B Effects on time by gravity or motion

[Suppaman]

TL;DR Summary

I have read that depending on your distance from a gravitational source your flow of time is different.

I read that just a few feet difference in height would show a different rate of time and that was measurable. However, If the clock that is at a more elevated position it will be traveling through more space, is that what is responsible for the difference in the rate of time for that clock? I understand that clocks at different heights from the Earth's gravity field do experience different rates of the flow of time. I wish to know what is the cause of this difference.

 

[Gaussian97]

This difference is the gravitational field itself. It's a main result of General Relativity. In the Newtonian limit you have $$\Delta t_{1}=\left(1-g\left(\phi_2-\phi_1\right)\right)\Delta t_{2}$$ where $\phi_i$ is the gravitational potential.

 

[Suppaman]

OK. And our clock, if it was far from any source of gravity but was traveling the same path through space would also, because of this travel, experience an effect on the rate of time it experiences?

 

[Ibix]

General relativity regards space and time as parts of a single thing, imaginatively named "spacetime". It turns out that, just as in space there is a notion of "distance" there is a notion of "distance" in spacetime, usually called interval. For massive objects, such as ourselves and clocks, the interval along our path through spacetime turns out to be equal to the amount of elapsed time we measure - called proper time (proper in the Latin sense of "your own", like property, not in the modern English sense of "correct").

As you say, it turns out that if I arrange to start two clocks simultaneously, one on the floor and one on a shelf, and stop them simultaneously (I'm glossing over a few subtleties in the meaning of simultaneous), they will show different times. This is because the elapsed time (the "distance") between the "clock start" and "clock stop" events was different along the paths followed by the two clocks. This is a consequence of spacetime being curved - one of the other consequences of which is what we call gravity.

So I wouldn't actually describe the effect as "the rate of time being different", although you certainly do see that written. I'd rather say that the curvature of spacetime is such that clocks at different altitudes follow different paths through spacetime, and if one compares the clock readings at two different times, the lengths of the paths between the first and second comparison are different.

 

[Ibix]

OK. And our clock, if it was far from any source of gravity but was traveling the same path through space would also, because of this travel, experience an effect on the rate of time it experiences?

"Travelling through space" isn't something you can define without reference to something else. So I'm not sure what you are trying to ask here. What object are you thinking of as "not moving"?

 

[Suppaman]

So, it is not the gravitational field that is causing the difference? I am not arguing, I just do not see what is causing the time difference as both answers I get to my question say it is one thing or another thing. Which is it?

 

[Ibix]

So, it is not the gravitational field that is causing the difference?

The curvature of spacetime is the gravitational field.

The point is that if you set up a way to compare the reading on two clocks now, and wait a little bit and compare them again, you will find that the higher clock has advanced slightly less. The reason for that is that spacetime is curved, and the path through spacetime followed by the higher clock is of a different "length", which turns out to be equivalent to elapsed time. It doesn't have anything to do with traveling through space (again, glossing over some subtleties here). This effect would happen if the Earth did not rotate or orbit anything.

 

[Suppaman]

What "path" is the higher clock traveling, where does "length" of this pat come from? I can think that the higher clock is experiencing a different space-time because the gravity field for it is different but do not see any traveling involved. If this is just semantics are there different words that can be used?

 

[Gaussian97]

What "path" is the higher clock traveling, where does "length" of this pat come from? I can think that the higher clock is experiencing a different space-time because the gravity field for it is different but do not see any traveling involved. If this is just semantics are there different words that can be used?

Is traveling in time, not in space.

 

[Suppaman]

OK. Not to confuse things but if we have our spaceship going at a good percentage of C then it is experiencing a different rate of time because it is moving through space-time? Our clock in a different gravity field is experiencing a different rate of flow only because of the strength of the gravity field and not because it is moving through space. So we have different ways to change the rate at which time passes, adjust gravity or speed of motion through space. Does anyone know what causes the rate of time to change in either case?

