B Newtonian vs Relativistic Mechanics

[Grimble]

A frame of reference is a mathematical object, not a physical object. In GR it is formalized as something called a tetrad, in SR it is less formally used to refer to a coordinate system. Either way it is a mathematical object, not a physical object. A reference frame is part of the analysis, not part of the experiment.

You can use a frame of reference regardless of whether or not there is any substance located at the origin. Many times it is convenient to use reference frames where no object is at rest (e.g. the center of momentum frame) so no object is located at the origin other than "in passing".

From a theoretical standpoint proper time is the spacetime interval along a given timelike worldline, which is experimentally measured by a clock traveling on that given worldline. Neither the theoretical nor the experimental meaning of proper time mentions a frame of reference nor even "inertial".

Yes, I understand the Frame of Reference is not a physical thing. It is essentially the map of Spacetime from the perspective of a real or virtual observer at the origin of the frame.

If one takes that map of Spacetime, for a virtual observer at the origin (or null point if you prefer) of the map, then that observer's worldline - plotted on the map depicted by that frame of reference - would be a straight vertical line as that observer is permanently at rest in that frame, because the null point is his position.

Surely if,

proper time is the spacetime interval along a given timelike worldline, which is experimentally measured by a clock traveling on that given worldline

then isn't that the definition of the time axis of the map of spacetime that is a frame of reference, whatever motion that virtual, real or imaginary object at the origin has relative to any other object, particle, substantial point or body in Spacetime? For the Time axis of a Frame of Reference IS the path of a virtual clock at rest at the origin of that Frame of Reference.

Please, I am not trying to redefine anything but seeing a relationship, something that fulfils the definition - I will try to understand if that is wrong, but there must be someway in which it doesn't work...

The most confusing thing for me - and, I can only guess, for others is using a term like Proper Time because it implies a fixed and rigid scale. That Proper Time implies a specific rate that is the same for every observer; that time passes at the same rate for any observer on a clock that is at rest relative to them. (And maybe only for those which are non accelerating, with zero gravity)

As I say I am not trying to redefine anything I am just commenting on what the term Proper Time seems to mean vide:

Yes, the original German term is "Eigenzeit". The word "eigen" translates also as "private", "own", "separate", "distinctive". I think every of these translations would have been more appropriate than "proper", which has also the association of true, correct, genuine, appropriate, adequate. Associations which are misleading, and not present in the German "eigen", which also has "peculiar" as a possible translation. http://www.dict.cc/?s=eigen

It there would have been a better translation of "Eigenzeit", I would guess there would have been less confusion about the twin "paradox". Of course, if the two twins compare their clocks, and see different results, one would not wonder very much if this clock time would have been named "private time" or "distinctive time" or "peculiar time". But if what is compared is strongly associated with "true time" or "correct time", once it is named "proper time", the situation is quite different.

 

[Dale]

Please, I am not trying to redefine anything but seeing a relationship, something that fulfils the definition - I will try to understand if that is wrong, but there must be someway in which it doesn't work...

Ah, OK. I got the mistaken impression that you were trying to redefine things. It seemed that you were trying to define proper time in terms of an inertial frame, instead of the other way around. Sorry I misunderstood.

The time measured by *any* clock is the proper time along its worldline. So a clock measures it's proper time regardless of whether or not it is at rest at the origin.

There is indeed a relationship between coordinate time (in an inertial frame) and proper time. That is $d\tau^2=dt^2-dx^2-dy^2-dz^2$. For a clock at rest anywhere in the frame (not just the origin) we have $0=dx=dy=dz$ so $d\tau=dt$.

 

[PeroK]

Yes, the original German term is "Eigenzeit". The word "eigen" translates also as "private", "own", "separate", "distinctive". I think every of these translations would have been more appropriate than "proper", which has also the association of true, correct, genuine, appropriate, adequate. Associations which are misleading, and not present in the German "eigen", which also has "peculiar" as a possible translation. http://www.dict.cc/?s=eigen

It there would have been a better translation of "Eigenzeit", I would guess there would have been less confusion about the twin "paradox". Of course, if the two twins compare their clocks, and see different results, one would not wonder very much if this clock time would have been named "private time" or "distinctive time" or "peculiar time". But if what is compared is strongly associated with "true time" or "correct time", once it is named "proper time", the situation is quite different.

As an English speaker, "proper time" sounds right to me. "Private time", for example, would be time a particle spends alone, without any interference from observers! And "peculiar time" or "distinctive time" don't sound right at all. Possibly, "intrinsic" or "natural" time would be alternatives. But, really, if someone is blaming the term "proper" for a lack of understanding of SR, they are really clutching at straws! (I'm not sure how you'd say that in German!)

"Too many words, not enough maths" is the problem here, IMHO.

 

[Ilja]

But, really, if someone is blaming the term "proper" for a lack of understanding of SR, they are really clutching at straws! (I'm not sure how you'd say that in German!)

My point is not about blaming something, but about improving the understanding some common misunderstandings. If you don't understand what someone is misunderstanding, you cannot help him to reach a better understanding.

I prefer to name proper time "clock time". This avoids the misunderstanding inherent in proper time as well.

 

[stevendaryl]

It there would have been a better translation of "Eigenzeit", I would guess there would have been less confusion about the twin "paradox". Of course, if the two twins compare their clocks, and see different results, one would not wonder very much if this clock time would have been named "private time" or "distinctive time" or "peculiar time". But if what is compared is strongly associated with "true time" or "correct time", once it is named "proper time", the situation is quite different.

I'm not sure if that gets at the reason people are confused by the twin paradox. To me, the misconception is that (apparently) each twin can view the other one as aging faster, which seems like a logical contradiction. I don't think that the phrase "proper time" is relevant in explaining why people find it confusing---the first introduction to the twin paradox usually doesn't even mention proper time.

 

[robphy]

I like Taylor & Wheeler's "wristwatch time".

 

[vanhees71]

Well, I also never understood why somebody has a problem with the twin paradox. At least after you have understood the necessity for Minkowski spacetime instead of Galilean spacetime (in SRT), it should be very clear that the difference in proper times of particles following different world lines is a logical consequence. It's not more mysterious than the fact that traveling between two places along different routes means to travel a different distance. There's a "proper" distance defined as the length of the shortest geodesic connecting the two points. The same you have here for "distances" in the sense of the Minkowski pseudo-metric.

 

[pervect]

Well, I also never understood why somebody has a problem with the twin paradox. At least after you have understood the necessity for Minkowski spacetime instead of Galilean spacetime (in SRT), it should be very clear that the difference in proper times of particles following different world lines is a logical consequence. It's not more mysterious than the fact that traveling between two places along different routes means to travel a different distance. There's a "proper" distance defined as the length of the shortest geodesic connecting the two points. The same you have here for "distances" in the sense of the Minkowski pseudo-metric.

