A better way of talking about time?

[Yuras]

Mentors’ note : This thread was split off from https://www.physicsforums.com/threads/time-dilation-for-two-observers-in-relative-motion.1068098/

If two people walk away from each other, they both look smaller relative to the other. Each might say that eventually the other is only about as big as his or her own thumb.

Excellent analogy. Note though that we don't talk about "height contraction" in this case.

Time dilation as a concept is a pedagogical disaster. IMO it should have not been introduced in the first place. There is only one time - the proper one. It's too late probably, but on the other hand we somehow managed to (mostly) get rid of "relativistic mass" - another miss-invention of the past.

 

[Sagittarius A-Star]

There is only one time - the proper one.

And there is another kind of time - the coordinate time. I think, this is a pedagogically important part of SR.

on the other hand we somehow managed to (mostly) get rid of "relativistic mass" - another miss-invention of the past.

We have it still, it is only called energy, which is a coordinate dependent quantity.

 

[Yuras]

And there is another kind of time - the coordinate time. I think, this is a pedagogically important part of SR.

I understand why you are saying that, but... I came from the geometric camp. In my world coordinates are just labels, they can't be important.

We have it still, it is only called energy, which is a coordinate dependent quantity.

It's actually a 4-scalar :
$$E=g_{\alpha\beta}p^\alpha u^\beta$$
I mean, it obviously depends on the observer via $u$, but proper time does that as well.

 

[Sagittarius A-Star]

I understand why you are saying that, but... I came from the geometric camp. In my world coordinates are just labels, they can't be important.

It's actually a 4-scalar :
$$E=g_{\alpha\beta}p^\alpha u^\beta$$
I mean, it obviously depends on the observer via $u$, but proper time does that as well.

Proper time is invariant. The energy I mean (=relativistic mass), the time-component of the 4-momentum, is not invariant.

 

[Orodruin]

Excellent analogy. Note though that we don't talk about "height contraction" in this case.

That’s because it does not describe the Euclidean analogy of time dilation, which is two odometers moving from a common point in different directions. Each odometer also projects a line orthogonal to its motion at fixed distances from the start. Both odometers will note that its own 1 m line is crossed by the other odometer when that one shows a number greater than one.

What was described was simply that effects may be symmetrical between observers.

Time dilation as a concept is a pedagogical disaster. IMO it should have not been introduced in the first place. There is only one time - the proper one. It's too late probably, but on the other hand we somehow managed to (mostly) get rid of "relativistic mass" - another miss-invention of the past.

It has the very same issues as using Cartesian coordinates on Euclidean space. The analogies are abundant. The only reason people “get” Euclidean effects better is that they have more intuition in Euclidean geometry than in Minkowski geometry.

I understand why you are saying that, but... I came from the geometric camp. In my world coordinates are just labels, they can't be important.

The thing is that they are important for how people learn - for better or worse. This is as true in Euclidean space as in Minkowski space.

It's actually a 4-scalar :
$$E=g_{\alpha\beta}p^\alpha u^\beta$$
I mean, it obviously depends on the observer via $u$, but proper time does that as well.

I mean, yes, that is a scalar. But changing coordinates changes what observer is at rest in those coordinates and therefore which scalar you are referring too. This is somewhat unavoidable if you introduce coordinates. You might as well say that the coefficients of a vector relative to a basis are scalars (because in one sense they are - just contract with the dual basis), but that doesn’t mean that you will have exactly the same numerical coefficients in a different basis. It is definitely a way forward, but not how most people learn.

 

[Orodruin]

Proper time is invariant. The energy I mean (=relativistic mass), the time-component of the 4-momentum, is not invariant.

You are missing their point. Their point is that energy relative to an observer is an invariant quantity. Obviously it depends on the observer because the observer 4-velocity is in there.

 

[Sagittarius A-Star]

You are missing their point. Their point is that energy relative to an observer is an invariant quantity. Obviously it depends on the observer because the observer 4-velocity is in there.

