Time Dilation: Concepts of Modern Physics by Beiser

[jinksys]

I'm reading Concepts of Modern Physics by Beiser, and the chapter example says:

A spacecraft is moving relative to earth. An observer finds that in one hour, according to her clock 3601s elapse on the spacecraft 's clock. What is the craft's velocity relative to earth? (This is not a homework question, I can get the correct answer.)

So is this saying that for every 3600 seconds on earth, the spacecraft 's clock moves 3601 seconds? Wouldn't this mean that time runs faster on the craft and not slower?

 

[Doc Al]

Seems like an ambiguously worded example. Are you quoting it exactly, word for word?

In any case, moving clocks are observed to run slow. So if she is observing a moving clock, then when she sees 1 hour pass on the observed clock, her clock may show 3601 seconds having passed.

 

[jinksys]

Here is the example, verbatim:

A spacecraft is moving relative to the earth. An observer on the Earth finds that, between 1 and 2 pm according to her clock, 3601s elapse on the spacecraft 's clock. What is the space craft's speed relative to the earth?

Solving for v, I have:

v = c \sqrt{1-\frac{t_0^2}{t^2}}

Where t0 is the clock at rest, and t is the clock in motion.

t0 has to be more than t or else the square produces an imaginary number. So from this it seems like the clock in motion always has to read faster than the stationary one.

Furthormore, here is the solution it gives:

Here, t0 = 3600s is the proper time interval on the Earth and t = 3601 s is the time interval in the moving frame as measured from the earth. From here they provide the math solution which is 7.1 x 10^6 m/s.

 

[Doc Al]

Here is the example, verbatim:

A spacecraft is moving relative to the earth. An observer on the Earth finds that, between 1 and 2 pm according to her clock, 3601s elapse on the spacecraft 's clock. What is the space craft's speed relative to the earth?

This is Example 1.1 in Beiser, right? (I just looked it up on amazon.com preview.) In any case, Beiser has it backwards. (Someone should tell him!)

 

[jinksys]

This is Example 1.1 in Beiser, right? (I just looked it up on amazon.com preview.) In any case, Beiser has it backwards. (Someone should tell him!)

Yep, the first example :( not a good start to a book I have to use all semester!

 

[Doc Al]

Yep, the first example :( not a good start to a book I have to use all semester!

That's for sure. Please point this out to your instructor.

 

[jinksys]

That's for sure. Please point this out to your instructor.

Luckily I was able to get an international version for $20, I feel for those who bought it from the bookstore for $176. I have another question...

This is the author's formula and explanation of time dilation...

t = \frac{t_0}{\sqrt{1-\frac{v^2}{c^2}}}

t0 = rest clock
t = moving clock
c = speed of light
v = speed of clock in motion

An Amazon review says that he has definitions of t and t_0 backwards, is this true?

 

[Doc Al]

An Amazon review says that he has definitions of t and t_0 backwards, is this true?

I'd say so. That formula is OK if t0 means the time elapsed on the moving clock and t means the time elapsed on the "rest" clocks. Better read that book critically! (Get another book as a backup.)

 

[jinksys]

I'd say so. That formula is OK if t0 means the time elapsed on the moving clock and t means the time elapsed on the "rest" clocks. Better read that book critically! (Get another book as a backup.)

I am definitely going to Borders or Barnes and Noble and getting a backup. Do you have any recommendations?

 

[Doc Al]

I suggest you post something in the Science Book discussion forum asking for opinions. We have quite a few active instructors who might be able to recommend a good one. (Someone who's taught from a book would be able to give a useful opinion.)

 

[jinksys]

Ok, well now that I have that sorted out, I actually do have other questions regarding time dilation.

Suppose there are two GIANT clocks, one perched upon the Earth and one perched upon a spacecraft . The clocks can be seen by both observers, one on Earth and one in the craft.

Are the following true?

