What, exactly, are invariants?

[PeterDonis]

If I look at a spacetime diagram, I still have to be careful about what is an invariant and what is not.

Lengths along of curves and angles between curves where they intersect. In other words, geometric invariants.

 

[Freixas]

And that means you need to read carefully and not just take what you read at face value.

Here is the longer response I promised.

At least to start, I'm going to use the Taylor/Wheeler lattice as the equivalent of training wheels (or of @Ibix's radar gun). What can be observed using the lattice will be an invariant.

In order to do predictive physics (where I have some set of initial conditions and want to predict some ending conditions), I need to state a problem so that values are relative to the lattice, which defines the rest frame. I also need to define how the lattice's clocks are initialized (I could initialize them in any of a number of ways, but let's limit the choices to ones that implement "valid" simultaneity conventions).

If I want to start and end with invariants, I need to begin with initial conditions defined in terms of observable events. Relative speed might be an "angle in spacetime between two timelike worldlines at the point where they intersect," but I don't know how to observe that. Instead, I might state that, say, Alice is moving inertially and she was spotted at coordinate $c_1$ and time $t_1$ and coordinate $c_2$ at time $t_2$. Alternately, I could create a special lattice observer whose job is to measure Alice's relative velocity and display it on a billboard for all to see. Since I'm talking about initial conditions, I could provide the number seen on the billboard.

The problem in the OP asked about the elapsed time in one location based on an event happening elsewhere. Using the billboard method, I could place a special observer in a position such that light from both locations reach it at the same time. If the lattice's clocks were initialized using Einsteinian synchronization, then we could place the observer midway between the two events; if some other synchronization was used, the location might be closer to one event or the other. The observer could then view an event occurring in one place, view the time on a clock elsewhere on the lattice, and display the result on a billboard.

This is a slow and crude technique, but it establishes the simultaneity of two events as an invariant physical observable. This is one of the pieces I was missing.

Once I can do this, I can use the same technique for other things that depend on a simultaneity convention: length, speed, and clock rates. In each case, I need to understand how someone can use the raw event data captured by the lattice to measure these quantities. If I go through the mental exercise, then I can verify that I start and end with invariants.

The process may be cumbersome, but at least I can understand it.

I notice that everyone seems to drop the simultaneity convention requirement. Yes, I gather that if nothing is mentioned, Einsteinian synchronization is universally assumed. But no one can predict the billboard numbers displayed unless the calculation method takes the lattice's simultaneity convention into account, so I keep including it.

Lengths along of curves and angles between curves where they intersect. In other words, geometric invariants.

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.

 

[vanhees71]

While vectors and tensors are great for some things (especially efficient calculation),
they may not be great in some other things
(like giving a geometric intuition of what the tensor calculation is saying).

Well, you always have to physically understand what your model is saying. Manifestly covariant formulations have the advantage that they provide you with models that are at least not contradicting the very foundations of relativity, i.e., the spacetime model it is based on. Also I've the impression that geometry is overemphasized (e.g., in endless discussions of Minkowski diagrams which are not so simple to understand as one might think).

So, I'd suggest that one also use spacetime-vectors on a planar (1+1)-Spacetime diagram, together with its trigonometry… since it’s not too much harder than Euclidean geometry, when properly guided in it.
The angle between two given vectors is an invariant.

It's not too much harder than Euclidean geometry. What's hard is to give up ones Euclidean interpretation of the "paper plane"!

Since relativity is about the "geometry of spacetime",
I think it's good to be able to connect with the "geometry" visually,
and not just [vector- and tensor-]algebraically or "as a set of components that transforms as ...".

Relativity provides a spacetime model, and that's of course "geometric", but then you have to go further and understand the "geometry" in the sense of Riemann, Klein, et al. i.e., providing a symmetry group, which can be used to find the dynamical laws of Nature, which can be compared with observations. E.g., you can argue that gravitation is just spacetime curvature, but at the same time it's an interaction as the other fundamental interactions too. One should also have in mind the approach as given in, e.g., Weinberg's book of 1971 or Feynman's Lectures on Gravitation and not stick only to the standard geometric interpretation. Particularly it becomes clear that "general covariance" is a "local gauge symmetry", which is very important. One of the problems Einstein had to derive his field equations was that this important feature was not clear to him and his collaborators and that's why it took him about 10 years to finally get GR.