My belief is that the main issue people have with understanding the twin paradox is that they don't understand that simultaneity is relative, and that they also don't understand what one mean if/when one says that simultaneity is relative, nor do they understand what one means if/when one says that "time is not absolute".

This may not be the only issue. Sheer, Shaefer and Vokos talk about the issue of understanding the relativity of simultaneity in their paper "The challenge of changing deeply-held student beliefs about the relativity of simultaneity", https://arxiv.org/abs/physics/0207081. While the title of the paper focuses on the issue that I've mentioned, reading the paper shows other gaps in student understanding that make it difficult for them to reach a proper understanding. So one may need to fill in those OTHER gaps in student understanding , before one can effectively address the issue of understanding the relativity of simultaneity, after which point one is finally ready to talk about the twin paradox.

Some other issues the authors have identified are very simple ones related to students not understanding aspects of non-relativistic physics, rather basic issues such as how reference frames work, how to properly account for propagation delays of signals that move at a finite speed, and even the idea that the order of two events that happen to a pointlike observer does not depend on the reference frame one chooses.

Given the lack of understanding of these points about non-relativistic physics, students have a difficult time dealing with the twin paradox. They mainly rely on their intuition, which does not work at all for relativity, and they have difficulty following the formal steps needed to work the problem in order to change their incorrect intuitions. I'm not aware of any good solution to the problem - I think Scherr's paper gives some good advice based on practice and observation in the context of a classroom, but the methods that work in a classroom do not necessarily work in a forum such as PF. A rather general remark by the autors about student errors is "However, in many cases, conceptual difficulties seemed to prevent students from answering correctly". It may be the case that untangling these various conceptual errors may simply require more effort and thought and study than most casual readers are able to give. It also suggests that the solution may have to be done one student at a time, i.e. individual attention to the student is required.

 

[robphy]

Given the lack of understanding of these points about non-relativistic physics, students have a difficult time dealing with the twin paradox. They mainly rely on their intuition, which does not work at all for relativity, and they have difficulty following the formal steps needed to work the problem in order to change their incorrect intuitions. I'm not aware of any good solution to the problem - I think Scherr's paper gives some good advice based on practice and observation in the context of a classroom, but the methods that work in a classroom do not necessarily work in a forum such as PF. A rather general remark by the autors about student errors is "However, in many cases, conceptual difficulties seemed to prevent students from answering correctly". It may be the case that untangling these various conceptual errors may simply require more effort and thought and study than most casual readers are able to give. It also suggests that the solution may have to be done one student at a time, i.e. individual attention to the student is required.

One solution is to provide the student with the right tool: the spacetime diagram.
In my opinion, "a spacetime diagram is worth a thousand words".
To me, an obvious follow-up study is to see
if students reason better with "worldlines on spacetime diagrams" (presuming they have been appropriately developed)
rather than "diagrams of space and spatial trajectories in one frame of reference" (which are featured prominently in their study).
Most introductory textbooks seem to avoid the spacetime diagram... or else merely mention it in passing
(maybe because it seems that Einstein did not reason with them...
in fact, he initially hated the idea of spacetime as something the superfluous that the mathematicians dreamt up...
until, of course, he realized that he needed it for general relativity).

 

[vanhees71]

Thanks, pervect. This comes very timely, because right now the semester started, and I teach a lecture called "Mathematical additions to theoretical physics 2", which was introduced some years ago to help the students with the mathematics needed for the theory lecture, which start at our university already in the very first semester, which is challenging for both teachers and students, because the German high-school education in mathematics is a desaster. What's called math at high school is not what you understand under that name as a physicist (let alone a mathematician). Now in Theoretical Physics 2 the professor starts with special relativity. So I'll give in my math class an introduction to Minkowski space. So perhaps this physics-didactic papers give me some ideas how to make the business easier to understand ;-)).

@robphy: Yes, that's what I thought. So yesterday I started right away with Minkowski space and draw the usual space-time "plane". I think here the greatest difficulty for the students is that the axes of (at least) one of the inertial observers is not "orthogonal in the Eucildean sense" and to forget about the Euclidean structure of the plane you usually associate with the plane you draw the space-time diagram on. You have to substitute it in your thinking by the "Minkowski geometry", which is pseudo-Euclidean rather than Euclidean. I hope this becomes clear by drawing the time- and spacelike hyperbolae defining the "unit mesh" on this Minkowski plane. On the other hand, the Lorentz transformation in this plane (a Lorentz boost of course), becomes very intuitive from the 2nd Einstein postulate (constancy of the speed of light for all inertial observers): You construct the world line of Bob in Alice's referene frame, which must be in the forward light cone of the origin (velocity of Bob relative to Alice less than the speed of light). Then the Bob's spatial vector must point such that the world line of the light front is the bisecting between Bob's time-like and the space-like angle. Then you just need to normalize these two vectors in the sense of the Minkowski pseudo-metric.

 

[robphy]

You have to substitute it in your thinking by the "Minkowski geometry", which is pseudo-Euclidean rather than Euclidean. I hope this becomes clear by drawing the time- and spacelike hyperbolae defining the "unit mesh" on this Minkowski plane.

Yes, that is the usual dilemma... and hyperbolas are not very intuitive.
If only there was another way...

 

[Dale]

One solution is to provide the student with the right tool: the spacetime diagram.
In my opinion, "a spacetime diagram is worth a thousand words".

This and four-vectors were what finally made SR fall into place in my mind. All the thought experiments and formulas just left me with a mind full of disconnected and unconvincing facts.

 

[robphy]

Although I was familiar with 4-vectors and the dot-product (which was preserved by the Lorentz transformations) in component form,
things didn't click with me until I saw the radar method, which established operationally what an inertial observer measured...
as well as directly showed how the signature of the metric arises.
Then, with the k-calculus (based on the doppler factor), it is easy to derive the lorentz transformations
(as well as noting that the k-calculus' simplicity comes from it working in the eigenbasis of the lorentz transformation).

 

[Grimble]

The time measured by *any* clock is the proper time along its worldline. So a clock measures it's proper time regardless of whether or not it is at rest at the origin.

Yes, I understand that, but what I am associating is that for a clock at rest at the origin, because it is permanently and continuously at rest at the origin, then its worldline is coincident the worldline of the origin, so the timescale for a frame of reference is the proper time of the virtual clock at the origin of the frame.

But a bigger problem for me - and I guess for others too, is that the term proper time implies a particular time scale that would be the same foe proper time in every case... yet it seems too that it is not...

 

[peety]

Just a brief comment on why the twins appear to present a paradox: we confound equivalence with symmetry.

 

[PeroK]

Yes, I understand that, but what I am associating is that for a clock at rest at the origin, because it is permanently and continuously at rest at the origin, then its worldline is coincident the worldline of the origin, so the timescale for a frame of reference is the proper time of the virtual clock at the origin of the frame.