To an observer on Earth and spaceship's clock ticks slower than the Earth's clock.
To the observer in the ship, the Earth's clock moves slower than his. (assuming he is at a constant speed).

 

[Doc Al]

To an observer on Earth and spaceship's clock ticks slower than the Earth's clock.
To the observer in the ship, the Earth's clock moves slower than his. (assuming he is at a constant speed).

Yes, both "see" the other's clock as running slow. I put "see" in quotes since relativistic effects like time dilation are what's observed after taking light travel time into account. (In order to make sense of what you see, you must factor in the time it takes for the light to reach you. You don't just go by raw observations--what you literally see.)

 

[Naty1]

Yes each observer sees the other fellows time as running slow. A good way to visualize this is via a "mirror reflecting photon clock"...a simple example which reflec the longer path taken by a photon in relative motion...maybe someone can post an online diagram source...

 

[atyy]

I am definitely going to Borders or Barnes and Noble and getting a backup. Do you have any recommendations?

I learned from an old edition of Beiser - it's very good. But typos are indeed irritating and make things hard, so it's good to also have eg. AP French's Special Relativity, and French and Taylor's An Introduction to Quantum Mechanics, Schroeder's An Introduction to Thermal Physics. Schaum's series is usually very reliable.

 

[jinksys]

I asked my physics adviser about the time dilation equation and she said that T should be larger than T0 because the moving clock moves slower. She said the larger number represents the amount of local time it will take for moving clock to show T0.

 

[Doc Al]

I asked my physics adviser about the time dilation equation and she said that T should be larger than T0 because the moving clock moves slower.

Huh? Slower that what?

She said the larger number represents the amount of local time it will take for moving clock to show T0.

Huh? The number shown on the clock itself is the local time (in the clock's frame)!

Ask her this question: A rocket travels past Earth (point A) to planet X (point B). During that trip, the rocket clock shows a time t₀. Will Earth clocks show a larger or smaller time for that rocket to go from A to B?

 

[jinksys]

This is Example 1.1 in Beiser, right? (I just looked it up on amazon.com preview.) In any case, Beiser has it backwards. (Someone should tell him!)

So, for the problem to be correct with Special Relativity the 'proper time' should be t0=3600, and t should be 3601, since it was stated that the ship was moving relative to Earth and the observer was at rest on earth. It looks like they set up the problem correct, since they get the correct answer, but in the problem and setup they have the times reversed.

Is this right?

 

[Doc Al]

So, for the problem to be correct with Special Relativity the 'proper time' should be t0=3600 and t=3601, since it was stated that the ship was moving relative to Earth and the observer was at rest on earth. It looks like they set up the problem correct, since they get the correct answer, but in the problem and setup they have the times reversed.

Is this right?

Yes. It's not just a typo, though, it's more of a sloppily constructed problem. What they meant to say was something like: "Earth observers find that from 1:00:00 pm to 2:00:01pm on their clocks only 3600 seconds have passed on the ship's clock."

They compound the confusion with the first sentence of the solution: "Here t₀ is the proper time interval on the earth..." huh? They should have said, "Here t₀ is the proper time interval as recorded by the ship's clock..."

 

[jinksys]

Yes. It's not just a typo, though, it's more of a sloppily constructed problem. What they meant to say was something like: "Earth observers find that from 1:00:00 pm to 2:00:01pm on their clocks only 3600 seconds have passed on the ship's clock."

They compound the confusion with the first sentence of the solution: "Here t₀ is the proper time interval on the earth..." huh? They should have said, "Here t₀ is the proper time interval as recorded by the ship's clock..."

Again, thanks for the all the help.

Here is an email I sent out to my classmates regarding this example. Was I correct in my statements?

If you were one of the lucky ones who've been able to acquire a textbook, I would like you to take a look at example 1.1 in the text.

Example 1.1 is set-up incorrectly. It should read:

A spacecraft is moving relative to the earth. An observer on the Earth finds that between 1:00:00 PM and 2:00:01 PM, according to her clock, 3600 seconds elapse on the spacecraft 's clock. What is the spacecraft 's speed relative to earth?

