Talk:Four-gradient

Importance

Importance to math students: One of the first steps on graduate level. Importance in General: Between important fill & specialized knowledge (mid/lowmid) Two References have been added. The corresponding german wikipage has a non-english link. Maybe a link to Wolfram would be better

                                Phi0618 (talk) 17:55, 21 November 2007 (UTC)

Generalization of what?

The first sentence gets it backwards: it is gradient which is the more general concept. We take gradient, retrict it to four dimensions and space-time metric and only then get the usual four-gradient used in physics. — Preceding unsigned comment added by 149.156.47.235 (talk) 13:00, 7 September 2012 (UTC)

It means the four-gradient is a space-time analogue of the Gibbs-Heaviside gradient (as the gradient-link makes clear). Though both these may logically (for mathematicians) derive from 1-forms in some abstract space, it would probably be safe to assume that historically the Gibbs-Heaviside gradient came first. --catslash (talk) 14:56, 7 September 2012 (UTC)

Box symbol

As there seems to be some doubt about this, here are four works plucked from Google books which use $\Box$ and $\Box ^{2}$ for the 4-gradient and d'Alembertian respectively.

• Clifford algebras in analysis and related topics By John Ryan
• Differential forms in electromagnetics By Ismo V. Lindell
• Introduction to relativistic quantum chemistry, Volume 2006 By Kenneth G. Dyall, Knut Faegri
• The theories of chemistry By Jan C. A. Boeyens

Admittedly these authors do seem to be in a minority. --catslash (talk) 12:49, 23 February 2010 (UTC)

This also goes against the notation I've seen elsewhere in Wikipedia (e.g. D'Alembert operator, Klein-Gordon equation). It seems to me we should add a parenthetical to clarify this. What do you think? Gneisss (talk) 15:52, 25 July 2016 (UTC)

Notational Differences

There seems to be a bit more notational difference in the use of 4-vectors than in a lot of other physics. I will simply add the same note here that is in the article... It applies to the use of the box symbol as well.

Regarding the use of scalars, 4-vectors and tensors in physics, various authors use slightly different notations for the same equations. For instance, some use $m$ for invariant rest mass, others use $m_{o}$ for invariant rest mass and use $m$ for relativistic mass. Many authors set factors of $c$ and $\hbar$ and $G$ to dimensionless unity. Others show some or all the constants. Some authors use $v$ for velocity, others use $u$. Some use $K$ as a 4-wavevector (to pick an arbitrary example). Others use $k$ or $\mathbf {K}$ or $k^{\mu }$ or $k_{\mu }$ or $K^{\nu }$ or $N$, etc. Some write the 4-wavevector as $({\frac {\omega }{c}},\mathbf {k} )$, some as $(\mathbf {k} ,{\frac {\omega }{c}})$ or $(k^{0},\mathbf {k} )$ or $(k^{0},k^{1},k^{2},k^{3})$ or $(k^{1},k^{2},k^{3},k^{4})$ or $(k_{t},k_{x},k_{y},k_{z})$or $(k^{1},k^{2},k^{3},ik^{4})$. Some will make sure that the dimensional units match across the 4-vector, others don't. Some refer to the temporal component in the 4-vector name, others refer to the spatial component in the 4-vector name. Some mix it throughout the book, sometimes using one then later on the other. Some use the metric (+---), others use the metric (-+++). Some don't use 4-vectors, but do everything as the old style E and 3-vector p. The thing is, all of these are just notational styles, with some more clear and concise than the others. The physics is the same as long as one uses a consistent style throughout the whole derivation.208.104.19.227 (talk) 15:57, 1 August 2016 (UTC)

A good working convention for 4-vector notation based on:

Rindler, Wolfgang Introduction to Special Relativity (2nd edn.) (1991) Clarendon Press Oxford ISBN 0-19-853952-5

4-vector:

$\mathbf {A} =(a^{0},\mathbf {a} )=A^{\mu }=(a^{0},a^{i})=(a^{\mu })=(a^{0},a^{1},a^{2},a^{3})==>(a^{t},a^{x},a^{y},a^{z})$ *Rindler allows $a^{i}=A^{i}$, pg.56*

