Different Clock Rates Throughout Accelerating Spaceship

[1977ub]

I have been reading a lot of relativity-related material and clearing up a few gaps in my general knowledge. I read something that struck me as off. Perhaps I am missing something.

Usenet Physics FAQ -> The Relativistic Rocket
http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html

In the set-up: "If a rocket accelerates at 1g (9.81 m/s2) the crew will experience the equivalent of a gravitational field with the same strength as that on Earth."

Presumably this means the entire crew will experience the same acceleration - at the leading and trailing ends of the ship, for instance.

Later, we are told that: "inside the rocket, a clock attached to the rocket's ceiling (i.e. furthest from the motor) ages faster than a clock attached to its floor."

Then in the next paragraph: "it tells us something fundamental about gravity, via Einstein's Equivalence Principle. Einstein postulated that any experiment done in a real gravitational field, provided that experiment has a fairly small spatial extent and doesn't take very long, will give a result indistinguishable from the same experiment done in an accelerating rocket. So the idea that the rocket's ceiling ages faster than its floor (and that includes the ageing of any bugs sitting on these) transfers to gravity: the ceiling of the room in which you now sit is ageing faster than its floor; and your head is ageing faster than your feet. ... This difference in ageings on Earth has been verified experimentally. In fact, it was absolutely necessary to take into account when the GPS satellite system was assembled."

Now, I had been under the impression that differences in clock speeds at different altitudes were due to the gravitational field being weaker at higher altitudes.

I understand that once we start expanding the width or height of our accelerating laboratory, we can make measurements to tell whether we are in acceleration vs gravitational field.

Does the clock at the "top" of this accelerating frame indeed tick faster than one at the floor?

Is the reason for this indeed analogous to clocks ticking *faster* at higher altitudes above the Earth's surface?

Is the reason for gravitational time dilation closer to a massive body related or unrelated to the *greater* strength of the gravitational field there?

Does a local measurement of acceleration/g-forces at the top of the accelerating frame differ from a measurement at the bottom?

I would not have thought so. Even if it is true, I would have taken the gravitational difference in clock speeds near Earth to be something that could NOT be noticed or simulated for an observer in an accelerating box, such as by measuring clock speed at the leading vs trailing end.

Thank you.

 

[mfb]

In the set-up: "If a rocket accelerates at 1g (9.81 m/s2) the crew will experience the equivalent of a gravitational field with the same strength as that on Earth."

Presumably this means the entire crew will experience the same acceleration - at the leading and trailing ends of the ship, for instance.

This is a very good approximation, but it is not exact (unless the ship can change its length).

Now, I had been under the impression that differences in clock speeds at different altitudes were due to the gravitational field being weaker at higher altitudes.

No, it is the potential difference, not the local gravitational attraction.

Does the clock at the "top" of this accelerating frame indeed tick faster than one at the floor?

As seen by the spaceship, or as seen from earth? And with a flexible spaceship or not?

I think the other questions are answered or depend on my own questions about the setup.

 

[1977ub]

First - thank you.

Let's take a modern rocket with conventional materials. I imagine it will "settle" slightly but not undergo any further deformation - similar to if sitting on the surface of the earth.

Engine set so that a person sitting at the bottom experiences 1g.

1) Do people at the top experience 1g force, same as those at the bottom do?

2) Do all crew agree that clocks at the top are moving more quickly than those at the bottom?

 

[PAllen]

First - thank you.

Let's take a modern rocket with conventional materials. I imagine it will "settle" slightly but not undergo any further deformation - similar to if sitting on the surface of the earth.

Engine set so that a person sitting at the bottom experiences 1g.

1) Do people at the top experience 1g force, same as those at the bottom do?

2) Do all crew agree that clocks at the top are moving more quickly than those at the bottom?

1) The top will feel very slightly less acceleration than the bottom, under reasonable rigidity assumption. This, however, is not the main reason:

2) All crew will will agree top clocks go faster (by a very small amount). This would be true even if you arranged (by slowly stretching the rocket per rocket crew) for the top and the bottom to experience identical g force.