The solution set-up should read:

Here t0 = 3600s is the proper time interval on the spacecraft and t=3601s as measured by a clock at rest with the observer.

Since it was stated that the spaceship was moving relative to earth, a clock in the ship's frame of reference measures the 'proper time'.

The plugging-in of numbers and solving for v part of the example is correct, however.

 

[Doc Al]

Sounds good to me (and much improved over the original).

 

[avalanchesj]

Not to stray too far from the original post, but i have a question. Please keep in mind that I am a computer geek not a physics major, professor, or any other large brained person sporting a white coat. Does time itself slow down or just our relitive perception of time?

lets try this scenerio:

If I'm looking at my watch and it is syncronized with a clock aboard a spaceship before it departs from Earth and travles thru space at close to the speed of light, and the ship is transmitting it's clock time to me. When the ship returns will my watch read a different time then the ships clock? or does it only depend on the time it takes the signal to travel to me from it's location? in other words does time actually change for something travleing at at or near the speed of light or does time just APPEAR to change to an observer?

I hope I didnt confuse anyone. lol

 

[yuiop]

lets try this scenerio:

If I'm looking at my watch and it is syncronized with a clock aboard a spaceship before it departs from Earth and travles thru space at close to the speed of light, and the ship is transmitting it's clock time to me. When the ship returns will my watch read a different time then the ships clock? or does it only depend on the time it takes the signal to travel to me from it's location? in other words does time actually change for something travleing at or near the speed of light or does time just APPEAR to change to an observer?

This is the famous "Twin's paradox" of which there a thousand threads here. In a nutshell, your watch will show more time elapsed than the clock on the ship. If the journey is long enough and the ship travels fast enough then you will have aged noticeably more than the ship pilot. All physical processes (chemical, nuclear, biological etc) are affected equally by time dilation. The twin's paradox requires one of the twins to turn around and so that traveller experiences proper acceleration and cannot be considered an inertial observer. This breaks the symmetrical time dilation relationship observed in purely inertial reference frames, where each observer considers the other observer's clock to be running slower than their own.

It is only when clocks are brought back together and compared side by side that you can give a definitive definition of which clock was ticking slower.

I hope I haven't confused you now ;)

 

[avalanchesj]

It is only when clocks are brought back together and compared side by side that you can give a definitive definition of which clock was ticking slower.

Pardon my ignorance, but you used a lot of complicated phrasing (for me at least) but i think i got the jist of what you were saying, let me reitterate in my own words so i can be sure i understand.

When traveling at or near the speed of light, an observer from a relitivly stationary object (such as Earth), would notice that the ships clock seems to be running slower than his own. So time does change, not just our perception of time.

Is this because Space and Time are one in the same (Spacetime) and since the space between the Earth and the ship is growing more and more as the ship moves further away, the difference in time is also growing more and more? or did i miss the mark entirely? If so, could you explain it in a more dumbed down context? Ppossibly a pop-up book? lol jk

Because if that is the case then why do objects in orbit also show signs of time dialation? The space between the objects doesn't change, just the speed of the orbiting object. Is speed the only factor in time dialation?

I guess I'm just trying to wrap my head around WHY time seems to dialate when an object is moving at high speeds now lol.

Thank you in advance.

 

[yuiop]

When traveling at or near the speed of light, an observer from a relitivly stationary object (such as Earth), would notice that the ships clock seems to be running slower than his own. So time does change, not just our perception of time.

This is slightly different than the example in the last post, because the clocks have not been brought back alongside each other and compared side by side. While clocks moving relative to each other are spatially separated there is a symmetrical relationship. The pilot on the ship thinks the Earth clock is ticking slower than the ship clock. Remember I said "It is only when clocks are brought back together and compared side by side that you can give a definitive definition of which clock was ticking slower"?

