# Are two line derivations harmful in teaching special relativity

#### bernhard.rothenstein

Are "two line derivations" harmful in teaching special relativity

My experience on the forum showed me that some participants do not agree with the simple, “two line” derivations, of the equations that account for relativistic effects like time dilation, length contraction and Lorentz transformations, incriminating papers published by American Journal of Physics and probably by European Journal of Physics. The discussion with them is difficult taking into account the way in which they defend their point of view, rejecting start from the beginning the point of view of the opponents.
I propose an experiment, presenting a “two line” derivation of the Lorentz transformations, inviting those who agree with it and those who do not agree to present theirs punctual point of view.
Consider two inertial reference frames I and I’ in the standard arrangement and theirs relative position at a time t when detected from I and at a time t’ when detected from I’. The involved events are E(x=ct,y=0,t=x/c) in I and E’(x’=ct’,y’=0,t’=x’/c) in E’ the events being generated by the light signal that performs the synchronization of the clocks in the two frames. Because we can add only physical quantities measured in the same inertial reference frame we have in I
f(V)x’=x-Vt=x(1-V/c) (1)
f(V)x=x’+Vt’=x’(1+V/c. (2)
From (1) and (2) we obtain
f(V)=(1-V2/c2)1/2 (3)
and so
x=(x’+Vt’)/(1-V2/c2)1/2 (4)
x’=(x-Vt)/(1-V2/c2)1/2 (5)
Combining (4) and (5) we obtain
t=(t’+Vx’/c2)/(1-V2/c2)1/2 (6)
t’=(t-Vx/c2)/(1-V2/c2)1/2. (7)
Of course during the derivations the instructor could provide supplementary information.
Probably as an old fashioned teacher I fully agree with the derivation presented above. I invite teachers at all levels and learners to present theirs opinion.
sine ira et studio

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#### nakurusil

My experience on the forum showed me that some participants do not agree with the simple, “two line” derivations, of the equations that account for relativistic effects like time dilation, length contraction and Lorentz transformations, incriminating papers published by American Journal of Physics and probably by European Journal of Physics. The discussion with them is difficult taking into account the way in which they defend their point of view, rejecting start from the beginning the point of view of the opponents.
I propose an experiment, presenting a “two line” derivation of the Lorentz transformations, inviting those who agree with it and those who do not agree to present theirs punctual point of view.
Consider two inertial reference frames I and I’ in the standard arrangement and theirs relative position at a time t when detected from I and at a time t’ when detected from I’. The involved events are E(x=ct,y=0,t=x/c) in I and E’(x’=ct’,y’=0,t’=x’/c) in E’ the events being generated by the light signal that performs the synchronization of the clocks in the two frames. Because we can add only physical quantities measured in the same inertial reference frame we have in I
f(V)x’=x-Vt=x(1-V/c) (1)
f(V)x=x’+Vt’=x’(1+V/c. (2)
From (1) and (2) we obtain
f(V)=(1-V2/c2)1/2 (3)
and so
x=(x’+Vt’)/(1-V2/c2)1/2 (4)
x’=(x-Vt)/(1-V2/c2)1/2 (5)
Combining (4) and (5) we obtain
t=(t’+Vx’/c2)/(1-V2/c2)1/2 (6)
t’=(t-Vx/c2)/(1-V2/c2)1/2. (7)
Of course during the derivations the instructor could provide supplementary information.
Probably as an old fashioned teacher I fully agree with the derivation presented above. I invite teachers at all levels and learners to present theirs opinion.
sine ira et studio

Bernhard,

Several of us tried (and apparently failed) to explain to you that the "two line" derivations of Lorentz transforms are wrong since they contain circular logic. Would you please look up the explanations given in the 3 or 4 threads that you have opened on the same subject?

#### Hurkyl

Staff Emeritus
Gold Member
Consider two inertial reference frames I and I’ in the standard arrangement and theirs relative position at a time t when detected from I and at a time t’ when detected from I’. The involved events are E(x=ct,y=0,t=x/c) in I and E’(x’=ct’,y’=0,t’=x’/c) in E’ the events being generated by the light signal that performs the synchronization of the clocks in the two frames. Because we can add only physical quantities measured in the same inertial reference frame we have in I
f(V)x’=x-Vt=x(1-V/c) (1)
f(V)x=x’+Vt’=x’(1+V/c. (2)
From (1) and (2) we obtain
f(V)=(1-V2/c2)1/2 (3)
and so
x=(x’+Vt’)/(1-V2/c2)1/2 (4)
x’=(x-Vt)/(1-V2/c2)1/2 (5)
Combining (4) and (5) we obtain
t=(t’+Vx’/c2)/(1-V2/c2)1/2 (6)
t’=(t-Vx/c2)/(1-V2/c2)1/2. (7)
I'm going to have to object for purely formal reasons.

