Age Relative? Time, Speed and Paradox Explored

[Fredrik]

I just know that two inertial frames of reference move at constant speed relative to each other and that speed of light c is the same for both observers.

This is correct.

With these two conditions it is enough to show that both observers reach contradictory conclusions...

This is wrong.

 

[ramollari]

This is wrong.

Ok let me clarify something. If each of the observers correctly concludes that the other's clock runs slower does this necessarily mean that there's a contradiction/paradox? This conclusion results from the fact that the frames move relative to each other at constant speed and c is the same for both observers.
Will be any of the observers ever be able to deduce the other's age without the need to return to Earth?

 

[Fredrik]

Ok let me clarify something. If each of the observers correctly concludes that the other's clock runs slower does this necessarily mean that there's a contradiction/paradox?

This is not a paradox. The reason is that these guys actually make the same prediction about spacetime, and SR is a theory of spacetime, not of space and time separately.

Will be any of the observers ever be able to deduce the other's age without the need to return to Earth?

They will be able to calculate the other observer's age at any point in spacetime.

 

[Fredrik]

They will be able to calculate the other observer's age at any point in spacetime.

This answer isn't very good, so I'll provide a better one. All observers will agree about a person's age at any point on his world line, but not at other points in spacetime. They can calculate the person's age at any point in spacetime, but in general they will not get the same result. The reason is that they may disagree about which point on the world line is simultaneous with the point they're considering (the point at which they're supposed to calculate this person's age).

This comment also answers the question that started this thread. Age is relative at most points in spacetime, but not on the world line (which is the set of events that this person is experiencing).

 

[4newton]

Ramollari:
You have the right idea in the first place. Don’t be put off by convoluted answers.
As proved by experiment. It is a fact that two clocks do not agree on accumulated time when one transitions away and returns to the first clock. The difference has nothing to do with acceleration or observation. The difference between clocks is only dependent on the rate and length of time of the transition. The answer given has no mechanism of the change.

This answer isn't very good, so I'll provide a better one. All observers will agree about a person's age at any point on his world line, but not at other points in spacetime. They can calculate the person's age at any point in spacetime, but in general they will not get the same result. The reason is that they may disagree about which point on the world line is simultaneous with the point they're considering (the point at which they're supposed to calculate this person's age).

This comment also answers the question that started this thread. Age is relative at most points in spacetime, but not on the world line (which is the set of events that this person is experiencing).

The answer does not address the reason the clock ticks slower. The change of the tick of the clock is not a result of observation. The change is real, as proven when the clocks come back together.

The age of all observers may always be calculated at any point by any observer. Any observer may calculate the reading on any other clock. This is the basic idea of SR. If you deny this ability then SR has no use at all.

 

[4newton]

Fredrik:
Now that we are in TD we can work from observation and develop a theory works in all cases including QM.

A good place to start is the Big Bang. Again I ask you did the BB create all that there is including space, time, spacetime, or is there a basic frame of dimensions that exceeds the universe.

 

[SamCJ]

I found this exhaustive thread in archives because the question interests me. I grew tired of the differences of opinion that confused me. I would like to pose the question a little differently.

Two twins are called Earthbound and Spacestation, each is wearing perfectly accurate radioactivity decay watches and each has a pulsar pulse counter. When Spacestation gets into orbit he believes he is standing still and that Earthbound is spinning at a high rate of speed. Earthbound believes that he is standing still and Spacestation is orbiting at great speed. These respective observations go on for 20 years, after which Earthbound joins Spacestation on the space station in orbit and he takes his perfectly accurate watch and his perfectly accurate pulse counter with him.

Question: When the two compare watches and pulse counters, what will the see? Could Einstein prove your answer is correct using mathematics?
Hasn't this experiment already been done and where can I find the results?

 

[pervect]

The experiment has actually been done as a side-effect of the implementation of the GPS system.

The result is that the satellite clocks run faster than the ground clocks. The type of clock - quartz, atomic, pulse counter, radioactive decay, etc. is totally irrelevant in determining this result, except that the accuracy one can expect from the clock varies significantly among clock types, and some clocks may not be accurate enough to reliably measure the results because of uncontrolled varaitions in how well they keep time. (Pulse counting in particular is subject to large enviromental variation).

The satellite clocks run faster, and not slower, because of a factor that you did not realize was significant but was nonetheless very important - the altitude of the clock. To clarify this a bit, what turns out to be important is the gravitational potential energy of the clock - for more details, see some of the threads on gravitational time dilation.

For more on the GPS clocks, see

http://www-astronomy.mps.ohio-state.edu/~pogge/Ast162/Unit5/gps.html

Because an observer on the ground sees the satellites in motion relative to them, Special Relativity predicts that we should see their clocks ticking more slowly (see Lecture 32). A straightforward calculation using Special Relativity predicts that the on-board atomic clocks on the satellites should fall behind clocks on the ground by about 7 microseconds per day because of the slower ticking rate.

Further, the satellites are in high orbits, where the curvature of spacetime due to the Earth's mass is less than it is at the Earth's surface. A prediction of General Relativity is that clocks closer to a massive object will seem to tick more slowly than those located further away (see Lecture 20 on Black Holes). As such, when viewed from the surface of the Earth, the clocks on the satellites appear to be ticking faster than identical clocks on the ground. A calculation using General Relativity predicts that the clocks in each GPS satellite should get ahead of ground-based clocks by 45 microseconds per day.

