Twin Paradox (by acceleration)

[starthaus]

In that case, perhaps two additional ones may also be helpful:
http://xxx.lanl.gov/abs/physics/9810017 [Am.J.Phys. 67 (1999) 1007]
http://xxx.lanl.gov/abs/gr-qc/9909035

Thank you :-)

 

[Jonnyb42]

But that is really easy. It is only Newtonian mechanics, no Einstein relativity is involved. From their point of view, one is rotating around the other (e.g. Moon around Earth or vice versa). They miss each other because the direction of force does not need to coincide with the direction of velocity. I do not see what is your problem with that.

Yeah but, the coordinate system you are talking about does not rotate with the particles. The frame is centered around the center and both particles lie on the same axis, the x-axis for example. In that system, what explains the particles not colliding? I know you might say it is because the frame of reference is non-inertial, but what truly makes it non-inertial? Only a third observer would have to say that the reference frame is non-inertial.

 

[Demystifier]

I know you might say it is because the frame of reference is non-inertial, but what truly makes it non-inertial? Only a third observer would have to say that the reference frame is non-inertial.

That is not correct. Being in a non-inertial frame is an absolute statement. If you are in a non-inertial frame for one observer, then you are in a non-inertial frame for ANY observer. In particular, you can determine experimentally whether you are accelerating or not. Velocity is relative, but acceleration is absolute.

Now you will probably ask: But if I accelerate, then I accelerate with respect to what? In Newtonian mechanics it was not completely clear. For example, Mach thought that acceleration is defined with respect to (all) distant stars in the Universe. But in the relativity theory, this question can be answered: You accelerate with respect to the metric field (metric tensor).

 

[JesseM]

That is not correct. Being in a non-inertial frame is an absolute statement. If you are in a non-inertial frame for one observer, then you are in a non-inertial frame for ANY observer. In particular, you can determine experimentally whether you are accelerating or not. Velocity is relative, but acceleration is absolute.

To put it simply, inertial observers feel weightless, non-inertial observers feel G-forces (which can be measured with an accelerometer)

 

[Mike_Fontenot]

[...] It seems that the twin that ages the most, is the one that does the short final burst of acceleration to come to rest in the other twin's rest frame.

No, it's the opposite. The twin who does the accelerating, WHEN THEIR SEPARATION IS NON-ZERO, will be the younger, after they are again stationary with respect to one another.

The initial acceleration by B (when their separation is zero) has no effect: you can reformate the problem as two unrelated newborns, who are moving at a constant speed with respect to one another, and who happened to be momentarily adjacent at the instant they are both born, with neither one accelerating then.

As the two twins initially move apart (say, at a relative speed of 0.866c, to give a simple value of gamma = 2), they EACH will (correctly) conclude that the other is ageing more slowly (by a factor of 2). They are BOTH correct. Neither can adopt any other conclusion, without contradicting their own elementary measurements.

So, given the complete symmetry of this initial phase, you can actually consider B to be "the home twin", and A is "the traveler". If you do that, you can apply the CADO equation that I gave earlier, to get their two conclusions about their corresponding ages at various instants.

Suppose that the traveler (A) is 10 years old at the instant that he accelerates (call it point P), and they were both zero years old when they were co-located at birth.

The home twin (B) concludes that she is 20 years old when the traveler accelerates at point P, so she concludes that their separation is 20 (0.866) = 17.32 lightyears.

The CADO equation then says that CADO_T immediately before the acceleration is

CADO_T = CADO_H - L*v/(c*c)

or

CADO_T = 20 - (17.32)(0.866) = 20 - 15 = 5 years old.

This is B's age right before the acceleration by A, according to A. So, right before A accelerates, B says she is 20, but A says she is 5. (And they both agree that A is 10 then).

Immediately after the acceleration, the CADO equation says

CADO_T = 20 - (17.32)(0) = 20 years old.

So now, the twins AGREE on their relative ages (as they always will whenever their relative speed is zero). They both agree that when A is 10 (immediately AFTER his acceleration), B is 20.

And thereafter, they both agree that their rates of ageing are equal, and that B is always 10 years older than A.

 

[Al68]

Now you will probably ask: But if I accelerate, then I accelerate with respect to what? In Newtonian mechanics it was not completely clear. For example, Mach thought that acceleration is defined with respect to (all) distant stars in the Universe. But in the relativity theory, this question can be answered: You accelerate with respect to the metric field (metric tensor).

Isn't this equivalent to Mach's principle, if the metric field is determined by distant masses?

