Special Relativity/Lorentz Transformation Derived in two minutes

[yuedongxiao]

I was trying to write some "popular science" story on special relativity and explain it to kids a couple of days ago, but find the postulate of speed of light is too hard to be accepted as a fact. So I started working from the Galilean transformation, and realized that Lorentz transformation is just a simple extension of it.

The full derivation is at
http://netbula.com/ydx-SR-lorentz-transformation.pdf

Put it simply, let's assume the general form of the transformation of event (x_A, t_A)
from reference A to B (where B moves at v relative to A) is

x_B = r * (x_A – v * t_A )

switching the references around, A moves at -v relative to B, we must have

x_A = r * (x_B + v * t_B )

From these two equations, we get

t_B = r * t_A + x_A * (1 – r*r) /(r*v)

Now, if we want space and time be treated symmetrically by our nature, we have

- r * v = (1-r*r)/ (r*v)

And we find

r = 1 /sqrt (1- v * v)

For analysis, see http://netbula.com/ydx-SR-lorentz-transformation.pdf

 

[bcrowell]

This seems like an interesting approach to me. It would be nicer IMO to provide a little better motivation for the step where you say '...if time and space are to be treated "equally"...' If you don't point to some input from empirical observations, I don't think the reader will necessarily appreciate why the relativistic transformation is to be preferred over the Galilean one.

A good treatment of this topic, at a level appropriate for young people, is given in Hewitt's Conceptual Physics. He uses Einstein's 1905 axioms.

Like you, I prefer approaching the subject from the point of view of symmetry rather than Einstein's axioms, but my reason is different from the one you stated. I feel that using the Einstein axioms gives a very old-fashioned point of view, which is that light has some special foundational status. The symmetry approach dates back to a 1910 paper by Ignatowski, and has been resurrected in textbooks by Morin and Rindler. My own presentation in that spirit is given here: http://www.lightandmatter.com/html_books/6mr/ch01/ch01.html

Ignatowsky, Phys. Zeits. 11 (1910) 972.
Morin, Introduction to Classical Mechanics, Cambridge, 1st ed., 2008
Rindler, Essential Relativity: Special, General, and Cosmological, 1979, p. 51

 

[zhermes]

The lorentz transformation is by no means a 'simple' extension of the galilean transformation. Einstein postulated that the speed of light was constant (consistent with recent evidence), and from it deduced that time is not absolute, and that space and time can be treated similarly--neither of those deductions had any experimental evidence at the time, and are FAR more of a stretch than the speed of light being constant.

I applaud your derivation of the lorentz transformation in a novel manner, but its premises are far more complicated and--from a historical, or education perspective--far less obvious or clear.

 

[yuedongxiao]

This seems like an interesting approach to me. It would be nicer IMO to provide a little better motivation for the step where you say '...if time and space are to be treated "equally"...'

Thank you for the comment. My starting point in fact did not assume any empirical knowledge, but merely asks the question, what if NATURE treats time and space in the same way in the transformation between reference frames? We humans perceive space and time. Does NATURE make the distinction in this context? What if it does not? My assumption seems to be more fundamental than a postulate on the speed of light.

 

[yuedongxiao]

I applaud your derivation of the lorentz transformation in a novel manner, but its premises are far more complicated and--from a historical, or education perspective--far less obvious or clear.

I don't think my premises are more complicated. There was no assumptions on speed of light and no assumptions on dimensionality. So, Euclid could have derived those equations (the LT) 2000 years ago if he made the leap in treating space and time alike.

 

[zhermes]

Instead of just assuming a constant speed of light (experimentally observed over 100 years ago) you assume that 1) space-dimensions and time are interchangeable*, & 2) differing measurements of space and time are NOT equivalent (only observed in the last ~half-century).

* In regards to '1,' this is not entirely true. It has been known to scientists for hundreds of years that time is indeed different from space (e.g. the arrow of time).

I'm just saying, I think its easier to except (or in other words, explain to a child) that the speed of light is constant than that size, time, energy etc change when things move.

 

[DaTario]

Within you deduction you set no place for light velocity. How do you expect this result to resemble LT ?

