Talk:Four-vector

message: 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, .--  F = q(E + v × B) 13:52, 12 January 2012 (UTC)
replies: text: 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)
replies: text: 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)

text: 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)
replies: text: 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)
replies: text: 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)

text: 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)
thread_title: 4-position

message: The definition currently given for heat flux  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)
replies: text: 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)
replies: text: 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)

text: 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)

text: 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)
replies: text: 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)
replies: text: 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)
thread_title:  Four-heat flux 

message: 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∧Ŝc2ħεИτlk 10:38, 17 July 2013 (UTC)
thread_title:  still in the process of rewriting... 

message: 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)
replies: text: Yes, incorrectly worded (can't remember if I or someone else wrote these things, probably a mixture). Fixed anyway. M∧Ŝc2ħεИτlk 23:49, 11 January 2015 (UTC)
replies: text: I've reworded it somewhat, to emphasize the magnitude-preserving origin of the Lorentz transforms. —Quondum 03:51, 12 January 2015 (UTC)
thread_title:  Basis transforms are general linear, not Lorentz 

message: 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 , and . 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)
replies: text: 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∧Ŝc2ħεИτlk 20:49, 21 May 2015 (UTC)
thread_title:  Respecting variance 

message: in Germany we call it "relativistic Pythagoras". Ra-raisch (talk) 17:59, 24 August 2017 (UTC)
thread_title:  energy–momentum relation: