What, exactly, are invariants?

[Dale]

(*) for example, what does "orthogonal" fundamentally mean? Does it really mean "90-degrees"?

This is a good example of a concept that gets generalized. Well before I started doing relativity I had to generalize the notion of "orthogonal" when I learned the Fourier transform and found out that cos functions formed an orthogonal basis for functions. There doesn't even exist any word to distinguish this notion of orthogonality from the less general notion of orthogonality as a 90 degree angle. But nevertheless the presentation of this new concept of orthogonality was sufficiently clear that I didn't get confused.

 

[DrGreg]

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.

There are some ways of introducing relativistic concepts without the use of coordinates, but these don't seem to be widely used. Bondi k-calculus is one method where the introduction of coordinates can be delayed (you can work with proper time only, to start with), and (as far as I can remember) Geroch's General Relativity from A to B also emphasises geometry and de-emphasises coordinates.

You do need to decompose spacetime into space and time in order to compare relativity with Newtonian physics, but the decomposition can be described with geometrical language instead of coordinates. (The key point, of course, is that there are multiple ways to decompose.)

 

[vanhees71]

(bolding mine)

I'm not sure what counts as "overemphasizing".

(Regarding tensors: I think Einstein pursued tensor-calculus because the classical pseudo-Euclidean geometries (hyperbolic and elliptic space) are too simple [being constant-curvature spaces] to describe the observable universe. He needs more general spaces that are better handled in the style of tensor calculus, rather than classical nonEuclidean geometry. For introducing special relativity, we should delay tensor methods for the student... especially if they haven't done Euclidean geometry with tensors.) ​

I mean Minkowski's analysis first enabled the formulation of the theory using tensors in the pseudo-Euclidean affine manifold, which indeed describes the special-relativistic spacetime model most adequately, and the great benefit is the "manifestly covariant formalism", which enables us to find physical laws that are compatible with this spacetime model. For me the main feature is also "geometry", but not in the sense that it is in many respects analogous to Euclidean affine space but that it provides the Poincare group as its symmetry group (i.e., $\text{ISO}(1,3)^{\uparrow}$ as the Lie group of the symmetry-transformation group that is connected smoothly with the identity). That is of course also analogous to the Euclidean case (the closest analogy would be $\text{SO}(4)$).

It is nevertheless very important stress the very important difference of the Minkowski product with signature (1,3) or (3,1), because that difference enables to establish a causality structure making this specific affine spaces apt to define a spacetime model. That's why I consider it so important to emphasize the differences between Minkowski space and Euclidean space, particularly when you draw a Minkowski diagram on a sheet of paper, which we are trained from elementary school on to interpret in the sense of a Euclidean affine plane and then easily misread the Minkowski diagram, which has to be intepreted strictly as a Lorentzian affine plane.

The step from SR to GR is then not that difficult anymore: It just makes the notion of Poincare symmetry local and in this way realized the strong equivalence principle to describe the gravitational interaction. That's why Einstein got a Lorentzian (i.e., pseudo-Riemannian) spacetime model, and the dynamics of the gravitational field implied the dynamical nature of spacetime.

To me, the geometrical interpretation is an attempt
to equip the student with tools for computation and reasoning.

(Why it works is not clear... but it seems to work. ​

The geometrical structures seem to encode a lot of the physics... ​

and many of the geometric constructions and calculations lead to consistent results ​

that agree with experiment. And yes we can of course tensorialize these constructions.)​

As with any tool,
we show the student "how to use it" and (to avoid overemphasis) "how not to use it".

I never avoid four-vectors and -tensors. To the contrary, I introduce them as soon as possible, even before I draw the first Minkowski diagram. Many introductory textbooks avoid four-vectors in the beginning and make SR more complicated than necessary.

Why this works is very clear then: It works for the same reason, why in Newtonian physics you work with Euclidean affine 3D space and an additional "time line" (a fibre bundle) and that's why for Newtonian physics the corresponding Euclidean tensors and tensor fields (in QM in addition of course also spinors) are the natural way to formulate the physical laws: It's because they lead to laws that are compatible with the Galilei-Newton space-time model and realize the Galilei group as its symmetry group.

In the same sense it is natural to work with four-vectors, -tensors, and -fields to describe the physical laws within special and general relativity and in this way realize the (global or local) Poincare symmetry of relativistic physics.

It's amazing, how much the choice of a spacetime model determines how the physical laws look but are at the same time flexible enough to describe a huge realm of phenomena.

It might be helpful to give a "storyline from some set of first principles"(*)
showing HOW these structures arise
and not just sort-of jump in the middle pointing out
analogies and non-analogies here and there.