But a bigger problem for me - and I guess for others too, is that the term proper time implies a particular time scale that would be the same foe proper time in every case... yet it seems too that it is not...

I used to have a book called "The Tyranny of Words". It was actually about the dangers in politics of letting words dictate how you think. The same is true in science. You mustn't let specific words dictate how you can and can't think.

In this case it it always "the proper time/length of" something. In fact, the German word "eigen" carries this association that it belongs to a specific thing.

Each inertial reference frame has its own coordinate time (which is the proper time of a particle at rest in that frame). But, a particle traveling with changing velocity has its own proper time that is not the coordinate time of any inertial reference frame.

 

[Grimble]

This and four-vectors were what finally made SR fall into place in my mind. All the thought experiments and formulas just left me with a mind full of disconnected and unconvincing facts.

Well, I have just had a look at 'four-vector' in Wiki - sheesh! It seems one must have a good understanding of the topic to understand the terms in the explanation! Some Wiki entries seem to be written by the cognoscenti for the cognoscenti!
Certainly not in my remit! (groan!)

 

[Grimble]

There are many references above to the Twin Paradox, but I am unsure as to whether you all mean the same thing.

When I first came across it I understood the paradox was that each twin had the same experience of the other - reciprocating what the other experienced.

Something that I am sure I read - either on Wiki or in some forum - that in Special Relativity the relative movement of two bodies was always reciprocal - that one could swap the roles of A and B and the result would be the same (with only the labels A and B swapped).

Extending this the the twin paradox though brought in the fact that the traveller was subject to acceleration - which has nothing to do with Special Relativity - which is where the twin paradox is introduced.

But this leads me confused about just what is claimed to be the real state of things - yes I find the paradox confusing - or rather the explanations that some get so involved in.

It seems to me (with Occam's razor in my hand) that if the journey is treated as a single outward movement, with the traveller passing the stationary twin at a constant speed, then we have pure SR. No acceleration - and allowing for the journey time of any measurements, then the twins will have reciprocal measurements of the others motion and clocks. Movement in SR is, after all all relative.

Then each twin will measure the other's time dilated and clock slowed.

Adding in the deceleration/acceleration and the return only complicates the experiment.

Which raises the question of whether each clock slows, or is it just read differently by a moving observer? Reading coordinate time as opposed to proper time?

I have seen this addressed by means of a third twin (triplet/virtual twin? hehehe) who is traveling back and synchronizes their clock with the outbound twin's; leading to the claim that it is changing frame that makes the difference but no explanation of just what that means or implies...

Another little point that confuses me in the twin paradox explained as a result of only one twin accelerating is that the total time difference when the traveller returns is dependent on the speed and the duration of the journey; not upon any factor related to the acceleration - neither to the rapidity of that acceleration nor to its duration.

(I'm sorry, Dale, but it wasn't me who brought the twin paradox into this thread! - And I am not arguing with it merely saying what I find confusing about it)

 

[Dale]

Well, I have just had a look at 'four-vector' in Wiki - sheesh! It seems one must have a good understanding of the topic to understand the terms in the explanation! Some Wiki entries seem to be written by the cognoscenti for the cognoscenti!
Certainly not in my remit! (groan!)

All you have to do is take a usual vector with components $(x,y,z)$ and add a fourth component $(t,x,y,z)$. And voila, you have a four vector.

There are only a couple of small differences. The first is that you have to use units where c=1. Otherwise you have to throw in factors of c to get the units to match. The second is that the dot product changes to

$(t,x,y,z)\cdot(T,X,Y,Z)=-tT+xX+yY+zZ$

 

[PeroK]

Which raises the question of whether each clock slows, or is it just read differently by a moving observer? Reading coordinate time as opposed to proper time?

The book from which I learned SR doesn't mention proper time until page 121 (at the same stage as four-vectors). By that time, it has already covered inertial reference frames, time dilation, lengths, simultaneity, paradoxes, the Lorentz Transformation and Spacetime Diagrams (in that order).

There is, of course, no set order for these things, but I suggest you are trying to digest the whole of SR in this thread, and before you have really understood the basics. I suggest you need to focus on the basics one step at a time.

 

[Grimble]

The book from which I learned SR doesn't mention proper time until page 121 (at the same stage as four-vectors). By that time, it has already covered inertial reference frames, time dilation, lengths, simultaneity, paradoxes, the Lorentz Transformation and Spacetime Diagrams (in that order).

There seems to be no way I can refer to Proper Time and have what I say understood without someone or several misinterpretations of what I am trying to say - this may be due to my mis-phrasing, lack of understanding or to others reading between the lines and determining meanings that weren't there. Such is the baggage attatched to that term.

So may I plead with you, one and all, and change my terminology?

Allow me to separate, as least as far as this thread goes, measurements of time made by an observer at the origin of a frame on the clock that he is holding, which I will call Local clock time and measurements made by an observer in another frame as Remote clock time.

Now it seems to me that such Local clock time, measured on a standard universal clock that, obeying the same physical Laws in each and any Inertial Frame, ought to keep the same time - or what else is the first postulate for?

The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of coordinates in uniform translatory motion.

All Inertial frames are the same, they all appear to be at rest in spacetime from their own perspective.

So how can the speed of their clock be affected by their speed? It will be affected by the relative speed of an observer, measuring from another frame. One frame will be moving at different speeds relative to different observers and each will measure the observed clock slow - but by different amounts depending on their relative speeds. That one clock that is observed cannot physically run at different rates - as measured by the local observer holding that clock it can only be ticking at one rate, surely?
The different speeds must be how the various observers calculate (Lorentz Transformations), yes, how they calculate the clock to have slowed, as they measure it.

Please believe me that I am not trying to rewrite anything but looking to understand how these different lines of thought and logic fit together.

Can you explain to me where reciprocity fits in - it seems such a very basic property that is at the very heart of how relativity works...

 

[Grimble]

Einstein's principle of velocity reciprocity (EPVR) reads:

We postulate that the relation between the coordinates of the two systems is linear. Then the inverse transformation is also linear and the complete non-preference of the one or the other system demands that the transformation shall be identical with the original one, except for a change of v to −v

Or: 'As B is to A, so A is to B'.

Now I know I found this under an the topic of Wigner Rotation - and I went no further down that route! Yet the principle seems to be a simple one concerning the fact that relative movement is reciprocal - only the sign of v changes.

