Confusion about four vector notation

AI Thread Summary
The discussion highlights the confusion surrounding the various notations used for four-vectors in special relativity, noting that there is no universally accepted standard. Participants emphasize the importance of choosing a single textbook to follow consistently, with recommendations including Taylor and Wheeler's "Spacetime Physics" and Morin's "Special Relativity for the Enthusiastic Beginner." The preference for different conventions, such as the sign conventions (+--- vs. -+++) and the representation of time components, is acknowledged as a matter of personal choice. The conversation also touches on the diminishing use of the complex notation (ict) in modern texts. Overall, the key takeaway is to remain flexible and adapt to the notation used in the chosen learning resource.
patric44
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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
 
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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.
 
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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 I'd exclude any textbook using the ##\mathrm{i} c t## convention, because this is really outdated nowadays.
 
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can you suggest an introductory level book that doesn't use the "ict"
 
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.
 
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My favorite is Landau&Lifshitz vol. 2.
 
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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​
Spacetime Geometry and Cosmology​
and his unfinished draft of "Div Grad Curl are Dead"​
I think the source of this viewpoint comes from​
Jan Schouten - Tensor Analysis for Physicists​
which is a more readable summary with physical applications of his Ricci Calculus book.​
 
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