High School Special relativity - frame of reference

[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

 

[vanhees71]

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.

Interesting. I should read Rindler's book in more detail. I was aware about this derivation only through the artikel by Frank et al from 1911, which was cited in some Am. Jour. Phys. article (I forgot which one). Thanks for the hint anyway.

 

[Orodruin]

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.

Certainly, that practice should be frowned upon. A common way of a more "light" introduction is to first derive time dilation and length contraction and then use them to derive the Lorentz transformations using different constructions. We do it that way in an introductory online course that I developed for the department a few years back.

Interesting. I should read Rindler's book in more detail.

If I remember correctly, he starts with a (rather hand-waving) argument to conclude that the transformation must be linear and the relative velocity to be $v$ to further reduce it somewhat. Then he first assumes $t' = t$ to get the Galilei transformation and later that $c$ is invariant to get the Lorentz transformation. (This is what he does in "Introduction to Special Relativity", I do not know what he does in "Relativity: Special, General, and Cosmological".)

 

[m4r35n357]

Certainly, that practice should be frowned upon. A common way of a more "light" introduction is to first derive time dilation and length contraction and then use them to derive the Lorentz transformations using different constructions. We do it that way in an introductory online course that I developed for the department a few years back.

I would strongly disagree about TD and LC. We are witness pretty much daily here to casualties of that approach. You end up with people who don't know what (me or you) is dilated, think that clocks really change their speed, and try to do SR and GR problems with Newtonian equations and $\gamma$.

The problem IMO is premature simplification; the Lorentz Transform is the product of $\gamma$ with a 2x2 matrix. Lose the matrix and it doesn't surprise me in the slightest that newbies find the concept of simultaneity a problem.

That is the motivation behind my approach in #98. It gives the right answers (eg. the twin paradox, which is nothing more than three spacetime intervals joined up) most simply, which enables the "tricky" stuff like TD, LC to be postponed (perhaps indefinitely, as they are rarely used to calculate anything but muons!) until students have the mathematics to see what they really mean (again, not much IMO), and to get the right answers for themselves.

 

[vanhees71]

The trick by Einstein in 1907 was not to first derive time dilation and length contraction first (which is of course also a nice probability, I remember from my high school textbook "Metzler Physik") and then the Lorentz transformation, but the other way around. The math is much simpler in Einstein's derivation and straighter to the point.

In my SRT FAQ article I shortened this even further by just using Minkowski space-time geometry right away.

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

 

[SiennaTheGr8]

... which enables the "tricky" stuff like TD, LC to be postponed (perhaps indefinitely, as they are rarely used to calculate anything but muons!)

Length contraction must be accounted for when transforming densities between frames.

 

[m4r35n357]

Length contraction must be accounted for when transforming densities between frames.

I don't think there are that many SR "newbies" who are looking to transform densities ;) As I said, that other stuff can all be done after the LT and spacetime interval are assimilated, as these enable the student to check their own answers and understanding.

 

[Orodruin]

I would strongly disagree about TD and LC. We are witness pretty much daily here to casualties of that approach. You end up with people who don't know what (me or you) is dilated, think that clocks really change their speed, and try to do SR and GR problems with Newtonian equations and γγ\gamma.

This depends a lot on how it is presented and in what context. I agree that, if it is done improperly, it will certainly lead to issues. With such an approach it is necessary that the relativity of simultaneity is treated properly in connection to it. The fact remains that you can derive the Lorentz transformations using these arguments and that you need to delve a bit deeper to understand why length contraction and time dilation works both ways.

That is the motivation behind my approach above. It gives the right answers (eg. the twin , is nothing more than three spacetime intervals) most simply, which enables the "tricky" stuff like TD, LC to be postponed (perhaps indefinitely, as they are rarely used to calculate anything but muons!) until students have the mathematics to see what they really mean (again, not much IMO), and to get the right answers for themselves.

I think omitting it altogether is to do the students a disservice. You can bet your hat on at least 90% of the students having read about it before and those students will likely either feel cheated you did not treat it or maintain their misconceptions because you did not (or both).

Length contraction must be accounted for when transforming densities between frames.