As someone who doesn't know the "two line derivation" by name, your presentation leaves me baffled. I could probably figure it out, since I do know SR, and am reasonably good at this sort of reconstruction process. But a poor student just learning this stuff... (1) Aesthetic point -- I could understand using only one spatial coordinate, or all three, but I don't understand why you would choose to have two spatial coordinates. I'm going to edit the y-coordinate out for my comments.

(2) What is "standard arrangement"? I'm going to assume you mean that the origins of I and I' should be coincident. Should we expect a student to know that?

(3) What the heck is E(x=ct,t=x/c) supposed to mean? I suspect, from context, that you mean to consider the line x = ct in I, or maybe you mean a certain set of points on that line. But I am not creative enough to see how E(x=ct,t=x/c) could resemble any sort of standard way to express that.

(4) "the light signal that performs the synchronization of the clocks in the two frames" -- clocks in relative motion cannot be synchronized. Since you are talking about deriving the Lorentz transforms, I would assume you would have two clocks in relative motion. So, I'm baffled.

(5) What the heck is V, and f(V), and why would f(V) x' = x? And f(V) x = x' + Vt'?

#### bernhard.rothenstein

two line derivations

Bernhard,

Several of us tried (and apparently failed) to explain to you that the "two line" derivations of Lorentz transforms are wrong since they contain circular logic. Would you please look up the explanations given in the 3 or 4 threads that you have opened on the same subject?
Thank you but you are not the single person I have invited to discussions. Are you so sure that all of them contain circular logic?

#### bernhard.rothenstein

two line derivations

I'm going to have to object for purely formal reasons.

As someone who doesn't know the "two line derivation" by name, your presentation leaves me baffled. I could probably figure it out, since I do know SR, and am reasonably good at this sort of reconstruction process. But a poor student just learning this stuff... :frown
I invited to discussion teachers of physics at different levels. I learned the expression "two line derivation" and as I see you guessed its meaning. The poor student could understand: simple, time saving derivation.

(1) Aesthetic point -- I could understand using only one spatial coordinate, or all three, but I don't understand why you would choose to have two spatial coordinates. I'm going to edit the y-coordinate out for my comments. is a two space dimensions vorbiden?

(2) What is "standard arrangement"? I'm going to assume you mean that the origins of I and I' should be coincident. Should we expect a student to know that?
Strange enough I (European since 1.01.2007) have learned the expressionm standard arrangement from the American literature.
(3) What the heck is E(x=ct,t=x/c) supposed to mean? I suspect, from context, that you mean to consider the line x = ct in I, or maybe you mean a certain set of points on that line. But I am not creative enough to see how E(x=ct,t=x/c) could resemble any sort of standard way to express that.E(x=ct, t=x/c) stands for an event generated by a light signal emitted at t=0 from the origin O that performs the synchronization of the clocks located at the different points of the OX axis

(4) "the light signal that performs the synchronization of the clocks in the two frames" -- clocks in relative motion cannot be synchronized. Since you are talking about deriving the Lorentz transforms, I would assume you would have two clocks in relative motion. So, I'm baffled. Don't be. The light signal synchronizes the stationary clocks in I and in I' respectively the problem being to establish a relationship betweem the readings of such two clocks when they are located at the same point in space and not to synchronize clocks in relative motion/.

(5) What the heck is V, and f(V), and why would f(V) x' = x? And f(V) x = x' + Vt'?
V is the relaive velocity of the involved reference frames and f(V) transforms a proper length in a measured one because,what the heck, we can add only physical quantities of the same nature measured in the same inertial reference frame.
Thank you for teaching me a new expression "what the heck". Using it I could ask you what the heck did you not notice that I mentioned in my message that during the derivations the instructor could provide more information. I have provided them in bold above. Please let me know where is the derivation presented above circular. As I see your objections are more aesthetic in nature.

#### lalbatros

bernhard,

I think your starting point should be more like this:

f(V)x’=x-Vt=x(1-V/c) (1)
g(V)x=x’+Vt’=x’(1+V/c) (2)

and by simple arguments you could proceed, indeed.

Note that I don't see why the number of lines in the derivation should indicate the clarity and the quality of the course.

