I The Energy - Momentum Equation vs the Energy - Mass Equation

[vanhees71]

I'm still puzzled about the question, what all this effort should be good for.

Of course, some of the theorems (or rather lemmas) about the Minkowski geometry are useful, e.g., the "causality property" that the temporal order of events is unchanged under proper orthochronous Lorentz transformations if and only if the events are spacelike or lightlike separated or that a vector Minkowski-orthogonal to a time-like vector is spacelike and a Minkowski orthogonal vector to a light-like vector is parallel to this light-like vector (maybe there are more), but all these are more easily formulated with Minkowski products of the various vectors rather than with any square roots of them.

 

[Sagittarius A-Star]

I'm still puzzled about the question, what all this effort should be good for.
...
but all these are more easily formulated with Minkowski products of the various vectors rather than with any square roots of them.

The norm of a 4-vector is the square-root. It is, except for photons, a directly measureable quantity in the object's rest frame. That could be for example the proper time of a time-like twin, which is measured by that twin's wristwatch or my proper energy, also called my "invariant mass", measured by the bathroom scale, if I am standing still on it.

The instrument for measuring this norm directly must have the same worldline as the object, to that the related 4-vector applies. For example is the wristwatch of a tachyon also a tachyon (from the standpoint of us, the tardyons).

 

[vanhees71]

I stick to the mathematically sound and solid usual definitions. What you declare as norm is no norm and it is overly complicated and pretty useless.

 

[Sagittarius A-Star]

I stick to the mathematically sound and solid usual definitions.

That's O.K. Only, in SR it is not "unusual" to use the term "norm" also for 4-vectors, according to some examples, which were linked or mentioned here.

What you declare as norm is no norm

I meant "pseudo-norm". I omitted the "pseudo", because I assumed, that it's clear from the context.

it is overly complicated

I find it easy.

and pretty useless.

I find, in the context of a geometry-based theory like SR, a concept of 4-vectors without a (pseudo-)norm is incomplete.

 

[Dale]

The instrument for measuring this norm directly must have the same worldline as the object

I wouldn’t read too much into directness. Most measurements are rather indirect. For example, the current mise en pratique for measuring mass is very indirect, involving measurements of voltage, current, speed, and frequency, along with previous measurements of acceleration. Even the more common, most direct technique, is actually measuring torque.

 

[Dale]

I stick to the mathematically sound and solid usual definitions. What you declare as norm is no norm and it is overly complicated and pretty useless.

Nonetheless this particular generalization of terminology is well established and is not just personal “What you declare” terminology, but standard and well accepted.

If you wish merely to grumble then that is fine. But I f you wish to persuade others to adopt your viewpoint then you will need to come up with persuasive arguments. That it doesn’t meet the formal definition of a norm is not persuasive, because people using this terminology already know that and are content with the generalization.

So far your most persuasive argument was about confusion engendered in students. I would develop or expand that type of argument. Unless you are interested only in grumbling.

 

[vanhees71]

I find, in the context of a geometry-based theory like SR, a concept of 4-vectors without a (pseudo-)norm is incomplete.

To the contrary, it is utmost important for the consistency of the theory that the geometry of relativistic spacetimes is pseudo-Riemannian and not Riemannian!

 

[vanhees71]

Nonetheless this particular generalization of terminology is well established and is not just personal “What you declare” terminology, but standard and well accepted.

If you wish merely to grumble then that is fine. But I f you wish to persuade others to adopt your viewpoint then you will need to come up with persuasive arguments. That it doesn’t meet the formal definition of a norm is not persuasive, because people using this terminology already know that and are content with the generalization.

So far your most persuasive argument was about confusion engendered in students. I would develop or expand that type of argument. Unless you are interested only in grumbling.

It's the first time in this thread that I've seen that somebody defines $\| \cdot \|$ as a "norm" based on an indefinite fundamental form. It's not needed and even more confusing than the standard use of the word "metric" for the bilinear form. The standard terminology is already confusing enough for students, as you can see on the example of this useless discussion.