(*) for example, what does "orthogonal" fundamentally mean?​

Does it really mean "90-degrees"?​

How do we construct the observer's x-axis (her "spaceline") given her t-axis (her "timeline"),​

starting from a "circle" (which is what defines orthogonality)?​

Is relativity really "so unlike anything we have ever seen"​

or can it be responsibly shown to be "similar" and maybe "hauntingly familiar" to something we've already seen?​

​

Exactly. That's why for SR I start with Einstein's two postulates and introduce the four-vector and -tensor formalism with the Lorentzian fundamental form first. Then, specializing one-selves to two-dynamical point-particle motion, I introduce the Minkowski diagrams in a time-space plane, and then you don't come to the idea to misinterpret a Minkowski diagram by drawing any Euclidean circles in it but the hyperbolae (space-like and light-like as well as their "degenerate" case of the lightlike light-cone) to construct the correct "unit tics" on the axes of different inertial frames. You also don't introduce "angles" but "rapidities" in this plane (of course there's a great analogy between the Euclidean "angles" and the Minkowskian "rapidities", but they are clearly not the same).

 

[Freixas]

It's certainly been fascinating listening in on the opinions of how best to introduce people to relativity. It's pretty clear that I'm missing a lot by not knowing more about 4-vectors, tensors, and a number of other things.

For what it's worth, I once trained to be a cross-country ski instructor. The lesson that stuck with me most was that some students learned best if you demonstrated a technique, some of you described it, and some if you guided their movements. We're all different and we learn in different ways.

Personally, I'm visual and I prefer approaches that I can picture as I going for a daily walk. Until I can understand something visually, concepts don't really sink in. Take the spacetime interval. It should be an easy one to grasp using the analogy to a spatial interval but I've yet to figure out what it's good for. I could wander through a lot of explanations; someday, someone might use just the right words that will create my aha! moment.

Speaking of understanding, language is imprecise. The word "orthogonal" apparently has a meaning other than 90°; used without qualification, it can confuse rather than enlighten. There are some even more basic words that are unclear. For some explanations I get in this forum, I think I understand what was said, only to figure out later that I didn't understand at all.

For another example of language confusion, let's take "proper length". The word length can be a synonym for "distance" or it can be considered a property of an object. Wikipedia uses the latter sense in a common definition: "Proper length or rest length is the length of an object in the object's rest frame."

If an object is under acceleration does it still have a rest frame (and thus a proper length)? If the endpoints are accelerating uniformly, then its "proper length" is not invariant. If we insist on maintaining an invariant "proper length", then clocks at the endpoints of the length must be moving at different velocities, and so cannot be in the same rest frame. It looks like the object must stop accelerating for the definition to apply (but oddly, if you were on, say, an accelerating spaceship, it seems you could use Einstein's technique of laying down rods to measure its "proper length").

"Proper distance" seems clearer. We're no longer talking about an object, but about the separation of two events, and we can apply a more precise definition.

"Proper time" is confusing since "time" can be used in the sense of "what time is it?" (a single value) and also a duration (an interval formed from two time values). When Wikipedia says "Proper distance is analogous to proper time," I believe they are using time in the latter sense.

Would it be clearer to talk about "proper duration" and "proper distance"? But "time" and "length" are what has been historically used, and those who understand what these very simple words mean mean might not realize they could be misinterpreted.

I'm not actually asking for answers to any questions posed here (although I suspect I will hear some ). I'm just musing on the problems with learning (and teaching) relativity. Their are a lot of barriers and no one approach will work for everyone. It sounds like you've all taken unique paths.

 

[PeterDonis]

If an object is under acceleration does it still have a rest frame (and thus a proper length)?

If it is under the right kind of acceleration (the technical name for which is "Born rigid acceleration"), yes.

If the endpoints are accelerating uniformly, then its "proper length" is not invariant.

First, do you mean "invariant" or "constant"? An object's proper length can be invariant--the same in all frames--but still change as we move along the object's worldline. I'm going to assume you actually meant "constant", which means, not only that we can identify an appropriate invariant (which will turn out to be arc length along a particular spacelike curve), but that this invariant remains the same as we move along the object's worldline (so there is actually a whole series of spacelike curves, each describing the proper length of the object at successive instants of the object's proper time).

By "accelerating uniformly", do you mean both endpoints have the same proper acceleration (i.e., the same reading on an accelerometer)? If so, then your statement is true, but I don't think it's true for the reason you think it is. (The reason it is true, in relativity, is called the "Bell spaceship paradox", which has prior PF posts and an Insights article that you can look up.)

However, "accelerating uniformly" is often used to refer to a different acceleration profile, the one I referred to above (i.e., "Born rigid"), in which the front of the object has a smaller proper acceleration than the back, and the object's proper length does stay the same.

If we insist on maintaining an invariant "proper length", then clocks at the endpoints of the length must be moving at different velocities

Different velocities in what frame?

and so cannot be in the same rest frame

Yes, they can, in two senses:

(1) The momentarily comoving inertial frame of the object at some instant of its proper time;

(2) The non-inertial "rest frame" we can construct if the object is undergoing Born rigid acceleration. (This frame is called "Rindler coordinates" for the case of linear acceleration in flat spacetime.)

 

[robphy]

I'm not actually asking for answers to any questions posed here (although I suspect I will hear some ). I'm just musing on the problems with learning (and teaching) relativity. Their are a lot of barriers and no one approach will work for everyone. It sounds like you've all taken unique paths.

Not to disappoint...
yes, we all have taken different worldlines.

For what it's worth, I once trained to be a cross-country ski instructor. The lesson that stuck with me most was that some students learned best if you demonstrated a technique, some of you described it, and some if you guided their movements. We're all different and we learn in different ways.

