Help with special relativity mathematics

[Santilopez10]

I am having a hard time trying to understand this transformation from lorentz:https://imgur.com/a/WYWMO

(You should ignore the spanish part and just focus on the math). I can’t understand well why they turn into what you can see in the second picture, when taking really small values of x.
Also, I tried to approximate to the fourth equation, but I get minus partial derivative of tau respect of x prime, instead of plus.
I would appreciate your help, thanks. I hope this is understandable, it is my first post, and I have no knoweldge of latex.

 

[PeterDonis]

I am having a hard time trying to understand this transformation from lorentz

What you're showing isn't a Lorentz transformation; at best, it's a small piece of one particular attempt to teach the Lorentz transformation. Without more context, I don't think anyone is going to be able to help. What book is this from? What chapter?

 

[Santilopez10]

What you're showing isn't a Lorentz transformation; at best, it's a small piece of one particular attempt to teach the Lorentz transformation. Without more context, I don't think anyone is going to be able to help. What book is this from? What chapter?

“Great illusion”, greatest works of alber einstein” Stephen hawking edition, chapter 3

If you want I can write the things stated before the equations, it is all things about references systems, with more than 1 system.

Also, I would really appreciate how to get from the second equation to the third, thanks.

 

[PeterDonis]

Do you mean this?

https://www.amazon.com/dp/076243564X/?tag=pfamazon01-20

 

[Ibix]

The text is a translation of Einstein's original paper "On the electrodynamics of moving bodies", part I section 3.

The third equation to the fourth seems to me to follow correctly. Post your working if you can't get it (you can use LaTeX for maths - there is a guide linked below the reply box).

The second equation to the third is simply applying the total derivative d/dx' to both sides, applying the chain rule and cancelling some terms, I believe.

 

[Ibix]

Incidentally, the Spanish title of Hawking's book is indeed "The great illusion" ("La gran ilusion" - amazon.es link) which seems like a terrible translation of an allusion to Einstein's "distinction between past, present and future is only a stubbornly persistent illusion" quotation. I realize that book titles are often changed by publishers who know nothing about the subject, but I'd worry slightly about the quality of this translation from its title.

 

[Santilopez10]

The text is a translation of Einstein's original paper "On the electrodynamics of moving bodies", part I section 3.

The third equation to the fourth seems to me to follow correctly. Post your working if you can't get it (you can use LaTeX for maths - there is a guide linked below the reply box).

The second equation to the third is simply applying the total derivative d/dx' to both sides, applying the chain rule and cancelling some terms, I believe.

Could you show me by latex how to get from the second to the third please?

 

[DrGreg]

See this thread from 2005:

https://www.physicsforums.com/threads/understanding-einsteins-math.95942/

 

[Santilopez10]

The text is a translation of Einstein's original paper "On the electrodynamics of moving bodies", part I section 3.

The third equation to the fourth seems to me to follow correctly. Post your working if you can't get it (you can use LaTeX for maths - there is a guide linked below the reply box).

The second equation to the third is simply applying the total derivative d/dx' to both sides, applying the chain rule and cancelling some terms, I believe.

\begin{equation}
\frac {1}{2} \left( \frac 1{c-v} + \frac 1{c+v} \right) \frac {\partial \tau} {\partial t} = \frac {\partial \tau} {\partial x´} + \frac 1 {c-v} \frac {\partial \tau} {\partial t}

\end{equation}
then by summing and assuming partial tau over partial t is 0 I get:
\begin{equation}

\frac {v} {c^2 - v^2} \frac {\partial \tau}{\partial t} = \frac {\partial \tau}{\partial x´}

\end{equation}

which obviously comes into:

\begin{equation}
\frac {v} {c^2 - v^2} \frac {\partial \tau}{\partial t} - \frac {\partial \tau}{\partial x´} = 0
\end{equation}

And thus, I cannot realize how to get it as a plus instead of a minus. (the $\frac {\partial \tau}{\partial x´}$ term)

 

[DrGreg]

assuming partial tau over partial t is 0

Try again without assuming that.

 

[Santilopez10]

Try again without assuming that.

Would you mind showing me? I thought a partial derivative of a constant was just 0.

 

[DrGreg]

Would you mind showing me? I thought a partial derivative of a constant was just 0.

In equation (1) $\partial \tau / \partial t$ appears on both sides. Why would you set it zero on one side but not on the other, which is what you seem to have done?

 

[Santilopez10]

In equation (1) $\partial \tau / \partial t$ appears on both sides. Why would you set it zero on one side but not on the other, which is what you seem to have done?

I don’t really know, please teach me! Jajaja

 

[PeterDonis]

I thought a partial derivative of a constant was just 0.

Why do you think $\partial \tau / \partial t$ is a partial derivative of a constant?

I don’t really know, please teach me!

You need to do the work yourself. We can give you some help, but we're not going to just hand you the solution. If you want to learn, you need to do your part.

 

[Ibix]

I don’t really know, please teach me! Jajaja

My advice: don't try to be clever. This is just algebra.

Edit: and re-read Dr Greg's post #10.