The twin paradox

[Gobbleston]

Hello,

 

I originally had a fairly basic question about the twin paradox but when attempting to look it up, I clicked the Wikipedia article and it was totally chock full of equations relative to speed, and the Doppler effect, and other such things.

 

I was led to believe that the twin paradox has to do more so with the two twins and their relation to one another based on the bending of space time because of their acceleration/mass.

 

Einstein’s theory of general relativity basically means (as far as I know) that the faster an object moves, the greater its mass increases, and because objects of larger mass create a greater gravitational pull on space time, that would account for the differences in age. Is that wrong?

 

Forgive me if this is laden with complete ignorance. :/

 

 

[pmb]

I was led to believe that the twin paradox has to do more so with the two twins and their relation to one another based on the bending of space time because of their acceleration/mass.

The twin paradox is about two identical twins, one who stays on Earth while the other one accelerates off to a distant place. Comes to a stop and then returns. The one that remained behind will have aged more than the twin that went away and came back. It has nothing to do with the bending of space time or their mass. It has only to do with on of the twins acceleration.

 

Einstein’s theory of general relativity basically means (as far as I know) that the faster an object moves, the greater its mass increases, ...

You can look at it that way, yes.

 

and because objects of larger mass create a greater gravitational pull on space time, that would account for the differences in age. Is that wrong?

The faster the body the greater the mass. The greater the mass the greater the gravitational pull. There are two factors at play here. One is the increase in mass of the moving body. The second is the increase in gravitational force as a function of speed.

 

The rest is not not quite right. It is true that time passes slowed for those near gravitating body than those far away though.

[Gobbleston]

But isn't acceleration directly linked to mass and mass's effect on space/time? So the twin's mass from their acceleration should have at least some relation there, shouldn't it? The faster you go, the more mass you accumulate, the more mass you accumulate, the more gravity you generate, the more gravity you generate, the more space/time is bent.

 

Like the whole thing about clocks at different altitudes running differently from one another. I understand that the acceleration would be faster at higher altitudes, but wouldn’t the gravity also have less effect the further you were from the bulk of Earth’s mass, thusly making the curvature of time less as well? The theory of special relativity with space/time factored in (which is the basis of relativity, right?) states that acceleration and gravitational pull should have the same effect on objects if the forces equal each other.

 

Is the way I’m understanding this just a different angle than how it is usually viewed?

 

 

[pmb]

But isn't acceleration directly linked to mass and mass's effect on space/time?

Not that I'm aware of, no.

 

So the twin's mass from their acceleration should have at least some relation there, shouldn't it?

No.

 

The faster you go, the more mass you accumulate, the more mass you accumulate, the more gravity you generate, the more gravity you generate, the more space/time is bent.

Yes. In the twin paradox that very small field generated by the moving twin is small enough to ignore. In any case, relative to the twin, there is no mass increase of himself or the ship he's in.

 

Is the way I’m understanding this just a different angle than how it is usually viewed?

Yes. But keep in mind that its the wrong way.

[swansont]

But isn't acceleration directly linked to mass and mass's effect on space/time? So the twin's mass from their acceleration should have at least some relation there, shouldn't it? The faster you go, the more mass you accumulate, the more mass you accumulate, the more gravity you generate, the more gravity you generate, the more space/time is bent.

 

The mass of the object is not one of the variables. A large mass will feel the same kinematic time dilation as a small one, if the speeds are the same. The acceleration is not the source of the difference in the time (or is not typically a significant source; you could create a scenario where it is). The time difference is due to the speed and the effect is integrated over the duration of the travel. The acceleration is what changes you from one inertial frame to another but in most treatments of the problem the details of the acceleration are ignored as being insignificant.

[Gobbleston]

@pmb

 

You do know that you're not being helpful, right? You're just telling me I'm wrong. Great, thanks for that.

 

@swansont

 

I didn't mean the mass of the object itself as being a factor, I meant the mass increasing proportional to the speed. As for the use of the word acceleration, I was just referring to what was said earlier, in that the more you accelerate, the faster you are, the more mass you have.

 

I don't understand how something with a much larger mass (a ship traveling close to the speed of light), can't have an effect on the way time behaves toward it.

 

 

[mathematic]

There is a much simpler explanation, based on special relativity (not general) which ignores acceleration. Example: twin 1 stays on earth and twin 2 goes to alpha centauri (4.3 light years away) and back. Assume twin 2 can get to near the speed of light very fast, travel to the star, stop fast, turn around, get up to near the speed of light, and stop fast upon returning to earth.

