Talk:Four-velocity

Magnitud of Four-Velocity
I think it is wrong to claim that it is always c or -c. If we have a stationary spacetime with Killing Vector $$ \xi_t={\partial \over \partial t} $$, then the four-velocity of a stationary observer should be proportional to $$\xi_t$$, i.e. it should be $$g_{tt}$$.

unnamed thread
Simplified presentation a little while making notation more consistent with other relativity articles. ---Mpatel 19:06, 14 Jun 2005 (UTC)

Alternative approach using complex numbers

 * A work in progress


 * First Harmonic 22:48, 28 December 2006 (UTC)

Removed contentious phrase
I have removed the following sentence:


 * This observation, though trivial (as we will see from the formulas below), has a rather nice interpretation: in spacetime, an object is always in motion (at the speed of light!); it's just that in a rest frame, this motion is all in the time direction.

In my view this is misleading, although statements to this effect do appear in some "coffee-table" popularisations of relativity. An object moves in space. In spacetime it is a worldline and does not move at all. The four-velocity is just the normalised tangent vector to the worldline.--Dr Greg (talk) 16:58, 3 October 2008 (UTC)

Interpretation
Therefore light, and anything else traveling at light speed, do not experience the "flow" of time.

Well, if we, at rest, or uniformly moving (our four-velocity pointing in direction of our proper time axis), experience the flow of (our proper) time, couldn't we conjecture that anything traveling at speed of light along arbitrary spatial (spatial to us) axis would probably experience the flow of its own proper time just as well, like we do? OTOH, we know that speed of light remains the same in all frames of reference, which means that all observers, regardless of where their respective four-velocity vectors point will experience the light as moving perpendicular to their respective four-velocity vectors. What that tells us about the path or shape of photons and other massless particles? Or, more bluntly, do we need more dimensions then four to explain it? --147.91.1.43 (talk) 13:59, 9 October 2009 (UTC)

Does the "Theory of Relativity" section assume constant velocity?
It seems that everything in the "Theory of Relativity" section assumes constant velocity, without saying so. Otherwise, would we have $$t=\gamma \tau$$? Also, I find writing things like $$x^1(\tau)=x^1(t)$$ confusing. Isn't it bad form? I don't feel I know enough about this area to edit anything. — Preceding unsigned comment added by 66.188.89.180 (talk) 18:26, 8 January 2014 (UTC)


 * The Special Theory of Relativity is formulated with a definition of Four-velocity such that, for any object, it is always of constant magnitude, although its direction can change. It is a unit tangent vector.  With this definition the Theory can be interpreted as a (multi-dimensional) geometry. For objects in free-fall the integral of the four-velocity is a geodesic (think "straight line" in flat space) in space-time. Cloudswrest (talk) 23:13, 17 October 2014 (UTC)

This was an error. I replaced the 3 ts with taus. 200.83.115.207 (talk) 19:51, 17 October 2014 (UTC)

Four-velocity is not a four vector under time inversion.
The proper time is not an invariant. It is a pseudo-invariant, with time-reversed observers disagreeing with each other on the sign of the proper time. You can see this because if t'=-t, dτ'=-dτ. The 4-velocity is the derivative of the 4 position (a 4-vector) with the proper time (a pseudo scalar under time reversal). This means that the 4-velocity is a pseudo 4-vector under time reversal. — Preceding unsigned comment added by 124.170.85.44 (talk) 07:46, 7 January 2015 (UTC)


 * The whole concept is a little murky; in particular, your statement is not valid until proper time has been suitably defined, and then only under a strange convention. I have yet to see proper time defined so as to be dependent on the reference frame of some arbitrary observer, as you have implicitly assumed. If you assume that the direction of proper time is assigned by some observer-independent convention to the world line (conceptually as the direction of time experienced by the entity that the world line), a time-reversed observer still sees the same proper time as a function of position along the worldline. A sensible convention would assign a direction of proper time to every (time-like) worldline so that the directions are all consistent with continuous deformations of any such worldline into another (all the while keeping the tangents time-like). Without having explicitly addressed which convention is being used, your statement above does not make sense. And as I indicated in my edit comment, this kind of thing does not belong in the lead in any event. As to it not being a four-vector, I agree that its interpretation as one must be highly constrained, but that is already clarified further down in the lead. —Quondum 14:47, 7 January 2015 (UTC)


 * I agree with Quondom that proper time has to be defined to be coordinate-independent; a change of coordinates cannot change the direction in which proper time increases. So, either you impose a restriction on allowable coordinates that prevents coordinate time being reversed (the article seems to be mostly written under an assumption of Minkowski coordinates rather than general coordinates), or else you allow the possibility that coordinate time could go backward relative to proper time (so $$t = \pm \gamma \tau$$, depending on choice of coordinates, and the temporal component of 4-velocity can be negative).


 * User:124.170.85.44's rewrite is inappropriate for several reasons:
 * the issues raised don't belong in the lead;
 * the terminology "vector with respect to boosts, ..." is technically incorrect; something either is or isn't a vector, without qualification.
 * --  Dr Greg   talk  19:59, 7 January 2015 (UTC)

"I agree with Quondom that proper time has to be defined to be coordinate-independent; a change of coordinates cannot change the direction in which proper time increases." The problem is that I don't think the direction of proper time can be defined in a co-ordinate-independent way, because special relativity exhibits T-symmetry. If one observer sees a subluminal particle travelling along a wordline, passing first through four-position A, and then through B, there's nothing to stop another observer (with a clock running backward compared with the first), to see the same particle going first from B, and then to A. Who is right? It's not decideable under the postulates of special relativity. It can be proved that the proper time squared is an invariant for all inertial observers (the infinitesimal proper time squared is proportional to the infinitesimal space-time interval between two infinitesimally separated events), but not the proper time itself.

