B Special relativity - frame of reference

[PeterDonis]

I'm not asking about philosophical viewpoints, I am asking for scientific evidence.

I already gave you an answer as regards astronauts: we can't measure biological aging precisely enough. So if that's the only evidence that will satisfy you, it doesn't exist.

 

[richrf]

Not in any meaningful way you aren't. Whether one uses "measured aging" or "real biological aging" one gets the same answer. On my 50th birthday the "measured aging" is 50 years, and the "real biological aging" is 50 years. You have a distinction without a difference.

Two twins can be measured to be of different ages, but are biological the same age, e.g. both have gray hair and aged skin. The question is whether clock measurement observations determine the biological aging process?

 

[PeterDonis]

Aren't you asking a philosophical/metaphysical question, or has physicalism be proven as a scientific fact?

No, and no. You can't "prove" physicalism. So if a proof of physicalism is the only thing that will satisfy you, it doesn't exist.

However, if such proofs are the only thing that will satisfy you, further discussion of the issues you are raising here is pointless, since you refuse to accept reasonable arguments based on the evidence we do have--the same reasonable arguments that convince everybody else that, if you send a clock along with an astronaut, the astronaut ages at the same rate that the clock ticks. And clocks, whose elapsed times we can measure precisely enough, have indeed been sent on the same kinds of trips that astronauts have, and the elapsed times on the clocks match precisely the predictions of GR. All of this is strong indirect evidence that biological aging goes at the same rate as everything else. If you are unwilling to accept that, then, again, further discussion is pointless.

 

[richrf]

I already gave you an answer as regards astronauts: we can't measure biological aging precisely enough. So if that's the only evidence that will satisfy you, it doesn't exist.

Thank you. Can you provide me with a link describing the case of the astronauts. Presumably they were able to determine how to measure biological aging in a spacecraft vs. remaining in Earth. This is what I am most interested in.

 

[PeterDonis]

Two twins can be measured to be of different ages, but are biological the same age, e.g. both have gray hair and aged skin.

You can't measure biological age precisely enough by things like gray hair and aged skin. People vary a lot in when, during their biological aging, they develop such features. As I've already said, we do not have any way of measuring biological aging precisely enough to directly test for differences between people who follow the kinds of paths through spacetime that we can achieve, such as astronauts spending extended periods in low Earth orbit as compared with people on Earth--or even astronauts going to the Moon or Mars as compared with people on Earth. All of the elapsed time differences in these cases, as measured by very accurate clocks, are extremely small--from a few parts per billion to at best a few parts per million. We have no way of detecting differences that small in the biological ages of people or other living things.

 

[PeterDonis]

Can you provide me with a link describing the case of the astronauts. Presumably they were able to determine how to measure biological aging in a spacecraft vs. remaining in Earth.

Did you even read my posts? I have said several times that we cannot measure biological aging that precisely.

You are now banned from this thread since you are evidently unable or unwilling to address the actual points others are making, or even read them.

 

[Ibix]

@richrf - I already showed you a way to directly link aging to clock measurements. You have a clock whose pendulum will crash into a gate and stop if the gate is closed. You open and close the gate rhythmically based on some biological process like your heartbeat, and the clock keeps running because the gate is always open just when the pendulum needs to pass. But according to relativity, any observer may consider themself to be at rest, so you, your heart, and the clock are moving at different speeds according to different observers. Thus your heart rate must slow the same amount as the clock, or else observers would disagree about direct observables like "is the clock running", "did you hear a clang from the pendulum hitting the gate".

The only way out of that is to assume that relativity is fundamentally wrong: there is some privileged frame where mechanical clocks and biological clocks tick at the same rate, but they don't in others. I'm not sure the concept really makes sense, though, since we're left wondering what two clocks that aren't synchronised but aren't wrong are even doing. How are they both clocks? And anyway, since Michelson and Morley in the 1890s we've been failing to find evidence of such a privileged frame.

Then there's the point that all macroscopic structures are made of atoms. And atoms, we do know, obey relativity (the Hafele-Keating experiment). So claiming that biological systems don't do so is a bit like the claims perpetual motion machine designers make - every component part can be shown easily to obey conservation of energy, but somehow you can combine them and they don't. Similarly, every part of a biological system is made of atoms that obey relativity, but somehow the combination doesn't. Is that at all plausible?

