Does Light's Speed Depend on the Motion of Its Source?

[bernhard.rothenstein]

The second postulate is stated as:
Light is always propagated in empty space with a definite velocity c that is independent of the state of motion of the emitting body.
It could be rephrased as:
Light is always propagated in empty space with a definite velocity c independent of the fact that the source is in a state of rest or in a state of motion relative to the observer who measures its speed.
1. Is "the source is in a state of motion" a direct consequence of the first postulate?
2. If so, then a result obtained from a scenario in which the observers are not obliged to measure the speed of light emitted by a moving source could be considered as a conseuence of the first postulate?
Thanks for your answer.

 

[bernhard.rothenstein]

The second postulate is stated as:
Light is always propagated in empty space with a definite velocity c that is independent of the state of motion of the emitting body.
It could be rephrased as:
Light is always propagated in empty space with a definite velocity c independent of the fact that the source is in a state of rest or in a state of motion relative to the observer who measures its speed.
1. Is "the source is in a state of rest" a direct consequence of the first postulate?
2. If so, then a result obtained from a scenario in which the observers are not obliged to measure the speed of light emitted by a moving source could be considered as a conseuence of the first postulate?
Thanks for your answer.

Please read
1. Is "the source in a state of rest" a direct consequence of the first postulate?

 

[Fredrik]

I would say that Einstein's first postulate is too ill-defined to have any direct consequences. Einstein's statement of it doesn't really make sense. See e.g. the discussion in posts #32, #35, #37, #38 in this thread.

If you're asking for the reason why it's always possible to find an inertial frame in which the straight line representing the motion of a physical observer whose accelerometer reads zero is parallel to the time axis, I would say that it's a consequence of the definition of an inertial frame. But even both of Einstein's postulates taken together are insufficient to define what an inertial frame is.

 

[bernhard.rothenstein]

I would say that Einstein's first postulate is too ill-defined to have any direct consequences. Einstein's statement of it doesn't really make sense. See e.g. the discussion in posts #32, #35, #37, #38 in this thread.

If you're asking for the reason why it's always possible to find an inertial frame in which the straight line representing the motion of a physical observer whose accelerometer reads zero is parallel to the time axis, I would say that it's a consequence of the definition of an inertial frame. But even both of Einstein's postulates taken together are insufficient to define what an inertial frame is.

Thanks

Please let me know how would you start teaching special relativity theory.

 

[Fredrik]

I haven't had the opportunity to teach a class on SR, so I haven't worked out a plan in detail, but this is an outline of what I think I would do:

I'd start with spacetime diagrams. I'd tell them that I'm going to explain the theory to them at least twice, first in a sloppy non-mathematical way, and then in an axiomatic way. I would explain the synchronization convention and show them how to draw the coordinates used by a second inertial observer in the spacetime diagram that represents our own point of view, assuming that we have accepted the result of the Michelson-Morley experiment as valid and therefore agree that the speed of light is 1 (in the appropriate units).

Then I'd discuss length contraction and time dilation on a qualitative level. I wouldn't worry so much about numbers at this stage, and instead focus on explaining what these phenomena are and why they occur. I would show them a few examples, e.g. the barn and pole paradox.

Then I'd show them the explicit form of a 1+1-dimensional Lorentz transformation and explain that it's the only linear transformation that has the properties we need. I would consistently use matrix notation, and units such that c=1. If my students aren't familiar with matrices, I'd teach them.

Then I'd define Minkowski space, and say a few things about the relationship between Minkowski space and the Poincaré group. I would talk about the differences between inertial frames in SR and pre-relativistic physics. I'd talk about coordinate time and proper time, and coordinate distance and proper length.

Then I'd talk about Eistein's postulates. I'd have to mention them at some point just so that my students will know what other people are talking about when they mention them but I would say that their significance is mainly historical. I would emphasize very strongly that they aren't mathematical axioms, but rather loosely stated guidelines that are meant to help us guess what mathematical model to use in the new (in 1905) theory of space, time and motion that we're trying to find.

I would explain that Minkowski space is that model, and that the physics is in the postulates that tell us how to interpret the mathematics of Minkowski space as predictions about the results of experiments. I would talk about some of the postulates here, but wait until later before I present a complete list of postulates. (I actually don't have a complete list at this time ).

The twin paradox is next. I'd show them a bunch of different ways to resolve it.