To look at the problem another way, think about how the four-velocity is calculated in co-ordinate time. Consider how 4-position transforms under time reversal T: $$T:X=(t,\vec{x}) => X^\prime = (-t,\vec{x}) = diag(-1,1,1,1) X$$.

The 4-velocity under time reversal transforms like this:

$$T:U=\gamma(v)(1,\vec{v}) => U^\prime = \gamma(v)(1,-\vec{v})=diag(1,-1,-1,-1)U$$.

So if you choose to define a four-vector as a quantity that transforms the same way as the four-position under all transformations in $$O(1,3)$$, the four-velocity is not a four vector, because it doesn't transform the same way under time reversal, it acquires an extra negative sign. It is however, a pseudo four-vector (it transforms the same way up to a sign under all transformations). I'm happy not to go into detail in the lead, so long as we don't call it a four-vector. — Preceding unsigned comment added by 124.170.85.44 (talk) 04:18, 8 January 2015 (UTC)


 * IMO, you are violating multiple WP guidelines, such as WP:OR, WP:RS and WP:CON, no matter how convinced you are of your own argument. Please start acting within the guidelines of the community, rather than pushing your own convictions. And start signing your posts. —Quondum 04:41, 8 January 2015 (UTC)

I'm new to editing (but not to reading!) wikipedia, so I apologize if I'm violating guidelines. However, I don't think this is original research; it follows quite immediately from definitions. Surely you don't want incorrect statements in the lead? If you can find anything actually wrong with my argument above, I'd like to hear it, but I think I've laid out the problem quite well. Either special relativity doesn't exhibit T-symmetry, or four-velocity is not a four-vector as that term is currently defined in wikipedia. - Sam — Preceding unsigned comment added by 124.170.85.44 (talk) 05:04, 8 January 2015 (UTC)
 * Can you provide any reputable sources which explain how and why 4-velocity is a pseudo-4-vector rather than a 4-vector? Zueignung (talk) 05:36, 8 January 2015 (UTC)

I haven't checked for other sources. I'll do that now. But first, do other editors see the sense of my argument? Does anyone disagree with the logic? Further, has anyone got a reputable source for the contention that proper time itself (rather than proper time squared) is an invariant? 124.170.85.44 (talk) 05:53, 8 January 2015 (UTC)

OK, I've found a couple of sources. The first is http://philosophyfaculty.ucsd.edu/faculty/wuthrich/PhilPhys/MalamentDavidB2004StudHistPhilModPhys_TimeReversalInv.pdf. Read from page 306, "time reversal invariance." The second source is perhaps a little more clear http://philsci-archive.pitt.edu/3280/1/arntzenius_greaves_TRCE.pdf. Read from page 8. This passage especially; "There are two fundamental types of objects in a classical electromagnetic world. There are the world-lines of charged particles, and there is the electromagnetic field. Now, the dynamics happens to be such that it will be convenient, mathematically, to represent the motions of particles by means of four-velocities, where the four-velocity at any point on the worldline is tangent to the worldline at that point. The crucial fact now is that a world-line does not have a unique tangent vector at a point: at each point on a world-line, there is a continuous infinity of four-vectors that are tangent to the world-line at the point in question. We can narrow things down somewhat by stipulating that four-velocities are to have unit length, but this still does not quite do the trick: one can associate two unit-length four-vectors that are tangent to the world-line at the point in question (if $$v_a \in T_p$$ is one, then $$-v_a$$ is the other."

Also this: "If $$v_a$$ is the four-velocity, i.e. is the unit-length future-directed tangent, to a given worldline at some point $$p$$ relative to our original choice of temporal orientation, then $$-v_a$$ will be the four-velocity relative to the opposite choice of temporal orientation."

Sorry, retroactively signing 124.170.85.44 (talk) 10:42, 8 January 2015 (UTC)


 * If you were the experienced with Wikipedia (or any other encyclopedia), then you would be aware that it matters little what you think is logically correct. What matters what is in reliable sources. Proper time is the Lorentz-invariant distance (in any Lorentz frame, modulo a positive constant, the speed of light) between two time-like separated events on a world-line. The concept of time-like is an Lorentz-invariant one. Proper time is easily seen to be a scalar, see the formulae in proper time, the relevant one for this discussion being
 * $$\Delta \tau = \sqrt{\left(\Delta t\right)^2 - \frac{\left(\Delta x\right)^2}{c^2} -  \frac{\left(\Delta y\right)^2}{c^2} - \frac{\left(\Delta z\right)^2}{c^2}}.$$
 * Any minus-sign introduced by a non-orthocronous Lorentz transformation (of the two events in the definition of proper time between them), say $∆t → ∆t' = −∆t$, cancel when differences are squared. You will not find proper time presented in the literature as a pseudo-scalar. YohanN7 (talk) 10:45, 8 January 2015 (UTC)

And if you were experienced with special relativity, you'd know it was spelled "Lorentz." The formula you've referenced shows that all inertial observers agree on the the magnitude of the proper time difference between two time-like separated events, which is not in dispute. Two observers who disagree on the direction of time could both take the positive square root of $$(\Delta \tau)^2$$, and agree on the number they get. However, they will be in dispute as to which event occured first, and which second. As to sources, I've already cited two. How many do you want? 124.170.85.44 (talk) 11:03, 8 January 2015 (UTC)

Also this, https://books.google.com.au/books?id=DobLa3hjAl0C&pg=PA487&lpg=PA487&dq=time+reversal+four-velocity&source=bl&ots=qzINJa4DeA&sig=gDMM1nYAhBDd2XEMxMrF9DPb6-o&hl=en&sa=X&ei=mo6uVLrmJ4ai8QWTsILoAg&ved=0CCcQ6AEwAzgK#v=onepage&q=time%20reversal%20four-velocity&f=false 124.170.85.44 (talk) 14:09, 8 January 2015 (UTC)


 * (ec) Yeah, "Lorentz". Thought something was funny, but couldn't put my finger on it I have, after all, written a mile-long article about Lorentz transformations in Wikipedia.


