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In printed paper 3-vectors can be denoted bold italic while 4-vectors can be denote just bold.

While handwriting 3-vectors are denoted by arrows above letters.

Is there a similar way to denote 4-vectors?

I mean except indexed Einstein notation.


I am just reading Misner Thorn Willer and wish to conserve their notation (p. 20):

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As you proceed with MTW the "boldness" decreases. The large majority of eqs are written in index notation, for the simple reason that it is the most precise one. There is a paragraph somewhere explaining the advantages and pitfalls of different notations. For serious work the index notation is unavoidable, except when working with forms only. So just be patient and keep on reading. –  magma 11 hours ago

4 Answers 4

What's wrong with abstract index notation? It's pretty nearly universal. Otherwise, I've only just seen people use boldface. If you have spaces of multiple dimensions coexisting, it's really hard to get a decent notation without using index conventions. The most common one I've seen is:

4D: $v^{\mu}, g_{\mu\nu}$


$v^{I}, \gamma_{IJ}$


$v^{A}, q_{AB}$

Though this is something that is not universal by a longshot, either. But index conventions are also the only clean way I know of of distinguishing a vector/form from its pullbacks/push-fowards

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Could you maybe show explicit what you mean with the last sentence? –  ungerade Nov 28 '12 at 21:39
@ungerade: explicit about what push-forward and pull-back are? –  Jerry Schirmer Nov 28 '12 at 22:23
No. I mean how you use it to distinguishing f.e. a vector from its pull-back in a clean way. –  ungerade Nov 28 '12 at 22:58
@ungerade: Ah... the idea would be to use the indices to indicate what space you're in and the symbols to indicate what quantity you're working with. Take the metric tensor in a 3+1 formulation. The 4-metric is $g_{\mu\nu}$ and the 3-metric is $\gamma_{ij}$. For some applications, it is useful to talk about the quantity in the 4-manifold whose pullback is the 3-metric, and which is normal to everything else. it is not murderous to notation to call this quantity $\gamma_{\mu\nu}$ –  Jerry Schirmer Nov 28 '12 at 23:03
Nice. I have the "use the symbols where you have the best chance to decipher it again"-rule at my handwritten calculations ;). –  ungerade Nov 28 '12 at 23:24

In my lectures, and also in my Theoretical Physics FAQ at http://www.mat.univie.ac.at/~neum/physfaq/physics-faq.html, I am using $\backslash p$ to denote a bold $p$, etc..

This is very convenient and never leads to confusion.

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In typing I have seen 4-vectors denoted in the same italic manner as scalars ($x$), with the case ostensibly clear from context. In these cases, often bold, Roman typesetting ($\mathbf{x}$) denotes 3-vectors.

Another method I have seen used, especially by people on the relativity side of the relativity/quantum divide, is arrows for 4-vectors ($\vec{x}$) and underlines for 3-vectors ($\underline{x}$). I personally like this notation because it distinguishes both from scalars and can be used in writing just as well as in typing.

However, I draw the line (no pun intended) at trying to extend this to higher-rank situations. Some people (I see this more on the engineering side of things) will use a double arrow or two over/underlines to denote 4- or 3-tensors of rank 2. At this point it's probably best to switch to index notation.

Of course, in pure mathematics often all such distinctions are dropped, and it is not uncommon for everything to be typeset in unadorned italic type.

Edit: Referring to MTW in particular, I will say this. The book is many things, but a lesson in good typesetting it is not. I've never seen so much use of bold - on italics and frakturs, on Greek letters, on nablas and differentials - and I've certainly never seen different levels of bold-ness being employed in the same text.

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I've seen underlines to denote one-forms before though, so be careful. –  Jerry Schirmer Nov 29 '12 at 0:17
@JerrySchirmer: I've seen under-tildes for four-vectors and over-arrows for 3/2 vectors. I guess there's no fixed convention for it (besides index notation), better to specify which notation you are using every time :) –  Manishearth Nov 29 '12 at 7:17

You can use any notation you like. Only thing is that it YOU should find it convenient. After learning index notation most people find it more convenient than other notations, and that's why they use it.

A four vector is an ordered set of four real numbers. In index notation the $\mu $ th component of a vector $x$ is denoted as $x^\mu$. So for example for the 4-vector $x=(1,2.5,4,6)$ we have $$x^0=1,x^1=2.5,x^2=4,x^3=6$$. However there is no harm if you denote this vector as $x=1\hat t+2.5\hat i + 4\hat j +6 \hat k$, where $\hat t=(1,0,0,0)$ is unit vector in time direction and similarly $\hat i=(0,1,0,0)$ etc. In fact while learning index notation one should better try to translate quantities expressed in index notation in his/her own favorite notation.

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