On the one way speed of light....

[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.