The Logical solution to the Twin Paradox Explained comprehensively

[TakenItSeriously]

A month or so ago, I was asked to provide a logical explanation to the Twin Paradox that showed why symetrical views of time was not a paradox or how was symmetrical views of time completely consistent with the time deviation experienced between the twins. So as promised I fully explain:

1. Why length contraction is the reason why the twins experience time deviation. 
2. Why time deviation was not a paradox with the twins symmetrical views of time because length contraction is the asymetrical component that was never considered.
3. Finally, I show how their symetrical views of time really are completely consistent for the entire journey using a trasponder solution with each other.

 

As always, I will be using the example given in Wikipedia so that you can verify the results with those given in that example and I can skip the math for simplicity sake. I will only be providing the logical models that fully explains the paradox.

 

BTW, this is a good example of my previous statements that while logic may be prone to intuitive errors of false premise, such as the Earth is the center of the Universe because everything appears to revolve around the Earth, math is just as prone to intuitive errors of false conclusions.

 

The key points of the problem are:

• The ship carrying one of the twins goes straight to Alpha Centauri and back.
• Acceleration is assumed to be an insignificant factor so velocity is a constant 80% of the speed of light in both directions.
• Alpha Centauri is assumed to be in relativisticly static motion relative to the Earth with a proper distance of 4 light years.
• Each twin is equiped with a powerful transponder that pings with a source frequency of exactly once per second or 1Hz.

 

The Earth twin sees that Alpha Centauri is a static 4 lightyears away. Therefore, he calculates the trip will take 4/0.8 = 5 years each way or 10 years total.

 

The ships twin has a different perspective of the trip when moving at 0.8c due to length contraction, the distance is only 60% of the proper distance or 2.4 light years away. Therefore from his point of view, the trip will only last 3 light years each way or 6 years total.

 

So when he returns, he experiences 6 years while the Earth Twin experiences 10 years, however that is not the paradox.

 

The paradox is based on the fact that each twin should have symmetrical points of view of their brothers time which is true:

 

When the ships twin is on the outbound leg moving away from Earth at 0.8c then each twin sees their brothers time as moving at 1/3 of normal, or they would each be receiving a transponder ping only once every 3 seconds.

 

When the ships twin is on the return leg, then each twin sees their brothers time as moving at 3x normal or they would each be recieving 3 pings/sec.

 

These time shifts are due to the relativistic redshift which I didnt bother working out the math again, but the formulas are pretty simple and include time dialaion plus normal doppler effect due to lagtime, so that you can verify the results yourself or just refer to the Wikipedia example which uses the exact same problem.

 

The logical resolution to this paradox is the fact that while their views of each others time is symetrical, their views of the distance traveled is asymetrical.

 

The reason why is that the Earth, Alpha Centari, and the space in between the two are all in the Earths inertial reference frame, while the ship plus what is inside the ship is all that is in the ship’s inertial reference frame.

 

The Earth twin sees the ship is length contracted by 60% which has no bearing on the trip

 

The ships twin sees the Earth’s inertial frame as length contracted and as we said, the Earth, AC and the distance inbetween is all included within that inertial reference frame. Therefore, from the ship’s twin’s point of view the distance is length contracted by 60% of 4 light years or 2.4 light years.

 

So with 60% less distance to trave, then the trip takes 60% less time to travel from the ship twins point of view.

 

Another words while their point of view of time is symmetrical, their point of view of distance is asymtrical which accounts for their deviation in time experienced.

 

We can confirm this by correlating their point of view with lagtime. Another words, from the Earth twins point of view, the ships twin would take 5 years plus it would take 4 years for the light (or transponder signal) to get back to Earth from Alpha Centauri:

5 years + 4 years = 9 years

 

That means the Earth twin would expect to witness the ship actually execute the turnaround 9 years after the ships departure. Or when the transponder signal recieved back on Earth would change from 1 ping every 3 seconds to 3 pings/second then the ship would have executed the turn which would happen 9 years after the origional launch.

