On the one way speed of light....

[ghwellsjr]

In SR, at least, relative velocity computed in any coordinates is invariant (given two specific events on two world lines), while separation speed is frame(coordinate dependent).

Is the reason you say "given two specific events" to allow for non-inertial worldlines? If the two worldlines are inertial, do you still need two specific events? And are there two events on each worldline for a total of four?

I'm really having trouble understanding this so I would appreciate any elaboration you could provide.

 

[PAllen]

Is the reason you say "given two specific events" to allow for non-inertial worldlines? If the two worldlines are inertial, do you still need two specific events? And are there two events on each worldline for a total of four?

I'm really having trouble understanding this so I would appreciate any elaboration you could provide.

Yes, it is to allow arbitrary world lines. For inertial world lines, the 4-velocity is constant on each, so choice of event on each is irrelevant. However, for arbitrary world lines you have to talk about an event on each. You don't need 4 events; not sure where that comes from. Given a world line, there is well defined unit tangent vector at every point that we call 4-velocity. In SR, comparing vectors at a distance is well defined, unlike GR. Given the two 4-velocity vectors (working in any frame or even general coordinates with Minkowski metric converted to those coordinates), you just take the dot product (via metric) of the two 4 velocities. That gives you gamma corresponding to their relative speed.

 

[ghwellsjr]

Yes, it is to allow arbitrary world lines. For inertial world lines, the 4-velocity is constant on each, so choice of event on each is irrelevant. However, for arbitrary world lines you have to talk about an event on each. You don't need 4 events; not sure where that comes from. Given a world line, there is well defined unit tangent vector at every point that we call 4-velocity. In SR, comparing vectors at a distance is well defined, unlike GR. Given the two 4-velocity vectors (working in any frame or even general coordinates with Minkowski metric converted to those coordinates), you just take the dot product (via metric) of the two 4 velocities. That gives you gamma corresponding to their relative speed.

Thanks. Is this a standard technique that you could read about in most any good textbook or did you figure this out on your own?

 

[PAllen]

Thanks. Is this a standard technique that you could read about in most any good textbook or did you figure this out on your own?

It would be covered in books which treat SR in curvilinear coordinates (typically those that review it for general relativity). For pure SR books that cover this material, by reputation (I don't have copies of these) I would guess Rindler's book ( https://www.amazon.com/dp/0198539525/?tag=pfamazon01-20 ) and this one: https://www.amazon.com/dp/3642372759/?tag=pfamazon01-20

Of course, if you are willing to tackle a book that does SR as a prelude to GR, Carroll's book (http://preposterousuniverse.com/spacetimeandgeometry/) would have it as well as the first chapter of online lecture notes (http://preposterousuniverse.com/grnotes/).

[edit: However, I can give a simple explanation. In Minkowski coordinates, the 4-velociy at any point on a world line is given by: γ(1,v) where v is the ordinary 3-vector speed. For a world line at rest in such coordinates it is then (1,0,0,0). The dot product of these using the minkowski metric is trivially γ. However, a dot procuct (also called a scalar product, and various other names) is an invariant scalar. So it necessarily will come out the same in all coordinates. Since you can always introduce coordinates such that one the tangent vectors is (1,0,0,0), it follows that universally, the dot product of two 4-velocities gives gamma corresponding to their relative speed.]

 

[harrylin]

If I have two clocks in space at rest wrt each other and just a meter apart, I could synchronise them. If they were far enough away from any other mass so gravitational forces are nullified, then if I just let these clocks sit there for a few million years, expansion will separate them but without effecting synchronisation.

Then at some time in the future we have pre-set clock A for example, to send a light signal to clock b. We compare the two times and we would have a measure for the one way speed of light.

Everything I have read says the one way speed of light is impossible to measure. So where does the above thought experiment break down?

Would gravity between the clocks be enough to stop them receding from each other with expansion? I can't think of any thing else.

As your question is about cosmology, it's interesting to note that Markus gives an answer that is somewhat different from what we said here (not that I think that it has anything to do with one way speed though) - #2

 

[PAllen]

As your question is about cosmology, it's interesting to note that Markus gives an answer that is somewhat different from what we said here (not that I think that it has anything to do with one way speed though) - #2

Markus does note (weakly) that his description is characteristic of the use of cosmologist's preferred coordinates. This description is what I have noted is folation dependent. His contrast to flat spacetime (that is, his implication that this requires curvatures) is factually incorrect. You can construct cosmological coordinates in flat spacetime that have homogeneously expanding distances between coordinate stationary world lines that are inertial, and for which the separation speed is arbitrarily > c for those far enough apart. These are called Milne coordinates and they demonstrate the falsity of the very often repeated claim that these features require curved spacetime. They show that these features follow from finding a congruence of timelike geodesics that are in relative motion that is homogeneous and isotropic, and then building orthogonal coordinates from this congruence. Curvature is not required, and expanding space is a coordinate/foliation dependent notion, not an intrinsic feature of the manifold. Further, there is no coordinate independent meaning to statements like space growing between two world lines versus world lines moving through space.

