Einstein tried to prove LT back to 1905 and 1920 in vain. On page 1209 of University Physics (the Book 1 on page 12 of this book) the authors adopted the easiest way to prove LT. The way they failed is the same as the way Einstein failed in 1920 which is related to the function theory of mathematics. Why LT is so hard to prove? The answer depends on how much do you care about physics. I know that the author of this book do care about physics.
The author of this book explained in detail about what Einstein was missing in each of two proofs and how the easiest way was trapped within function theory of mathematics, all in one of author's earlier books. After that, he thought, if he was able to point out the key errors in some famous proofs of LT, why didn't he try to prove LT by himself? The author has spent most of his leisure time on LT for about nine years and this book is the answer to above question.
In this book, the author used some algebra and a few Cartesian coordinates. I know that both of them are all within high school mathematics. The result of the proof is that:
When v = 0, where v is the relative speed between S and S', then LT is (t', x', y', z') = (t, x, y, z) for all events.
When v > 0, then LT is (t', x', y', z') = (γt, -γvt, y, z).
Yes, mathematically speaking, LT is so simple. I believe that you will find no mistake in any step of the proof. The proof is absolutely worth the price of this book. If you really care about physics, you will love the proof! It is very neat.
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Lorentz Transformation for high school students
By Sauce Huang
Trafford PublishingCopyright © 2014 Sauce Huang
All rights reserved.
The velocity of Light
The first knowledge relative to Lorentz Transformation (LT) is the velocity of light. A velocity has a speed and a direction.
Scientists already proved that the speed of light is independent to the speed of the source of light. For sources with low speed, you may refer to Sagnac effect (1913 by George Sagnac) and for sources with high speed you may refer to the experiment reported on 8/20/1964 by T. Alvager, F. Farley, J. Kjellman and I. Wallin.
1-1. How about the velocity?
Let me start this knowledge from a simple setup. Let a train moves at a constant speed v along a straight segment of railroad. I will need your help to point a flashlight to outside of the train from an open window and let the flashlight perpendicular to the plane containing the frame of that window.
Then I want you to turn the flashlight on and off once in the night time so that there should be a ray of light rushes out into the dark space. Now, I need you to imagine the speed of the ray and the direction of it.
About the speed, scientists already assure us that the speed of photons will not be influenced by the speed of the flashlight so that it will not be (c^2+v^2)^(1/2). The speed of photons is a constant c, in vacuum.
I hope you understand this ugly way to display the wrong speed of (c^2+v^2)^(1/2). The symbol of "^" is to represent the function of exponents, c^2 is for the square of c and (c^2+v^2)^(1/2) is for the positive square root of (c^2+v^2).
How about the direction of the ray? Let me ask you to turn the flashlight on and off when you see the center line of a road perpendicular to the railroad. In that situation I will say the direction of that ray is away from the train and along the center line of that road. Do you think so?
I hope you do. Then, as the middle train on the diagram of next page, I will assume that I already put a huge mirror face to the train at the center line of the road about 100 meter away to reflect the ray back to the train.
Now, please imagine this situation and decide if the returning ray will hit the center point of the front circular, square or rectangular surface of your flashlight.
If your answer is will not, which is what I believe, then your idea is showed on the bottom train of the diagram on next page. If your answer is yes then I will talk about it later.
1-2. A Natural Selection
If you select that the return ray will not hit the center point of the face of your flashlight then the velocity of light is independent to the velocity of the source of light. I call it the natural selection of the velocity of light and mark that kind of ray by a Cn vector.
A Cn vector will go by its own direction. No matter how fast the source of light moves or how fast the source of light spins, a Cn vector will represent the speed c and the direction like the ray is emitted from a stationary source in the universe.
When technology is good enough to record nanosecond and nanometer accurately, scientists could use the setup in 1-1 to design an experiment. I hope that Cn vectors will be verified.
Under the definition of Cn vectors, the figure 39-6 on page 1200 of "University Physics" tenth edition by Young & Freedman published in year 2000 (Book 1) will need some modifications.
