How to synchronize clocks in accelerating frame?

[Adel Makram]

I don`t understand how synchronization of clocks can be made in an accelerating frame.
let`s put a source of light at the center of an accelerating rocket. From it, many pairs of pulses are emitted one at a time in two opposite directions.
Because the source is located at an equidistant from two ends. and because the speed of light is constant, the pulses should reach two observers, at front and back ends of the rocket, at the same time.

At the source and at any time, the distance between the wave fronts of 2 successive waves.
$$\Delta x=c \Delta \tau$$
$$\Delta \tau=\frac{\Delta x}{c}$$
where $$\Delta \tau$$ is the time separation between two waves.
However, because the front observer, will see the rate of ticking is relatively slower (red shifted) and the back observer will observe the rate of ticking is faster (blue shifted),
$$\Delta \tau_F=\frac{\Delta x}{c} \neq \Delta \tau_B=\frac{\Delta x}{c}$$
which implies c must not be constant.
$$\Delta \tau_F$$ and $$\Delta \tau_B$$ are time lapse between the two waves as observed by front and back observer, respectively.

 

[Orodruin]

Who says they can be synchronised?

 

[Adel Makram]

Who says they can be synchronised?

But signals are reaching them at the same time, so they are synchronized by definition.

 

[Orodruin]

But signals are reaching them at the same time, so they are synchronized by definition.

Define "same time".

 

[PAllen]

There is no uniquely preferred simultaneity convention for an non-inertial frame of reference. The only reason there is for inertial frames is that the Einstein convention has many advantages and can be shown to uniquely put important formulas in their simplest form, without using tensors or connections. With accelerated frames, every plausible choice for synchronization gives up one or another property of inertial frame synchronization. Which you like better depends on your purpose, as all have defects.

 

[Dale]

I don`t understand how synchronization of clocks can be made in an accelerating frame

The Einstein synchronization convention doesn't work in an accelerating frame. You have to choose another convention.

 

[Adel Makram]

Define "same time".

For the source, the time taken by the signal to reach any end is the distance to that end divided by c.

 

[Adel Makram]

There is no uniquely preferred simultaneity convention for an non-inertial frame of reference. The only reason there is for inertial frames is that the Einstein convention has many advantages and can be shown to uniquely put important formulas in their simplest form, without using tensors or connections. With accelerated frames, every plausible choice for synchronization gives up one or another property of inertial frame synchronization. Which you like better depends on your purpose, as all have defects.

What if the source sends two pulses to both ends, reflected back from mirrors and then received again at the center. In this way, the frequencies of pulses received from ends are the same, so the observer at the center may conclude that he can be able to synchronize both clocks!

 

[Ibix]

Are they the same? I think the pulses will be red or blue shifted either from bouncing off the advancing or retreating mirror, or due to the "gravitational field".

 

[stevendaryl]

I don`t understand how synchronization of clocks can be made in an accelerating frame.
let`s put a source of light at the center of an accelerating rocket. From it, many pairs of pulses are emitted one at a time in two opposite directions.
Because the source is located at an equidistant from two ends. and because the speed of light is constant, the pulses should reach two observers, at front and back ends of the rocket, at the same time.

It depends on what you mean by synchronizing clocks. What you can do is come up with a time standard throughout an accelerating rocket, but this time standard will not be the time kept by a standard clock. To compute the coordinate time $T$, you have to perform a linear (or more accurately, an affine) transformation on the time showing on a clock. Clocks at the front of the rocket will tend to run fast compared to this coordinate time, and clocks at the rear of the rocket will tend to run slow compared to this coordinate time.

 

[Adel Makram]

Are they the same? I think the pulses will be red or blue shifted either from bouncing off the advancing or retreating mirror, or due to the "gravitational field".

The amount of the red shift is compensated by the amount of blue shift after reflection and vice versa.

 

[A.T.]

so the observer at the center may conclude that he can be able to synchronize both clocks!

What do you mean by "synchronize clocks", for clocks that run at different rates? You can start them simultaneously according to some frame, but in any frame their times will start to drift apart immediately after that.

 

[pervect]

I don`t understand how synchronization of clocks can be made in an accelerating frame.
let`s put a source of light at the center of an accelerating rocket. From it, many pairs of pulses are emitted one at a time in two opposite directions.
Because the source is located at an equidistant from two ends. and because the speed of light is constant, the pulses should reach two observers, at front and back ends of the rocket, at the same time.

At the source and at any time, the distance between the wave fronts of 2 successive waves.
$$\Delta x=c \Delta \tau$$
$$\Delta \tau=\frac{\Delta x}{c}$$
where $$\Delta \tau$$ is the time separation between two waves.
However, because the front observer, will see the rate of ticking is relatively slower (red shifted) and the back observer will observe the rate of ticking is faster (blue shifted),
$$\Delta \tau_F=\frac{\Delta x}{c} \neq \Delta \tau_B=\frac{\Delta x}{c}$$
which implies c must not be constant.
$$\Delta \tau_F$$ and $$\Delta \tau_B$$ are time lapse between the two waves as observed by front and back observer, respectively.

The coordinate speed of light is constant in any inertial frame of reference, but it is not constant in an accelerated frame of reference. So if one takes "speed" to mean distance/time in some particular coordinate system, the "speed" of light defined in this manner just isn't constant in the coordinate system of an accelerated frame.

If you consider two clocks at different heights in your accelerated frame (say SI standard clocks, based on cesium transitions),when you compare them they won't even run at the same rate. This can be described informally as being due to "gravitational frequency shift", and is an effect that is noticable in atomic clocks on Earth. It's rather hard to come up with a meaningful notion of "synchronizing" clocks that run at different rates, though there is a notion of how to define a coordinate time standard on the Earth - or in an accelerated frame. The "standard" isn't necessarily the only way to define coordinate time, but it's the one most commonly used. If one wants to define a non-standard coordinate time this isn't wrong in principle, but it will require detailed explanations of how the non-standard coordinates are defined and used.

The usual coordinate time for an accelerated frame is called "Rindler coordinates". But I'm not going to try to get into the details of how they work in this post.

 

[Adel Makram]

Thank you all for enlightenment.

 

[kvantti]

This might help (3rd page, "Red shift").

Uniformly accelerated observers in special relativity