We have not done a twin paradox with real twins, no. We can't accelerate them to the speeds needed to see anything with a clock as imprecise as "how much grey hair do I have". But why would you expect a special exception to physics for biological systems?

 

[Herman Trivilino]

Two twins can be measured to be of different ages, but are biological the same age,

No, that's not possible, as far as I know. Do you have a reference that supports this claim?

The question is whether clock measurement observations determine the biological aging process?

Clock measurements do not determine the biological aging process, but clock readings do measure the biological aging process.

 

[PeterDonis]

All, please be aware that @richrf has been thread banned, so he cannot respond to further posts.

 

[Dale]

I need to read more to understand this. Is not clear to me why acceleration makes the differece

Acceleration, by itself, doesn’t make the difference. However, it does break the symmetry.

Relativity, like a lot of modern physics, is largely about symmetries. The principle of relativity, in particular, is talking about a symmetry between inertial reference frames.

An inertial reference frame is one where Newton’s first law (inertia) holds: an object not subject to any force moves in a straight line at a constant velocity. If you draw a spacetime diagram in such a frame, then inertial objects are represented by straight lines. There are many different inertial frames, but anything that is a straight line in one is a straight line in any other. Conversely, anything which is a bent line in one inertial frame is a bent line in any other. Bent lines in spacetime represent objects which are accelerating.

The principle of relativity asserts that all inertial frames are equally valid. That is a symmetry principle. But the traveling twin experienced acceleration so his worldline is bent in any inertial frame. But when we talk about “his frame” we specifically mean a frame where his line is straight and at rest along the t axis. So this frame cannot be inertial, and so the symmetry is broken, and his frame is not equivalent to an inertial frame.

 

[DanMP]

Everyone ages at one second per second along their own path through spacetime. The difference between the two twins is that Bob follows a shorter path through spacetime--one that has fewer total seconds along it--than Alice does. That is why Bob is younger when they meet up again.

Ok, but how Bob/Alice/any clock are physically affected by the "path through spacetime"? In the odometer analogy we know how "the path" (the road) makes the odometer to record more/less kilometers, but how does the spacetime actually affect the clocks/people?

 

[Nugatory]

Ok, but how Bob/Alice/any clock are physically affected by the "path through spacetime"? In the odometer analogy we know how "the path" (the road) makes the odometer to record more/less kilometers, but how does the spacetime actually affect the clocks/people?

Different paths through spacetime have different lengths, and the length of a timelike path through spacetime is the amount of time that a clock moving along that path ticks off. We use clocks to measure "distances" in spacetime the same way we use odometers to measure distances in space.

 

[DanMP]

Different paths through spacetime have different lengths, and the length of a timelike path through spacetime is the amount of time that a clock moving along that path ticks off. We use clocks to measure "distances" in spacetime the same way we use odometers to measure distances in space.

This is more like a definition than an explanation. It does not explain the action, if any, made by the spacetime to affect the "ticking" of the clock. For odometers we have such an action: the road makes the wheels turn/rotate due to friction. How is spacetime affecting the clocks/people?

 

[Dale]

For odometers we have such an action: the road makes the wheels turn/rotate due to friction. How is spacetime affecting the clocks/people?

For every clock we have such an action too. For an atomic clock the energy level difference for the hyperfine transition makes the oscillator oscillate at a specific frequency.

 

[phinds]

This is more like a definition than an explanation. It does not explain the action, if any, made by the spacetime to affect the "ticking" of the clock. For odometers we have such an action: the road makes the wheels turn/rotate due to friction. How is spacetime affecting the clocks/people?

Time is inherently a part of space-time and clocks measure the time part of space-time. That's all there is to it. Time is what clocks measure. All clocks tick at one second per second in their own frame of reference. There is no other "action" performed by space-time on time. What matters is, as Nugatory pointed out, the path through space-time. This is exactly analogous to a space path where distance is measured in meters instead of seconds. Don't be confused by the action of an odometer --- there are other ways to measure distance but they all measure the same distance (assuming they are working correctly).

 

[Dale]

Don't be confused by the action of an odometer --- there are other ways to measure distance but they all measure the same distance (assuming they are working correctly).

That is an excellent point. It isn’t about the odometer, it is about the geometry.

Yes, friction rolls the wheel and yes the hyperfine transition has a specific energy, but the point is that the geometry makes it so that a bent path in space always requires more wheel rotations and a bent path in spacetime requires fewer hyperfine transitions.

