in the twin paradox, it applies to the properties of one object seen from two different reference frames. So to say that a proper time is invariant does not mean that two different observers must experience the same proper time between two events, it means that any observer must conclude that a single other observer or object experiences the proper time that a clock moving with them would read. The invariant is essentially the instructions to transliterate the concepts of space and time between different observers following different paths. If the observer-paths connect the same events, so are closed, you only need to correct for acceleration and gravity differences between the paths. If the paths do not connect the same events, so are open, the transliterations involve another correction, and that correction is a construct we call "distance". The proper time along any particular path, seen from any other path, is an invariant, as it is a measurable that any correct theory would have to get right. If someone wants to use a different clock that was not at both events, yet interpret what a clock would read along a particular path joining the events, they'll have to make the transliteration into the invariant.

Lengths, on the other hand, aren't like that, because the "proper length" between all causally connected events is always zero. And acausal events cannot have one observer at both ends of the path, so there can never be an invariant measurable involved. Since it's not a measurable, it then becomes a matter of convention, and there are many ways to define distance in general relativity. Even in special relativity, one can choose the minimum distance between acausal events, which happens in a reference frame where the events are deemed to be simultaneous-- but the word "deemed" is crucial there, it is a construct of the theory-- not a measurable. Measurements are local, as they are carried out by instruments at particular events in spacetime and hence can only reflect the conditions at the time and place of the measurement, so can only measure causally connected intervals, i.e., proper times.

[hhEb09'1]

First, I want to thank you for staying with this. Personal knocks aside, I've enjoyed it, and I'm learning a lot just putting myself in your position, and understanding where you're coming from.

So to say that a proper time is invariant does not mean that two different observers must experience the same proper time between two events,

Which is why I asked about which clock was doing the measuring. Different clocks, different paths, give different proper times between events--that was the point.

Lengths, on the other hand, aren't like that, because the "proper length" between all causally connected events is always zero.

Are you saying that because proper length is only defined for simultaneous events? If so, then the proper time for such events would also be zero.

Whatever "transliterations" have to be done, can be done for distance as well as time.

Since it's not a measurable, it then becomes a matter of convention, and there are many ways to define distance in general relativity. Even in special relativity, one can choose the minimum distance between acausal events, which happens in a reference frame where the events are deemed to be simultaneous-- but the word "deemed" is crucial there, it is a construct of the theory-- not a measurable.

The minimum proper time between any events is always zero.

Measurements are local, as they are carried out by instruments at particular events in spacetime and hence can only reflect the conditions at the time and place of the measurement, so can only measure causally connected intervals, i.e., proper times.

But which clock, that is the question.

As you pointed out, earthquakes along the path can screw up the measurement. Certainly, there are pitfalls for clocks as well.

[Ken G]

First, I want to thank you for staying with this. Personal knocks aside, I've enjoyed it, and I'm learning a lot just putting myself in your position, and understanding where you're coming from.

Yes, it is always interesting to follow a thread with an intelligent and skeptical mind. I don't think there's been any personal knocks?!

Which is why I asked about which clock was doing the measuring. Different clocks, different paths, give different proper times between events--that was the point.

Well I agree with the statement, but don't see the point.

Are you saying that because proper length is only defined for simultaneous events? If so, then the proper time for such events would also be zero.

I'm saying that proper length, being the length between two events as measured by an observer at both events, is always zero-- nothing to measure. No other lengths can be measured, only inferred based on various assumptions about what is happening somewhere other than where the observer is. Proper time, on the other hand, suffers from no such limitation. Ergo, the only "proper" measurement is of time. Furthermore, the "speed of light" is a critical concept, but all speeds are rates, and when we use proper time as our fundamental measurement of an interval, then we see that "speeds" are rates of time, and the claim that the speed of light is always a constant reduces to the observation that the rate of proper time is necessarily unity. All that is left for experiment is the actual determination of c in some useful distance scale drawn from the way we conceptualize our experience (a "foot", for example). That last step is the only one that is not scale invariant in relativity. The units of time are always completely arbitrary, for all that matters is the rate of proper time, which must be unity for any unit of time.