 

In twin 1's reference frame, twin 2 has been gone over 8.6 years, so twin 1 has aged over 8.6 years. Twin 2's experience is quite different, ignoring the acceleration and deceleration, he has traveled to the star and back at near the speed of light. Applying Lorentz transformation to the distance and the time of flight, the distance will be close to zero and so will the travel time, so he will have aged very little.

[Gobbleston]

@mathmatic

 

Hmm. I feel like I'm almost there. I haven't even heard of Larentz transformation before, so there's a big blob of ignorance. I'm unclear as to why twin 2 would experience it much quicker. If he is going light speed, and it is 4.3 light years away, going at light speed should still take 4.3 years. For both twins. I mean, if some event on Alpha Centauri happens, (it blows up, whatever) it will take 4.3 years for us to be able to see that event, and likewise, the time whatever event happened will have long happened there when we see it.

 

Like, if person A on Alpha Centauri turns on a christmas tree, and we see it 4.3 years later, we turn ours on back, and they will see it 8.6 years after they turned on there's. I don't understand how the same wouldn't be for an object, or person inside of said object, taking the same trip.

 

 

[juanrga]

Hello,

 

I originally had a fairly basic question about the twin paradox but when attempting to look it up, I clicked the Wikipedia article and it was totally chock full of equations relative to speed, and the Doppler effect, and other such things.

 

I was led to believe that the twin paradox has to do more so with the two twins and their relation to one another based on the bending of space time because of their acceleration/mass.

 

Einstein's theory of general relativity basically means (as far as I know) that the faster an object moves, the greater its mass increases, and because objects of larger mass create a greater gravitational pull on space time, that would account for the differences in age. Is that wrong?

 

Forgive me if this is laden with complete ignorance. :/

 

First, as the Wikipedia link emphasizes there is not such paradox in the sense of a logical contradiction.

 

A complete description of the twins can be made in general relativity, but it can be also treated with special relativity, which deals only with inertial frames (constant velocity) and where spacetime does not bend but is flat . This would convince you that the twin behaviour has nothing to do with acceleration or mass. (Moreover, mass in general relativity is a fixed quantity denoted by [math]m[/math]: See A no nonsense introduction to general relativity).

 

The difference in ages can be shown using special relativity, and its expression for time-dilation factor [math]\sqrt{1 - v^2/c^2}[/math]. Time goes slow for moving objects [math]v \neq 0[/math] than for objects at rest [math]v=0[/math].

 

Suppose that Ann stays at home and Bob rockets away at 3/5 light speed. Introduce [math]v=3/5 c[/math] in the above special relativistic formula and you get a time dilation of 80% for Bob. Bob lets 4 years pass. Bob returns at 3/5 light speed (time dilation is again 80% for him), taking another 4 years. Ann thinks 10 years have passed according to her clock, and both Ann and Bob agree that Bob is two years younger (10 years * 80% = 8 years).

Edited June 30, 2012 by juanrga

[pmb]

@pmb

 

You do know that you're not being helpful, right? You're just telling me I'm wrong. Great, thanks for that.

Dear Gobbleston,

 

I appologize for not being helpful. It wasn't like I was intentionally paying you short shrift or anything like that. I've been very ill this week and the energy has been saapped from my body. That's why my responses haven't been as good as could have. I've just been too week to give it my best. Please cut me a little slack, okay? Thanks.

 

I'll try to do better. Let me start from scratch.

 

I originally had a fairly basic question about the twin paradox but when attempting to look it up, I clicked the Wikipedia article and it was totally chock full of equations relative to speed, and the Doppler effect, and other such things.

 

I was led to believe that the twin paradox has to do more so with the two twins and their relation to one another based on the bending of space time because of their acceleration/mass.

The Twin Paradox is defined as follows. Note: What follows occurse in flat spacetime. Two twins are initially at rest at home, i.e. at the spatial location R = (x, y, z). One twin speeds off and later comes back home only to finds that he's aged less than his twin who remained at home. This is a result of time dilation, i.e. time in his moving frame of reference runs more slowly than time as measured at home. That's the Twin Paradox. Spacetime does not curve during all of this. Once spacetime is flat its flat unless you place matter in the space. In the Twin Paradox the mass of the twins is so small as to be neglected for this purpose. Mass is not a factor in all of this.

 

Einsteins theory of general relativity basically means (as far as I know) that the faster an object moves, the greater its mass increases, and because objects of larger mass create a greater gravitational pull on space time, that would account for the differences in age. Is that wrong?

Yes. It's wrong. The difference in aging of the twins is accounted for by regular time dilation as accounted for in special relativity, not gravitational time dilation as accounted for in general relativity. While its true that each twin has mass that isn't accounted for when we calculate the difference in aging. As a matter of fact we can make the mass of the twin as small as we wish. The twins could simply be a small amount of radioactive material whose mass is so small that it can be neglected and the material being radioactive can act like a clock.