 

If you do the math and count the pings received during those 9 years at 1 ping/3 seconds adds up to:

9 years x 1/3 = 3 years 

which is actually what the ships clock would read at the turn around by both twins.

 

Events that include both a time and a place must always be consistent to any inertial frame.

 

 

On the trip back to Earth, the journey would only seem to take 1 year as perceived by the Earth twin so:

9 years + 1 year = 10 years total time as expected by the Earth twin. 

 

However, during that 1 year, the transponder is pinging 3 times per second so it adds up to 3 years

1 year x 3 = 3 years

3 years + 3 years = 6 years total.

 

The Ships twin experiences something different. He hears the transponder received from Earth ping once every three seconds, and since the outbound leng only takes 3 years, he sees the Earth clock as counting only 1 year.

3/3 = 1 year

 

On the return leg, the ship’s twin experiences three years worth of pings that are pinging at 3 pings per seconds, therefore the Earth clock advances 9 years during his 3 year return leg. 

3 years x 3 = 9 years

1 year + 9 years = 10 years 

which is the time elapsed back on Earth.

 

So not only is the math consistent, it sould not even be a surprise to either twin that their brother has aged differently.

 

 

 

Edited March 12, 2018 by TakenItSeriously

[pavelcherepan]

On 3/12/2018 at 8:03 AM, TakenItSeriously said:

transponder is pinging 3 times per second

I'm confused about this part the most. If the transponder on the Earth is pinging at 1 Hz, the twin on the ship in his own FoR should experience 1 ping per second, although at a higher frequency than the source due to blue shift. Why would the ship twin actually experience time as going 3x faster?

[TakenItSeriously]

52 minutes ago, pavelcherepan said:

I'm confused about this part the most. If the transponder on the Earth is pinging at 1 Hz, the twin on the ship in his own FoR should experience 1 ping per second, although at a higher frequency than the source due to blue shift. Why would the ship twin actually experience time as going 3x faster?

Both twins would hear their own transponder at 1 ping/second and that never changes.

When the ship is on the return leg, the earth twin would be recieving the ship’s twin’s signal at three pings per second and the ship’s twin would be recieving the Earth twin’s signal at three pings per second. So it’s still symmetrical.

Both would be due to relativistic blueshift, but when frequency increases, it doesnt just mean in pitch, but in cycles per second which in this case a cycle is a ping. That doesn't mean that time would actually be sped up, for anybody. it’s only a timelag illusion where the ship is kind of racing its own light.

Another words as the ship is leaving Alpha Centauri at 80% c, the light and radio signal is leaving Alpha Centauri at 100% c.

So from the Earth FoR the light takes 4 years to reach Earth while the ship takes 5 years to reach Earth. With only 1 year inbetween the two. That means the ship must transmit 3 years worth of pings received in only 1 years time.

On the other hand at the turn around, the ship has experienced only 1 year of pings from Earth due to that same lag time because the “now” time is still four years away back on Earth. So now the ship is racing opposite that light coming from earth from 4 years in the past plus the 5 years experienced by the Earth for the second leg of the trip so 4+5 or 9 years of Earth’s pings are crammed into 3 years of travel time for the twin on the ship.

It’s confusing, I know, but I hope that makes sense.

Edited March 13, 2018 by TakenItSeriously

[Rob McEachern]

On 3/11/2018 at 8:03 PM, TakenItSeriously said:

Acceleration is assumed to be an insignificant factor so velocity is a constant 80% of the speed of light in both directions.

That is a rather dubious assumption. The out-going twin can never return unless he or she reverses direction, which requires acceleration, to slow down and reverse course. If this acceleration is too great, it will kill the twin. If it is too small, the twin will die before returning, because it will take too long to slow down and reverse course.

A one g acceleration is about 10m/s². The speed of light is about 3x10⁸ m/s. So at 1g acceleration (earth gravity), how many seconds do you have to accelerate in order to reach 80% of the speed of light? How many do you have to accelerate in the opposite direction, to return at 80% of c? How many to slow down and stop, upon arrival?

It's going to be a long trip, even if you accelerate the entire time.

