Why isn't length contraction permanent even though time dilation is?

[swansont]

I have not done the calculations as yet but if allowed we will go through them step by step. Here I will propose a thought experiment.

 

It is similar to the Twin Paradox situation. The twins ages are synchronized because they start off on Earth together. To know anything about rates of velocity the space travelers would need to at some stage synchronize their clocks and calibrate their measurements of distance and speed so we would give them both a calibrated radar speed camera recorder before they set off on the long journey to the closest star.

To test out the equipment one twin (Bob) flies out to the Moon and back and on the return pass he buzzes Alice on the ISS at 0.8 the speed of light.

 

Do both of them measure the pass as being at the same speed? Lets see if we can use the speed cameras to find out.

 

According to relativity, yes they do.

[Strange]

does he have to take his speed measurement to be the same as ours?

 

Yes. Speed is measured relative to something else. Therefore either can consider themselves stationary and the other moving. Therefore they will both determine the same relative speed.

[Robittybob1]

 

Yes. Speed is measured relative to something else. Therefore either can consider themselves stationary and the other moving. Therefore they will both determine the same relative speed.

Regardless as to who is moving relative to the Earth? Remember all instruments were calibrated such that the Earth was at rest. Even the ISS is so close to at rest we will ignore the time differences between Houston and the ISS. (No need to say "Houston we have a problem".)

[Strange]

Regardless as to who is moving relative to the Earth?

 

You can consider the Earth stationary. Or you can consider the other as stationary and the Earth moving. It is symmetrical. It is the theory of relativity.

[Robittybob1]

 

You can consider the Earth stationary. Or you can consider the other as stationary and the Earth moving. It is symmetrical. It is the theory of relativity.

Is there an effect due to where the synchronization and calibrations took place? I still feel an alien craft would have different standards and so we can't compare two complete strangers. Relativity will still be tested for we want to measure the closing speeds of each of them measured by each of them.

Will the radar gun on both sites measure the same reading or a different one.

We also need to see how these guns work at some stage.

OK we could use LIDAR machines http://en.wikipedia.org/wiki/LIDAR_speed_gun#How_police_LiDAR_guns_work

 

So the LiDAR measures time-in-flight of each pulse and requires only 2 pulses over a period of time of as little as 3 ms (theoretically) to determine the velocity of a vehicle.

The pulses may have to be separated by a bit more or else the return signal will be too late.

 

We know even at the speed of light a return trip from the Moon will be a little over a second and I'd say we could look at the relative motion both going away and coming toward the ISS so we have a good 3 second window to measure the speeds.

The time of flight of each pulse will depend on distance so if we want a pulse to go out and come back before the next and final pulse is sent the timing has to be right for the craft needs only travel one distance but the light pulse has to go twice the distance (there and back). So if Bob has just come past the Moon heading toward the ISS at 0.8 c he is only going to take another 1.25 seconds to travel that distance, but the light pulse will take a second to get there and a second back, so the LiDAR will need to be fired when Bob is still making his approach to the Moon so the light pulse is already there at the moon just as he is passing there, so the light pulse will return just in time for Alice to send another pulse before he passes crazily close to the ISS.

 

I get the feeling for ease of calculation we had better use slower speeds so we can get the timing of the pulses right.

Edited December 23, 2014 by Robittybob1

[Robittybob1]

It seems if the light pulses have a delay for 2.6 seconds the pulse will be out there near the Moon and the reflection back at the ISS in time to put up another pulse even at the higher relativistic speeds. Plenty of time is available at the slower speeds but will the effects show up?

It might be a couple of days before the maths is completed - Christmas you know!

 

Do we all agree that the LiDAR on the craft returning from the Moon's direction will be time dilated? So even though the ISS positioned LiDAR will be set on 2.6 second pulse delay, the one on Bob's craft will need pre-adjustment. But we need to know ahead of time by how much for we will only get one run at this before Bob is off to Alpha Centauri.

 

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

If Bob's LiDAR was set up such that 3 ms (calibrated on Earth, but dilated at speed) after receiving a signal from Alice it would activate a pulse, this way it is known that the photons will travel back to the ISS at the speed of light so the time difference will indicate the magnitude of gamma .

Edited December 23, 2014 by Robittybob1

[Robittybob1]

There are endless variations but for Bob's view of the situation when Alice recorded Bob doing 0.6 c we have the following. The 3 millisecond pulse return was measured by Alice as having dilated to 3.75 ms which is consistent with a clock moving at 0.6 c.

