Talk:Four-velocity

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 τ₁, τ₂ 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**, τ₁ = τ*₁ = τ**₁, τ₂ = τ*₂ = τ**₂. That is, τ₁, τ₂ 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 = -x₀y₀ + x₁y₁ + x₂y₂ + x₃y₃ 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,

${\begin{aligned}\tau '_{2}-\tau '_{1}&=\int _{t'_{1}=-t_{1}}^{t'_{2}=-t_{2}}-dt'{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\\&=-(-t_{2}--t_{1}){\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\\&=(t_{2}-t_{1}){\sqrt {1-{\frac {v^{2}}{c^{2}}}}}\\&=\tau _{2}-\tau _{1}\end{aligned}}.$

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)