 * You have not supplied any reliable sources. A philosophers pdf doesn't count even if it would support your argument (these two don't). I suggest you take this to the reference desk (either in math or physics (natural sciences?)) if you have further questions. This page is for discussion about improvement of the article, not arguments about what is right or wrong.


 * Your mistakes are three. Proper time is defined in terms of more basic entities, namely two events in space-time being time-like separated. It is these you need to Lorentz transform and make your calculation. Proper time is a scalar period. The next mistake is that you treat proper time as the zeroth component of a 4-vector (like energy). It isn't, except in particular frames (related by pure rotations in space). While it is true that you can use proper time as the time-coordinate in certain frames, in the transformed the zeroth component isn't proper time of anything any more in general. The 4-velocity is defined in terms of proper time ( a scalar ), not the zeroth component of a 4-vector. The third mistake is that you claim two observers will disagree on which event occurred first. You need proper time to decide, not the time coordinate. All observers agree on this in terms of proper time. Both can (and should to settle questions of this sort) transform to a frame where proper time is the zeroth coordinate. YohanN7 (talk) 14:14, 8 January 2015 (UTC)

Okay, I think at least the point has been made that this discussion belongs on the talk page without edits being made to the article, which should make it more relaxed; it is interesting after all. For now I'll mention that none of the quotes above from sources should be interpreted as corroboration of four-velocity as a pseudo-vector. For example, "a continuous infinity of four-vectors that are tangent to the world-line at the point in question" is best interpreted as a reference to a projective space, and makes no reference to any sign change under transformation. Also, "original choice of temporal orientation" makes no reference to an observer's time coordinate. A choice of temporal orientation along a worldline is arbitrary, but nevertheless observer-independent. A parallel example is a choice of orientation: the choice is arbitrary, but is coordinate/transform-independent. The orientation should be considered to be additional structure of the manifold rather than depending on a choice of coordinates. Temporal orientation can be global on a Minkowski space (and many other Lorentzian manifolds), and hence may similarly be defined as additional observer-independent structure associated with the space. There are several consistent ways of approaching this choice, so it is necessary to be explicit about the choices made (or definitions). And though proper time along a worldline can be defined as a pseudo-scalar, this introduces IMO valueless mathematical ugliness, and we have yet to find a source that does this. It may also be noted that four-velocity can be defined without any reference to proper time, and that examining spacelike worldlines and their tangents can be quite illuminating for this discussion. And just to further confuse things, one can coordinatize Minkowski space with four spatial coordinates. :) —Quondum 16:04, 8 January 2015 (UTC)

Proper time cannot be defined as a pseudo-scalar because it wouldn't be proper time. If it helps, always think of Lorentz transformations as passive transformations, i.e. change of coordinates. Proper time evolves in the right direction (forward as defined in the rest frame!) for all world lines of massive particles, no matter what coordinates you use. If $τ_{1}, τ_{2}$ are the proper times for two events in the rest frame, then in any two frames obtained from the rest frame by a Lorentz transformation (possibly internally related by a "time-reversing" transformation) L* and L**, $τ_{1} = τ*_{1} = τ**_{1}, τ_{2} = τ*_{2} = τ**_{2}$. That is, $τ_{1}, τ_{2}$ are not to be confused with $t*_{1}, t*_{2}$ and $t**_{1}, t**_{2}$. Proper time is a geometric object, just like 4-vectors and higher order things, not to be treated as a coordinate in this context, though it happens to equal zero component of the coordinates in the rest frame. What I'm saying is that the direction of proper time along a world line isn't an arbitrary choice, while the "orientation", i.e. your coordinates are. YohanN7 (talk) 17:08, 8 January 2015 (UTC)
 * YohanN7, I'd say you have this exactly wrong. Given a space R4, a Lorentzian metric tensor g on this space, and a time-like world line in this space (and no further structure), how do you find the direction of "proper time"? There is no forward direction of time, any more than there is a forward direction of space. There are two ways of assigning a proper time parameter to any time-like world line (or proper distance to a space-like curve), up to addition of a constant, and this is perfectly symmetric. A rest frame does not define a "forward direction". And temporal orientability of a Lorentzian space is a coordinate-independant concept, just like the orientability of a vector space. —Quondum 18:02, 8 January 2015 (UTC)
 * Rest frame of particle tracing out world line and a clock brought along with it. The clock measure proper time. It is not arbitrary. YohanN7 (talk) 18:39, 8 January 2015 (UTC) YohanN7 (talk) 18:39, 8 January 2015 (UTC)
 * That's circular (in effect: "I'm assigning a direction of time by defining a clock that has a given forward direction, therefore the choice of time direction is not arbitrary"). A clock is, in its simplest incarnation, an assignment of proper time to a world line in such a way as to be compatible with the metric tensor. This can be done in two ways. Alternately, if you wish to refer to a physical universe, you are again externally breaking the symmetry of the system I specified by imposing a pre-decided temporal orientation (additional structure that I had excluded). Given only the structure that I mentioned, there is no way to distinguish two clocks with the same world line, running antiparallel. Re-examine the structure that I defined: the definition incorporates nothing that breaks the temporal symmetry. The microscopic laws of physics as we know them (and especially special relativity) are time-reversal-symmetric (CPT symmetry). finding a time arrow implied by physics is a problem that has vexed many; in every case some external factor has been invoked (e.g. the direction in which we observe entropy increasing, the direction in which we observe the universe expanding; if one assumes objective collapse of the wavefunction, this phenomenon would provide a direction of time). —Quondum 19:14, 8 January 2015 (UTC)