 

speed in C => 0.6 c (as recorded by Alice)

beta=v/c => 0.6

gamma = 1/sqrt(1-beta^2) => 1.25

speed of light m/s => 299792458

Earth Moon distance m => 384,400,000 m

Bob's time (dilated by gamma) (travel time in sec get less as speed increases) => 1.709627176 s

3ms time dilated by gamma (in milliseconds) => 3.75 ms (as recorded by Alice)

Now to try and work out Bob's relative velocity as measured by the LiDAR on board.

 

Now Bob can see the ISS straight ahead, the Moon is to the left and the Earth will be to the right as they pass by at an estimated speed of 0.6 c. Bob has his doubts about what he will measure as his own relative velocity as for he has surmised that since his clock was calibrated against Alice's clock and Alice recorded his relative speed at 0.6 c the onboard clock will be time dilated as he had to change direction and reapply the thrusters to regain top speed (he is well aware of previous acceleration). This was previously discovered by Bob's grandfather, one of the travelling twins, (the so called Paradox Twins) and they ended up different ages because the symmetry was broken. (It was rumoured that the cosmic radiation got to him in the end so he tragically died sooner than the older twin.)

 

Now Bob has to figure out when to turn on the LiDAR to get two pulses recorded before the ISS passes to the right.

 

"Bob's time (dilated by gamma) (travel time in sec get less as speed increases) => 1.709627176 s" what that means is that since Bob's clock has slowed down it will take fewer of his seconds to go from the Moon to the Earth, so by his calculation of velocity = distance over time, his velocity is greater for he covers the same distance in less time. Because the trip will take less time he has more issues with fitting it all in.

Bob's LiDAR has been pre-adjusted to fire at an interval of exactly 2 seconds (That is like 2 ticks of the clock) so it is irrespective of how fast the clock is moving. Remember to Alice Bob's clock is ticking slower so those 2 seconds is measured by her as been dilated by gamma, so 2 seconds of Bob's time measures 2.5 of Alice's Earth seconds.

 

[if we can do without length contraction to worry about a position in space measured in meters is the same place wrt all parties. I'll leave it there for a day or so and think it through. Merry Christmas folks. ]

Edited December 24, 2014 by Robittybob1

[Janus]

It is very simple to show that you cannot do away with length contraction by having the frames measure their relative speeds differently.

 

All is takes is one LiDAR pulse.

 

First we consider things from the frame of the ISS. Bob fires off a pulse towards the ISS as he passes the Moon and when his clock reads 0. It takes ~1.282 sec for that pulse to reach the ISS. In which time, Bob has traveled ~ 231,000,000m and is now ~153,000,000 m from the ISS. The pulse bounces back towards Bob at c while Bob continues to travel towards the ISS at 0.6c and they meet in another ~0.32 sec. Total time between firing of pulse and return to Bob, ~1.602 sec. Bob's clock of course runs at a rate of 0.8 that of the ISS clock, so he ticks off ~1.28 sec between the firing and reception of the pulse. Bob continues on and passes the ISS when the ISS has recorded ~2.14 sec and Bob's clock reads ~1.71 sec.

 

Now consider things from Bob's frame. Without length contraction, for him to travel from the Moon to the ISS in 1.71 sec, the relative speed between them has to be .8 c (384400000/1.71 = 0.8), So when he fires his pulse at c, it meets the ISS rushing toward him at 0.8 c in ~0.71 sec, by which time, the distance to the ISS will have closed to by ~171,000,000m and is ~213,600,000m away. The pulse returns at c, taking another ~0.71 sec to return to Bob, for a total time of 1.42 sec.

 

But wait, by the ISS's observations, only 1.28 sec ticked off for Bob between sending and receiving the pulse. This is an physical contradiction. If we had set things up so that Bob's clock started running when he sends the pulse and stops running when he receives the reflection, Then assuming Bob stops when he reaches the ISS (or passes it, slows down and returns) so that he, the clock and the ISS are all in the same frame, You will have two observers in the same frame at the same place saying that the same clock reads two different things.

 

It gets worse if Bob is traveling even closer to c relative to the Earth, if he is moving at 0.866 c, then it takes ~1.48 sec for him to reach the ISS in ISS time and ~0.74 sec in Bob time, for Bob to cross the Moon-ISS distance in 0.74 by his clock, the relative speed has to be 1.732 c. So now, when Bob bounces his LiDAR pulse off of the ISS, according to Bob, he reaches the ISS at 1.732c before the reflected LiDAR pulse (traveling at c relative to him) returns to him.

