Confusion about four vector notation

[patric44]

Homework Statement

what is the correct formula of the gradient in four vector notation

Relevant Equations

x_{mu}=(ct,-r)

hi guys

I am trying to learn special relativity and relativistic quantum mechanics on my own and just very confused about the different conventions used for the notation!?, e.g: the four position 4-vector some times denoted as
$$
x_{\mu}=(ct,-\vec{r})\;\;or\;as\;x_{\mu}=(ict,\vec{r})
$$
or for the contra-variant case
$$
x^{\mu}=(ct,\vec{r})\;or\;as\;x^{\mu}=(ict,\vec{r})
$$
the 4-gradiant also this way with 1/ic or 1/c, and sometimes the "time" component as x4 or as x0 , sometime with an "i" or without it, i tried to learn from different sources and most of them are different, what is the standard notation used for the position 4-vector and the gradient 4-vector? what is the easiest book to learn the subject for a beginner?

thanks in advance

 

[PeroK]

There isn't a standard notation. Personally, I've never used a textbook that uses the complex $ict$ notation.

At this level it shouldn't make too much difference whether a book uses the convention $+---$ or $-+++$. You should be able to handle either without difficulty.

 

[vanhees71]

I strongly recommend to choose one textbook at the beginning and work with it through the basics. I'd exclude any textbook using the $\mathrm{i} c t$ convention, because this is really outdated nowadays.

 

[patric44]

can you suggest an introductory level book that doesn't use the "ict"

 

[Ibix]

For SR, I like Taylor and Wheeler's Spacetime Physics (there's a free-to-download version on Taylor's website). @PeroK usually recommends Morin's Special Relativity for the Enthusiastic Beginner, the first chapter of which is free online.

As noted above, there's no agreed standard for anything in relativity, but $ict$ is all but vanished as far as I'm aware (and about time too, in my opinion), although I haven't read enough QFT texts to have a view of the state of play there. +--- versus -+++ is a matter of choice. I prefer +--- because I've usually found it leads to fewer "forgot to take the modulus, so the sqrt function complained" incidents, but this may depend on your personal interests. I prefer to write time as the zeroth component because often you suppress a spatial dimension or two by a careful choice of coordinates, and it feels more natural to say "$(t,x,y,z)$ can be reduced to $(t,x,y)$" than "$(x,y,z,t)$ can be reduced to $(x,y,t)$". But that's definitely personal preference. You have to be flexible so that you can work with what the book you are working from uses.

 

[vanhees71]

My favorite is Landau&Lifshitz vol. 2.

 

[PeroK]

I like Helliwell's book as a solid introduction to SR.

https://www.goodreads.com/book/show/6453378-special-relativity

 

[robphy]

Generally, I'd suggest Taylor and Wheeler (as @Ibix does) and Bondi for introductions to relativity.
However, for your stated goal and question on notation,
I'd suggest Woodhouse - Special Relativity.
https://www.amazon.com/dp/1852334266/?tag=pfamazon01-20

It is a more mathematically advanced introduction to special relativity
with emphasis on spacetime geometry, more careful attention to tensor notation,
and application to electromagnetism.

A position 4-vector is most naturally a vector $V^\mu$ (drawn as an arrow or vector or an ordered pair of points),
which, in the presence of a metric, can have its index lowered to the dual-vector, covector, or 1-form $V_\mu=g_{\mu\nu}V^\nu$ (drawn as an ordered pair of hyperplanes).

soapbox mode:

While convenient for calculations, sometimes index-gymnastics hides​

the more fundamental nature of the objects involved.​

​

For more on this viewpoint of the more fundamental nature of objects, look at​

Burke's Applied Differential Geometry​

https://www.amazon.com/dp/0521269296/?tag=pfamazon01-20​

Spacetime Geometry and Cosmology​

https://www.amazon.com/dp/0486845583/?tag=pfamazon01-20​

and his unfinished draft of "Div Grad Curl are Dead"​

https://people.ucsc.edu/~rmont/papers/Burke_DivGradCurl.pdf​

​

I think the source of this viewpoint comes from​

Jan Schouten - Tensor Analysis for Physicists​

https://www.amazon.com/dp/0486655822/?tag=pfamazon01-20​

which is a more readable summary with physical applications of his Ricci Calculus book.​

​

​