Talk:Four-vector/Archive 1

Archive 1

Untitled

Er... "four-lots of stuff"? -- Wapcaplet 19:15 19 May 2003 (UTC)

Well, in case I missed some four-things that I haven't heard of yet.

Not even supposed to learn that there is something called a four-force until Thursday... كسيپ Cyp 19:22 19 May 2003 (UTC)

I figured as much. Though, it shouldn't really be an article link, since we're not going to write an article on "four-lots of stuff" :) Usually it's good to put things like that either in italics on the article, like:

This is a stub, please feel free to fix it

or better yet, so as not to interfere with the article itself, inside an HTML comment in the page content, so it only shows up to those editing it. -- Wapcaplet 19:28 19 May 2003 (UTC)

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...this is not a proper inner product, since its value can be negative...

Wups! that's a non sequitur. An inner product can in general be negative. I guess what Michael Hardy meant to say was that it's not a proper inner product because $x\cdot x$ can be negative for some $x$. Changed text accordingly.
—Herbee 23:02, 2004 Feb 25 (UTC)

By the way I notice an inconsistency between this article and Lorentz transformation. In this article, Lorentz transformation is defined as including 3-space rotations, where the article on LT is confined to boosts. In physics, both meanings are used at times. The articles should be homogenized, however, for good encyclopedic practice. -- Decumanus 23:18, 25 Feb 2004 (UTC)

Most of the time I've heard "Generalized Lorentz transformation" used to include both Lorentz boosts and rotations in order to disambiguate from simple Lorentz boosts. --Laura Scudder 16:26, 6 Apr 2005 (UTC)

Actually a Lorentz transformation consists of boosts (or special lorentz transformations), 3D rotations, invertions and compositions of the three types. There is however a subgroup of Lorentz transformations, called the group of proper orthochronic Lorentz transformations. It consists of L s.t.: $detL=+1$, $L_{00}\geq 1$ (positive time orientation). Such transformations can be decomposed by the polar decomposition into a boost and a 3D rotation.

Lurco (talk) 08:34, 11 February 2012 (UTC)

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As Minkowsky metric is not a proper metric I feel this is not the right point of view to look at these things.

Note that the Minkowsky metric is equivalent to the fact that any Lorentz transformation has determinant equal to one, thus conserving spacetime volumes.

Does anyone have more information on this subject?

So far as I know the Minkowsky metric is a perfectly proper metric that is only special in that it indicates flat, cartesian spacetime.

With regards to volume conservation, that's just a consequence of length conservation, which for proper definition of length will be true for any metric since length is a Lorentz invariant:

$ds^{2}=dx^{\alpha }dx_{\alpha }=dx^{\alpha }dx^{\beta }g_{\alpha \beta }$

It might sound like I'm weaseling out of it by saying "proper definition of length", but it is nontrivial to decide how to define length when spacetime is curved. --Laura Scudder 20:52, 6 Apr 2005 (UTC)

To be specific, the definition of a Lorentz transformation is that it preserves the metric, so it's determinant is equal to +\-1. And yes, Minkowski metric satisfies the general condition to be a metric. Positive definiteness is what distigushes the Euclidean metric from the general definition. Therefore I am changing the "As Minkowsky metric is not a proper metric" to "Notice, that the Minkowski metric is not a Euclidean metric beacause it is indefinite, and vector can have - in general - nonpositive length." Lurco (talk) 08:34, 11 February 2012 (UTC)

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I changed a = 1, 2, 3, 4 to 0, 1, 2, 3 because this is how I've always seen it, and sometimes in earlier physics courses x, y, z are labeled as x₁, x₂, x₃, so I thought it's clearer.