Yes. The combined information about a frame-dependent quantity and the related reference frame is invariant. In the case of the proper time, the reference frame (rest frame of the object) is implicitly included.

 

[Orodruin]

Yes. The combined information about a frame-dependent quantity and the related reference frame is invariant. In the case of the proper time, the reference frame (rest frame of the object) is implicitly included.

There is no reference to any reference frame when it comes to proper time. Proper time is a geometrical quantity that refers to the path length along a time-like world line. Just as path length in Euclidean space is perfectly well defined as a line integral along the path. No reference to a particular coordinate system is needed.

However, just as in Euclidean space - some coordinates may make expressions easier than others.

 

[Sagittarius A-Star]

Proper time is a geometrical quantity that refers to the path length along a time-like world line.

Yes, you are right. In SR, proper time is the norm of a 4-vector and therefore invariant. Frame-dependent are the components of a 4-vector (if a reference frame is not included in the information).

 

[Yuras]

The analogies are abundant. The only reason people “get” Euclidean effects better is that they have more intuition in Euclidean geometry than in Minkowski geometry.

People think about space geometrically, while we explain them space-time in terms of coordinates. No wonder they come back and ask strange questions again and again.

The thing is that they are important for how people learn - for better or worse.

My point is that we should change how we teach people.

I'm actually a bit surprised to encounter an opposition here. Physics doesn't require coordinates in a sense that everything we measure at the end of the day is a proper 4-dimentional object. While time dilation is an effect of coordinates, so putting it into a center of the theory is... unphysical?
For any nontrivial calculation we need coordinates, so we can't avoid them. But they should be secondary. And IMO pop-sci should stop emphasizing things like time dilation, relative simultaneousness etc.

BTW we are hijacking the thread and probably confusing the OP.

 

[robphy]

People think about space geometrically, while we explain them space-time in terms of coordinates.

I’m a proponent of spacetime diagrams… since we often refer to the “spacetime geometry”. Sadly, I think Einstein thought Minkowski made it (to Einstein) unnecessarily complex… and so we have many physicists who regard the spacetime diagram as “too difficult”.

I think a spacetime diagram of the situation and its Euclidean analogue would help the OP.
For example, look at
https://www.physicsforums.com/threa...procity-of-time-dilation.1057792/post-6973393

 

[Orodruin]

My point is that we should change how we teach people.

I'm actually a bit surprised to encounter an opposition here. Physics doesn't require coordinates in a sense that everything we measure at the end of the day is a proper 4-dimentional object. While time dilation is an effect of coordinates, so putting it into a center of the theory is... unphysical?
For any nontrivial calculation we need coordinates, so we can't avoid them. But they should be secondary. And IMO pop-sci should stop emphasizing things like time dilation, relative simultaneousness etc.

You misread me. I do try to teach relativity without going too much into coordinates, time dilation, length contraction, etc. It is difficult for a reason though: People generally have a lot of baggage here before they come to an actual course where they are ready to be given the geometric treatment. Even in Euclidean geometry, people are taught to compute the length of lines using Pythagoras' theorem, which is essentially breaking the line down into its coordinate components. There is also an often deeply rooted sense of universal time which does not make things any easier.

I do cover time dilation, length contraction, etc towards the end of my SR course for one very simple reason: If I did not, then people would not unlearn what they already "learned" so they can learn what is actually going on. There is simply too much of these concepts in popular media and lower level courses that unless you cover it, students will still get it wrong (I have tried not covering it, it just results in students trying to apply it to the exam problems anyway and failing miserably).

BTW we are hijacking the thread and probably confusing the OP.

Feel free to open a new thread if you want to discuss it further.
[Mentors’ note: after exercising the mentorly superpowers that I acquired when I was bitten by a radioactive doodlebug as an adolescent, we are now in that different thread]

 

[Nugatory]

Time dilation as a concept is a pedagogical disaster. IMO it should have not been introduced in the first place. There is only one time - the proper one. It's too late probably, but on the other hand we somehow managed to (mostly) get rid of "relativistic mass" - another miss-invention of the past.
… I came from the geometric camp. In my world coordinates are just labels, they can't be important.