It seems that your approach still needs the constancy of c postulate in order to be identical to LT.

Best wishes

DaTario

 

[DaTario]

I will turn my last comment into a question,:

How do you enter the light velocity c in you expression for r ?

 

[yuedongxiao]

I will turn my last comment into a question,:

How do you enter the light velocity c in you expression for r ?

If you read the link to the full paper in the starting post, you will see a section on how to determine the unit speed. The procedure is to derive relativistic dynamics, and from there we will a particle with 0 rest mass will have to move at the unit speed. Whether a photon has a zero mass is a experimental question, but so far, photon is found have a mass smaller than 10^-54 kg, making the speed of light a good candidate for the unit speed.

 

[DaTario]

Ok I saw the paper. Now I get it. Interesting.

But in your paper just before equation (4) there is an allegation the supports the factor gamma as the only way to produce alternatives to Galilean transformation. Could you ellaborate a little more on this argument? Explaining it.

Best wishes

DaTario

 

[yuedongxiao]

Ok I saw the paper. Now I get it. Interesting.

But in your paper just before equation (4) there is an allegation the supports the factor gamma as the only way to produce alternatives to Galilean transformation. Could you ellaborate a little more on this argument? Explaining it.

In A at t_A, the position of B's origin is well defined, and the distance from B's origin to the x_A is some physical length. Imagine A puts a ruler along the x, |(x_A - vt_A)|=L is some length of a segment of the ruler in A. Now, consider how the physical length will transform. It should not change simply because x_A or t_A are shifted (translation symmetry), it has to be just a function of L. And if the space-time is linear, the only thing one can have is r*L.

 

[stevmg]

The constancy of light and the symmetry of the "expansion" x - vt = 0 and x + vt = 0 with a linear relationship from the Galilean concept is the core of Einstein's derivation LT.

Then he does some weird algebraic manipulations after that.

Interestingly, once you dilate time from one FR to another, you contract distance to the same degree making one concomitant with the other.

 

[stevmg]

The full derivation is at
http://netbula.com/ydx-SR-lorentz-transformation.pdf

I read the article, now and is quite interesting. I've seen what is essentially that derivation before by a Dr. Shankar at Yale in his lectures of modern physics on an "elementary" level. It is much simpler than Einstein's derivation and, unwittingly, I stated above what you state in your article about the symmetry between space and time so that the t's and the x's can be interchanged. I guess I knew more than I thought.

Unresolved question from another thread...

Can one exceed the speed of light? Can light exceed the speed of light (or is light speed not constant is another way of saying it.)

On a sphere as a surface mimicking a 2D world (plus time) in curved format, s₁ = s₂ at 90^o to each other would yield s at the equator = both s₁ and s₂ but on a flat plane, the equivalent to the "equator," s would be s₁$$\sqrt{2}$$ = s₂$$\sqrt{2}$$. In other words, curvature of spacetime makes distances shorter and would "slow" near light speed even slower.

 

[yuedongxiao]

I read the article, now and is quite interesting. I've seen what is essentially that derivation before by a Dr. Shankar at Yale in his lectures of modern physics on an "elementary" level. It is much simpler than Einstein's derivation and, unwittingly, I stated above what you state in your article about the symmetry between space and time so that the t's and the x's can be interchanged. I guess I knew more than I thought.

Unresolved question from another thread...

Can one exceed the speed of light? Can light exceed the speed of light (or is light speed not constant is another way of saying it.)

On a sphere as a surface mimicking a 2D world (plus time) in curved format, s₁ = s₂ at 90^o to each other would yield s at the equator = both s₁ and s₂ but on a flat plane, the equivalent to the "equator," s would be s₁$$\sqrt{2}$$ = s₂$$\sqrt{2}$$. In other words, curvature of spacetime makes distances shorter and would "slow" near light speed even slower.

As I read it, Professor Shankar's derivation in your link seems to use the light speed in nailing down \gamma. I think my derivation was simpler and cleaner. I intended to show it to kids who don't know much about "speed of light".

From my derivation, one cannot exceed the unit speed if one comes from a word where speed of 0 is the starting point.