I've also been considering changing this equation:

$x\cdot y=x^{a}\eta _{ab}y^{b}=\left({\begin{matrix}x^{0}&x^{1}&x^{2}&x^{3}\end{matrix}}\right)\left({\begin{matrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{matrix}}\right)\left({\begin{matrix}y_{0}\\y_{1}\\y_{2}\\y_{3}\end{matrix}}\right)=-x^{0}y_{0}+x^{1}y_{1}+x^{2}y_{2}+x^{3}y_{3}$

to this one:

$x\cdot y=x^{a}\eta _{ab}y^{b}=\left({\begin{matrix}x^{0}&x^{1}&x^{2}&x^{3}\end{matrix}}\right)\left({\begin{matrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{matrix}}\right)\left({\begin{matrix}y^{0}\\y^{1}\\y^{2}\\y^{3}\end{matrix}}\right)=-x^{0}y^{0}+x^{1}y^{1}+x^{2}y^{2}+x^{3}y^{3}$

since that operation should be equivalent to dotting the elements of x^α with y_α

$x\cdot y=x^{\alpha }y_{\alpha }=\left({\begin{matrix}x^{0}&x^{1}&x^{2}&x^{3}\end{matrix}}\right)\left({\begin{matrix}y_{0}\\y_{1}\\y_{2}\\y_{3}\end{matrix}}\right)=\left({\begin{matrix}x^{0}&x^{1}&x^{2}&x^{3}\end{matrix}}\right)\left({\begin{matrix}-y^{0}\\y^{1}\\y^{2}\\y^{3}\end{matrix}}\right)$

In other words y₀ = -y⁰ so the equation currently in the article is technically misleading if you look at it long enough. I guess I'll just be bold and let someone correct me if they feel strongly about it.

Also, on the issue of units. I changed it mostly because I wanted to include in the article on the electromagnetic four-potential the equation which works in Gaussian units:

$\Box A={\frac {4\pi }{c}}J$,

which since I've never done four-vector equations in SI units, I can't seem to make work in them. I didn't spend much time on it, but I thought I'd have to tweak with the units in J₀ to make it work, which makes no physical sense, since all the elements have the same units to start with. Any insights are appreciated. --Laura Scudder 07:56, 10 Apr 2005 (UTC)

I blundered in defining the four-potential. Having checked the units and discovering that Rindler defines the four-potential in SI, the expression for the four-pot. should be:

${\tilde {A}}^{a}=\left(\phi ,\mathbf {A} c\right)$

(althouth Rindler uses the Minkowski metric with trace -2 instead of +2). Better check signs.

Mpatel

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So now I'm confused about the use of K for the four-frequency. Normally one sees K defined as the four-wavevector (a generalization of a vector and not a scalar being the main difference in terminology - ie four-momentum versus four-energy), which makes the use of the symbol K more intuitive:

$K^{\alpha }\equiv ({\frac {\omega }{c}},\mathbf {k} )$

They're the same to within a constant, but I'm simply worrying about consistency and clarity. --Laura Scudder 02:24, 13 Apr 2005 (UTC)

Agreed. I changed to 4-frequency as it's simpler to work with (no need to unravel omega and and k to get freq. and wavelength). I've changed the K to an N.

Had a brainwave yesterday and discovered a much slicker proof of Planck's relation, although some1 out there will probably find a flaw with it. Mpatel 07:47, 13 Apr 2005 (UTC)

Proof of Planck's Law in GR ?

Can some1 tell me what I've done wrong (if anything) in claiming that the proof of Planck's Law in this article is valid in GR ? Maybe something to do with energy causing spacetime curvature ?It was the arbitrariness of the partial derivative that got me thinking that it was valid in GR. Hmmm... --Mpatel 16:48, 26 July 2005 (UTC)

The purported proof isn't a proof at all, not in SR, and not in GR.--24.52.254.62 04:44, 22 November 2006 (UTC)

subscripts and superscripts

hi

when you write U(superscript a) U (subscript a) =-c*c in the fourth formula line of 'examples of four-vectors in dynamics' what exactly do you mean? is this an inproduct? what does U(subscript a) mean?