After that I guess I'd talk about four-velocity, momentum and energy, about forces and acceleration. I'd talk about Born rigidity, the "rigid" rotating disc and that kind of stuff. I'd go through lots of examples of relativistic calculations with them.

Then I'd show them the complete list of postulates that define the theory (assuming that I have figured it out by then).

If there's enough time, and the students are going to study GR later, I'd also show them how state the theory in the language of differential geometry. I'd show them that the geometric definition of the Poincaré group (the isometry group of the Minkowski metric) is equivalent to the algebraic definition (the set of functions x\mapsto\Lambda x+a with \Lambda^T\eta\Lambda=\eta).

 

[bernhard.rothenstein]

I haven't had the opportunity to teach a class on SR, so I haven't worked out a plan in detail, but this is an outline of what I think I would do:

I'd start with spacetime diagrams. I'd tell them that I'm going to explain the theory to them at least twice, first in a sloppy non-mathematical way, and then in an axiomatic way. I would explain the synchronization convention and show them how to draw the coordinates used by a second inertial observer in the spacetime diagram that represents our own point of view, assuming that we have accepted the result of the Michelson-Morley experiment as valid and therefore agree that the speed of light is 1 (in the appropriate units).

Then I'd discuss length contraction and time dilation on a qualitative level. I wouldn't worry so much about numbers at this stage, and instead focus on explaining what these phenomena are and why they occur. I would show them a few examples, e.g. the barn and pole paradox.

Then I'd show them the explicit form of a 1+1-dimensional Lorentz transformation and explain that it's the only linear transformation that has the properties we need. I would consistently use matrix notation, and units such that c=1. If my students aren't familiar with matrices, I'd teach them.

Then I'd define Minkowski space, and say a few things about the relationship between Minkowski space and the Poincaré group. I would talk about the differences between inertial frames in SR and pre-relativistic physics. I'd talk about coordinate time and proper time, and coordinate distance and proper length.

Then I'd talk about Eistein's postulates. I'd have to mention them at some point just so that my students will know what other people are talking about when they mention them but I would say that their significance is mainly historical. I would emphasize very strongly that they aren't mathematical axioms, but rather loosely stated guidelines that are meant to help us guess what mathematical model to use in the new (in 1905) theory of space, time and motion that we're trying to find.

I would explain that Minkowski space is that model, and that the physics is in the postulates that tell us how to interpret the mathematics of Minkowski space as predictions about the results of experiments. I would talk about some of the postulates here, but wait until later before I present a complete list of postulates. (I actually don't have a complete list at this time ).

The twin paradox is next. I'd show them a bunch of different ways to resolve it.

After that I guess I'd talk about four-velocity, momentum and energy, about forces and acceleration. I'd talk about Born rigidity, the "rigid" rotating disc and that kind of stuff. I'd go through lots of examples of relativistic calculations with them.

Then I'd show them the complete list of postulates that define the theory (assuming that I have figured it out by then).

If there's enough time, and the students are going to study GR later, I'd also show them how state the theory in the language of differential geometry. I'd show them that the geometric definition of the Poincaré group (the isometry group of the Minkowski metric) is equivalent to the algebraic definition (the set of functions x\mapsto\Lambda x+a with \Lambda^T\eta\Lambda=\eta).

Thank you. Consider please that you have presented special relativity that way and a student puts the following question:
If I state that measuring the speed of a light signal emitted by a source at rest relative to the observer, all of them obtain the same value c, is the consequence of what.

 

[Fredrik]

Thank you. Consider please that you have presented special relativity that way and a student puts the following question:
If I state that measuring the speed of a light signal emitted by a source at rest relative to the observer, all of them obtain the same value c, is the consequence of what.

I'm not sure I understand the question. All of what? If you're asking why the speed of light is the same in all inertial frames, I'd have to say that relativity doesn't explain it. I would explain that if we interpret "the speed of light" as the slope of a null line in an inertial frame, then this fact is just a part of the definition of "inertial frame" on Minkowski space. I would explain how how inertial frames on pre-relativistic spacetime are different. (See e.g. #3 here).

I would also explain that if we interpret "the speed of light" to be a reference to actual light, then the question can only be answered by a theory of electrodynamics.

I'm quoting myself from the other thread:
(By the way, you asked me some time ago if I had summarized my views on SR somewhere. Post #3 in the other thread is a pretty good summary).