I think this simple diagram is a good representation of reciprocity:
https://ac0077b2-a-62cb3a1a-s-sites.googlegroups.com/site/specialrelativitysimplified/home-1/minkowski-diagrams/Chess%20Pieces.jpg?attachauth=ANoY7cpYvRnGrNbH-8vkwzdH0O_R6bwfd2drCpIqXJbLRBLEQRRaCtnjFA2GgiwgEIY6wWRkA_CpuhHnXS0YbGbnnfn33G0l7iThkJKAsPC0W7lpUxEha9-h1sStHOa9q5e5uKSbCsQPK98S8vutgwnznysYdxddjzhj1oSaW--avxYm4iAvPGEH7BX1LP-4N8-FbeTnDjl9vErRhNfFvsSULoj6pXXeRgOxnAqraZPoGLZbI-t-0PcigtAImjUeej6FBXfWJ11WmZYLRNAyBnGXJRSGxW51Eg%3D%3D&attredirects=0

 

[Simon Bridge]

Now it seems to me that such Local clock time, measured on a standard universal clock that, obeying the same physical Laws in each and any Inertial Frame, ought to keep the same time - or what else is the first postulate for?

... "same time" as what? Not each other - since they may have different relative velocities with respect to each other.
However - observers who are stationary wrt a clock will observe the clock tick off one second per second (i.e. the reference clock for that frame keeps the same time as all the other stationary clocks in that frame. Some care is needed to make sure of this due to the finite speed of light.
Generally it is not useful to make a local/distant distinction like this (local and non-local are technical terms that you need later anyway.)

All Inertial frames are the same, they all appear to be at rest in spacetime from their own perspective.

Everything is at rest in it's own frame. It does not "appear" to be at rest: it is at rest. "Appear" implies there is some true or absolute state motion - there is no such thing. Frames do not have their own perspective: that requires an observer. Frames are what observers use to make observations (of time and distance) in.
This is a kind-of mental discipline.

So how can the speed of their clock be affected by their speed? It will be affected by the relative speed of an observer, measuring from another frame. One frame will be moving at different speeds relative to different observers and each will measure the observed clock slow - but by different amounts depending on their relative speeds. That one clock that is observed cannot physically run at different rates - as measured by the local observer holding that clock it can only be ticking at one rate, surely?

That is correct - ones own reference frame is stationary but other's may be moving.
You will observe a moving clock to run slow, without any (lorentz) calculation involved ... you just use your clock to time events on the other clock just like you use it to time anything.

The clocks are not physically affected by relative speed: it is the observation of the clocks that are affected.

The observations are affected by relative speed in much the same way as lengths are affected by relative distance away ... "farther objects are smaller" would be the rule for perspective. The "proper length" of an object being what you measure when you are right next it. SR extends the rules of perspective to include relative velocity as well as relative distance... so now the proper length is measure right next to the object, while it is stationary.

[We generally think of distant objects as "appearing" smaller though... this is because we are used to thinking of the ground or something as providing an absolute reference frame.]

Can you explain to me where reciprocity fits in - it seems such a very basic property that is at the very heart of how relativity works...

It means that if someone is moving past you at speed v, then you are moving past them at speed -v. It's not just relativity.
[On the chessboard analogy, black see white move one square to the right, so white sees black move one square to the left... not happy with that analogy since one of them sees their own square change colour.]

In SR it means that if you see a passing clock tick slowly, then it's observer sees your clock tick just as slowly.
Since the twins can spend an arbitrarily long time when neither is accelerating, and they both notice the other's clock is slow, then how come the accelerated twin always ends up younger?
It's pretty easy to see how in space-time diagrams.

The toughest part is getting rid of ideas that rely on absolute motion.
I usually find that the following primer is accessible at HS level:
http://www.physicsguy.com/ftl/html/FTL_intro.html
... you want the bits about space time diagrams but it is probably worth plowing through the rest too.
There is a specific treatment of several solvable paradoxes in there ... tldr: different observers agree about the overall effect (like which twin ends up younger) but disagree about how it came about.

 

[Simon Bridge]

You originally asked about what is different...? Skimming - you seem to have got a lot of stuff about language and thought experiments to highlight the fun stuff.
It may be, however, that you may appreciate a different tack that does not need much beyond HS level understanding. vis:

To replace Galilean relativity, SR has to include it and then extend it to cover phenomena where Galilean relativity is unhelpful.
This gets checked by experiment - there is a list of experiments here:
http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html

The classic Mt Washington muon experiment is a teaching demo often used to show how time dilation looks in practise.


It is also possible to go through the HS physics course and show you the relativity relations next to all the ones you are used to.
ie. if we accelerate a particle, from rest, through potential U (i.e. you may release it from a spring so $U=\frac{1}{2}kx^2$) then it's final speed is given by $v=\sqrt{2U/m}$ (remember the equation for kinetic energy and conservation of energy?) ... but this suggests that a sufficiently big U will get v>c. You can work out how big that is: $U>\frac{1}{2}mc^2$ ... then do the experiment.
The improved relation is: $v = c\sqrt{1-1/(1+U/mc^2)^2}$ ... which is harder on the eyes, but is a good description of Nature.
It also gives answers close to the one you learned if the speed is a lot smaller than c.

... and so on.

 

[stevendaryl]

So how can the speed of their clock be affected by their speed? It will be affected by the relative speed of an observer, measuring from another frame. One frame will be moving at different speeds relative to different observers and each will measure the observed clock slow - but by different amounts depending on their relative speeds. That one clock that is observed cannot physically run at different rates - as measured by the local observer holding that clock it can only be ticking at one rate, surely?

You should keep in mind that the "rate" of a clock as it proceeds along a journey is a ratio of two numbers: \frac{\delta \tau}{\delta t}, where \delta \tau is the elapsed time on the clock from the beginning of its journey to the end, and \delta t is the difference in time coordinates between the beginning of the journey and the end. \delta t is a physically meaningful quantity. In contrast, \delta t is conventional: It depends on how we set up our coordinate system, and there are many different ways, and each of them has its own notion of \delta t.

I like to remind people of the analogies between the so-called paradoxes of Special Relativity and the good-old Euclidean geometry that people take for granted. Most paradoxes have a very close analogy.

Suppose we have a system of highways that cross each other, and bend around. Each highway has a system of "road markers", which are just signs beside the road with real numbers on them, and the numbers increase as you move down the highway (or decrease if you're going in the opposite direction). Those road markers give you a "local" view of your progress down a highway. You don't have to compare your progress with anyone else, you can just say: "I've traveled 50 units down highway A", and that uniquely describes where you are (well, assuming we know the starting point).

But now, if we want to compare two different journeys, this local view is not good enough. We have to set up a coordinate system. So here's a way that we can set up a coordinate system for our highways. We pick one highway, highway A, to be our standard, and we define the x-coordinate of any point P on any highway in this way: You move along highway A until you can get to a point P' such that the straight line between P and P' is perpendicular to highway A. Then you define the x-coordinate of P to be just equal to the value of the closest roadmarker for point P'. Now, in terms of this x-coordinate, we can define a "rate" for any highway:

rate = \dfrac{\delta s}{\delta x}

where \delta s is the change in the roadmarker number as you move down the highway, and \delta x is the change in the x-coordinate.