 

[Ibix]

What "path" is the higher clock traveling,

It's usually called a worldline. Imagine drawing a picture of a planet with two clocks sitting on it on a piece of paper. Then draw another picture of the planet and clocks a second later, then another a second after that, and another and another. Stack all your pictures up, then imagine dissolving the paper and leaving just the ink floating there. You'd have a cylindrical lump of ink, the planet, with two tubes on the side of it, the clocks. The length I'm talking about is essentially the height of the cylinder.

Now imagine that each sheet of paper was slightly thicker at one edge than the other, so my stack would curve slightly to one side. This time, when I dissolve the paper, one of the clock tubes will be slightly longer than the other because it was going through thicker paper. If the top of each sheet is "the universe at a given time", now I've got slightly more time for one clock than the other, due to curvature.

That's only an analogy - it's usually called the block universe (google with care - there's a lot of nonsense out there). It uses Euclidean geometry instead of the rather more complex pseudo-Riemannian geometry that underlies relativity, so there's quite a lot wrong with it. Nevertheless, it hopefully conveys a sense of what I'm getting at.

 

[Ibix]

Not to confuse things but if we have our spaceship going at a good percentage of C then it is experiencing a different rate of time because it is moving through space-time?

As I asked before, what is it traveling at high speed with respect to?

Actually, the underlying mechanism is the same - two clocks in relative motion are following different paths through spacetime, and those paths have different "lengths". However, in the case of motion (far from a gravitational field, just to keep things simple) the reason is closely analogous to why the hypotenuse of a right angled triangle doesn't have the same length as the other sides of the triangle. Loosely speaking, the worldline of one clock is one side of the triangle, the worldline of the other clock is the hypotenuse, and the distance it traveled through space according to the first clock is the third side.

 

[Suppaman]

"As I asked before, what is it traveling at high speed with respect to? "

Just to step aside for a second, if my two space ships traveling at nearly C relative to their homeworld decide to say that ship one is just a 1/4 mph faster than its partner so it should be able to accelerate away from its partner very fast because the speed between the two is nowhere near C. Now if you say, wait, it is relative to where it left together and I point out that homeworld was traveling through space relative to something else at nearly C and our two ships cannot be going anywhere C relative to homeworld which is already too speedy. Sorry for the detour but I do not understand your reference to speed respect to.

 

[Ibix]

I do not understand your reference to speed respect to

There's no absolute speed. When you say you are traveling at 30mph, you mean 30mph with respect to the surface of the Earth (which is to say, as measured by someone for whom the Earth's surface is stationary). But you are also doing up to 1000mph relative to a point at the north pole as the Earth turns, and roughly 20km/s relative to the Sun as the Earth orbits it. All of these speeds are correct - but all are meaningless unless I tell you what I'm regarding as "at rest".

That's why you need to specify what you are considering to be "at rest" when you tell me how fast something is going.

if my two space ships traveling at nearly C relative to their homeworld decide to say that ship one is just a 1/4 mph faster than its partner so it should be able to accelerate away from its partner very fast because the speed between the two is nowhere near C. Now if you say, wait, it is relative to where it left together and I point out that homeworld was traveling through space relative to something else at nearly C and our two ships cannot be going anywhere C relative to homeworld which is already too speedy.

Speeds don't add. If I'm doing a speed $u$ in one direction and you are doing $v$ in the opposite direction, you will say that my speed is $$u'=\frac{u+v}{1+uv/c^2}$$If $u$ and $v$ are very small compared to lightspeed then $uv/c^2$ is very nearly zero and $u'$ is very nearly the sum of our speeds, which is why you can add every day speeds - the error from neglecting the $uv/c^2$ for car speeds (even supersonic jet speeds) is smaller than neglecting the effect of bugs splattering on the front of your car by several orders of magnitude. But as you go faster and approach the speed of light, you can no longer neglect that term. Its effect is such that $u'$ is always less than $c$, regardless of $u$ and $v$. For example, if your ship is doing 0.99c relative to your homeworld and your homeworld is doing 0.99c relative to me, then your ship is doing (0.99+0.99)/(1+0.99*0.99)=0.99995c relative to me.

 

[Dale]

Does anyone know what causes the rate of time to change in either case?

