Talk:Four-vector

4-position

We should add the statement:

$\mathbf {X} \cdot \mathbf {X} =X^{\mu }X_{\mu }=(c\tau )^{2}=s^{2}\,\!$

implying

${\rm {d}}\mathbf {X} \cdot {\rm {d}}\mathbf {X} ={\rm {d}}X^{\mu }{\rm {d}}X_{\mu }=c^{2}{\rm {d}}\tau ^{2}={\rm {d}}s^{2}\,\!$

$\rightarrow {\frac {{\rm {d}}X^{\mu }{\rm {d}}X_{\mu }}{{\rm {d}}\tau ^{2}}}={\frac {{\rm {d}}X^{\mu }{\rm {d}}X_{\mu }}{{\rm {d}}\tau {\rm {d}}\tau }}=V^{\mu }V_{\mu }=c^{2}\,\!$

or lines to that effect. This would make the relation beteen the magnitudes of the 4-position and velocity clearer. There is currently no statement of this. Referance to add: Dynamics and Relativity, J.R. Forshaw, A.G. Smith, Wiley, 2009, ISBN 978 0 470 01460 8.-- F = q(E + v × B) 13:52, 12 January 2012 (UTC)

No response? Geuss i'll just add it, and while at it clean up the article here and there. Ideally equations shouldn't be titles, it looks funny, so those subsections should become one.-- F = q(E + v × B) 15:53, 29 January 2012 (UTC)

I 4-got to mention: there is not even an upfront mention 4 what a 4-vector generally is. It kicks off with 4-position + spacetime, but then the inner product of two general 4-vectors, then 4-vectors in dynamics. It’s almost on the right track, but should state the general 4-vector before the scalar product. In the lead, the 4-position is there anyway, so shifting the 4-position after talking about how the scalar product leads to invariant quantities.-- F = q(E + v × B) 16:06, 29 January 2012 (UTC)

A four-position isn't a four-vector because it doesn't transform like a like vector under a general transformation e.g. when you change your origin. See for example the definition of a vector in Gravitation (Misner, Wheeler, Thorne) etc.The simplest vector is a difference in position, not the position itself. A vector is easiest to be defined using a differential, i.e. a velocity. Could someone please rewrite the article section to clarify this? 2001:690:21C0:F097:E564:D20D:C76F:1A2B (talk) 18:56, 27 January 2016 (UTC)

A 4-vector is most commonly defined as a set of quantities transforming the same way as the position vector under homogeneous Lorentz transformations. I'll check MTW though when I get home. YohanN7 (talk) 11:47, 28 January 2016 (UTC)

In flat spacetime, the spacetime manifold can be identified with the tangent space at any event, so in that sense 4-position relative to some specified origin event could be regarded as a 4-vector. (4-vectors exist only within a tangent space.) In curved spacetime, no such identification is possible. That probably needs to be explained in the article using simpler language for the benefit of readers who don't know about manifolds or GR. -- Dr Greg  talk  20:40, 28 January 2016 (UTC)

Yes. In Minkowski space this is made somewhat clear (I hope). The last paragraph in the lead limits the scope of this article to SR. Could we just link (or copy) the explanation in Minkowski space (provided it stands up for inspection)? YohanN7 (talk) 11:15, 29 January 2016 (UTC)

It is linked, and I believe in the right place where the position four-vector is introduced in the lead. YohanN7 (talk) 12:17, 29 January 2016 (UTC)

Four-heat flux

The definition currently given for heat flux Q = (0,q) is incomplete. It can't possibly be true in all frames, so in which frame is it true? -- Dr Greg  talk  01:56, 14 September 2012 (UTC)

I agree. The timelike component would have to be something like the heat energy density to be Lorentz-invariant, at a guess. Does someone have the reference given to be able to check? I unfortunately do not. — Quondum 05:01, 14 September 2012 (UTC)

I agree as well that we would expect a heat energy density for the timelike component but the reference actually gives everything exactly as I presented it. The Q is in the rest frame of the fluid, apologies for that I'll put that statement in. Maschen (talk) 08:52, 14 September 2012 (UTC)

Thanks, that's better, though the mention of "fluid" rather than "solid" means that the rest frame is only defined locally, i.e. there's no global rest frame. For what it's worth, I've just discovered the article Relativistic heat conduction which seems to define heat flux as a covector:

$Q_{\alpha }=-k{\frac {\partial \theta }{\partial x^{\alpha }}}$

where k is thermal conductivity and θ is temperature. -- Dr Greg  talk  11:15, 14 September 2012 (UTC)

Yeah, I just discovered the article now that you pointed to the link! Maybe this should replace what I added? Maschen (talk) 11:25, 14 September 2012 (UTC)

Since this article (Four-vector) is explicitly about relativistic vectors, and there is reason to doubt its correctness, it is pointless simply quoting the reference verbatim. If some reference used in Relativistic heat conduction gives the relativistic 4-vector (and the article seems to agree that there is a density term as the time component), then it should be used by preference. — Quondum 15:29, 14 September 2012 (UTC)

Agreed, and partially implied above. MTW is the only in-depth book on GR I have, which is why I keep adding it everywhere that happens to be relevant. I'll change it now. Maschen (talk) 15:45, 14 September 2012 (UTC)

still in the process of rewriting...