 

[John Cuthber]

Ignoring the acceleration is ignoring what is usually given as the actual answer to the paradox.

It's absurd.

[Eise]

14 hours ago, John Cuthber said:

Ignoring the acceleration is ignoring what is usually given as the actual answer to the paradox.

It's absurd.

Not so absurd as it seems. The acceleration is not essential. Essential is that the spaceship changes its inertial frame. That makes the situation asymmetrical. Of course this implies, when you try to make the example more realistic, that the spaceship accelerates (at least at the turning point).

Adaptions to the twin paradox have been made where nobody returns to the earth, but that one spaceship flies past the earth (with its 0.8c) to Alpha Centauri, where it meets a spaceship that flies in the opposite direction, also with 0.8c (compared to the inertial frame of Alpha Centauri anf the earth). So no acceleration is involved, but if you add the times of the two flights (according to the spaceships themselves) then during their flight times more time has passed on earth. 

If you look at the explanations at Wikipedia, you see that often acceleration is neglected, e.g. this one:

But the difference in time still exists. Later in the article more realistic scenarios, with acceleration, are presented (which makes the math a bit more complicated, of course).

Edited March 14, 2018 by Eise

[Rob McEachern]

2 hours ago, Eise said:

often acceleration is neglected

So is the equivalence principle and the acceleration due to gravity. If a spacecraft accelerates outbound at 1g and back inbound at 1 g, etc, the space traveler experiences the same acceleration (except from a brief reversal of thrust) as someone remaining on earth. However, if you leave one atomic clock on earth and have another circle the earth in an aircraft, the two clocks experience different accelerations and thus display differing elapsed times when they are finally brought back together and compared.

[swansont]

1 hour ago, Rob McEachern said:

However, if you leave one atomic clock on earth and have another circle the earth in an aircraft, the two clocks experience different accelerations and thus display differing elapsed times when they are finally brought back together and compared.

The difference is due to the difference in gravitational potential, not the acceleration. In fact, in the Hafele-Keating experiment, g was assumed constant in the analysis, and that works because the variation in g was small compared to the precision of the measurement.

[Rob McEachern]

1 hour ago, swansont said:

The difference is due to the difference in gravitational potential, not the acceleration.

A difference in gravitational potential, is a difference in acceleration (the acceleration due to gravity) - that was my point and that is the equivalence principle. But I'm sure you already know this. What causes the difference in acceleration, gravity or engine-thrust, is irrelevant to the effect.

[Janus]

1 hour ago, Rob McEachern said:

So is the equivalence principle and the acceleration due to gravity. If a spacecraft accelerates outbound at 1g and back inbound at 1 g, etc, the space traveler experiences the same acceleration (except from a brief reversal of thrust) as someone remaining on earth. However, if you leave one atomic clock on earth and have another circle the earth in an aircraft, the two clocks experience different accelerations and thus display differing elapsed times when they are finally brought back together and compared.

To build on what swansont has already alluded to.  The fact that our Earth observer and space traveler experience the same local acceleration is not the determining factor in terms of the time dilation each would measure in other clocks.    As swansont said, gravitational time dilation is due to the difference in gravitational potential, or in other words, the total effect of the gravitational field between the position of the two clocks.    One way to visualize  this is to consider how fast would something dropped from the higher altitude be moving when it reached the lower altitude.  That would be a measure of the difference in gravitational potential.

So if we take an object and drop it from a altitude 1 earth radius above the surface, when it starts its fall it will experience 1/4 the acceleration it does at the surface and it will hit the ground moving at some speed. A clock placed at this altitude will run faster than one on the surface.

Now if we put a clock on the surface of the world with 4 times the mass of the Earth and twice the radius, it will experience 1g just like a clock on the surface of the Earth. If we put an object 1 earth radius above the surface, it will experience 4/9 the acceleration as the surface.   If we drop an object from this height, it will hit the surface moving faster than one dropped from an equal height on the Earth.   In addition, a clock placed here will run faster than one on the surface by a greater factor than the difference in rate between the two clocks separated by the same altitude in the Earth scenario, even though the difference in acceleration experienced by the Earth clocks is larger than the difference for the second set of clocks.

