One-Way Speed of Light

[cianfa72]

It's given in post #54.

Ah ok, from #54

In this frame we can compute the proper velocity, by dividing the distance by the proper time elapsed along the worldline.

Therefore to define such "proper velocity" it is required to pick a spatial reference frame and, w.r.t. it, calculate the above ratio.

And yes, as you pointed out, "proper velocity" would be undefined for light, since the " elapsed proper time" along it is.

 

[cianfa72]

Just to keep it simple let's drop two dimensions and consider the frame representation of inertial motion. W.r.t. the Einstein's synchronized frame it has the form $x= a + bt$. Transforming to Anderson's frame we get also a linear form $X= A +BT$. Since w.r.t. Anderson's frame inertial motion occurs at constant coordinate velocity then, by definition, it is an inertial frame as well.

Following this line of reasoning, every linear transformation with constant coefficients from standard inertial Einstein's synchronizated frame gives rise to a (global) inertial frame as well (every inertial motion occurs with constant coordinate velocity w.r.t. it). Do you agree?

 

[Dale]

It depends how you define an inertial frame. That is not my preferred definition. I prefer to define an inertial frame by the form of the metric: $ds^2=-c^2 dt^2+dx^2+dy^2+dz^2$. But other definitions are certainly possible.

 

[robphy]

Sorry, maybe I missed the definition of "proper velocity" given by @pervect. Is it basically related to the "spacetime angle" between a test object w.r.t. a reference body at the point where the two worldlines cross ?

A useful spacetime-trigonometry dictionary:

rapidity	$\theta$
dimensionless-velocity	$\beta=(v/c)=\tanh\theta$
time-dilation factor	$\gamma=\cosh\theta$
proper velocity (also called celerity)	$\beta\gamma=\sinh\theta$
doppler factor	$k=\exp\theta$
proper-time	Minkowski-arclength along a timelike worldline

 

[pervect]

Which means that the "proper velocity", as you are calling it, is not an invariant (because it's calculated from a frame-dependent quantity), so it's (a) misnamed (the adjective "proper" is usually reserved for invariants) and (b) not physically meaningful.


I don't see how that can be true if it's a frame-dependent quantity, per the above.

There's a 1:1 mapping from this quantity to the 4-velocity, which is of course manifestly frame independent. By the way, I got the term "proper velocity" from a paper which discussed various notions of velocity in SR, namely proper velocity, celerity, and rapidity. I forget the name, but I'm sure I could dig it up if needed. There's some small but probably non-zero possibility that my memory of the terminology could be slightly off.

To give a specific but coordinate dependent description of this procedure, if we use coordinates t,x,y,z, then the quantity in question can be written as dx/dtau, dy/dtau, dz/dtau. To get the fourth component and find the 4-velocity, we do need to compute the fourth component of the 4-velocity, dt/dtau from the other components using the rule that -dt/dtau^2 + dx/dtau^2 + dy/dtau^2 + dz/dtau^2 = -1, i.e. the fact that the length of the 4-velocity is -1 (using my usual choice of sign conventions and using units where c=1). But this is easily done.

Note that the 4-velocity also gives us the 4-momentum for a point particle.

So, I think I've demonstrated that there is a meaningful notion of velocity without worrying about clock synchronization. Clock synchronization has other uses, but we don't need two clocks to have some concept of velocity, we can easily make do with one and avoid the whole mess in the case of physical objects which travel timelike worldlines..

I haven't seen this discussed anywhere in detail.

The thought experiment that set me down this path ages ago was considering airplanes and time-zones. I'm idealizing the experiment to avoid complicating factors such as prevailing winds and wind resistance.

The trip time, using wall clock (which are set via local time zones), depends on the direction of travel. The same airplane travelling west-east and east-west take a different amount of time to transverse the same path. Do we take seriously the idea that the airplanes have different velocities depending on their direction of travel? Would we claim that they had different momenta? In general, we do not, it's pretty obvious it's just an artifact of how we chose to synchronize the clocks. And there's an very easy and straightforwards way to tell that the travel time of the (idealized, no-wind) airplane really doesn't depend on the direction. Just put a clock on the plane and notice that the elapsed time doesn't depend on direction.

