I What, exactly, are invariants?

[vanhees71]

Why not just call it "proper time" as usual in the literature?

 

[PeterDonis]

Why not just call it "proper time" as usual in the literature?

It is also usual in the literature to describe proper time as "arc length along timelike curves".

It is not usual in the literature to use the term "arc length" to refer to the Euclidean arc length, which, as I noted, has no physical meaning in relativity physics.

 

[vanhees71]

Ok, let it at this. There are of course textbooks explaining things more consistent than others, and it's of course just a matter of taste which nomenclature you prefer.

 

[DrGreg]

Although my contributions to this thread so far have been to defend the use of geometrical language, I think there is something for us pro-geometers to take heed of from @vanhees71's criticisms, and that is that I think we (including myself) may sometimes be guilty of launching into geometrical language without adequately explaining to our readers why we do it, and what the differences are between the geometries of Euclid and spacetime.

So when we use terms such as "geometry", "length", "curvature", "geodesic", "orthogonal", "angle", etc, we are making an analogy: the equations that describe a concept in 4D spacetime look very similar, but not necessarily identical, to the equations that describe a geometrical concept in 3D Euclidean space, so we choose to extend the meaning of a word from 3D Euclidean geometry into 4D spacetime, even though they are not quite the same.

(Of course, some or all of these concepts also extend into more general mathematical abstract theories such as differential geometry or infinite-dimensional Hilbert spaces, but most of our readers in this relativity forum probably don't care about that.)

 

[Dale]

So when we use terms such as "geometry", "length", "curvature", "geodesic", "orthogonal", "angle", etc, we are making an analogy: the equations that describe a concept in 4D spacetime look very similar, but not necessarily identical, to the equations that describe a geometrical concept in 3D Euclidean space, so we choose to extend the meaning of a word from 3D Euclidean geometry into 4D spacetime, even though they are not quite the same.

I wouldn't call this an analogy, but rather a generalization. This is not just analogous to geometry, it is actually geometry but a generalization of geometry that contains our usual Euclidean geometry as one special case.

Any time you generalize something there are going to be differences from the specialized concept that we are generalizing, and it is important to teach those differences. But there are also going to be many similarities that carry over from the special case to the general case. Much of the experience that is built up on the special concept will also apply to the general concept. That is indeed the power of generalizations.

Perhaps we don't do enough teaching students about those differences, but if you can connect to the similarities then students get a tremendous advantage. How many twin paradox threads come from people that understand the geometry? 0. The relativity of simultaneity is the most difficult concept for students, and once a student gets the geometric understanding that conceptual hurdle disappears too.

 

[Sagittarius A-Star]

Interestingly, the adjectives "circular" and "hyperbolic" were in the past not only assigned to angles, but also to trigonometric/hyperbolic functions.

Hyperbolic functions were introduced in the 1760s independently by Vincenzo Riccati and Johann Heinrich Lambert.[13] Riccati used Sc. and Cc. (sinus/cosinus circulare) to refer to circular functions and Sh. and Ch. (sinus/cosinus hyperbolico) to refer to hyperbolic functions. Lambert adopted the names, but altered the abbreviations to those used today.[14] The abbreviations sh, ch, th, cth are also currently used, depending on personal preference.
...
The hyperbolic functions represent an expansion of trigonometry beyond the circular functions. Both types depend on an argument, either circular angle or hyperbolic angle.

Source:
https://en.wikipedia.org/wiki/Hyperbolic_functions

 

[robphy]

Maybe, I'm too pedantic, but I don't like to call this integral an "arc length".

Why not just call it "proper time" as usual in the literature?

It is also usual in the literature to describe proper time as "arc length along timelike curves".

We call it "proper time" because that gives its physical interpretation.

We also refer to it as (as @PeterDonis says) "arc length along timelike curves" because that gives a mathematical/geometrical interpretation...
so that we may TRY to import aspects of (say) the classical differential geometry of curves into relativity.

For example, https://en.wikipedia.org/wiki/Frenet–Serret_formulas
were introduced in 3-dimensions and
extended to n-dimensions https://en.wikipedia.org/wiki/Frenet–Serret_formulas#Formulas_in_n_dimensions.