Yes, that's why it's good to be multi-modal.
Words, algebra, geometry, coordinates, tensors, analogies, limiting cases...

Personally, I'm visual and I prefer approaches that I can picture as I going for a daily walk. Until I can understand something visually, concepts don't really sink in. Take the spacetime interval. It should be an easy one to grasp using the analogy to a spatial interval but I've yet to figure out what it's good for. I could wander through a lot of explanations; someday, someone might use just the right words that will create my aha! moment.

The spacetime interval [and the squared-interval] between two events is the fundamental "invariant" in special relativity, just like the distance [or squared-distance] between two points is fundamental in Euclidean geometry.
While everyone decomposes a displacement vector in the plane into coordinate-dependent components,
they agree on the distance. By analogy, a similar thing is true for the spacetime interval.

• For timelike-related events in Minkowski spacetime, it's the "wristwatch time" (Minkowski's "proper time") along the inertial observer that meets both events.
• For nearby spacelike-related events, it could be the proper-length (the distance between two parallel inertial worldlines in the frame of those worldlines).
• For lightlike-related events, it's the indication that the events are lightlke-related.
• (In an energy-momentum diagram, the analogous quantity for a timelike or lightlike 4-momentum vector is the invariant-mass of the object.)

Speaking of understanding, language is imprecise. The word "orthogonal" apparently has a meaning other than 90°; used without qualification, it can confuse rather than enlighten. There are some even more basic words that are unclear. For some explanations I get in this forum, I think I understand what was said, only to figure out later that I didn't understand at all.

In any technical discussion, one has to learn the vocabulary and the definitions to fully participate in the discussion. For me, rather than just vague words, having a mathematical definition helps, particularly ones I can draw.

( "unionized" might be interpreted one way by many people, but a very different way by a chemist.)
We tell students that terms in physics have specific meanings, that are different from casual conversation.
(How many times have you heard a sportscaster use "force", "energy", "momentum", "power", etc... interchangeably? )

"Orthogonal" as "90-degrees" is an elementary characterization of perpendicular.
But mathematics is about finding structure among special cases...
it was decided that "orthogonal" could be more generally a statement that the dot-product or inner-product is zero, as @Dale suggested in #106 when talking about orthogonal polynomials. Geometrically, I also like the characterization of "being tangent to a radius vector" as I suggested in #77. "90-degrees" turns out to be a special case, which doesn't work in the general case.

"Proper time" is confusing since "time" can be used in the sense of "what time is it?" (a single value) and also a duration (an interval formed from two time values).

"proper time" is defined by Minkowski as "eigenzeit" to be one's own time.
Bondi uses "private time". Taylor&Wheeler use "wristwatch time" [my favorite].

To me, language is imperfect... so, I often prefer to write "proper-time" as if it were a new word.. a new noun, not to be interpreted as "an adjective with a noun".. but inseparable. (I might write wristwatch-time.)

The technical language (with hopefully algebraic and geometric definitions) is needed for clarity.
It's not meant to exclude people.
If something is unclear, one has to ask for the definition.
(One might question it or be puzzled by it... but it should be accepted as the working definition.)

 

[Dale]

That's why for SR I start with Einstein's two postulates and introduce the four-vector and -tensor formalism with the Lorentzian fundamental form first.

That works well because it is easy to connect the 2nd postulate to the fundamental form.

That's why I consider it so important to emphasize the differences between Minkowski space and Euclidean space, particularly when you draw a Minkowski diagram on a sheet of paper,

I just think you can do all of that without needing to say that angles in spacetime make no sense.

 

[Freixas]

By "accelerating uniformly", do you mean both endpoints have the same proper acceleration (i.e., the same reading on an accelerometer)?

That was one option. Under that option, the object's proper length increases (i.e. the string breaks).

Different velocities in what frame?

Hmm... I tried to model a case where the proper length of an accelerating object remained the same (using Gamma). Using the viewpoint of the initial rest frame, I assigned a constant acceleration to the rear endpoint. I then calculated the position of the front endpoint so that, adjusting for the length contraction given the speed of the rear endpoint, the length of the object was constant. The front endpoint does not move as far as the rear one, so it's velocity must be less, which implies that the front clock moves faster than the rear.

Given this, there might be an inertial frame relative to which the rear and front clocks are synchronized. Relative to that frame, the length of the object could not be equal to it's length at the start (the one instant in which it was at rest).

But maybe this was not the right way to model this. If the object at rest measured L, then maybe I should have used the comoving frame of the rear endpoint and then found the point L distance away and at the same time relative to that frame. I tried that and came up with this diagram:

Here, the starting length is 1. So I calculate the comoving frame at various points in time for the rear of the object (so that (0, 0) of the comoving frame is at that point), transform the point (1, 0) back to the rest frame, and draw a line connecting the two points.

From the point of view of the rear endpoint, the length is constant. Are the clocks synchronized?

If I connect the endpoints of the right sides of the lines, I get a rough idea of the motion of the front of the object. Just eyeballing this, the front appears to move slower than the rear (from the viewpoint of the rest frame), so the front clock moves faster and for longer, and so the clocks would not seem to be in synch along the lines of simultaneity.