 

[PeterDonis]

There's a 1:1 mapping from this quantity to the 4-velocity

But the mapping is frame-dependent. So even though 4-velocity is frame independent, proper velocity is not.

I think I've demonstrated that there is a meaningful notion of velocity without worrying about clock synchronization.

No, you haven't, since, as noted above, proper velocity is frame-dependent.

The meaningful notion of velocity that is independent of clock synchronization, because it's frame independent, is simply 4-velocity itself.

 

[robphy]

The 4-velocity vector $\hat u$ is the unit-tangent vector to the timelike worldline of a particle.
Using $(+---)$ we have $\hat u\cdot \hat u=1$.

Given an observer's inertial frame with 4-velocity $\hat t$ with $\hat t\cdot \hat t=1$,

• the temporal-component of that particle's unit-tangent vector (think hyperbolic cosine)
  is the time-dilation factor times the observer 4-velocity : $\gamma \hat t$
• the spatial-component of that particle's unit-tangent vector (think hyperbolic-sine)
  is the proper-velocity (a.k.a celerity) of that particle: $\tilde S =\hat u-\gamma \hat t$
• the slope of that particle's unit-tangent vector (think hyperbolic-tangent)
  is the spatial-velocity

$\hat u=\gamma \hat t + \tilde S$,
where $\hat t\cdot \hat u=\gamma=\cosh\theta$
and (since $\tilde S =\hat u-\gamma \hat t$) we have $\hat t\cdot \tilde S=0$.
So,
$\tilde S \cdot \tilde S= \hat u\cdot \hat u -2 \hat u\cdot \gamma \hat t+ \gamma^2 \hat t \cdot \hat t=(1)-2\gamma (\gamma) + \gamma^2(1)=1-\gamma^2$,
which is non-positive in my signature-convention... thus "$\tilde S$ is spacelike".
[This is $-\sinh^2\theta$. So, $\tilde S$ is the proper-velocity or celerity of the particle according to the observer.]

The ratio $\frac{\sinh\theta}{\cosh\theta}=\frac{S}{\gamma}=V=\tanh\theta$ (the spatial velocity according to the observer).

 

[cianfa72]

The 4-velocity vector $\hat u$ is unit-tangent vector to the timelike worldline of a particle.
Using $(+---)$ we have $\hat u\cdot \hat u=1$

Given an observer's inertial frame with 4-velocity $\hat t$ with $\hat t\cdot \hat t=1$

I believe the above definition of 4-velocity extends as it is to General Relativity since it is a (well-defined) geometric notion.

 

[Dale]

There's a 1:1 mapping from this quantity to the 4-velocity, which is of course manifestly frame independent.

Isn’t that true of ordinary 3 velocity too?

 

[cianfa72]

It depends how you define an inertial frame. That is not my preferred definition. I prefer to define an inertial frame by the form of the metric: $ds^2=-c^2 dt^2+dx^2+dy^2+dz^2$. But other definitions are certainly possible.

Therefore in the realm of SR you reserve the term inertial frame only for (global) frames w.r.t. the following conditions holds:

1. inertial motion (zero proper acceleration) occurs with zero coordinate acceleration
2. the one-way speed of light is isotropic and is the frame-invariant $c$

 

[pervect]

Isn’t that true of ordinary 3 velocity too?

Yes. In my opinion, its simply an example of how, for things other than light, a one-way velocity can be meaningfully defined. While the formulation for relativistic momentum is slightly simpler in terms of the proper velocity (m v_proper) as opposed to (gamma m v) I don't think that's really that important.

I would group all the various slightly different ways of measuring velocity as part of the same family at a conceptual level, even though the nitty-gritty details are different.

 

[pervect]

The spacetime interval along the worldline of a light ray is zero. The term "proper time" doesn't even apply to light because that term only applies to timelike intervals, not lightlike intervals.


That's correct.


The definition @pervect gave for "proper velocity" would be undefined for light, yes.