That is a defensible proposition (I’ve recommended Taylor and Wheeler’s “Spacetime Physics” so often that I think I deserve a cut of the sales) but I find that many people are unable to appreciate the power of the geometric approach until they have ground through the coordinate-based approach so familiar from high school classical mechanics.

The pedagogical disaster, at least as measured by observed confusion in various PF threads (such as the one from which this was split) is not teaching relativity of simultaneity alongside time dilation and length contraction and showing how the three together create a consistent whole.

 

[Yuras]

You misread me.

Sorry if that's the case. Though by opposition I mostly meant the initial response from @Sagittarius A-Star

People generally have a lot of baggage here before they come to an actual course where they are ready to be given the geometric treatment.

That's 100% true. But it's made by our own hands, isn't it? And it's very hard to fix at this point, that's why I'm calling it a pedagogical disaster.

I do cover time dilation, length contraction, etc towards the end of my SR course for one very simple reason: If I did not, then people would not unlearn what they already "learned" so they can learn what is actually going on.

Yes, it's a reasonable practical approach. The question is how would you teach SR if there would be no such baggage.

 

[Yuras]

Since we are in a separate thread now, I can actually answer the original question:

how is this possible when both observers can claim that it is the other's time that is passing more slowly

It's possible because there is nobody there to check their claims until they actually come into one point and compare their clocks locally.

But I'm afraid that would I post this answer, the OP would conclude that physicists don't agree even among themselves :)

 

[Orodruin]

That's 100% true. But it's made by our own hands, isn't it? And it's very hard to fix at this point, that's why I'm calling it a pedagogical disaster.

Depends what meaning you put into ”own”. I certainly did not make it. People who died before I was born did to a large extent. So large that it is now everywhere. High-school teachers are parroting it at an alarming rate and even more alarming is all the popular media - not in the least YouTube videos - embracing it. If you go on reddit physics subs you are more than likely getting downvoted because people simply do not understand any other way.

If you have any ideas on how to fix this I am all ears.

Yes, it's a reasonable practical approach. The question is how would you teach SR if there would be no such baggage.

A fair question, but one I have not spent too much time thinking about as it is an ideal case not likely to occur any time soon.

 

[Yuras]

Depends what meaning you put into ”own”. I certainly did not make it.

I didn't mean you of course. And from the very beginning I have said that it was probably too late to fix.

If you have any ideas on how to fix this I am all ears.

Sorry, I'm only able to complain :)
I'm not teaching anyone, and I'm not even in the academia.
Though I think we have to at least acknowledge the issue.

 

[Nugatory]

Since we are in a separate thread now, I can actually answer the original question:

It's possible because there is nobody there to check their claims until they actually come into one point and compare their clocks locally.

Well, it’s another way of reinforcing that simultaneity is not directly measurable. But yes…

I'm afraid that would I post this answer, the OP would conclude that physicists don't agree even among themselves :)

Your instinct that it doesn’t belong in the other thread is pedagogically sound

 

[Yuras]

I find that many people are unable to appreciate the power of the geometric approach until they have ground through the coordinate-based approach so familiar from high school classical mechanics.

I have only anecdotal evidence, but it's kinda opposite (survival bias?) I remember people (me included) struggling with constantly accelerating observer (the most "complicated" topic most of them ever encounter) until they see the coordinate-free one-liner derivation.
It hurts every time I remember how many hours I wasted trying to get sense of time and space "switching roles" inside a blackhole and similar things.