Whatever "transliterations" have to be done, can be done for distance as well as time.

Of course, but what I'm saying is that the meaning of any transliteration, when complete, rests in the meaning of the final step. If someone translates Greek into Swahili for me, they have done nothing, but if they translate into English, then the translation has actually done something. Any language may be transliterated into any other-- but proper time is the only one that actually connects to experience, because it is the only one that can actually be measured in anything but the infinitesmal tangent space (where we can pretend we have rigid rulers). No meaning emerges from relativity until the transliteration is done into proper time, this is what I am saying. The clock is the fundamental measuring tool, for it is the instrument the observer takes along. Rulers, to get distances, would need to be wielded by an array of observers, chosen arbitrarily to connect the events of interest. (Laying the same ruler down over and over is not a length measurement, without all kinds of arbitrary assumptions that may or may not hold.)

The minimum proper time between any events is always zero.

Again, I don't see why that is relevant. Relativity begins with the choice of an observer, and all its results allow that observer to apply physics. Thus it makes no difference what the "minimum" proper time is between events, it matters what the proper time is for some observer at both events. That's how we do physics, we are an observer at various events, there are no exceptions.

But which clock, that is the question.

And the answer is simple-- the clock of the observer who is doing physics. You see, in my view relativity is not physics, physics is done by an observer. Relativity is the means for transliterating between observers, so we can apply that same physics to other observers and understand how it will look to us when transliterated. It means you don't have to do everything yourself to know what will happen to you. Basically, physics is time, relativity allows the extension of physics into a concept of space, with the useful assumption that the physics will be the same "elsewhere".

[hhEb09'1]

I came back to check on this old thread, and found that a response that I thought I had posted had not posted.

I was wondering why I wasn't banned.

I'm saying that proper length, being the length between two events as measured by an observer at both events, is always zero-- nothing to measure.

Proper length is no always zero, but the proper length in the instances you are talking about is zero, but the proper time is always zero then too.

That's my point, I think. Time and space are not the same thing exactly, but they're sorta duals of each other. As in projective geometry, where whatever you can say about lines and points, you can say the same about points and lines. Two unique points always determine a unique line, and so two unique lines always determine a unique point, lines and points are duals of each other.

It just occurred to me that you reject the meter bar as not having an observer at both ends of the meter bar, but the meter bar is at both ends. That's not much different than a clock that extends from one point in time to another. As you say, transliterations have to be made--and the path of the clock determines its measurement just as much as the path of the meter bar determines its.

You see, in my view relativity is not physics, physics is done by an observer. Relativity is the means for transliterating between observers, so we can apply that same physics to other observers and understand how it will look to us when transliterated. It means you don't have to do everything yourself to know what will happen to you.

Exactly. So why insist that the single same observer be at both ends of the measurement--especially when that observer has to endure a time span and not really be the "same"?

I'm not the same observer I was thirty years ago

[Ken G]

Proper length is no always zero, but the proper length in the instances you are talking about is zero, but the proper time is always zero then too.

Obviously if that were true, nothing I've said would make any sense. But it isn't true. What "proper" means is, an interval (in space or time) between two measurements done by the same observer in their own reference frame. As such, proper length measurements are always zero (no observer ever moves anywhere in their own frame), and proper time measurements are never zero. Note that you will see the phrase "proper length" used, but if you are paying attention, my entire point is that this is always actually proper time described in length units-- because a true proper length is never measured to be anything but zero.

That's my point, I think. Time and space are not the same thing exactly, but they're sorta duals of each other.

They have a very important relationship as useful concepts in physics, that is true. But what I'm saying is that the primary difference between them is that one can actually be measured, whereas the other can only be conceptualized beyond the tangent space in which it actually has meaning. So there is an important physical difference between them, in addition to the mathematical sign difference, that makes it appropriate to think of time as more fundamental, and proper intervals are proper time intervals.

