The Twin Paradox and the Equivalence Principle

[haroldholt]

I'm having a little trouble understanding the equivalence principle explanation of the twin paradox.

I understand that the resolution to the paradox according to the equivalence principle is that the non-traveling twin has a higher gravitational potential energy in the pseudo-gravitational field created when treating the traveling twin's reference frame as an inertial one, but I'm not sure that I understand why this is so.

Doesn't gravitational potential energy at a particular point depend on the strength of the gravitational field at that point? If the twins have different gravitational potential energies, does this mean that the gravitational field is a different strength for each of them? And if this is the case, where is the pseudo-gravitational field at its strongest? For a massive body, the gravitational field is the strongest at the centre of the body. But being a flat pseudo-gravitational field in the case of the twin paradox, there obviously is no centre.

 

[Simon Bridge]

The resolution of the paradox is that one twin is not in an inertial frame - so there is an asymmetry between them that has the traveling twin end up younger.

An accelerating frame is indistinguishable from gravity.
If you treat the traveler's frame as inertial, then you get the appearance of a gravitational field much as you get a centrifugal force when you go around a corner. This is not a "real" gravitational field like you are used to thinking about. Its there to make the math come out in agreement with observation and the assumption of which frame is inertial.

Imagine you are in a closed room that is lit by a bulb hanging by a cord from the ceiling. You observe that the bulb is hanging at an angle so you conclude that there must be some sort of force deflecting the bulb right?

This is provided the reference frame of the room is inertial ... the room could be accelerating.

 

[haroldholt]

The resolution of the paradox is that one twin is not in an inertial frame - so there is an asymmetry between them that has the traveling twin end up younger.

An accelerating frame is indistinguishable from gravity.
If you treat the traveler's frame as inertial, then you get the appearance of a gravitational field much as you get a centrifugal force when you go around a corner. This is not a "real" gravitational field like you are used to thinking about. Its there to make the math come out in agreement with observation and the assumption of which frame is inertial.

Imagine you are in a closed room that is lit by a bulb hanging by a cord from the ceiling. You observe that the bulb is hanging at an angle so you conclude that there must be some sort of force deflecting the bulb right?

This is provided the reference frame of the room is inertial ... the room could be accelerating.

I could probably simplify my question by asking the following: if the traveling observer has a lower gravitational potential energy and is therefore further down the "gravitational well", then where is the bottom of the well?

Taking the gravitational field of the earth, for example, the bottom of the gravitational well is the point at the centre of the mass.

Or is my question irrelevant since the gravitational field being created in the case of the twin paradox isn't real?

 

[Mentz114]

I could probably simplify my question by asking the following: if the traveling observer has a lower gravitational potential energy and is therefore further down the "gravitational well", then where is the bottom of the well?

Taking the gravitational field of the earth, for example, the bottom of the gravitational well is the point at the centre of the mass.

Or is my question irrelevant since the gravitational field being created in the case of the twin paradox isn't real?

The twin scenario in flat spacetime can be resolved without gravity being introduced. Are you talking about twins in the Earth's field or in flat spacetime ?

 

[HallsofIvy]

The equivalence principle says that acceleration is locally the same as an inertial frame with gravity. Since that is local the questions "where is the center" or "where is the bottom of the gravity well" do not arise.

 

[haroldholt]

The equivalence principle says that acceleration is locally the same as an inertial frame with gravity. Since that is local the questions "where is the center" or "where is the bottom of the gravity well" do not arise.

If the equivalence principle only applies locally, how do we conclude that the non-traveling twin has a higher gravitational potential energy?

 

[harrylin]

I could probably simplify my question by asking the following: if the traveling observer has a lower gravitational potential energy and is therefore further down the "gravitational well", then where is the bottom of the well?

Taking the gravitational field of the earth, for example, the bottom of the gravitational well is the point at the centre of the mass.

Or is my question irrelevant since the gravitational field being created in the case of the twin paradox isn't real?