(We can [in principle] solve high-school geometry problems vectorially and tensorially.
But could we easily and intuitively explain the geometry that is involved from those vectors and tensors,
or from transformations (rotations)?
Many situations in relativity can be treated geometrically, but we sadly don't seem to do so.

For me the analytical approach to geometry via vectors was a revelation. Unfortunately we did not learn about tensors but simple transformations like rotations we learned about. Most situations in relativity can be treated with great advantage using manifestly covariant tensor (or spinor) formulations, which of course is also geometry but without the obstacle of easily misunderstood drawings in a non-Euclidean plane ;-).

 

[robphy]

Well, you always have to physically understand what your model is saying. Manifestly covariant formulations have the advantage that they provide you with models that are at least not contradicting the very foundations of relativity, i.e., the spacetime model it is based on. Also I've the impression that geometry is overemphasized (e.g., in endless discussions of Minkowski diagrams which are not so simple to understand as one might think).

As you say, one has to understand physically what is happening.
My point is that vectors and tensors are great, but not enough.
Although they provide and display the invariance of relativity,
especially in complicated calculations,
it's not so easy to connect them to one's physical and mathematical intuition.

That's why I said (new red bolding mine)
"one also use spacetime-vectors on a planar (1+1)-Spacetime diagram".
That can help with the visualization of the geometry

Even with vectors, tensors, and spacetime diagrams,
that's not enough for physical intuition.

As an undergrad, I had 3 courses in relativity using texts by Skinner, Lawden, and Landau.
(I also had Taylor-Wheeler and Misner-Thorne-Wheeler as side references.)
I'm certainly not claiming to have learned anywhere-near-everything from these books.
I could do enough of the work to get A's in them, but I didn't feel I "got it".

In grad school, learning more tensorial methods and the abstract-index notation from Wald's text helped.
But it wasn't until I learned about operational definitions of distance and time measurements via radar-methods on a spacetime diagram from Geroch, did things finally click for me. For me, I could now tie together the verbiage of introductory texts, the notations of vectors and tensors in coordinate form and in abstract-index form, and physics connected to observation using light signals and clocks (which is more relativistic in spirit than rods and clocks).

On my own in grad school, I stumbled upon Yaglom's "A simple non-Euclidean geometry and its physical basis", which introduced me to Klein and the Cayley-Klein geometries. I also tumbled upon Schouten's "Ricci Calculus" and "Tensor Analysis for Physicists", which introduced me to visualizing tensors.

Ideally, one really should try to be fluent in (and fluid in, in the sense of "being able to inter-connect") all of these methods.

My $0.03.

 

[Dale]

My $0.03.

Hmm, inflation hits PF also?

For me, the spacetime diagrams and the concept of four-vectors built my intuition. But I always use explicit four-vector or tensor-based math to check my intuition. I think that it is important to have both. As you said "also". Sometimes our intuition is wrong and needs to be retrained. So it is necessary to have a solid foundation that you can trust to get the right answer even when intuition fails.

 

[vanhees71]

In grad school, learning more tensorial methods and the abstract-index notation from Wald's text helped.
But it wasn't until I learned about operational definitions of distance and time measurements via radar-methods on a spacetime diagram from Geroch, did things finally click for me. For me, I could now tie together the verbiage of introductory texts, the notations of vectors and tensors in coordinate form and in abstract-index form, and physics connected to observation using light signals and clocks (which is more relativistic in spirit than rods and clocks).

That's exactly my point, but remember the very heated discussion we had in these forums, when I dared to make the point that a reference frame is not simply an abstract "coordinate patch" in a pseudo-Riemannian manifold but something made of real things in the lab ;-)).

On my own in grad school, I stumbled upon Yaglom's "A simple non-Euclidean geometry and its physical basis", which introduced me to Klein and the Cayley-Klein geometries. I also tumbled upon Schouten's "Ricci Calculus" and "Tensor Analysis for Physicists", which introduced me to visualizing tensors.

Ideally, one really should try to be fluent in (and fluid in, in the sense of "being able to inter-connect") all of these methods.

My $0.03.

FACK.

