A Question Concerning Time Dilation

[Janus]

 

 

 

 

See, that makes sense to me. If the remaining time dilation in the Sam-Mary scenario, once gravitational difference has been accounted for, is due only to acceleration, then it makes sense.

 

Again, you must be careful here. Acceleration, in of itself, does not cause time dilation. Example, if you put a clock on a centrifuge, and compare it rate to one at the center, it will run more slowly, Now let's say that you change the parameters of the centrifuge so that it has a different length of arm and spins at a different angular velocity in such a way that the clock experiences the same acceleration force as it did before. You will find that the clock will run at a different rate than it did before. Conversely, You could change things so that the clock travels at the same speed on the end of the arm but experiences a different acceleration, and it will run at the same rate as it did before.

[swansont]

Again, you must be careful here. Acceleration, in of itself, does not cause time dilation. Example, if you put a clock on a centrifuge, and compare it rate to one at the center, it will run more slowly, Now let's say that you change the parameters of the centrifuge so that it has a different length of arm and spins at a different angular velocity in such a way that the clock experiences the same acceleration force as it did before. You will find that the clock will run at a different rate than it did before. Conversely, You could change things so that the clock travels at the same speed on the end of the arm but experiences a different acceleration, and it will run at the same rate as it did before.

 

And this experiment has been carried out. You can solve for it by using acceleration and a pseudo-gravitational potential or just the straight kinematic term, but in the end the dilation is still v^2/2c^2. (i.e. they are equivalent, as they have to be)

 

Last section of

http://blogs.scienceforums.net/swansont/archives/1426

references included

 

See, that makes sense to me. If the remaining time dilation in the Sam-Mary scenario, once gravitational difference has been accounted for, is due only to acceleration, then it makes sense.

A Janus has pointed out, it's due to the speeds, which are not the same.

[Elen Sila]

Alright. Just answer this question then, I guess.

 

If, when time dilation occurs, both parties view the other as moving slower, and difference in time elapsed only results when, through acceleration of one or both parties, the two reference frames are reunited, why should there be a difference in time elapsed if neither party accelerates to reunite with the other, and if they both maintain the same positions relative to each other, communicating remotely?

[swansont]

Alright. Just answer this question then, I guess.

 

If, when time dilation occurs, both parties view the other as moving slower, and difference in time elapsed only results when, through acceleration of one or both parties, the two reference frames are reunited, why should there be a difference in time elapsed if neither party accelerates to reunite with the other, and if they both maintain the same positions relative to each other, communicating remotely?

They can't maintain the same position relative to each other and be in different inertial frames. One of them has to be accelerating, or they are at rest with respect to each other.

[Elen Sila]

And that makes sense to me. What I don't understand is these statements.

 

"Again, you must be careful here. Acceleration, in of itself, does not cause time dilation."

"A Janus has pointed out, it's due to the speeds, which are not the same."

 

I will explain exactly what's causing the confusion on my part.

 

Velocity is not absolute. You cannot simply say "Mary is travelling faster than Sam", because she's not. Nothing travels "faster" than anything else, in absolute terms. You can only determine relative velocity, in which case, in this scenario, Mary and Sam have zero relative velocity, compared to each other. Mary does have greater acceleration, and Sam is standing deeper inside a gravitational well; so those are two absolute reasons why you could say that one or the other is experiencing time dilation relative to the other. But it seems to me that, in this scenario, the only time dilation that can be happening is acceleratory or gravitational. Yet the statements I quoted, from you and Janus, seem to indicate that time dilation is occurring for a reason pertaining to velocity, which, as far as I can see, cannot be the case.

Edited August 31, 2011 by Elen Sila

[swansont]

And that makes sense to me. What I don't understand is these statements.

 

"Again, you must be careful here. Acceleration, in of itself, does not cause time dilation."

"A Janus has pointed out, it's due to the speeds, which are not the same."

