# Gravity slows down time.

Then, how is this situation different than one where there is no time dilation and the clock at the top is just running at different rate than the clock at the bottom?

Let's call the frame of reference located at the surface of the earth O, and the one at a height above the earth's surface O'. Both observers agree to record the number of observed sunsets over a particular interval of time. Over some interval of time, O records N sunsets for a total elapsed proper time of τ. O' measures N' sunsets over a proper time τ'. We'll say that the day has a length d in the coordinates of O, and d' in the coordinates of O'. Then, the number of sunsets recorded in O is N = τ/d, and in O', N' = τ'/d'. For simplicity, we'll represent the effect of gravitational time dilation by ξ, so that τ' = ξτ, and d' = ξd. Then, the total number of recorded sunsets in O' is N' = τ'/d' = ξτ/ξd = τ/d = N. So, they record the same number of sunsets.

Why? Because, ultimately, they make their observations over different intervals of time (because of gravitational time dilation), and the length of the day also varies (because of gravitational time dilation). Janus accurately stated this.

O records N sunsets for a total elapsed proper time of τ. O' measures N' sunsets over a proper time τ'. We'll say that the day has a length d in the coordinates of O, and d' in the coordinates of O'. Then, the number of sunsets recorded in O is N = τ/d, and in O', N' = τ'/d'. For simplicity, we'll represent the effect of gravitational time dilation by ξ, so that τ' = ξτ, and d' = ξd. Then, the total number of recorded sunsets in O' is N' = τ'/d' = ξτ/ξd = τ/d = N. So, they record the same number of sunsets.

Yeah, each observer records the same number of sunsets per unit time. That is the paradox. Time for the higher observer is going faster, so the number of sunsets will start pulling ahead of the bottom observer. How can that be if they are on the same spot on the earth?

Then, how is this situation different than one where there is no time dilation and the clock at the top is just running at different rate than the clock at the bottom?

Again, how is this situation different than one where there is no time dilation and the clock at the top is just running at different rate than the clock at the bottom?
That IS the difference! In the OP's scenario, the clocks are identical and therefore tick at the same local rate!

Yeah, each observer records the same number of sunsets per unit time. That is the paradox.
Not a paradox, an error. They do NOT measure the same time interval between sunsets.

Are you a sock puppet for the (now banned) OP?

Yeah, each observer records the same number of sunsets per unit time.

No. If a sunset occurs once every d units of proper time in O, then it occurs once every ξd units in O'. However, the total number of sunsets observed is equal, as they are recorded over different intervals of time.

No, there are no paradoxes in relativity, only misunderstandings. It's elementary to demonstrate that special relativity is mathematically consistent, and therefore cannot contain a logical contradiction. If you wanted to demonstrate that relativity is wrong, you would need to perform an experiment that contradicted its predictions (Hint: none exist), as no logical paradoxes or mathematical errors occur in SR. Markus Hanke has a rather elegant proof of the self-consistency of relativity here:

http://www.thescienceforum.com/physics/29958-general-proof-special-relativity-self-consistent.html

Let me posit the following two scenarios...
There are 4 observers, we'll call them A, B, C, and D. A is at the bottom of a tall building, B is at the top. C is at the bottom of a tall building D is at the top.
A clock maker gives a clock to A and a clock to B. He says, "I have designed the clock for B to run twice as fast as the clock for A." We'll assume for the moment
that time for B is passing as the same rate as the time for A. The clock maker then gives a clock to C and a clock to D. He says, "These two clocks run at the same rate."
We'll assume that time dilation is so great, that time for D is passing twice as fast as time for C. A, B, C, and D all synchronize their clocks at 6:00 am. Sunset is at 6:00 pm.
At 12:00 p.m., according to A's clock, observer B looks at his clock and out his window.
What time does B's clock show and is there a sunset out his window?
At 12:00 p.m., according to C's clock, observer D looks at his clock and out his window.
What time does D's clock show and is there a sunset out his window?

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Zeno, your question has a fundamental problem that makes it impossible to answer. That problem is that you cannot make statements like this in relativity:

At 12:00 p.m., according to A's clock, observer B looks at his clock and out his window.

You aren't clear on what "12:00 pm" means here. Does it mean that, in the frame of reference of A, A's clock measures this time? Or, does it mean that in B's frame of reference, A's clock reads 12:00 pm? You need to be absolutely precise about whose measurements are being compared. If you can clarify, I'll try my best to explain the scenario.

Also, please dispose of the fact that A and B are given faulty clocks. It only confuses the situation. We use clocks to visually represent the time coordinate in each frame reference.

I think it is perfectly clear.
At 12:00 p.m., according to A's clock, observer B looks at his clock and out his window.
Does it mean that, in the frame of reference of A, A's clock measures this time? Or, does it mean that in B's frame of reference, A's clock reads 12:00 pm?
When the clock that is in A's frame of reference shows 12:00 pm, observer B looks at his own (B's) clock and out the window.
Are you implying that when A and B look at A's clock they see two different times coming from the same clock?
It seems that if A's clock shows 12:00 pm then B must agree that it also shows 12:00 pm.

