I What, exactly, are invariants?

[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, Schrodinger, 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!

 

[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.

 

[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".