 

[rede96]

Further, there is no coordinate independent meaning to statements like space growing between two world lines versus world lines moving through space.

If there is no distinction between those two statements, e.g. no distinction between separation due to velocity between two objects and separation due to space expanding between two objects, then doesn't having a coordinate system where expanding distances between stationary world lines that are inertial where separation speed is arbitrarily > c lead to predictions that would violate relativity?

For example how would the Lorentz transform work between such two world lines as they are separating at speeds > c? Wouldn't this predict a time between these objects going backward relative to the other?

 

[Dale]

For example how would the Lorentz transform work between such two world lines as they are separating at speeds > c?

It wouldn't. The Lorentz transform only works for transformations between inertial frames in flat spacetime.

 

[rede96]

It wouldn't. The Lorentz transform only works for transformations between inertial frames in flat spacetime.

Ok, I guess I must have misunderstood PAllen's statement as below:

You can construct cosmological coordinates in flat spacetime that have homogeneously expanding distances between coordinate stationary world lines that are inertial, and for which the separation speed is arbitrarily > c for those far enough apart. These are called Milne coordinates and they demonstrate the falsity of the very often repeated claim that these features require curved spacetime.

Doesn't this suggest that this > c separation is between inertial frames in flat spacetime?

 

[Dale]

No, Milne coordinates are not inertial.

 

[rede96]

No, Milne coordinates are not inertial.

Right, well that has confused me. So what does this statement below mean?

You can construct cosmological coordinates in flat spacetime that have homogeneously expanding distances between coordinate stationary world lines that are inertial,

 

[PAllen]

Right, well that has confused me. So what does this statement below mean?

Each world line of the congruence (= constant position in the coordinates) is inertial, but the each such world line has a different 4-velocity. They all intersect at the pseudo-big bang in the past. The coordinates themselves are curvilinear (but orthogonal, in that the flat metric has no off diagonal elements, but the non-constant metric terms are functions of cosmological time). Each t=constant surface, restricted to 2d, represented in a 2 x 1 subset of Minkowski coordinates looks like a type of hyperboloid.

 

[Ibix]

The worldlines are inertial, the coordinates are not. Einstein originally imagined a literal coordinate frame made of rigid rods with clocks at the junctions. Non-inertial coordinates use an infinite array of rockets under thrust as their reference points, rather than the junctions of an unpowered scaffolding.

An inertial worldine describes a free-falling body, independent of the coordinate system used.

 

[PAllen]

For example how would the Lorentz transform work between such two world lines as they are separating at speeds > c? Wouldn't this predict a time between these objects going backward relative to the other?

Note that two world lines going in opposite spatial directions in Minkowski coordinates, at .9999..c, are separating at 1.9999...c. This separation speed is what corresponds to cosmological recession velocity. Milne coordinates just allow producing a complete flat analog cosmology in a similar way to how Rindler coordinates model a large BH horizon up to what is possible without curvature or global topological considerations. Milne coordinates show the limit of how much of cosmology is or is not related to curvature.

Note that that 1.9999..c separation speed corresponds to a relative velocity of .999..c. Similarly, the world lines with e.g. 5c separation speed in Milne coordinates still have a relative velocity of < c. Finally, I have already pointed out previously, that while relative velocity is fundamentally ambiguous in GR because it depends on the path along which you parallel transport 4-veloties, it is still true that no matter what path you pick, you always get a relative velocity <c for distant galaxies. Recession velocity is a separation speed, which is a perfectly well defined notion in SR as well as GR, and has no upper bound in either SR or GR (and is coordinate dependent in both SR and GR). Relative velocity is unambiguous in SR and always < c. Relative velocity in GR is fundamentally ambiguous, but still always < c.

 

[Ibix]

Einstein originally imagined a literal coordinate frame made of rigid rods with clocks at the junctions. Non-inertial coordinates use an infinite array of rockets under thrust as their reference points, rather than the junctions of an unpowered scaffolding

Apologies - an unpowered scaffolding makes an inertial coordinate system in Minkovski space-time, not FLRW space-time. In FLRW space-time, an unpowered array of space stations and their clocks makes an inertial coordinate system. An array of rockets under thrust can be used to define a non-inertial coordinate system in either.