A lower gravitational potential just corresponds to a lower height; gravitational potential energy at a particular point does not depend on the strength of the gravitational field at that point. But the equivalence principle explanation (at least the strong one, pretending a "real" field) isn't widely accepted.

- Original (and "strong") equivalence principle explanation:
https://en.wikisource.org/wiki/Dialog_about_Objections_against_the_Theory_of_Relativity

- Physics FAQ commentary:
http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_gr.html

Harald

 

[haroldholt]

A lower gravitational potential just corresponds to a lower height; gravitational potential energy at a particular point does not depend on the strength of the gravitational field at that point. But the equivalence principle explanation (at least the strong one, pretending a "real" field) isn't widely accepted.

- Original (and "strong") equivalence principle explanation:
https://en.wikisource.org/wiki/Dialog_about_Objections_against_the_Theory_of_Relativity

- Physics FAQ commentary:
http://math.ucr.edu/home/baez/physics/Relativity/SR/TwinParadox/twin_gr.html

Harald

Which explanation, from the point of view of the traveling twin, is the most widely accepted?

 

[Simon Bridge]

Just a side note:

if the traveling observer has a lower gravitational potential energy and is therefore further down the "gravitational well", then where is the bottom of the well?

You are aware that the choice of a zero potential is arbitrary?
If you put zero at infinity - then all potentials are negative and some more than others. OPs question then becomes - where is the smallest potential to be found?

One would expect to find this at the center of mass for the Universe right?
But the Universe does not have a center...

But this sort of question does not depend on the equivalence principle.

The starting point to understanding this would be that the infinite distance = zero potential is a convention only - not actually real. The question is actually about the structure of space-time.

Assuming that's where he was headed.

 

[Mentz114]

Which explanation, from the point of view of the traveling twin, is the most widely accepted?

The Einstein argument is rather dated and the problem is best analysed ( as the Baez page suggests) using the proper length argument

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

 

[harrylin]

Which explanation, from the point of view of the traveling twin, is the most widely accepted?

I think that the special relativistic explanation is the most widely accepted - thus perhaps not really "from the point of view of the traveling twin".

 

[PeterDonis]

I understand that the resolution to the paradox according to the equivalence principle is that the non-traveling twin has a higher gravitational potential energy in the pseudo-gravitational field created when treating the traveling twin's reference frame as an inertial one, but I'm not sure that I understand why this is so.

Check out the Usenet Physics FAQ entry on the Twin Paradox:

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

In particular, the "Equivalence Principle Analysis" page:

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

 

[PeterDonis]

But being a flat pseudo-gravitational field in the case of the twin paradox, there obviously is no centre.

There is no center, but there is still a definite "up-down" direction, defined by the direction of the "force" you have to exert to stay stationary in the field. (Which is ultimately derived from the direction the traveling twin's rockets push him when he turns around.) The traveling twin is "below" the stay-at-home twin when he turns around because the direction from the traveling twin to the stay-at-home twin is "up".

 

[haroldholt]

The Einstein argument is rather dated and the problem is best analysed ( as the Baez page suggests) using the proper length argument

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

When you say that the Einstein argument is rather dated, do you mean that the idea of treating the traveling twin's frame of reference as an inertial one is dated in general?

 

[stevendaryl]

The Einstein argument is rather dated and the problem is best analysed ( as the Baez page suggests) using the proper length argument

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

I've always been slightly annoyed by the "gravitational time dilation" explanation for the twin paradox for two reasons.

First, it's kind of circular. You say that the accelerating twin is equivalent to a stationary twin in a gravitational field, and then you apply the formula for gravitational time dilation. But how do you know that clocks in a gravitational field experience gravitational time dilation? By transforming the problem to an equivalent case involving acceleration in flat spacetime, and using SR. You can skip the "gravitation" step and just use SR, plus calculus to transform to a noninertial coordinate system, if that's more convenient.