I would say it is the metric. It “changes” for the same reason that the hypotenuse of a triangle is longer than one of the legs. That is a function of the metric.

 

[Suppaman]

I just finished my nap, you do those at my age. Now, saying it changes because it does, well, not useful if you want to create a test of the "how." Yes, I know, a lot of things are not yet known, but there must be some speculation. It would seem the fabric, vacuum of space, how this could interact on the very small scale of the things that make "matter" might explain something. And I am interested in thinking about such speculation but I know I lack the science to even do that, but some people must have.

 

[Ibix]

Now, saying it changes because it does, well, not useful

I don't think @Dale is saying that it changes because it does. I think he's saying that time is a consequence of the metric - so the "rate of time" measured by a clock changes in different circumstances because of the structure of the metric. The metric is the mathematical object that leads to the notions of distance and elapsed time - in a sense, it's the mathematical formalisation of the stack-of-paper model I was talking about earlier.

Actually, that's all GR says about this - Einstein's equations simply let you define a metric that is consistent with the mass and energy that's in play, and to work out their future dynamics. If you want a more fundamental understanding, you need to study our theories of quantum gravity. Unfortunately, we don't yet know which (if any) of them is correct. You're running up against the limits of what we know here.

 

[Dale]

Yes, I know, a lot of things are not yet known, but there must be some speculation.

I don’t think this is in the unknown category. The metric is determined by the distribution of energy, momentum, pressure, and stress.

 

[Janus]

I just finished my nap, you do those at my age. Now, saying it changes because it does, well, not useful if you want to create a test of the "how." Yes, I know, a lot of things are not yet known, but there must be some speculation. It would seem the fabric, vacuum of space, how this could interact on the very small scale of the things that make "matter" might explain something. And I am interested in thinking about such speculation but I know I lack the science to even do that, but some people must have.

Try thinking of it this way:
You and someone else are standing next to each other on a flat featureless plane. You start walking North at a fixed pace, and your companion starts walking Northeast at the same pace.
It is obvious that your companion is not traveling North at the same rate as you are. He is dividing up his motion between Northward and Eastward movement.

Now swap "moving North" with "moving through time" and "moving East" with "moving in space"

Now with this view, you are moving forward through time at a fixed rate, but not moving in space. You are "at rest". Your companion is also moving through time, but also is moving through space. But he still has to "divide up" his over all "motion" between these two. Thus he doesn't move through time as fast as you do.

This is basically the idea behind time dilation due to motion. Space and time have been combined into a single idea of Space-time that are no longer independent of each other.

However, there is an additional rub. With the above example, both parties involved agree as to who is progressing "North" or through "time" at the greater rate.

But there is no equivalent to North-South or East-West in space-time. Only the equivalents of forward-back and Left-Right.

So in our examples above, each of you judges that the direction you are walking in as being "forward", and that the other person is making slower progress in the "forward" direction than you are.

If forward movement is the space-time equivalent of moving through time, and left-right as space, then each of you judge yourself as moving through time, but not space and the other as moving through both time and space. Each considers himself as being at "rest". It also means that each of you judges the other as moving more slowly through time ( aging slower).

The gist of it is that the relationship between time and space is more complicated than it is assumed to be in Newtonian physics, and everyone measures their movement through it according to their own reference, as there is no fixed "direction" to time or space.

 

[Suppaman]

Relative... OK, it is the end of time, everything has gone over the horizon and all matter has evaporated. It is just me and my spaceship. I start accelerating in any direction, but with nothing left in the universe, there is nothing to be relative to. I am accelerating so I know I am going faster, time slows for me, what happens, do I reach the ultimate speed limit, but that is relative to something and there is nothing in the universe to be relative to. Is it just time that is the limit, if I go so fast time stops incrementing and I can not go faster because the clock has stopped?

 

[Ibix]

I am accelerating so I know I am going faster

Faster than what? You've nothing to judge it by. And there's no way to tell (without something to define as stationary) whether you're accelerating or decelerating.

time slows for me

No it doesn't - your clock continues to tick at one second per second.

do I reach the ultimate speed limit, but that is relative to something and there is nothing in the universe to be relative to

If there's nothing to measure your speed relative to, you can't really define your speed.