* To define a theory of matter in the framework of SR, we either add matter "manually" (e.g. by associating a mass and other properties with a curve that represents the motion of a particle), or by adding the principle of least action as an additional postulate, and let each Lagrangian define a theory of matter.

Note that I'm not making a distinction between "matter" and "radiation" here, or between "particles" and "fields". I'm calling anything described by a Lagrangian "matter".

A really thorough answer to the question you asked includes a discussion about this stuff, and a proof that electromagnetic waves propagate along null lines. A really, really thorough answer would also prove that the probability that a photon emitted at event A will be detected at event B has its maximum value when B is on the future light cone of A. (Most SR students wouldn't understand such a proof, but it can still be a good idea to at least mention that this is the sort of calculation we'd have to make to really answer the question).

I would of course also mention the M-M experiment, and other experiments that demonstrate that SR is a better theory than pre-relativistic classical mechanics.

 

[bernhard.rothenstein]

Consider that observers from different inertial reference frames are equipped with identical machine guns and laser guns. Measuring the speed of the light emitted by the laser gun and the speed of the bullet emitted by the machine gun all observers obtain the same values c and u respectively as long as the two guns are in a state of rest relative to them. That is the consequence of what?
Thanks for your answers.

 

[bernhard.rothenstein]

I haven't had the opportunity to teach a class on SR, so I haven't worked out a plan in detail, but this is an outline of what I think I would do:

I'd start with spacetime diagrams. I'd tell them that I'm going to explain the theory to them at least twice, first in a sloppy non-mathematical way, and then in an axiomatic way. I would explain the synchronization convention and show them how to draw the coordinates used by a second inertial observer in the spacetime diagram that represents our own point of view, assuming that we have accepted the result of the Michelson-Morley experiment as valid and therefore agree that the speed of light is 1 (in the appropriate units).

Then I'd discuss length contraction and time dilation on a qualitative level. I wouldn't worry so much about numbers at this stage, and instead focus on explaining what these phenomena are and why they occur. I would show them a few examples, e.g. the barn and pole paradox.

Then I'd show them the explicit form of a 1+1-dimensional Lorentz transformation and explain that it's the only linear transformation that has the properties we need. I would consistently use matrix notation, and units such that c=1. If my students aren't familiar with matrices, I'd teach them.

Then I'd define Minkowski space, and say a few things about the relationship between Minkowski space and the Poincaré group. I would talk about the differences between inertial frames in SR and pre-relativistic physics. I'd talk about coordinate time and proper time, and coordinate distance and proper length.

Then I'd talk about Eistein's postulates. I'd have to mention them at some point just so that my students will know what other people are talking about when they mention them but I would say that their significance is mainly historical. I would emphasize very strongly that they aren't mathematical axioms, but rather loosely stated guidelines that are meant to help us guess what mathematical model to use in the new (in 1905) theory of space, time and motion that we're trying to find.

I would explain that Minkowski space is that model, and that the physics is in the postulates that tell us how to interpret the mathematics of Minkowski space as predictions about the results of experiments. I would talk about some of the postulates here, but wait until later before I present a complete list of postulates. (I actually don't have a complete list at this time ).

The twin paradox is next. I'd show them a bunch of different ways to resolve it.

After that I guess I'd talk about four-velocity, momentum and energy, about forces and acceleration. I'd talk about Born rigidity, the "rigid" rotating disc and that kind of stuff. I'd go through lots of examples of relativistic calculations with them.

Then I'd show them the complete list of postulates that define the theory (assuming that I have figured it out by then).

If there's enough time, and the students are going to study GR later, I'd also show them how state the theory in the language of differential geometry. I'd show them that the geometric definition of the Poincaré group (the isometry group of the Minkowski metric) is equivalent to the algebraic definition (the set of functions x\mapsto\Lambda x+a with \Lambda^T\eta\Lambda=\eta).

You would not mention Galileo at all?
On which base could I state that if you move relative to me with speed V I move relative to you with speed -V?

 

[Fredrik]

You would not mention Galileo at all?
On which base could I state that if you move relative to me with speed V I move relative to you with speed -V?