 

[Yuras]

I’m a proponent of spacetime diagrams… since we often refer to the “spacetime geometry”

Spacetime diagrams should be taught in a kindergarten! Though it should not contain the $x$ axes and instead of $t$ axes there should be just the world line of an observer.
(Disclaimer: I didn't actually think it through, so it might be a bad idea at the end of the day :) )

 

[Orodruin]

Spacetime diagrams should be taught in a kindergarten! Though it should not contain the $x$ axes and instead of $t$ axes there should be just the world line of an observer.
(Disclaimer: I didn't actually think it through, so it might be a bad idea at the end of the day :) )

The inherent problem with removing the axes is that we ultimately have Minkowski geometry represented on Euclidean. You’ll need to at least draw a light coneor two, which I guess is easy enough in Minkowski space.

 

[PeroK]

Spacetime diagrams should be taught in a kindergarten! Though it should not contain the $x$ axes and instead of $t$ axes there should be just the world line of an observer.
(Disclaimer: I didn't actually think it through, so it might be a bad idea at the end of the day :) )

I have to work hard to see things geometrically. I'm much happier with algebra. Spacetime diagrams do nothing for me, I'm afraid. I have no mental mechanism to transform the actual Euclidean geometry of a 2D image to Minkowski geometry. Whereas, I used to be able to do a lot of algebra in my head.

In fact, Terrence Tao is a bit like this (although on a different plane):

https://mathstodon.xyz/@tao/113465947720694301

 

[Sagittarius A-Star]

I remember people (me included) struggling with constantly accelerating observer (the most "complicated" topic most of them ever encounter) until they see the coordinate-free one-liner derivation.

How does this one-liner look like?

 

[Yuras]

How does this one-liner look like?

I was totally sure I'll find it on wikipedia, but it's not there... Had to come up with something on the fly, so don't expect too much.
Basically you calculate hyperbolic angle (aka rapidity) between the initial 4-velocity $u_0$ and the current one $u$. Start with their inner product $cosh(w)=u_0\cdot u$ and differentiate it by the proper time:
$$sinh(w)\dot w=u_0 \cdot \dot u = |\dot u| sinh(w)$$
The last step is analogous to $cos(\frac{\pi}{2}+w)=-sin(w)$ (note that $u$ and $\dot u$ are orthogonal). I don't know hyperbolic trigonometry enough to justify it right now, so I'm hand waving here, but in practice you of course just calculate $u_0\cdot\dot u$ in the reference frame of $u$:
$$u_0\cdot\dot u=
\begin{pmatrix}
cosh(w) &
sinh(w) &
0 &
0
\end{pmatrix}
\cdot
\begin{pmatrix}
0 \\
|\dot u| \\
0 \\
0
\end{pmatrix}
$$
It's a bit of a cheating, but the statement itself is coordinate independent anyway.
One way or another, after cancelling $sinh$ we get the following:
$$\dot w = |\dot u|$$
If the (magnitude of the) 4-acceleration is constant, then we just have uniform hyperbolic rotation, i.e. a hyperbola.
Note that this method handles variable acceleration as long as it stays in the same plane as $u_0$ and $u$. For more general case you have to stop before the last step:
$$sinh(w)\dot w=u_0 \cdot \dot u$$
(I really hope I didn't make any silly mistake, and all the above actually makes sense :) )

 

[Yuras]

I have to work hard to see things geometrically. I'm much happier with algebra. Spacetime diagrams do nothing for me, I'm afraid. I have no mental mechanism to transform the actual Euclidean geometry of a 2D image to Minkowski geometry. Whereas, I used to be able to do a lot of algebra in my head.

Yeah, it's useful only for very basic things (aka almost never). Still I think it's an important tool to teach students to formulate their problems in geometric terms. I mean, there are different ways to do algebra: you can calculate the time component of 4-momentum in a particular frame or it's projection on observer's 4-velocity. The result is the same either way, but physical significance is huge. I think spacetime diagrams help to internalize more geometric approach. Though I'm not teaching SR, so my opinion on this is probably just a laughable nonsense.

 

[Sagittarius A-Star]

One way or another, after cancelling $sinh$ we get the following:
$$\dot w = |\dot u|$$
If the (magnitude of the) 4-acceleration is constant, then we just have uniform hyperbolic rotation, i.e. a hyperbola.