One could play a similar game and try to extend it from the Euclidean geometry
into relativity (e.g. https://arxiv.org/abs/gr-qc/0601002 "On the differential geometry of curves in Minkowski space" by J. B. Formiga, C. Romero (AJP 74,1012 (2006)).

If we did not recognize it as an arc-length,
then maybe it would have taken a long time to make the connection.
Could we have now GPS without this connection?

• Suppose Einstein was so against Minkowski and the mathematicians
  that he rejected the geometrical interpretation of spacetime and invariant ways of thinking.
  Could he still develop general relativity?
• Without the invariant viewpoint extended into relativity,
  would we still be stuck with the coordinate singularity in Schwarzschild?

 

[PeterDonis]

Suppose Einstein was so against Minkowski and the mathematicians
that he rejected the geometrical interpretation of spacetime and invariant ways of thinking.
Could he still develop general relativity?

Einstein's answer to this question, IIRC, was no: he said that adopting the geometric viewpoint of Minkowski (which he had initially ignored) was essential to developing GR.

 

[martinbn]

What is interesting to me is that in school, geometry is introduced more or less they way it was devolpoed. The coordinates and their use comes after the pupils have learned quite a bit of geometry. So there isn't really any confusion of what coordinates are for. On the other hand realtivity seems to be taught backwards. Starting with coordinates and transformations, even to define notions that can be explain without coordinates. Then the long strugle.

 

[vanhees71]

I'm not against to draw analogies between Euclidean and Minkowski geometry but I prefer a somewhat more careful language. What was important for Einstein in his formulation of GR was not so much this overemphasizing of the analogies between Euclidean and pseudo-Euclidean geometry but the tensor calculus on pseudo-Riemannian manifolds.

Of course, I also teach my teachers students (1+1)D Minkowski diagrams, but I emphasize the important differences to the Euclidean 2D plane rather than to overemphasize the similarities, and that helps them to read the Minkowski diagrams properly. Particularly the construction of the unit lengths on the axes of different inertial frames with Lorentzian basis vectors (the analogues of the Cartesian basis vectors in Euclidean vector spaces) using the hyperbolae is utmost important to read off length contraction and time dilation properly. Of course, the diagrams helps to understand the "relativity of simultaneity" and the "twin paradox".

The latter is of course even easier to understand by just comparing the two proper times of the time-like worldlines of the twins. As I said before, I'd never call it an "arc length", because this is too easily mixed up with the Euclidean notion of "length".

 

[DrGreg]

I'd never call it an "arc length", because this is too easily mixed up with the Euclidean notion of "length".

I suppose you could call it an "arc interval", but somehow that doesn't sound right!

 

[vanhees71]

As I said, I simply call it "proper time". An "arc interval" doesn't make sense to me, and I've never heard this expression anywhere.

 

[robphy]

As I said, I simply call it "proper time". An "arc interval" doesn't make sense to me, and I've never heard this expression anywhere.

I also call it "proper time" to give it its physical interpretation.
Then I say that it's represented by the spacetime version of the "arc length" on a spacetime diagram,
giving it a geometric translation together with a set of tools for making computations.

When I say "proper length",
I will soon say that it's represented by the spacetime version of the "distance between parallel lines" on a spacetime diagram.

I may get lazy and eventually skip the word "represented".

To me, these are akin to saying "the work done by the force"
is [represented by] the "area under the force-vs-displacement graph".I think what Minkowski's reformulation and reinterpretation does
is to reveal to folks a somewhat familiar structure [that one has to get used to]
to give them a handle on relativity.
I think it made relativity more accessible to others.

Without the geometric interpretation, we might be doing a lot more talking
using a vague and verbose vocabulary.

 

[Dale]

I emphasize the important differences to the Euclidean 2D plane rather than to overemphasize the similarities, and that helps them to read the Minkowski diagrams properly

I think that is a good approach. Your students should still get the benefits and should “easily” avoid the risks. Maybe they get the benefits slightly slower than a less cautious approach, but what is an extra week or two on concepts like these?

 

[robphy]

(bolding mine)

I'm not against to draw analogies between Euclidean and Minkowski geometry but I prefer a somewhat more careful language. What was important for Einstein in his formulation of GR was not so much this overemphasizing of the analogies between Euclidean and pseudo-Euclidean geometry but the tensor calculus on pseudo-Riemannian manifolds.