 

[Dale]

The standard terminology is already confusing enough for students, as you can see on the example of this useless discussion

Have you ever seen actual students actually struggle with the concept of the Minkowski norm? What indication do you have that it is truly confusing to students? If you have not seen that then exactly what struggles would you anticipate they would have and why? Perhaps the reason that the standard terminology seems confusing to students is insufficient emphasis of the similarities and so adopting this terminology would help. What makes you think it wouldn’t?

Personally, for me it is still about geometry. Anything that we can do to emphasize the geometry is a good thing. Yes, it is important to explain that Minkowski geometry is not the same as Euclidean geometry, but most Euclidean geometric concepts have a close analogue in Minkowski geometry. The similarities are every bit as important as the differences because the similarities allow them to reason geometrically.

How many students who understand the Minkowski norm geometrically will be fooled by the twin paradox?

 

[vanhees71]

My experience is that it is hard for students (including myself when I was a student) to forget about the Euclidean heuristics built from long training in Euclidean (plane) geometry when it comes to read a Minkowski diagram. My students told me that it was important for them that I emphasized the important difference between Euclidean and Minkowski and that instead of circles you need time- and space-like hyperbolae to determine the "unit tick marks" on the axes of different inertial reference frames depicted by them. The said, it's much easier to understand in this way than it was presented to them at high school. After some thinking I came to the conclusion that the way I treat the kinematics in

https://itp.uni-frankfurt.de/~hees/pf-faq/srt.pdf

is the shortest and simplest way without too high-level math.

In Minkowski geometry there's no norm induced by the fundamental form (Minkowski product), and that makes all the difference. There are also no angles in the Eulidean sense (though there are rapidities which are similar but not the same).

I think, the right didactics is a good mixture of algebraic/analytical and geometric concepts. The geometry of spacetime, as used in physics, is analytic geometry anyway!

 

[Dale]

My experience is that it is hard for students (including myself when I was a student) to forget about the Euclidean heuristics built from long training in Euclidean (plane) geometry when it comes to read a Minkowski diagram. My students told me that it was important for them that I emphasized the important difference between Euclidean and Minkowski and that instead of circles you need time- and space-like hyperbolae to determine the "unit tick marks" on the axes of different inertial reference frames depicted by them. The said, it's much easier to understand in this way than it was presented to them at high school. After some thinking I came to the conclusion that the way I treat the kinematics in

https://itp.uni-frankfurt.de/~hees/pf-faq/srt.pdf

is the shortest and simplest way without too high-level math.

In Minkowski geometry there's no norm induced by the fundamental form (Minkowski product), and that makes all the difference. There are also no angles in the Eulidean sense (though there are rapidities which are similar but not the same).

I think, the right didactics is a good mixture of algebraic/analytical and geometric concepts. The geometry of spacetime, as used in physics, is analytic geometry anyway!

This is a much more persuasive argument. Instead of just complaining about not meeting a particular definition (which we already know and don’t find objectionable) you have provided personal experience and expertise (which we didn’t know and which explains your position).

When discussing semantics, simply quoting a definition is not persuasive precisely because the best definition is in dispute. Your explanation here is far better because it explains why you prefer the non-generalized definition.

 

[Sagittarius A-Star]

It's the first time in this thread that I've seen that somebody defines $\| \cdot \|$ as a "norm" based on an indefinite fundamental form.

Do you mean the https://courses.maths.ox.ac.uk/node/view_material/44180 #119? If, yes, then I don't understand, how you come to "indefinite fundamental form".

In the paper, they define a pseudo-norm only for the limited scope of positive $A \cdot A$.

Proposition 8 (Minkowski triangle inequality). If $U$ and $V$ are future-pointing, timelike four-vectors ...
...
The pseudo-norm of a future-pointing timelike vector is ...

It was only me, who tried in #120 to extend the scope, based on Wikipedia, which describes an extended scope.

I just found another paper. There, the "norm" is not defined as the square-root. I think, "norm squared" would be a more logical name for that:

The norm of a vector is defined to be inner product of the vector with itself; unlike in Euclidean space, this number is not positive definite
...
(A vector can have zero norm without being the zero vector.)

Source (at end of page 15):
https://preposterousuniverse.com/wp-content/uploads/grnotes-one.pdf