Yes, they can, in two senses:

Well, I gave it two tries. A diagram might help in understand the two ways you mention. I can look up "Born Rigid".

Note that I am leaving on a trip tomorrow and so may disappear from this conversation for random periods of time over the next few weeks.

 

[Freixas]

For timelike-related events in Minkowski spacetime, it's the "wristwatch time" (Minkowski's "proper time") along the inertial observer that meets both events.

This bullet point "clicks". The other points I'll have to think about.

In any technical discussion, one has to learn the vocabulary

Your right, of course. Since I'm not learning any of this formally, I sometimes think I understand what was said. And one can note a formal definition without it always sinking in (until much later).

Thanks for your help. Note that I am leaving on a trip tomorrow and so may disappear from this conversation for random periods of time over the next few weeks.

 

[Freixas]

"proper time" is defined by Minkowski as "eigenzeit" to be one's own time.
Bondi uses "private time". Taylor&Wheeler use "wristwatch time" [my favorite].

I meant to comment on this. "Wristwatch time" for me means that I look at my watch and it displays a value. But if Wikipedia is to trusted ("Proper distance is analogous to proper time"), "proper time" is a duration, an interval between two wristwatch numbers (in the same way that distance is measured between two spatial coordinates).

So which is it (or is it both)?

 

[PeterDonis]

I tried to model a case where the proper length of an accelerating object remained the same

That's simple. Assume the object is accelerating in the positive $x$ direction. The front and rear of the object, in any inertial frame, will have worldlines that are concentric hyperbolas, of the form $x^2 - t^2 = x_r^2$ (for the rear) and $x^2 - t^2 = x_f^2$ (for the front), where $x_r < x_f$. The proper accelerations of the two ends will be $c^2 / x_r$ (for the rear) and $c^2 / x_f$ (for the front). And the "surfaces of constant time" for the object--the surfaces in which the line segments of constant proper length lie--are simply straight lines through the origin of the inertial frame, with gradually increasing slopes, i.e., lines of the form $t = k x$, where $0 \le k \lt 1$ (the $k = 0$ line is just the line $t = 0$, which we can take as the instant at which the acceleration begins).

I strongly suggest taking the time to draw a spacetime diagram of the above. Then, for extra points, look up Rindler coordinates and show how the hyperbolas are "grid lines" of the Rindler time coordinate, while the "surfaces of constant time" I described above are "grid lines" of the Rindler space coordinate.

 

[vanhees71]

That works well because it is easy to connect the 2nd postulate to the fundamental form.

I just think you can do all of that without needing to say that angles in spacetime make no sense.

I simply don't introduce angles in an (1+1)D Minkowski diagram. Only rapidities make sense and they have a geometrical meaning as the areas in connection with hyperbolas as discussed above although this area visualization doesn't help much, at least for me. The areas make more sense when working with light-cone coordinates aka @robphy 's "rotated graph paper".

 

[Dale]

Only rapidities make sense

And rapidities are generalized angles.

 

[vanhees71]

Sigh, yes, I don't deny that, but why must one draw this analogy which confuses beginners of the subject?

 

[Dale]

why must one draw this analogy which confuses beginners of the subject?

I think you are right that it is not necessary. I don’t object to that.

Not everything that makes sense is necessary or even advisable.

 

[Freixas]

That's simple. Assume the object is accelerating in the positive x direction.

One question first.

In Gamma, I plot constant acceleration using a formula from https://math.ucr.edu/home/baez/physics/Relativity/SR/Rocket/rocket.html. Assuming $c = 1$, then $t = \sqrt{x^2 + 2x / a}$. Is this the same as proper acceleration?

You gave the proper acceleration for the rear as $1/x_r$. If I substitute this for $a$, I get $x^2 - t^2 = -2xx_r$, instead of $x^2 - t^2 = x_r^2$. I'm not sure how to relate the formulas.

 

[PeterDonis]

Is this the same as proper acceleration?

The $a$ on the web page you linked to is the proper acceleration of the rocket, yes.

I'm not sure how to relate the formulas.

Note that on the web page you linked to, it says $d$, not $x$. $d$ is the distance traveled from the starting point (in the inertial rest frame of the starting point). For the two hyperbolas describing the front and rear of the object, the starting points are $x_r$ (for the rear) and $x_f$ (for the front), in the coordinates of the inertial rest frame of the starting point. So for those two hyperbolas, you would have $d = ( x - x_r )$ (for the rear) and $d = ( x - x_f )$ (for the front).

If you plug those two formulas for $d$ into the equation from the web page, you should get the same formulas for $t$ that you get if you take the two equations for the hyperbolas that I gave you and solve them for $t$ (i.e., rearrange them algebraically to give you a formula for $t$).

 

[Freixas]

If you plug those two formulas for into the equation from the web page, you should get the same formulas for that you get if you take the two equations for the hyperbolas that I gave you and solve them for (i.e., rearrange them algebraically to give you a formula for ).

Thanks. I'll get back to you on the diagram. I've drawn it, but want to think about this a bit more before posting or commenting further.