Second, it has bizarre consequences. Let me describe a variant of the twin paradox: There are two planets many light-years apart. Two twins were separated at birth and one went to live on one planet, and the other went to live on the other planet. The two planets are at rest relative to each other, and the twins are the same age, according to the reference frame of the planets. When one twin reaches age 20, he accelerates to nearly the speed of light. and travels to the other twin. When he arrives, he has only 21 years old, but the other twin is 40 years old. The traveling twin explains this using gravitational time dilation: during the time of acceleration, the distant twin aged rapidly, gaining at least 20 years during the brief time of acceleration.

What's wrong with that explanation? Well, imagine that traveling twin, after accelerating, changes his mind; he decelerates and returns to his own planet. If the whole process of accelerating and decelerating was quick (much less than one year) then the twins will still be approximately the same age. But what that means is that the distant twin gained 20 years during acceleration, and then lost 20 years during deceleration. Gravitational time dilation can work to youthen a distant twin as well as age him prematurely.

 

[Dale]

imagine that traveling twin, after accelerating, changes his mind; he decelerates and returns to his own planet. If the whole process of accelerating and decelerating was quick (much less than one year) then the twins will still be approximately the same age. But what that means is that the distant twin gained 20 years during acceleration, and then lost 20 years during deceleration. Gravitational time dilation can work to youthen a distant twin as well as age him prematurely.

The problem mentioned here isn't a problem of the gravitational explanation, but one of poorly specified non inertial coordinate systems. There is no one unique meaning to "the reference frame" of a non inertial observer. The most naive frame that you describe simply doesn't cover regions of spacetime sufficiently "low" for time to run backwards. You cannot use a chart to make claims about a region of spacetime that it doesn't cover. In any valid coordinate chart which does cover the other twin, his age will never go backwards.

 

[lalbatros]

... The resolution of the paradox is that one twin is not in an inertial frame - so there is an asymmetry between them that has the traveling twin end up younger. ...

It is possible to script another scenario where the "second twin" doesn't return home and just sends "somebody else" back and in such a way that there is no acceleration anywhere in the scenario.

In this way, the geometry appears to contain the full effect.

 

[stevendaryl]

The problem mentioned here isn't a problem of the gravitational explanation, but one of poorly specified non inertial coordinate systems. There is no one unique meaning to "the reference frame" of a non inertial observer. The most naive frame that you describe simply doesn't cover regions of spacetime sufficiently "low" for time to run backwards. You cannot use a chart to make claims about a region of spacetime that it doesn't cover. In any valid coordinate chart which does cover the other twin, his age will never go backwards.

I would think that the point of the "gravitational time dilation explanation" is to explain the differential aging from the point of view of the traveling twin. So what you're saying now is that if the traveling twin has too complicated a path, then you can't explain it from the point of view of the traveling twin. For sufficiently complicated situations, there is no "coordinate system of the traveling twin" from which can "explain" the differential ages.

As you say, you can use several different coordinate charts for different parts of the trip. But then the explanation "The inertial twin is older because of gravitational time dilation" really doesn't make sense. Once you have multiple charts, you're abandoning the idea of looking at things "from the point of view of the traveling twin".

 

[stevendaryl]

The problem mentioned here isn't a problem of the gravitational explanation, but one of poorly specified non inertial coordinate systems.

My other point stands. The gravitational explanation isn't an explanation at all,
it's a calculation. If in inertial coordinates, the invariant interval is given by:

ds² = dt² - dx²/c²

and you change coordinates to X,T, then it follows that in terms of X and T

ds² =
(A² - C²/c²) dT²
+ 2(AB - CD/c²) dX dT
+ (B² - D²/c²) dX²

where A = ∂t/∂T,
B = ∂t/∂X
C = ∂x/∂T
D = ∂x/∂X

This is just calculus, there is no new insight about gravitation or the way it affects clocks required.

 

[A.T.]