That is not explained by the physical mechanisms since, as you point out, we can replace them with any other mechanisms. It is explained by the geometry.

 

[PeroK]

Ok, but how Bob/Alice/any clock are physically affected by the "path through spacetime"? In the odometer analogy we know how "the path" (the road) makes the odometer to record more/less kilometers, but how does the spacetime actually affect the clocks/people?

They are fundamentally not physically affected by the travelling. It is simply that less physical time has passed.

 

[Janus]

This is more like a definition than an explanation. It does not explain the action, if any, made by the spacetime to affect the "ticking" of the clock. For odometers we have such an action: the road makes the wheels turn/rotate due to friction. How is spacetime affecting the clocks/people?

You have two men, they start off walking from the same point but in slightly different directions. They have the same stride. At some point, one of the men turns and changes the direction he is walking in so that he is now headed back towards the other man's path. Once he crosses it, bot men measure their present distance from the starting point. The man who changed direction will be shorter distance from the starting point. This was not caused by anything affecting him or the pace at which he walked, just the fact that he took a different route.
As long as you keep looking for something that "affects" the ticking of clocks as the cause for time dilation, you are barking up the wrong tree. As Dale has said, it is a geometry issue; the geometry of space-time.

 

[Orodruin]

Also, see the link in post #2 of this thread.

 

[DanMP]

For every clock we have such an action too. For an atomic clock the energy level difference for the hyperfine transition makes the oscillator oscillate at a specific frequency.

Yes, I know that, but still, where is the role of spacetime in that energy level difference and/or in the oscillation?

Time is inherently a part of space-time and clocks measure the time part of space-time.

How exactly they do that? According to Dale, clocks are counting oscillations. I can't see the direct connection between that and the spacetime, or "the time part of space-time".

Yes, friction rolls the wheel and yes the hyperfine transition has a specific energy, but the point is that the geometry makes it so that a bent path in space always requires more wheel rotations and a bent path in spacetime requires fewer hyperfine transitions.

That is not explained by the physical mechanisms since, as you point out, we can replace them with any other mechanisms. It is explained by the geometry.

The geometry is good in explaining how to understand and apply the theory, but it is as good in "explaining" what is really happening as contour lines in a topographical map: we can say that we get more tired walking between A and B because we cross contour lines (climbing a hill), but we need to explain why/how contour lines are responsible for this. The same is valid, in my opinion, for world lines / paths through spacetime: we need to show the direct connection, if there is any, between them and the clocks.

As long as you keep looking for something that "affects" the ticking of clocks as the cause for time dilation, you are barking up the wrong tree.

No, I'm pretty sure that this is "the best tree", if we want to solve the mystery of dark matter and to finally/really understand relativity.

 

[phinds]

No, I'm pretty sure that this is "the best tree", if we want to solve the mystery of dark matter and to finally/really understand relativity.

Dude, you are SERIOUSLY barking up the wrong tree. I am constantly astounded by the patience of the mentors here but I predict that this thread has run its course.

 

[DanMP]

Dude, you are SERIOUSLY barking up the wrong tree. I am constantly astounded by the patience of the mentors here but I predict that this thread has run its course.

I think that my questions are legitimate and deserve to be answered.

 

[Ibix]

The geometry is good in explaining how to understand and apply the theory, but it is as good in "explaining" what is really happening as contour lines in a topographical map: we can say that we get more tired walking between A and B because we cross contour lines (climbing a hill), but we need to explain why/how contour lines are responsible for this.

This analogy is wrong. The relevant one is why is it a longer distance over the hill than through a straight tunnel. What answer would you consider acceptable to that question?

 

[m4r35n357]

No, I'm pretty sure that this is "the best tree", if we want to solve the mystery of dark matter and to finally/really understand relativity.

Wow. That came from nowhere. Are you really telling all these scientists how to do their job?

 

[DanMP]

This analogy is wrong. The relevant one is why is it a longer distance over the hill than through a straight tunnel. What answer would you consider acceptable to that question?

My analogy is not wrong, because it is about causality. The one with the tunnel is, again, just geometry.

 

[m4r35n357]

My analogy is not wrong, because it is about causality. The one with the tunnel is, again, just geometry.

You are confusing doing work against gravity (countours) with traveling a further distance (route).

 

[phinds]

I think that my questions are legitimate and deserve to be answered.

But they have BEEN answered. You just don't seem to LIKE the answers.