I would definitely mention Galilei, and use his principle of relativity to motivate that particular detail. This would be during the "sloppy" part of the presentation. To be more precise, it would be during this step: "Then I'd show them the explicit form of a 1+1-dimensional Lorentz transformation and explain that it's the only linear transformation that has the properties we need." I would show them how to obtain the explicit form of the 1+1-dimensional Lorentz transformation from the assumption that we're looking for a linear transformation that maps the light cone at the origin onto itself (which we can think of as a more well-defined version of Einstein's second postulate) and satisfies

\Lambda(-v)=\Lambda^{-1}(v)

(which is obviously consistent with the idea behind the principle of relativity, but not implied by it, because it's stated in a sloppy non-mathematical way). The condition above is needed to show that the \gamma that appears in

\Lambda=\gamma\begin{pmatrix}1 & -v \\ -v & 1\end{pmatrix}

is the one we're used to.

Consider that observers from different inertial reference frames are equipped with identical machine guns and laser guns. Measuring the speed of the light emitted by the laser gun and the speed of the bullet emitted by the machine gun all observers obtain the same values c and u respectively as long as the two guns are in a state of rest relative to them. That is the consequence of what?

This can of course also be described as a consequence of Galilei's principle of relativity. The challenge is to explain how that fits into special relativity. I would say that the explicit form of the relativity principle that appears in SR is the fact that if two physical observers whose accelerometers read zero use the usual synchronization procedure to assign coordinates to events, the function that describes a change from one of these coordinate systems to the other will be a Poincaré transformation, i.e. an isometry of the Minkowski metric.

I could try to explain which postulates of SR imply this result, but I would have a hard time doing that without a complete list of postulates. (I'm of course talking about the postulates that I think are needed to properly define the theory, and not about Einstein's postulates).

 

[bernhard.rothenstein]

I would definitely mention Galilei, and use his principle of relativity to motivate that particular detail. This would be during the "sloppy" part of the presentation. To be more precise, it would be during this step: "Then I'd show them the explicit form of a 1+1-dimensional Lorentz transformation and explain that it's the only linear transformation that has the properties we need." I would show them how to obtain the explicit form of the 1+1-dimensional Lorentz transformation from the assumption that we're looking for a linear transformation that maps the light cone at the origin onto itself (which we can think of as a more well-defined version of Einstein's second postulate) and satisfies

\Lambda(-v)=\Lambda^{-1}(v)

(which is obviously consistent with the idea behind the principle of relativity, but not implied by it, because it's stated in a sloppy non-mathematical way). The condition above is needed to show that the \gamma that appears in

\Lambda=\gamma\begin{pmatrix}1 & -v \\ -v & 1\end{pmatrix}

is the one we're used to.


This can of course also be described as a consequence of Galilei's principle of relativity. The challenge is to explain how that fits into special relativity. I would say that the explicit form of the relativity principle that appears in SR is the fact that if two physical observers whose accelerometers read zero use the usual synchronization procedure to assign coordinates to events, the function that describes a change from one of these coordinate systems to the other will be a Poincaré transformation, i.e. an isometry of the Minkowski metric.

I could try to explain which postulates of SR imply this result, but I would have a hard time doing that without a complete list of postulates. (I'm of course talking about the postulates that I think are needed to properly define the theory, and not about Einstein's postulates).

In [1] the relativity principle is stated as
All physical laws are the same in any IRF. No inertial reference is "privileged", i.e. distinguisahable from the other IRF's by means of "internal" empirical evidences.
Does that statement satisfy you?
Do you consider that each "internal" emprical evidence should be mentioned separately like:
distances measured perpendicular to the direction of relative motion have the same magnitude for all inertial observers?
[1] Guido Rizzi, "Synchronization Gauges and the principle of Special Relativity," arXiv:gr-qc/0409105v2 18 Oct 2004
(also published Physics Foundation)
Thanks for your help.

 

[Fredrik]

In [1] the relativity principle is stated as
All physical laws are the same in any IRF. No inertial reference is "privileged", i.e. distinguisahable from the other IRF's by means of "internal" empirical evidences.
Does that statement satisfy you?

I'm satisfied with it as long as we only think of it as an idea that's supposed to help us guess what mathematical model to use. I don't think we should treat it as a mathematical axiom. (The term "IRF" hasn't been defined in advance, it isn't 100% clear what a "physical law" is, it isn't 100% clear what "internal empirical evidence is", etc.).