I've always been slightly annoyed by the "gravitational time dilation" explanation for the twin paradox for two reasons.
First, it's kind of circular. You say that the accelerating twin is equivalent to a stationary twin in a gravitational field, and then you apply the formula for gravitational time dilation. But how do you know that clocks in a gravitational field experience gravitational time dilation? By transforming the problem to an equivalent case involving acceleration in flat spacetime, and using SR. You can skip the "gravitation" step and just use SR, plus calculus to transform to a noninertial coordinate system, if that's more convenient.

So, you don't like invoking the word "gravity" and the EP, but instead stick to the SR non-inertial frame as such, where clocks run at different rates too.

Second, it has bizarre consequences. Let me describe a variant of the twin paradox: There are two planets many light-years apart. Two twins were separated at birth and one went to live on one planet, and the other went to live on the other planet. The two planets are at rest relative to each other, and the twins are the same age, according to the reference frame of the planets. When one twin reaches age 20, he accelerates to nearly the speed of light. and travels to the other twin. When he arrives, he has only 21 years old, but the other twin is 40 years old. The traveling twin explains this using gravitational time dilation: during the time of acceleration, the distant twin aged rapidly, gaining at least 20 years during the brief time of acceleration.

What's wrong with that explanation? Well, imagine that traveling twin, after accelerating, changes his mind; he decelerates and returns to his own planet. If the whole process of accelerating and decelerating was quick (much less than one year) then the twins will still be approximately the same age. But what that means is that the distant twin gained 20 years during acceleration, and then lost 20 years during deceleration. Gravitational time dilation can work to youthen a distant twin as well as age him prematurely.

Yes, but don't this bizarre consequences appear in the non-inertial frame of the traveling twin in any case, even if you skip that gravity part, as you suggest above?

 

[stevendaryl]

So, you don't like invoking the word "gravity" and the EP, but instead stick to the SR non-inertial frame as such, where clocks run at different rates too.

Yes, but don't this bizarre consequences appear in the non-inertial frame of the traveling twin in any case, even if you skip that gravity part, as you suggest above?

If you use noninertial coordinates, as Dale says, there is no such thing as the coordinate system of the accelerated observer. Instead, you have several "coordinate charts", and you have to piece together several of them to compute elapsed time for a clock. One of the criteria for being a legitimate coordinate chart is that the same spacetime point cannot have two different coordinates.

It's actually like trying to compute the length of a path on the surface of the Earth using maps. If you look at a flat map, distances are completely distorted near the North Pole and South Pole (the North Pole itself is stretched out into a line, instead of being just a point). It's not very good for computing distances. If you look at maps for smaller regions, say 100 km x 100 km, then a flat map gives a good approximation to distances. So a way to compute the length of a very long route over the Earth is to start with a 100x100 map of the starting point. Compute the length up until the route passes out of the region. Then get another map, and figure out which point on the second map corresponds to the exit point on the first map. Keep adding up the distances until the route passes outside of the region for the second map. Then get a third map, etc.

So you can compute the total distance as follows:
Let P₁ be the starting point. Find some map that covers P₁.
Let P₂ be the last point on the first map. Pick a second map that covers P₂. Let P₃ be the last point on the second map. Etc.

Let D_n be the distance between P_n and P_n+1, as measured using the nth map. Then D = sum over n of D_n.

The same sort of thing works to find elapsed time for a complicated path through spacetime.

 

[TSny]

A nice explanation of the twin paradox using the GR point of view is in the old book "Relativity, Thermodynamics, and Cosmology" by Tolman.

If you go to Google Books at books.google.com/?hl=EN and search "Tolman Cosmolgy" you will find the book. The explanation starts on page 194.

 

[Dale]

For sufficiently complicated situations, there is no "coordinate system of the traveling twin" from which can "explain" the differential ages.

No, I said there was no unique coordinate system. Meaning there are a variety of ways of constructing a non inertial coordinate system, and you have to specify which one you are using in a lot of detail.