 

[Freixas]

(on the diagrams anyway--I realize the proper lengths are invariant)

I don't seem to be able to edit my post. This parenthetical comment is incorrect. Proper lengths are invariant, but I don't think there is such as thing as "the proper length of a worldline". From the point of view of the observer whose worldline we draw, they are always at position 0 and the length of their journey is also 0. Sorry for the error.

 

[PeterDonis]

Proper lengths are invariant, but I don't think there is such as thing as "the proper length of a worldline".

The corresponding invariant along a worldline (i.e., a timelike curve) is proper time.

 

[PeterDonis]

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.

You're thinking of it backwards.

Take an ordinary object, say a coin, and look at it from different angles. Its appearance will be different, but its physical properties will not change. The physical properties are invariants.

Your two spacetime diagrams are the analogue of looking at an ordinary object from two different angles. The appearances are different, but the physical properties do not change. The physical length along a given curve does not change even though its apparent length changes--just as the coin's physical size and shape does not change even though its apparent size and shape does. Similar remarks apply to angles.

 

[Freixas]

Your two spacetime diagrams are the analogue of looking at an ordinary object from two different angles. The appearances are different, but the physical properties do not change. The physical length along a given curve does not change even though its apparent length changes--just as the coin's physical size and shape does not change even though its apparent size and shape does. Similar remarks apply to angles.

My diagrams were in response to this exchange:

If I look at a spacetime diagram, I still have to be careful about what is an invariant and what is not.

Lengths along of curves and angles between curves where they intersect. In other words, geometric invariants.

If I look at a spacetime diagram, all I have is a projection with respect to a particular frame of reference. My point (and yours, apparently) is that looking at this projection doesn't give me any insights into what is and isn't an invariant.

The only way I know of to look at objects in spacetime geometrically is through one or more projections. Is there an alternate geometrical representation for spacetime curves in which the lengths and angles are invariant?

 

[PeterDonis]

My point (and yours, apparently) is that looking at this projection doesn't give me any insights into what is and isn't an invariant.

No, that isn't my point. My point is the opposite: you already know that the lengths along worldlines and the angles between them are invariants. You don't learn that by looking at the diagram. You learn that by understanding what relativity says as a physical theory and understanding what actual measurements you make to know lengths along worldlines and angles between them. You don't make those measurements just by looking at a diagram, any more than you measure the size and shape of a coin by just looking at it from some angle.

Is there an alternate geometrical representation for spacetime curves in which the lengths and angles are invariant?

You are chasing a phantom here. Basically, what you're looking for is a diagram of 4-dimensional locally Lorentzian geometry that represents all lengths and angles exactly. This is a fool's errand. You can't even have such a diagram for the surface of the Earth, which is a 2-dimensional locally Euclidean geometry and so is much closer to what we can visualize than 4-d spacetime is.

What you need to understand is that you can do physics without such an exact diagram. And then learn how.

 

[robphy]

Draw a circle with two radii and label the smaller angle between the radii,
then draw the tangent through the tip of one of the radii to form a triangle [after suitable extending the segments].
Can you identify invariants and non-invariants in this figure?

You could check by rotating the page and
seeing if you get the same results by using the same procedures.

You could also use suitable geometrical tools to measure various quantities.

I think that once this is understood, then you can ask the same kinds of questions for special relativity.
(You might even translate your "frame of reference" and projection notions into their
Euclidean analogues... to see re-interpret what you are asking yourself.)

 

[Freixas]

No, that isn't my point. My point is the opposite: you already know that the lengths along worldlines and the angles between them are invariants.

Sorry, there are a lot of posts here. I think @robphy was the one suggesting that I could use geometry; he clarifies his approach in #42.

This is a fool's errand.

Agreed. I didn't think this was possible, but mathematicians have a lot of tricks and I'm hardly the one to provide a theorem showing this is impossible.

 

[robphy]

The only way I know of to look at objects in spacetime geometrically is through one or more projections. Is there an alternate geometrical representation for spacetime curves in which the lengths and angles are invariant?

My light-clock diamonds approach is an attempt to represent the proper-time along piecewise inertial worldlines... and do so with a physical mechanism in the spirit of relativity.
Rapidity ("spacetime-angles" between timelike tangents) can be seen as
areas of sectors in unit-hyperbolas ("circles").