I don't think I quite agree with your point, but I'll think about it more when I get more time. The thing is - I'm not actually personally all that invested in the non-tensor version, I think the tensor version of the argument is cleaner. For an A-level treatment, there's really no need not to include the fourth component right from the start - it doesn't affect the argument much.

However, my goal was not to give an A-level discussion, but to present a very simple argument about an alternate formulation of velocity that does not require clock synchronization. This formulation is not applicable to light, but it's applicable to everything else.

The idea is that if people are getting unnecessarily hung up on the clock synchonization issue, it might be helpful to point out to them that one can do enough physics to do useful things, such as find the momentum of a point particle, without needing to even define a clock synchronization.

In order to make the idea more accessible to a wide audience, I chose to omit the fourth component or any discussion of tensors.

As an aside, I think it is possible to do physics without tensors. So I wouldn't agree with an argument that any discussion of physics that doesn't use tensors is necessarily wrong. It's just not manifestly covariant so it has some constraints on the coordiantes used. I think I did a reasonable job of discussing what sort of coordinates are needed.

 

[PeterDonis]

I don't think I quite agree with your point

I'm not sure why. See below.

an alternate formulation of velocity that does not require clock synchronization

Since your formulation is frame-dependent, it does require clock synchronization; picking a frame includes picking a clock synchronization.

one can do enough physics to do useful things, such as find the momentum of a point particle, without needing to even define a clock synchronization.

To be clear, I'm not disputing that one can do physics without having to pick a clock synchronization; I just don't think your particular definition of proper velocity does that.

 

[pervect]

I had a few more thoughts , though I haven't dug up any references on the topic. Some of these thoughts are philosophical in nature.

A non-linear 1 to 1 map of a tensor to something else isn't a tensor, because tensors must be linear. But I still think of them as representing "reality", because of the fact that if you know one, because of the mapping, you can compute the other, therefore they must contain the same information about the "real" physics.

Thus, I think that three velocities and four velocities represent the same physical quantity. When we start to do relativity, 4-velocites are useful but we do not need to throw away the notion of three velocities entirely. For instance, nobody that I know of claims that you must learn 4-velocities to do relativity, that you simply cannot do relativity without them. People can and do learn relativity using only 3-velocities.

Thus, I don't think using a slightly different notion of velocity, the "proper velocity", breaks anything important. The advantage and reason for doing it is hat it doesn't need simultaneity.

As far as the 1:1 mapping goes. There might or might not be issues with non-time orientable manifolds which makes it a 2:1 mapping, because one needs to find the sign of the time compoonent. I regard this as a minor nit, but I thought I'd mention it for completeness. I lack intuitoin for non-time orientable manifolds.

I also didn't mention the rest of the context that supports some of the ideas I alluded to, such as the notion of "at rest". The structure I used (but didn't explain that I was using it) is called a time-like congruence. It can be defined by definining a field of time-like 4-velocities at every event which can intuitively be thought of as defning a "state of rest" at every point. This structure doesn't have any inherent notion of simultaneity. Worldlines, called integral curves, exist which have the property that the tangent of the worldlines is equal to the 4-velocity at every event. These can be thought of as the worldlines of "observers at rest" making up the congruence.

The notion of a congruence doesn't demand any notion of simultaneity. By identifying each worldline in the congruence as a point in an abstract space, one can define a quotient manifold of four dimensoinal space-time that represent a three dimensional "space". I could probably dig up the papers that I'm thinking of that do this for rotating frames if there is enough interest, but I suspect that there is not the interest and it'd take some work to dig the references up. The fact that the congruence doesn't inherently require any notion of simultaneity makes it a powerful tool for sidestepping most simultaneity issues. A discussion of any counterexamples of the sort of problem that needs simultaneity, the sort of problem that can't be handled with the techniques of congruences just cant handle, would be interesting. However, I don't have any to offer at the moment. I'd be interested if anyone thinks of one.

To summarize - the notion of a congruence guides much of my thinking , providing some of the framework I alluded to without explaining what framework I was using. The notion of a congruence does not inherently need or define a notion of simultaneity. The notion of using proper time to compute proper velocities works well within this framework to do most physical problems without even defining a simultaneity convention. I'd be interested if anyone has a specific problem that doesn't conceptually "fit" within this framework, I feel it may be an overstatement to state that it can do everything, but I can't think of anything that it really can't do offhand.

