# The Principles of Relativity - Help Kindly Requested

• Bongos
In summary, Einstein's derivation for Tau is based on partial differentiation and uses the gamma function.
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

#### Attachments

• einstein.jpg
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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'.

Ich said:
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 said:
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.

Bongos said:
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.

Bongos said:
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.

## What is the theory of relativity?

The theory of relativity is a foundational concept in physics that explains the relationship between space and time. It was first developed by Albert Einstein in the early 20th century and is divided into two main parts: the special theory of relativity and the general theory of relativity.

## What is the difference between special and general relativity?

The special theory of relativity deals with objects moving at constant speeds in a straight line, while the general theory of relativity accounts for objects moving in more complex ways, such as those affected by gravity. The general theory of relativity also provides a more complete and accurate description of the universe compared to the special theory.

## How does relativity impact our understanding of the universe?

The principles of relativity have revolutionized our understanding of the universe by showing that space and time are not absolute, but rather relative to an observer's frame of reference. This theory has also helped to explain many phenomena, such as the bending of light around massive objects and the existence of black holes.

## What evidence supports the theory of relativity?

There is a wealth of evidence that supports the theory of relativity, including observations of the bending of starlight during a solar eclipse, the slowing down of time for objects in motion, and the accuracy of predictions made by the theory. The discovery of gravitational waves in 2015 also provided strong evidence for the general theory of relativity.

## How is relativity relevant to everyday life?

While relativity may seem like a complex and abstract concept, it has practical applications in our everyday lives. For example, GPS systems rely on the principles of relativity to accurately calculate and adjust for time differences between satellites and receivers on Earth. Additionally, the development of nuclear power and nuclear weapons would not have been possible without the understanding of relativity.

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