 * Only one direction along a worldline is physically realizable in that you can attach a clock to it. That direction is not arbitrary. What this clock shows is the definition of proper time. And yes, physics, and proper time in particular, refers to a physical universe. Nothing is circular here.


 * Yes, time is a mystery, but not in the context of special relativity and not in the context of this article. YohanN7 (talk) 20:45, 8 January 2015 (UTC)


 * All you're doing is providing a motivation for using a particular definition of proper time, one which happens to be equivalent to a global temporal orientation choice. I happen to think that this is probably the most useful definition, but it is helpful to be clear on our definitions. Since this definition asymmetric with respect to time-reversal, care must be taken if it is used in this context. Using this definition, the question of pseudo-anything becomes clear. —Quondum 22:45, 8 January 2015 (UTC)

YohanN7, let's forget about the clock time of an arbitrarily accelerating particle, and just restrict our thinking to the "lab" frame. In this frame, you could have two observers (by definition at rest with respect to one another), whose clocks run backwards relative to each other. They both sit there and record events in the universe, and they both think the other's time is running backwards. Who is right? Both and neither; it's all relative within special relativity.

Now, you may argue the first postulate of relativity only imposes symmetry in displacements, rotations, and boosts, not to spatio-temporal inversions. In this case, I suggest the symmetry group of special relativity is the restricted Lorentz group, $$SO+(1,3),$$ rather than $$O(1,3).$$ Then there would be no such things as pseudo-tensors of any rank. Alternatively, you could allow time reversal transformations, but impose a particular temporal orientation, as Quondum discusses. Overal CPT symmetry, combined with both broken P, and CP symmetry, suggests that the universe may have a preferred temporal orientation (there's also thermodynamics at the macro-scale). But then we have to bring in some pretty advanced mid-20th century physics, which I don't think belongs in a discussion of a 110 year old theory. The third possibility is you just impose an arbitrary temporal orientation, in the knowledge that the opposite could have been chosen.

There may be other possibilities I haven't considered. Can anyone provide a link as to how this was originally done? If the question wasn't important, and time inversions were just dismissed as unphysical, why does special relativity even discuss non-orthochronous transformations in the first place? Perhaps Mincowski had a reason? 124.170.85.44 (talk) 20:33, 8 January 2015 (UTC)


 * No, let us not forget about the clock of the particle. It is needed for the definition of proper time. Then two observers at rest, each with a clock will find matching times. Proper time simply does not run backwards for any of them. YohanN7 (talk) 20:45, 8 January 2015 (UTC)

OK, but why can't the particle have two clocks, each running backwards relative to each other?124.170.85.44 (talk) 20:57, 8 January 2015 (UTC)
 * Because no clock (except in some bars I've been to) run backwards. A reasonable requirement is to require ( before the experiment is performed, in the lab frame) that they show the same time so we know they are working. YohanN7 (talk) 21:21, 8 January 2015 (UTC)

Then why even talk about time reversal in special relativity? Why not just the restricted Lorentz group as the symmetry of inertial space-time?124.170.85.44 (talk) 21:26, 8 January 2015 (UTC)


 * This is because the full Lorentz group $O(3, 1)$ (or $O(1, 3)$ if you want) is the group that leaves the bilinear form $x ⋅ y = -x_{0}y_{0} + x_{1}y_{1} + x_{2}y_{2} + x_{3}y_{3}$ invariant. This form is what mathematically pops out from demanding the constancy of the speed of light (one could enlarge to the conformal group, but no physics relevant to this has been found in this context).


 * It is interesting to know which laws of physics have time-reversal as a symmetry. It used to be taken for granted that both parity ($x → -x$ etc) and time reversal ($t → -t$) are symmetries (are "conserved" under the respective inversions) of all theories. In the 50's it was discovered that they are not symmetries of the electroweak interaction. There is no conflict with special relativity, it is (at least for parity I think) explained by the fact that certain particles are "left-handed", meaning that in the rest frame in which the particles are created, the spin projection along the direction of momentum can be only one. But don't press me on this because I have only a layman's (if even that) knowledge about it. The combined operations of CPT is a symmetry. YohanN7 (talk) 22:10, 8 January 2015 (UTC)


 * The question of time-reversal symmetry is not directly influenced by the definition of proper time, and the latter most certainly does not influence the symmetry group of physical laws. With an asymmetric definition (e.g. by imposing a global choice of temporal orientation), one naturally has to be careful about its use in the examination of symmetries. —Quondum 22:51, 8 January 2015 (UTC)


 * Global choice of temporal orientation? We don't make these choices, nature does. Edit: The choice of direction for proper time is not "global" (whatever you mean by that), it applies to world lines. End edit. Proper time has, as opposed to space or time coordinates, a preferred direction. YohanN7 (talk) 23:33, 8 January 2015 (UTC)
 * What I mean is that even before applying the "preferred direction" inferred from nature, we can show that any local choice of temporal direction at one point of any world line immediately implies a consistent choice throughout the manifold on every timelike world line, subject only to a temporal orientability constraint on the manifold. —Quondum 00:22, 9 January 2015 (UTC)

