What, exactly, are invariants?

[robphy]

when you guys said "angle", you were talking about relative velocity.

Technically, "relative velocity" is analogous to "relative slope".
Since in Euclidean trigonometry we have $m=\tan\phi$, in special relativity we have $(v/c)=\tanh\theta$.

• Relative slope would be like $m_{rel}=\tan(\phi_B-\phi_A) =\frac{m_B-m_A}{1+m_B m_A}$ (which is invariant of choice of grid orientation)... note that slopes don't "add" or "subtract".
• Relative velocity would be like $v_{rel}/c=\tanh(\theta_B-\theta_A)=\frac{1}{c}\frac{v_B-v_A}{1-v_Bv_A}$ (which is invariant of choice of inertial frame).
• In Galilean spacetime trigonometry, using Yaglom's Galilean trig functions ($\mbox{cosg}\eta=1$; $\mbox{sing}\eta=\eta$, and $\mbox{tang}\eta=\frac{\mbox{sing}\eta}{\mbox{cosg}\eta}=\eta$, relative velocity is $v_{rel}/c=\mbox{tang}(\theta_B-\theta_A)=\frac{1}{c}\frac{v_B-v_A}{1-0v_Bv_A}=\frac{1}{c}(v_B-v_A)$ (which is invariant of choice of inertial frame), where $c$ is any convenient velocity scale used to make the left-hand side dimensionless [it could be $c=1 \rm m/s$].
  Note that additivity [or subtractivity] of velocities in Galilean physics is an exceptional case... not the norm, when viewed in this generalized viewpoint (in comparison to Euclidean geometry and Special Relativity).

 

[Nugatory]

but relative velocity looks to me like a type III invariant: the velocity of any object with respect to a given frame of reference is a relative velocity.

That given frame of reference is unnecessary baggage here.

The velocity with respect to a given frame (which is best considered a coordinate velocity) has no physical significance unless there is something at rest in that frame so that it is the relative velocity between two objects. But if we have two objects we don’t need the frame at all; the relative velocity is measured by bouncing a light signal from one object off of the other and measuring the Doppler shift of the reflected signal, a locally measured and coordinate-independent result.

So either the result calculated with the aid of the frame has no physical significance or it can be calculated without the aid of the frame. Either way the frame is superfluous when we’re thinking in terms of invariants.

 

[Ibix]

Here are two Minkowski spacetime diagrams. Both show the same worldlines relative to different frames, but the lengths (on the diagrams anyway--I realize the proper lengths are invariant) and the angles are not obviously invariant. I'm not sure how looking at a diagram clarifies what is and isn't an invariant.

I seem to have missed a few posts, apologies. On this topic, though, the relationship between two Minkowski diagrams is analogous to the relationship between two maps of the same area with north pointing in different directions. The maps are related by a rotation, the Minkowski diagrams by a hyperbolic "rotation" (aka a Lorentz boost). Perhaps of interest is that, written in matrix notation, a rotation is$$
\left(\begin{array}{cc}
\cos(\theta)&\sin(\theta)\\
-\sin(\theta)&\cos(\theta)
\end{array}\right)$$and a boost is
$$
\left(\begin{array}{cc}
\cosh(\psi)&-\sinh(\psi)\\
-\sinh(\psi)&\cosh(\psi)
\end{array}\right)$$where $\psi$ is the rapidity associated with the $v$ in the more familiar form of the Lorentz transform.

You can smoothly rotate a map - pin it to the table at one point and rotate, and each point will sweep out a circle. Similarly you can smoothly boost a Minkowski metric, in which case each point sweeps out a hyperbola. I wrote a little Javascript program to demonstrate this, here. I'm afraid it predates the wide uptake of touch screens and works better with a device with a mouse, but the "canned" diagrams available further down the page draw several scenarios for you. Then you can either select a timelike line or enter a velocity and the diagram will smoothly boost to that frame. I think this is helpful to "see" what the relationship between two Minkowski diagrams is (the "Hyperbolae and spokes" diagram is worth a look).