Greek index {0..3}, Latin index {1..3}

[UPPERCASE] for 4-vectors and tensors: $\mathbf {A} =A^{\mu }$ or $F^{\mu \nu }$, exception Minkowski metric $\eta _{\mu \nu }$

[lowercase] for scalars, 3-vectors, and individual components: $(a^{0},\mathbf {a} )=(a^{0},a^{i})==>(a^{t},a^{x},a^{y},a^{z})$, exception energy $E$

Individual components will tend to have tensor indices or dimensional basis labels: $(a^{\mu })=(a^{0},a^{i})==>(a^{t},a^{x},a^{y},a^{z})$ or $(a^{t},a^{r},a^{\theta },a^{z})$

[non-bold] for tensor index notation and individual components: $A^{\mu }=(a^{0},a^{i})$

[bold] for vectors of either sort or tensor component groups: 4-vector $\mathbf {A}$, 3-vector $\mathbf {a}$, tensor component group: the EM fields $\mathbf {e}$ and $\mathbf {b}$, which act like 3-vectors in classical EM

$F^{\mu \nu }=\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }={\begin{bmatrix}0&-e_{x}/c&-e_{y}/c&-e_{z}/c\\e_{x}/c&0&-b_{z}&b_{y}\\e_{y}/c&b_{z}&0&-b_{x}\\e_{z}/c&-b_{y}&b_{x}&0\end{bmatrix}}$

3-electric field $\mathbf {e} =(e_{x},e_{y},e_{z})$, 3-magnetic field $\mathbf {b} =(b_{x},b_{y},b_{z})$ *Note these are not the spatial components of 4-vectors however*

4-EM vector potential $\mathbf {A} =A^{\mu }=\left({\frac {\phi }{c}},\mathbf {a} \right)$ *Rindler had used 4-potential $\mathbf {\phi ^{\mu }} =(\phi ,c\mathbf {w} )=cA^{\mu }$, pg.107*

4-gradient $\mathbf {\partial } =\partial ^{\mu }=\left({\frac {\partial _{t}}{c}},-\mathbf {\nabla } \right)$ *Rindler had used $E_{\mu \nu }=\mathbf {\phi } _{\nu ,\mu }-\mathbf {\phi } _{\mu ,\nu }=cF_{\mu \nu }$ comma gradient notation, pg.104*

These are all easy to implement in HTML or Wikipedia and show up well in the various browsers, and you are not limited to a certain font which may or may not work on some browsers. Also, the meaning of each type of object is very clear, whether you mean a tensor, a 4-vector, a 3-vector, a scalar, an individual component, etc.

And to these I usually add the following:

$\mathbf {\partial } \cdot \mathbf {X}$ is a 4-vector style, which is typically more compact and can use dot notation, always using bold uppercase to represent the 4-vector.

$\partial ^{\mu }\eta _{\mu \nu }X^{\nu }$ is a tensor index style, which is sometimes required in more complicated expressions, especially those involving tensors with more than one index, such as $F^{\mu \nu }=\partial ^{\mu }A^{\nu }-\partial ^{\nu }A^{\mu }$.

'c' factor in the temporal component, which allows the entire 4-vector (and its individual components) to have consistent dimensional units and for the spacetime 4-vector name to match the spatial 3-vector name, which is helpful for Newtonian limiting cases:

4-velocity $\mathbf {U} =\gamma (c,\mathbf {u} )$ {SI units [m/s]}

4-momentum $\mathbf {P} =m_{o}\mathbf {U} =\left({\frac {E}{c}},\mathbf {p} \right)=(mc,\mathbf {p} )$ {SI units [kg m/s]}

4-acceleration $\mathbf {A} =\gamma (c{\dot {\gamma }},{\dot {\gamma }}\mathbf {u} +\gamma {\dot {\mathbf {u} }})$ {SI units [m/s^2]}

4-wavevector $\mathbf {K} =\left({\frac {\omega }{c}},\mathbf {k} \right)$ {SI units [rad/m]} a temporal angular frequency/c and spatial 3-wavevector

4-acceleration $\mathbf {A} =\gamma (c{\dot {\gamma }},{\dot {\gamma }}\mathbf {u} +\gamma {\dot {\mathbf {u} }})$ {fully relativistic}

4-acceleration $\mathbf {A} ==>(0,{\dot {\mathbf {u} }})=(0,\mathbf {a} )$ {in the Newtonian limiting case v<<c}

Also, always denote the rest case senario with a naught, so there is no chance of misinterpretation.