If you think about it, the rate of a highway will be different, depending on which highway you choose as your standard for setting up your coordinate system. (If the roadmarkers are evenly spaced, then the rate will be given by: rate = \frac{1}{\sqrt{1 + m^2}}, where m is the slope of the highway, relative to the standard highway, where slope = tan(\theta), where \theta is the angle between them). This just means that this rate, while it might be useful for calculations, has no absolute physical meaning, because the physical meaning can't depend on the arbitrary choice of which highway is the standard.

 

[Zayl]

Allow me to ask a question here...
When I created this thread, the system insisted I insert a prefix at the start for the level. I put a B as when you start talking of:then I am lost - I had a 'high school education' I guess you would call it - I am from the UK.

I am thinking of this in a very basic way currently. I understand Newtonian Mechanics - that is what we were taught at school - very simple and straight forward. What I am looking for now, is some help in finding out which of Newton's laws are changed by Relativity.
Newtons Mechanics are represented in Cartesian coordinates with a constant time factor that is the same throughout.
In Relativistic Mechanics, I understand we are considering Time as the fourth dimension, rather than as a common standard, but how is this depicted in diagrams? In the Minkowski diagrams it seems to be the rotation of the moving frame's time access? OK but what of the other three axes? Are they not still orthogonal?

Newton's Laws are simply incorrect. It is nothing but an approximation which also fails to be an approximation at large enough relative motion. Then when the Lorentz factor is applied it becomes correct.

 

[Grimble]

... "same time" as what? Not each other - since they may have different relative velocities with respect to each other.
However - observers who are stationary wrt a clock will observe the clock tick off one second per second (i.e. the reference clock for that frame keeps the same time as all the other stationary clocks in that frame. Some care is needed to make sure of this due to the finite speed of light.

YES, every stationary clock in a frame will keep the same time, one second per second, because they are all at rest in that frame. They will tick at one second per second, measured internally with in that frame without any involvement of anything external to that frame. You said it - one second per second, the SAME as the time in every other frame measured internally without recourse to anything external to that frame. Each frame considered on its own which cannot be moving as we are making no reference to anything outside the individual frames. Each frame on its own. No reference to ANY other frame.

It is the relative speed between the observer and the clock that results in time dilation/clocks being measured to run slow. As you said:

The clocks are not physically affected by relative speed: it is the observation of the clocks that are affected.

[On the chessboard analogy, black see white move one square to the right, so white sees black move one square to the left... not happy with that analogy since one of them sees their own square change colour.]

Oh, for goodness sake! hehehe! The chess board is only there for reference! How about if we make the squares just lines, without colour... Why do you have to divert the topic all the time - not just you Simon, but so many of you 'experts' will divert away from the topic to go into the subtle meanings of how things are said! Sometimes it seems one can't make a statement without being told you are using the wrong words! Give us some room please - you know my level of science, for goodness sake, why expect me to use all the terminology as you would? That is unreasonable. It also means that at times I cannot say what I want for however I try - even inventing a phrase as I have done here - for me to use in explaining what I mean and even that is torn to pieces by scientific grammar n...

I am sorry, I apologise, but it is extremely stressful when anything I say is turned round to mean something different.

Another little point that confuses me in the twin paradox explained as a result of only one twin accelerating is that the total time difference when the traveller returns is dependent on the speed and the duration of the journey; not upon any factor related to the acceleration - neither to the rapidity of that acceleration nor to its duration.

Since the twins can spend an arbitrarily long time when neither is accelerating, and they both notice the other's clock is slow, then how come the accelerated twin always ends up younger?
It's pretty easy to see how in space-time diagrams.

It is because it is also the one that is moving relative to the observer who ends up younger. Remember - time dilation only affects the traveller

 

[Grimble]

Newton's Laws are simply incorrect. It is nothing but an approximation which also fails to be an approximation at large enough relative motion. Then when the Lorentz factor is applied it becomes correct.

Really? I understood that they were seen as a limiting case dealing solely with speeds much less than c; that they are limited in their application...

 

[Ibix]

If you take Zayl's view literally, relativity is also incorrect because it fails at singularities, and is presumably not giving exactly the same predictions as quantum gravity will elsewhere. A better view is that Newtonian mechanics is, as you say, an approximation to relativity. Where it's predictions are indistinguishable from those of relativity to the precision you are able to measure, it's as "right" as relativity. It does go completely off the handle with large velocities or displacements, as you are aware

 

[Grimble]

Oh, for goodness sake! hehehe! The chess board is only there for reference! How about if we make the squares just lines, without colour... Why do you have to divert the topic all the time - not just you Simon, but so many of you 'experts' will divert away from the topic to go into the subtle meanings of how things are said! Sometimes it seems one can't make a statement without being told you are using the wrong words! Give us some room please - you know my level of science, for goodness sake, why expect me to use all the terminology as you would? That is unreasonable. It also means that at times I cannot say what I want for however I try - even inventing a phrase as I have done here - for me to use in explaining what I mean and even that is torn to pieces by scientific grammar n...

I am sorry, I apologise, but it is extremely stressful when anything I say is turned round to mean something different.

May I apologise for the acerbic tone of that post. Ageing bones can be somewhat unrelenting when they stress the passing years...
which is no excuse! I ought to leave replies to my morning time when my patience is not tried by aches and pains.
On re-reading your post this morning, I find it very reasonable and very helpful, if I could I would remove the above critical passage - (Admins?)

 

[Grimble]

As for Time dilation and Relativistic Mechanics, the effect of the invariance of c is easily demonstrated; including exactly what it is, how and why it occurs - when I draw it with Occam's razor in hand! (that was a joke...)

https://ac0077b2-a-62cb3a1a-s-sites.googlegroups.com/site/specialrelativitysimplified/home-1/minkowski-diagrams/Time%20diagrams%20%281%29.png?attachauth=ANoY7cp139iymu-8u-3t2YLYd-HgpGmmjme09LwHg3GlazYAWPN_3-FWURhbfgwzknY5Xs_4LuKGjQK9Xzeau2UEmEOObcbTtNoxyG2GRk1VTo7S6IeRSqnD5zp8gAyWcCuDkLl2rsh-_DVQDcrr7RD36Zp3rn0feheAA340kBquXZaX4kKKT3IXybolAz4XUgZnHFD8Let8yXJtJEWS5bPB4OIycOwOMIzhwf_vmn2rZyI8iWGH_hChGyjstAD7K77GlpFOOB0AcfNZBd_uYNYT7bhrHIkMjlLpUEtVvSs1NMff_z1R-nI%3D&attredirects=0

The Spacetime Interval as shewn in this link:

Armed with this insight, I recommend the following YouTube video:

is (ΔS)² = (Δt)² - (¹/_cΔx)²

When we draw this out in detail in the drawing on the right, we have the
Spacetime Interval measured, t = 0.8seconds
The time measured on the moving clock, t' = 1 second
The time it would take Zach and clock B to travel distance x = ¹/_cvt' = 0.6 seconds

This demonstrates that the Spacetime Interval measured in resting Alice's Frame, Δt, comprises the travel time of the light in the moving frame , Δt' less the time component of the translation of Zach and his Clock B which is ¹/_cΔvt' or making that calculation by means of Pythagoras:
(Δt)² = (Δt')² - (^v/_cΔt')²
or Δt = Δt'√1 - v²/c²

Let me say this is no new theory or new interpretation it is simply reading the spacetime diagram and reading what it is saying as a description of what is happening, using simple Euclidean geometry.