 

I will explain exactly what's causing the confusion on my part.

 

Velocity is not absolute. You cannot simply say "Mary is travelling faster than Sam", because she's not. Nothing travels "faster" than anything else, in absolute terms. You can only determine relative velocity, in which case, in this scenario, Mary and Sam have zero relative velocity, compared to each other. Mary does have greater acceleration, and Sam is standing deeper inside a gravitational well; so those are two absolute reasons why you could say that one or the other is experiencing time dilation relative to the other. But it seems to me that, in this scenario, the only time dilation that can be happening is acceleratory or gravitational. Yet the statements I quoted, from you and Janus, seem to indicate that time dilation is occurring for a reason pertaining to velocity, which, as far as I can see, cannot be the case.

 

Mary and Sam are not at rest with respect to each other when measured by an inertial observer; in this case a convenient observer would be one somewhere on the axis of rotation so that Mary and Sam are both traveling at a constant speed. Regardless, an inertial observer will not see them moving at the same speed.

[csmyth3025]

...Velocity is not absolute. You cannot simply say "Mary is traveling faster than Sam", because she's not. Nothing travels "faster" than anything else, in absolute terms. You can only determine relative velocity, in which case, in this scenario, Mary and Sam have zero relative velocity, compared to each other. Mary does have greater acceleration, and Sam is standing deeper inside a gravitational well; so those are two absolute reasons why you could say that one or the other is experiencing time dilation relative to the other. But it seems to me that, in this scenario, the only time dilation that can be happening is acceleratory or gravitational. Yet the statements I quoted, from you and Janus, seem to indicate that time dilation is occurring for a reason pertaining to velocity, which, as far as I can see, cannot be the case.

I'm no expert on special or general relativity, but I'm beginning to see the nature of your dilemma.

 

First, we can dispense with further discussion of why it is that the clock on a geostationary satellite runs faster than a clock on the ground. I take it that the previous replies have satisfactorily explained the general relativistic notion that a clock in a weaker gravitational field (the clock in the geostationary satellite) is not slowed down as much as a clock in a stronger gravitational field (the clock on Earth). Thus, when comparing the two clocks, both Mary and Sam will agree that Mary's clock is running faster than Sam's clock. The situation on the satellite and on Earth is asymmetric in regard to the gravitational field strength at their respective distances from the Earth's center.

 

Now we get to the special relativistic notion that two observers - each in uniform inertial motion, but both moving relative to each other - can both detect that the other's clock is running slower than their own by communicating with signals back and forth between them. How is it possible for them both to say that the other's clock is running slow in this experiment?

 

Once we find an answer to this question we can move on to the question of how it is that an observer on the Earth would agree that a clock in geostationary orbit (always directly overhead at the same distance) is running slower than the observer's Earth-bound clock, provided that we cancel out the aforementioned effect of general relativity. The fact that both the observer on the geostationary satellite and the Earth-bound observer agree on this difference in their respective clocks implies that there is not a symmetrical relationship between the two observers, as is required by the purely special relativistic example given above.

 

Does this pretty much describe your questions?

 

I would love to give you an answer, but I'm as baffled as you are by these questions.

 

Chris

[Janus]

And that makes sense to me. What I don't understand is these statements.

 

"Again, you must be careful here. Acceleration, in of itself, does not cause time dilation."

"A Janus has pointed out, it's due to the speeds, which are not the same."

 

I will explain exactly what's causing the confusion on my part.

 

Velocity is not absolute. You cannot simply say "Mary is travelling faster than Sam", because she's not. Nothing travels "faster" than anything else, in absolute terms. You can only determine relative velocity, in which case, in this scenario, Mary and Sam have zero relative velocity, compared to each other.

 

 

As Swansont pointed out, Mary and Sam do not have a zero relative velocity according to everyone. If they were in an inertial frame, (traveling in a straight line at constant speed), then all observers that are themselves in inertial frames would also agree that they had zero velocity with respect to each other. As it is, they wouldn't.

