Exploring Time Dilation Due to Velocity in Special Relativity

[sisoev]

KEV, the more I learn from you the more I got confused.

What I understand is that in this specific experiment the clocks (not the time) run different because the ground clock is in rotating frame and the air-craft clocks are in accelerating frames.

How that relates to inertial frames where SR claims the time to be different?

And by the way, stevmg talks about the cars in your experiment which was supposed to explain my vision about the difference in the clocks. And it was:

Every orbiting around the Earth object tends to "free fall" on the surface.
If two objects are free falling and pass through a point at certain altitude with different speed they'll measure different gravitational force at that point.
The two airplanes have to maintain same altitude with different speed(velocity).
That should result (at least I see it that way) in different gravitational force on both airplanes.

Can you try to use some mathematics on this idea of mine, KEV?

 

[yuiop]

KEV, the more I learn from you the more I got confused.
What I understand is that in this specific experiment the clocks (not the time) run different...

Time is a what a clock measures. Saying "the clocks (not the time) run different" makes no sense. Are you trying to make a distinction between elapsed time and instantaneous clock rates or between coordinate time and proper time?

...because the ground clock is in rotating frame and the air-craft clocks are in accelerating frames

Actually, the ground clock and the aircraft clocks are all in rotating frames or you could also say they are all in accelerating frames. You seem to trying to make a distinction between rotating frames and accelerating frames when there isn't one in this particular case.

In your original post you expressed an interest in the time dilation due to velocity in the Hafele Keating experiment. To learn about the velocity effect you should try and isolate the gravitational effects from the velocity effects. You can do that by keeping the altitude of all the the clocks equal and that is why I introduced the car version. You can also do it by by considering a similar experiment in flat space far from any gravitationally significant massive bodies. All you need is for two rockets to fly in opposite directions around a circular path and a reference clock that stays at the start point. In order to travel in a circle, the rockets need to direct a certain amount of thrust outwards to provide the necessary centripetal force to prevent them traveling in a straight line. The pilots on the rockets will feel and measure "centrifugal force" but the time dilation of the rocket clocks relative to the reference clock will be a function of velocity only (i.e. 1/sqrt(1-v^2)) and the forces involved in traveling in a circle, will have no effect on the time dilation at all. Let us say the reference clock is truly inertial and accelerometers and gyroscopic devices attached to it detect no acceleration at all. This reference clock is analogous to the imaginary clock in the ECIF that you referred to as the inertial clock "in a frame of reference at rest with respect to the centre of the earth".
The rocket clocks are not inertial and their rest frame is a rotating reference frame. Now let us introduce a third rocket. This extra rocket travels around the circle slower than the two and it has its own clock. The rest frame of this additional clock is also a rotating reference frame. This fourth clock is analogous to the ground clock in the Hafele Keating style experiment where altitude differences are eliminated. It experiences time dilation due to velocity relative to the the truly inertial reference clock just like the other rocket/aircraft clocks.

And by the way, stevmg talks about the cars in your experiment which was supposed to explain my vision about the difference in the clocks. And it was:

I do not think Steve was trying to explain your vision about the difference in the clocks. Everything he said is correct and he was basically stating what relativity says, which is not necessarily the same as what your vision says.

What Steve said was:

At an instant in time, whatever an "instant" is, assuming the ground speed of both aircraft are the same but in opposite directions, the eastbound plane for that instant is traveling faster than the westbound plane (the Earth moves at a thousand miles an hour near the equator) in an eastbound direction. Thus, by time dilation, the 1/gamma factor is greater going eastward rather than westward and thus a greater slowing.

On the Earth's surface, there is no difference in velocity and with reference to the Earth, the 1/gamma's would be the same and the time dilation would be the same.

You seemed to have latched onto his last sentence and glossed over his first paragraph. If you understood his first paragraph, you would not still be asking the question that you are.

Let me try and put his statements another way. Assuming the aircraft fly at approximately 600mph relative to the ground, the Eastbound aircraft clock is moving at 1000+600 = 1600mph in the ECIF, the ground clock is traveling at 1000 mph in the ECIF and the Westbound aircraft clock is traveling at 1000-600 = 400 mph on the ECIF. Therefore the Eastbound aircraft clock time dilates the most and the Westbound clock time dilates the least, relative to the ECIF clock. The time dilation of the ground clock relative to the ECIF clock is somewhere between the two.

Every orbiting around the Earth object tends to "free fall" on the surface.
If two objects are free falling and pass through a point at certain altitude with different speed they'll measure different gravitational force at that point.
The two airplanes have to maintain same altitude with different speed(velocity).
That should result (at least I see it that way) in different gravitational force on both airplanes.

Can you try to use some mathematics on this idea of mine, KEV?

You seem to more focused on getting other people to explain what you mean by your vision, rather than trying to understand what they are trying to tell you.
Mathematics is a precise language and no on can put your statements into a mathematical form unless you put them in a precise form. For example:

Every orbiting around the Earth object tends to "free fall" on the surface.