Of course, I also teach my teachers students (1+1)D Minkowski diagrams, but I emphasize the important differences to the Euclidean 2D plane rather than to overemphasize the similarities, and that helps them to read the Minkowski diagrams properly...

I'm not sure what counts as "overemphasizing".

(Regarding tensors: I think Einstein pursued tensor-calculus because the classical pseudo-Euclidean geometries (hyperbolic and elliptic space) are too simple [being constant-curvature spaces] to describe the observable universe. He needs more general spaces that are better handled in the style of tensor calculus, rather than classical nonEuclidean geometry. For introducing special relativity, we should delay tensor methods for the student... especially if they haven't done Euclidean geometry with tensors.) ​

To me, the geometrical interpretation is an attempt
to equip the student with tools for computation and reasoning.

(Why it works is not clear... but it seems to work. ​

The geometrical structures seem to encode a lot of the physics... ​

and many of the geometric constructions and calculations lead to consistent results ​

that agree with experiment. And yes we can of course tensorialize these constructions.)​

As with any tool,
we show the student "how to use it" and (to avoid overemphasis) "how not to use it".

It might be helpful to give a "storyline from some set of first principles"(*)
showing HOW these structures arise
and not just sort-of jump in the middle pointing out
analogies and non-analogies here and there.

(*) for example, what does "orthogonal" fundamentally mean?​

Does it really mean "90-degrees"?​

How do we construct the observer's x-axis (her "spaceline") given her t-axis (her "timeline"),​

starting from a "circle" (which is what defines orthogonality)?​

Is relativity really "so unlike anything we have ever seen"​

or can it be responsibly shown to be "similar" and maybe "hauntingly familiar" to something we've already seen?​

​

 

[Dale]

(*) for example, what does "orthogonal" fundamentally mean? Does it really mean "90-degrees"?

This is a good example of a concept that gets generalized. Well before I started doing relativity I had to generalize the notion of "orthogonal" when I learned the Fourier transform and found out that cos functions formed an orthogonal basis for functions. There doesn't even exist any word to distinguish this notion of orthogonality from the less general notion of orthogonality as a 90 degree angle. But nevertheless the presentation of this new concept of orthogonality was sufficiently clear that I didn't get confused.

 

[DrGreg]

What is interesting to me is that in school, geometry is introduced more or less they way it was devolpoed. The coordinates and their use comes after the pupils have learned quite a bit of geometry. So there isn't really any confusion of what coordinates are for. On the other hand realtivity seems to be taught backwards. Starting with coordinates and transformations, even to define notions that can be explain without coordinates. Then the long strugle.

There are some ways of introducing relativistic concepts without the use of coordinates, but these don't seem to be widely used. Bondi k-calculus is one method where the introduction of coordinates can be delayed (you can work with proper time only, to start with), and (as far as I can remember) Geroch's General Relativity from A to B also emphasises geometry and de-emphasises coordinates.

You do need to decompose spacetime into space and time in order to compare relativity with Newtonian physics, but the decomposition can be described with geometrical language instead of coordinates. (The key point, of course, is that there are multiple ways to decompose.)

 

[vanhees71]

(bolding mine)

I'm not sure what counts as "overemphasizing".

(Regarding tensors: I think Einstein pursued tensor-calculus because the classical pseudo-Euclidean geometries (hyperbolic and elliptic space) are too simple [being constant-curvature spaces] to describe the observable universe. He needs more general spaces that are better handled in the style of tensor calculus, rather than classical nonEuclidean geometry. For introducing special relativity, we should delay tensor methods for the student... especially if they haven't done Euclidean geometry with tensors.) ​

I mean Minkowski's analysis first enabled the formulation of the theory using tensors in the pseudo-Euclidean affine manifold, which indeed describes the special-relativistic spacetime model most adequately, and the great benefit is the "manifestly covariant formalism", which enables us to find physical laws that are compatible with this spacetime model. For me the main feature is also "geometry", but not in the sense that it is in many respects analogous to Euclidean affine space but that it provides the Poincare group as its symmetry group (i.e., $\text{ISO}(1,3)^{\uparrow}$ as the Lie group of the symmetry-transformation group that is connected smoothly with the identity). That is of course also analogous to the Euclidean case (the closest analogy would be $\text{SO}(4)$).