As in projective geometry, where whatever you can say about lines and points, you can say the same about points and lines. Two unique points always determine a unique line, and so two unique lines always determine a unique point, lines and points are duals of each other.

There is a mathematical structure of spacetime that does not care at all whether time or distance are viewed as more fundamental. But the mathematics of spacetime sits on the back of the physics that uses it-- and in that physics, there is an important difference. Furthermore, recognizing the difference has conceptual value. For example, it becomes clear what "causality" means: one event can cause another if they can happen to the same observer, i.e., if they are separated by a real (rather than imaginary) proper time. That reason alone would justify using proper time nomenclature over proper distance. The other good reason is that "proper" should mean "actually measured by an observer".

It just occurred to me that you reject the meter bar as not having an observer at both ends of the meter bar, but the meter bar is at both ends. That's not much different than a clock that extends from one point in time to another.

It's not the clock that extends, it's the observer using the clock. So it's very different-- especially when considering the scales on which relativity matters most: cosmological scales. There is one observer at "both ends of clock", not so at both ends of the ruler. So if you need two different observers communicating to each other, you are going to have to interpret their relationship in some way that is not a measurement.

As you say, transliterations have to be made--and the path of the clock determines its measurement just as much as the path of the meter bar determines its.

No transliteration is needed for the clock-- it just reads out the time interval.

I'm not the same observer I was thirty years ago

None of us are.

[Tensor]

I'm not the same observer I was thirty years ago

Well, I can be if I put my reading glasses on.

[publius]

I'm pretty much with Ken here. Reading this, I'm pondering the notion of proper length as it is defined. Basically, in space-time, the norm s^2 is the invariant. If that sucker is positive (using a positive metric signature), then the separation, norm, interval, whatever you call it, is time-like, and that means some inertial observer would could see those events occuring at the same "place" (constant spatial coordinate) at different times. |s| is then the time interval according to his clock. That observer's world line is just the "straight line" between those events. (And note other non-inertial observers could join those events via a curved path and experience less proper time -- this is flipping the triangle inequality backwards, which is what time-like separations do).

If s^2 is negative, that means some observer could "see" those events occuring *simultaneously* at different places. |s| is then the distance between those events according to that observer.

And to that, big whoop, really. There is something special only about the time-like |s|.

Now, go to curved space-time, and things get even more strange, or better, things are less meaningful than you hope they would be. Our notion of "straight lines" has to become geodesics. In flat space-time, the notion of the straight line path is straightfoward. But go to curved space-time, and we've got to find the geodesics to get any notion of the "separation between events". Time-like paths there, which could be followed by a real observer, are really the only ones that mean anything, although space-like geodesics can be defined.

Beyond this, "proper distance" is sometimes used to mean integrating a space-like ds between events that occur at the same coordinate time according to some observer.

That is, it's the |s| we get by integrating a path along a spatial hyperslice for some observer (and, as the Ehrenfest Paradox mess illustrates, these spatial hyperslices can go awry).

And this shows that a proper distance defined this way is not the same thing thing as the geodesic path between events. It all depends on whose time coordinate, which spatial hyperslices you are using. Well, what I'm saying here can be confusing. If "kosher" global spatial hyperslices exist in some coordinate system, then events at constant time are all space-like as described above. That is, our coordinates pick these events for us. However, go the other way and choose two events at random, and I don't think you're guaranteed to be able to construct a "kosher" spatial hyperslice they lie on in all cases.

In Cosmology, the details of which I'm not clear on and will defer to more informed commentary, the "proper distance" between galaxies is sometimes spoken of. That's just integrating ds between co-movers at constant coordinate time t, using the scale factor (and that would NOT BE THE SAME "distance" along the tangent space of a co-mover, such as with the static deSitter coordinates!) That is, we're just picking spatial hyperslices in the co-moving coordinatization and declaring those space-like intervals to be something "proper". {And since the co-moving spatial coordinates don't change there, only the metric factor increases with time, we say "space is expanding". Big whoop, again, although that whoop is taken quite literally by many, too many IMHO}.

