Help with Lorentz Transformation Derivation

[Curiatron]

I am trying to teach myself about the Lorentz Transform. I have been using the derivation in [1] as my resource*. I am able to follow the derivation pretty well, but have some questions on how they got from equation (25) to equation (28). Here is a snippet of the derivation that I'm stuck on:

Here are my questions:

1. Regarding equation (26), when would parameter a ever be less than 0?

2. Why is parameter a associated with the speed of light? I don’t understand their explanations for this association, including:

a) The derivation explains that parameter a is invariant. I realize that it is well stated that the speed of light is invariant. However, are there any other invariant speeds? If so, then why couldn’t parameter a be identified with one of those invariant speeds?

b) The derivation also says a is identified with the speed of light because of Maxwell's equations and electromagnetic waves. What particular properties of Maxwell's equations and electromagnetic waves explain why a should be identified with the speed of light?

 

*Note: I'm not sure if it's appropriate to attach someone else's pdf to posts, so I didn't do so. (This is my first posting here so still trying to figure out the rules.) If someone can kindly inform me on if it is/is not proprer to do so, that would be appreciated. I'll happily post the pdf if it is allowed.

 

[1] http://www2.physics.umd.edu/~yakovenk/teaching/Lorentz.pdf

[xyzt]

I am trying to teach myself about the Lorentz Transform. I have been using the derivation in [1] as my resource*. I am able to follow the derivation pretty well, but have some questions on how they got from equation (25) to equation (28). Here is a snippet of the derivation that I'm stuck on:

Here are my questions:

1. Regarding equation (26), when would parameter a ever be less than 0?

2. Why is parameter a associated with the speed of light? I don’t understand their explanations for this association, including:

a) The derivation explains that parameter a is invariant. I realize that it is well stated that the speed of light is invariant. However, are there any other invariant speeds? If so, then why couldn’t parameter a be identified with one of those invariant speeds?

b) The derivation also says a is identified with the speed of light because of Maxwell's equations and electromagnetic waves. What particular properties of Maxwell's equations and electromagnetic waves explain why a should be identified with the speed of light?

 

*Note: I'm not sure if it's appropriate to attach someone else's pdf to posts, so I didn't do so. (This is my first posting here so still trying to figure out the rules.) If someone can kindly inform me on if it is/is not proprer to do so, that would be appreciated. I'll happily post the pdf if it is allowed.

 

[1] http://www2.physics.umd.edu/~yakovenk/teaching/Lorentz.pdf

1. a is a parameter, as such , it can be >0 or <0 or [math]\infty[/math] (see the author's discussion)

2. a has dimensions of speed square, the author does not (initially) associate it with the speed of light, he could have written:

 

If a>0 then [math]a=\sigma^2[/math]

If a<0 then [math]a=-\sigma^2[/math]

 

2a. Speed of light (in vacuum ) is frame invariant

Speed of gravitational waves (in anything) is frame invariant and equal to speed of light in vacuum.

 

2b. The equation of electromagnetic waves is what produces the identification.

[xyzt]

BTW, you are welcome!

[Curiatron]

BTW, you are welcome!

Thanks for your helpful responses. It took me a bit to digest your answers, so that's why I didn't respond right away .

 

2a. Speed of light (in vacuum ) is frame invariant

Speed of gravitational waves (in anything) is frame invariant and equal to speed of light in vacuum.

This is probably a naive question, but are there any other frame invariant speeds that we can apply the Lorentz Transform to?

[Janus]

Thanks for your helpful responses. It took me a bit to digest your answers, so that's why I didn't respond right away .

 

This is probably a naive question, but are there any other frame invariant speeds that we can apply the Lorentz Transform to?

You can have only one frame invariant speed. One of the consequences of an frame invariant speed is that becomes the natural speed limit for the universe. Once you establish a speed as invariant, you cannot have any speeds greater than it.