Landau and Lifshitz, Classical Theory of Fields: Proper time is defined using moving clocks tracing out a world line. They calculate (quite trivially)
 * $$\tau_2 - \tau_1 = \int_{t_1}^{t_2}dt\sqrt{1 - \frac{v^2}{c^2}} = (t_2 - t_1)\sqrt{1 - \frac{v^2}{c^2}}$$

relating proper time to the time coordinate in an arbitrary Lorentz frame, the last step assuming for simplicity that the clocks aren't accelerated. Now reverse the time-coordinate,
 * $$t' = -t, x' = x, y = y' = y, z = z'.$$

Then,
 * $$\begin{align}\tau'_2 - \tau'_1 &= \int_{t'_1 = -t_1}^{t'_2 = -t_2}-dt'\sqrt{1 - \frac{v^2}{c^2}}\\

&= -(-t_2 - -t_1)\sqrt{1 - \frac{v^2}{c^2}}\\ &= (t_2 - t_1)\sqrt{1 - \frac{v^2}{c^2}}\\ &= \tau_2 - \tau_1\end{align}.$$

Thus proper time as defined in a very reliable source is a scalar. You can find this (including the verbatim statement that proper time is a scalar) in a score of books. YohanN7 (talk) 23:33, 8 January 2015 (UTC)


 * Did you copy the part from "Now reverse the time-coordinate ..." from the book? It contains a sign error. —Quondum 00:00, 9 January 2015 (UTC)
 * More specifically, the manipulation used is a variable substitution (which proves nothing), whereas what is needed is a reapplication of the definition to the new set of coordinates. —Quondum 00:07, 9 January 2015 (UTC)
 * A variable substitution is exactly what the time-reversal amounts to here. I'm not going to continue discussing here. See your talk page. YohanN7 (talk) 01:03, 9 January 2015 (UTC)

Yeah why is the sign reversed in the dummy variable time within the integral (Confusingly called t')?124.170.85.44 (talk) 00:09, 9 January 2015 (UTC)

Let's go back to an ordinary curve in euclidean space. Regardless of parameterisation, everyone agrees on the arc length from A to B, (let's call it s). However, to use s as a special parametrisation to uniquely describe a point on the curve, you still need a direction of orientation.124.170.85.44 (talk) 00:17, 9 January 2015 (UTC)

Actually looking at your formula above, I think I see the problem. By all means, let's consider your unaccelerated clock; $$\tau_2 - \tau_1 = \int_{t_1}^{t_2}dt\sqrt{1 - \frac{v^2}{c^2}} = (t_2 - t_1)\sqrt{1 - \frac{v^2}{c^2}}$$

Now change the sign of t


 * $$\tau'_2 - \tau'_1 = (t'_2 - t'_1)/ \gamma=(t_1 - t_2)/ \gamma=\tau_1-\tau_2$$

The tau's are swapped. So there was an odd number of sign errors made in the integral.124.170.85.44 (talk) 00:47, 9 January 2015 (UTC)


 * Be careful not to confuse proper time with proper time interval (which is akin to proper distance along a spacelike curve). I'm going to bring this back to the beginning, since this is not clarifying much: there are many ways to look at this, and this is sufficient reason not to single out one definition for use in the article. Add to this that we have no reference mentioning of pseudo-vectors. —Quondum 01:07, 9 January 2015 (UTC)

OK, Quondum. In that case, how do you say a four-velocity transforms under time reversal? Also, YohanN7, do you agree there was a sign error?124.170.85.44 (talk) 01:31, 9 January 2015 (UTC)

The integral formula is, while correct, not Lorentz invariant under time reversal (which is why it has to be used the way I did - by a coordinate change). Use instead from above
 * $$\Delta \tau = \sqrt{\left(\Delta t\right)^2 - \frac{\left(\Delta x\right)^2}{c^2} -  \frac{\left(\Delta y\right)^2}{c^2} - \frac{\left(\Delta z\right)^2}{c^2}}.$$

and set $∆τ = τ_{2} - τ_{1}$. Then you get,
 * $$\tau_2 - \tau_1 = \sqrt{\left(\Delta t\right)^2 - \frac{\left(\Delta x\right)^2}{c^2} -  \frac{\left(\Delta y\right)^2}{c^2} - \frac{\left(\Delta z\right)^2}{c^2}}(*).$$

for a Lorentz frame related to the rest frame of the particle by a proper orthochronous Lorentz transformation (so that $t_{2} > t_{1})$, right?
 * $$\tau_2 - \tau_1= (t_2 - t_1)\sqrt{1 - \frac{v^2}{c^2}}$$

But the formula (*) is manifestly invariant under $t ↔ −t$ (so that $t_{2} < t_{1})$, right?
 * $$\tau'_2 - \tau'_1 = (t'_1 - t'_2)\sqrt{1 - \frac{v^2}{c^2}} = (t_2 - t_1)\sqrt{1 - \frac{v^2}{c^2}} = \tau_2 - \tau_1$$

YohanN7 (talk) 01:52, 9 January 2015 (UTC)