it's probably just the notation I'm not familiar with

thx

$U^{a}$ just means a four-vector - think of it as a row or column vector. The abstract index notation is being used here. It's a little like $u^{i}$ representing the velocity in Newtonian mechanics. The tricky bit comes when trying to make sense of U_a which just means $U^{b}\eta _{ba}$ ($\eta$ is the Minkowski metric), or , in English, 'multiply the vector $U$ by the matrix $\eta$'. To answer your question, yes, it's an inner product: $U^{a}U_{a}=\eta _{ab}U^{a}U^{b}$. ---Mpatel (talk) 16:05, August 18, 2005 (UTC)

Is their any chance this notation can be clarified within the article? I understand SR somewhat, but reading through this article spins me in circles because it throws around a lot of mathematical notations that are unique to relativity without explaining them or providing links. --144.92.240.78 06:25, 17 November 2005 (UTC)

Inner product

inner product ? I don't think it's an inner product. There are issues. That is suppose we include both time like and space like then $x\cdot y$ can be less than zero.

Now, If we only restrict ourselves to one of them and choose the metric(ie multiply by -1) accordingly. Then the question is it still a vector space? The answer is no. By these counter examples.

First, let's consider $<x^{0},x^{1},x^{2},x^{3}>$ s.t. :${x^{0}}^{2}-{x^{1}}^{2}-{x^{2}}^{2}-{x^{3}}^{2}>0$

But :$<-x^{0},0,0,0>$ clearly is timelike. But

$<x^{0},x^{1},x^{2},x^{3}>+<-x^{0},0,0,0>=<0,x^{1},x^{2},x^{3}>$ is space like. The same goes for the other choice( ie choosing $<0,-x^{1},-x^{2},-x^{3}>$ and the same trick works). So, restricting ourselves to one of them doesn't help either.

Therefore, there is no way to make the it an inner product-- Piti 128.237.241.252 07:01, 19 April 2006 (UTC)

Describing $\mathbf {X} ^{\alpha }X_{\alpha }$ as inner product is not the quite accurate. The most precise definition would be to call it a scalar product. This emphasises on the fact that the quantity being described remains invariant under any kind of transformation. This also addresses your concern about the numerical value being negative for timelike intervals.

To address this issue I am editing out the pseudo before the inner product in the section, Mathematics of four-vectors. I am also rephrasing inner product to scalar product linking to the article on dot product. However a re-write of the article dot product is in order to include the definition of a scalar product in the context of relativity in particular and tensor analysis in general. I would also suggest a name change of the article from dot product to scalar product.--Fatka (talk) 00:51, 27 October 2008 (UTC)

Position vector versus displacement vector

Of course the coordinates of an event are the components of a four-vector! The position is the precisely the displacement from the origin. A displacement vector is precisely the difference between two position vectors. Alfred Centauri 19:11, 17 October 2006 (UTC)

Confusion

Which is correct?

$\eta _{ab}=\left({\begin{matrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{matrix}}\right)$

or

$\eta _{ab}=\left({\begin{matrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{matrix}}\right)$

this is the only place I've seen the second way..... according to http://en.wikipedia.org/wiki/Minkowski_space it is the first way. If they are supposed to be different then possibly mention why they are different. If it is just a convention, it should mention that. 128.95.3.72 (talk) 18:57, 2 June 2008 (UTC)

It is a convention. Lorentz transformation uses the -+++ metric while this article and spacetime at least uses the +--- metric. Unfortunately Special relativity uses an even different metric because it defines the contravariant displacement four-vector as (t,x,y,z). TStein (talk) 04:06, 19 June 2008 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Four-vector/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

==High importance== 4-vectors simplify the understanding of e&m and relativity quite a bit and is necessary to understand quantum field theory. TStein (talk) 04:07, 15 June 2008 (UTC)

Last edited at 04:07, 15 June 2008 (UTC). Substituted at 14:48, 1 May 2016 (UTC)