 

[robphy]

I simply don't introduce angles in an (1+1)D Minkowski diagram. Only rapidities make sense and they have a geometrical meaning as the areas in connection with hyperbolas as discussed above although this area visualization doesn't help much, at least for me. The areas make more sense when working with light-cone coordinates aka @robphy 's "rotated graph paper".

Only rapidities make sense

And rapidities are generalized angles.

Sigh, yes, I don't deny that, but why must one draw this analogy which confuses beginners of the subject?

Do you introduce hyperbolic-trigonometric methods to find components of [4-]vectors,
akin to using [circular-]trigonometric methods with forces on a free-body diagram?
(I think few beginning students will use a dot-product or [Pythagorean] invariant methods.)

I think $E=\frac{ADJ}{HYP}=m\cosh\theta$ and $p=\frac{OPP}{HYP}=m\sinh\theta$ [ and $v=\frac{OPP}{ADJ}$]
is more intuitive and more familiar
than $E=\gamma m$ and $p=\gamma mv$,
not to mention $E=\frac{m}{\sqrt{1-v^2}}$ and $p=\frac{mv}{\sqrt{1-v^2}}$.

(Many collision problems in energy-momentum space
can be solved using methods similar to those
used in statics [equilibrium] problems on a free-body diagram.)

 

[vanhees71]

I introduce of course hyperbolic functions and rapidities, but I don't call them "trigonometric". This name is everywhere in the literature reserved for the functions sin, cos, tan, etc. Of course, I introduce Minkowski space with its indefinite fundamental form first, before drawing any diagrams.

 

[robphy]

I introduce of course hyperbolic functions and rapidities, but I don't call them "trigonometric". This name is everywhere in the literature reserved for the functions sin, cos, tan, etc. Of course, I introduce Minkowski space with its indefinite fundamental form first, before drawing any diagrams.

Yes, I know you introduce the hyperbolic functions.
But do use them (like the circular-trigonometric functions) to solve problems involving triangles,
e.g. finding components of vectors in the given frame or in another frame.
Do you use the terms opposite, adjacent, and hypotenuse?

 

[vanhees71]

Of course not. I use vector algebra to solve kinematical problems. The greatest progress in the understanding of geometry since Euclid was the introduction of analytical methods by Descartes et al.

 

[robphy]

Of course not. I use vector algebra to solve kinematical problems. The greatest progress in the understanding of geometry since Euclid was the introduction of analytical methods by Descartes et al.

So,

• in a first introduction to free-body diagrams [for introductory students],
  do you use vector-algebra and dot-products to solve (say) an inclined-plane problem?
• in a first introduction to relativity [for introductory students],
  do you use vector-algebra and dot-products to solve (say) a time-dilation problem?

 

[vanhees71]

For the inclined plane I use sometimes a free-body diagram, and there of course Euclidean geometry and language is adequate. For me the liberation from free-body diagrams by using generalized coordinates and Hamilton's principle was a great relief ;-)).

For the kinematical effects in SR I use the Lorentz transformation, which I introduce from the two postulates, motivating the Minkowski product (or rather the corresponding quadratic form). Only then I introduce Minkowski diagram for illustrative purposes making clear from the first moment on that there are no Euclidean notions left on the corresponding "paper plane".

 

[robphy]

I meant to comment on this. "Wristwatch time" for me means that I look at my watch and it displays a value. But if Wikipedia is to trusted ("Proper distance is analogous to proper time"), "proper time" is a duration, an interval between two wristwatch numbers (in the same way that distance is measured between two spatial coordinates).

So which is it (or is it both)?

An important feature of elapsed “proper time” (“wristwatch time”) is that it is a function of the timelike-path between the endpoint-events.
So, the difference of two readings of the wristwatch worn by the astronaut measures the elapsed proper time for that astronaut.

(To me, “Interval” suggests “magnitude of a displacement”, which is associated with a straight path. So I avoid it in this context.)

“Proper distance” as an analogue of proper time is of limited value for physics. While proper time can measured by a wristwatch along a timelike curve (the worldline of the wristwatch), a general spacelike curve doesn’t have much physical meaning, e.g, those spacelike curves that are not on a constant time slice associated with an observer.

We have examples of spacelike curves that are not achronal…. so there could be a causal curve joining two events on a “curve with an everywhere-spacelike tangent-vector.”

However “proper time along a worldline with time measured by a wristwatch” is a good analogy to “distance along a path in Euclidean space with distance measured by an odometer “.

 

[Freixas]

I strongly suggest taking the time to draw a spacetime diagram of the above.

It turns out I had already done this in #113. However, I combined the two approaches and here is the result:

First, I drew a red and green line using your formulas, with $x_r = 1$ and $x_f =2$, stepping through rest time from 0 to 10 in steps of 0.1. Here is the Gamma code:

xr = 1; xf = 2; lastR = (xr, 0); lastF = (xf, 0); for t = 0 to 10 step .1 { r = (sqrt(xr^2 + t^2), t); f = (sqrt(xf^2 + t^2), t); path [path lastR, r], style: "color: red"; path [path lastF, f], style: "color: green"; lastR = r; lastF = f; }

Then I created observers whose acceleration was 1/1 and 1/2. I drew the former in blue, but didn't draw the latter. The blue line overlays the red, as I would expect.

