I What, exactly, are invariants?

[robphy]

Some old posts on "[pseudo-]angles" in special relativity:

• https://www.physicsforums.com/threa...-like-and-time-like-vectors.74115/post-554189
• https://www.physicsforums.com/threads/angles-in-minkowski-space.991133/post-6364136

I mentioned that the Gauss-Bonnet theorem might be the way to nail down definitions of
angles [with signs, if needed] between 4-vectors (not-necessarily of the same type).
Here are some papers that pursue this approach:

• A relativistic version of the Gauss-Bonnet formula
  Garry Helzer
  J. Differential Geom. 9(4): 507-512 (1974). DOI: 10.4310/jdg/1214432546
  https://projecteuclid.org/journals/...ss-Bonnet-formula/10.4310/jdg/1214432546.full
• The Gauss–Bonnet theorem for Cayley–Klein geometries of dimension two
  Alan S. McRae
  New York J. Math. 12 (2006) 143–155.
  https://nyjm.albany.edu/j/2006/Vol12.htm
  https://nyjm.albany.edu/j/2006/12-8p.pdf

 

[Dale]

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

Rapidity is an “angle” in the same sense that the spacetime interval is a “length”. I have seen you post enough about relativity that I know you understand both this concept and also the typical misuse of terminology involved.

Since we want to encourage a geometric understanding of relativity it is unavoidable to use geometric words. The words are appropriate because they represent a reasonable Lorentzian generalization of the corresponding Euclidean concept. The fact that they are generalizations means that care must be taken and appropriate caveats/warnings should be given. It does not mean that the generalizations do not make sense as you claimed

 

[vanhees71]

Well, yes, I'm always trying to avoid this abuse of Euclidean language, which is very misleading.

 

[Dale]

Well, yes, I'm always trying to avoid this abuse of Euclidean language, which is very misleading.

There are recent cases where the misleading terminology has indeed led to long discussions with some novices here, where they seem to be unable to understand the ways that the generalization differs from the Euclidean concept. So I do see that risk too.

However, I also see the benefit in leveraging student’s existing geometrical knowledge. It is a risk-benefit trade off. Some students will be harmed by the risks and some will be helped by the benefits.

 

[robphy]

Well, yes, I'm always trying to avoid this abuse of Euclidean language, which is very misleading.

• 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.

 

[Sagittarius A-Star]

Well, yes, I'm always trying to avoid this abuse of Euclidean language, which is very misleading.

I think it becomes is clear, if the expressions "circular angle" and "hyperbolic angle" are used. But they still contain both the word "angle".

In other words, this means just as how the circular angle can be defined as the arclength of an arc on the unit circle subtended by the same angle using the Euclidean defined metric, the hyperbolic angle is the arclength of the arc on the "unit" hyperbola subtended by the hyperbolic angle using the Minkowski defined metric.

Source:
https://en.wikipedia.org/wiki/Hyperbolic_angle#Relation_To_The_Minkowski_Line_Element

 

[Freixas]

When most people say “geometry”, the drawings done in the style of Euclid come to mind.

Novice here.

Yes, that's what I assume. At some point, it sunk in that when you guys said "angle", you were talking about relative velocity.

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.

When someone says "geometry", I think "diagram". When someone uses geometric terms in an advanced way, it leads to my confusion at the bottom of my post #32.

I hate to bring up my classification scheme, 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. My approach may be clunky because I don't have the advanced geometrical knowledge of the people in this group, but it let's me limp along.

 

[Freixas]

By the way, the talk of relative velocity leads me to another question I've had. Are there any conditions in which we talk about the relative velocity of two observers when they are not colocated?

If we have two frames of reference (even comoving frames), I picture them as a sort of "field" in which every point of one frame has the same relative velocity with respect to the the colocated point of the other frame. More precisely, I might say that if I have two colocated objects, each at rest with one of the two frames, their relative velocity will be the same. But, given the same set of objects, can I say anything about their relative velocities if they are not colocated?

In novice problems, we are given examples where two object are both moving inertially. Their relative velocity is generally presented as an invariant independent of their location.

If you have a problem in which one observer is accelerating, there is even a formula that tells me the relative velocity of that observer to one in the initial rest frame of the accelerating observer. I give the formula $t$ (with respect to the rest observer) and it gives me $v$, the relative velocity of the two observers.

If we have two accelerating observers, we could pick the comoving frame of one and determine the relative velocity. We have a problem, of course; if we reverse whose comoving frame we use, we don't get the same lines of simultaneity, so the relative velocity could be different.

In all three cases, actually, the lines of simultaneity for the two objects are never the same unless they are at rest with respect to each other. But because at least one object is moving inertially, we get the same relative speed regardless of how we angle each object's line of simultaneity.

When I talk about the relative velocities of two non-colocated objects, am I always dealing with what @PeterDonis calls "an artifact of the choice of coordinates" or am I ever dealing with invariants? It seems to me that the answer should all be one way or the other.

 

[Dale]

Are there any conditions in which we talk about the relative velocity of two observers when they are not colocated?

As long as spacetime is flat, that is fine. If spacetime is curved then there is no unique way to compare the relative velocity of spatially separated worldlines.

 

[robphy]

I hate to bring up my classification scheme, 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. My approach may be clunky because I don't have the advanced geometrical knowledge of the people in this group, but it let's me limp along.

As I suggested earlier,
it might be best to start with Euclidean geometry because the notions of invariance are already there and are likely more familiar.
Then, proceed to special relativity which adds additional likely non-intuitive notions to the situation.

Draw any diagram ( with circles, lines, points, angles,etc…) in the Euclidean style… lay down a grid (Cartesian style).

Describe features of that diagram with your classification scheme.
(A concrete labeled diagram might help.)
Once you do that, it will be easier to consider the special relativity version of the situation.

(This is a strength of the geometric analogy. Many times issues raised about special relativity are also featured in Euclidean geometry and Galilean relativity. However, it seems that special relativity is the one of three that has to defend itself.)

 

[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.