It is nevertheless very important stress the very important difference of the Minkowski product with signature (1,3) or (3,1), because that difference enables to establish a causality structure making this specific affine spaces apt to define a spacetime model. That's why I consider it so important to emphasize the differences between Minkowski space and Euclidean space, particularly when you draw a Minkowski diagram on a sheet of paper, which we are trained from elementary school on to interpret in the sense of a Euclidean affine plane and then easily misread the Minkowski diagram, which has to be intepreted strictly as a Lorentzian affine plane.

The step from SR to GR is then not that difficult anymore: It just makes the notion of Poincare symmetry local and in this way realized the strong equivalence principle to describe the gravitational interaction. That's why Einstein got a Lorentzian (i.e., pseudo-Riemannian) spacetime model, and the dynamics of the gravitational field implied the dynamical nature of spacetime.

To me, the geometrical interpretation is an attempt
to equip the student with tools for computation and reasoning.

(Why it works is not clear... but it seems to work. ​

The geometrical structures seem to encode a lot of the physics... ​

and many of the geometric constructions and calculations lead to consistent results ​

that agree with experiment. And yes we can of course tensorialize these constructions.)​

As with any tool,
we show the student "how to use it" and (to avoid overemphasis) "how not to use it".

I never avoid four-vectors and -tensors. To the contrary, I introduce them as soon as possible, even before I draw the first Minkowski diagram. Many introductory textbooks avoid four-vectors in the beginning and make SR more complicated than necessary.

Why this works is very clear then: It works for the same reason, why in Newtonian physics you work with Euclidean affine 3D space and an additional "time line" (a fibre bundle) and that's why for Newtonian physics the corresponding Euclidean tensors and tensor fields (in QM in addition of course also spinors) are the natural way to formulate the physical laws: It's because they lead to laws that are compatible with the Galilei-Newton space-time model and realize the Galilei group as its symmetry group.

In the same sense it is natural to work with four-vectors, -tensors, and -fields to describe the physical laws within special and general relativity and in this way realize the (global or local) Poincare symmetry of relativistic physics.

It's amazing, how much the choice of a spacetime model determines how the physical laws look but are at the same time flexible enough to describe a huge realm of phenomena.

It might be helpful to give a "storyline from some set of first principles"(*)
showing HOW these structures arise
and not just sort-of jump in the middle pointing out
analogies and non-analogies here and there.

(*) for example, what does "orthogonal" fundamentally mean?​

Does it really mean "90-degrees"?​

How do we construct the observer's x-axis (her "spaceline") given her t-axis (her "timeline"),​

starting from a "circle" (which is what defines orthogonality)?​

Is relativity really "so unlike anything we have ever seen"​

or can it be responsibly shown to be "similar" and maybe "hauntingly familiar" to something we've already seen?​

​

Exactly. That's why for SR I start with Einstein's two postulates and introduce the four-vector and -tensor formalism with the Lorentzian fundamental form first. Then, specializing one-selves to two-dynamical point-particle motion, I introduce the Minkowski diagrams in a time-space plane, and then you don't come to the idea to misinterpret a Minkowski diagram by drawing any Euclidean circles in it but the hyperbolae (space-like and light-like as well as their "degenerate" case of the lightlike light-cone) to construct the correct "unit tics" on the axes of different inertial frames. You also don't introduce "angles" but "rapidities" in this plane (of course there's a great analogy between the Euclidean "angles" and the Minkowskian "rapidities", but they are clearly not the same).

 

[Freixas]

It's certainly been fascinating listening in on the opinions of how best to introduce people to relativity. It's pretty clear that I'm missing a lot by not knowing more about 4-vectors, tensors, and a number of other things.

For what it's worth, I once trained to be a cross-country ski instructor. The lesson that stuck with me most was that some students learned best if you demonstrated a technique, some of you described it, and some if you guided their movements. We're all different and we learn in different ways.