I located events on the rear worldline at intervals of 0.1 rest time. I found the comoving frame for this event and transformed the point (1, 0) in the comoving frame back to the rest frame. I connected the event to the transformed point. The right endpoint of this line falls on the green line generated using your formulas.

I then used the formulas on the rocket page to calculate tau for each worldline at each of the lines connecting the two curves. I had formerly eyeballed that the front clock would have more time than the rear clock and this seems to be the case.

The lines represent the lines of simultaneity for the rear worldline, so to an observer at the rear endpoint, everything on the line has the same time value. But the observer on the front endpoint would not agree. Here's what I originally said:

If an object is under acceleration does it still have a rest frame (and thus a proper length)? If the endpoints are accelerating uniformly, then its "proper length" is not invariant. If we insist on maintaining an invariant "proper length", then clocks at the endpoints of the length must be moving at different velocities, and so cannot be in the same rest frame.

It looks like the phrase "cannot be in the same rest frame" is incorrect. I should have said something like "the two ends of the object will not agree on a common rest frame (except at the start)."

Would this mean that if someone at the front of an accelerating object shined a light at someone at the rear, the light would be blue-shifted?

 

[PeterDonis]

The lines represent the lines of simultaneity for the rear worldline, so to an observer at the rear endpoint, everything on the line has the same time value. But the observer on the front endpoint would not agree.

Careful. The lines of simultaneity are lines of simultaneity for both observers--for both observers, everywhere on the line does have the same time value. But the elapsed proper time between two particular lines of simultaneity is not the same for both observers; the front observer has more elapsed proper time than the rear.

In other words, for any given line of simultaneity, you can construct an inertial frame for which that line is the $x'$ axis (this frame is what I referred to as the momentarily comoving inertial frame), and both observers will be at rest in that inertial frame at the events on their worldlines that intersect the line of simultaneity you chose. But you will have to adjust both observers' clocks by different amounts to make them (momentarily) synchronized in that inertial frame.

What this means is that if you try to construct a single non inertial frame that includes both observers (which is what Rindler coordinates is), the metric coefficient $g_{00}$ in this frame will depend on spatial position; it will not be constant.

I should have said something like "the two ends of the object will not agree on a common rest frame (except at the start)."

No, that's not correct either. See above.

Another invariant way of showing that there is a "common rest frame" for both observers is to look at round-trip light travel times. If a light signal bounces back and forth between the two observers, each observer will measure its round-trip travel time to be constant, indicating that the distance between the two observers is not changing. The actual time they measure will be different for the two observers (the rear observer will measure a smaller round-trip travel time than the front), but for both observers, the time will not change.

Would this mean that if someone at the front of an accelerating object shined a light at someone at the rear, the light would be blue-shifted?

Yes.

 

[Freixas]

Careful. The lines of simultaneity are lines of simultaneity for both observers--for both observers, everywhere on the line does have the same time value.

Ok, I partially understand this, although I can see the gaps in my knowledge.

Let's select three points: $(0, 0)$, $(x_r, 0)$, $(x_f, 0)$. We'll say we are viewing these points in an instantaneous rest frame: a comoving frame. If we boost these points to view them relative to any other frame, $(0, 0)$ stays where it is and the other two trace hyperbolic paths. Lines connecting the points remain colinear regardless of the boost.

The boosted views are just different views of the original setup. The lines connecting the points represent proper lengths as well as common lines of simultaneity.

But that's boosting. If we create worldlines that match the boosted hyperbolic paths, we
can pick points that can be "reverse boosted" back to the original setup. So any worldlines set up this way will have points that maintain proper length and simultaneity.

Boosting the $(x_r, 0)$ and $(x_f, 0)$ points and connecting them with a line yields the same line as finding the comoving frame of the hyperbolic worldlines (from either end). To find the line for the comoving frame, I pick an event on the worldline, find the tangent, and then find the line mirroring the tangent around a 45 degree line that goes through that event. Why this is the same line as we get by boosting the original two points (a Lorentz transformation of each point using a common relative velocity) requires some mathematical magic than I don't know. For the moment, I can accept that these are equivalent methods.

Another missing gap is why creating worldlines that match boosted points results in proper acceleration (that it results in acceleration is not a mystery).

Then there's the mystery of how two observers can maintain simultaneity with each other at every instant yet have different elapsed time. If observers move inertially and have different clock rates, they also have different lines of simultaneity.

Because "simultaneity" is a little woo-woo in the sense that the selection of a line of simultaneity is arbitrary, I thought I would look for things that could be directly observed, which is why I mentioned the blue shift.

So here are some of the other freaky things that I think happen in an accelerating ship that maintains its proper length:

• An observer looking toward the front sees a blue shift, indicating relative motion (in the past) toward the observer. Since the blue shift increases with time, the observer could conclude that the front of the ship is approaching at ever-greater velocity.
• Looking toward the rear, the effects are reversed: there is an increasing red shift, showing that the rear of the ship in receding at an ever-greater rate.
• On the other hand, you mentioned that the round-trip travel time of light would be constant for observers at each end, so they would each conclude that the other end of the ship remained a constant distance away. I gather that the frequency of the reflected light would be unchanged.
• If an observer at the rear of the ship pulls out a tape measure and begins to lay it down while walking toward the front, the length reading on the tape will be the same as when the ship was at rest and will not change at any time during acceleration. Any other equivalent method will work, too: the observer could count steps moving from one end to the other and compare the count to one obtained moving in the opposite direction.