When applying this to our accelerating space traveler, to use the equivalence principle, we have to consider what the equivalent gravitational field to his acceleration would be like.  In this case, it would be a uniform gravity field that extends to infinity along the line of acceleration that does not diminish in strength with distance.  Clocks that are in the direction he is accelerating will run fast, and clocks in the opposite direction will run slow. The greater the distance between them and these clocks, the larger the difference in their tick rates.

Thus for our space traveling observer, As he accelerates at 1 g away from the Earth, not only does his speed relative to the Earth increase causing him to measure a time dilation in the Earth clock, but the Earth is getting increasingly further away in the direction opposite to his acceleration vector. As a result, he would measure an additional increase in the slowing of the Earth clock tick rate.

When he changes his acceleration in order to decrease his relative velocity and then accelerate back towards the Earth, the magnitude of the acceleration remains unchanged. However, the Earth's position relative to the acceleration vector does change.  Now it is in the direction of the acceleration, and thus according to our traveler, the Earth clock runs fast.

This is how the equivalence principle would be applied in this case. One thing to note is that this "equivalent gravity" due to acceleration is only measured by our accelerating observer. Our Earth observer would only measure time dilation due to the difference in relative velocities.  (Both observers would also measure any difference due to relative positions in Earth's gravity field, but depending on the exact scenario, this can be insignificant.  If you are using scenarios of high fractions of c over light year distances,  this additional factor will likely be smaller than the rounding errors in your calculations. )

 

 

 

[Rob McEachern]

11 minutes ago, Janus said:

As swansont said, gravitational time dilation is due to the difference in gravitational potential

See my response to swansont.

[Strange]

6 minutes ago, Rob McEachern said:

See my response to swansont.

Janus's post explains why your response was wrong ...

[Janus]

19 minutes ago, Rob McEachern said:

A difference in gravitational potential, is a difference in acceleration (the acceleration due to gravity) - that was my point and that is the equivalence principle. But I'm sure you already know this. What causes the difference in acceleration, gravity or engine-thrust, is irrelevant to the effect.

No, it isn't.

To put numbers to my earlier example:

Acceleration due to gravity is GM/r²

Gravitational potential is -GM/r

For the Earth, the difference in acceleration at the surface and 1 earth radius above the surface is 7.35 m/s²

The difference in potential is 31255879.6  joules/kg

For our 4x Earth mass planet, the difference in acceleration between surface and the same altitude above the surface is 5.45 m/s²

The difference is potential is 41674506.2 joules/kg

This is 33% larger than that for the same altitude difference for the Earth, while the difference in acceleration is only 74% as much.

You can even have a difference in potential without any difference in acceleration. In a uniform gravity field, the difference in gravitational potential would be found by gh, where g is the acceleration due to gravity throughout the whole field and h is the height difference between them. In this case, there is no difference in acceleration over the region being measured (gravitational potential over small height differences approach this ideal, as g changes insignificantly over the region considered.)

 

 

[swansont]

46 minutes ago, Rob McEachern said:

A difference in gravitational potential, is a difference in acceleration (the acceleration due to gravity) - that was my point and that is the equivalence principle. But I'm sure you already know this. What causes the difference in acceleration, gravity or engine-thrust, is irrelevant to the effect.

No, it's not. Even if g is constant, i.e. there is no difference in acceleration, a different vertical position will be a different potential, and cause a frequency change in a clock.

 

(edit: xpost with Janus)

[Rob McEachern]

2 minutes ago, swansont said:

a different vertical position will be a different potential

Obviously. Please read what I said.  My original comment stated that the space-traveler experiences 1g, total, not 1g from gravity plus 1g from thrust or anything else. The whole point is to arrange for the traveler to experience the EXACT same experience as the non-traveler. That cannot be done unless the thrust is continually adjusted to compensate for the change in the gravitational potential, due to increasing distance from the earth.