 

The simple fact is that you cannot do away with length contraction and still keep the(experimentally verified)invariant speed of light.

 

The invariant nature of the speed of light automatically requires length contraction.

 

This can shown by the classic train experiment:

 

We start with a train traveling at some speed relative to the embankment. At the midpoint of the train is an observer. There is an observer on the embankment also.

 

There are also two points marked off on the tracks an equal distance away from the embankment observer (the red dots in the following animation). It is also set up so that the length of the train as measured by the embankment frame is equal to the distance between the red dots, and that when the each end of the train aligns with a red dot(left end with left dot and right end with right dot), a flash of light is emitted.

 

Thus we get the following events:

 

 

Train aligns with dots, flashes go off, light from flashes travel at c, reaching embankment observer simultaneously.

Also note that since the train is moving from left to right, the observer at the midpoint of the train meets the right flash before the left flash catches up with him, and is next to two different points of the embankment when this happens.

 

Now we consider how things occur in the train's frame. Keeping in mind that the following must hold:

1. The speed of light relative to the train must be c (c is invariant)

2. Events that are co-located (happen at the same place) must be agreed upon by both frames. Thus, the fact that the two flashes arrive at the embankment observer at the same time and that the flashes originate when the ends of the train and red dots align.

 

Now, from the train frame, it is the embankment observer that is moving from right to left, and he is rushing to meet the left flash while the right flash chases after him. So the only way that the two flashes can meet at the embankment observer is for the left flash to start after the right flash. Since the flashes are initiated by the alignment of the ends of the train and the red dots, this means that the right end of the train must hit its dot before the left end hits it, in order to assure that the right flash starts first. But this means that the distance between the ends of the train and red dots can't be equal, and in fact, the distance between dots must be shorter than the train so that the right end can hit its dot before the left end reaches it.

 

what you end up with is this

 

 

Note that every co-located event matches with the first animation. you will even note that the train observer is next to the same point of the embankment when each flash arrives in both.

 

However, the distance between red dots has to be length contracted as measured by the train in order for these events to be consistent between frames.

 

Now I need to point out one more thing. In the first animation, I show the train fitting perfectly between the red dots. But the train is moving, so this is actually a length contracted frame. In other words, this train, if at rest with respect to the embankment, would be longer than the distance between red dots. But this extra length is not enough to make things work out in the second animation. It takes the the combination of the its longer "rest length" and the length contraction of the embankment to do this. Reciprocity of length contraction between frames is needed to be consistent with light speed invariance.

 

It seems like you are trying to shoehorn Relativity's size 12 feet into your size 10 preconceptions.

[Robittybob1]

Janus - I appreciate that someone is reading it. What I would ask you to do is, please, is it possible you work with 0.6 c rather than 0.8c because then I will be able directly compare your answer with mine (that I'm working on)?

 

The only other thing I am trying to imagine is that at all times we talk of the situation from the perspective of the person doing the measuring, so I wouldn't say like you seem to have "Bob fires off a pulse towards the ISS as he passes the Moon and when his clock reads 0. It takes ~1.282 sec for that pulse to reach the ISS. In which time, Bob has traveled ~ 231,000,000m and is now ~153,000,000 m from the ISS. The pulse bounces back towards Bob at c while Bob continues to travel towards the ISS at 0.6c and they meet in another ~0.32 sec."

From Bob's perspective the ISS is coming toward him, not him going toward the ISS. Also are your clock times corrected for time dilation?

Can Bob use the signal from Alice as the time to turn on the LiDAR? Bob first senses Alice's LiDAR when he is at the Moon distance from the ISS, 0.3 ms after the LiDAR will sent the first pulse, it has been figured since the things ahead are coming toward him at less that the speed of light, there should be some point where two pulses can be fitted in, but it has to be done in less time than the travel time for Bob to the ISS (1.709627176 sec) 1.7 seconds doesn't seem enough but the light waves will hit the ISS at a position, well somewhere in between, but where and when.

If light is going at the speed of light out from Bob and Alice's ISS was able to travel at the speed of light toward bob the reflection would have been at the halfway point, so if at the speed of light it is half way, at 0.6 the speed of light (or thereabouts (doubt)) one could imagine the reflection being at 0.6 the distance to the half point.