Before anyone says the first section on mathematics is too long, it probably is, and will be trimmed, but it is necessary content for a full coverage of four-vectors. Before the version currently talked about, the article discussed differentials, inner products, etc. without a sound section developing these ideas applied specifically for any and all four vectors, which is silly. M∧Ŝc²ħεИ_τlk 10:38, 17 July 2013 (UTC)

Basis transforms are general linear, not Lorentz

The lead makes the claim that "four-vectors transform by the Lorentz transformations with a change of basis". Their components (not the vectors themselves!) transform under the general linear transform with a change of basis. If one restricts bases to "standard" bases, the transformations of the components are indeed the Lorentz transforms, but we would then need to define a standard basis first. The distinguishing feature of a four-vector is that it is an element of a Minkowski vector space, and this is all that needs to be said in the lead. —Quondum 18:47, 9 January 2015 (UTC)

Yes, incorrectly worded (can't remember if I or someone else wrote these things, probably a mixture). Fixed anyway. M∧Ŝc²ħεИ_τlk 23:49, 11 January 2015 (UTC)

I've reworded it somewhat, to emphasize the magnitude-preserving origin of the Lorentz transforms. —Quondum 03:51, 12 January 2015 (UTC)

Notation

I've just noticed that the article's notation convention is immediately broken after its statement: the 4-vectors $\mathbf {e} _{\alpha }$ are lowercase, not the uppercase of the convention! -- Dr Greg  talk  23:53, 9 January 2015 (UTC)

Agreed. Although it is not unusual to use a different convention for the basis vectors, I think we should make it consistent in this article. Also the use of boldface should be restricted to only 3-d vectors, as it is (in my limited memory) unusual to use boldface for 4-vectors. I'll look at making this consistent. —Quondum 00:01, 10 January 2015 (UTC)

Not too obvious what to do. For now, I think we could change the basis 4-vectors into upper case, leaving 4-vectors bold. I'd also prefer to make all bold vectors italic, though this is a personal preference. —Quondum 00:15, 10 January 2015 (UTC)

Ack, yes... My fault as usual. One solution: all 3-vectors in usual bold or bold-italic with their components in usual italic

$\mathbf {x} =(x_{1},x_{2},x_{3})\,,\quad {\boldsymbol {x}}=(x_{1},x_{2},x_{3})$

and all 4-vectors in bold serif with their components in serif

${\boldsymbol {\mathsf {x}}}=({\mathsf {x}}_{0},{\mathsf {x}}_{1},{\mathsf {x}}_{2},{\mathsf {x}}_{3})$

(unfortunately it does not seem possible to create sans serif italic on Wikipedia... it certainly is in real LaTeX). M∧Ŝc²ħεИ_τlk 23:49, 11 January 2015 (UTC)

Given this constraint, perhaps we should leave the upper case as the notation of a four-vector. Using sans-serif for this distinction is also (I think) very unusual. It is quite common to use upper case for tensors, including vectors, so I'm happy leaving that part as is. For me, the only question would then be whether we should change the roman bold to italic bold. —Quondum 00:05, 12 January 2015 (UTC)

In some sources sans serif is used to differ from a 3-vector (e.g. Goldstein's Classical mechanics, or MTW Gravitation), but then again sans serif may be used for tensors...

I don't mind what notation is used, feel free to change for consistency. Maybe make the 4-vectors bold italic and capital, for clearer difference, but someone could straighten them.

One quibble (again, partly my fault), the basis vectors E could conflict the E-field in EM. Not a problem for this article since the EM field tensor is not a 4-vector, but the notation is not so ideal. This is why I'm inclined for bold sans serif for 4-vectors. M∧Ŝc²ħεИ_τlk 00:36, 12 January 2015 (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 67.197.233.24 (talk) 13:48, 10 April 2016 (UTC)

Respecting variance

This revert does not fully honour the variance implied by the notation. It should probably read something like

$\mathbf {P} =P_{\alpha }\gamma ^{\alpha }=P_{0}\gamma ^{0}+P_{1}\gamma ^{1}+P_{2}\gamma ^{2}+P_{3}\gamma ^{3}={\dfrac {E}{c}}\gamma ^{0}-p_{x}\gamma ^{1}-p_{y}\gamma ^{2}-p_{z}\gamma ^{3}$

or

$\mathbf {P} =P^{\alpha }\gamma _{\alpha }=P^{0}\gamma _{0}+P^{1}\gamma _{1}+P^{2}\gamma _{2}+P^{3}\gamma _{3}={\dfrac {E}{c}}\gamma _{0}+p_{x}\gamma _{1}+p_{y}\gamma _{2}+p_{z}\gamma _{3},$

though I have not checked the signs for non-numeric subscripts (rightmost expression) properly. The position of the numeric subscript can change the sign. For example, we can have P² = −P₂, and γ² = −γ₂. But the summation over indices of opposite variance as here never takes negative signs, even in Feynman slash notation (alphabetic subscripts here are not an indication of variance). —Quondum 14:17, 21 May 2015 (UTC)

I am the one who originally added the section, and maybe it goes off a tangent and could be deleted altogether. The original motivation was the alternative expression of four-vectors using matrices, which may be of interest to some. To answer your question, you Quondum are correct that the metric is not needed in a contraction between upper and lower indices (only if one or other has an index lowered/raised by the metric). User:DrGreg correctly back-reverted the IP. M∧Ŝc²ħεИ_τlk 20:49, 21 May 2015 (UTC)

energy–momentum relation:

in Germany we call it "relativistic Pythagoras". Ra-raisch (talk) 17:59, 24 August 2017 (UTC)