Yes, this is trivially correct. The hyperbolic angle $w$ is ${\alpha \over c}$ times the elapsed proper time of the Rindler observer along the hyperbola via definition of the hyperbolic angle. Therefore $\dot w = \alpha/c$.

Source posting:
https://www.physicsforums.com/threa...-without-delving-into-gr.1055627/post-6964768

 

[Sagittarius A-Star]

Spacetime diagrams should be taught in a kindergarten! Though it should not contain the $x$ axes and instead of $t$ axes there should be just the world line of an observer.
(Disclaimer: I didn't actually think it through, so it might be a bad idea at the end of the day :) )

I think is is important to teach the Minkowski metric. I see no good reason to remove it and the related definition of TD i.e. from the following paper. Only that it may be to difficult to teach / understand would be for me i.e. not a good reason.

Source:
http://www.faculty.luther.edu/~macdonal/EGR/EGR.html

 

[Yuras]

I think is is important to teach the Minkowski metric. I see no good reason to remove it and the related definition of TD i.e. from the following paper.

Sorry, but I don't understand your point. I suggested to drop the axis, not metric.
Most likely I'm just misreading your comment, but are you implying that metric tensor is somehow related to coordinates? The whole point of dropping the axis is to prevent students from picking up such misconceptions.

 

[robphy]

Yeah, it's useful only for very basic things (aka almost never)

Here are some non-basic situations where a diagram is quite useful.

• Here's a spacetime diagram for the clock-effect/twin-paradox.
  (from my PF insight https://www.physicsforums.com/insights/spacetime-diagrams-light-clocks/ )
  
• Here's an energy momentum diagram to do an elastic collision
  (from https://www.physicsforums.com/insig...vial-example-a-relativistic-elastic-collision )
  
  
• Here I used an energy-momentum diagram to help me derive and generalize a formula someone asked about
  

  $|\vec{p}^{\text{com}}_{\pi}|=\frac{M_{p}\sqrt{T^{L}_{\pi}(T^{L}_{\pi}+2M_{\pi})}}{\sqrt{(M_{p}+M_{\pi})^{2}+2T^{L}_{\pi} M_{p}}}$

  https://physics.stackexchange.com/q...-out-of-knowledge-kinetic-energy-in-lab-frame
  It turns out in can be interpreted in terms of the analogue of the law of sines.
  
  
• See also
  https://physics.stackexchange.com/q...imit-calculation-subtleties-with-four-vectors
  and
  https://physics.stackexchange.com/q...r-when-a-relativistic-particle-partially-abso

 

[Sagittarius A-Star]

are you implying that metric tensor is somehow related to coordinates?

No, I do not refer to the metric tensor alone. My above screenshot shows the product of the metric tensor with the squared differences-position 4-vector. It results in the (squared) norm of the differences-position 4-vector. Squared coordinate differences are shown.

See also:

Source:
http://www.faculty.luther.edu/~macdonal/EGR/EGR.html

 

[Orodruin]

I was totally sure I'll find it on wikipedia, but it's not there... Had to come up with something on the fly, so don't expect too much.
Basically you calculate hyperbolic angle (aka rapidity) between the initial 4-velocity $u_0$ and the current one $u$. Start with their inner product $cosh(w)=u_0\cdot u$ and differentiate it by the proper time:
$$sinh(w)\dot w=u_0 \cdot \dot u = |\dot u| sinh(w)$$
The last step is analogous to $cos(\frac{\pi}{2}+w)=-sin(w)$ (note that $u$ and $\dot u$ are orthogonal). I don't know hyperbolic trigonometry enough to justify it right now, so I'm hand waving here, but in practice you of course just calculate $u_0\cdot\dot u$ in the reference frame of $u$:
$$u_0\cdot\dot u=
\begin{pmatrix}
cosh(w) &
sinh(w) &
0 &
0
\end{pmatrix}
\cdot
\begin{pmatrix}
0 \\
|\dot u| \\
0 \\
0
\end{pmatrix}
$$
It's a bit of a cheating, but the statement itself is coordinate independent anyway.
One way or another, after cancelling $sinh$ we get the following:
$$\dot w = |\dot u|$$
If the (magnitude of the) 4-acceleration is constant, then we just have uniform hyperbolic rotation, i.e. a hyperbola.
Note that this method handles variable acceleration as long as it stays in the same plane as $u_0$ and $u$. For more general case you have to stop before the last step:
$$sinh(w)\dot w=u_0 \cdot \dot u$$
(I really hope I didn't make any silly mistake, and all the above actually makes sense :) )