These are taking advantage of the fact that
the equality of areas on a plane in euclidean geometry
also holds in Minkowski geometry---this is an affine notion, not a metrical one.

This tries to follow the spirit of various "proof without words" demonstrations.

 

[Freixas]

Draw a circle with two radii and label the smaller angle between the radii,
then draw the tangent through the tip of one of the radii to form a triangle [after suitable extending the segments].
Can you identify invariants and non-invariants in this figure?

I have a problem with the question. The most basic definition of an invariant is "a function, quantity, or property which remains unchanged when a specified transformation is applied." Things might be invariant with respect to one transformation and not invariant with respect to another.

In S.R., I believe invariants are invariant with respect to changing the frame of reference. If my math is right (always suspect), S.R. invariants aren't invariant with respect to changing the simultaneity convention.

With the problem you gave, it's not clear what transformation(s) you are picturing. You mentioned rotation; the answer to your questions might be different if I applied a shearing transform. Given specific radii, the angles of the triangle are invariant with respect to rotation, translation, and scaling, but not with respect to shearing.

 

[robphy]

I have a problem with the question. The most basic definition of an invariant is "a function, quantity, or property which remains unchanged when a specified transformation is applied." Things might be invariant with respect to one transformation and not invariant with respect to another.

In S.R., I believe invariants are invariant with respect to changing the frame of reference. If my math is right (always suspect), S.R. invariants aren't invariant with respect to changing the simultaneity convention.

With the problem you gave, it's not clear what transformation(s) you are picturing. You mentioned rotation; the answer to your questions might be different if I applied a shearing transform. Given specific radii, the angles of the triangle are invariant with respect to rotation, translation, and scaling, but not with respect to shearing.

I'm thinking about a high-school geometry problem.
We draw the figure and we study it,
without laying down a grid, or rotating it into some standard position, or
sequentially rotating it into standard positions to analyze certain segments.
We use Euclidean geometry, the Pythagorean theorem, trigonometric definitions, etc...
(Maybe we have gotten used to rotating the diagram to see that its orientation doesn't matter.)

For such a problem, we won't be shearing.
(For Galilean spacetime diagrams, I might be shearing... but that's a "rotation", a Galilean transformation.
One can learn to read such diagrams like a diagram for Euclidean geometry,
without having to transform to a particular reference frame to analyze.)

---

Concerning change the simultaneity convention... i don't know.
I haven't really thought about it.
My focus hasn't ventured into variants of special relativity,
but rather on how Minkowski space is one of a family Cayley-Klein geometries.

Presumably, the choice of simultaneity convention will imply that
either the hyperbola as "circle" is not representative of the metric and/or
the tangent line to that hyperbola is not used to define orthogonality,
which mimics the scheme that applies Euclidean geometry and Galilean spacetime geometry.

I would consider looking into how to formulate such simultaneity conventions
in terms of invariant structures in special relativity. What gets perturbed?

 

[Dale]

the angles are not obviously invariant. I'm not sure how looking at a diagram clarifies what is and isn't an invariant.

This is where it is worth writing things in a manifestly covariant formulation. The length is invariant because it can be written as $$\int_P \sqrt{g_{\mu\nu} dx^\mu dx^\nu}$$ and an angle is invariant because it can be written $$\cos(\theta)=\frac{g_{\mu\nu} x^\mu y^\nu}{\sqrt{g_{\mu\nu} x^\mu x^\nu} \sqrt{g_{\mu\nu} y^\mu y^\nu}}$$

Once you write a quantity in a manifestly covariant formulation then you immediately know that it is invariant under any coordinate transformation.

 

[robphy]

Of course, @Dale 's calculation shows the invariance, when you are given $x^\mu$ and $y^\mu$.

If however, the $y^\mu$ is tied to an arbitrary choice of axis.
Then the angle is a scalar tied to choice of axis.
This angle, however, is not an invariant, independent of the choice of axis.

In your diagram, I think you were referring to the angle made in the diagram of your various frames.
So, such an angle isn't independent of the choice of axis, as you said.

 

[vanhees71]

Note that, of course, angles make only sense in Euclidean metrics, not in Lorentzian pseudo-metrics.

 

[Dale]

Note that, of course, angles make only sense in Euclidean metrics, not in Lorentzian pseudo-metrics.