Personally, I'm visual and I prefer approaches that I can picture as I going for a daily walk. Until I can understand something visually, concepts don't really sink in. Take the spacetime interval. It should be an easy one to grasp using the analogy to a spatial interval but I've yet to figure out what it's good for. I could wander through a lot of explanations; someday, someone might use just the right words that will create my aha! moment.

Speaking of understanding, language is imprecise. The word "orthogonal" apparently has a meaning other than 90°; used without qualification, it can confuse rather than enlighten. There are some even more basic words that are unclear. For some explanations I get in this forum, I think I understand what was said, only to figure out later that I didn't understand at all.

For another example of language confusion, let's take "proper length". The word length can be a synonym for "distance" or it can be considered a property of an object. Wikipedia uses the latter sense in a common definition: "Proper length or rest length is the length of an object in the object's rest frame."

If an object is under acceleration does it still have a rest frame (and thus a proper length)? If the endpoints are accelerating uniformly, then its "proper length" is not invariant. If we insist on maintaining an invariant "proper length", then clocks at the endpoints of the length must be moving at different velocities, and so cannot be in the same rest frame. It looks like the object must stop accelerating for the definition to apply (but oddly, if you were on, say, an accelerating spaceship, it seems you could use Einstein's technique of laying down rods to measure its "proper length").

"Proper distance" seems clearer. We're no longer talking about an object, but about the separation of two events, and we can apply a more precise definition.

"Proper time" is confusing since "time" can be used in the sense of "what time is it?" (a single value) and also a duration (an interval formed from two time values). When Wikipedia says "Proper distance is analogous to proper time," I believe they are using time in the latter sense.

Would it be clearer to talk about "proper duration" and "proper distance"? But "time" and "length" are what has been historically used, and those who understand what these very simple words mean mean might not realize they could be misinterpreted.

I'm not actually asking for answers to any questions posed here (although I suspect I will hear some ). I'm just musing on the problems with learning (and teaching) relativity. Their are a lot of barriers and no one approach will work for everyone. It sounds like you've all taken unique paths.

 

[PeterDonis]

If an object is under acceleration does it still have a rest frame (and thus a proper length)?

If it is under the right kind of acceleration (the technical name for which is "Born rigid acceleration"), yes.

If the endpoints are accelerating uniformly, then its "proper length" is not invariant.

First, do you mean "invariant" or "constant"? An object's proper length can be invariant--the same in all frames--but still change as we move along the object's worldline. I'm going to assume you actually meant "constant", which means, not only that we can identify an appropriate invariant (which will turn out to be arc length along a particular spacelike curve), but that this invariant remains the same as we move along the object's worldline (so there is actually a whole series of spacelike curves, each describing the proper length of the object at successive instants of the object's proper time).

By "accelerating uniformly", do you mean both endpoints have the same proper acceleration (i.e., the same reading on an accelerometer)? If so, then your statement is true, but I don't think it's true for the reason you think it is. (The reason it is true, in relativity, is called the "Bell spaceship paradox", which has prior PF posts and an Insights article that you can look up.)

However, "accelerating uniformly" is often used to refer to a different acceleration profile, the one I referred to above (i.e., "Born rigid"), in which the front of the object has a smaller proper acceleration than the back, and the object's proper length does stay the same.

If we insist on maintaining an invariant "proper length", then clocks at the endpoints of the length must be moving at different velocities

Different velocities in what frame?

and so cannot be in the same rest frame

Yes, they can, in two senses:

(1) The momentarily comoving inertial frame of the object at some instant of its proper time;

(2) The non-inertial "rest frame" we can construct if the object is undergoing Born rigid acceleration. (This frame is called "Rindler coordinates" for the case of linear acceleration in flat spacetime.)

 

[robphy]

I'm not actually asking for answers to any questions posed here (although I suspect I will hear some ). I'm just musing on the problems with learning (and teaching) relativity. Their are a lot of barriers and no one approach will work for everyone. It sounds like you've all taken unique paths.

Not to disappoint...
yes, we all have taken different worldlines.

For what it's worth, I once trained to be a cross-country ski instructor. The lesson that stuck with me most was that some students learned best if you demonstrated a technique, some of you described it, and some if you guided their movements. We're all different and we learn in different ways.