I came up with this scenario to try and illustrate the difference between a clock that is simply running too fast (no time dilation)
and actual, real time dilation.

The problem is the definition of 'now' is a sticky thing in relativity. B will look out his window when the light from A's clock, showing the 12:00 display, reaches him. Sync'ing clocks is a non-trivial problem, as illustrated by the slightly convoluted way in which the GPS network works, compared to such a network which would exist within a Newtonian universe. In the time it takes the light to go from A to B A will have been able to do various tasks, will have seen the sunset progress etc. As chinglu's other threads illustrate (specifically the lengthy post of mine where I go through some calculations), the motion of observers can result in disagreement about the timing of events, their location, even the order in which they occurred. This is why it is particularly important to be specific when making statements about "The time when...." in relativity.

I think it is perfectly clear.

Its not. It only appears to be if you aren't familiar with the failure of simultaneity in relativity.

When the clock that is in A's frame of reference shows 12:00 pm, observer B looks at his own (B's) clock and out the window.

This is the problem. If you're using the measurements of time taken by A, you can only describe what A observes. You can never say, "when A measures that a time t has elapsed, B then looks and sees......". I strongly recommend reading over the Wikipedia page on the topic of simultaneity first:

http://en.wikipedia.org/wiki/Relativity_of_simultaneity

As AlphaNumeric points out, assigning a meaning to "now" that spans several frames of reference is impossible. There is no time coordinate independent of any frame of reference.

How can you guys not be able answer questions on such a simple scenario?
Is B somehow forbidden from looking at his own clock and out the window when A's clock displays 12:00 pm?
Can you explain why? No, of course not because there is no reason why.
Maybe Albert Einstein suddenly appears and says "A's clock reads 12:00 pm, hold still while I put this blindfold on you."
Maybe you are suggesting that when A's clock displays 12:00 pm, it displays some other time than 12:00 pm
according to observer B? Obviously, that's impossible because if A and B both look at A's clock they must
agree on the time that is displayed there.
At the moment that A's clock displays 12:00 pm, B must agree that A's clock reads 12:00 pm. Correct?
How does relativity of simultaneity apply to the scenario I have described?
What are the two events that are spacially separated?
Who is at rest relative to these two events and who is moving?

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At the moment that A's clock displays 12:00 pm, B must agree that A's clock reads 12:00 pm. Correct?

You're doing it again. Exactly what AlphaNumeric and I explicitly stated makes no sense. You're writing "The moment A's clock displays 12:00 pm" as if that concept makes any independent sense. You could, for example, write this: "When A observes that his clock reads a time T, he then observes that B's clock measures a time T'", or "When B measures his clock to read a time T, he observes A's clock to read a time T'", or even "When B observes that A's clock reads a time T, B also observes......". But you can not mix the measurements together. By simply asserting that there is a moment that "A's clock displays 12:00 pm", you've failed to provide any indication to who observes that A's clock displays 12:00 pm. Each observer has their own timeline.

How does relativity of simultaneity apply to the scenario I have described?

In the sense that we're dealing with multiple frames of reference who disagree on the passage of time. The concept of the failure of simultaneity applies in general relativity too, as global frames of reference are impossible to define. The following is from MTW:

Gravitation by Misner said:
"In Newtonian theory or special relativity, one chooses hypersurfaces of constant time. But in dynamic regions of curved spacetime, no naturally preferred time coordinate exists. This situation forces one to make a totally arbitrary choice of hypersurfaces to use in visualizing the time-development of geometry, and to keep in mind how very arbitrary that choice was."

It seems that you may need to familiarize yourself with relativity before posing these scenarios first. I say this not as an insult, but simply as a recommendation - if you're not familiar with the predictions of relativity, then all you are going to do is lead yourself (and those trying to explain the situation) on a wild goose chase.

How can you guys not be able answer questions on such a simple scenario?
The problem is that notions about space and time which we view as 'simple' due to our intuition and experience developed in our everyday lives no longer apply within relativity in a great many ways. Depending on the position and motion of different observers some events happen in different orders or perhaps never at all!

You keep saying "at the moment the clock displays 12" but from the perspective of whom? A? B? Someone else? Are they moving relative to A?

At the moment B sees A's clock read 12:00 then indeed B sees A's clock read 12:00, that is a tautology. But is that the same time as A saw his clock read 12:00? Depends. What if we did it with a referee C. C just happens to be at a particular place at a particular instant that he sees A's clock tick over to 12:00 just as he sees B take off his blindfold. But referee D, who is somewhere else in the universe and/or moving in a different way sees B take off his blindfold before he sees 12:00 on A's clock. Another referee E sees the reverse, the clock ticks over to 12:00 and then later B is seen to take his blindfold off.

A and B could sync their clocks when they meet, go away from one another and then when B's clock gets to 12:00 he looks at A's clock to check they agree. They probably won't.