I guess the bottom line is this: Proper time is defined in a coordinate-independent way along a world path. You might as well write what the clock shows along a a line piece of paper (representing space time). It is coordinate independent by the way in which it is defined. Whether the coordinates for an event along the line is (t, x, y, z) or (T, X, Y, Z) doesn't matter. At that event is written 42 for the proper time in both cases. All the math above is really irrelevant too for the discussion. It only says how the proper time relates to particular coordinates. It is the way proper time is defined that make it a scalar. YohanN7 (talk) 02:06, 9 January 2015 (UTC)

I don't agree with this. I think you're making elementary sign errors to justify a preconceived position. I think you put it very well yourself "The integral formula is, while correct, not Lorentz invariant under time reversal."124.170.85.44 (talk) 02:56, 9 January 2015 (UTC)

I say your equation * should actually read

$$(\tau_2 - \tau_1)^2 = \left(\Delta t\right)^2 - \frac{\left(\Delta x\right)^2}{c^2} -  \frac{\left(\Delta y\right)^2}{c^2} - \frac{\left(\Delta z\right)^2}{c^2},$$

$$\tau_2 - \tau_1 = \pm \sqrt{\left(\Delta t\right)^2 - \frac{\left(\Delta x\right)^2}{c^2} -  \frac{\left(\Delta y\right)^2}{c^2} - \frac{\left(\Delta z\right)^2}{c^2}} $$ or

$$|\tau_2 - \tau_1| = \sqrt{\left(\Delta t\right)^2 - \frac{\left(\Delta x\right)^2}{c^2} -  \frac{\left(\Delta y\right)^2}{c^2} - \frac{\left(\Delta z\right)^2}{c^2}}.$$

Those quantities are the only ones we can say are invariant to all frames124.170.85.44 (talk) 03:12, 9 January 2015 (UTC)


 * The integral is derived, under the assumption that the Lorentz frame is related to the res frame by a proper orthochronous Lorentz transformation, from a similar integral which is invariant under time reversal. This is why it must be used the way I did. Without any assumptions, the latter formulas obtain, which are invariant under all Lorentz transformations.


 * If you get any other result than what I obtained, it only shows your math is off. You still have no clue how proper time is defined, do you? To make it simple enough to understand: Proper time are fixed numbers assigned to events along a timelike path in spacetime. No coordinates! If you in your calculation of what the expression for proper time is in coordinates arrive at anything else than what gives the assigned numbers, then you have screwed up. YohanN7 (talk) 11:38, 9 January 2015 (UTC)


 * I said at the start that "your statement is not valid until proper time has been suitably defined". Quoting from Landau and Lifshitz. p.8: "The time read by a clock moving with a given object is called the proper time for this object."  This definition is Lorentz-invariant. Implicit in this definition (use of the concept of a clock) also implies a specific direction for the increase of the proper time. Even though texts get sloppy about tracking the sign correctly in illustrations, you'll be hard-pressed to find a text that does not consider proper time to be an invariant. The kicker is this though: too many other formulae depend on the invariance (including under time-reversal) of both proper time and of four-velocity, such as relations involving momentum. If proper time is a pseudo-scalar, and four-velocity is a pseudo-vector, either proper mass becomes a pseudo-scalar or momentum becomes a pseudo-vector ... and for all I know half the tensors we deal with would have to be redefined as pseudo-tensors, with no benefit whatsoever. Defining proper time as a observer-dependent quantity is unworkable. —Quondum 15:16, 9 January 2015 (UTC)


 * Thank you Q for saving us from some further trouble. You still cannot possibly define it as a pseudo-scalar, even if you reverse its sign ignoring natures advice. It would still be a scalar. Mathematically, one defines
 * $$ds^2 = \eta_{\mu\nu}dx^\mu dx^\nu, \quad ds = \sqrt{\eta_{\mu\nu}dx^\mu dx^\nu},$$
 * give or take a convention-dependent sign on the far right (and in the metric). This is manifestly Lorentz invariant by the very definition of the Lorentz group in terms of bilinear forms, see classical group, and generalizes to general relativity in the obvious way. In any frame,
 * $$ \Delta \tau \equiv \frac{1}{c} \int ds.$$
 * This is the mathematical version of the clock. Then you play on from there. You need to parametrize your worldline in order to compute the integral. One such way is to use the time-coordinate in your Lorentz frame, perhaps the frame in which the particle is at rest in which case the parameter is proper time itself. Whether you can extract $dt$ or $−dt$ from the resulting square root depends on whether your Lorentz frame is related to the rest frame of the particle (clock) via an orthochronous LT or not. The resulting expression (to be found above in the case of a non-accelerated particle) is thus not Lorentz invariant under time-reversal. This explains why I said you have screwed up mathematically if you obtain any other result than the invariant formula gives in the rest frame, namely $τ_{2} − τ_{1}$.
 * I'm not going to go into further detail, it can be found in textbooks (Barton Zweibach, Sting theory: a first course is very detailed), and I have the feeling that you (the ip) need to convince yourself rather than getting convinced by someone else. YohanN7 (talk) 16:27, 9 January 2015 (UTC)

"You still cannot possibly define it as a pseudo-scalar, even if you reverse its sign ignoring natures advice." This is quite wrong. To repeat myself for the last time, all you've proved is that proper time squared is invariant, and that the magnitude of proper time on a worldline is invariant. Observers in frames time reversed from each other disagree on the direction of proper time, as I showed. I don't think I can productively engage further with you, YohanN7.

