[Pmb]

If I read you correctly, you are saying that the frequency of an atomic clock depends on the speed with which things are approaching it.

No. By the way, it's redundant to speak of atomic clock rather than just using the word clock since that term implies a clock which keeps accurate and precise time regardless of its environment. I.e. the rate of the "clock" does not depends on it's temperature of the clock, it.  When you speak of a special kind of clock, such as an atomic clock, the reader often forms the opinion that time dilation only applies to special kinds of clocks and not to time itself.

What I've said is that the rate of time passing will depend on the position of an inertially moving clock is located in an observer's frame of reference when that observer is accelerating in addition to the rate at which it runs due to its speed. This is because according to an accelerating observer this clock is in a gravitational field. The relationship between the time as recorded by the accelerating observer dt and the proper time measured by the clock dT is

dt/dT = 1/sqrt[1 + 2*Phi/c² - (v/c)²]

This is the general Lorentz factor for an observer in a time orthogonal spacetime.

Whew! May I have my cup of hot cocoa now?

[Pmb]

But in my example A, B and the observer are all in a straight line. What PmB implied was that B would see A's clock blue-shifted simply because someone was travelling form A to B, which seems at the very least improbable.

No. That is not at all what I was saying. Please read if over again more carefully and my new post too.

[jeffreyH]

But in my example A, B and the observer are all in a straight line. What PmB implied was that B would see A's clock blue-shifted simply because someone was travelling form A to B, which seems at the very least improbable.

This is pythagorean as in the equivalence of squared values, not triangles.

[alancalverd]

But in my example A, B and the observer are all in a straight line. What PmB implied was that B would see A's clock blue-shifted simply because someone was travelling form A to B, which seems at the very least improbable.

No. That is not at all what I was saying. Please read if over again more carefully and my new post too.

What piqued my skeptical interest was

However the earth clocks will run at a rate which depends not only on the speed of the ship but also according to the position of the earth in the ships gravitational field.

- no mention of acceleration.

The reason for considering atomic clocks is because they are presumed to be reproducible everywhere and unaffected by acceleration, unlike mechanical clocks.

[Pmb]

What piqued my skeptical interest was

However the earth clocks will run at a rate which depends not only on the speed of the ship but also according to the position of the earth in the ships gravitational field.

- no mention of acceleration.

Think of the earth as being a clock, or having a clock on it. Call this clock earthclock. Now consider applying the equivalence principle to the ship's clock. Call this clock shipclock. Let each clock be an idea clock (which is what you were trying to do by using atomic clocks). The speed of the earth can change as measured from the ship's frame of reference by accelerating towards the earth. This happens when the ship's rockets fire.

I was saying that the rate at which the earthclock runs depends on the speed of the earthclock and the location of the earthclock in a gravittional field (from gravitational time dilation). The rate of the earthclock is as measured by the ship clock.

Note: the only reason I said that you don't need the qualifier "atomic" on the earthclock is because it's already assumed. All clocks in SR are assumed to be ideal clocks unless otherwise stated.

Are we clear now?

[Ignorant Enthusiast (OP)]

Now how do we know how fast the car was travelling? Our experimenter on earth will have told us the length of the test track, and will transmit a signal every second. To make it simple, let's say the car travels 100m in 100s. So we receive 100 time pulses, after which we see that the car has travelled 100m. The effect of the blue shift has indeed been to increase the frequency with which we perceive the time signals c ompared with our clock, but whilst the car may appear to have been travelling faster than 100m/s by our clock, it hasn't done so by his clock - and that is the one we use to measure speed.

This is close to what I was asking, if it could appear to us that something could be travelling faster than light due to it being in a different frame of reference to us, without actually travelling faster than light in reality.  Which I am assuming is the case judging by your answer of "whilst the car may appear to have been travelling faster than 100m/s by our clock".

[alancalverd]

The rate of the earthclock is as measured by the ship clock.

Ah, it's all semantics. Being a feet-of-clay experimental scientist, I would call that the apparent rate of the earthclock, knowing that the "true" rate is exactly the same as the ship clock because they are both ideal clocks from my workshop! 

And I'm afraid that not every reader of this forum knows that we are only talking about ideal clocks unless we say so!

[Pmb]

Ah, it's all semantics. Being a feet-of-clay experimental scientist, I would call that the apparent rate of the earthclock, knowing that the "true" rate is exactly the same as the ship clock because they are both ideal clocks from my workshop! 

When you make a statement like the that you're in effect saying that time dilation is only apparent since the clock isn't "really" slowing down, the slowing down is only "apparent." What is real or not is what you can measure and you can measure the earthclock to be running slower so this is not apparent at all.

And I'm afraid that not every reader of this forum knows that we are only talking about ideal clocks unless we say so!

So instead of saying "atomic clock" use the term "ideal clock" instead. I think it will make things much clearer. Otherwise readers will want to know why you have to use an atomic clock instead of an ordinary clock.

[alancalverd]

Au contraire! If I'm on the ship I can see pulses from the earthclock arriving slightly further apart than the pulses from the shipclock, but knowing they are both atomic clocks and being wise to relativity, I know that the earthclock is not really running slower. Indeed by knowing that fact and no other, I can calculate my speed away from earth by measuring the apparent rate of the earthclock.

If we didn't distinguish between real and apparent rates, we would believe that the universe contains an infinite number of different elements! Fortunately we can identify the hydrogen spectrum from its pattern and deduce from the shift of its apparent center frequency that a galaxy is moving with respect to us, which makes astrophysics a lot simpler.

I'm perfectly happy to tell anyone why I'm using an atomic clock! From my present point of view it's very far from ideal because I can't wear it on my wrist, and less than 100 years ago an ideal clock would have been a ship's chronometer, but neither my wristwatch nor a chronometer could be relied on to behave like a standard cesium clock once the rocket engine starts.   

[Pmb]

Au contraire! If I'm on the ship I can see pulses from the earthclock arriving slightly further apart than the pulses from the shipclock, but knowing they are both atomic clocks and being wise to relativity, I know that the earthclock is not really running slower. Indeed by knowing that fact and no other, I can calculate my speed away from earth by measuring the apparent rate of the earthclock.

You're confusing the effects due to Doppler shift and those due to time dilation. The arrival rate is caused by both of them acting together. However if you were to have an object moving so close to a wall that they are almost touching then you can exchange signals with that wall and track time by the rate at which spots were left on the wall by the moving clock. The rate at which spots are left will be soley due to the rate at which the clock is ticking and the speed it is moving. Doppler effects won't play a role. I'm surprised that you don't know this by now.

I'm perfectly happy to tell anyone why I'm using an atomic clock!

And waste time in each post doing so for no good reason. If you were to do a survey of SR texts you'll notice that almost all of them will state what an ideal clock is, using an atomic clock as an example, then they never mention it again. And for very good reason.

[jeffreyH]

If there was a theoretical point in the spacetime continuum where a global rest frame could be found then would an object residing there be older than the apparent age of the universe? A conundrum.