What do you mean by "orbiting"? Orbiting generally means a particle is being accelerated by gravitational forces only. The aircraft are not orbiting. They maintain altitude by thrust and lift forces. Do you mean ' tends to "free fall" on the the surface' or did you mean to say "tends to free fall towards the ground"?

If two objects are free falling and pass through a point at certain altitude with different speed they'll measure different gravitational force at that point.

I wonder why you talking about gravitational force when force has nothing to do with time dilation? Gravitational time dilation is related to gravitational potential, not gravitational force. Are you aware of that? Do you really mean force? Are you aware that free falling objects at any velocity always measure zero gravitational force? Are you talking about horizontal or vertical speeds?

The two airplanes have to maintain same altitude with different speed(velocity).

Different speed(velocity) relative to what? Relative to the ground they have about the same speed.

That should result (at least I see it that way) in different gravitational force on both airplanes.

The gravitational force acting on an object with motion relative to the gravitationally massive body is a slightly tricky subject, which is probably more complex than you want to get into at this stage. Your vision of velocity dependent gravitational force seems to be a form of "effective force" where you mean the gravitational force minus the centrifugal force. Here is a thought experiment that I hope will demonstrate that force is not a useful concept where time dilation is concerned. Imaging the Earth is a perfect spinning sphere made of something like titanium that resists being deformed by rotation. A clock on the equator runs slower than a clock at the North pole. You could reason that this is because the clock on the equator feels less downward force because of the rotational speed at the equator and therefore has a lower effective gravitational force acting on it. Now we could put a clock on a high tower at the North pole so that the "effective gravitational force" acting on the tower clock is equivalent to the effective gravitational force acting on the clock at the equator. Now it can be demonstrated that the tower clock runs faster than the pole ground clock and the equator clock runs slower than the pole ground clock even though they have the same amount of total force acting on them. Now if you really mean force rather than potential, do you want me to mathematically prove to you that your velocity dependent gravitational force concept does not work as far as time dilation is concerned. If I have misinterpreted your vision, then you will have to clarify a lot of things. The best way to do that is with a specific thought experiment that is kept at simple as possible.

Can you try to use some mathematics on this idea of mine, KEV?

Now that I think of it, I gave you all the equations you need back in post #28. For example you can compare all the relative time dilation factors in the titanium Earth experiment to a reasonable accuracy (ignoring frame dragging etc) using this single equation:

$$t_0 = t\sqrt{1-v^2/c^2)}\sqrt{1-2GM/r}$$

where $t_0$ is the proper time of a given clock and using units such that c=1.

 

[sisoev]

Thanks, KEV.
You gave me a lot to think about :)

 

[sisoev]

The gravitational force acting on an object with motion relative to the gravitationally massive body is a slightly tricky subject, which is probably more complex than you want to get into at this stage. Your vision of velocity dependent gravitational force seems to be a form of "effective force" where you mean the gravitational force minus the centrifugal force. Here is a thought experiment that I hope will demonstrate that force is not a useful concept where time dilation is concerned. Imaging the Earth is a perfect spinning sphere made of something like titanium that resists being deformed by rotation. A clock on the equator runs slower than a clock at the North pole. You could reason that this is because the clock on the equator feels less downward force because of the rotational speed at the equator and therefore has a lower effective gravitational force acting on it. Now we could put a clock on a high tower at the North pole so that the "effective gravitational force" acting on the tower clock is equivalent to the effective gravitational force acting on the clock at the equator. Now it can be demonstrated that the tower clock runs faster than the pole ground clock and the equator clock runs slower than the pole ground clock even though they have the same amount of total force acting on them. Now if you really mean force rather than potential, do you want me to mathematically prove to you that your velocity dependent gravitational force concept does not work as far as time dilation is concerned. If I have misinterpreted your vision, then you will have to clarify a lot of things. The best way to do that is with a specific thought experiment that is kept at simple as possible.

Hi :)
I was quite busy in the last days but I am back again.

The above is very close to what my concern is and I'd like to focus on it for now.
If we build a tower on the North pole, we will increase the centrifugal force but will decrease the gravitational force of the earth.
There is no way to equalize the "effective gravitational force" on both clocks in your "rough experiment".

As for the clocks representing different time, my concern is quite straight forward; a clock in electromagnetic field will run slower, but that does not affect the time, it is inaccuracy.
All clocks (no matter their precession) are material bodies and the processes in them depend on the forces applied on them and the energy change due to the different forces.

Two clocks next to each other do not represent different time, only because one of them is spading in a circle with 2000 mph.
And my right hand does not live in different time, only because the clock on it is close to electromagnetic field

 

[stevmg]

Sports Fans!

My post (#33 on this topic) did not bring into account General relativity. I haven't fully digested SR. My concept was strictly SR.

Sort of like the twin (non) paradox. You can show that there is a difference in age between the staionary twin and the one that goes away and comes back (he/she will be younger) just using SR principles alone (you know, Minkowski time-space and all that) without resorting to GR and its acceleration/deceleration.

So, before we build a 10 million mile high North Pole tower, I think we answered the basic question... I think.