 

[pervect]

First - thank you.

Let's take a modern rocket with conventional materials. I imagine it will "settle" slightly but not undergo any further deformation - similar to if sitting on the surface of the earth.

Engine set so that a person sitting at the bottom experiences 1g.

1) Do people at the top experience 1g force, same as those at the bottom do?

No. But as mentioned before, for short rockets it's a good approximation.

2) Do all crew agree that clocks at the top are moving more quickly than those at the bottom?

Some care is needed to specify how you measure the relative motion between the top and the bottom.

For instance, the round-trip signal time of a light beam between the top and the bottom will be constant.

The rocket doesn't have a "true" frame, but it's got something that comes close to it. In this almost-frame, there is no relative motion between the top and the bottom of the rocket. Which is necessary if the rocket is to have a constant length in the "almost-frame".

 

[1977ub]

If you read an explanation of GR predictions and results, you will see one of the central ones is that the clock further from massive body will run more quickly. This prediction is not made by SR. The "rocket" example goes out of its way to focus on results which can be seen with SR+acceleration without GR. What surprised me in the page I referenced is this section: "the idea that the rocket's ceiling ages faster than its floor (and that includes the ageing of any bugs sitting on these) transfers to gravity: the ceiling of the room in which you now sit is ageing faster than its floor; and your head is ageing faster than your feet.:"

"transfers to gravity" ?

Presumably the effects have different magnitude and occur for different reasons.

The calculation of the slower clock at the top of my room uses the distance from that ceiling to the center of the Earth vs the distance from the floor to the center of the Earth in order to compare the gravitational potentials, correct? And for the accelerating rocket, there is no 3rd point "focus" analogous to the center of the Earth, so the effect and its calculation are going to be very different. Aren't these phenomena so different that we wouldn't think of the one as "transferring" to the other?

 

[PAllen]

Some care is needed to specify how you measure the relative motion between the top and the bottom.

I assumed question two's ambiguous wording 'clocks moving more quickly' referred to the rate of the clocks, not their state of motion. Normally, this is what is of interest about a clock.

 

[PAllen]

Presumably the effects have different magnitude and occur for different reasons.

The calculation of the slower clock at the top of my room uses the distance from that ceiling to the center of the Earth vs the distance from the floor to the center of the Earth in order to compare the gravitational potentials, correct? And for the accelerating rocket, there is no 3rd point "focus" analogous to the center of the Earth, so the effect and its calculation are going to be very different. Aren't these phenomena so different that we wouldn't think of the one as "transferring" to the other?

Actually, the magnitude is the same for comparable 'every day' magnitudes. Further, if you take compute the time dilation difference between two nearby radial values, for radii much larger than the Schwarzschild radius, and only keep leading Taylor expansion terms, you get that the difference is just proportional to g * (r2-r1).

 

[1977ub]

Ok. Well that's something for me to chew on. The difference in the tick-rates of two clocks, one at sea level vs one raised by a mile - this is not too far off from the difference in the tick-rates of two clocks, one at the base and one at the nose of a mile-high rocket accelerating in space with roughly 1g.

 

[Fredrik]

Presumably the effects have different magnitude and occur for different reasons.

The reasons are the same.

The accelerating rocket in SR: The world line of the top of the rocket is more like a geodesic than the world line of the bottom of the rocket.

A house standing firmly on the ground in GR: The world line of the ceiling is more like a geodesic than the world line of the floor.

 

[1977ub]

I guess I'm still stuck thinking of:
1) gravitational force (Newtonian) decreasing as the square of the distance from the body's center, and

2) a clock's rate slows as the gravitational force/acceleration that it experiences increases.

Let's take an example where instead of the Earth we have a much more massive and smaller body, such that our rocket is sitting on the surface, 1 mile from the center of the body, and at the bottom of the sitting rocket, gravity is experienced as 1g. Presumably at the top of the rocket, we are twice as far from the center of the massive body as the rocket bottom is, and the top clock is much faster relative to the bottom as we would have on earth. The acceleration due to gravity - and thus the extent of decrease in time dilation - is much smaller up there than it would be a mile off the Earth's surface.