Is it just time that is the limit, if I go so fast time stops incrementing and I can not go faster because the clock has stopped?

No. Your clock always ticks at one second per second. Other people - who you've specifically excluded from this scenario - might regard your clock as ticking slowly, but you would also regard their clock as ticking slowly. That may seem paradoxical but it isn't - not any more than Janus' example where both people regard the other as advancing more slowly because they disagree about forwards. In this case, you disagree about which direction in spacetime is "time".

As you go faster relative to me, I regard your clock as ticking more slowly. But nothing happens to your clock from your perspective, because you can always regard yourself as stationary and that will always be the natural choice of perspective for you. There is no absolute speed.

 

[Suppaman]

Try thinking of it this way:

I can do that, as I read I comprehend the words and their meaning and the image it causes in my mind. There is a "but", if I look at (stay with me) a very tasty cake and then I read through the recipe, all the necessary steps, a careful list of ingredients (with no background how they came into existence) of the proper measure, how to mix, the pans to use, temp and bake times. Perhaps a frosting and all the rules for that. Finish the cake, put it on the table. Look at it. None of the recipes is apparent, It just is.

What I want is the recipe for time, all the little things that must be just so for it to work as it works. I know I could put the bits of the cake in a mass spectrometer, I was a developer for one (2001 to 2015) but that would only tell me what was there, not how it got constructed. We are learning to make clocks that are much smaller and more accurate than ever so we should be able to design some tests to learn the secrets of time. I am getting old, science needs to do better.

 

[Ibix]

What I want is the recipe for time

In GR, it falls out of the metric. Future theories may say other things, but as noted above, we don't know which, if any, of our current candidates is the correct one.

I am getting old, science needs to do better.

I'm afraid we can't make these things run to your personal timetable. Discoveries will happen when they happen.

 

[Dale]

It is just me and my spaceship. I start accelerating in any direction, but with nothing left in the universe, there is nothing to be relative to.

You would be accelerating relative to your rocket’s exhaust and your rocket+exhaust center of mass.

time slows for me,

Time never slows for you in your own reference frame. It only slows for you in someone else’s frame.

What I want is the recipe for time

Again, that is the metric. Or maybe the Einstein field equations which tell you how to determine the metric from the energy, momentum, pressure, and stress distribution

Edit: Having thought about it a bit, in your analogy I think the cake is the metric and the recipe is the EFE

 

[Janus]

Relative... OK, it is the end of time, everything has gone over the horizon and all matter has evaporated. It is just me and my spaceship. I start accelerating in any direction, but with nothing left in the universe, there is nothing to be relative to. I am accelerating so I know I am going faster, time slows for me, what happens, do I reach the ultimate speed limit, but that is relative to something and there is nothing in the universe to be relative to. Is it just time that is the limit, if I go so fast time stops incrementing and I can not go faster because the clock has stopped?

Imagine you are carrying an endless supply of clocks. as you accelerate your drop clocks overboard (carefully so you don't give them any extra velocity. Yo want them to have the same velocity as the ship at the moment of release.
Since you are accelerating, each clock will fall "behind you" . Now remember that addition of velocities equation from post #14? You'll need to apply it here. So let's say that you drop off clock 1 and then continue to accelerate until you are moving at 0.1c relative to clock 1. You now drop of clock 2 and then continue accelerating until you are moving at 0.1c relative to clock 2. How fast are you moving relative to clock 1?
Not 0.2c, but just 0.198c. Even though you measure your speed relative to clock 1 as being 0.1c and Clocks 1 and 2 measure their speeds realtive to each other as being 0.1c, you measure your relative speed with respect to clock 1 as being less than 0.1c+0.1c.
If you now drop clock 3 and accelerate to 0.1c relative to it, you'll measure your speed as being 0.296 c relative to clock 1. Nothing will ever stop you from accelerating to 0.1c relative to the last clock you dropped, but no matter how many clocks you drop, you will never measure your speed relative to clock 1 as being other than less than c.