Under the definition of Cn vectors, the figure\38-5 on page 963 of "Fundamentals of Physics extended" fifth edition by Halliday, Resnick and Walker published in year 1997 (Book 2) will also need some modifications and the proved result after the modification will be "time acceleration", not "time dilation" any more.
1-3. The Mainstream Selection
Yes, logically speaking, a ray may adjust its direction according to the velocity of the source of light while maintaining its speed c. Why? Because scientists don't know a ray thoroughly yet.
I mark that kind of rays by Cm vectors. This kind of vector is a confusing vector. It will partially follow the operational rules of vectors. The Cm vectors will follow the operational rule for directions but not the operational rule for scales.
Yes, regarding following the operational rules of vectors, a Cn vector is worse than a Cm vector because a Cn vector will ignore both of the scale and direction of the velocity vector of the source of light.
The mainstream selection in physics is that the return ray in the section 1-1 will hit the center point of the face of your flashlight. That means Cm vectors is the choice of most scientists of physics in representing a ray of light wave.
Remember that, neither Cn vectors nor Cm vectors are mathematical vectors. Both of them violet the operational rules of mathematical vectors. They are for physics only.
I believe that someday scientists should be able to put this train experiment into test. When that day comes you or your children may see the result of the competition between model of Cn vectors and model of Cm vectors.
1-4. The Cm vectors
Since photons is an important object of physics let me talk more about the character of Cm vectors.
Mathematically, when scientists decide the direction of a ray emitting from a moving source at constant velocity V, it will be like the combined vector, the Cc vector, in the following diagram.
The length of Cc is |Cc| = (c^2+v^2)^(1/2), where |V| = v. The length of Cn vectors and Cm vectors are |Cn|=|Cm|= c.
As you can tell, when |V| > 0, we always have |Cm| < |Cc| but the direction of Cm is always the same as it of Cc.
The idea of mainstream professors of physics makes photons behave in between of independent (the spped) and dependent (the direction) to the velocity of the source of light.
The photons in a ray will follow the direction of vector Cc, which is the direction of combined vector from vectors of Cc and V. That is the dependent portion, the direction of photons depend on the velocity of the source of light.
But photons in a ray will not always accept the speed portion of the vector Cc. When |V| = 0, we do have |Cc| = |Cm| = c but when |V| > 0, |Cc| > |Cm| = c.
I hope you have a clear picture of how the velocity of the source of light influence the behavior of photons it emits. We don't know which model of Cn and Cm is matching the nature, the fact, but currently most professors of physics believe that Cm vectors is the right model.
The model of Cn vectors let the velocity of light totally independent of the velocity of the source of light.CHAPTER 2
Michelson Morley experiment
Scientists use the model of Cm to explain the famous Michelson Morley experiment (MMX, 1887) so that the null result of MMX causes scientists to rely on Lorentz Transformation (LT) for a logical explanation of MMX.
However, following the model of Cn vectors, you will understand that when the reflecting ray returns back to the beam splitter, either in the figure 37-16 on page 1154 of (Book 1) or figure 36-45 on page 927 of (Book 2), it will not reach the center point of the splitter as the model of Cm vectors expected.
When the apparatus turns as designed procedure of MMX, the model of Cn vectors shows the reflecting ray and the penetrating ray will meet at the center area of the beam splitter at 45 degree and 225 degree only. That means, within Cn model, MMX is self-explained to expect for the null result so long as the area around the center point of the light bulb is flat.
The minor experimental results in all MMX related experiments were due to the curvature of the light bulb.
If you like, you may study the MMX with this new point of view base on the model of Cn vectors.
According to the Cn model, when the penetrating ray returns back to the center area of the beam splitter, it will not combine with the original reflecting ray. It will combine with the reflecting ray of another emitting ray which is emitted from a location away from the center point of the light bulb.
A diagram of how the MMX can be explained properly is provided on page 114 of the book "Special Relativity of Roses & Happiness".
You may refer to it if you like to dig into more details of the MMX. But for the purpose of this book, I will stop talking about MMX and introduce LT in as much details as I can offer.CHAPTER 3
Scientists in the mainstream of physics use LT to explain the null result of MMX. This is definitely a valid option so long as LT is well learned, at least, as good as what you will learn from this book.