 

[PeterDonis]

I think that three velocities and four velocities represent the same physical quantity.

You keep ignoring the crucial point that 3-velocity is frame dependent and 4-velocity is not. That simple fact means that your statement just quoted above cannot be correct.

The notion of a congruence doesn't demand any notion of simultaneity.

This is true. But it's also true that congruences, as you yourself note, are defined by a 4-velocity field. Not by a 3-velocity field. This is an obvious consequence of the fact that congruences, like 4-velocities, are not frame dependent, whereas 3-velocities are.

the notion of a congruence guides much of my thinking

If this is true, I think you really need to reconsider your claims about 3-velocities, since they simply don't match what you're (correctly) saying about congruences.

 

[pervect]

You keep ignoring the crucial point that 3-velocity is frame dependent and 4-velocity is not. That simple fact means that your statement just quoted above cannot be correct.


This is true. But it's also true that congruences, as you yourself note, are defined by a 4-velocity field. Not by a 3-velocity field. This is an obvious consequence of the fact that congruences, like 4-velocities, are not frame dependent, whereas 3-velocities are.


If this is true, I think you really need to reconsider your claims about 3-velocities, since they simply don't match what you're (correctly) saying about congruences.

One of the points I'm trying to establish is that given a congruence, and a three velocity, one can construct the 4 velocity. Thus, specifying a congruence and a 3-velocity specifies a 4-velocity by construction.

Letting the 4 velocity be $\vec{u}$ and the 3 velocity be $\vec{v}$ with a magnitude of v. Let the 4-velocity of the congruence at the point in question be \vec{c}. Then we can write:

$$\vec{u} = \frac{1}{\sqrt{1-v^2}} \vec{c} + \vec{v}$$

As far as congruences being defined by 4-velocities, while that's the standard approach, it is backwards to say that one must use 4-velocites to define a congruence. The congruence exists as a geometric object, they don't have to be defined in terms of the 4-velocity, it's simply convenient for communciation. Saying that we "have to" do things in some particular manner is just not true - though it's a mistake I've made myself. Better to say "we need to do something in this manner or something equivalent". I am trying to establish the equivalence of this approach, and to accomplish that goal I am using concepts from the 4-vector approach. So I am resisting being told that I "have to" do things in some specific way, an instead showing that the way I am suggesting to do things is equivalent to that way.

To recap, I think we got off on an argu-discussion about how to measure distance, when I pointed out that proper velocities do not need to have a clock synchronization to measure, all they need are a measurement of distance and proper time, the later of which does not require clock synchronization. I *think* we agree on the fact that we do not need clock synchronization to measure proper time intervals.

Of course, we do need to be able to measure distances as well as times to measure velocities, though.

Note that in the absence of isotropy, the relationship between proper velocity and non-proper velocity does depends on the clock synchronization. Thus my observation that the proper velocity can avoids the whole messy business of talking about how to syncrhonize clocks. We do have to agree on how to measure distances, but once we've got that down we do NOT need clock synchronization to measure proper velocity, and we DO need it to measure the standard "two-clock" version of velocity.

I would also say that a good experimental approach to explaining isotropy is to compare the measurement of proper velocity to the measurements of two-clock velocity. If the relation between the two does not depend on direction, we have an isotropic or fair clock syncrhonization. Note that due to time dilation one cannot expect the two to be equal - the best one can hope for is there is some proportionality constant which depends only on the magnitude of the velocity but not the direction of the velocity. A clock synchronization scheme that makes this happen is an isotorpic or fair clock synchronization - and we can experimentally test any given clock synchronizaton scheme to determine if it is in fact 'fair' as long as we are able to measure proper time.

 

[Dale]

Of course, we do need to be able to measure distances as well as times to measure velocities, though.

I think in the mathematical formulation of differentiable manifolds that velocities are first, and distances are defined by integrating velocities. So you need velocity and time to define distance. But I am not very certain about that, I could be mis remembering

 

[PeterDonis]

Let the 4-velocity of the congruence at the point in question be \vec{c}.