$E=mc^{2}=\gamma m_{o}c^{2}=\gamma E_{o}$

$E$ = relativistic energy

$E_{o}$ = rest energy

This way it is also easier to spot the Lorentz scalar invariants:

$\mathbf {P} =m_{o}\mathbf {U}$

$\mathbf {K} =\left({\frac {\omega _{o}}{c^{2}}}\right)\mathbf {U}$

$\mathbf {P} =\hbar \mathbf {K}$

$\mathbf {P} =i\hbar \mathbf {\partial }$, the QM Schrödinger relations $E=i\hbar \partial _{t}$ and $\mathbf {p} =-i\hbar \mathbf {\nabla }$

John Wilson (Scirealm) 10 April 2016

http://www.scirealm.org/4Vectors.html — Preceding unsigned comment added by 208.104.19.227 (talk) 16:01, 1 August 2016 (UTC)

There are several references in the article to the use of the "gradient" in QM. IMO, this (a) is completely out of place in an article that is intrinsically about a classical operator, and (b) is making a leap based on the similarity of notation, but in QM it is not appropriate to call it a gradient operator, as it operates on a (wave)function that is not a scalar function of the point in spacetime. Aside from the WP:OR aspects. I'm not likely to engage much here (if I did, I would delete it all, considering my opinion), but I thought I'd bring some attention to this. —Quondum 01:42, 5 August 2017 (UTC)

A quick skim of "Modern Elementary Particle Physics: The Fundamental Particles and Forces", by Gordon Kane, a textbook about relativistic quantum mechanics, makes extensive use of the 4-gradient $\partial ^{\mu }$ in chapters 2 - 8. Those chapters are: (2) Relativistic Notation, Lagrangians, Currents, and Interactions. (3) Gauge Invariance. (4) Non-Abelian Gauge Theories. (5) Dirac Notation for Spin. (6) The Standard Model Lagrangian. (7) The Electroweak Theory and Quantum Chromodynamics. (8) Masses and the Higgs Mechanism. The 4-gradient is not just a classical SR 4-vector. It has great utility in modern quantum theory. Please do your research before you talk about deleting stuff. Otherwise, I have great respect for your wiki edits.

Also, from Quantum Field Theory, pg. 39, by John W. Norbury: "A quick route to the KGE is with the relativistic formula $p^{2}=p_{\mu }p^{\mu }=m^{2}$ giving $p^{2}-m^{2}=0$ and $(p^{2}-m^{2})\phi =0$ and $p^{2}->-\Box ^{2}$ giving $(\Box ^{2}+m^{2})\phi =0$. The KGE can be written in terms of 4-vectors, $(p^{2}-m^{2})\phi =0$ and is therefore manifestly covariant..." — Preceding unsigned comment added by 208.104.84.162 (talk) 12:47, 8 August 2017 (UTC)

References

Covariant derivative?

Something is amiss. I found a website claiming that although $\partial _{\mu }$ is allowed, $\partial ^{\mu }$ isn't and in principle, one is actually using a covariant derivative in Minkowski space. Some precision or clarification on that issue would be helpful.TonyMath (talk) 15:17, 11 July 2015 (UTC)

What is the website? Or the info from the website? There are quite a few 4-Gradient Usage examples that give valid physics using either a lower or upper index on the 4-gradient. Everything I have ever read shows that index raising/lowering is valid on physical 4-vectors using the Minkowski flat spacetime metric $\eta ^{\mu \nu }$ in SR, and using the more general $g^{\mu \nu }$ in GR. My understanding is that covariant derivative is only necessary in curved spacetime GR.

208.104.19.227 (talk) 15:32, 2 August 2016 (UTC)