I disagree. Angles between spacelike vectors in a Lorentzian manifold are exactly the same as angles between vectors in Euclidean manifolds. Angles between timelike vectors in a Lorentzian manifold are related to relative velocities and there is no reason that they should not be considered to make sense. Null vectors clearly don't work, and I am not sure about combinations of timelike and spacelike vectors, but to broadly say that angles don't make sense in Lorentzian manifolds goes too far, IMO.

 

[DrGreg]

I agree with Dale. If anyone doesn't like using the word "angle" between timelike vectors, you can call it "rapidity" instead, but it's essentially the same concept geometrically.

 

[robphy]

I disagree. Angles between spacelike vectors in a Lorentzian manifold are exactly the same as angles between vectors in Euclidean manifolds.

One needs to be careful here.
When a set of a spacelike vectors is orthogonal to a given timelike vector (so these spacelike vectors are parallel to a hyperplane of simultaneity, and are thus "spatial vectors" to that timelike observer),
then "Angles between these spacelike vectors in a Lorentzian manifold are exactly the same as angles between vectors in Euclidean manifolds."

However, on a (1+1)-spacetime diagram.
The t- and t'-axes are timelike vectors, and x- and x'- axes are spacelike vectors.
Note: the "angle" between the x- and x'-axes is not a Euclidean angle...
it's numerically equal to the rapidity between the timelike t- and t'-axes.

 

[DrGreg]

One needs to be careful here.
When a set of a spacelike vectors is orthogonal to a given timelike vector (so these spacelike vectors are parallel to a hyperplane of simultaneity, and are thus "spatial vectors" to that timelike observer),
then "Angles between these spacelike vectors in a Lorentzian manifold are exactly the same as angles between vectors in Euclidean manifolds."

However, on a (1+1)-spacetime diagram.
The t- and t'-axes are timelike vectors, and x- and x'- axes are spacelike vectors.
Note: the "angle" between the x- and x'-axes is not a Euclidean angle...
it's numerically equal to the rapidity between the timelike t- and t'-axes.

This is something I hadn't considered before, but after thinking about this, I think we deal with this as follows:

Given two spacelike 4-vectors $\textbf{K}$ and $\textbf{L}$ evaluate$$
\lambda = \frac{ g( \textbf{K}, \textbf{L} ) } { \sqrt{| g( \textbf{K}, \textbf{K} ) |} \, \sqrt{| g( \textbf{L}, \textbf{L} ) |}} \, .
$$If $|\lambda| \leq 1$, there is an associated "Euclidean" angle $\cos^{-1} \lambda$.

If $|\lambda| \gt 1$, there is an associated rapidity $\cosh^{-1} |\lambda|$, in the sense indicated in the second half of robphy's post.

That seems to make sense, or have I overlooked something?

---

Footnote added Sun 18 Jul: The "Euclidean" angle could be $\cos^{-1} (-\lambda)$, depending on which sign convention you use for the metric.

 

[Dale]

One needs to be careful here.
When a set of a spacelike vectors is orthogonal to a given timelike vector (so these spacelike vectors are parallel to a hyperplane of simultaneity, and are thus "spatial vectors" to that timelike observer),
then "Angles between these spacelike vectors in a Lorentzian manifold are exactly the same as angles between vectors in Euclidean manifolds."

However, on a (1+1)-spacetime diagram.
The t- and t'-axes are timelike vectors, and x- and x'- axes are spacelike vectors.
Note: the "angle" between the x- and x'-axes is not a Euclidean angle...
it's numerically equal to the rapidity between the timelike t- and t'-axes.

Agreed, and I have no problem with appropriate caveats and highlighting things that make spacetime angles different. I just think that geometrically they are “angles” in the same sense that geometrically spacetime intervals are “lengths” and it is worth bringing in that sort of reasoning and intuition (with appropriate caveats etc.)

 

[robphy]

Agreed, and I have no problem with appropriate caveats and highlighting things that make spacetime angles different. I just think that geometrically they are “angles” in the same sense that geometrically spacetime intervals are “lengths” and it is worth bringing in that sort of reasoning and intuition (with appropriate caveats etc.)

There is definitely an intuition to be imported.
One just needs some care
and one has to allow a weakening strict Euclidean thinking to embrace these analogous variations.
(I guess one could pseudo- prefix a bunch of terms.)