To repeat, I'm not doing anything other than highlighting the gaps in my understanding. I understand some bits, but not others.

 

[PeterDonis]

Let's select three points: $(0, 0)$, $(x_r, 0)$, $(x_f, 0)$. We'll say we are viewing these points in an instantaneous rest frame: a comoving frame

Yes, that's obvious from the coordinates you give.

If we boost these points to view them relative to any other frame, $(0, 0)$ stays where it is and the other two trace hyperbolic paths.

Yes, that's one way of looking at it. In somewhat more technical language, the hyperbolas are integral curves of the boost Killing vector field.

Lines connecting the points remain colinear regardless of the boost.

I'm not sure what you mean here. A single line is always collinear with itself. Perhaps you mean to say that the three points always lie along a single line regardless of the boost.

The boosted views are just different views of the original setup.

One can view them that way. But one can also view the hyperbolas as the worldlines of two accelerating observers who remain at rest relative to each other, with the proper distance between them remaining constant, and the boost moves the two observers along their respective worldlines, i.e., it represents their time evolution. (These two different ways of viewing the action of a Lorentz boost are sometimes referred to as "passive" vs. "active" transformations in the literature.)

The lines connecting the points represent proper lengths as well as common lines of simultaneity.

More precisely, the arc lengths along the lines between the points are the proper lengths between them; the lines themselves, as a whole, are common lines of simultaneity.

If we create worldlines that match the boosted hyperbolic paths, we
can pick points that can be "reverse boosted" back to the original setup. So any worldlines set up this way will have points that maintain proper length and simultaneity.

See above regarding "passive" vs. "active" transformations.

Boosting the $(x_r, 0)$ and $(x_f, 0)$ points and connecting them with a line yields the same line as finding the comoving frame of the hyperbolic worldlines (from either end).

Yes.

To find the line for the comoving frame, I pick an event on the worldline, find the tangent, and then find the line mirroring the tangent around a 45 degree line that goes through that event.

Yes.

Why this is the same line as we get by boosting the original two points (a Lorentz transformation of each point using a common relative velocity) requires some mathematical magic than I don't know.

It has to do with the hyperbolas being integral curves of the boost Killing vector field.

Another missing gap is why creating worldlines that match boosted points results in proper acceleration (that it results in acceleration is not a mystery).

Proper acceleration is the same thing, geometrically, as path curvature. Since we are working in flat Minkowski spacetime, any curve that is not a straight line in an inertial frame has nonzero path curvature. So in this special case, proper acceleration and coordinate acceleration are in perfect correspondence.

Then there's the mystery of how two observers can maintain simultaneity with each other at every instant yet have different elapsed time.

This isn't a mystery. It's a simple consequence of spacetime geometry and the geometry of the hyperbolas.

If observers move inertially and have different clock rates, they also have different lines of simultaneity.

Obviously this is false since we have just gone to considerable lengths to describe a counterexample, so why are you asserting it?

Because "simultaneity" is a little woo-woo in the sense that the selection of a line of simultaneity is arbitrary

In general, yes, but you have described a particular method of selecting lines of simultaneity: pick lines that are orthogonal (in the Minkowski sense) to the hyperbolic worldlines at every point. This is a common method of defining lines of simultaneity, and it often picks out a set of lines of simultaneity with useful properties.

• An observer looking toward the front sees a blue shift

Yes.

• indicating relative motion (in the past) toward the observer.

From the viewpoint of an inertial frame, yes. But not from the viewpoint of the non-inertial frame (Rindler coordinates). From the viewpoint of that frame, the blueshift is analogous to "gravitational" blueshift. (In fact, Einstein's derivation of this result was one of his early results from the equivalence principle.)

• the blue shift increases with time

No, it doesn't. The blueshift is constant. So is the redshift observed by the front observer, with respect to light signals from the rear observer.

• you mentioned that the round-trip travel time of light would be constant for observers at each end,

Yes.

• so they would each conclude that the other end of the ship remained a constant distance away. I gather that the frequency of the reflected light would be unchanged.

Yes.

• If an observer at the rear of the ship pulls out a tape measure and begins to lay it down while walking toward the front, the length reading on the tape will be the same as when the ship was at rest and will not change at any time during acceleration.

Assuming that the walking is done slowly enough, yes.

 

[Freixas]

I'm not sure what you mean here. A single line is always collinear with itself. Perhaps you mean to say that the three points always lie along a single line regardless of the boost.

Yes.

(These two different ways of viewing the action of a Lorentz boost are sometimes referred to as "passive" vs. "active" transformations in the literature.)

Interesting.

This isn't a mystery. It's a simple consequence of spacetime geometry and the geometry of the hyperbolas.

It's a mystery to me. I've verified that this is true with Gamma. But that's different from understanding it.

If observers move inertially and have different clock rates, they also have different lines of simultaneity.

