The Principles of Relativity - Help Kindly Requested

[Bongos]

Hello,

I'm new to these forums so I hope this first post is ok.

I'm currently reading & (slowly) working my way through the book "The Essential Einstein - His Greatest works" published by Penguin books but I'm stuck on a derivation on page 11 !

I've attached the derivation as a jpg image to this post (I currently don't know how to work latex as yet) and it's just two lines of mathematics.

My question is, how does he get from the first line to the second ?

It looks to me as thought it's partial differentiation as there are partial derivatives on the second line but when I try doing so it just doesn't work out right. Am I right in thinking it's partial derivatives ? Also he quotes a function, Tau but does not define the function only its parameters so that confuses me slightly as to how he differentiates this and ends up with constants.

Your help is appreciated

Bongos

 

[Ich]

Yes, these are partial derivatives.
$$\tau$$ is the time as measured by the moving observer. Generally, it is a function of t,x,y,z.
You know that for any function
$$\tau(x,t+dt) \simeq \tau(x,t) + \frac{\partial \tau}{\partial t}dt$$?
Do these steps, subtract $$\tau(0,0,0,t)$$ and divide by x'.

 

[Bongos]

Yes, these are partial derivatives.
$$\tau$$ is the time as measured by the moving observer. Generally, it is a function of t,x,y,z.
You know that for any function
$$\tau(x,t+dt) \simeq \tau(x,t) + \frac{\partial \tau}{\partial t}dt$$?
Do these steps, subtract $$\tau(0,0,0,t)$$ and divide by x'.

Thanks Ich,

It took a bit of working out. I originally started applying the approximation you gave to the WHOLE of the LHS of the first line, but I ended up with a half of Tau in the result. I then realized I just apply it the the second argument of the LHS and again on the right (after a bit of re-arranging) and the Tau function drops out.

Thanks for that !

 

[DrGreg]

As luck would have it, I answered this very question 4 years ago in post #6 of the thread Understanding Einstein's Math.

 

[Bongos]

As luck would have it, I answered this very question 4 years ago in post #6 of the thread Understanding Einstein's Math.

Thanks DrGreg.

I remember actually working this out before but second time round I completely forgot ! The PDF is useful as well as I needed a hint to understand the next line in einstein's paper.

I wonder why Einstein never put a few words into explain how he got from one equation to the next, even if he put 'an approximation' it might give people a hint at how he arrived at the second line.

 

[DrGreg]

I wonder why Einstein never put a few words into explain how he got from one equation to the next, even if he put 'an approximation' it might give people a hint at how he arrived at the second line.

Well, he could have added a few words of explanation, but I guess he just assumed his readers would follow the step. Experienced mathematicians (and physicists) can become so familiar with applying the chain rule in situations like this, they can do it in their heads and assume the reader can, too. Bear in mind he was (I believe) writing a technical paper aimed at the physicists and mathematicians of the time, not the general public.

 

[yuiop]

I've attached the derivation as a jpg image to this post (I currently don't know how to work latex as yet) and it's just two lines of mathematics.

$$\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$$

Left click on the above equation and a small window pops up showing the latex code that was used to generate it. Enter the code exactly as shown including the red text in square brackets. You can also left click on the equation posted by Ich to see how the symbols he used are generated. That should be enough to get you started.