Nothing effects your time rate (as measured by you), but if you measure the rate of the clocks you leave behind, they will run slow compared to your own. The further behind, the slower they run. (It's a bit more complicated if you consider yourself under constant acceleration as you make the measurements. So for simplicity's sake we will assume that you cut your engine just before you drop off each clock and make your measurements, and then fire it up again)

On the other hand, an observer left with clock 1 would note that your rate of acceleration decreases with time as you drop off the clocks. So for example, while for you, upon dropping off clock three, you will have measured yourself as having accelerated to 0.1c relative to Clock 1, while the difference between clocks 1 and 2 has deceased to 0.096c, The observer with clock 1 would say that the relative velocity between himself and clock 2 is and always has been 0.1c, and you have only accelerated by 0.096c between dropping off clocks 2 and 3. he would also say that your clock runs slower and slower as you accelerate.

The point is that both viewpoints are equally valid and you can't choose one over the other as being the "correct" one.

 

[pervect]

Summary: I have read that depending on your distance from a gravitational source your flow of time is different.

I read that just a few feet difference in height would show a different rate of time and that was measurable. However, If the clock that is at a more elevated position it will be traveling through more space, is that what is responsible for the difference in the rate of time for that clock? I understand that clocks at different heights from the Earth's gravity field do experience different rates of the flow of time. I wish to know what is the cause of this difference.

I assume your question is based on the rotating Earth?

On a non-rotating planet, a higher clock (at rest relative to the surface of the planet) would always tick faster than a lower clock, pre-supposing one has the usual synchronization scheme so that we can talk about comparing clock rates at all.

By "at rest", I basically mean having a constant latitude and lognitude relative to the surface of the planet.

On a rotating planet, if we assume that the clock is still at rest relative to the surface of the planet by keeping a constant lattitude and longitude, there may be an additional effect that counters this. This effect goes in the opposite direction if it is present, and may even become large enough to counter the first effect if the altitude is high enough.

The opposing effect will be largest for a clock on the equator, and totally absent for a clock at the poles.

It's possible to be more quantitative, but the original question wasn't phrased very precisely to I think this may be an adequate answer. Assuming I have guessed the question correctly, of course.

 

[Suppaman]

I do follow your thoughts and would like to ask/comment on the first point. When something takes off on a journey to C it appears that the starting point is a fixed reference point. Almost like it is entangled, the point and the ship. These starting points may not have any physical location that is specified by some map coordinate, it is just where the journey started. And you can never get too far from this point that you lose track of where you started. You might not ever find it again and I do not know when the link is broken, if ever. If you built a ship from matter that was sent flying from the big bang at near C then you might not be able to get this ship to go very fast as it would always be referenced to whence it started. If I have this wrong, please tell me what I misunderstand. I believe I followed the balance of your reply.

 

[Dale]

When something takes off on a journey to C it appears that the starting point is a fixed reference point. Almost like it is entangled, the point and the ship.

Wow! That is a truly impressive intuitive leap in a completely wrong direction.

Points don’t get entangled, particles do. All of the rest of your post reads like one of those train wrecks where one small defect causes one car to skip the track and then the whole train follows. I don’t even know where to start. Best if you just go back and delete this whole line of reasoning and ask some clarifying questions about whatever point was said above that triggered this departure.

 

[Ibix]

When something takes off on a journey to C it appears that the starting point is a fixed reference point.

This is a matter of choice. If you are willing to pay the price in terms of increased mathematical complexity, you can regard yourself as stationary even if you accelerate.

These starting points may not have any physical location that is specified by some map coordinate, it is just where the journey started.

Of course the start points have a physical location. It might be specified in different ways (and may mean different things) for different frames, but it absolutely has a location.

If you built a ship from matter that was sent flying from the big bang at near C then you might not be able to get this ship to go very fast as it would always be referenced to whence it started.

I have no idea what you are trying to communicate here. You can always regard yourself as "at rest", so of course you can always accelerate.