 

[Passionflower]

The full derivation is at
http://netbula.com/ydx-SR-lorentz-transformation.pdf

Nice, keep up the good work!

 

[stevmg]

As I read it, Professor Shankar's derivation in your link seems to use the light speed in nailing down \gamma. I think my derivation was simpler and cleaner. I intended to show it to kids who don't know much about "speed of light".

From my derivation, one cannot exceed the unit speed if one comes from a word where speed of 0 is the starting point.

I have downloaded your .pdf and will go over it more slowly to fully grasp it. I must admit that I could never get the "feel" of Einstein's derivation in his book Relativity. I really appreciate the time you took to go over Dr. Shankar's presentation (in .pdf format.) Thus, I will be able to compare his and your interpretations to see the differences.

CONJECTURES ON THE SPEED OF LIGHT
As far as the speed of light being the limiting factor in a universal sense, I have seen no evidence anywhere that this is not the case (but I am certainly NO authority in this area!) I have heard "arguments" presented that talk about adding proper times over long distances and since a "proper" velocity = distance/proper time this could inflate the so-called proper c as well as other cosmological stuff that I do not understand or even know if they have any truth to them. The only way that I could comprehend light speeds greater than the 300,000,000 m/sec would be if light speed actually varies itself in different parts of the universe. That is not unheard of since light speed does slow down in media such as water or oil (Fitzeau experiment.) In that sense, why couldn't we be in an area where it is the 3X10⁸ m/sec as advertised. Another concept would be a curvature in space (non-spherical) in which there are two geodesics: one a long way and the other a shorter way, but in that case we would see two stars instead of one and not know it.

All this latter discusssion is PURE CONJECTURE and I have seen no evidence to support anything. Your comment on c being the limiting factor doesn't actually rely on what c is. Your argument would hold up for any c. Do you know of any evidence that c is not constant in the universe? Can one exceed our advertised c?

Many thanks,

stevmg

PS - I am still "enthralled" with Michelson's speed of light determination in the 1920s in which he bounced light off a rotating octagonal mirror 22 miles down stream and back off a second mirror. The measurement between the the two mirrors was done by triangulation and was within 1/4 inch of tolerance. I can't even draw a line on a pepr with a six inch ruler within 1/4"of tolerance. That was an absolutely amazing piece of work!

 

[yuedongxiao]

The speed of light seems to be an experimental question.

 

[yossell]

The speed of light seems to be an experimental question.

Most of physics is.

 

[DaTario]

The speed of light seems to be an experimental question.

Could you help me understand the way you took in order to apply F = dp/dt to obtain the condition that m = 0 implies v = 1 ? I am trying but it seems quite tricky.

Best Wishes

DaTario

 

[yuedongxiao]

Could you help me understand the way you took in order to apply F = dp/dt to obtain the condition that m = 0 implies v = 1 ? I am trying but it seems quite tricky.

DaTario

Once you derived the energy-mass relation for a non-zero rest mass, you realize that any particle with zero rest mass at v<1 would have zero energy. Therefore, an energetic object with zero rest mass must be moving at v=1.

 

[DaTario]

Once you derived the energy-mass relation for a non-zero rest mass, you realize that any particle with zero rest mass at v<1 would have zero energy. Therefore, an energetic object with zero rest mass must be moving at v=1.

Is it just to state E = m0 c^2 / ( 1 - v^2)^(1/2) ??

If it is so, then you have a quite simple demonstration in deed.

BTW, concerning my last doubt, when you say that a factor on the Euclidean version is the only acceptable option in modifying this transform, it seems that the principle of relativity is implicitly being applied, as this attitude (inclusion of a factor in Galilean transform) preserves the equality between distances. If for reference frame A, Xa - Xb = Xd - Xe then in another reference frame B, X´a - X´b = X´d - X´e. Then, gravitational forces and electrical forces will be balanced, for instances, in every RF, avoiding the need for changes in the laws. And this claim is exactly the principle of relativity; Every reference frame does the same physics.

Could you comment on that?