 

[Ibix]

My analogy is not wrong, because it is about causality. The one with the tunnel is, again, just geometry.

So "it's just geometry" is, to you, an acceptable answer for "why are the distances different", but not for "why are the times different". Do I understand you correctly?

 

[russ_watters]

How exactly they do that? According to Dale, clocks are counting oscillations. I can't see the direct connection between that and the spacetime, or "the time part of space-time".

The geometry is good in explaining how to understand and apply the theory, but it is as good in "explaining" what is really happening as contour lines in a topographical map: we can say that we get more tired walking between A and B because we cross contour lines (climbing a hill), but we need to explain why/how contour lines are responsible for this. The same is valid, in my opinion, for world lines / paths through spacetime: we need to show the direct connection, if there is any, between them and the clocks.

No, I'm pretty sure that this is "the best tree", if we want to solve the mystery of dark matter and to finally/really understand relativity.

You're fixated on the idea of objects interacting with space(/time) probably because most of our experiences happen on Earth, where we are physically attached to an object with a certain geometry, so it makes for a convenient example. Please understand: this analogy is leading you astray because you are not, in fact, interacting with space (you are not interacting with geometry) you are interacting with an object.

If you use spaceship travel instead, the geometry is harder to visualize, but the fact that you are not interacting with space becomes obvious.

 

[DanMP]

Are you really telling all these scientists how to do their job?

No. I obtained interesting results doing that, so I offered a hint. Nothing more. Take it or leave it (to me) :-)

 

[DanMP]

So "it's just geometry" is, to you, an acceptable answer for "why are the distances different", but not for "why are the times different". Do I understand you correctly?

Yes. Distances in space are easy to deal with, but with time is different:

the geometry makes it so that a bent path in space always requires more wheel rotations and a bent path in spacetime requires fewer hyperfine transitions.

Why (if I understand correctly) a clock measures less time in a bent (longer) path in spacetime? This (at least) requires an explanation, but also:

clocks are counting oscillations. I can't see the direct connection between that and the spacetime, or "the time part of space-time".

 

[Orodruin]

Yes. Distances in space are easy to deal with, but with time is different:

It really isn't.

Why (if I understand correctly) a clock measures less time in a bent (longer) path in spacetime? This (at least) require an explanation, but also:

Because Minkowski geometry does not work like Euclidean geometry. Rather than having the Pythagorean theorem $c^2 = a^2 + b^2$, you have the corresponding relation $(c\tau)^2 = (ct)^2 - x^2$. This geometry is required in order for the speed of light to be invariant.

 

[SiennaTheGr8]

Why (if I understand correctly) a clock measures less time in a bent (longer) path in spacetime? This (at least) requires an explanation

Because it's a shorter path in spacetime.

 

[DanMP]

... Minkowski geometry does not work like Euclidiean geometry. Rather than having the Pythagorean theorem $c^2 = a^2 + b^2$, you have the corresponding relation $(c\tau)^2 = (ct)^2 - x^2$. This geometry is required in order for the speed of light to be invariant.

Ok, so one question was answered. Thank you!

What about the other one?

 

[Ibix]

This (at least) requires an explanation, but also:

To measure distance you see how many times a thing of known constant length fits between point A and point B. To measure duration you see how many times a thing of known constant time fits between event A and event B. We call the first thing a ruler and the second a clock.

 

[Nugatory]

How exactly they do that? According to Dale, clocks are counting oscillations. I can't see the direct connection between that and the spacetime, or "the time part of space-time".

We can measure the distance along a path in space with meter sticks: place the meter sticks end to end and then count the number of meter sticks needed. Equivalently we can use just one meter stick, repeatedly picking it up and putting it down again with the near end where the far end had been, and count the number of times we placed the stick before we reached the end of the path.

Counting oscillations is the analgous procedure for measuring the interval along a timelike path through spacetime. Just as a meter stick represents one unit of distance, one cycle of an oscillator represents one unit of time.

 

[Orodruin]

Ok, so one question was answered. Thank you!

What about the other one?

Your post only contained one question.

 

[DanMP]

To measure distance you see how many times a thing of known constant length fits between point A and point B. To measure duration you see how many times a thing of known constant time fits between event A and event B. We call the first thing a ruler and the second a clock.

Yes, this is a nice explanation. I like it. Thank you!