Yes, that's why it's good to be multi-modal.
Words, algebra, geometry, coordinates, tensors, analogies, limiting cases...

Personally, I'm visual and I prefer approaches that I can picture as I going for a daily walk. Until I can understand something visually, concepts don't really sink in. Take the spacetime interval. It should be an easy one to grasp using the analogy to a spatial interval but I've yet to figure out what it's good for. I could wander through a lot of explanations; someday, someone might use just the right words that will create my aha! moment.

The spacetime interval [and the squared-interval] between two events is the fundamental "invariant" in special relativity, just like the distance [or squared-distance] between two points is fundamental in Euclidean geometry.
While everyone decomposes a displacement vector in the plane into coordinate-dependent components,
they agree on the distance. By analogy, a similar thing is true for the spacetime interval.

• For timelike-related events in Minkowski spacetime, it's the "wristwatch time" (Minkowski's "proper time") along the inertial observer that meets both events.
• For nearby spacelike-related events, it could be the proper-length (the distance between two parallel inertial worldlines in the frame of those worldlines).
• For lightlike-related events, it's the indication that the events are lightlke-related.
• (In an energy-momentum diagram, the analogous quantity for a timelike or lightlike 4-momentum vector is the invariant-mass of the object.)

Speaking of understanding, language is imprecise. The word "orthogonal" apparently has a meaning other than 90°; used without qualification, it can confuse rather than enlighten. There are some even more basic words that are unclear. For some explanations I get in this forum, I think I understand what was said, only to figure out later that I didn't understand at all.

In any technical discussion, one has to learn the vocabulary and the definitions to fully participate in the discussion. For me, rather than just vague words, having a mathematical definition helps, particularly ones I can draw.

( "unionized" might be interpreted one way by many people, but a very different way by a chemist.)
We tell students that terms in physics have specific meanings, that are different from casual conversation.
(How many times have you heard a sportscaster use "force", "energy", "momentum", "power", etc... interchangeably? )

"Orthogonal" as "90-degrees" is an elementary characterization of perpendicular.
But mathematics is about finding structure among special cases...
it was decided that "orthogonal" could be more generally a statement that the dot-product or inner-product is zero, as @Dale suggested in #106 when talking about orthogonal polynomials. Geometrically, I also like the characterization of "being tangent to a radius vector" as I suggested in #77. "90-degrees" turns out to be a special case, which doesn't work in the general case.

"Proper time" is confusing since "time" can be used in the sense of "what time is it?" (a single value) and also a duration (an interval formed from two time values).

"proper time" is defined by Minkowski as "eigenzeit" to be one's own time.
Bondi uses "private time". Taylor&Wheeler use "wristwatch time" [my favorite].

To me, language is imperfect... so, I often prefer to write "proper-time" as if it were a new word.. a new noun, not to be interpreted as "an adjective with a noun".. but inseparable. (I might write wristwatch-time.)

The technical language (with hopefully algebraic and geometric definitions) is needed for clarity.
It's not meant to exclude people.
If something is unclear, one has to ask for the definition.
(One might question it or be puzzled by it... but it should be accepted as the working definition.)

 

[Dale]

That's why for SR I start with Einstein's two postulates and introduce the four-vector and -tensor formalism with the Lorentzian fundamental form first.

That works well because it is easy to connect the 2nd postulate to the fundamental form.

That's why I consider it so important to emphasize the differences between Minkowski space and Euclidean space, particularly when you draw a Minkowski diagram on a sheet of paper,

I just think you can do all of that without needing to say that angles in spacetime make no sense.

 

[Freixas]

By "accelerating uniformly", do you mean both endpoints have the same proper acceleration (i.e., the same reading on an accelerometer)?

That was one option. Under that option, the object's proper length increases (i.e. the string breaks).

Different velocities in what frame?

Hmm... I tried to model a case where the proper length of an accelerating object remained the same (using Gamma). Using the viewpoint of the initial rest frame, I assigned a constant acceleration to the rear endpoint. I then calculated the position of the front endpoint so that, adjusting for the length contraction given the speed of the rear endpoint, the length of the object was constant. The front endpoint does not move as far as the rear one, so it's velocity must be less, which implies that the front clock moves faster than the rear.