This seems unnecessarily complicated. The quick way is to note that the world-line of an observer with constant proper acceleration is necessarily a hyperbola because anything else would not have constant curvature (just as a curve of constant curvature in Euclidean space is a circle). We therefore have (up to spacetime translations)
$$
t^2 - x^2 = - 1/a^2
$$
(the RHS must be negative for the hyperbola to be time-like)
It is simple to verify that $a$ is the proper acceleration:

Parametrize the hyperbola using the hyperbolic one*
$$
t = \sinh(as)/a, \quad x = \cosh(as)/a
$$
This gives $dX/ds = (\cosh(as),\sinh(as))$ which has norm 1 and is therefore the 4-velocity $V$ with $s$ being the proper time. It follows that
$$
A = \frac{dV}{ds} = a (\sinh(as), \cosh(as))
$$
which squares to $\alpha^2 = -A^2 = a^2$.

* Here I obviously chose $as$ as the argument of the hyperbolic functions because I know that makes $s$ the proper time. However, it is also easy to make this a reasonable guess or just use a rapidity $\theta$ as the argument. In the latter case it immediately follows that rapidity is $as$ by rescaling to normalize 4-velocity ($\theta = as$ this is the relativistic generalization of $v = at$ for constant proper acceleration).

The bottom line is you don’t have to delve deep into differentiating gamma factors and solving an ODE in $t$ to solve the case of constant proper acceleration.

Edit: $\LaTeX$ versions of the hyperbolic functions exist: \sinh -> $\sinh$ and \cosh -> $\cosh$ etc.

 

[Yuras]

Here are some non-basic situations where a diagram is quite useful.

These are great diagrams!
But "basic" is a subjective term. Your examples are exercises, very useful for studding, while practical problems are too messy to draw, harder to interpret graphically (e.g. 4D) and easier to handle algebraically. That's my limited experience anyway.

No, I do not refer to the metric tensor alone.

Sorry, probably I'm just too tired right now, but I don't see your point. Let's hope someone will jump in and address it.

 

[Yuras]

This seems unnecessarily complicated.

But it's more general since it handles varying acceleration.

The bottom line is you don’t have to delve deep into differentiating gamma factors and solving an ODE in to solve the case of constant proper acceleration.

That's right. But for some reason it's always presented as a deep delve into differentiating gamma factors.
In general coordinate-free derivation can't beat coondinate-based one since you can always convert the former to the latter. Every non-trivial coordinate-free one contains a hidden coordinate frame.

 

[Orodruin]

But for some reason it's always presented as a deep delve into differentiating gamma factors.

"Always" seems like a too strong statement. It is actually a false statement as the way I just presented is how I do it in my SR course. "Usually" or "often" may be more accurate, and mainly in very introductory courses. At the core it is also just geometry used to infer that the shape must be a hyperbola. It is the direct analogy of a circle in Euclidean space. I typically spend the entire first lecture (45'+45') just introducing Minkowski geometry and drawing parallels to Euclidean space.

I also have a very pragmatic reason for not digging deep into differentiating gamma factors: Apart from the fact that it just loses students' attention - I don't want to do it because it is just messy and you're bound to get something wrong at some point.

 

[Yuras]

"Always" seems like a too strong statement.

I'll try to use hyperbola only in its strict mathematical sense from now on.

the way I just presented is how I do it in my SR course

I'm jealous of your students. When I was studying SR we were still recovering from "SR can't handle accelerating observers" thing.