 

[Herman Trivilino]

I am not arguing, I just do not see what is causing the time difference as both answers I get to my question say it is one thing or another thing.

Suppose you have two clocks. You place them next to each other and synchronize them. Then you do something, like move one of the clocks relative to the other. You might slowly move it to the top of a mountain, leave it there for awhile, and then slowly return it. Or, you could fly one of them around the world. Or send one away at a high speed relative to you and then have it return (twin paradox). At the end, the clocks are next to each other again. They won't still be synchronized. If they were still synchronized what would be the cause of that?

 

[pervect]

I suspect the usual confusion here, though I can't follow the original poster's argument well enough to be positive I'm right. (There are other issues that cause confusion than the one I'll present, but the one I'll present is very common).

The "usual issue" is that people assume, both explicitly and/or implicitly, that the notion of synchronizing clocks is independent of the observer.

The notion of "synchronizing clocks" is needed, for instance, to determine the idea of "now". When we have a model of space-time, "now" is the set of points that occur "at the same time".

The notion of synchronizing clocks, the notion of "now" in special relativity, is observer dependent. Trying to understand special relativity without realizing this issue inevitably leads to confusion. The bad assumption that causes much confusion is to assume that the notion of "now" independent of the observer, that everyone agrees on what "now" is.

An implicit assumption of the existence of "now" is needed to talk about the rate at which clocks tick when they are at different locations. We compare the time on one clock "now" to the time on a clock at a different location "now".

This usually first shows up in the twin paradox, in flat space-time - a much easier topic to talk about than gravity.

Basically, the only way that in A's frame of reference that B's clock can run slow, and that in B's frame of reference, A's clock runs slow, is when A and B have different notions of "now".

There's a name for this issue, it's called the "relativity of simultaneity". However, just giving the name of the issue doesn't explain it enough so that people who are not already aware of the issue understand it. In general, it seems very hard to talk about this issue in a way that will be understood, but I keep trying.

Things get very complicated if someone tries to understand general relativity without understanding this feature of special relativity. Special relativity is much easier to talk about.

 

[PAllen]

Some of this will be beyond the OP, but I thought it would be useful to quantify 3 effects in question in this thread:

1) Time dilation between two clocks, each with constant proper acceleration, maintaining fixed distance from each other.

2) time dilation (in comparison to a system of stationary clocks as in (1)) due to change in tangential velocity with altitude of clocks comoving with the surface of the earth.

3) Tidal gravity corrections to (1), that is how much the time dilation from top to bottom of a building is diffferent from in a uniformly accelerating rocket of the same altidude.

The relevant quantity I compute is (d/dx)(dτ/dt) for physically appropriate coordinates. This gives a rate of time rate change per meter (in the units I use).

For (1), the exact value of this quantity is simply g/c². This is about 10^-16, thus time rate changes about 1 part in 10¹⁶ per meter. Note that this is the only one of these effects that has been observed on the surface of earth. Even the latest research clocks are not yet precise enough to measure either of the other effects. Let us call (1) the Rindler time dilation as distinct from (3).

For (2) the exact value of this quantity is -vϒω/c², where omega is radians per second for Earth's rotation. If you compare this with (1), specifically (1)/(2), with the approximation that gamma near Earth is close to 1, you get g/rω², which is about 300. Thus this effect is 300 times smaller than the Rindler time dilation. This is currently undetectable, but another order of magnitude improvement research clocks should make this detectable.

For (3), assuming near earth, the time rates are near 1 compared to Schwarzschild t, and that changes in r are very close to physical distance, then the derivative quantity is given by:

(R/2r²) (1 - R/r)^-.5

where R is the Schwarzschild radius and r is the SChwarzschild r coordinate. Noting that with Newtonian approximation, g=GM/r²; and R = 2GM/c², we can write this as:

(g/c²) (1 + gr/c²), using one term taylor expansion of the square root.

This shows the tidal correction to Rindler dilation is about 6*10^-10 times the Rindler dilation. This also means it is about 5 million times smaller than the tangential velocity dilation change with altitude. There is no likelihood of directly detecting this correction near the Earth's surface, in the foreseeable future.