 

[PAllen]

2) a clock's rate slows as the gravitational force/acceleration that it experiences increases.

This is your key misunderstanding. In a static field in GR that can be approximated by a Newtonian potential, clock rate differences for static clocks is a function of potential difference not acceleration. Thus, GR says, in the limit if a uniform field, clock difference rate is proportional to g*h, where h is the separation between them; thus proportional to the potential difference. I explained earlier how you can show, for large r, and small difference in r, the clock rate difference is also a function of potential difference: g(r1-r2).

 

[1977ub]

That seems to be splitting hairs a bit for what I'm trying to find out...

Ok perhaps you could take a look at this?

3 cases - do these 1g-experiencing clocks - tick at the same rate?

a) clock on surface of Earth experiencing 1g.
(neglect rotation/orbit of the earth)

b) clock in space in our solar system,
far enough from the Sun to experience 1g,
held in place by rocket engine,
(neglecting presence of other planets)

c) clock in space inside rotating ring,
such that clock experiences 1g from 'centrifugal' force.

 

[mfb]

In general, all those will tick at different rates.
As mentioned before, gravitational acceleration itself is not relevant, gravitational potential is.

 

[PAllen]

That seems to be splitting hairs a bit for what I'm trying to find out...

Ok perhaps you could take a look at this?

3 cases - do these 1g-experiencing clocks - tick at the same rate?

a) clock on surface of Earth experiencing 1g.
(neglect rotation/orbit of the earth)

b) clock in space in our solar system,
far enough from the Sun to experience 1g,
held in place by rocket engine,
(neglecting presence of other planets)

c) clock in space inside rotating ring,
such that clock experiences 1g from 'centrifugal' force.

A couple of issues here:

It is a misconception to imagine an objective tick rate for clock compared to some 'standard'. Instead, you should think only in terms of comparing tick rates on two clocks that exchange signals. As you know from SR, you can have the situation where each finds the other one slower.

If the three clocks you describe compare rates, as mfb noted, they would all be different from each other, and the relation would be rather complex.

What is simple is comparing two clocks at rest relative to a static field (that is, each finds their proper acceleration is constant over time), and they are in the same static field (e.g. rocket; near earth; etc.). Then the difference in rate between those clocks will proportional to the potential difference between them.

 

[DrGreg]

In general, all those will tick at different rates.
As mentioned before, gravitational acceleration itself is not relevant, gravitational potential is.

1977ub,

Another way of putting this is that if you want to calculate the difference in rates between two clocks, it's not enough to know the gravitational acceleration at the locations of the two clocks, you need to know it at every point in between as well.

(Under the assumption of a static spacetime.)

 

[1977ub]

PAllen,

My attempt was to describe 3 situations where the clocks were not moving relative to one another. (#c obviously wobbles). Observers near the clocks could compare notes to see if they are ticking at the same rate.

DrGreg,

Ahh. interesting. Very often GR examples are compared to a default situation of being out away in empty space. Thus, here http://en.wikipedia.org/wiki/Gravitational_time_dilation we read that "a clock on the surface of the Earth (assuming it does not rotate) will accumulate around 0.0219 seconds less than a distant observer over a period of one year. In comparison, a clock on the surface of the sun will accumulate around 66.4 seconds less in a year."

My question related to the 3 cases would treat each in isolation from other factors and compare to "a distant observer."

I understand that these cases are different... for some reason my mind is still balking... i guess getting further into GR math at some point might cure me... ok - let's take case a vs b, are there any back-of-envelope calculations that could figure out which clock ticks slower wrt "a distant observer" ?

Thanks.

 

[Fredrik]

Observers near the clocks could compare notes to see if they are ticking at the same rate.

Each observer could compare one of those clocks to his own clock, but what would be the point of that? The results will also depend on how those observers are moving.

