What is the effect of the relativity of simultaneity?

[gonegahgah]

If you had a tall tower on the equator and the tower had a clock at its base and a clock at the top would those clocks travel through time differently too each other?

Let's say for example that one clock is at 1r from the centre of the Earth and the other is at 2r - just to get a significant distance between them..

The clock at 1r would be at 1g and the clock at 2r would be at 1/4g (approximately).
The clock at 1r would also be traveling at 1s while the clock at 2r would be traveling at 2s.

Would both these SR+GR effect the relative time for both clocks?
What effect would they have?

 

[bcrowell]

The effect relates to the gravitational potential, not the gravitational field. http://www.lightandmatter.com/html_books/genrel/ch01/ch01.html#Section1.5 See subsection 1.5.5 for the equation you need for calculating the effect.

 

[gonegahgah]

Thanks bcrowell. I understand that we look at it in terms of potential and not the actual gravity. So that's okay.
Can you or someone please tell me if these two clocks will travel through time differently to each other?

 

[bcrowell]

Can you or someone please tell me if these two clocks will travel through time differently to each other?

Yes. Have you read the link in #2?

 

[Naty1]

Yes they pass time differently...there are two effects, gravitational potential differences and velocity v = wr differences...
Quickly skimming the reference, I did not pick up a direct explanation.

If you search Wikipedia for GPS system or similar you can find that time corrections to offset the different passage of time on orbiting satellites and GPS Earth stations is substantial. This is analogous to your posted situations...clcoks at different heights...

 

[Parkur]

If you had a tall tower on the equator and the tower had a clock at its base and a clock at the top would those clocks travel through time differently too each other? Let's say for example that one clock is at 1r from the centre of the Earth and the other is at 2r - just to get a significant distance between them.. The clock at 1r would be at 1g and the clock at 2r would be at 1/4g (approximately). The clock at 1r would also be traveling at 1s while the clock at 2r would be traveling at 2s. Would both these SR+GR effect the relative time for both clocks? What effect would they have?

General relativity contains special relativity, so it's not strictly correct to say that you are adding the effects of the latter to the effects of the former. I think what you mean to ask is, will the difference in location within the gravitational field and the difference in velocity both contribute to the difference in the clock rates. General relativity covers both of these effects very simply. You can read about this on many online resources, such as: http://www.mathpages.com/rr/s6-06/6-06.htm

The answer to your question is that the clock at the top of the tower would gain about 7E-10 seconds per second relative to the clock on the ground. Almost all of this difference is due to the difference in height, which tends to make the higher clock faster. The higher clock also has greater speed, which tends to make it slower, but this effect is only about 3.5E-12 seconds per second, so by far the dominant effect is due to the height (in this example).

 

[gonegahgah]

Thank you people for those answers.
So I would guess from this that the top clock is more like the twin who stays at home and is aging faster due to time dilation effects and the bottom clock is more like the twin who travels away and ages slower due to time dilation effects and who upon return finds that the stay at home twin is much older than they are?
Is that correct?

 

[Parkur]

I would guess from this that the top clock is more like the twin who stays at home and is aging faster due to time dilation effects and the bottom clock is more like the twin who travels away and ages slower due to time dilation effects and who upon return finds that the stay at home twin is much older than they are? Is that correct?

No, that's not correct (unless when you say "more like" you simply mean that the higher clock runs faster overall, without regard to the reason). The twins situation doesn't usually involve gravity, it simply considers the effects of motion, and in the example you described the clock at the top of the tower is moving faster, so it is slowed due to that effect. However, in the situation you described the clocks are also at different altitudes in a gravitational field, and the higher clock is faster due to the gravitational effect. So it doesn't make any sense to say "the top clock is more like the twin who stays at home". It's a completely different situation, but if you focus on just the velocity effects, the higher clock is like the traveling twin.

 

[gonegahgah]

Sorry, I meant are the time dilation effects the same; not are they parallel situations.

Would the same aging effect occur (whether it is SR or GR related) due to a correlating direction of time dilation? So would the higher altitude clock travel faster through time due to a similar direction of time dilation (albiet mostly due to relative GR) as the twin who stays at home (albiet due to relative motion). Does time dilation have the same effect in both cases? Will the higher altitude clock age faster and travel through time faster than the lower altitude clock; as also occurs for the twins?

 

[yuiop]

If you had a tall tower on the equator and the tower had a clock at its base and a clock at the top would those clocks travel through time differently too each other?

Let's say for example that one clock is at 1r from the centre of the Earth and the other is at 2r - just to get a significant distance between them..

The clock at 1r would be at 1g and the clock at 2r would be at 1/4g (approximately).
The clock at 1r would also be traveling at 1s while the clock at 2r would be traveling at 2s.

Would both these SR+GR effect the relative time for both clocks?
What effect would they have?

Although as others have pointed out, we shouldn't call these SR+GR effects, we can split the time dilation into velocity and gravitational potential components. For a spherical non rotating massive body you can use:

$$t = t_0 \sqrt{1-\frac{v^2}{c^2}} \sqrt{1-\frac{2GM}{rc^2}}$$

where v is the instantaneous tangential velocity as measured by a non rotating observer at altitude r. The Earth is not exactly spherical and neither is it non-rotating, but even so, the above formula is good enough to work out the time dilation of GPS satellite clocks (for example) to a reasonable degree of accuracy.