Best wishes

DaTario

 

[yuedongxiao]

Is it just to state E = m0 c^2 / ( 1 - v^2)^(1/2) ??

If it is so, then you have a quite simple demonstration in deed.

BTW, concerning my last doubt, when you say that a factor on the Euclidean version is the only acceptable option in modifying this transform, it seems that the principle of relativity is implicitly being applied, as this attitude (inclusion of a factor in Galilean transform) preserves the equality between distances. If for reference frame A, Xa - Xb = Xd - Xe then in another reference frame B, X´a - X´b = X´d - X´e. Then, gravitational forces and electrical forces will be balanced, for instances, in every RF, avoiding the need for changes in the laws. And this claim is exactly the principle of relativity; Every reference frame does the same physics.

Could you comment on that?

Best wishes

DaTario

No. When you transform a physical LENGTH=L in one reference to another, the transformation does not change with respect to location and time.

 

[jason12345]

I think this idea of space time symmetry was exploited in:

Space-time exchange invariance: Special relativity as a symmetry principle
American Journal of Physics -- May 2001 -- Volume 69, Issue 5, pp. 569-575
Issue Date: May 2001

http://arxiv.org/PS_cache/physics/pdf/0012/0012011v2.pdf

I don't see the advantage, since the LTs can be derived from the single relativity postulate that all inertal frames are equivalent. There must also be a limiting velocity regardless, with the postulate making it the same for all frames. This undetermined limiting velocity is c in the LTs.

Space and time symmetry in the LTs follows from the above single postulate, yet you've added it as an additional postulate in your derivation. I don't see the advanatge in your derivation.

Regards,

Jason

 

[yuedongxiao]

I think this idea of space time symmetry was exploited in:

Space-time exchange invariance: Special relativity as a symmetry principle
American Journal of Physics -- May 2001 -- Volume 69, Issue 5, pp. 569-575
Issue Date: May 2001

http://arxiv.org/PS_cache/physics/pdf/0012/0012011v2.pdf

I don't see the advantage, since the LTs can be derived from the single relativity postulate that all inertal frames are equivalent. There must also be a limiting velocity regardless, with the postulate making it the same for all frames. This undetermined limiting velocity is c in the LTs.

Space and time symmetry in the LTs follows from the above single postulate, yet you've added it as an additional postulate in your derivation. I don't see the advanatge in your derivation.

Regards,

Jason

It's the other way around. We don't know there was a max speed. So the starting point was just an extension of the Galiliean transformation. Then as I showed, we actually have two choices. In one choice, the space-time transformation is symmetric. In such a universe, we find there is a max speed, as shown by the velocity addition formula derived from the transformations. With the other choice, one would have no such properties. The speed can be infinite.

 

[DaTario]

No. When you transform a physical LENGTH=L in one reference to another, the transformation does not change with respect to location and time.

Sorry, but I didn't understand the above. Could you explain ?

DaTario

 

[DaTario]

It's the other way around. We don't know there was a max speed. So the starting point was just an extension of the Galiliean transformation. Then as I showed, we actually have two choices. In one choice, the space-time transformation is symmetric. In such a universe, we find there is a max speed, as shown by the velocity addition formula derived from the transformations. With the other choice, one would have no such properties. The speed can be infinite.

But you confirm that you are using some sort of relativity principle/postulate when you say that a factor is the only possible solution for this extension of the Galilean transfomation, don't you?

DaTario

 

[jason12345]

It's the other way around. We don't know there was a max speed. So the starting point was just an extension of the Galiliean transformation. Then as I showed, we actually have two choices. In one choice, the space-time transformation is symmetric. In such a universe, we find there is a max speed, as shown by the velocity addition formula derived from the transformations. With the other choice, one would have no such properties. The speed can be infinite.

As Dadario has pointed out, by introducing the lorentz contraction factor in equation (4), you have made the maximum speed finite. You state that the coefficient of XA in (6) can be expanded as a Taylor series, without mentioning the interval over which it converges. You then decide to eliminate terms of order 2 and above because the theory is "linear" which looks to me wrong because the transformation is still linear in x and t as long as their coefficients are constant.