 

[DrGreg]

Another related analogy between Euclidean and Minkowskian geometry is the decomposition of a timelike 4-vector into components relative to a Minkowski coordinate system, e.g. for particle 4-momentum $\textbf{P}$$$
\textbf{P} = (E, \, \textbf{p}) = (m \cosh \psi, \, m \textbf{e} \sinh \psi)
$$where $\textbf{e}$ is a unit 3-vector parallel to 3-velocity, $c=1$, $m = ||\textbf{P}|| = \sqrt{E^2 - p^2}$, and $\psi = \tanh^{-1}(p/E)$.

This is the Minkowskian equivalent of Euclidean rectangular-to-polar conversion.

 

[Freixas]

You can smoothly rotate a map - pin it to the table at one point and rotate, and each point will sweep out a circle. Similarly you can smoothly boost a Minkowski metric, in which case each point sweeps out a hyperbola. I wrote a little Javascript program to demonstrate this, here. I'm afraid it predates the wide uptake of touch screens and works better with a device with a mouse, but the "canned" diagrams available further down the page draw several scenarios for you. Then you can either select a timelike line or enter a velocity and the diagram will smoothly boost to that frame. I think this is helpful to "see" what the relationship between two Minkowski diagrams is (the "Hyperbolae and spokes" diagram is worth a look).

My software program, Gamma, can boost a Minkowski diagram. It can do this interactively or through animation. The two diagrams in the comment you were responding to were drawn by specifying two worldlines relative to a rest frame and then (smoothly) redrawing the wordlines relative to a new rest frame (given as a relative velocity to the original rest frame) by dragging a slider.

This provides less insight about invariants than one would expect. Per Peter's comment, for example, these diagrams are projections of spacetime. Things that are invariant in spacetime aren't necessarily invariant in their various projections—not unless you bring in knowledge from outside the diagram, such as that a line whose length changes while boosting is actually maintaining an invariant spacetime interval because the endpoints are moving along specific hyperbolic paths.

 

[Dale]

Things that are invariant in spacetime aren't necessarily invariant in their various projections—not unless you bring in knowledge from outside the diagram

Yes, you definitely need to bring in that outside knowledge. Minkowski geometry is different from Euclidean geometry.

Again, (maybe the 3rd time now) the easiest way is to write it in a manifestly covariant form. It is worth learning the math just for this exact feature.

 

[robphy]

This provides less insight about invariants than one would expect. Per Peter's comment, for example, these diagrams are projections of spacetime. Things that are invariant in spacetime aren't necessarily invariant in their various projections—not unless you bring in knowledge from outside the diagram, such as that a line whose length changes while boosting is actually maintaining an invariant spacetime interval because the endpoints are moving along specific hyperbolic paths.

And as I have suggested earlier,
similar claims would have to apply to any diagram from Euclidean geometry.

Practically any feature that is invariant in Euclidean geometry [suitably formulated*]
will have an analogous feature that is invariant in Minkowski spacetime geometry, and vice versa.

*For example, orthogonality in Euclidean geometry isn't really about a "90-degree angle".
It's about being tangent to circle where the radius meets the circle.

Per Peter's comment, for example, these diagrams are projections of spacetime.

I'm not sure what you mean here.
If what you say [whatever you mean by it] is true, then the same thing can be said about Euclidean diagrams [as projections of a 4d-space?].

 

[Freixas]

Again, (maybe the 3rd time now) the easiest way is to write it in a manifestly covariant form. It is worth learning the math just for this exact feature.

FWIW, I got it the first time. This is not something that will happen overnight. I've looked at some pages on 4 vectors and listened to a YouTube lecture introducing them (by a Yale physics professor).

There is a long (and not easy) path to get from where I am now to where it is easy. If this is considered the easiest path, then it looks a little grim. My crutch of using a Taylor/Wheeler lattice and making sure I dealt only with observables might not be at all elegant, but might at least guarantee that I can start and end a problem with invariants. From where I stand, this looks easier than taking a couple of years of math/physics classes.