 

[1977ub]

The ambiguities of SR comparisons of tick-rates do not come up since all 3 observers find they are not moving relative to one another, or to a 4th "distant observer" for that matter.

 

[Fredrik]

The ambiguities of SR comparisons of tick-rates do not come up since all 3 observers find they are not moving relative to one another, or to a 4th "distant observer" for that matter.

I still don't see a meaningful way to interpret the results. Consider a GR spacetime that contains two spherical distributions of mass that are far enough from each other that the spacetime is approximately a Schwarzschild spacetime* in the vicinity of each of them. (This is just to make sense of your (a) and (b)). Where is the distant observer located? Do you want to put him at a distance from both objects that's far greater than the distance between the objects? In that case, the signal from a clock near one of the spherical objects will first have to travel through a region of spacetime that's a lot like the Schwarzschild spacetime associated with that object, but as it moves closer to the distant observer, the properties of the spacetime around it keep changing. It sounds very difficult to do calculations.

And even if you could do a calculation (of e.g. the interval between arrivals of signals sent 1 s apart from the clock near one of the spherical objects), how would you interpret the result? It seems that you want to know something about a specific point in spacetime, but the result will depend on the properties of spacetime along the entire path the signal takes to get to the distant observer.

*) A Schwarzschild spacetime describes a universe that's completely empty except for one spherically symmetric non-rotating distribution of mass.

 

[Fredrik]

By the way, I think it's a bad idea to think of what's going on in these scenarios as clocks having different ticking rates in different situations. The way I see it, clocks always tick at the same rate. The numbers they display tell you nothing about the clock, and something about the path through spacetime that the clock has taken.

 

[1977ub]

Frederik,

It sounds like you have a big problem with phrases such as those that are used in the wiki page. If you were an editor there, would you remove them? Is there no value to these types of comparisons? Is yours a common belief?

 

[Nugatory]

Frederik,

It sounds like you have a big problem with phrases such as those that are used in the wiki page. If you were an editor there, would you remove them? Is there no value to these types of comparisons? Is yours a common belief?

Fredrik is accurately describing the math and the science.

The page is not outright wrong, but it is consistently sloppy about remembering that when we say that a clock is "running slower" there's an implied something else that we're comparing against.
(If you haven't looked at the talk section of that wiki page, you will find it interesting)

 

[pervect]

By the way, I think it's a bad idea to think of what's going on in these scenarios as clocks having different ticking rates in different situations. The way I see it, clocks always tick at the same rate. The numbers they display tell you nothing about the clock, and something about the path through spacetime that the clock has taken.

I agree with this view - unfortunately there are plenty of sources that talk about clocks ticking at different rates. It's not really a huge problem which view you use UNLESS the person is imagining some sort of "universal clock" relative to which the rate of ticking of all others can be measured. In which case it's a huge problem.

Unfortunately, I have a strong suspicion that many, if not most, lay readers implicitly assume the existence of such a "universal clock" in spite of perhpas having a few lectures about "the relativity of simultaneity" float by over their heads. I hope I'm just being pessimistic...

 

[1977ub]

Presumably, as one gets out away from the Sun (or the Earth, or a black hole, whatever) the further away into relatively empty space, there is a limit which is approached for the ratio between the high gravity clockspeed and nearly zero gravity clockspeed. the multiplier is 1 right next to the clock near the body, and asymptotically approaches some exact figure the further we get away. This is what I presume is being discussed on the wiki page, i.e 66.4 sec / year does not apply exactly at any distance from the ideal isolated sun but is approached asymptotically as we head away.

 

[Nugatory]

I agree with this view - unfortunately there are plenty of sources that talk about clocks ticking at different rates. It's not really a huge problem which view you use UNLESS the person is imagining some sort of "universal clock" relative to which the rate of ticking of all others can be measured. In which case it's a huge problem.

Unfortunately, I have a strong suspicion that many, if not most, lay readers implicitly assume the existence of such a "universal clock" in spite of perhpas having a few lectures about "the relativity of simultaneity" float by over their heads. I hope I'm just being pessimistic...