I think a theory of physics should always be stated as a set of axioms that specify at least one mathematical model and tell us how to interpret the mathematics as predictions about the results of experiments. Every statement that we intend to use as the starting point of some mathematical proof must be stated in mathematical language, using only well-defined mathematical terms. But unfortunately it isn't possible to state all of the axioms that define the theory using only well-defined mathematical terms. For example, it's immediately obvious that at least one of them will have to mention the word "clock". So we have to accept some "normal" language in the axioms, but it should be kept to a minimum.

Do you consider that each "internal" emprical evidence should be mentioned separately like:
distances measured perpendicular to the direction of relative motion have the same magnitude for all inertial observers?

I think we should include all the axioms we need to get predictions about the results of experiments, and no more. If we include everything we can think of, we will probably have some axioms that are completely redundant.

 

[JustinLevy]

I am just a gradstudent, but I have helped teach SR at the university level. In the US, this is usually lumped in with a classical mechanics class.

Roughly speaking, there is a tradeoff between two things:
1) building intuition without destroying previous (correct) intuition
2) mathematical details

With enough time, both can be done. The preferred method (based on discussions of teaching at several universities) is, as Fredrik suggested, to teach things twice. Introducing more detail (and mathematical tools) with each pass.

This is done for almost all subjects: E&M, classical mechanics, etc. Unfortunately, in the US, usually the "second" going around of QM waits till gradschool. SR is usually too short of a topic to fill an entire class and is included in other things (usually E&M and/or classical mechanics) and they try to go over it roughly, then in more detail all in one course. The few schools that teach (and require) GR for undergrads probably doesn't have that problem, but I don't know of any such university offhand.

Prof. Rothenstein, you often posts obtuse proofs of things here, that if they give any insight into your teaching or personal way of thinking about SR, I feel would leave a student very lacking on the physical intuition and instead makes things seem like just a series of math tricks (which incidentally is roughly how Lorentz viewed relativity for a long time). You seem to favor "quickness of mathematical result" over "understanding" of result. I feel this is probably not very helpful to the students. Similarly, if we started students with lagrangian mechanics instead of Newton's mechanics, they would problably leave with a feeling that it is nothing more than a series of mathematical procedures.

Yes students need to work (many) problems and get used to the mathematical machinery, so that they can eventually look through the math to see the physics. But it is not the mathematical steps of deriving the Lorentz tranformations, or the velocity addition formula that matter. There are many ways of doing those. Some professors don't even prove such things (and leave it to the textbooks), so they can focus on more things.

but I would say that their [Einstein's postulates] significance is mainly historical. I would emphasize very strongly that they aren't mathematical axioms, but rather loosely stated guidelines

Yes! I wish that was made clearer in many textbooks.
Somewhat related to this, is that one of the largest problems students have I noticed, is that they have trouble breaking from their old notions of a coordinate system. Our shorthand of saying someone is in a particular inertial frame, or the time of some event for each person, doesn't help. Stating historical versions of the postulates and just inserting "inertial frame" just prolongs this problem. At some point a discussion of what a coordinate system is (and isn't) usually needs to happen. Introducing frank discussions of coordinate systems early helps with mathematical rigor (tradeoff item #2), but can seriously hurt intuition (tradeoff item #1) since they can feel they are restarting from scratch.

Sometimes professors just don't discuss it (since it can confuse more than help), and the times I helped teach this is how it was done ... hoping they would figure it out from all the spacetime diagrams, etc. But some students would still be left thinking things like "measuring one way velocity of light" was coordinate independent, etc.

I would talk about some of the postulates here, but wait until later before I present a complete list of postulates. (I actually don't have a complete list at this time ).

For the handwavy ones, we used the usual two postulates referring to inertial coordinates. For precision, (after they saw the math with matrices, etc) there was a homework where they showed two boosts used together were not necessarily another boost matrix... we need rotations as well. So without going into "group theory" they get most of the idea. It is then explained that the mathematical statement of SR is the requirement that the fundamental laws of physics have poincare symmetry ... translations, rotation, boosts. Depending on where they inserted SR in the classical mechanics course affected how they phrased "fundemental laws of physics", but you get the idea: SR = requirement of poincare symmetry. (That space-time had a metric, ie. the so called "clock hypothesis" (wiki or math.ucr.edu) was taken for granted, but I guess it could be stated explicitly ... I'm not sure I would have appreciated the need for it if it was presented to me when I was learning as a student the first time.)