 

[rede96]

Note that two world lines going in opposite spatial directions in Minkowski coordinates, at .9999..c, are separating at 1.9999...c. This separation speed is what corresponds to cosmological recession velocity.

Note that that 1.9999..c separation speed corresponds to a relative velocity of .999..c. Similarly, the world lines with e.g. 5c separation speed in Milne coordinates still have a relative velocity of < c.

I think this is the thing I am having difficultly to understand. Is what you are saying that the 1.9999..c is the separation rate as measured by the coordinate system being used but the 'real' relative velocity as measured between the two moving bodies would be 0.9999..c? So for example if A and B are moving in opposite spatial directions in Minkowski coordinates at 0.9999..c and A sends a light signal to B, B will still receive the light signal as relative velocities are <c even though the separation rate, as measured using the coordinate system, is 1.9999..c. which is the separation rate as being measured relative to some imaginary fixed centre point?

 

[harrylin]

I think this is the thing I am having difficultly to understand. Is what you are saying that the 1.9999..c is the separation rate as measured by the coordinate system being used but the 'real' relative velocity as measured between the two moving bodies would be 0.9999..c? So for example if A and B are moving in opposite spatial directions in Minkowski coordinates at 0.9999..c and A sends a light signal to B, B will still receive the light signal as relative velocities are <c even though the separation rate, as measured using the coordinate system, is 1.9999..c. So is the separation rate is being measured relative to some imaginary fixed centre point?

A light ray "moves relatively to the initial point of [moving system] k, when measured in the stationary system, with the velocity c - v " (Einstein 1905, in the derivation section of the LT). If the max. speed is c in any direction, then the max. speed difference is c - (-c) = 2c. It's as simple as that. There is nothing special going on or weird about it.

1.99999.. c is thus their "separation speed" (which Einstein and some textbooks call "relative speed") according to the reference system in which the two objects are moving in opposite directions, each at 0.9999.. c wrt that reference system.
0.99999.. c is the separation speed (or "relative speed") according the reference system in which A is in rest, and the same according to the reference system in which B is in rest.
This is simply because in the last two systems, either A or B is taken to be in rest so that the max. difference is not 2c but just 1c.

 

[rede96]

1.99999.. c is thus their "separation speed" (which Einstein and some textbooks call "relative speed") according to the reference system in which the two objects are moving in opposite directions, each at 0.9999.. c wrt that reference system.

wrt to the reference system that makes sense to me. The issue I am having is transforming the relative velocities from the reference system to the rest frame of A or B

0.99999.. c is the separation speed (or "relative speed") according the reference system in which A is in rest, and the same according to the reference system in which B is in rest.
This is simply because in the last two systems, either A or B is taken to be in rest so that the max. difference is not 2c but just 1c.

Looking at this classically for a moment, I can see that if I choose a reference system where A moves 1 unit of distance per 1 unit of time, and B the same in the opposite direction, then I know the 'separation' speed is 2 units of distance per unit of time.

If I was then to attach a very long ruler onto A that was in the same units of length as the reference system when at rest wrt to that reference system, and this imaginary ruler was behind A, so that A could read the rate B was moving away according to this ruler, then A can say classically that B was moving away from it at 2 units of distance per unit of time. (Time is A's FOR)

If the unit of distance was just under a light year and the unit of time 1 year, then in A's frame A can say B is moving away from him at a rate of just under 2 light years per year, which is of course is nearly 2c, which violates relativity.

So there must be a way to transform this situation from the classical interpretation in order to keep relative velocities between A and B <c

 

[PAllen]

I think this is the thing I am having difficultly to understand. Is what you are saying that the 1.9999..c is the separation rate as measured by the coordinate system being used but the 'real' relative velocity as measured between the two moving bodies would be 0.9999..c? So for example if A and B are moving in opposite spatial directions in Minkowski coordinates at 0.9999..c and A sends a light signal to B, B will still receive the light signal as relative velocities are <c even though the separation rate, as measured using the coordinate system, is 1.9999..c. which is the separation rate as being measured relative to some imaginary fixed centre point?