[swansont]

3 minutes ago, Rob McEachern said:

Obviously. Please read what I said.  My original comment stated that the space-traveler experiences 1g, total, not 1g from gravity plus 1g from thrust or anything else. The whole point is to arrange for the traveler to experience the EXACT same experience as the non-traveler. That cannot be done unless the thrust is continually adjusted to compensate for the change in the gravitational potential, due to increasing distance from the earth.

You made an incorrect comment about a Hafele-Keating type experiment; I made no remarks about the other scenario. To me it's irrelevant.

You say "obviously" but that contradicts your earlier claims. Why not just acknowledge (and learn from, if need be) the error, instead of trying to defend or whitewash it?

[Rob McEachern]

2 minutes ago, swansont said:

You made an incorrect comment about a Hafele-Keating type experiment

No. You made an incorrect interpretation of my comment. You seem to have interpreted my use of the word "acceleration" as implying only to a change in velocity (or speed?), rather the the more inclusive use of the term to imply the existence of a force (such as a gravitational potential). You seem to have assumed that I believed that the effect is solely due to relative motion. But I said no such thing.

[swansont]

4 hours ago, Rob McEachern said:

No. You made an incorrect interpretation of my comment. You seem to have interpreted my use of the word "acceleration" as implying only to a change in velocity (or speed?), rather the the more inclusive use of the term to imply the existence of a force (such as a gravitational potential).

You said the time difference was due to the difference in acceleration ("the two clocks experience different accelerations and thus display differing elapsed times"), and that is simply not true. The clocks would give a time difference even if the accelerations were the same.

A gravitational potential is not a force.

Quote

You seem to have assumed that I believed that the effect is solely due to relative motion. But I said no such thing.

I made no such determination. I simply took you at your word and corrected a mistake.

[studiot]

Back to the OP

 

First off TakenItSeriously (please get a shorter handle) I am going to say +1 for encouragement.

On 12/03/2018 at 12:03 AM, TakenItSeriously said:

BTW, this is a good example of my previous statements that while logic may be prone to intuitive errors of false premise, such as the Earth is the center of the Universe because everything appears to revolve around the Earth, math is just as prone to intuitive errors of false conclusions.

Iam am impressed by the reasoning of your case, this is best chain of reasoning I have seen you present.

But you should beware avoiding mathematics because the best of reasoning is useless if founded on shaky premises.

It is possible to reach the wrong conclusion from them or it is possible for two (or heaven forbid more) errors to 'cancel out', thus reaching the right conclusion for the wrong reasons.

 

Looking at your statements of symmetry, the difficulty is that the symmetry of the Physics relies on a common variable.

That is the symmetry is in the the same variable (one variable) in both aspects.

The common variable in this case is the relative velocity.

You have taken time from one twin's frame and compared it with time from the other twin's frame.

So you are comparing two different situations.

The actual symmetry works like this:

 

Twin A sees twin B receeding at 0.8c

Twin B sees twin A receeding at 0.8c

 

You have, however correctly identified that what happens to the rest of the universe is the basis of the logical resolution of the paradox.

 

Note that the travelling twin (B) has no means of measuring the distance to his destination, once he has set off.

 

 

 

Edited March 14, 2018 by studiot

[Rob McEachern]

46 minutes ago, swansont said:

A gravitational potential is not a force.

I never said it was. I said "imply the existence of a force". Do you really wish to maintain the a gravitational potential does not imply the existence of a gravitational force, and that that in turn implies that objects at different heights from the center of gravity would experience different accelerations, if not preventing from falling by some other force?

[Janus]

1 hour ago, Rob McEachern said:

No. You made an incorrect interpretation of my comment. You seem to have interpreted my use of the word "acceleration" as implying only to a change in velocity (or speed?), rather the the more inclusive use of the term to imply the existence of a force (such as a gravitational potential). You seem to have assumed that I believed that the effect is solely due to relative motion. But I said no such thing.

5 minutes ago, Rob McEachern said:

I never said it was. I said "imply the existence of a force". Do you really wish to maintain the a gravitational potential does not imply the existence of a gravitational force, and that that in turn implies that objects at different heights from the center of gravity would experience different accelerations, if not preventing from falling by some other force?