We can measure the distance along a path in space with meter sticks: place the meter sticks end to end and then count the number of meter sticks needed. Equivalently we can use just one meter stick, repeatedly picking it up and putting it down again with the near end where the far end had been, and count the number of times we placed the stick before we reached the end of the path.

Counting oscillations is the analgous procedure for measuring the interval along a timelike path through spacetime.

Another very nice explanation. Thank you!

Ok, thank you all. Best regards!

 

[Dale]

The geometry is good in explaining how to understand and apply the theory, but it is as good in "explaining" what is really happening as contour lines in a topographical map: we can say that we get more tired walking between A and B because we cross contour lines (climbing a hill), but we need to explain why/how contour lines are responsible for this. The same is valid, in my opinion, for world lines / paths through spacetime: we need to show the direct connection, if there is any, between them and the clocks.

Let me ask you a closely related question here. If we draw a triangle on a piece of paper and then measure the length of one side with a ruler and compare it to the length of the other two sides measured with similar rulers, then we always find that the two sides together are longer than the one side. How would you explain that?

The reason I ask is because I think that the geometry is a complete answer, but you do not. So if I can understand what sort of answer you would find satisfying for the triangle question then I can probably make a similar answer for time that would be satisfying for you.

 

[DanMP]

Let me ask you a closely related question here. If we draw a triangle on a piece of paper and then measure the length of one side with a ruler and compare it to the length of the other two sides measured with similar rulers, then we always find that the two sides together are longer than the one side. How would you explain that?

The reason I ask is because I think that the geometry is a complete answer, but you do not. So if I can understand what sort of answer you would find satisfying for the triangle question then I can probably make a similar answer for time that would be satisfying for you.

As you can see, I finally accepted geometry / spacetime as a good/valid answer. My problem was that I thought a bent path should be longer than a straight one. I received 2 answers explaining that "Minkowski geometry does not work like Euclidean geometry" and "it's a shorter path in spacetime". With this info, it was easy to see the logic of the mainstream explanation.

However, I still don't think that this is the only explanation possible (I actually found one myself, in agreement with the current one, but different in the understanding of reality). Considering the above topographic map analogy, if someone would be able to successfully explain gravity, work, etc. using such a model, with contour lines, would you believe that we live in a 2D world? Wouldn't you prefer a 3D approach, equally successful? The same is with my explanation: it is a 3D one, but in agreement with 4D spacetime explanation. And there are ways to experimentally test the new model. Unfortunately, I can't tell you more, unless I'm allowed to offer the link in a private message.

PS: If someone has an accident, we can safely say that his path through spacetime led to it. Would you accept "path through spacetime" as an explanation in this case?

 

[Dale]

I finally accepted geometry / spacetime as a good/valid answer.

Excellent!

I still don't think that this is the only explanation possible (I actually found one myself, in agreement with the current one, but different in the understanding of reality)

Agreed. It is well known that it is not the only explanation possible. These explanations are called “interpretations” and all use the same math and make the same experimental predictions. They just differ in how the variables are interpreted and what parts are considered “real”.

However, do be aware that it is the answer which directly generalizes to GR. For that reason it is by far preferred by the community. I personally encourage you to know and use all interpretations as mental/organizational aids without getting bogged down in debating the philosophical superiority of one or the other.

And there are ways to experimentally test the new model.

Then it would be a new theory and not a new interpretation. Almost certainly it would be experimentally falsified already. However, we cannot discuss it here until after it is published in the scientific literature.

 

[Grinkle]

However, we cannot discuss it here until after it is published in the scientific literature.

As far as I know, though, its allowed to ask if some idea has historically been proposed / published / discarded etc. That said, it doesn't seem this thread is headed that way.

 

[Sorcerer]

Yes. Distances in space are easy to deal with, but with time is different:

Why (if I understand correctly) a clock measures less time in a bent (longer) path in spacetime? This (at least) requires an explanation, but also:

Because distance in Minkowski space has a different formula. It's not a² + b², it's a² - b²

EDIT- sorry Orodruin already addressed this.

 

[Sorcerer]

Excellent!

Agreed. It is well known that it is not the only explanation possible. These explanations are called “interpretations” and all use the same math and make the same experimental predictions. They just differ in how the variables are interpreted and what parts are considered “real”.

However, do be aware that it is the answer which directly generalizes to GR. For that reason it is by far preferred by the community. I personally encourage you to know and use all interpretations as mental/organizational aids without getting bogged down in debating the philosophical superiority of one or the other.