 

Here's another example of such a case. Mary and Sam are in a rocket that is accelerating (according to them) at a constant rate. Mary is in the nose and Sam is near the tail. According to them, they maintain a constant distance from each other and are at rest with respect to each other. However, according to someone in an inertial frame, the rocket is constantly changing velocity. As a result, the length of the rocket is always changing due to length contraction. This means that Mary and Sam do not maintain a constant distance from each other and are not at rest with respect to each other according to this frame.

 

Another thing to note is that, as measured by anyone in the rocket frame, Mary's clock runs faster, despite the fact that Mary and Sam are undergoing the same acceleration and the only difference between them is their spatial separation.

[Elen Sila]

Alright. I suppose that makes sense. I still have a lot to learn, of course. But I guess my question appears fairly cleared up now.

 

Also, while I was thinking about this scenario at school the other day, I came up with the following diagram, which actually kind of demonstrates that Mary and Sam must have a non-zero relative velocity. Let me know if this diagram is accurate, and really does have anything to do with the scenario and its solution.

 

Again, in case you can't read my handwriting, it says, "Because Sam and Mary are sending off their signals at non-uniform angles, Sam having to return his signals at a wider angle than Mary, it can be determined that Sam and Mary are NOT in the same reference frame, and that it is impossible for two parties to even be in the same reference frame unless they share the same linear acceleration."

[J.C.MacSwell]

Alright. I suppose that makes sense. I still have a lot to learn, of course. But I guess my question appears fairly cleared up now.

 

Also, while I was thinking about this scenario at school the other day, I came up with the following diagram, which actually kind of demonstrates that Mary and Sam must have a non-zero relative velocity. Let me know if this diagram is accurate, and really does have anything to do with the scenario and its solution.

 

Again, in case you can't read my handwriting, it says, "Because Sam and Mary are sending off their signals at non-uniform angles, Sam having to return his signals at a wider angle than Mary, it can be determined that Sam and Mary are NOT in the same reference frame, and that it is impossible for two parties to even be in the same reference frame unless they share the same linear acceleration."

 

What do you mean by this?

 

NOT share the same inertial rest frame? (then yes)

[Elen Sila]

What do you mean by this?

 

I'm not even sure.

I do know this, though: if the earth suddenly disappeared, leaving Sam and Mary floating in space, Mary would go flying off into the distance way faster than Sam would. That, at least, shows that they do not have the same inertial rest frame.

[swansont]

I'm not even sure.

I do know this, though: if the earth suddenly disappeared, leaving Sam and Mary floating in space, Mary would go flying off into the distance way faster than Sam would. That, at least, shows that they do not have the same inertial rest frame.

 

They are not in an inertial frame if there is an acceleration and gravity is not present.

[J.C.MacSwell]

They are not in an inertial frame if there is an acceleration and gravity is not present.

 

 

I don't understand what you mean when you say this. Can you not describe accelerated movement with respect to an inertial frame?

Edited September 3, 2011 by J.C.MacSwell

[csmyth3025]

They are not in an inertial frame if there is an acceleration and gravity is not present.

I think what Elen Sila is saying is that if Earth were suddenly taken out of the picture, Sam would be traveling in some direction with a linear velocity of ~465 m/s (assuming he is at the equator). Mary, who is in geostationary orbit directly above Sam, would be travelling in the same direction with a linear velocity of ~3,070 m/s.

 

In this scenario the difference in (linear) velocity between Sam and Mary would be ~2,605 m/s. If one were to ignore the gravity of the Sun and the galaxy as a whole, they could both be considered to be in their own inertial frame of reference. Sam would see that Mary is moving away from him at 2,605 m/s, and Mary would see that Sam is moving away from her at 2,605 m/s (only in the opposite direction).