Is this because Space and Time are one in the same (Spacetime) and since the space between the Earth and the ship is growing more and more as the ship moves further away, the difference in time is also growing more and more?

It is not about the space or distance between the observer and the observed clock growing. (See below).

Because if that is the case then why do objects in orbit also show signs of time dialation? The space between the objects doesn't change, just the speed of the orbiting object. Is speed the only factor in time dilation?

You are right about orbiting object showing time dilation relative to a clock at the centre, despite the distance from the central clock not growing. It is more about motion relative to a "grid" or reference frame rather than relative to an individual observer. If you are at the centre of that orbit, (and sticking to 2 spatial dimensions for now) you can imagine a grid of synchronised clocks spread out at regular intervals along x and y lines forming a map with coordinates. The orbiting object is a bit like a race car on circular track on this map. Its velocity is relative to the track (or grid or map) rather than to you standing at the centre and it is that "grid" velocity that determines its relative time dilation. If you are interested in learning more about the "grid" I might elaborate at a later time ;)

I guess I'm just trying to wrap my head around WHY time seems to dialate when an object is moving at high speeds now lol.

There is no real explanation for "why". Nature just does what it does and we puzzle over it. Time dilation does however seem to be one the required ingredients for a universe where all the laws of physics are the same in any inertial reference frame. On the other hand, I am not sure why a universe has to have that property, but our universe seems to have it.

 

[avalanchesj]

Ahhhh... the gears in my head are still smoking but I'm starting to get the picture. so it's all relitive. it all depends on witch frame of reference you're are observing, from my point of view his clock is running slow, and from his point of view mine is running slow, it's not until we are both moving at the same speed (relitive to each other) that we can determine which clock was running slow. so a satalite's velocity is relitive to space, not to me on the planet?

 

[avalanchesj]

I think Naty1 explained it in a different post and it makes sense this way for me:

with special relativity in which velocities depend on the choice of reference frame. Time and distance vary in different partsof spacetime; any frame is as good as any other; different frames give different results.

 

[yuiop]

Ahhhh... the gears in my head are still smoking but I'm starting to get the picture. so it's all relitive. it all depends on witch frame of reference you're are observing, from my point of view his clock is running slow, and from his point of view mine is running slow, it's not until we are both moving at the same speed (relitive to each other) that we can determine which clock was running slow. so a satalite's velocity is relitive to space, not to me on the planet?

Let's take an inertial reference frame (S) in which the Earth is at rest. A rocket takes from Earth and travels 0.8c for 1 year (Earth time). Now the Earth accelerates to 0.8c in the same direction as the rocket. To an observer that remains at rest in frame S (i.e. they never accelerated) less time has elapsed on the Earth clocks than on the rocket clocks. To an observer on the rocket (which is still moving at 0.8c relative to frame S) less time has elapsed on the rocket clocks than on the Earth clocks. Note that there is a dispute about which clocks are showing less elapsed time despite both the rocket and the Earth are moving at the same speed and at rest with respect to each other. As I said before, the question of elapsed time is relative while clocks are spatially separated and can only be resolved when the clocks are alongside each other. It is not even necessary for two clocks to be moving at the same speed to compare elapsed times of clocks with relative motion, as long as they are initially (and possibly briefly) alongside each other and then finally (and possibly briefly) alongside each other again. The indicated times can be compared as the clocks pass each other without coming to rest with respect to each other. The moving at the same speed bit is not important for comparing elapsed times, but the being alongside each other bit is important.

As for the satellites, your statement is basically correct although there some that would choke on the phrase "relative to space". Basically circular or rotational motion has an absolute nature and there is no ambiguity about who is turning. For example if you do a 360 degree spin it is physically and measurably different to the staying still and the Earth rotating around you.

Earlier you asked "Is speed the only factor in time dialation?". The answer is no. Gravitational potential also affects time dilation, but technically that is outside the domain of Special Relativity and I'm not sure if you want to get into that now.