For a very tall tower, it is easy to see that if it was tall enough the velocity at the top of the tower would be approaching light speed and this places a limit on how tall the tower can be, for anybody with non-zero rotation. This indicates that the velocity component that increases the time dilation with increasing altitude, overwhelms the gravitational time dilation which decreases with increasing altitude.

For a small orbiting object the the orbital velocity is given by $v=\sqrt{(GM/r)}$ and the above time dilation formula can be written in terms of altitude only as:

$$t = t_0 \sqrt{1-\frac{GM}{rc^2}} \sqrt{1-\frac{2GM}{rc^2}} = \sqrt{1-\frac{3GM}{rc^2}+\frac{2G^2M^2}{r^2c^4}$$If I recall correctly, it works out that whether or not a given satellite clock ticks faster than a clock on the surface of the Earth, depends on the altitude of the satellite.

P.S. Short answer to your question is... yes

If you use the first formula, plug in the values for G, c, M (the Mass of the Earth), r (radius of the Earth at the equator) and v (the velocity at the equator of the Earth) to obtain the time dilation for a clock at the base of the tower. For the clock at the top of a tower of height h, replace v with v*(r+h)/r.

 

[yuiop]

Does time dilation have the same effect in both cases? Will the higher altitude clock age faster and travel through time faster than the lower altitude clock; as also occurs for the twins?

As you have probably figured out by now from my previous post, you can only assume that the clock at the top ages faster if you assume the Earth (or other massive body) is not rotating. If the massive body is rotating and the tower is tall enough, the clock at the top will age slower than the clock at the base.

 

[gonegahgah]

lol. Thx yuiop. I was thinking there was probably a transition point where the effects given to 'SR' would begin to be greater than the effects given to 'GR' and that above a certain altitude the time dilation would change from being slower to being faster. Is that what you are meaning yuiop?

 

[gonegahgah]

If I get an answer for the last question that is cool. But the main thing I wanted to check was that the clock at the top of the tower would animate a little faster and travel through time a little faster than would the clock at the bottom of the tower. Thanks for that.

And although the twin example is not exactly parallel that there is still time dilation involved and that the effect itself is the same where one moves through time faster than the other.

With the twins what we see is that the twin who travels away animates slower and travels through time more slowly. So by the time they get back they experience less time elapsed for themselves than does the twin remaining at home.

In the same respect, the clock at the top of the tower animates faster and travels through time slightly faster. So the top clock experiences more time elapsed for itself relative to what the bottom clock experiences. And the bottom clock experiences less time elapsed for itself relative to what the top clock experiences.

In this way the two examples both show the results of time dilation differences.
Would somebody be able to confirm that I have this correct; or rectify me if I don't?

 

[yuiop]

If I get an answer for the last question that is cool. But the main thing I wanted to check was that the clock at the top of the tower would animate a little faster and travel through time a little faster than would the clock at the bottom of the tower. Thanks for that.

This assumption is fine if we assume a hypothetical non rotating gravitational body that the tower stands on.

And although the twin example is not exactly parallel that there is still time dilation involved and that the effect itself is the same where one moves through time faster than the other.

With the twins what we see is that the twin who travels away animates slower and travels through time more slowly. So by the time they get back they experience less time elapsed for themselves than does the twin remaining at home.

In the same respect, the clock at the top of the tower animates faster and travels through time slightly faster. So the top clock experiences more time elapsed for itself relative to what the bottom clock experiences. And the bottom clock experiences less time elapsed for itself relative to what the top clock experiences.

In this way the two examples both show the results of time dilation differences.
Would somebody be able to confirm that I have this correct; or rectify me if I don't?

Yes, this is essentially correct given the condition I gave above.

For a gravitational twin experiment you can do this. Two twin siblings are the same age and both at the top of the tower. One is dropped down by parachute to the base. Some long time later the second twin drops to joins his brother. The one that spent the most time at the top of the tower will be younger than the one that spent the most time at the base of the tower when they meet again. Since they both experience the same trip down the tower, the difference in their ages can be attributed entirely to gravity.

 

[yuiop]

lol. Thx yuiop. I was thinking there was probably a transition point where the effects given to 'SR' would begin to be greater than the effects given to 'GR' and that above a certain altitude the time dilation would change from being slower to being faster. Is that what you are meaning yuiop?

Yes, this is sort of correct, but the other way around.Above a certain altitude the time dilation would change from being faster to being slower. See this Wolfram Alpha plot.

For a tower that is that less than about 5 times the radius of the Earth, the clock at the top is faster than the clock at the base, but for a tower that is higher than about 5 times the radius of the Earth, the clock at the top is slower than the one at the the base due to the SR velocity induced time dilation caused by the rotational velocity of the Earth exceeding the GR gravitationally induced time dilation. Strictly speaking, GR takes both gravitational and velocity effects into account, as well as charge, stress and everything else, so it not correct to call the gravitational component of time dilation the GR component.