 

I think we can apply the formula for special relativistic time dilation to their new situation:

 

 

For small values of v/c, by using binomial expansion this approximates to:

 

 

(ref.

http://en.wikipedia....f_time_dilation )

In this case we would have:

dilation = ~1-[(2605 m/s)²/(2)(2.998x10⁸)²] = ~1-[(6.8x10⁶)/(1.8x10¹⁷)] = ~1-(3.8x 10^-11)

This difference below 1 of about 3.8×10⁻¹¹ represents the fraction by which Mary's and Sam's clocks differ. If we then multiply by the number of nanoseconds in a day we get:

-(3.8x10^-11)(60)(60)(24)(10⁹) = ~-3283 ns per day, or about -3.3 microseconds per day.

It's entirely possible that I may have this calculation all wrong, so corrections are welcome.

Chris

Edited to insert minus sign in final result

Edited September 3, 2011 by csmyth3025

[Elen Sila]

Hmm. I just came up with another scenario.

 

Let's say Sam and Mary are both in space stations, orbiting the earth at the same altitude; Sam is several degrees ahead of Mary.

In this scenario, Sam and Mary are not travelling in the same direction linearly, in the galilean/newtonian sense; however, they are travelling in the same direction geodesicly, in the einsteinian sense (at least, based on my understanding of the geodesy of spacetime according to Einstein, which I'm willing to bet is probably even worse than my understanding of time dilation).

 

It seems to me that, in the curvature of spacetime by the earth's mass, Mary is following directly in Sam's path, and therefore her experience of time relative to him ought to be the same as if they were both travelling in a straight line, one after the other. Are they in the same reference frame in this scenario? Or am I totally misinterpreting the concept of spacetime geodesy?

[swansont]

I think what Elen Sila is saying is that if Earth were suddenly taken out of the picture, Sam would be traveling in some direction with a linear velocity of ~465 m/s (assuming he is at the equator). Mary, who is in geostationary orbit directly above Sam, would be travelling in the same direction with a linear velocity of ~3,070 m/s.

 

True. But if for some reason we were just ignoring gravity yet the were moving in a circle, there would still be acceleration.

 

Hmm. I just came up with another scenario.

 

Let's say Sam and Mary are both in space stations, orbiting the earth at the same altitude; Sam is several degrees ahead of Mary.

In this scenario, Sam and Mary are not travelling in the same direction linearly, in the galilean/newtonian sense; however, they are travelling in the same direction geodesicly, in the einsteinian sense (at least, based on my understanding of the geodesy of spacetime according to Einstein, which I'm willing to bet is probably even worse than my understanding of time dilation).

 

It seems to me that, in the curvature of spacetime by the earth's mass, Mary is following directly in Sam's path, and therefore her experience of time relative to him ought to be the same as if they were both travelling in a straight line, one after the other. Are they in the same reference frame in this scenario? Or am I totally misinterpreting the concept of spacetime geodesy?

 

Here the clocks will be running at the same rate, but signals sent would be affected. Since they are still in an accelerating frame (rotation), the speed of light is not constant. Light sent along the orbital path will travel at c+v or c-v, depending on the direction it goes. This is known as the Sagnac effect.

 

I don't understand what you mean when you say this. Can you not describe accelerated movement with respect to an inertial frame?

 

Yes, you can describe it, but if you are the one being accelerated you are not in that frame.

[J.C.MacSwell]

True. But if for some reason we were just ignoring gravity yet the were moving in a circle, there would still be acceleration.

 

 

 

Here the clocks will be running at the same rate, but signals sent would be affected. Since they are still in an accelerating frame (rotation), the speed of light is not constant. Light sent along the orbital path will travel at c+v or c-v, depending on the direction it goes. This is known as the Sagnac effect.

 

 

 

Yes, you can describe it, but if you are the one being accelerated you are not in that frame.

 

To me this seems more than a little ambiguous and a source of confusion (it can't be just me)

 

Is there a link or source for that definition?