Then it would be a new theory and not a new interpretation. Almost certainly it would be experimentally falsified already. However, we cannot discuss it here until after it is published in the scientific literature.

Hey Dale, regarding the actual math, correct me if I'm wrong, but there is only one possibility that is consistent with isotropy, the principle of relativity, and a finite universal speed limit, right?

I've seen several general derivations of coordinate transformations between inertial reference frames, and even managed to do a fairly general one myself, and I don't see how there is any other possibility. Granted, you could use the same math and interpret an invisible, impossible to detect absolute reference frame for example, but wouldn't the math always work out the same?

 

[DanMP]

As far as I know, though, its allowed to ask if some idea has historically been proposed / published / discarded etc. That said, it doesn't seem this thread is headed that way.

I already insisted too much with this, so I think it is better to stop, at least for a while. I may come back, in a new thread, with questions about the experiments. Until then, thank you all for your patience and for your useful answers.

 

[Dale]

regarding the actual math, correct me if I'm wrong, but there is only one possibility that is consistent with isotropy, the principle of relativity, and a finite universal speed limit, right?

Yes, that is correct.

 

[vanhees71]

Hey Dale, regarding the actual math, correct me if I'm wrong, but there is only one possibility that is consistent with isotropy, the principle of relativity, and a finite universal speed limit, right?

I've seen several general derivations of coordinate transformations between inertial reference frames, and even managed to do a fairly general one myself, and I don't see how there is any other possibility. Granted, you could use the same math and interpret an invisible, impossible to detect absolute reference frame for example, but wouldn't the math always work out the same?

Indeed, for me this is the most beautiful derivation, but it's very mathematical, and that's why you usually don't find it in textbooks about SRT:

Assuming only the special principle of relativity, the existence of inertial frames, and the "Euclidicity" of the space wrt. to any inertial observer, leads to two possible spacetimes, i.e., Galilei-Newton or Einstein-Minkowski spacetime.

A nice paper treating the Lorentz transformation in 1+1 dimensions in this way, is
https://doi.org/10.1119/1.4901453

For the general treatment I know only an old German paper

Ann. Phys. 339, 825 (1911)
doi:10.1002/andp.19113390502

My own attempt to derive the LT in this way is also available in German only:

https://th.physik.uni-frankfurt.de/~hees/faq-pdf/mech.pdf

 

[m4r35n357]

Indeed, for me this is the most beautiful derivation, but it's very mathematical, and that's why you usually don't find it in textbooks about SRT

As a "non-professional" I like to start by addressing the differences between Galilean and Special Relativity by direct comparison. So, starting from the Galilean transform ("assume Newton's laws hold good"), we ask why is there a zero element in it, and what would be the consequences of allowing it to be non-zero? Well straight away you see that it would destroy the notion of universal time. so that is already dealt with in advance ;) Then we ask what other properties does the Galilean Transform have, and the only one I think you need is that the determinant is one (which eliminates annoying scaling issues going forward and back).

The derivation then proceeds as per the last third of my most quoted source (worth reading it all afterwards I think!), although I like to fill in a coupe of steps for clarity. So now we have the Lorentz Transform, and the velocity addition formula. At this point I like to demonstrate explicitly that the spacetime interval is invariant by doing the algebra, and that is it for the basics.

I think that is the shortest and simplest (for a beginner not into advanced mathematics).

 

[Orodruin]

Indeed, for me this is the most beautiful derivation, but it's very mathematical, and that's why you usually don't find it in textbooks about SRT:

Assuming only the special principle of relativity, the existence of inertial frames, and the "Euclidicity" of the space wrt. to any inertial observer, leads to two possible spacetimes, i.e., Galilei-Newton or Einstein-Minkowski spacetime.

Really? I have seen it in several textbooks (at least in one of Rindler's books that I remember off the top of my head) and it is the way I usually like to introduce it - although I gloss over some points rather quickly - in class. However, I would agree that this is not the approach taken in most "modern physics" treatments, which would be in the first two years of university and a more superficial treatment.

 

[vanhees71]

Well, if you just tell your students how the Lorentz transformation looks without any physical nor mathematical argument, I'm pretty sure they'll miss the whole point why to introduce relativity to begin with. As I said, for the introductory course, I'd use a less mathematical more physical derivation like the one by Einstein of 1907:

https://einsteinpapers.press.princeton.edu/vol2-trans/266