That the Spacetime Interval measured in the moving frame from the resting frame, comprises the total time passed in the moving frame, ct' less the time factor related to the translation of the moving frame. So the resting observer measures less time to have passed for the moving clock, it runs slow.

It seems to me that is due to simple Relativistic Mechanics because of the invariance of c. Once that is added into the equation the rest, time dilation, proper time, coordinate time and everything else falls out of it using no more than Euclidean Geometry.

After all, the 'principle of the invariance of the speed of light' is the postulate that changed everything.

 

[Dale]

Hmm, I don't know about this software. The lines should be hyperbolas.

 

[Grimble]

Yes, it seems to me that that is a fundamental conclusion developed in Minkowski's great concept of Spacetime, which as he states in his introduction is what this concept is built upon:

I would like to show you at first, how we can arrive – from mechanics as currently accepted – at the changed concepts about time and space, by purely mathematical considerations.

A natural line of reasoning coming from his equation c²t² - x² = 1, which plotted against cartesian coordinates produces a hyperbola.

Yet that is holding fast the Newtonian mechanics, where we can happily plot the time against the displacement with no qualms about exceeding the speed of light.
Indeed that leads directly (when we use seconds against light seconds to the familiar 45° limit for the speed of light. Yet when we add two speeds, as we do in the moving Light clock (the speed of the clock and the speed of the light in the clock, the result is >c.

This is the reason(?) that the plot becomes a hyperbola.

All absolutely right and correct.

Yet if we give due attention and gravity to the second postulate in Relativity, it seems to me that we should accept that we are given two facts. That light will always travel one light second per second and that the clock will have a horizontal displacement = vt and that the time axis for the moving clock (in this case) must pass through the point where the x coordinate crosses the 1 second point on the rotated time axis. (0.6,0.8 in the diagram).

Time dilation is shewn where the rotated, moving time scale, crosses the 1.0 coordinate of the observer (0.6c - Lorentz Factor = 1.25) where t' = 1.25 = γt and x' = 0.75, which by length contraction becomes 0.6 for the observer. x = ^x'/_γ

I first thought along these lines when considering the path of a fast moving body and I realized that after it had been traveling for time t, it would have traveled a distance of vt and the furthest a photon could travel being 1 light second per second would be along the x-axis where the time passed would be zero.

Again let me emphasise this is not changing anything, only how it is plotted. The only difference is that because light travel is the best measure of time as it comes mathematically from the second postulate, we measure the distance imaginary light would have traveled from the initial event, the departure of the moving body from a point of contact or crossing of paths and calculate the slowing of time measured in the observed body by the observer, i.e the moving clock slowing. (Literally in this case!).

https://ac0077b2-a-62cb3a1a-s-sites.googlegroups.com/site/specialrelativitysimplified/home-1/minkowski-diagrams/Light%20clocks%20%281d%29small.png?attachauth=ANoY7cqN4iyplnt75M01m0saod0TBN71ldfM1KNUQ2GOj7uMWkrE1r75-RvifPuwOSf9mLYtNJcsXLhgGQNsGH9bukC4JDck5HFF170ijAOoGCfm4lC7uOQQQ8pE6UCTAsA-qolr-pHMMTZ3S_dSwHxvBO52GltUilvCJiFsWxXUOGVdacGap4EGdZTXV34BdUsVIAcEpArH1WpYi47JHfPhYTOd2Cn1eMiELWq_em2AQ9Cwm1hHZUroC47C860Ui5n-WtX0_r2tNsm2YSK4Utk7AlfYX2BSgEYPQ1-2fTOXvIZOm57TIOY%3D&attredirects=0

I am not trying to rewrite any mathematics. All I have done here is to stop adding two vectors to find a third; and instead subtracting one vector from the combined vector to determine the reduction in the third vector, Vector subtraction rather than addition.

As I say I am not trying to rewrite anything or change relativity or anything like that, but to check my understanding, and the way I visualise it.

 

[saim_]

Nobody explains relativity better than Einstein himself; at least for those not very mathematically sophisticated.

Relativity: The Special and General Theory by Einstein
http://www.schloss-sihlberg.ch/dl/6e29795ff0c56f00fb50a75e83a0eb47/relativity.pdf

I read this in eighth grade and it took me almost a month, it'll probably take you less than that. I would advise you to stop wrestling with four-vectors and try to understand the basics from this book first.

 

[Dale]

As I say I am not trying to rewrite anything or change relativity or anything like that, but to check my understanding, and the way I visualise it.

Your description here does not clarify what you are doing. Whatever this is it is not a spacetime diagram. It looks wrong to me and if this is your understanding then it seems wrong to me also.

Do you have a professional reference explaining what this type of diagram is and does. To me it looks flat out wrong, but I am willing to entertain the idea that it is simply unfamiliar if there is a good reference.

 

[the_emi_guy]

Your description here does not clarify what you are doing. Whatever this is it is not a spacetime diagram. It looks wrong to me and if this is your understanding then it seems wrong to me also.

Do you have a professional reference explaining what this type of diagram is and does. To me it looks flat out wrong, but I am willing to entertain the idea that it is simply unfamiliar if there is a good reference.

The OP is not intending for this to be a space-time diagram. Read posts 47 and 48. It is the very common pictorial used to show how the consistency of c leads to time dilation. If you want a reference: Feynman Lectures on Physics Vol 1, 15-4 "Transformation of Time" (or probably any other elementary text on SR).

Grimble,
Keep in mind that both axes in this pictorial are distance (there is no time axis) . We are intending to measure the time between two "events" by measuring the distance, horizontal and vertical, that light must travel between the events. (The first event is the flashlight being switched on, the second event is the light reaching the mirror).

On the other hand, the space-time diagram (video referenced in post 49), has time as the vertical axis, thus the hyperbolas and 45 degree "speed of light" line.

 

[Dale]

The OP is not intending for this to be a space-time diagram. Read posts 47 and 48. It is the very common pictorial used to show how the consistency of c leads to time dilation.

Maybe you are right. The vertical axis should be labeled "y" and not "ct" if that is the case. Let's see what he says.

 

[vanhees71]

The space-time diagram is obviously wrong. The lightcone is defined by $x=\pm ct$ in this (1+1)-dimensional diagrams. The units on the time-like axes of different inertial observers is determined by the hyperbola $(ct)^2-x^2=1 \text{unit}^2$ and not by the circle drawn in the diagram. It's a time-space plane in Minkowski rather than Euclidean space!