For a tower that is about 645,000 times as high as the radius of the Earth the top of the tower would be moving at light speed and the clock at the top will have stopped, but of course nature would not allow that to actually happen and the rotational velocity of the Earth would slow down first. Another plot here.

 

[gonegahgah]

Cool. Thanks yuiop. Yeh I mixed up that bit on the transition from faster to slower. Silly me. Thanks for correcting that.

What do you mean when you say 'hypothetical non-rotating gravitational body'? Wouldn't a tower at the equator be on a rotating gravitational body? Does that change the answer?

 

[yuiop]

What do you mean when you say 'hypothetical non-rotating gravitational body'? Wouldn't a tower at the equator be on a rotating gravitational body? Does that change the answer?

I meant that a clock on the top of tower on a 'hypothetical non-rotating gravitational body' will always be faster than a clock at the base of the same tower, no matter how high the tower is.

If the tower is on the equator of a rotating body such as the Earth, then whether or not the clock at the top of the tower is faster than a clock at the base of the tower, depends on the height of the tower.

 

[gonegahgah]

Thanks yuiop.

So they will age at different rates which is cool. I thought that might be the case but I was wondering what the effect is on something like the tower itself too. I assume that the top of the tower will age differently to the bottom of the tower just as the two clocks tick and age differently at the top and bottom. And in our example where the top clock ages faster than the bottom clock; the top of the tower will also age faster than the bottom of the tower. That is the case; isn't it? It doesn't just apply to the clocks?

 

[gonegahgah]

I'll go onto the next bit yuiop. I am wondering about the different journeys of the top of the tower and the bottom of the tower through time.

To begin with let's first take a look at the effects for the twins. Can you confirm whether I have the following correct?

We can see that the traveling twin remains younger. Inversely, the stay at home twin grows older. If the distance to cover there and back is d light years then the stay at home twin will look at their own clock and think it took the traveling twin a multiple of d years to get there. The traveling twin will look at their own clock and think it only took them a fraction of d years to get there and back.

The traveling twin might think they traveled the full distance to the destination and back at faster than the speed of light. However, when they step out they find that the universe has aged much more and that they must have got there and back at much slower than the speed of light.

But it is not that the traveling twin got there by their own clock faster than the speed of light. Instead the universe contracted around them and so the distance became much shorter for them to travel. Everything else became smaller, the planets, the galaxies and the distance between things. Those things also became faster so that the Earth spun faster (in line with the stay-at-homer's clock) and orbited the sun quicker and the galaxies spun faster. Or so it appears to the traveling twin so that by the time they get back the universe around them is a lot older than they were expecting and certainly compared to how they aged themselves.

I'm really not sure how the Earth twin experiences the travel of the traveling twin so I'll leave that. Suffice it to say that the traveling twin arrives back much later than traveling at the speed of light would directly suggest but will also be found to be much younger; because they traveled through time slower.

Is that correct so far?

 

[yuiop]

Thanks yuiop.

So they will age at different rates which is cool. I thought that might be the case but I was wondering what the effect is on something like the tower itself too. I assume that the top of the tower will age differently to the bottom of the tower just as the two clocks tick and age differently at the top and bottom. And in our example where the top clock ages faster than the bottom clock; the top of the tower will also age faster than the bottom of the tower. That is the case; isn't it? It doesn't just apply to the clocks?

Sorry about the delay getting back to this, Almost forgot about it! That is correct. Time dilation applies to everything that experiences time, so that includes towers and people.

I'll go onto the next bit yuiop. I am wondering about the different journeys of the top of the tower and the bottom of the tower through time.

To begin with let's first take a look at the effects for the twins. Can you confirm whether I have the following correct?

We can see that the traveling twin remains younger. Inversely, the stay at home twin grows older. If the distance to cover there and back is d light years then the stay at home twin will look at their own clock and think it took the traveling twin a multiple of d years to get there. The traveling twin will look at their own clock and think it only took them a fraction of d years to get there and back.

The traveling twin might think they traveled the full distance to the destination and back at faster than the speed of light. However, when they step out they find that the universe has aged much more and that they must have got there and back at much slower than the speed of light.

But it is not that the traveling twin got there by their own clock faster than the speed of light. Instead the universe contracted around them and so the distance became much shorter for them to travel. Everything else became smaller, the planets, the galaxies and the distance between things.

This is basically correct. As long as time and distance are measured in the same reference frame both observers will agree on their relative velocity. Whether or not the universe length contracted depends on the reference frame.

Those things also became faster so that the Earth spun faster (in line with the stay-at-homer's clock) and orbited the sun quicker and the galaxies spun faster. Or so it appears to the traveling twin so that by the time they get back the universe around them is a lot older than they were expecting and certainly compared to how they aged themselves.

This is where things get a little complicated. The traveling twin will measure the stay-at-homer's clock to be ticking slower and will measure the Earth to be rotating slower. This seems counter intuitive and to really understand what is going on you need to understand the "relativity of simultaneity".