As the signal returns to Bob's LiDAR Alice keeps on approaching, the light will take the same amount of time to return.

[some replies have come in so I'll break and read them. Dratt! One of them was my own.]

 

@ Janus - what I find difficult is to read someone else's figures and make instant sense of them, I'm using an Excel spreadsheet where all graduations of speed can be analysed (I am increasing the relative velocity by 0.01 c at each step)

So once I have got my head around how to do the calculations of when to fire the LiDAR I will be able to tell where you and I see the issue differently.

 

What I need is the logic to work out the steps so I can enter the formulas into the cells.

 

Note: Bob and Alice can't read each other's clock and the firing times have to be preset. So statements like this are forbidden "But wait, by the ISS's observations, only 1.28 sec ticked off for Bob between sending and receiving the pulse. This is an physical contradiction."

Edited December 24, 2014 by Robittybob1

[Robittybob1]

It is unreasonable to be hard and fast about not saying Bob is heading toward the ISS for he well knows he started off from stationary on the Earth some weeks ago and he headed out, stopped, turned around and now is heading back to Alice that he knows is/was stationary when he left.

He also knows that as he sped up his time was be dilated.

Even though he knows he is moving he finds it is so easy to view the situation as the Earth and the Moon speeding up towards him.

He had been taught classical relativity and very easily slips back into that mindset.

Nothing short of a catastrophe would made the Earth and the Moon change direction so right or wrong he finds himself thinking he is moving.

And I tend to agree with him.

Edited December 25, 2014 by Robittybob1

[Robittybob1]

It is very simple to show that you cannot do away with length contraction by having the frames measure their relative speeds differently.

 

All is takes is one LiDAR pulse.

 

First we consider things from the frame of the ISS. Bob fires off a pulse towards the ISS as he passes the Moon and when his clock reads 0. It takes ~1.282 sec for that pulse to reach the ISS. In which time, Bob has traveled ~ 231,000,000m and is now ~153,000,000 m from the ISS. The pulse bounces back towards Bob at c while Bob continues to travel towards the ISS at 0.6c and they meet in another ~0.32 sec. Total time between firing of pulse and return to Bob, ~1.602 sec. Bob's clock of course runs at a rate of 0.8 that of the ISS clock, so he ticks off ~1.28 sec between the firing and reception of the pulse. Bob continues on and passes the ISS when the ISS has recorded ~2.14 sec and Bob's clock reads ~1.71 sec.

 

Now consider things from Bob's frame. Without length contraction, for him to travel from the Moon to the ISS in 1.71 sec, the relative speed between them has to be .8 c (384400000/1.71 = 0.8), So when he fires his pulse at c, it meets the ISS rushing toward him at 0.8 c in ~0.71 sec, by which time, the distance to the ISS will have closed to by ~171,000,000m and is ~213,600,000m away. The pulse returns at c, taking another ~0.71 sec to return to Bob, for a total time of 1.42 sec.

 

But wait, by the ISS's observations, only 1.28 sec ticked off for Bob between sending and receiving the pulse. This is an physical contradiction. If we had set things up so that Bob's clock started running when he sends the pulse and stops running when he receives the reflection, Then assuming Bob stops when he reaches the ISS (or passes it, slows down and returns) so that he, the clock and the ISS are all in the same frame, You will have two observers in the same frame at the same place saying that the same clock reads two different things.

 

It gets worse if Bob is traveling even closer to c relative to the Earth, if he is moving at 0.866 c, then it takes ~1.48 sec for him to reach the ISS in ISS time and ~0.74 sec in Bob time, for Bob to cross the Moon-ISS distance in 0.74 by his clock, the relative speed has to be 1.732 c. So now, when Bob bounces his LiDAR pulse off of the ISS, according to Bob, he reaches the ISS at 1.732c before the reflected LiDAR pulse (traveling at c relative to him) returns to him.

 

The simple fact is that you cannot do away with length contraction and still keep the(experimentally verified)invariant speed of light.

 

The invariant nature of the speed of light automatically requires length contraction.

 

....

"It is very simple to show that you cannot do away with length contraction by having the frames measure their relative speeds differently." Well that is the purpose of the experiment, to find out whether this is true.

"All it takes is one LiDAR pulse" I disagree with that as LiDAR takes two pulses to make a speed measurement.