Keep in mind that it's not just a matter of learning 4 vectors or whatever. It is a matter of learning it well enough to reach that "now I understand" moment. @robphy, for example, listed a bunch of books he waded through until everything clicked for him—and it sounds like this happened after his undergrad time.

For some background, I'm 68 and a retired software engineer. I work on a variety of projects. My little venture into S.R. physics is for entertainment and not my main thing, and I'm unlikely to show up at a university class any time soon. It's unlikely I will ever understand invariants as you do, but I can probably understand them enough to get by in my little corner of physics.

BTW, in case it's not clear, I appreciate your help. And I am listening.

And as I have suggested earlier, similar claims would have to apply to any diagram from Euclidean geometry.

An invariant is something that is unchanged by some transformation. When we're talking about S.R., that transformation is defined as the Lorentz transformation. When we're talking about Euclidean geometry, I don't know which transformation you have in mind. As I mentioned, some things are invariant with respect, say, to rotation and translation, and not invariant with respect to shearing or scaling. You might be thinking of some mapping of Galilean relativity to Euclidean geometry; I don't know.

In any case, it sounds like we agree that looking at a diagram by itself lends no insight about invariants.

I'm not sure what you mean here.
If what you say [whatever you mean by it] is true, then the same thing can be said about Euclidean diagrams [as projections of a 4d-space?].

I had two diagrams (projections of worldlines with respect to different rest frames) which contained invariant elements. Inspecting the diagrams did not provide any insights what elements were invariant (and I already knew what some of the elements were—inspecting the diagrams didn't add anything).

I'm sure you could put together a great lecture that would work through a set of examples and, assuming you allowed me a lot of time to ask questions, would make your Euclidian-to-Minkowski analogies clear and useful. That, of course, would go way beyond any reasonable expectations for a PhysicsForum discussion. I'm not sure exactly what you're thinking, and you don't know what I know and don't know, so it's difficult for me to follow the exercise you have in mind.

I appreciate that you are trying to point me in a useful direction, and, at some point, maybe what you've said will become clear.

 

[robphy]

An invariant is something that is unchanged by some transformation. When we're talking about S.R., that transformation is defined as the Lorentz transformation. When we're talking about Euclidean geometry, I don't know which transformation you have in mind. As I mentioned, some things are invariant with respect, say, to rotation and translation, and not invariant with respect to shearing or scaling. You might be thinking of some mapping of Galilean relativity to Euclidean geometry; I don't know.

In any case, it sounds like we agree that looking at a diagram by itself lends no insight about invariants.

The Lorentz Transformation (a "boost") is the Minkowski spacetime analogue of a
rotation in Euclidean space, as @Ibix posted in #73.

The rendering of spacetime from a different inertial reference frame in Minkowski spacetime
is the analogue of rotating a sheet of graph paper on the Euclidean plane.

(Scaling and shearing in Minkowski spacetime and Euclidean space
are not acceptable transformations of interest because these are not isometries.
In particular, for scaling, the determinant is not identically equal to 1.)

@robphy, for example, listed a bunch of books he waded through until everything clicked for him—and it sounds like this happened after his undergrad time.

This is true.
One difference between back then and now
is that we have a place for active discussion on the internet,
as well as graphical tools at our fingertips,
plus many more insightful presentations than
the few really-good books with spacetime diagrams that I had access to back then.

I should add that the thing that made things finally click was not the books…
but it was the lectures I sat in on by Bob Geroch. (I had already taken the relativity course for credit with Bob Wald… where I did learn a lot.) But it was Geroch’s course that made it “click”…. He is an advocate of geometric thinking (including efficient coordinate-free tensorial calculations), spacetime diagrams, and operational definitions. (Old course notes [written way before I was there] are available at http://home.uchicago.edu/~geroch/Course Notes . A polished version is available at http://www.minkowskiinstitute.org/mip/books/geroch-gr.html )

 

[Dale]

There is a long (and not easy) path to get from where I am now to where it is easy. If this is considered the easiest path, then it looks a little grim

Yes, it is the easiest path, and it will pay many other dividends if you continue to study relativity.

this looks easier than taking a couple of years of math/physics classes

You shouldn’t need any classes. You know algebra and calculus already, I believe. So this is just some notation (Einstein’s summation convention) and maybe a couple of chapters of differential geometry, like chapters 1 and 2 of Carroll’s “Lecture Notes on General Relativity”.