 

[the_emi_guy]

The space-time diagram is obviously wrong. The lightcone is defined by $x=\pm ct$ in this (1+1)-dimensional diagrams. The units on the time-like axes of different inertial observers is determined by the hyperbola $(ct)^2-x^2=1 \text{unit}^2$ and not by the circle drawn in the diagram. It's a time-space plane in Minkowski rather than Euclidean space!

What space-time diagram are you referring to? The only space-time diagram I see in this entire thread is in the video referenced in post 49.

 

[vanhees71]

I'm referring to the diagram in #83. According to the axes labels it should be a space-time (Minkowski) diagram, but at least the determination of the units on the time-like axes "clock A" and "clock B" is not correct, because you draw a circle rather than a hyperbola.

 

[Ibix]

What space-time diagram are you referring to? The only space-time diagram I see in this entire thread is in the video referenced in post 49.

The diagram in #83 has axes labelled as if it were a space-time diagram but looks more like a diagram in the x-y plane. Both Dale and vanhees71 are expressing confusion about exactly what is being shown. You seem to be interpreting it as the latter, but it isn't clear that that is what Grimble intends. You are assuming that you know which mistake Grimble has made - which may or may not help him correct whatever his misunderstanding is depending on whether your assumption is correct.

Edit: beaten to it by vanhees, I see.

 

[the_emi_guy]

The diagram in #83 has axes labelled as if it were a space-time diagram but looks more like a diagram in the x-y plane. Both Dale and vanhees71 are expressing confusion about exactly what is being shown. You seem to be interpreting it as the latter, but it isn't clear that that is what Grimble intends. You are assuming that you know which mistake Grimble has made - which may or may not help him correct whatever his misunderstanding is depending on whether your assumption is correct.

Edit: beaten to it by vanhees, I see.

The OP, who indicated a high school level of education, has derived the Lorentz transformation from first principals in a manner typically employed in college physics classes and this seems to have gone completely unnoticed.

Have you gone back and read posts 47 and 48? I suggested this just 5 posts ago to clarify where the OPs diagrams originated.

 

[Grimble]

OK. Let us examine the diagram in #83 that is causing such confusion and I will explain exactly what it is intended to shew.

First things first, this is not a Minkowski Spacetime diagram. It is no more than a diagram to demonstrate how, by making the invariance of 'c' central there is no need to use hyperbolae. That is NOT to say there is anything wrong with Minkowski’s Spacetime; it having stood the test of time very robustly.

In both traditional Newtonian and in spacetime diagrams - as I understand them - the time axis is vertical because time is being measured by the distance light would have traveled along the y axis.

In the clock diagrams time for each observer, in clock A and in clock B, is measured parallel to the y/ct axis vertically for each clock at rest. Which due to the construction of said clocks is also the path of the light in each clock, as observed by an observer at rest with each clock. So time measured by either observer is the distance light traveled in their own clock, it reaches the mirrors, 1 light second away in 1 second. Identical time for identical clocks, in their own frames.

Yet for time in each clock measured from the other clock, we know that in one second that light will have traveled for one light second, in a clock that has has traveled vt light seconds, at 0.6c that will be 0.6 light seconds.

So measured from the stationary clock, the light in the moving clock will have two components to its motion, the movement of the light in the clock and the movement of the clock itself. Hence at 0.6c the light in the moving clock must reach point (0.6,0.8) in the frame of the rest clock after 1 second. Less time will be measured by the resting observer as the speed of the light has two components, the speed of light in the clock and the speed of the clock. So measured from the resting clock the time measured must be less that in the resting clock(?).

Now to me that is plain simple Newtonian Mechanics, working from what is known rather than assumed, that the speed of the light will be traveling at c. As that is a constraint placed upon this scenario by relativity.

Time is measured by virtual light emitted at the initial event, traveling at c in every direction - so an expanding sphere of virtual light, centred on that initial event; any radius of that sphere will be a measure of the interval from that initial event. (in this two dimensional view it will, of course, be a circle)

For a body at rest - such as the observer whose frame we are drawing it will be vertical as there is no displacement.

For a moving body, the line along which time is measured will be rotated because of the lateral displacement. In Newtonian mechanics that displacement is measured after the interval measured on the observer's time axis - the vertical axis of the diagram. In the mechanics of relativity - because of the invariance of c - the displacement means that time in the moving clock has to be measured where the displacement (0.6 light seconds) intersect the 1 light second circle of time because those are the two constraints which we know are true: the displacement of the clock (0,6 light seconds in my diagrams) and the one light second the light must have travelled.

The time interval is measured on the moving body's rotated time axis. In the clock diagram the light in the clock will have traveled 1 light second - to the virtual time sphere while moving clock has traveled vt from the observer in the stationary clock. That is why the time axis for the moving clock is rotated through angle β (sin β = ^v/_c rather than angle α (tan α = ^v/_c because the 1 second coordinate constitutes the hypotenuse rather than the adjacent side of the triangle; it has to be that way because of the invariance of c.

This is no more than simple euclidean geometry.

We are intending to measure the time between two "events" by measuring the distance, horizontal and vertical, that light must travel between the events. (The first event is the flashlight being switched on, the second event is the light reaching the mirror).

On the other hand, the space-time diagram (video referenced in post 49), has time as the vertical axis, thus the hyperbolas and 45 degree "speed of light" line.

This is a diagram showing three views of the mechanics of relativity

1. First, classical Newtonian Mechanics with no account taken of the second postulate. Resulting in a speed greater than 'c'.
2. Second, Minkowski's great Spacetime where taking heed of the hyperbolic function caters for the invariance of 'c'. Yet it is truly a mathematicians solution that leaves those of us with a less mathematical background, difficulties in appreciation. There is nothing wrong with it - it has been accepted for more than 100 years!
3. Thirdly, is but a simple view of the mechanics, centred on the invariance of 'c' and trying with the best will in the world to apply Occam's razor. Keeping to the facts that we are certain of: that light will travel at 'c', that the moving clock will travel the distance vt, and how that rotation affects the measurement by the resting observer.

https://ac0077b2-a-62cb3a1a-s-sites.googlegroups.com/site/specialrelativitysimplified/home-1/minkowski-diagrams/3%20mechanics%20compared.png?attachauth=ANoY7coaMPUDTIO6hy2MrLS6UA0jDfN3g7ZAN_d_dYGxxFrLFD1F93wbzGyClgdYgDCPnXBpWs34Sj6gnWmMjSkd5bfAlV9Lj7jKMPV5opPLquyBEphh_th791WMA0n1AUE4SZxL33Lm-03eT2SI2nsHIXYMlMt5Y7BzfaILnrBs8eGMUiXDo3bI54IMJsB-9kDI1GAscn0uKYnRBjAnALgAxoinCixSOdGZdQHeSYWXx70nhmaL7j9pT5wUaMdey5By8-N87nu99VB0xO7wRrq8cCvEWDNi2h7iLWIGsz7OQ-4-_ci1rOk%3D&attredirects=0
The best thing that can be said for this last attempt to understand what happens in simple terms is that time dilation and the Lorentz factor fall out of it without any effort.