The rest seems to be the confusion between reading other peoples clocks while you are very distant and moving relative to each other.

Alice and Bob don't know what time it was when the pulse is fired unless it was preplanned while they were calibrating and setting up their machines.

Edited December 25, 2014 by Robittybob1

[Robittybob1]

"It is very simple to show that you cannot do away with length contraction by having the frames measure their relative speeds differently." Well that is the purpose of the experiment, to find out whether this is true.

"All it takes is one LiDAR pulse" I disagree with that as LiDAR takes two pulses to make a speed measurement.

The rest seems to be the confusion between reading other peoples clocks while you are very distant and moving relative to each other.

Alice and Bob don't know what time it was when the pulse is fired unless it was preplanned while they were calibrating and setting up their machines.

So the two parameters that were agreed on pre-flight were:

1. The time intervals between the pulses, and

2. That Bob would send an additional signal 3 ms after receiving a pulse from Alice.

The time between the pulses for Bob was set at a shorter interval for it was known that his clock would be running slow compared to the calibrated master clock at Houston. Alice's clock onboard the ISS will also be slightly time dilated compared to the Houston one but not by any significant amount. Alice has already received a time dilated signal back from Bob so their prediction that Bob's time will be running slow has been verified.

Bob has just a split second to decide when to action his LiDAR pulses to check his velocity, Alice is anxiously waiting to detect these pulses. She hoping all is well and Bob hasn't run into any space debris or valuable assets for they had been pre-warned it was a dangerous and foolish place to run speed trials considering the number of satellites and the vast amount of space debris orbiting the Earth.

She knows there shouldn't be too many issues with the first pulse for as far as they are aware there are no satellites orbiting the Moon at the moment.

Edited December 26, 2014 by Robittybob1

[phyti]

My question is do you think length contraction really happens? What I see in the YT clip is that time is dilated and the ship goes further in dilated second .

Could length contraction be better explained better by showing differing relative speeds rather than than saying the distance to a star is altered by relativistic speeds?

The perception of an observer on earth: muons are created at (x,t)=(0, 0), and move at .9c until decaying at ground level at (x, t). Time dilation slows the muon clock to read t' at x.

The perception of an observer moving with the muons: the ground arrives at t', and x'=x(t'/t). Thus the ground is literally closer, as if the universe contracted in the direction of motion. Knowing that the muon motion cannot alter the dimensions of the universe, the effect is from the altered perception of the observer via time dilation.

He cannot fault his clock or his biological sense of time, and concludes the universe has contracted as it passes him. This satisfies the reciprocity of the principal of relativity.

It's my understanding that when something is going near the speed of light in reference to an observer, time dilation occurs and time goes slower for that fast-moving object. However, when that object goes back to "rest", it has genuinely aged compared to the observer. It's not like time goes slow for a while, and then speeds back to "normal," so that the age of the observer once again matches the object. The time dilation is permanent. Why wouldn't the same thing happen with length contraction? Since the two are so related, you'd think if one is permanent, the other would be also. And from everything I've read so far, length contraction is not permanent. An object will be at rest touching an observer, go far away near light speed, return to touching the observer, and be the same length it was at the beginning. It shortens, and then grows long again, as if its shrinkage was an illusion the whole time. Did I just not read the right things or what? Were my facts gathered incorrectly?

1. A clock produces, counts, and accumulates ticks (an arbitrary interval of activity). If clock A separates from an identical clock B, and later rejoins B, it will have ticked less. While in relative motion, each will appear to run slower than the other, via doppler effects, since clocks can be treated as frequencies. This supports the principle of relativity, in that observations are reciprocal, thus observers with the clocks see the same physical phenomenon. Because light speed is constant, any device requiring light interactions will occur slower, the faster the device moves relative to light. Since all objects are in motion, all clocks are losing time to various degrees. Any age difference between A and B requires a comparison of both at the same location before and after the excursion by A. In the simplest case, the one that changes speed to rejoin, will have aged less for the excursion. In SR there is no speed whereby the clock can regain lost time. If A rejoins B, its rate will change to match that of B.

2. Length contraction occurs when object motion affects the em fields that determine the separation of particles. This also results from constant light speed, and the calculations involve the same function of speed, the gamma factor, as used for time dilation. When the object slows to its original speed it regains the original length, i.e. it doesn't loose length.