In high school physics, students do not study LT because LT is beyond what people need to handle their daily affairs. There are too many practical subjects in physics and if some further knowledge of physics should be put into high school physics it would be solar energy, nuclear physics and quantum theory. I believe that the further knowledge for high school physics will not include LT because the main purpose of study for most students is for a better job, at least, more opportunity of employment and LT will not help.
However, if you study physics in university then you will have to learn LT. This book will explain the foundation of LT very clearly for you.
3-1. Galilean Transformation
First of all, what relative to LT and may be useful for high school students is the simple Galilean Transformation (GT).
This is the third item relative to LT which I have mentioned about up to now. The velocity of light and MMX are important in learning LT but GT is the soul of this book and the father of LT.
As a proverb states, like father like son, to understand LT well you must learn GT thoroughly. I said GT is simple however there are two hidden properties of GT can confuse students easily.
Let me show you both of them.
3-1-1. Two 3-dimensional Cartesian coordinates
In GT, there are two 3-dimensional Cartesian coordinates to represent the space portion of frame S and frame S' respectively. The frame S' moves at constant velocity V relative to S, in the common positive direction of their collinear horizontal axes, the x'-axis and the x-axis.
In GT and LT, scientists assume that the y-axis and z-axis of S are parallel to the y'-axis and z' - axis of S' respectively with same positive direction, like upward for y-axis and y'-axis. Scientists also assume that t'= t when the origins of S and S' coincide. Let me use Po to represent the origin point of S and Po' for S'.
If observers in S report space-time coordinates (t, x, y, z) for an event Eg and observers in S' report space-time coordinates (t', x', y', z') for the same event Eg then how are these two sets of numbers related? To relate these two reported coordinates is the main purpose of GT and LT. According to GT the relation will be (t', x', y', z') = (t, x-vt, y, z).
3-1-2. t'= t
The first equation of GT is t'= t. To Mr. Galilei the formula of time is theoretically simple if scientists let event Eg happen at time Tg and location Pg then think about it.
If scientists assume there are observers everywhere in S and S' then they can ask the observer in S at Pg to report event time t=Tg and ask the observer in S' at Pg to report event time t'=Tg to get the natural result of t'= t. It's simple and it explains that GT is theoretically correct.
However, practically speaking, it is impossible to arrange an observer for some possible event like "a bullet hits a flying bird" because observers are unable to fly along with the bird and it would be too dangerous to be that observer.
If GT is not practical then scientists need a practical solution for relativity and that is one of the reasons why LT was so welcomed by professors of physics when it showed up around the end of nineteenth century.
3-1-3. x'= x-vt
The next equation of GT is x'= x-vt --- (1). Because y'= y and z'= z are very obviously correct, the learning of GT will end after you understand the equation (1).
Now, I will need your imagination once again.
Let Ag be the project of Pg to the x-axis then x' is related to the distance of two points Po' and Ag.
We have either x'=Po' Ag or x'= -Po'Ag, it depends on if Ag is located at the right side or left side of Po' and x=PoAg or x= -PoAg. The last item PoPo' is decided by the moving point Po'. When observers in S' record event time t', the distance of PoPo' is v|t'| so that PoPo'=vt' or PoPo'= -vt'.
To combine the possible situations into a valid equation we need help of some diagrams like the one above.
According to that diagram, if Po-Po'-Ag or PoPo'=Ag we have Po'Ag= PoAg-PoPo' and Po'Ag= x', PoAg= x with PoPo'= vt' so that x'= x-vt'. Since in GT, we have t'= t so that the equation (1), x'= x-vt is true for these two situations.
In case of Po-Ag-Po', Ag=Po-Po' or Ag-Po-Po' I need you to use the same diagram with a little bit of imagination to move the project point Ag (of event point Pg) to new locations. You will find out the relations is Po'Ag= -PoAg+ PoPo', Po'Ag=PoAg+PoPo' and Po'Ag=PoAg+PoPo' respectively. All of the relationships are equal to x'= x-vt' and (1) is true.