Given this, there might be an inertial frame relative to which the rear and front clocks are synchronized. Relative to that frame, the length of the object could not be equal to it's length at the start (the one instant in which it was at rest).

But maybe this was not the right way to model this. If the object at rest measured L, then maybe I should have used the comoving frame of the rear endpoint and then found the point L distance away and at the same time relative to that frame. I tried that and came up with this diagram:

Here, the starting length is 1. So I calculate the comoving frame at various points in time for the rear of the object (so that (0, 0) of the comoving frame is at that point), transform the point (1, 0) back to the rest frame, and draw a line connecting the two points.

From the point of view of the rear endpoint, the length is constant. Are the clocks synchronized?

If I connect the endpoints of the right sides of the lines, I get a rough idea of the motion of the front of the object. Just eyeballing this, the front appears to move slower than the rear (from the viewpoint of the rest frame), so the front clock moves faster and for longer, and so the clocks would not seem to be in synch along the lines of simultaneity.

Yes, they can, in two senses:

Well, I gave it two tries. A diagram might help in understand the two ways you mention. I can look up "Born Rigid".

Note that I am leaving on a trip tomorrow and so may disappear from this conversation for random periods of time over the next few weeks.

 

[Freixas]

For timelike-related events in Minkowski spacetime, it's the "wristwatch time" (Minkowski's "proper time") along the inertial observer that meets both events.

This bullet point "clicks". The other points I'll have to think about.

In any technical discussion, one has to learn the vocabulary

Your right, of course. Since I'm not learning any of this formally, I sometimes think I understand what was said. And one can note a formal definition without it always sinking in (until much later).

Thanks for your help. Note that I am leaving on a trip tomorrow and so may disappear from this conversation for random periods of time over the next few weeks.

 

[Freixas]

"proper time" is defined by Minkowski as "eigenzeit" to be one's own time.
Bondi uses "private time". Taylor&Wheeler use "wristwatch time" [my favorite].

I meant to comment on this. "Wristwatch time" for me means that I look at my watch and it displays a value. But if Wikipedia is to trusted ("Proper distance is analogous to proper time"), "proper time" is a duration, an interval between two wristwatch numbers (in the same way that distance is measured between two spatial coordinates).

So which is it (or is it both)?

 

[PeterDonis]

I tried to model a case where the proper length of an accelerating object remained the same

That's simple. Assume the object is accelerating in the positive $x$ direction. The front and rear of the object, in any inertial frame, will have worldlines that are concentric hyperbolas, of the form $x^2 - t^2 = x_r^2$ (for the rear) and $x^2 - t^2 = x_f^2$ (for the front), where $x_r < x_f$. The proper accelerations of the two ends will be $c^2 / x_r$ (for the rear) and $c^2 / x_f$ (for the front). And the "surfaces of constant time" for the object--the surfaces in which the line segments of constant proper length lie--are simply straight lines through the origin of the inertial frame, with gradually increasing slopes, i.e., lines of the form $t = k x$, where $0 \le k \lt 1$ (the $k = 0$ line is just the line $t = 0$, which we can take as the instant at which the acceleration begins).

I strongly suggest taking the time to draw a spacetime diagram of the above. Then, for extra points, look up Rindler coordinates and show how the hyperbolas are "grid lines" of the Rindler time coordinate, while the "surfaces of constant time" I described above are "grid lines" of the Rindler space coordinate.

 

[vanhees71]

That works well because it is easy to connect the 2nd postulate to the fundamental form.

I just think you can do all of that without needing to say that angles in spacetime make no sense.

I simply don't introduce angles in an (1+1)D Minkowski diagram. Only rapidities make sense and they have a geometrical meaning as the areas in connection with hyperbolas as discussed above although this area visualization doesn't help much, at least for me. The areas make more sense when working with light-cone coordinates aka @robphy 's "rotated graph paper".

 

[Dale]

Only rapidities make sense

And rapidities are generalized angles.

 

[vanhees71]

Sigh, yes, I don't deny that, but why must one draw this analogy which confuses beginners of the subject?

 

[Dale]

why must one draw this analogy which confuses beginners of the subject?

I think you are right that it is not necessary. I don’t object to that.

Not everything that makes sense is necessary or even advisable.