I thought you were claiming that you could construct the 4-velocity of the congruence from the 3-velocity. That's not what you're doing here.

As far as congruences being defined by 4-velocities, while that's the standard approach, it is backwards to say that one must use 4-velocites to define a congruence. The congruence exists as a geometric object, they don't have to be defined in terms of the 4-velocity

You can also define it in terms of the worldlines that make it up, true. But the only invariant way to specify the worldlines is to specify their tangent vectors at every point, which means you're specifying 4-velocities anyway.

I am trying to establish the equivalence of this approach

I have not contested that, if you have chosen a frame, you can go back and forth between 3-velocities and 4-velocities, and you can choose to express everything in terms of 3-velocities, defined relative to that frame, if you find that easier for your intuition to grasp. Of course you can.

My point is that, to do this, you have to choose a frame. To define 3-velocities at all, you have to choose a frame. But you don't have to choose a frame to do physics. You can do physics entirely in terms of invariants. Choosing a frame at all is a convenience. So I think you are getting it backwards when you call using 4-velocities a convenience; 4-velocities are the invariants, in which the actual physics is contained. You're going to be dealing with them one way or another. The convenience is choosing a frame and using 3-velocities because your intuition can handle them more easily.

 

[PAllen]

Ok, if I understand what @pervect is getting at, it is that you can set up spatial coordinates in a physically realized inertial frame without using clocks (e.g. set up a network of bodies at mutual rest as define by no Doppler, and measure positions using a physical measuring body). Then proper velocity (a standard term, though not well chosen) can be measured for any test body (not light) with one clock on the body, distance determined by labels on the realized frame.

I claim that any such standard construction assumes isotropy (e.g., no Doppler = mutual rest, in all directions, implicitly assumes isotropy; so does the notion of Newtons laws holding in their simplest form).

But once you let isotropy slip in, there is simply no distinction between one way and two way speed, for light, or anything else. So, under assumption of isotropy, just measure two way speed of light with one clock and state it is also one way by assumption of isotropy.

 

[PAllen]

Ok, if I understand what @pervect is getting at, it is that you can set up spatial coordinates in a physically realized inertial frame without using clocks (e.g. set up a network of bodies at mutual rest as define by no Doppler, and measure positions using a physical measuring body). Then proper velocity (a standard term, though not well chosen) can be measured for any test body (not light) with one clock on the body, distance determined by labels on the realized frame.

I claim that any such standard construction assumes isotropy (e.g., no Doppler = mutual rest, in all directions, implicitly assumes isotropy; so does the notion of Newtons laws holding in their simplest form).

----

I am having second thoughts about the last claim above. Note that isotropy of 2 way light speed is verifiable, invariant measure. So the only room for assumed anisotropy is of the special type that preserves two way isotropy. However, on thinking about Doppler more, I am not sure that Doppler isotropy relies on anything other than two way light speed isotroppy. Further, there are other ways of setting up a physical inertial frame that explicity rely only on two way isotrpy - e.g. constancy of two way signal times = at mutual rest. Or even, just a rigid material framework. Once this is done, proper velicity per this frame is measurable with one clock, no simultaneity convention required.

So what am I missing? It now seems to me that celerity (proper veloclity) per an inertial set of spatial coordinates set up without any clock synchronization is measurable, and thus that the one way isotropy of celerity from some standard physical process becomes a measurable fact.

 

[PeterDonis]

Once this is done, proper velicity per this frame is measurable with one clock, no simultaneity convention required.

Yes, there is a simultaneity convention required: the one you imposed when you synchronized all the clocks. Doing that via two-way light signals doesn't get rid of the simultaneity convention; it imposes the simultaneity convention that says that the one-way speed is isotropic (same speed going out as coming back).

 

[A.T.]

Once this is done, proper velicity per this frame is measurable with one clock, no simultaneity convention required.

Yes, there is a simultaneity convention required: the one you imposed when you synchronized all the clocks.

What do you mean by "all the clocks"? @PAllen descibes a procedure to measure proper velocity with just one clock.