Any valid coordinate chart you choose can "explain" it correctly, but the explanation will differ. In none of them will the other twin age backwards.

 

[TSny]

Here’s a link to a derivation of the rate of an arbitrarily accelerating clock as observed by an arbitrarily accelerating observer assuming that the clock and observer are always moving along the same line.

//sdrv.ms/JPxY5H

The derivation is tedious, but the final result (equation 18) is quite simple. The rate of ticking of the clock according to the observer is just a product of a time dilation factor due to relative speed (u) and a factor due to the acceleration felt by the observer (g). (How u and g are defined for the accelerating clock and observer is discussed in the derivation.)

The factor due to the acceleration is interesting, it has the appearance of “gravitational time dilation” due to a uniform gravitational field of strength g.

 

[Underwood]

If you use noninertial coordinates, as Dale says, there is no such thing as the coordinate system of the accelerated observer.

That doesn't sound right. Shouldn't anyone be able to figure out how old the home twin is?

 

[stevendaryl]

That doesn't sound right. Shouldn't anyone be able to figure out how old the home twin is?

There are two different, but related questions about the age of a distant twin: (1) How old is that twin right now? (2) How old is that twin when such and such event happens? (where the event might be specified by: the two twins reunite).

To be able to talk about how old a twin is "right now" requires a choice about a time coordinate that covers both twins. For an inertial frame, there is a unique "best" time coordinate, but for an accelerated twin, there is no unique choice of a time coordinate.

 

[stevendaryl]

No, I said there was no unique coordinate system. Meaning there are a variety of ways of constructing a non inertial coordinate system, and you have to specify which one you are using in a lot of detail.

But the fact that there are a variety of ways to construct a noninertial coordinate system means to me that none of them is really telling things "from the point of view of the accelerated twin". The Rindler coordinates come close, but they cannot be used for a twin who "turns around" and starts accelerating in the opposite direction. Once you start using coordinate charts, you're really abandoning the idea of understanding things from any observer's point of view. The charts aren't necessarily from anyone's "point of view".

 

[Underwood]

To be able to talk about how old a twin is "right now" requires a choice about a time coordinate that covers both twins. For an inertial frame, there is a unique "best" time coordinate, but for an accelerated twin, there is no unique choice of a time coordinate.

If I was on a rocket trip, I would know that my sister back home always had some age at any time on my trip. If the rocket driver told me that she didn't have some age right then, I wouldn't believe him. I wouldn't believe anybody who told me that.

 

[Nugatory]

If I were on a rocket trip, I would know that my sister back home always had some age at any time on my trip. If the rocket driver told me that she didn't have some age right then, I wouldn't believe him. I wouldn't believe anybody who told me that.

A wise undergraduate physics student once said that the way to resolve all paradoxes of special relativity is to look for the evil and treacherous phrase "at the same time"... Sometimes it's well-hidden, but if you keep looking... Here that evil and treacherous concept is hiding in the text that I've bolded above.

Yes, of course your sister always has some definite age, just as her wristwatch will always record some definite time since you and she parted. However, different observers on different rocket ships moving at different speeds will have different ideas about what that age is; and that's not especially remarkable because they all have different ideas of what's "right then". They're checking her watch and her age at different times so of course they get different answers.

None of this has anything to do with what your sister experiences. She's happily living her life back at home, celebrating a birthday once a year and keeping track of her age in her home reference frame, unconcerned that you and the other rocket observers are seeing time pass at different rates.

 

[cosmik debris]

It is possible to script another scenario where the "second twin" doesn't return home and just sends "somebody else" back and in such a way that there is no acceleration anywhere in the scenario.

In this way, the geometry appears to contain the full effect.

Yes, an incoming triplet can transfer the time on the outgoing triplet's clock to hir own and carry that back to the stay at home triplet. No acceleration is necessary for the effect.