No, it must not. Take energy density for example, what needs to be considered is its proper interpretation as a component of the energy-momentum tensor. For charge density you must properly take into account its interpretation as the time-component of the 4-current density.

 

[m4r35n357]

I think omitting it altogether is to do the students a disservice. You can bet your hat on at least 90% of the students having read about it before and those students will likely either feel cheated you did not treat it or maintain their misconceptions because you did not (or both).

I said perhaps ;) You could make it a learning point at the end of a course to explain how various "pop science" scenarios really work, maybe.

 

[SiennaTheGr8]

No, it must not. Take energy density for example, what needs to be considered is its proper interpretation as a component of the energy-momentum tensor. For charge density you must properly take into account its interpretation as the time-component of the 4-current density.

I don't quite follow. Doesn't transforming a volume-density necessarily involve transforming a volume?

(If this is too off-topic, another thread would be fine with me.)

 

[Orodruin]

I don't quite follow. Doesn't transforming a volume-density necessarily involve transforming a volume?

(If this is too off-topic, another thread would be fine with me.)

This opens a whole new can of worms (such as the volume being a 3-form and that volume in the new frame not being a simultaneous volume) and is probably better suited for a new thread. Long story short is that a density is essentially the time component of a current. This is true also in classical mechanics.

 

[vanhees71]

To make the connection to 3-volume elements in this connection, you have to properly define them as hyper-surface elements and the proper "hypersurface-element normal", and all mysteries wrt. densities and the corresponding integrated quantities (charges) are gone, but I'd also say that's not a topic for the introductory SRT lecture (which at my university is taught in the 1st semester at the end of the theory 1 course lecture about classical mechanics, which is of course about point particles and no continuum mechanics or field theories yet; this comes only in the theory 3 course lecture about classical electrodynamics.

 

[David Lewis]

…which enables the "tricky" stuff like TD, LC to be postponed (perhaps indefinitely, as they are rarely used to calculate anything but muons!)

There will be some students don't plan to use their knowledge of Relativity Theory to do anything useful except pass tests. They're in it for personal enrichment and because they want to be well educated. Now if you are training professional scientists then the focus is on practical application.

 

[Dale]

There will be some students don't plan to use their knowledge of Relativity Theory to do anything useful except pass tests. They're in it for personal enrichment and because they want to be well educated.

In this case I think “well educated” is to be achieved the same way for both students that plan to use it later and for students who will not pursue it further.

 

[vanhees71]

There will be some students don't plan to use their knowledge of Relativity Theory to do anything useful except pass tests. They're in it for personal enrichment and because they want to be well educated. Now if you are training professional scientists then the focus is on practical application.

A good test should lead to let them fail... I now, I'm a bad guy, but any physicist who has not understood special relativity to some extent should not be able to earn a degree in physics. That's in the interest of everybody who wants to get a physicist since then the high reputation of physics degrees worldwide is also guaranteed in the future!

 

[pervect]

I don't quite follow. Doesn't transforming a volume-density necessarily involve transforming a volume?

(If this is too off-topic, another thread would be fine with me.)

More if you start a new thread, but I suggest drawing a space-time diagram of the 1-space + 1-time case. Draw the 1-d volume element for a stationary observer on the 2d (1+1) space-time diagram. The "volume" element here is just a line segment because there's only one spatial dimension in this case. Then draw the 1-d volume element for a moving observer, and note that the volume element of the stationary and moving observer are not the same set of points. It's not just length contraction, one must include the relativity of simultaneity. The 2-space + 1 time can be visuzlized with a bit more effort, above that and I think visualization is too difficult and would suggest an abstract approach.

If the diagrams would be helpful and you do start another thread, I could draw them, but I sincerely believe eople have better luck with diagrams they draw themselves.