Obviously this is false since we have just gone to considerable lengths to describe a counterexample, so why are you asserting it?

Hmm... I thought we had been talking about observers who were accelerating. Observers moving at constant velocity appear (to me) to have different lines of simultaneity except when they are moving at the same velocity.

From the viewpoint of an inertial frame, yes. But not from the viewpoint of the non-inertial frame (Rindler coordinates).

Interesting.

No, it doesn't. The blueshift is constant. So is the redshift observed by the front observer, with respect to light signals from the rear observer.

Ok, I think I see my mistake. I thought that since the starting velocity was 0, there would be no red/blue shift at the start so that it must increase with time. But light doesn't travel instantly, so by the time the light from an endpoint at the start of the worldline reaches the other end, the receiving end is moving. My initial assumption was wrong.

Assuming that the walking is done slowly enough, yes.

Clearly, there are problems that are way beyond my skills, but you just inspired another one. Let's simplify by stating that a ship is moving inertially (I don't think it matters). Assume a measuring tape is anchored to one end of a ship and wound on a spool. Someone picks up the spool and moves at a high relative velocity toward the other end.

The tape that is unwound has a relative speed of 0 and so it has no length contraction relative to the ship. The tape on the unwinding spool has a high relative velocity with small fluctuations, and so is length contracted. At the point where it unwinds onto the ship, it would experience a rapid change of velocity and so its length would rapidly expand. Would this cause a problem to the person unwinding the spool? I don't know.

In picturing the worldlines of points of the tape and of the person carrying it, I don't see a problem, but since you required walking slowly, there must be.

Thanks for all the cross-checking and the explanations!

 

[Freixas]

An important feature of elapsed “proper time” (“wristwatch time”) is that it is a function of the timelike-path between the endpoint-events. So, the difference of two readings of the wristwatch worn by the astronaut measures the elapsed proper time for that astronaut.

Thanks for clarifying. The point remains that the term "wristwatch time" is ambiguous. When you already know what it means, the ambiguity might not be obvious. No one uses the term "proper duration", but it seems unambiguous.

People might seem to understand "proper time", because problems often have a context that implies that one is to calculate a duration. To date, I've thought it could be used in both senses--the time at some instant and a duration.

 

[robphy]

Thanks for clarifying. The point remains that the term "wristwatch time" is ambiguous. When you already know what it means, the ambiguity might not be obvious. No one uses the term "proper duration", but it seems unambiguous.

I carry a ruler. The "measurement I read from it" is my measurement of a length.

I carry a wristwatch. The "measurement I read from it" (between two readings of it) is my measurement of the elapsed time that I and my wristwatch experienced... this is my elapsed proper time
or more-descriptively "wristwatch time"
(and everyone will agree that this is the elapsed time that I and my wristwatch experienced).

Practically any technical term is ambiguous... that's why it's a technical term.

I think part of the problem is that the word "proper" is what is ambiguous.
https://en.wikipedia.org/wiki/Proper_time is an invariant.
https://en.wikipedia.org/wiki/Proper_acceleration is an invariant
https://en.wikipedia.org/wiki/Proper_velocity is not invariant, it is relative to an observer.
It might be difficult for a novice to use "proper" properly.

 

[Freixas]

I carry a wristwatch. The "measurement I read from it" (between two readings of it) is my measurement of the elapsed time that I and my wristwatch experienced... this is my elapsed proper time (and everyone will agree that this is the elapsed time that I and my wristwatch experienced).

Yes, I understand what you mean. But I carry a wristwatch. Right now, it reads 7:55 AM. Unless I have a stopwatch mode and activate it, my wristwatch doesn't tell me elapsed time. When Taylor/Wheeler introduced and then used the term "wristwatch time" in their book, I thought they meant that most natural interpretation, not the interpretation that required two readings and a calculation, so some of what they wrote just didn't make much sense to me.

They adopted the term thinking it made things easier for beginners to to understand. I found it confusing. "Proper time" is only a little better, but it's the most commonly used term and so it's required learning. "Proper duration" is crystal clear and is the term I would prefer, but physicists aren't going to adopt my terminology. If you really like wristwatches, "wristwatch duration" would work. "Stopwatch time" might also work.

 

[PeterDonis]

I thought we had been talking about observers who were accelerating. Observers moving at constant velocity

Ah, sorry, I missed that you switched to talking about inertial observers. Yes, if you use the "orthogonal" method to define lines of simultaneity, inertial observers in relative motion will have different lines of simultaneity (they will not be parallel).

 

[PeterDonis]

It's a mystery to me.

As I said, it's a simple consequence of spacetime geometry and the geometry of the hyperbolas. The hyperbolas have the same asymptotes (the lines $t= x$ and $t = -x$ in an inertial frame), so they are "concentric" in a sense similar to concentric circles in Euclidean space, and the lines of simultaneity are "radial lines" from the "center" (the origin of the inertial frame) in a sense similar to radial lines in Euclidean space. Concentric circles in Euclidean space will be orthogonal to the same set of radial lines; similarly, "concentric" hyperbolas in Minkowski spacetime will be orthogonal to the same set of "radial" lines--i.e., they will have the same set of lines of simultaneity.