 

[Ibix]

Do we need to think about what we mean by "distance" in this discussion? Rulers measure interval orthogonal to the worldlines of their markings. If we use non-orthogonal coordinates we may define the spatial coordinates such that our rulers continue to measure coordinate differences, but this no longer corresponds (in general) to the interval between those worldlines along our spatial plane. So doing that is implicitly privileging the orthogonal system's synchronisation convention without quite using it.

So as long as you see proper velocity as coordinate difference over proper time and you are happy to keep an orthogonal system's spatial coordinates then I think you kinda-sorta don't need a synchronisation convention. If you see it as distance over proper time, though, it hinges on what you mean by distance and whether you go with the Einstein-esque "distance is what rulers measure" or define it as "interval along your spatial plane".

 

[cianfa72]

Rulers measure interval orthogonal to the worldlines of their markings. If we use non-orthogonal coordinates we may define the spatial coordinates such that our rulers continue to measure coordinate differences, but this no longer corresponds (in general) to the interval between those worldlines along our spatial plane. So doing that is implicitly privileging the orthogonal system's synchronisation convention without quite using it.

You mean consider the ruler's worldtube in spacetime. Then every mark on it is a timelike worldline within the worldtube. The spatial coordinate difference between them equals to the spacelike interval length between those worldlines along the spacelike plane only if we pick orthogonal coordinates in spacetime.

 

[PAllen]

Yes, there is a simultaneity convention required: the one you imposed when you synchronized all the clocks. Doing that via two-way light signals doesn't get rid of the simultaneity convention; it imposes the simultaneity convention that says that the one-way speed is isotropic (same speed going out as coming back).

No, I am proposing only using constancy of two way signal time as a definition of at mutual rest. The distance is then measured with a reference object.

 

[PAllen]

You mean consider the ruler's worldtube in spacetime. Then every mark on it is a timelike worldline within the worldtube. The spatial coordinate difference between them equals to the spacelike interval length between those worldlines along the spacelike plane only if we pick orthogonal coordinates in spacetime.

The key is that if they are at mutual rest, you can take each component distance measure (laying out your ruler one and over) without reference to timing or simultaneity.

 

[PAllen]

Do we need to think about what we mean by "distance" in this discussion? Rulers measure interval orthogonal to the worldlines of their markings. If we use non-orthogonal coordinates we may define the spatial coordinates such that our rulers continue to measure coordinate differences, but this no longer corresponds (in general) to the interval between those worldlines along our spatial plane. So doing that is implicitly privileging the orthogonal system's synchronisation convention without quite using it.

I guess this is the key point. How do you argue that simply laying a ruler between two bodies at mutual rest relies on 4-orthogonality or simultaneity? I guess one way to go is to note that if you want to use a different simultaneity convention it corresponds to using rulers that are moving relative to your mutually at rest bodies, at which point the simultaneity of a chosen moving frame is crucial. Thus, by process of elimination, the choice to use rulers at rest, without apparent timing constraints is implicitly assuming a simultaneity convention. This is very subtle. Any simpler argument available?

Saying the choice to use no clocks at all implies a simultaneity convention is pedagogically a tough sell. Underscoring why simultaneity is the toughest nut to crack in understanding SR.

 

[PAllen]

Ok, so a possibly better way to argue this is that if you set up spatial coordinates using reference objects and a mutually at rest framework in two different states of motion, you find that each sees that the distances assigned by the other between its own rest world lines corresponds to a specific choice of simultaneity - the orthogonal one.

 

[PeterDonis]

the distances assigned by the other to its own rest world lines corresponds to a specific choice of simultaneity - the orthogonal one.

Exactly.

 

[Ibix]

I guess this is the key point. How do you argue that simply laying a ruler between two bodies at mutual rest relies on 4-orthogonality or simultaneity?

I don't think jist two bodies does do that. But consider rotating the ruler about one end, thus defining a spherical group of worldlines all the same ruler measurement from the origin. We'd like them all to be the same distance in space from the origin, and it's at that point we've picked a foliation.

In essence I think we're saying "distance is what a ruler measures", and when we say that and try non-parallel measurements (especially of objects in motion wrt our rulers) we find we've picked out an orthogonal synchronisation convention.