 

[vanhees71]

Easier than drawing a diagram is to know that a density (e.g., electric charge per volume of a relativistic plasma) is always accompagnied by the correspondin current density. Both together build a Lorentz-vector field,
$$j^{\mu}(x)=(c \rho(x),\vec{j}(x)).$$
A "three-volume" is in fact a 3D space-like hypersurface with elements with a Lorentz-normal vector $\mathrm{d}^3 \sigma_{\mu}$. A space-like hypersurface is defined to have time-like hypersurface-element vectors only. The charge in this hypersurface is then invariant
$$Q_{\sigma}=\int_{\sigma} \mathrm{d}^3 \sigma_{\mu} \frac{1}{c} j^{\mu}.$$
The "naive" total charge
$$Q_{\text{tot,naive}}=\int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x} \rho(t,\vec{x})$$
is a Lorentz scalar only if it is conserved, i.e., if the continuity equation
$$\partial_{\mu} j^{\mu}=0$$
holds.

For a thorough discussion of this, see Jackson, Classical Electrodynamics.

 

[Orodruin]

Easier than drawing a diagram is to know that a density (e.g., electric charge per volume of a relativistic plasma) is always accompagnied by the correspondin current density. Both together build a Lorentz-vector field,
$$j^{\mu}(x)=(c \rho(x),\vec{j}(x)).$$
A "three-volume" is in fact a 3D space-like hypersurface with elements with a Lorentz-normal vector $\mathrm{d}^3 \sigma_{\mu}$. A space-like hypersurface is defined to have time-like hypersurface-element vectors only. The charge in this hypersurface is then invariant
$$Q_{\sigma}=\int_{\sigma} \mathrm{d}^3 \sigma_{\mu} \frac{1}{c} j^{\mu}.$$
The "naive" total charge
$$Q_{\text{tot,naive}}=\int_{\mathbb{R}^3} \mathrm{d}^3 \vec{x} \rho(t,\vec{x})$$
is a Lorentz scalar only if it is conserved, i.e., if the continuity equation
$$\partial_{\mu} j^{\mu}=0$$
holds.

For a thorough discussion of this, see Jackson, Classical Electrodynamics.

I want to stress again that this is not just the case in relativity, but also in classical mechanics - although it is perhaps not as evident there, since the time-component of 4-vectors are invariant under Galilei transformations. The current is still a spacetime 4-vector, but the transformation rule is instead based on the Galilei transformation
$$
\rho' = \rho, \quad \vec j' = \vec j - \vec v \rho.
$$
The current integrations over different spacetime hypersurfaces works exactly the same as in relativity, it is just that their relations between their interpretation in different frames are different due to there being an absolute simultaneity concept and the volume element is therefore the same between frames (as it should, the density does not transform!) and it is much more important how the spatial parts transform. The continuity equation can still be written on the form $\partial_\mu j^\mu = \kappa$, where $\kappa$ is the source.

The charge in classical mechanics is a scalar because simultaneity is well defined.

 

[SiennaTheGr8]

To be clear, I'm aware that charge density is the temporal component of the four-current-density.

What I was thinking of (but not expressing well) is the relation $V = V_0 / \gamma$ between a volume $V$ and the corresponding proper (rest-frame) volume $V_0$ (because of length contraction along the axis of observers' relative motion). This is relevant to the relation between a charge density and the corresponding proper charge density.

 

[Orodruin]

To be clear, I'm aware that charge density is the temporal component of the four-current-density.

What I was thinking of (but not expressing well) is the relation $V = V_0 / \gamma$ between a volume $V$ and the corresponding proper (rest-frame) volume $V_0$ (because of length contraction along the axis of observers' relative motion). This is relevant to the relation between a charge density and the corresponding proper charge density.

This is not due to length contraction. It has to do with the Lorentz transformation of the 4-current density.
$$
\rho' = \gamma(\rho - v \vec j).
$$
Letting $\vec j = 0$ defines the zero-current frame and therefore
$$
\rho' = \gamma \rho_0.
$$
To be clear, the only reason your heuristic works is because the current is conserved and there is no current in the frame where your charges are at rest. This has the same pitfalls as length contraction, in that what defines the length of an object in a frame where it is moving are not simultaneous events in the frame where it is moving.

 

[SiennaTheGr8]

I understand that the Lorentz transformation of the 4-current density leads to the same result, and I understand that "length" is a concept that must be handled carefully because of the relativity of simultaneity.

What I don't understand is how the length-contraction argument is "only" a heuristic. Have I been misled by (or simply misunderstood, more likely) several textbooks?