Personally, I would prefer to give several derivations, and I would start from the invariance of the a light spherical wave front:

x'²+y'²+z'²-c²t'² = x²+y²+z²-c²t² or d'²=d²

It should then be explained why the solution is a linear transformation.
Then, if the audience has a background in linear algebre, it should obvious that the general transformation is similar to a rotation, but in spacetime.

Derivations with mirrors and these old stuff are a waste of time, except from historical point of view.
Derivations from the Maxwell's equations are important and should be clearly liinked with the starting point d'²=d².

Finally, I think it is important that all pieces of the reasoning are discussed with the students. The maths are simple for special relativity, but the ideas are more subtile.

michel

#### bernhard.rothenstein

two line derivations

bernhard,

I think your starting point should be more like this:

f(V)x’=x-Vt=x(1-V/c) (1)
g(V)x=x’+Vt’=x’(1+V/c) (2)

and by simple arguments you could proceed, indeed.

Note that I don't see why the number of lines in the derivation should indicate the clarity and the quality of the course.

Personally, I would prefer to give several derivations, and I would start from the invariance of the a light spherical wave front:

x'²+y'²+z'²-c²t'² = x²+y²+z²-c²t² or d'²=d²

It should then be explained why the solution is a linear transformation.
Then, if the audience has a background in linear algebre, it should obvious that the general transformation is similar to a rotation, but in spacetime.

Derivations with mirrors and these old stuff are a waste of time, except from historical point of view.
Derivations from the Maxwell's equations are important and should be clearly liinked with the starting point d'²=d².

Finally, I think it is important that all pieces of the reasoning are discussed with the students. The maths are simple for special relativity, but the ideas are more subtile.

michel
thank you michele. with you and with the spirit of your message, I feel in Europe, remembering
Souvent, pour s'amuser, les hommes d'équipage
Prennent des albatros, vastes oiseaux des mers,
Qui suivent, indolents compagnons de voyage,
Le navire glissant sur les gouffres amers.
and with Baudelaire, I love so much.
Your first four lines are encouraging for me. As I mentioned in my message the instructor should guide the learner during the derivations. I think that well explained, (1) and (2) are in good relationship with the invariance of the interval, both a consequence of clock synchronization a la Einstein.bonne annee

#### robphy

Homework Helper
Gold Member
IMHO, there is some pedagogical value in short derivations... as long as it reflects and emphasizes the physics. Such attempts are not unlike the numerous attempts to derive the Pythagorean theorem.

FYI: Some philosophers of science are interested in various approaches to deriving the Lorentz Transformations from a minimal set of starting principles and assumptions. This attachment is appropriate for this topic: https://www.physicsforums.com/attachment.php?attachmentid=4406&d=1122686537
The [certainly longer than two-line] derivation by A.A. Robb using the causal structure is especially remarkable.

It seems to me that a true two-line derivation of the Lorentz Transformation should be modifiable into a two-line derivation of a Euclidean rotation... unless something very special for Minkowski spacetime is being used. So, what is this analogue?

Rather than "circular", it's probably more accurate to say there are probably additional assumptions that are not stated explicitly... suggesting that more "lines" are needed than the claimed "two". I haven't studied the proof presented in the first post... however, how obvious is the introduction of f(V)? Does its discussion imply more "lines"?

#### bernhard.rothenstein

two line derivations

IMHO, there is some pedagogical value in short derivations... as long as it reflects and emphasizes the physics. Such attempts are not unlike the numerous attempts to derive the Pythagorean theorem.

FYI: Some philosophers of science are interested in various approaches to deriving the Lorentz Transformations from a minimal set of starting principles and assumptions. This attachment is appropriate for this topic: https://www.physicsforums.com/attachment.php?attachmentid=4406&d=1122686537
The [certainly longer than two-line] derivation by A.A. Robb using the causal structure is especially remarkable.

It seems to me that a true two-line derivation of the Lorentz Transformation should be modifiable into a two-line derivation of a Euclidean rotation... unless something very special for Minkowski spacetime is being used. So, what is this analogue?

Rather than "circular", it's probably more accurate to say there are probably additional assumptions that are not stated explicitly... suggesting that more "lines" are needed than the claimed "two". I haven't studied the proof presented in the first post... however, how obvious is the introduction of f(V)? Does its discussion imply more "lines"?
Thank you for your answer. It is in the style I like so much.When I proposed the approach to LT I have mentioned that during the derivations the instructor could provide all the information required by the learners.That is the case with f(V) which is associated with linearity, reciprocity...My problem is with what to start the teaching of SR. IMHO we should start with Einstein's clock synchronization procedure.