And that is as arbitrary as the wind, really. So, a "proper distance" is not really proper at all!

That's something profound, I think, that comes from Uncle Al's Magnum Opus, here. Space and distance just don't have the meaning we want them to have. Proper time does have meaning.

Things can get even more fuzzy when you have more than one time-like geodesic path between events (think of two intersecting orbits for a simpe example), and |s| along those paths is different. The "shortest distance between two points(events)" is no longer unique! (Unless you cheat and invoke an embedding space, but that interval is then no longer "inside" your space-time, and then your embedding space might need two or more time-like coordinates anyway..............)

So, with that you realize the only thing you can hold onto is the proper time along a specified path.

-Richard

[Ken G]

But go to curved space-time, and we've got to find the geodesics to get any notion of the "separation between events". Time-like paths there, which could be followed by a real observer, are really the only ones that mean anything, although space-like geodesics can be defined.

That's the crux of what I'm saying. And I think the conceptual leap one can take from this is that physics is not about a whole bunch of observers telling each other what is going on, it is about one observer working everything out for themself, and another doing the same, and then the two comparing notes and saying "nothing seems to agree unless we choose the following way to transliterate our experiences, and then everything agrees perfectly". The physics of one observer plays out in time and borrows a sense of space from the tangent manifold to their own experience, but the connections across the curved spacetime between different observers requires relativity, so relativity is not a physical principle, it is a rule of translation that allows the same physical principles to be in effect. There's no need for any translation when dealing with time, it is what happens "to us".

The other advantage to this approach is that when we hear 'all objects move at c through spacetime', we say 'I prefer to say all objects move through proper time at rate unity', which sounds a lot more obvious.

However, go the other way and choose two events at random, and I don't think you're guaranteed to be able to construct a "kosher" spatial hyperslice they lie on in all cases.

And even if you can, note that it is not a measurable entity, it is a construct that requires collusion between observers. But which observers do you choose to use? Observers with no relative motion, observers in the local comoving frame, observers at infinity? The only non-arbitrary choice is to do all the observations yourself, but then you are observing time.

{And since the co-moving spatial coordinates don't change there, only the metric factor increases with time, we say "space is expanding". Big whoop, again, although that whoop is taken quite literally by many, too many IMHO}.

I agree, there's another thread about the over-interpretation of quantum mechanics, and here we have the over-interpretation of relativity. All following a few centuries after the over-interpretation of Newtonian determinism-- we seem to never learn.

Space and distance just don't have the meaning we want them to have. Proper time does have meaning.

Yes, that's how I see it too. There seems to be two roles for space, one is the instructions for mapping between two distant observers, and the other is more local, as though each "world line" was drawn along time, but drawn with a thick magic marker whose width actually means something. Space is weird. The way I'd sum it up is that we should not say
ds² = dx² - c² dt²
and we should not even say
dtau² = dt² - dx²/c²,
we should say
dx²/c² = dt² - dtau².
In other words, a metric is the instructions for defining what we mean by the spatial separation between two observers, given the observed differences in their proper times. Distance is a construct, time is a measurement-- and the distance construct is best expressed in units of time because it comes from measured times. Why other things seem to care about distance, like the strength of inverse-square forces for example, I can't begin to work out.

[publius]

Ken,

Right on, brother.

The other advantage to this approach is that when we hear 'all objects move at c through spacetime', we say 'I prefer to say all objects move through proper time at rate unity', which sounds a lot more obvious.

That's another one worthy of careful consideration. At the basic level,
s = c *tau, and that's just conversion between space and time units. The value of 'c' is therefore irrelevant, really. However, there is something fundamental there.

Is it that all observers move along world lines at this constant speed? Or our experience of what we call time, the frame rate of the movie so to speak, is the same for all observers? That is, we can move along whatever (time-like) tracks we like, but we move at the same speed.

Or is that more of a mathematical "game" devised to simply do the bookeeping? .................

-Richard