How about when t'<0 where Po' is approaching Po? Could you figure out x'= x-vt base on above diagram?
For the situation of Po=Po' we have t'=t=0 and x'=x so that the equation x'= x-vt is true. Now, we have considered all of possible situations of GT.
Do you think GT is mathematically correct even if the time equation, t'= t, is hard to prove in the real world?
3-2. Length Contraction
Lorentz Transformation (LT) let observer O stay at the origin of S and observer O' at the origin of S' so that LT can be more practical than GT. The starting point of LT is a hypothesis of length contraction. The hypothesis suggests that if O' use a ruler to measure two points on the x'-axis then the 1 unit length of the ruler will shrink to (1/ ) unit so that the measured result will enlarge by the factor of γ in S'.
However, both of S and S' use same unit like (meter/second) for velocity and it is very important that in LT both of O' and O know that O' moves at velocity V relative to O.
That means O' moves to the positive x-axis direction at speed v is a fact to both of O and O'. If we apply the γ factor to GT, then, the distance of two points measured in S' will be enlarged to γ times of the measurement for the same two points in S. Are you ready?
3-3. The Spatial Equation
Let me write the equation (1), x'= x-vt, and apply the length contraction hypothesis to (1).
The variables at right side of (1) are reported by O for the distance of two points Po'and Ag with PoAg-PoPo' while the left side, x', is reported by O' for the same distance between Po' and Ag directly; we should have x'= (x-vt) ---(2) under the length contraction.
3-4. The Time Equation
Since the length contraction hypothesis applies to the direction of x' axis (moving on x axis) only, we should have y'= y and z'= z in LT. How about the time equation?
I believe that Mr. Lorentz found his time equation by combining the spatial equation of LT and the spatial equation of inverse LT. If he did so, that action would make LT a useless transformation of (t', x', y', z') = (t, x, y, z).
But I will not assume that was the way Mr. Lorentz found his time equation of LT. I will assume that he found it by some hint of nature. I will let the time equation of LT be an independent equation. It is t'= γ(t-(vx/c^2)) ---(3)
Actually, the equation (2) is not perfectly accurate when Mr. Lorentz applied his hypothesis to GT. However, in this book, I will not discuss the detail of that issue. I will introduce LT by itself, without any background check on it.
That means, I will start to explain the details of equations (2) and (3) in LT to you. Most scientists of physics accept LT as true because they believe that LT can explain MMX under the model of Cm vectors.
3-5. Equation (2) and (3) in LT
Let me combine (2) and (3) by (2)+v(3), then we will get the following equation x = γ(x'+vt') ---(4) in LT. The key equation to get equation (4) is 1/(γ^2) = (1-(v^2/c^2)). It is complicated because γ is complicated. The symbol is named Lorentz factor and it is γ = 1/(1-(v^2/c^2))^(1/2).
Now, let us compare the (2) and (4) within LT. In (2) and (4), |x'| is the distance of Po'Ag measured by O' and |x| is the distance of PoAg measured by O. We also know that, v|t| is the distance of PoPo' measured by O so that the only measurement we have not defined now is v|t'|. Let me explain v|t'| step by step for you.
First of all, let me write (2) to show you the distance of Po'Ag in S is not measured directly. From the right side of (2), x'= γ(x-vt), the distance of Po'Ag is calculated by |PoAg-PoPo'| in S. Because Ag is a fixed point, O can measure the distance PoAg easily. But Po' is always moving, how does O measure PoPo'?
The key point is that the location of Po' can be marked on x-axis at the time the event happens, I name it Pe for the purpose of clarity.
Excerpted from Lorentz Transformation for high school students by Sauce Huang. Copyright © 2014 Sauce Huang. Excerpted by permission of Trafford Publishing.
All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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Table of Contents
1. The velocity of Light, 10,
2. Michelson Morley experiment, 21,
3. Lorentz Transformation, 24,
4. Ives-Stilwell experiment, 39,
I. APA, 45,
II. P132, 57,