Nope, not pessimistic. That assumption is so intuitive, so consistent with the experience of an entire life lived in non-relativistic conditions, that many people don't even realize that they're making that assumption.

It doesn't help any that it's much easier to say "ticking at different rates" than something rigorous involving simultaneity conventions and the ratio of proper time at the clock to proper time at the observer. Thus, even those who really do understand tend to speak in ways that do not explicitly reject the assumption of universal simultaneity; and as you say, no great harm is done as long as the listener understands as well.

 

[stevendaryl]

1) The top will feel very slightly less acceleration than the bottom, under reasonable rigidity assumption. This, however, is not the main reason:

2) All crew will will agree top clocks go faster (by a very small amount). This would be true even if you arranged (by slowly stretching the rocket per rocket crew) for the top and the bottom to experience identical g force.

There was a thread about this topic a year or so ago. There are two effects going on in the accelerated rocket that conspire to make the rear clock show an earlier time than the front clock:

1. Because of the relativity of simultaneity, a comoving inertial frame will find the front clock ahead of the rear clock, even if they show the same time in the "launch frame".
2. Because of length contraction of the rocket itself, the front clock is traveling at a slightly slower speed than the rear clock. So the time dilation factor effects the rear clock more.

The first effect is visible in the comoving frame. It is present even if you make the front and the rear experience equal accelerations.

The second effect is visible in the "launch" frame. If you "stretch" the rocket so that the front and the rear experience the same acceleration, then this effect disappears.

If you have a Born-rigid rocket (meaning that its length is always the same, when measured in the comoving inertial frame), then it's actually interesting: Effect number 1 is most important right after launch, but if you wait a really long time, then effect number 2 comes to dominate.

 

[1977ub]

Quote of stevendaryl :
"Because of length contraction of the rocket itself, the front clock is traveling at a slightly slower speed than the rear clock. So the time dilation factor effects the rear clock more."

The "length contraction" only applies for observers in the rest frame, correct?

I am primarily interested in the experiences of 2 observers on board - head vs tail of ship - how much acceleration they experience, and if they find their clocks to tick differently as they compare notes. (There is no simultaneity problem since they find each other to be relatively nonmoving.)

 

[stevendaryl]

Quote of stevendaryl :
"Because of length contraction of the rocket itself, the front clock is traveling at a slightly slower speed than the rear clock. So the time dilation factor effects the rear clock more."

The "length contraction" only applies for observers in the rest frame, correct?

I am primarily interested in the experiences of 2 observers on board - head vs tail of ship - how much acceleration they experience, and if they find their clocks to tick differently as they compare notes. (There is no simultaneity problem since they find each other to be relatively nonmoving.)

If the rocket ship is "rigid", meaning that its length doesn't change with time as measured by crew members, then the acceleration felt by someone at the front of the rocket will be less than the acceleration felt by someone at the rear of the rocket. Also, the time on the front clock will advance more quickly than the time on the rear clock. This is an operationally defined difference. Take a third clock to the rear, and synchronize the third clock to show the same time as the rear clock. Then move the clock up to the front of the rocket. Let it sit there for a year, or whatever. Then carry it back to the rear clock. The movable clock will be ahead of the rear clock.

 

[1977ub]

Stevendaryl,

Yes. Still struggling with parts of this. What if the nose is fully detached - forward and aft sections each having their own engines - and the base and nose synchronize in the initial rest frame such that they begin accelerating simultaneously? Would there then be any reason that they will later find that the clock in the nose module is ticking more slowly than the one at the base?

 

[jartsa]

Nope, not pessimistic. That assumption is so intuitive, so consistent with the experience of an entire life lived in non-relativistic conditions, that many people don't even realize that they're making that assumption.

It doesn't help any that it's much easier to say "ticking at different rates" than something rigorous involving simultaneity conventions and the ratio of proper time at the clock to proper time at the observer. Thus, even those who really do understand tend to speak in ways that do not explicitly reject the assumption of universal simultaneity; and as you say, no great harm is done as long as the listener understands as well.