I'm unlikely to show up at a university class any time soon. It's unlikely I will ever understand invariants as you do,

Sure you can. I have never taken a course that covered any relativity material. My background is engineering and medical imaging. Relativity is just for fun.

 

[Freixas]

You know algebra and calculus already, I believe.

I forgot almost all my calculus. I remember the basic concept behind it. That's about it. I never had any use for it and it's pretty much use it or lose it. I managed to hold on to some trig and some basics of matrices because I worked on some 3D graphics.

Sure you can.

Thanks for the vote of confidence!

 

[Dale]

I forgot almost all my calculus.

That will probably be the biggest challenge then, but since you previously learned it you should not find it difficult the second time around. You may want to get a symbolic computer software. SageMath is good and free. Mathematica is excellent, but expensive. That way you won’t need to re-memorize any of the integral formulas or mechanical details.

 

[vanhees71]

• When most people say “geometry”, the drawings done in the style of Euclid come to mind.
• But, some of us think of "geometry" in the sense of Felix Klein (as you say in #33, "providing a symmetry group"), possibly without a diagram in sight.

So, we have allowed the word "geometry" to be generalized beyond the more common Euclidean interpretation. (The alternative would be to use a prefix [Klein-, generalized- , pseudo- ] or a new word altogether.

Some of us think the notions of "length", "angle", "metric", "dot product", etc..
can also be generalized beyond the more common Euclidean/Riemannian interpretation.
Of course, a definition needs to be provided... and care must be taken.
But I think we do a pretty-good job in that
we don't need to always use a prefix [generalized- , pseudo- ] or a new word altogether
(except to make contact with already established terms: e.g. rapidity).

(Maybe not the best analogy but...
Maybe it's like some "re-boot" or "re-imagined" version of classic TV shows and movies.
One introduces a new viewpoint by relying on the classic version [likely, easier to sell the idea],
rather than introduce a new title altogether.)

From another point of view,
appropriately generalizing the meaning of words
is akin to a "unification" of previously disparate concepts.

• E.g. "gravity" in the sense of "mgh" near the Earth surface
  and "gravity" in the universal inverse-square law.
• I think hyperbolic geometry ( by Gauss, Lobachevsky and Bolyai)
  and Minkowski's spacetime geometry idea are examples... generalizing the geometry of Euclidean plane,
  all developed consistently and all as examples of Cayley-Klein geometries.

For me "geometry" in this context is to be understood in the sense of Klein's Erlangen program. In relativity we deal with a pseudo-Euclidean affine manifold with signature (1,3) or (3,1) in SR and a corresponding pseudo-Riemannian space.

Drawing a Minkowski diagram on paper is nice, but there's always the danger that particularly beginners in the subject misread it as depicting a Euclidean plane. In my opinion it is even counterproductive in learning the concepts and the kinematical effects. If it is introduced it must be made very clear that one has to use the pseudo-metric to construct the length units of the axes of one Lorentzian frame relative to another (whose units can be chosen arbitrarily of course) using hyperbolae rather than circles, and that the Euclidean angles have no geometrical meaning in this 1+1-d Lorentzian manifold. As an analogue, you have rapidities, which have the meaning of an area related to the hyperbolae, e.g., for time-like vectors
$$(x^0,x^1)=s (\cosh \eta,\sinh \eta),$$
or
$$(x^0,x^1)=s(\sinh \eta,\cosh \eta),$$
with $x \cdot x=\pm s^2$, respectively.