From a Cayley-Klein geometry ( https://en.wikipedia.org/wiki/Cayley–Klein_metric ) viewpoint,
angle and distance are "dual" concepts...
one is a measure of separation between lines that meet at a point, and
the other is a measure of separation between points that are joined by a line.
When expressed as a cross-ratio ( https://en.wikipedia.org/wiki/Cross-ratio ),
their similarities are clearer.

(On a sphere, angle and distance are more similar.
This is so-called double-elliptic in the Cayley-Klein geometries.
Anti-deSitter space is doubly-hyperbolic.
Galilean is doubly-parabolic.
Hyperbolic space has a hyperbolic-measure for distances but an elliptic/circular measure for angles. )

At the same time, one should be aware of some special properties that happen to arise in (say) the Euclidean case, that don't hold for the more general case. So, one should focus on the general case to develop the subject in a unified way... then make note of special situations that may arise.

 

[Hornbein]

In defense of Albert, I find his paper best at convincing the reader that special relativity corresponds to reality. It's very down to Earth. Once reader is convinced of this they can move on to formalisms that are easier to work with.

 

[robphy]

In defense of Albert, I find his paper best at convincing the reader that special relativity corresponds to reality. It's very down to Earth. Once reader is convinced of this they can move on to formalisms that are easier to work with.

In your sentence, instead of "best", I would use words like: insightful, breakthrough, groundbreaking, revolutionary, pioneering, first .

If I were to introduce relativity to an absolute beginner with the goal to understand the essence of relativity,
I would not start with Einstein.
I would probably start with Bondi... Relativistic kinematics that is distilled and more to the point.

If I were to introduce relativity to a more mathematically-mature reader
with the goal to suggest that relativity forces us
to rethink kinematics, dynamics, and better-understand electrodynamics,
I might suggest Einstein (among others).I wouldn't say, to an absolute beginner,
that Einstein's paper (by itself) would convince the reader that special relativity corresponds to reality.

I would say that
other people said that Einstein's paper suggests that special relativity corresponds to reality
and that I would try to read Einstein's paper because of that,
and that I would likely seek other formalisms that are easier to work with
(for me) to better understand Einstein's insights and the subsequent results from Einstein's insights.I would probably say similar things about the groundbreaking works by Newton, Maxwell, Schrödinger, etc...
For these great works,
I would seek other formalisms that are easier to work with
(for me) to better understand their insights and the subsequent results from their insights.

 

[vanhees71]

I disagree. Angles between spacelike vectors in a Lorentzian manifold are exactly the same as angles between vectors in Euclidean manifolds. Angles between timelike vectors in a Lorentzian manifold are related to relative velocities and there is no reason that they should not be considered to make sense. Null vectors clearly don't work, and I am not sure about combinations of timelike and spacelike vectors, but to broadly say that angles don't make sense in Lorentzian manifolds goes too far, IMO.

I can understand what you mean by angles between space-like vectors, i.e., you can define them as in Euclidean space as
$$\cos \theta = -\frac{a \cdot b}{\sqrt{-a \cdot a} \sqrt{-b \cdot b}},$$
where $-$ signs are due to my west-coast choice of the signature (+---).

What do you mean by an "angle" between time-like vectors? May be rapidities as in coordinates for Bjorken flow?

 

[DrGreg]

I can understand what you mean by angles between space-like vectors, i.e., you can define them as in Euclidean space as
$$\cos \theta = -\frac{a \cdot b}{\sqrt{-a \cdot a} \sqrt{-b \cdot b}},$$
where $-$ signs are due to my west-coast choice of the signature (+---).

What do you mean by an "angle" between time-like vectors? May be rapidities as in coordinates for Bjorken flow?

See my post #53 (including my quote from robphy's post #52).

When, in my notation, $\textbf{K}$ and $\textbf{L}$ (your $a$ and $b$) are both timelike, then $|\lambda| \geq 1$ and so $\cosh^{-1} |\lambda|$ is the rapidity between the 4-velocities that are parallel to $\textbf{K}$ and $\textbf{L}$.

 

[vanhees71]

Ok, but that's then an extension of the usual definition of an angle in Euclidean geometry. A rapidity is not an angle!