https://ac0077b2-a-62cb3a1a-s-sites.googlegroups.com/site/specialrelativitysimplified/home-1/minkowski-diagrams/Lorentz%20factor.png?attachauth=ANoY7cp8JpFJWuip-BS5Zt2Bxq3gX06yEMV6SAXi_uGKqO3cv3bOgFtwPl9xMQBeFcuc7nXX5Fk4KZ49UpdrCKak0k-LvYZaSQQ0gY29LbmSRByGzGTFrkhNGjCtR5SCH_v-SP_PE1Y-hiuF-TgZA2NfIjGaVHVW3dGAFXbpBoS2Isknzam8bhAoocoEEszL4knxcneXi8ANdiVi9JWvQYR73UokJPY6dZt43u7fzQp_2htoojq4E6sbSo0Iw5WPtMMqmnaqYPWdWyIwF3GPBTdsJtVQalLdvw%3D%3D&attredirects=0

 

[vanhees71]

I don't understand your diagrams. At least the $(ct,x)$-plane cannot be a Euclidean plane. This wouldn't enable us to define a causality structure. The correct construction of the Lorentz transformation can be found in

http://th.physik.uni-frankfurt.de/~hees/pf-faq/srt.pdf

pp. 8 and 9.

The Lorentz transformation implies all kinematic phenomena like time dilation and length contraction as is worked out in this manuscript too.

 

[bahamagreen]

"Nor must we overlook the fact that arguments which proceed from fundamental principles are different from arguments that lead up to them."
Aristotle

No doubt Einstein, Minkowski, et al knew this; and the difficulty when a change in the fundamental principles is itself the object of argument.

 

[Dale]

this is not a Minkowski Spacetime diagram

So then the vertical axis should not be labeled "ct", it should be labeled "y"

by making the invariance of 'c' central there is no need to use hyperbolae

There is still always going to be the need to use hyperbolas. The equation $\Delta s^2=-c^2 \Delta t^2+\Delta x^2+\Delta y^2+\Delta z^2$ is a hyperboloid.

 

[Grimble]

I don't understand your diagrams. At least the $(ct,x)$-plane cannot be a Euclidean plane. This wouldn't enable us to define a causality structure. The correct construction of the Lorentz transformation can be found in

http://th.physik.uni-frankfurt.de/~hees/pf-faq/srt.pdf

pp. 8 and 9.

The Lorentz transformation implies all kinematic phenomena like time dilation and length contraction as is worked out in this manuscript too.

Thank you, Vanhees71, I do understand that you are trying hard to help me, but with my level of High School Education (1960s) I am struggling with the first sentence of p8 and totally lost by sentence 2. Not by the wording but by the mathematical terms used, which I am unfamiliar with, in those terms at least.

On the other hand I am at a loss as to how you can fail to understand my diagrams!

I start with a simple diagram of two light clocks moving apart at 0.6c (to keep the diagram and any calculations simple. Perhaps if I take you through it again; starting with the difference between Newtonian mechanics with and without the invariance of 'c'.
https://ac0077b2-a-62cb3a1a-s-sites.googlegroups.com/site/specialrelativitysimplified/home-1/minkowski-diagrams/Mechanics%20compared.png?attachauth=ANoY7cp9GaHYoo-aiMlFyVhXmgpAWLw7Y4ZXy-MbINOoEK7NI6A2GzRvClzDgb_ZLF0H9cw9RzRFyiqXrOLPuoLcvDGafLnGPtswJdjScrSzAjyeiZ2YxTnI_vNHz5uBfVDlONZZCaokzsr7MLFKjBSKJEcgYp2A-Z-dtLnkAZLt1Rw3XBHscpycCyeSJP-taYPoccpRQYImLwDqZ8QvofhyrXNW6doxkt4STBS-nk5nIkTz1Fde-dNpL8g4TNChWLGA9gR5peECVEhVa9AWxctK_KrdCDdpQg%3D%3D&attredirects=0
In both diagrams the vertical axis can be read as both/either a ct (time) axis or as a y (spatial) axis. It is the path of a flash of light normal to the x axis. Being the passage of light it is possible to use it as a time axis for the observer in the clock at 0,0.

In the first diagram speed is unlimited and the moving light travels 1.166 light seconds in one second measured on Clock A's time axis.

In the second diagram, observing the 2nd postulate, the moving light of clock A (in the frame of Clock B, which is at rest at rest) travels a rotated path, but still only 1 light second (The curving red line) from 0,0 while clock A also travels 0.6 light years in that same 1 second. Hence the light will arrive at (-0.6,0.8) after traveling for one second, measured along Clock A's rotated time axis, as measured by Observer B.

I drew the second diagram that way round to emphasise the reciprocality of relativity. Either clock can take either role.

The diagrams in Post #93 reduce those diagrams to the fundamental items, time measured on the observers time line, time measured on the moving clock's time line and the distance traveled by the moving clock.

The new part I introduced was the second part which I included to shew how the invariant spacetime interval function can be seen as a hyperbola.

Is there any point in particular that is causing you difficulty?

 

[vanhees71]

Well, perhaps I'm just too used to the usual Minkowski diagrams. I cannot make sense of your "Euclidean" space-time plane. It's contradicting any intuition we have about the relativistic space-time structure.

 

[DrGreg]

In both diagrams the vertical axis can be read as both/either a ct (time) axis or as a y (spatial) axis. It is the path of a flash of light normal to the x axis. Being the passage of light it is possible to use it as a time axis for the observer in the clock at 0,0.

I see two problems.

1. The vertical axis is the y axis only. You can't treat it as the ct axis as well, because, although $y = ct$ is true for one path in each diagram, it's not true for other paths in the same diagram.
2. In each diagram the red parts and the green parts refer to different observers, i.e. different coodinate systems. It's misleading to superimpose both in the same diagram, so really you ought to split each diagram into separate green and red diagrams. And use different names for the red and green coordinates. Traditionally the two coordinate systems are written as $(ct, x, y)$ and $(ct', x', y')$.

 

[Dale]

In both diagrams the vertical axis can be read as both/either a ct (time) axis or as a y (spatial) axis.

This is wrong. Time is orthogonal to space so it cannot be represented by the same axis.

It is the path of a flash of light normal to the x axis. Being the passage of light it is possible to use it as a time axis for the observer in the clock at 0,0.

This would be a different axis, neither time nor space. It would be called a null axis. Something like this is used in radar coordinates, but usually with two null axes.