Because of these symmetries, there are a set of coordinate systems that are particularly useful if we are going to write out the laws of physics in coordinates dependent form. Inertial frames are the cartesian coordinate systems in which the laws of physics therefore look the "simplest". (Prof. Rothenstein, since you've brought up the 'arbitrariness' of clock synchronization before, if one uses this approach it also becomes clear why Einstein's synchronization is "better" and more "natural" than some scheme to have every 'inertial' frame agree on synchronization via sacrificing rotational symmetry.)

People of course can always bicker with correctness of phrasing, and presentation, but this seems to work well. It seems to be a good tradeoff, as it eventually gives them more foundation, yet still connects to previous ideas they have already used.

In the end, to each their own. But since I was taught similar to how Fredrik proposed (although spacetime diagrams weren't introduced immediately when I first saw it, it was introduced handwavy/main consequences qualitatively discussed first, then we used matrices, then Minkowski + four vectors), and since I have taught along those lines as well, I am partial to it. (However, I know one professor who loved spacetime diagrams and started with those. It had the advantage that it made all 'paradoxes' immediately moot, but many students had difficulty gaining intuition and connecting to previous intuition. While more difficult in my opinion, it could be a viable alternative.)

 

[bernhard.rothenstein]

It is surprising for me that the physicists who let me know how they would teach special relativity theory do not teach at all. So, even what they are arguing is correct, seems to me unrealistic.
I have taught physics for more then 30 years at a technical university where special relativity is a “Cinderella”, being present as a small chapter of a General Physics Course, short teaching time being allocated and so a “two step approach” should be avoided. Under such conditions, the teacher, who intend to present SRT could choose the following strategies:
1. Stating the two postulates on which SRT is based and deriving in a “quick trade” manner the Lorentz transformations. With them in hand the teacher could present the essential effects like time dilation, length contraction, Doppler Effect…
2. Starting with “thought experiments” the teacher could derive the Lorentz transformations. It would be interesting to start a discussion based on
J.M.Levy, “A simple derivation of the Lorentz transformation and the accompanying velocity and acceleration Am.J.Phys. 75, 615 (2007)
I would appreciate opinions about the pedagogical value of such an approach.
3. Journals like Am.J,Physics, Eur.J.Physics, The Physics Teacher present simple derivations of formulas that account for different relativistic effects, titles like “A two line derivation of…” being attractive for the teacher in time shortage.
4. Justin mentions teachers who present the equations without deriving them. I would avoid the lectures of such teachers and the students would consider such an approach offending.
I think teaching is a question of taste and conscience: De gustibus…
I story I like puts the question if the missionary who was eaten by the natives did his job well. The answer is no because he did not preach well or he has imposed his faith by force. I think the story could be extrapolated to teachers as well.
I offer the following piece of advice to Justin. Do not be so critical. Try to explain to your opponents where they are wrong, give them the opportunity to react and let them know the papers you have published in the field in order to learn from them.
I would highly appreciate the opinion of those who are involved in teaching.

 

[lalbatros]

I studied nuclear engineering at university.
We had a very complete course on electromagnetism.
The last chapter was about the Lorentz transformation, and it was an opportunity to get familiar with the usual notations and basis concepts like the metric. There was never an exam question on this.
I read "Gravitation" a few years later when I was done with exams.

Today, I believe that the first approach should indeed be based on electromagnetism.
But the topic needs to be approach from several side to be really understood.
After all, this is already apparent in OEMB.

 

[Mentz114]

I've taught quite a lot but never relativity, but I do remember doing a course in my last undergraduate year. The only thing I learned from that course was the Lorentz contraction formulae. No metric, no Minkowski space and no electrodynamics ! The garage and pole scenario actually came up in one of the finals papers.

What has subsequently been the most help is space-time diagrams, and electrodynamics. Most of the well known results in SRT can be derived geometrically quite easily from the ST diagram. Going from straight world-lines to curved ones is a natural extension that takes on into GR since the curved lines can be 'straightened' by 'curving' the axes.

 

[lalbatros]

Mentz114,

I enjoyed this book very much:

https://www.amazon.com/dp/067402611X/?tag=pfamazon01-20

Probably you will like it also.

 

[Mentz114]

Mentz114,

I enjoyed this book very much:

https://www.amazon.com/dp/067402611X/?tag=pfamazon01-20

Probably you will like it also.

Thanks.