The combination of these relativitic effects means that if not accounted for the clocks on-board each satellite would tick faster than clocks on the ground by about 38 microseconds per day (45-7=38)! This sounds small, but the high-precision required of the GPS system requires nanosecond accuracy,

Note that the GPS satellite system isn't necessarily one of the most precise tests of relativity that has been done - the GPS system was not specifically designed to test relativity. Rather, it simply illustrates that relativity is not just for textbooks anymore.

In evaluating the archive responses, you might want to note that the posts by users who eventually wound up being banned (and having their names crossed out) are generally not representative of current scientific thought.

Many times a few users can create a false sense of controversy or uncertanity about results that are really not controversial (mainly in the minds of those who are not aware of the facts, and hence falsely weight every opinion with equal value).

 

[SamCJ]

Thanks. I had not thought of the lesser amount of gravity at the higher altitude. Unfortunately, that oversight made you miss the point I was trying to understand. I have usually read the the ground-based twin will be the older one when they rejoin, but a time or two I read that each twin will think the other is younger. However, I have never read an explanation of how the spaceship twin can be younger when both twins believe that it is the other that is moving faster, and, as I understand relativity, they are correct in thinking that from their own perspective. By adding the watches, I was trying to remove subjective misperception by either twin as a factor. By adding the pulse counter, I was trying to create a time machine that was common to both twins. By having Earthbound join Spacestation in the space station, I was trying to remove acceleration as a factor. (I do not think Einstein mentioned subjective misperception or acceleration as a factor in determining their ages at the end of the trip, and I have not understood any of the discussion of simultanaety and frames of reference.)
I anticipate that you might respond the the space twin may be deceived about which one is moving, but we know it is he and not the earthbound twin because he went through acceleration for a relativity brief time at the beginning of the trip and he must decelerate at the end of the trip. But, if so, I say, Isn't acceleration the same as deceleration except in a different direction? If space twin's spaceship heads in the direction that the Earth came from, he would appear to all earthbound people to be accelerating, but in actuality, he would be decelerating. In that event, earthbound twin really would be moving faster until the space twin tried to turn around and come back, at which time he would really have to hurry.
In any case, my question is essentially the same one that was originally asked in this thread. If there was a good answer to it, I missed it in all the confusion. (Thanks also for the tip about the lines through the names.)

 

[jtbell]

When the two compare watches and pulse counters, what will the see? Could Einstein prove your answer is correct using mathematics?
Hasn't this experiment already been done and where can I find the results?

Time dilation for the "travelling twin" has been incidentally observed in studies of precise timekeeping for military applications (targeting bombs etc.). The required precision is such that relativistic effects are observable even on clocks in airplanes flying around in circles.

tycho.usno.navy.mil/ptti/ptti2002/paper20.pdf[/URL] (PDF file)

This isn't quite the same sort of situation you were asking about, but it's close enough that you might be interested in seeing these results.

 

[pervect]

Thanks. I had not thought of the lesser amount of gravity at the higher altitude. Unfortunately, that oversight made you miss the point I was trying to understand. I have usually read the the ground-based twin will be the older one when they rejoin, but a time or two I read that each twin will think the other is younger. However, I have never read an explanation of how the spaceship twin can be younger when both twins believe that it is the other that is moving faster, and, as I understand relativity, they are correct in thinking that from their own perspective.

I'd recommend doing the problem in flat space-time first, this means imagining that both spaceships are far from any planet and there is no gravity.

In that case, there is a very easy explanation. Each twin is correct in thinking that the other twin is moving faster, and each twin thinks that the other's clock is running slow.

However, in order for the twins to re-unite, one twin must accelerate.

The twin that accelerates is the twin that will experience the shortest amount of time. If you haven't read anything about it, there are only about a zillion articles on the twin paradox. I'd recommend the sci.physics.faq on the twin paradox as a good start

http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_paradox.html

To do the problem with gravity involved requires considering the metric of space-time. This is a lot harder job. For starters, even describing the metric of space-time from the perspective of the orbiting satellite would be quite a challenge. The "easy" coordiante system to use to describe the metric of space-time is a coordinate system anchored to the center of mass of the most significant massive body (in this case the Earth). If you have two massive bodies, the problem becomes almost insanely difficult (unless you use pertubative methods, in which case it becomes only extremely difficult). The difficulty/reward ratio is low for the problem with two massive bodies.

If you have only one massive body, though, the metric of space-time to use is the Schwarzschild metric, which you can also read about in the wikipedia

http://en.wikipedia.org/wiki/Schwarzschild_metric#The_Schwarzschild_metric

The process of computing the elapsed time on an observer consists of integrating dtau over the path followed by the body in Schwarzschild coordinates, where dtau^2 is given

dtau^2 =
(1-r/rs)*dt^2 - dr^2/c^2*(r-rs)-r^2/c^2*(dtheta^2+sin^2(theta) dphi^2

(Note dtau is just the metric, ds^2, from the Wikipedia, multiplied by a scaling factor of -1/c^2).

This is actually not that hard to do if you are familiar with basic calculus. If you work out this intergal for the orbiting satellite and for the ground based observer, you'll get the results I described.