You speak of 'a moment' as if it is some universal thing. The moment A sees his clock tick over to 12:00 is the moment A sees his clock tick over to 12:00 and the moment B sees A's clock tick over to 12:00 is the moment B sees A's clock tick over to 12:00. Beyond those two tautological statements you cannot say anything more without additional information about their positions, motions, gravitational fields etc.

Yes, this sounds like a convoluted answer to a 'simple scenario' but the fact is intuition and bias expectations based on our personal experiences of everyday life are terrible guides for relativity (and quantum mechanics). Unfortunately too many people cling to their intuition as if it couldn't possibly be wrong.

What a bunch of mumbo-jumbo.
You're writing "The moment A's clock displays 12:00 pm" as if that concept makes any independent sense. You could, for example, write this: "When A observes that his clock reads a time T, he then observes that B's clock measures a time T'", or "When B measures his clock to read a time T, he observes A's clock to read a time T'", or even "When B observes that A's clock reads a time T, B also observes......". But you can not mix the measurements together. By simply asserting that there is a moment that "A's clock displays 12:00 pm", you've failed to provide any indication to who observes that A's clock displays 12:00 pm. Each observer has their own timeline.

http://www.sciencemag.org/content/329/5999/1630.abstract

How then were scientists able to observe time dilation resulting from a gravitational difference of 1 meter at the Earth's surface?
I mean why didn't they look at each clock and say "I can't ascribe any meaning to what is displayed there because my head and both clocks
are in three different strengths of gravitational fields, so any time displayed is meaningless."
Or why didn't the scientist stand there and say, "I can't say for whom this clock displays this time, therefore any time displayed is meaningless and must be discarded because it has no meaning."

Zeno: you're missing the point
When the clock that is in A's frame of reference shows 12:00 pm, observer B looks at his own (B's) clock and out the window.
There is a problem with this scenario, which is that the time "when A's clock shows 12:00 pm" is different for A and B. How does B "know" when A looks at their clock? How does A communicate the information "I am looking at my clock, which says 12:00 pm" to B?
That's the problem.

What a bunch of mumbo-jumbo.
I'm sorry you're unfamiliar with the workings of the subject on which you wish to speak. Unfortunately that is your problem, not mine. Obviously you don't want to understand, you are already convinced your intuition/preconception/bias is sound and the problem lies somewhere other than between your ears.

As I said, I went through a special relativity calculation in another of chinglu's threads earlier this week. In it you'll see how the notion of 'now' and the order of events is not a universal thing set in stone. If you don't understand it and therefore decide to reject it, again it is your problem, not mine.

I'll leave it there, I'm sure you're needing to get back to being indignant about your own lack of comprehension. I shan't keep you.

The problem is the definition of 'now' is a sticky thing in relativity. B will look out his window when the light from A's clock, showing the 12:00 display, reaches him. Sync'ing clocks is a non-trivial problem, as illustrated by the slightly convoluted way in which the GPS network works, compared to such a network which would exist within a Newtonian universe. In the time it takes the light to go from A to B A will have been able to do various tasks, will have seen the sunset progress etc. As chinglu's other threads illustrate (specifically the lengthy post of mine where I go through some calculations), the motion of observers can result in disagreement about the timing of events, their location, even the order in which they occurred. This is why it is particularly important to be specific when making statements about "The time when...." in relativity.

The definition of now is within the frame. That is unless you can prove a frame to frame clock sync method.

So, the high observer has a definition of now and so does the low observer.

Anyway, you did not say. When the high observer climbs down very slowly, does it have a different time vs the low observer.

In that case, then must disagree on the earth's rotational position, which is a contradiction.

Are you going to support SR/GR or run?

I take it from your lack of any calculations that during the 7+ days you had off you were unable to do any of the calculations asked of you. You claim I run but I provided calculations in the other thread and you completely skipped over them and tried to change the subject to MMX/SR.

You claim a contradiction but despite being asked to provide the calculation and having more than a week to do it you have failed to provide quantitative justification. You are the one running. You obviously cannot do any calculation in relativity.

I take it from your lack of any calculations that during the 7+ days you had off you were unable to do any of the calculations asked of you. You claim I run but I provided calculations in the other thread and you completely skipped over them and tried to change the subject to MMX/SR.

You claim a contradiction but despite being asked to provide the calculation and having more than a week to do it you have failed to provide quantitative justification. You are the one running. You obviously cannot do any calculation in relativity.

Go to living reviews in relativity for the calculations. They are public.

Now, let's get back on task.

When the high person climbs back down very slowly with a different time, will they claim the earth is in a different rotational position than does the earth observer, yes or no?

Go to living reviews in relativity for the calculations. They are public.
Please link to the specific paper/article/page which computes the relevant geodesics and worldline lengths. Until such time as you do that I do not believe you.

When the high person climbs back down very slowly with a different time, will they claim the earth is in a different rotational position than does the earth observer, yes or no?
I asked you to provide the calculations for precisely that. My request is "on task". It is your inability to respond which is not on task.

Provide the calculations, which you now claim are freely available.