Quondum, yes a lot things would become pseudo-tensors, but this problem would only arise under time-reversal, which hardly anyone ever does. So it makes almost no practical difference. But to conclude, it's your opinion then that the time component of four-velocity goes $$\gamma => -\gamma?$$ Does only this gamma change its sign, or do all of them?124.170.85.44 (talk) 00:25, 10 January 2015 (UTC)


 * I put it so simply for you that a monkey would understand. You don't because you are trolling. You are simply a troll. Do the homework or don't post here again. YohanN7 (talk) 04:20, 10 January 2015 (UTC)

"The time read by a clock moving with a given object is called the proper time for this object." I still disagree that this is truly Lorentz invariant, Quondum. There could be two clocks moving with the object, each running backwards relative to the other. How do you decide which one is going in the "correct" direction? Does the clock have some complicated particle physics experiment onboard, to detect the direction of time from CP violations? Is it a macroscopic clock, which also plots whether entropy increases or decreases as clock time increases? It seems to me doing either of these things brings in a lot of extra complication. I realise that in special relativity this may be of simply philosophical significance, but in general relativity, temporal orientation is not globally definable in certain metrics. I therefore think this is a genuine problem.124.170.85.44 (talk) 04:57, 10 January 2015 (UTC)


 * This discussion has moved way beyond the purpose of the talk page of an article. It would be counterproductive for me to remain involved. —Quondum 16:25, 10 January 2015 (UTC)


 * Though I have been disqualified from this discussion due to incompetence, I'll chip in once more. I overheard a competent person saying that the definition,
 * $$ \Delta \tau \equiv \frac{1}{c} \int \sqrt{\eta_{\mu\nu}dx^\mu dx^\nu},$$
 * of proper time is fully, and manifestly so, Lorentz invariant. To see this, note that
 * $$\eta_{\mu\nu}dx'^\mu dx'^\nu = \eta_{\mu\nu}dx^\mu dx^\nu$$
 * is one of the equivalent conditions for
 * $$x' = \Lambda x$$
 * to be a Lorentz transformation.
 * Even if you manage to make a clock run backwards, the proper time thus defined (the negative of proper time as usually defined, modulo an additive constant) will still be a scalar by the above definition.


 * That is at least my recollection of what I heard. YohanN7 (talk) 18:31, 10 January 2015 (UTC)

"Magnitude" and "interval"
The terms "magnitude" and "interval" are often repurposed to mean a quadratic form applied to an underlying vector due to the lack of a better term. In a book, the terms can be defined in a way that is useful to the text and might even be uniform within a narrow field. However, in the encyclopaedic context where articles in many disciplines are heavily cross-referenced, this unfortunately falls foul of the principle of least astonishment. Perhaps we should debate how best to present these concepts that uses a more uniform terminology across WP, and less specialized use within a topic? In these examples, the usage is not uniform across texts, and there are no terms that are great in general use. Terms like "square interval" are reasonably common, and could be considered for use in WP (this example is already in use in WP). —Quondum 21:15, 13 March 2015 (UTC)


 * This is a late reply, but for the record... I thought that "magnitude" refers to the magnitude (modulus, norm, whatever...) of a vector e.g.


 * $$|X| = \sqrt{X_\mu X^\mu} = \sqrt{X_\mu g^{\mu\nu} X_\nu} = \sqrt{X^\mu g_{\mu\nu} X^\nu} $$


 * and "interval" is much more loosley applied, in the case of invariants and magnitudes in spacetime it would be the difference in proper time Δτ (or better yet in differentials dτ), then the square ("square interval") would be equal to the square modulus of the four position


 * $$|X|^2 = X_\mu X^\mu = X_\mu g^{\mu\nu} X_\nu = X^\mu g_{\mu\nu} X^\nu \,. $$


 * This may be "obvious" but what else are the most sensible uses of those terms in the context you describe? I haven't found them used differently elsewhere. M&and;Ŝc2ħεИτlk 21:31, 21 May 2015 (UTC)


 * Most people would expect something like you describe: a norm with the property that ||aX|| = |a| ||X||. However, this article says:
 * The magnitude of an object's four-velocity (the quantity obtained by applying the metric tensor to the four-velocity and itself) is always equal to the square of c, the speed of light.
 * and the link makes it clear that what is meant is something with the property that ||aX|| = a2 ||X||, in the form of ||X|| = gabXaXb. This is what many will find surprising.  OTOH, in the Minkowski context, the new (quadratic) meaning is actually more useful, is commonly used, and it just needs a name.  Unfortunately, many places overload "magnitude" or "interval" with this new meaning.  I'm suggesting that we might want to find better (and reasonably well-known) terms with this meaning ("quadratic magnitude", "square magnitude"?).  —Quondum 23:24, 21 May 2015 (UTC)


 * If you agree that most people expect to use what I wrote then it's also in agreement with most sources (especially established ones like LL, MTW, Goldstein). Where (which link) did you find ||aX|| = a2 ||X||? Using the above expressions with the Minkowski metric would always give ||aX|| = |a| ||X||.
 * About alternatives, "square magnitude" should be self-explanatory but "quadratic magnitude" may not for a typical reader (they may suspect it means something else). M&and;Ŝc2ħεИτlk 07:39, 22 May 2015 (UTC)


 * At the moment I am not trying to settle on an actual naming convention, just debating the point that WP has several articles (mainly related to relativity) that use terms such as these in the "unexpected" sense, and that we might want to change the terminology in use in WP. Proper time defines an "interval" in the "unexpected" sense, that is tosay, as what we might want to call the "square interval".  My quote above gives two examples: one in the lead of this article, where the term magnitude clearly is used in the sense of "square magnitude", and the link in that quote displayed as "magnitude" also defines it in the square sense. As a related point: we will have to distinguish clearly between "square magnitude" and "magnitude squared", since they can differ by a sign, the former being the more useful quantity, latter always being positive.  —Quondum 15:45, 22 May 2015 (UTC)

The quantity alluded to risks being any one of eight different numbers, well not all eight being different, but still. There is the ambiguity in the metric resulting in a factor of two. Then, whether should you take the square root or not is another factor. Additionally, some authors throw in an additional minus sign (one more factor of two) in the definition. Probably best is to chose one of these and remark that others exist. YohanN7 (talk) 16:52, 22 May 2015 (UTC)


 * I would prefer to avoid the question of "what is a magnitude?", but rather answer the question: "What given a vector $$X^\mu$$, what do we call the derived quantity $$X^\mu g_{\mu\nu} X^\nu$$? —Quondum 16:59, 22 May 2015 (UTC)


 * It amounts to choosing one of the eight possibilities. What do we call $$\pm X^\mu g_{\mu\nu} X^\nu$$?