Why is it that "lower clock is faster than upper clock" is more incorrect than "lower clock is slower than upper clock"?

I think it's because lower clock really is slower than upper clock.

 

[stevendaryl]

Stevendaryl,

Yes. Still struggling with parts of this. What if the nose is fully detached, and the base and nose synchronize in the initial rest frame such that they begin accelerating simultaneously? Would there then be any reason to reason that they will later find that the clock in the nose module is ticking more slowly than the one at the base?

If the front and the rear follow identical accelerations, then the two clocks will show the same time, as viewed from the "launch" frame, but the front clock will be ahead, according to crew members.

 

[1977ub]

Perhaps you can explain the flaw here:

I'll use Epstein's 3-rockets-in-a-row model.

* At regular intervals *, the central rocket sends a ping to front and back rockets.

Front and back rockets fire their boosters briefly when they get the ping. Also, clock ticks once.

Middle rocket delays appropriately after releasing the ping before boosting in order to stay midway between the other rockets.

We'll call them A, B, and C.
B releases first ping,
A, B, and C all boost together, as seen by them and the reference frame.
Now all are moving wrt the rest frame.
From rest frame perspective, 2nd ping hits A first, then B boosts, then ping hits C, which then boosts.
So with each ping, since rocket is moving faster wrt rest frame, A is boosting sooner and sooner wrt C. It appears to be accelerating faster than C.
Also with each ping, rest frame measures increasing length contraction between A & C.
This is basically the scenario I presented above, whereby nose/tail of rocket begin accelerating simultaneously.
It would appear from this reasoning that the clocks remain synchronized and therefore running at same rate as seen on the rockets,
while the rest observer is the one who observes the front and back clocks to run at different rates.

 

[stevendaryl]

Perhaps you can explain the flaw here:

I'll use Epstein's 3-rockets-in-a-row model.

The central rocket sends a ping to front and back rockets.

Front and back rockets fire their boosters briefly when they get the ping. Also, clock ticks once.

Middle rocket delays appropriately after releasing the ping before boosting in order to stay midway between the other rockets.

We'll call them A, B, and C.
B releases first ping,
A, B, and C all boost together, as seen by them and the reference frame.
Now all are moving wrt the rest frame.
From rest frame perspective, 2nd ping hits A first, then B boosts, then ping hits C, which then boosts.
So with each ping, since rocket is moving faster wrt rest frame, A is boosting sooner and sooner wrt C. It appears to be accelerating faster than C.
Also with each ping, rest frame measures increasing length contraction between A & C.
This is basically the scenario I presented above, whereby nose/tail of rocket begin accelerating simultaneously.
It would appear from this reasoning that the clocks remain synchronized and therefore running at same rate as seen on the rockets,
while the rest observer is the one who observes the front and back clocks to run at different rates.

Are you saying that A and C don't have their own clocks, but instead rely on counting the signals from B to know what the time is? That is, you want it to be the case that aboard the spaceship, time at C = number of "pings" received by C, and similarly for A?

In that case, aboard the spaceship they are using a coordinate time that is different from the time shown on a normal clock. Well, the two are the same for B, but for A, "clock time" is slower than "coordinate time", and for C, "clock time" is faster than "coordinate time".

If I understand correctly, then yes, as viewed by those aboard the rocket, everyone is using the same "coordinate time". But as viewed by those in the "launch" frame, the coordinate time for the front is ahead of coordinate time for the rear.

 

[1977ub]

As far as B is concerned, a ping is released once per second let's say. You don't reason that A & C will perceive it that way, I take it. Ok let's scale this back to a single ping/boost. While in rest frame, all agree that distances between ships will take light 1 second to reach from B to A or C, let's say. After the first brief round of boosting, Will the ships no longer find their own clocks to be synchronized? Will they no longer find their ships to be the same distance from one another as they did in the rest frame?