 

Does that help?

 

Question - How were you led to believe that the twin paradox has to do more so with the two twins and their relation to one another based on the bending of space time because of their acceleration/mass?

Edited June 30, 2012 by pmb

[swansont]

@swansont

 

I didn't mean the mass of the object itself as being a factor, I meant the mass increasing proportional to the speed. As for the use of the word acceleration, I was just referring to what was said earlier, in that the more you accelerate, the faster you are, the more mass you have.

 

I don't understand how something with a much larger mass (a ship traveling close to the speed of light), can't have an effect on the way time behaves toward it.

 

The acceleration does not have to change the speed, i.e. one can execute circular motion. It makes no difference. In fact, the experiment has been done for circular motion You can solve it with either special (speed) or general relativity (acceleration) approaches; they reduce to the same answer, as they must.

[Gobbleston]

As an aside, if some object of any mass was propelling through space, as it's speed increases, it's mass increases. Regardless of the initial question, it would generate it's own gravitational field due to its increased mass, right? So, while something inside of the mass might not be affected by the changes, if something were to come into contact with the speeding object, the space/time relevancy would be affected for it, correct?

 

 

[pmb]

As an aside, if some object of any mass was propelling through space, as it's speed increases, it's mass increases. Regardless of the initial question, it would generate it's own gravitational field due to its increased mass, right?

Right. What actually happens is that the gravitational field that is present in the rest from of the object becomes stronger as the body is moving.

 

So, while something inside of the mass might not be affected by the changes, if something were to come into contact with the speeding object, the space/time relevancy would be affected for it, correct?

Can you explain that in a different way. I'm totally lost.

[Gobbleston]

Say, if a space ship were traveling at close to the speed of light, and it hits a space bug (somehow doesn't splatter it). The bug on the outside of the ship should be affected by the gravity and the gravity's changes on space/time created by the speed, and so it's time frame would be different from what his bug friends are experiencing back at the hive. Say he were to hang out on the windshield for some years, when he is finally carted back to the hive, he should be a different age from the rest of them solely because of the gravity incurred by the mass generated by the high speed.

 

I mean, without specifically referring to the twin paradox thought experiment, if you were to have one twin living on planet a, and the second twin living on planet B which has several times the mass as planet A, they would age at different rates when finally reuniting, correct

 

 

[pmb]

First, as the Wikipedia link emphasizes there is not such paradox in the sense of a logical contradiction.

The term paradox as it is used here is defined as

http://www.merriam-webster.com/dictionary/paradox

a statement that is seemingly contradictory or opposed to common sense and yet is perhaps true

 

Say, if a space ship were traveling at close to the speed of light, and it hits a space bug (somehow doesn't splatter it). The bug on the outside of the ship should be affected by the gravity and the gravity's changes on space/time created by the speed, and so it's time frame would be different from what his bug friends are experiencing back at the hive. Say he were to hang out on the windshield for some years, when he is finally carted back to the hive, he should be a different age from the rest of them solely because of the gravity incurred by the mass generated by the high speed.

There are two sources of aging at play here. This is a compound problem since there is time dilation due to speed and time dilation due to gravity. Both contribute to the aging of the bug, In any casd, yes, his age would be different.

 

I mean, without specifically referring to the twin paradox thought experiment, if you were to have one twin living on planet a, and the second twin living on planet B which has several times the mass as planet A, they would age at different rates when finally reuniting, correct

If the planets were identical in both diameter and mass and each twin was lying on the surface then they'd age at the same rate. The twin on planet a's wristwatch would be running slower compared to twin b but twin b is also in a potential well too and rn faster than if it wasn't on a planet So the effect is that they cancel out leaving no aging. You can also look at this from a symmetry situation. They have to age at the same rate because one is not special over the other.

[Gobbleston]

I'm confused here, you say if both of the planets had the same mass and diameter, but I was asking about two different planets of greatly varying mass. I mean, if it would cause their clocks to be out of sync, wouldn't that follow that their lifespans would also become out of sync?

 

Oh, also, I know what a paradox is, just calling it that because that's the common used term for the situation.

 

 

[pmb]

I'm confused here, you say if both of the planets had the same mass and diameter, but I was asking about two different planets of greatly varying mass.

Oops! Sorry about that. My mistake.

 

I mean, if it would cause their clocks to be out of sync, wouldn't that follow that their lifespans would also become out of sync?

Yes. That's true.

 

Oh, also, I know what a paradox is, just calling it that because that's the common used term for the situation.

That was directed to whoever is reading this thread, not you.