 

[vanhees71]

Another related analogy between Euclidean and Minkowskian geometry is the decomposition of a timelike 4-vector into components relative to a Minkowski coordinate system, e.g. for particle 4-momentum $\textbf{P}$$$
\textbf{P} = (E, \, \textbf{p}) = (m \cosh \psi, \, m \textbf{e} \sinh \psi)
$$where $\textbf{e}$ is a unit 3-vector parallel to 3-velocity, $c=1$, $m = ||\textbf{P}|| = \sqrt{E^2 - p^2}$, and $\psi = \tanh^{-1}(p/E)$.

This is the Minkowskian equivalent of Euclidean rectangular-to-polar conversion.

$\psi$ is a rapidity and not an angle!

 

[DrGreg]

As an analogue, you have rapidities, which have the meaning of an area related to the hyperbolae

If you measure "distance" using the Lorentzian (pseudo-)metric and not the Euclidean metric, then rapidity is the ratio of Lorentzian arc length to Lorentzian radius (along a hyperbola, a curve of constant Lorentzian radius). I think that justifies the terminology "hyperbolic angle" for rapidity.

 

[Dale]

In my opinion it is even counterproductive in learning the concepts and the kinematical effects.

In my opinion it is essential. For me personally this geometric understanding was the key that made relativity “click” in my mind

 

[vanhees71]

If you measure "distance" using the Lorentzian (pseudo-)metric and not the Euclidean metric, then rapidity is the ratio of Lorentzian arc length to Lorentzian radius (along a hyperbola, a curve of constant Lorentzian radius). I think that justifies the terminology "hyperbolic angle" for rapidity.

The geometric meaning of the parameter $u$ ("rapidity") in the parametrization of a symmetric unit hyperbola
$$(x,y)=(\cosh u,\sinh u)$$
is not a length but an area as in the following picture

Proof: The green area can be parametrized by
$$\vec{x}(\lambda,u')=\lambda (\cosh u',\sinh u'), \quad \lambda \in [0,1], \quad u' \in [-u,u]$$
Then the area is
$$A=\int_0^1 \mathrm{d} \lambda \int_{-u}^u \mathrm{d} u \epsilon_{jk} \partial_{\lambda} x^{j} \partial_{u'} x^k.$$
The determinant under the integral is
$$\mathrm{det} \begin{pmatrix} \cosh u' & \lambda \sinh u' \\ \sinh u' & \lambda \cosh u' \end{pmatrix} =\lambda'$$
and thus finally
$$A=u.$$
The arc-length of the hyperbola is not that simple. The "arc" from $(1,0)$ to $P(u)$ is
$$L=\int_0^u \mathrm{d} u' \sqrt{\cosh^2 u' + \sinh^2 u'}$$
which is an elliptic function. According to Mathematica you get
$$L=-\mathrm{i} \mathrm{E}(\mathrm{i} u,2).$$

 

[DrGreg]

The arc-length of the hyperbola is not that simple. The "arc" from $(1,0)$ to $P(u)$ is
$$L=\int_0^u \mathrm{d} u' \sqrt{\cosh^2 u' + \sinh^2 u'}$$
which is an elliptic function. According to Mathematica you get
$$L=-\mathrm{i} \mathrm{E}(\mathrm{i} u,2).$$

That's the Euclidean arc length. The Lorentzian arc length is
$$L=\int_0^u \mathrm{d} u' \sqrt{\cosh^2 u' - \sinh^2 u'} = u$$

 

[vanhees71]

Maybe, I'm too pedantic, but I don't like to call this integral an "arc length". I guess that's semantics, but for me it was a revelation when I learned relativity from some book, where they don't made these strange statements about an indefinite "metric" as in my school book.

 

[PeterDonis]

I don't like to call this integral an "arc length".

Given that we are talking about a theory of physics, the Lorentzian arc length is the one that has physical meaning in the theory; the Euclidean arc length does not. So using the term "arc length" for the Euclidean one in the context of the physical theory is at least as confusing as using it for the Lorentzian one.

 

[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?​

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