[Robittybob1]

Bob and Alice had been over the years pre-flight discussing Relativity on the NASA approved science forums and had noticed one glaring mistake. There were these diagrams and animations of time clocks on board another craft where it was possible for both observers to see the workings of each other's time clocks. Many presenters had it so that one observer who was thinking his/her time wasn't dilated ending up observing the "one" photon traveling in a different direction as someone observing the same photon in a time dilated situation. Now in the words of Janus that is a "physical impossibility".

Bob and Alice had to work out a method so that in both situations the moment of reflection was always the same time, yet one photon seemed to be traveling a different length of path, if light speed never changes, and there is no length contraction in dimensions other than the direction of travel, so how can that be comprehended?

 

In clocks that have been calibrated, a moving clock will be showing slower time. But then can the person on the moving craft also look across and claim that the other guy is moving and their clocks are slowed?

Could that be "Yes, they are slowed but never slower than his"?

Both Bob and Alice are still confused. Muons are so interesting.

Edited December 26, 2014 by Robittybob1

[Robittybob1]

Going back over the threads in this forum on relativity this post struck me as being filled with wisdom:

http://www.scienceforums.net/topic/82735-special-relativity-twins-paradox-is-it-real/#entry801650

 

pzkpfw said "Basically: the twin who leaves and comes back has, by changing frames (by accelerating), travelled further in space. Because they travelled further in space, they travelled less in time."

I like that concept.

Next post it was considered incorrect - oops!

Edited December 27, 2014 by Robittybob1

[swansont]

Going back over the threads in this forum on relativity this post struck me as being filled with wisdom:

http://www.scienceforums.net/topic/82735-special-relativity-twins-paradox-is-it-real/#entry801650

I like that concept.

Next post it was considered incorrect - oops!

 

Because it is incorrect. v = d/t. If one travels further in less time, they have a higher speed. But each twin will see the other traveling at the same speed. You need a shorter distance in less time for the speeds to agree.

[Robittybob1]

 

Because it is incorrect. v = d/t. If one travels further in less time, they have a higher speed. But each twin will see the other traveling at the same speed. You need a shorter distance in less time for the speeds to agree.

I'm sure I've heard an expression like that before. It may have been slightly different. Would "because they travelled faster through space, they travelled slower through time" rather than "because they travelled further in space, they travelled less in time", be correct? (Minkowski spacetime diagrams)

Edited December 27, 2014 by Robittybob1

[swansont]

I'm sure I've heard an expression like that before. It may have been slightly different. Would "because they travelled faster through space, they travelled slower through time" rather than "because they travelled further in space, they travelled less in time", be correct? (Minkowski spacetime diagrams)

 

Possibly in terms of the four-velocity. The faster you travel through space, the slower you travel through time; the invariant quantity is c

[Robittybob1]

An inte

 

 

Possibly in terms of the four-velocity. The faster you travel through space, the slower you travel through time; the invariant quantity is c

Yes it is just one more of the many factors to consider when trying to comprehend relativity. A very interesting connection came up on another forum today and that has been confirmed by an internet search, and that is "the time dilation caused by gravity on the surface of a planet is equal to the time dilation for an object moving at the planet's escape velocity". So that said in another way is that the velocity that an object will gain by free falling into a gravitational field will be the velocity that will cause the same amount of time dilation as that object has while stationary in that gravitational field.

I wonder if I said that correctly, but it is such a new and not widely discussed idea (I could be wrong about that too, but I had not previously heard of it in the 4 years I've been on the science forums.)

"Since gravitational time dilation is tied to gravitational potential, and escape velocity is the speed at which you need to start with in order to climb to infinity against Earth's gravity with a ballistic trajectory." Very interesting that escape velocity and time dilation are linked and not just for things falling into a black hole but everything under the influence of gravity.

Edited December 28, 2014 by Robittybob1

[Robittybob1]

From Wikipedia on GPS

 

 

.... the satellites' clocks gain 45,850 nanoseconds a day due to general relativity effects.
and
... the satellites' clocks lose 7,214 nanoseconds a day due to special relativity effects
so .... "45850 – 7210 = 38640 ns

Hence the GPS satellite clocks gain approximately 38,640 nanoseconds a day or 38.6 μs per day due to relativity effects in total."

 

With Bob and Alice thought experiment the speed of the spacecraft will be the dominant effect so the slowing of time due to gravitation, as Bob approaches the Earth, will not be significant at the speed Bob is going (0.6 c).

Edited December 29, 2014 by Robittybob1