 

The use of acceleration due to gravity here has nothing to do with relative motion, it is a more universal expression for comparing the force of gravity, since it does not rely on the mass of the object undergoing the acceleration.

As far as your next post goes:

The first part holds in that a gravitational potential does imply a gravitational force,  however, this does not in turn imply that objects at different heights must experience different accelerations for there to be a difference in potential.   The difference in potential is the integral of the force over a distance and does not require that the force differs over the distance.

 

2 hours ago, Rob McEachern said:

Obviously. Please read what I said.  My original comment stated that the space-traveler experiences 1g, total, not 1g from gravity plus 1g from thrust or anything else. The whole point is to arrange for the traveler to experience the EXACT same experience as the non-traveler. That cannot be done unless the thrust is continually adjusted to compensate for the change in the gravitational potential, due to increasing distance from the earth.

In this case, the space traveler would have to combine two separate calculations to determine how fast the Earth clock ticks relative to his own at any given instant. One that factors in his position relative to the Earth's gravitational field, and another to determine its relative position in the "acceleration field".    The main difference being that the Earth's gravitational field strength decreases with distance from the Earth, and the strength of the "acceleration field does not.   This would be in addition to any time dilation due to relative velocity differences. (if you choose the instant in which the ship has just stopped its velocity away from the Earth and is just going to start back to Earth, At that moment it will be at rest with to the Earth, and you will only need to use the first two calculations.)

[swansont]

2 hours ago, Rob McEachern said:

Do you really wish to maintain the a gravitational potential does not imply the existence of a gravitational force, and that that in turn implies that objects at different heights from the center of gravity would experience different accelerations, if not preventing from falling by some other force?

Why would I wish to maintain that? (Don't bother answering, this is rhetorical; I never claimed it in the first place, so there's no way to maintain it)

[Strange]

2 hours ago, Rob McEachern said:

Do you really wish to maintain the a gravitational potential does not imply the existence of a gravitational force

The problem is that you appeared to equate gravitational force (or acceleration) with potential. Time dilation depends on the latter not the former. 

[TakenItSeriously]

On 3/14/2018 at 3:51 PM, studiot said:

Back to the OP

 

First off TakenItSeriously (please get a shorter handle) I am going to say +1 for encouragement.

Iam am impressed by the reasoning of your case, this is best chain of reasoning I have seen you present.

But you should beware avoiding mathematics because the best of reasoning is useless if founded on shaky premises.

It is possible to reach the wrong conclusion from them or it is possible for two (or heaven forbid more) errors to 'cancel out', thus reaching the right conclusion for the wrong reasons.

 

Looking at your statements of symmetry, the difficulty is that the symmetry of the Physics relies on a common variable.

That is the symmetry is in the the same variable (one variable) in both aspects.

The common variable in this case is the relative velocity.

You have taken time from one twin's frame and compared it with time from the other twin's frame.

So you are comparing two different situations.

The actual symmetry works like this:

 

Twin A sees twin B receeding at 0.8c

Twin B sees twin A receeding at 0.8c

 

You have, however correctly identified that what happens to the rest of the universe is the basis of the logical resolution of the paradox.

 

Note that the travelling twin (B) has no means of measuring the distance to his destination, once he has set off.

 

 

 

Thanks, I really appreciate that.

To answer your point, this is why I think that logical models and mathematical models are complimentary to each other. Where one is prone to intuitive error the other is not and vice-versa. Therefore you could say that math and logic are cross validating because they always follow different vectors of reasoning.

edit to add: BTW, sorry about the unwieldy username. It’s based on a bit of word play. On other forums such as twoplustwo.com which is fundamentally a poker and gaming forum, I go by TakenItEasy. Lots of people there called me Taken for short.

Edited March 20, 2018 by TakenItSeriously

[Bender]

On 14/3/2018 at 9:18 PM, Rob McEachern said:

Do you really wish to maintain the a gravitational potential does not imply the existence of a gravitational force,

Actually, if in a region the gravitational potential is constant, there is indeed no force. Force is the (negative of) derivative/gradient of potential.