 

Can you explain why something that is clearly there is said or implied not to be there?

 

My use of of the phrase "in the frame", would include everything of interest, including any Mack Truck in my vicinity accelerating or not. Can you tell me why this is wrong? I know that "in the frame of the truck" would imply something different but that is true whether the truck is accelerating or not.

 

Inertial frames are generally chosen as they simplify the physics, and not just linear transformations. Others are chosen because they simplify the math, at the price of adding pseudo forces to the physics.

 

If your definition is correct for physics there must be a reason for it. What is the use of it? Why is something that is accelerating said to be not in the frame?

[swansont]

To me this seems more than a little ambiguous and a source of confusion (it can't be just me)

 

Is there a link or source for that definition?

 

Can you explain why something that is clearly there is said or implied not to be there?

 

My use of of the phrase "in the frame", would include everything of interest, including any Mack Truck in my vicinity accelerating or not. Can you tell me why this is wrong? I know that "in the frame of the truck" would imply something different but that is true whether the truck is accelerating or not.

 

Inertial frames are generally chosen as they simplify the physics, and not just linear transformations. Others are chosen because they simplify the math, at the price of adding pseudo forces to the physics.

 

If your definition is correct for physics there must be a reason for it. What is the use of it? Why is something that is accelerating said to be not in the frame?

 

Actually it's not just accelerating. Anything moving is not in that frame, but in the context of relativity acceleration takes on added meaning. A frame of reference is a choice of coordinate system. If you are not at rest, you are moving with respect to that coordinate system. All measurements have to be made in reference to a coordinate system because quantifications are meaningless without it. But if the object has a constant velocity, we can assume it to be at rest, and all measurements made from another coordinate system will be reciprocally correct — there is no way to tell who is really moving and who is at rest, i.e. there is no preferred/absolute frame. However, that is not true of acceleration. You cannot simply put yourself into the frame of the accelerated object and have the measurements work out.

 

So you can measure an accelerating object with respect to any reference frame you choose, but it will not be in an inertial reference frame — we cannot choose the object to be at rest without adding extra corrections to make everything work out. You may, of course, still choose that frame and add those terms, but the presence of the additional terms means it is not an inertial frame.

[J.C.MacSwell]

Actually it's not just accelerating. Anything moving is not in that frame, but in the context of relativity acceleration takes on added meaning. A frame of reference is a choice of coordinate system. If you are not at rest, you are moving with respect to that coordinate system. All measurements have to be made in reference to a coordinate system because quantifications are meaningless without it. But if the object has a constant velocity, we can assume it to be at rest, and all measurements made from another coordinate system will be reciprocally correct — there is no way to tell who is really moving and who is at rest, i.e. there is no preferred/absolute frame. However, that is not true of acceleration. You cannot simply put yourself into the frame of the accelerated object and have the measurements work out.

 

So you can measure an accelerating object with respect to any reference frame you choose, but it will not be in an inertial reference frame — we cannot choose the object to be at rest without adding extra corrections to make everything work out. You may, of course, still choose that frame and add those terms, but the presence of the additional terms means it is not an inertial frame.

 

So "not in that frame" simply means not at rest and not accelerating in that frame?

 

Even while accelerating, there exists an inertial frame with respect to which I am at rest, though only instantaneously. So from your definition I am not in that frame?

 

"In that frame" means fixed in that frame?

 

Can you provide a link or source for that definition? Is it just the context you and your peers always use or is it strictly defined in physics?

Edited September 3, 2011 by J.C.MacSwell

[Elen Sila]

However, that is not true of acceleration. You cannot simply put yourself into the frame of the accelerated object and have the measurements work out.

 

Can you put yourself in the frame of an object that is accelerating gravitationally, through orbit? Wikipedia says that freefall is inherently inertial.