 * So, what do we call this? Not "the magnitude (or even the norm) of $$X^\mu$$", I hope. —Quondum 17:41, 22 May 2015 (UTC)


 * If you agree that it's sensible not to call $$\pm X^\mu g_{\mu\nu} X^\nu$$ a "magnitude", then why are you finding it necessary to differ between "square magnitude" and "magnitude squared"? In ordinary language and maybe even in mathematical language they amount to the same.
 * What do you expect to mean by "square magnitude", is the square of the norm of a vector X, $$|X|^2 = X^\mu g_{\mu\nu} X^\nu$$ referring to magnitude?
 * What about "magnitude squared", does magnitude refer to the norm $$|X| = \sqrt{X^\mu g_{\mu\nu} X^\nu}$$ and the square of this $$|X|^2 = X^\mu  g_{\mu\nu} X^\nu$$ the "magnitude squared"?
 * About the quote,
 * In a pseudo-Euclidean space, the magnitude of a vector is the value of the quadratic form for that vector.
 * it's a vaguely worded unreferenced one-sentence statement, making the claim that the magnitude of a vector is the square of what we expect ("value of the quadratic form" $$X^\mu g_{\mu\nu} X^\nu$$). How often is this definition of magnitude used throughout physics?
 * You're right I have not come across other examples from expected ones I wrote above. M&and;Ŝc2ħεИτlk 08:14, 23 May 2015 (UTC)
 * Using quotation marks at first mention together with a remark could offer an acceptable way out. The "norm"... or the "magnitude"... The "metric" b t w is, I believe, a full-fledged fully defined pseudo-metric (and the "inner product" a semi-inner product, confusing? absolutely!). What prevents defining a pseudo-norm (of a vector) for the purposes of WP? Note, I haven't read (in a while) what the article actually says. YohanN7 (talk) 08:54, 23 May 2015 (UTC)
 * At any rate, sticking to the squared quantity (whether with a minus sign and with whatever favorite metric signature) may be good in order to avoid delving into obscurities in the form of "imaginary intervals". There are the well-defined quantities "timelike" and "spacelike" in the description of a vector to take into consideration. YohanN7 (talk) 09:04, 23 May 2015 (UTC)
 * Fine, but in any case the terminology is still confusing, if we call the squared quantity $$X^\mu g_{\mu\nu} X^\nu$$ a magnitude of vector X instead of the norm (or equivalent names...) $$\sqrt{X^\mu  g_{\mu\nu} X^\nu}$$ of a vector X.
 * In 3d Euclidean space we don't call $$|\mathbf{r}|^2 = x^2 + y^2 + z^2$$ the "magnitude of the position vector $$\mathbf{r} = x \mathbf{e}_x + y \mathbf{e}_y + z\mathbf{e}_z$$", instead $$|\mathbf{r}| = \sqrt{x^2 + y^2 + z^2}$$ seems more natural. M&and;Ŝc2ħεИτlk 09:25, 23 May 2015 (UTC)
 * Yes, it is confusing. Another example is at, where we see a "spacetime interval" is the quadratic quantity, but "time-like interval" and "space-like interval" are linear.  There has been debate on this before (I forget on which page), and the literature is not helpful. For example, MTW p. 310 refers to the bilinear form defined by the metric applied to a differential form as an interval.  Not a very happy state of affairs.
 * M: why are you finding it necessary to differ between "square magnitude" and "magnitude squared"?' – There are two quantities to distinguish: $$X^2 = X^\mu g_{\mu\nu} X^\nu$$ and $$\left|X\right|^2 = \left|X^\mu g_{\mu\nu} X^\nu\right|$$.
 * Y: 'sticking to the squared quantity [...] may be good' – agreed, but we do not seem to have a simple name for this.
 * —Quondum 13:45, 23 May 2015 (UTC)


 * That's clearer, thanks, I was trying to pin down what quantities you actually meant, which is why I kept giving explicit expressions and asked what you meant by the terms you were trying to distinguish. I'll think about this before further posts. M&and;Ŝc2ħεИτlk 17:09, 23 May 2015 (UTC)


 * Looks like I'm wrong about this:
 *  The "metric" b t w is, I believe, a full-fledged fully defined pseudo-metric (and the "inner product" a semi-inner product, confusing? absolutely!)
 * So forget that terminology. (For every new definition I learn, I forget two old ones.) YohanN7 (talk) 08:03, 26 May 2015 (UTC)

Could someone please add a note of the non-standard definition of the four-velocity as defined by Landau-Lifshitz? This is still considered a classic text that is frequently cited, and has the definition u_i = dx_i /ds such that the normalisation is 1 and not c. — Preceding unsigned comment added by 220.253.241.149 (talk) 11:44, 7 May 2023 (UTC)