Harrylin's post makes it important to stress definitions. I am using a common set of modern definitions, and Harrylin's post notes that historically there has been variation on this. So let me start with definitions, in standard Minlkowski coordinates in SR, for two objects (A and B) moving at .9c in opposite directions:

1) In these coordinates, A and B both have a coordinate speed of .9c. Obviously, this is coordinate dependent quantity.
2) In these coordinates, the separation rate between A and B is 1.8c. This is also a coordinate dependent quantity.
3) The relative speed between A and B is about .9945c. This (as I define it) is an invariant (coordinate independent) quantity in SR. It can be computed to be the same even rotating, non-inertial frame (and corresponding coordinates) in SR. It is, of course, the same as the coordinate speed of A in an inertial frame in which B is at rest (and vice versa), but I am emphasizing the modern geometric definition of this quantity as a function of the scalar product of two unit tangent, 4-vectors - thus completely coordinate independent. [The scalar product is gamma corresponding to the relative speed].

To your specific question, yes - the fact that A can successfully send a light signal to B shows why the separation speed of 1.8c is not a relative velocity. Similarly, the fact that we see distant galaxies shows why their relative velocity (rather than recession rate = separation speed) is < c (but ambiguous in GR; unambiguous in the Milne analog cosmology in flat spacetime).

 

[harrylin]

wrt to the reference system that makes sense to me. The issue I am having is transforming the relative velocities from the reference system to the rest frame of A or B [..]
there must be a way to transform this situation from the classical interpretation in order to keep relative velocities between A and B <c

Yes there is; as I mentioned, this issue was part of the derivation of the Lorentz transformations - it's a feature of the difference between the Galilean transformations and the Lorentz transformations. From those also a simple direct velocity transformation was derived, as presented in a recent thread by Ibix: #41 However I think that his explanation will be better and clearer by deleting several sentences that start with "According to". I therefore copy my edited version of Ibis' example here (with a few more small changes):

Let's try a more every day example. Say your car has impact protection that's good for collisions up to 60mph. You are driving down a road doing 35mph according to your speedometer. You see someone coming straight towards you, also doing 35mph according to their speedometer.

According to someone crossing the road on foot, you are both coming at him at 35mph.
The pedestrian (scurrying out of the way) thinks that this is going to hurt you because 35+35>60.

What the pedestrian has done, probably without thinking about it, is to transform his speed measurements into the only frame that matters for your impact protection - yours. Although 35<60, he's realized that that doesn't matter and worked out the effect it will have on you.

The only thing the pedestrian has done wrong is to use an incorrect velocity transformation formula. The correct one is $$w=\frac{u+v}{1+uv/c^2}$$where u and v are 35mph, c is whatever the speed of light is in mph, and w is the speed you want to work out. This mistake really doesn't matter in this example - $1+uv/c^2$ is so close to one in this case that you need to worry about the braking effect from bugs hitting your windscreen before you worry about the error from using $w\simeq u+v$. But in your example this approximate formula (which is what you are using when you simply take the difference between velocities) is a very bad approximation indeed.

 

[rede96]

Harrylin's post makes it important to stress definitions. I am using a common set of modern definitions, and Harrylin's post notes that historically there has been variation on this. So let me start with definitions, in standard Minlkowski coordinates in SR, for two objects (A and B) moving at .9c in opposite directions:

1) In these coordinates, A and B both have a coordinate speed of .9c. Obviously, this is coordinate dependent quantity.
2) In these coordinates, the separation rate between A and B is 1.8c. This is also a coordinate dependent quantity.
3) The relative speed between A and B is about .9945c. This (as I define it) is an invariant (coordinate independent) quantity in SR. It can be computed to be the same even rotating, non-inertial frame (and corresponding coordinates) in SR. It is, of course, the same as the coordinate speed of A in an inertial frame in which B is at rest (and vice versa), but I am emphasizing the modern geometric definition of this quantity as a function of the scalar product of two unit tangent, 4-vectors - thus completely coordinate independent. [The scalar product is gamma corresponding to the relative speed].

To your specific question, yes - the fact that A can successfully send a light signal to B shows why the separation speed of 1.8c is not a relative velocity. Similarly, the fact that we see distant galaxies shows why their relative velocity (rather than recession rate = separation speed) is < c (but ambiguous in GR; unambiguous in the Milne analog cosmology in flat spacetime).

The only thing the pedestrian has done wrong is to use an incorrect velocity transformation formula. The correct one is

w=u+v1+uv/c 2​

w=\frac{u+v}{1+uv/c^2}where u and v are 35mph, c is whatever the speed of light is in mph, and w is the speed you want to work out. This mistake really doesn't matter in this example - 1+uv/c 2 1+uv/c^2 is so close to one in this case that you need to worry about the braking effect from bugs hitting your windscreen before you worry about the error from using w≃u+v w\simeq u+v. But in your example this approximate formula (which is what you are using when you simply take the difference between velocities) is a very bad approximation indeed.

That's great, and clears up the confusion I was having. Thanks very much for your help.