 

"Objects in free-fall really do not accelerate, but rather the closer they get to an object such as the earth, the more the time scale becomes stretched due to spacetime distortion around the planetary object (this is gravity). An object in free-fall is in actuality inertial, but as it approaches the planetary object the time scale stretches at an accelerated rate, giving the appearance that it is accelerating towards the planetary object when, in fact, the falling body really isn't accelerating at all. This is why an accelerometer in free-fall doesn't register any acceleration; there isn't any. By contrast, in newtonian mechanics, gravity is assumed to be a force. This force draws objects having mass towards the center of any massive body."

[swansont]

So "not in that frame" simply means not at rest and not accelerating in that frame?

 

Even while accelerating, there exists an inertial frame with respect to which I am at rest, though only instantaneously. So from your definition I am not in that frame?

 

"In that frame" means fixed in that frame?

 

Can you provide a link or source for that definition? Is it just the context you and your peers always use or is it strictly defined in physics?

 

I don't know if it is strictly defined, but it's a summary of how we apply Newton's laws of motion and relativity.

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

 

Can you put yourself in the frame of an object that is accelerating gravitationally, through orbit? Wikipedia says that freefall is inherently inertial.

 

Right. That was one of the epiphanies of general relativity — that being stationary in a gravitational field is like accelerating, and that freefall is inertial.

[J.C.MacSwell]

I don't know if it is strictly defined, but it's a summary of how we apply Newton's laws of motion and relativity.

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

 

 

I've seen that.

 

I don't see where it would support your definition of what is in or not in.

[swansont]

I've seen that.

 

I don't see where it would support your definition of what is in or not in.

 

The motion of a body can only be described relative to something else - other bodies, observers, or a set of space-time coordinates. These are called frames of reference. If the coordinates are chosen badly, the laws of motion may be more complex than necessary. For example, suppose a free body (one having no external forces on it) is at rest at some instant. In many coordinate systems, it would begin to move at the next instant, even though there are no forces on it. However, a frame of reference can always be chosen in which it remains stationary.

 

IOW, if you are stationary with respect to the coordinates, you are in that frame.

[J.C.MacSwell]

"However, a frame of reference can always be chosen in which it remains stationary"

 

Choice implies that there are others in which it does not remain stationary

 

Too me that example just states that you can choose an inertial rest frame for an object that is at rest (no external forces)

 

IOW, if you are stationary with respect to the coordinates, you are in that frame.

Not sure how that summarizes what you quoted, but in any case this contradicts your definition for an accelerating object not being in any inertial frame...

 

...since you can always choose an inertial frame that it would be stationary with respect to the coordinates of (zero velocity, regardless of the acceleration)

 

Sorry if this seems picky, that is not my intention, but to me "not in a frame" needs context, and I don't believe everyone automatically realizes you mean "not fixed in the coordinates of" or even "not at rest in". At times in can be inferred from the discussion (easier with your peers) but at times it seems misleading or implies something has somehow moved outside of a frame.

Edited September 4, 2011 by J.C.MacSwell

[swansont]

"However, a frame of reference can always be chosen in which it remains stationary"

 

Choice implies that there are others in which it does not remain stationary

 

Too me that example just states that you can choose an inertial rest frame for an object that is at rest (no external forces)

 

Relativity tells us that different inertial frames measure time and distance differently. So it doesn't make sense (to me, anyway) to speak of two observers being in the same frame and yet disagreeing on x and t.

 

Not sure how that summarizes what you quoted, but in any case this contradicts your definition for an accelerating object not being in any inertial frame...

 

...since you can always choose an inertial frame that it would be stationary with respect to the coordinates of (zero velocity, regardless of the acceleration)

 

Sorry if this seems picky, that is not my intention, but to me "not in a frame" needs context, and I don't believe everyone automatically realizes you mean "not fixed in the coordinates of" or even "not at rest in". At times in can be inferred from the discussion (easier with your peers) but at times it seems misleading or implies something has somehow moved outside of a frame.

 

An accelerating frame is by definition not inertial.