 

[Jonnyb42]

But in the relativity theory, this question can be answered: You accelerate with respect to the metric field (metric tensor).

WHAT!?

Why haven't I been told this before!?

Thank you very much Demistyfier, I really really need to learn general relativity.
I am still a bit confused, (because I have not learned about Mach's principle or general theory of relativity.) For instance, another question I have is, why isn't the metric field considered an 'absolute reference frame'?

 

[JesseM]

WHAT!?

Why haven't I been told this before!?

Thank you very much Demistyfier, I really really need to learn general relativity.
I am still a bit confused, (because I have not learned about Mach's principle or general theory of relativity.) For instance, another question I have is, why isn't the metric field considered an 'absolute reference frame'?

It's like an electromagnetic field, defined at every point in spacetime, but it doesn't have a rest frame (maybe Demystifier could clarify what it means to 'accelerate with respect to' something that has no rest frame?)

 

[Demystifier]

Isn't this equivalent to Mach's principle, if the metric field is determined by distant masses?

GR does not obey the Mach principle because the metric field is not determined ONLY by distant masses. Instead, when the matter is fixed, you still have a freedom to choose initial conditions for the metric field. In particular, even if there is no matter at all, there is still a metric field with respect to which acceleration is defined.

 

[Demystifier]

For instance, another question I have is, why isn't the metric field considered an 'absolute reference frame'?

Because the velocity is still relative. Only acceleration is absolute. In other words, you cannot talk about velocity of the metric field, but in a sense you can talk about acceleration of the metric field. This is related to the fact that metric field is not a first-rank tensor (i.e., a vector), but a second-rank tensor. This is just a rough, hopefully intuitive explanation, while a proper explanation requires more math.

 

[Demystifier]

It's like an electromagnetic field, defined at every point in spacetime, but it doesn't have a rest frame (maybe Demystifier could clarify what it means to 'accelerate with respect to' something that has no rest frame?)

Note that a photon moving with the velocity of light does have a rest frame, but that this frame is not a Lorentz frame. Instead, it is a light-cone frame.
Note also that the velocity of photon is related to the Poynting vector of the EM field. Finally, note that not all EM fields have a non-zero Poynting vectors.

 

[JesseM]

Note that a photon moving with the velocity of light does have a rest frame, but that this frame is not a Lorentz frame. Instead, it is a light-cone frame.
Note also that the velocity of photon is related to the Poynting vector of the EM field. Finally, note that not all EM fields have a non-zero Poynting vectors.

OK, but what about my question about how you define acceleration "relative to" a tensor field that has no rest frame of its own? I suppose by the equivalence principle, in the local neighborhood of any point in curved spacetime we can define locally inertial frames, so can we say that any worldline that passes through that point must have either zero or nonzero instantaneous acceleration relative to these local inertial frames?

 

[Al68]

Isn't this equivalent to Mach's principle, if the metric field is determined by distant masses?

GR does not obey the Mach principle because the metric field is not determined ONLY by distant masses. Instead, when the matter is fixed, you still have a freedom to choose initial conditions for the metric field. In particular, even if there is no matter at all, there is still a metric field with respect to which acceleration is defined.

Fair enough, but to my knowledge, Mach's Principle was never specific enough to say that GR doesn't "obey" it.

GR certainly seems very "Machian" to me, at least compared to Newtonian physics, even if it doesn't actually explain the source of inertia.

 

[yuiop]

The slightly surpising result, if I have worked it out correctly (and I might not of, because I am a bit tired) is that A (who has a spent a year lazing around while B hurtles off at 0.99c relative to him) has aged less than B. It seems that the twin that ages the most, is the one that does the short final burst of acceleration to come to rest in the other twin's rest frame.

No, it's the opposite. The twin who does the accelerating, WHEN THEIR SEPARATION IS NON-ZERO, will be the younger, after they are again stationary with respect to one another.

Yep, your right. I made a typo in the final sentence. Thanks for picking it up and have now edited the original post. The second sentence obviously contradicts the sentence immediately preceding it. Knew I shouldn't have been posting stuff when I was tired. Good to see someone is paying attention

 

[Demystifier]

Fair enough, but to my knowledge, Mach's Principle was never specific enough to say that GR doesn't "obey" it.

GR certainly seems very "Machian" to me, at least compared to Newtonian physics, even if it doesn't actually explain the source of inertia.

Fair enough. But there are even theories of gravity which are more Machian than GR. The Brans-Dicke theory for example.

 

[Demystifier]

OK, but what about my question about how you define acceleration "relative to" a tensor field that has no rest frame of its own? I suppose by the equivalence principle, in the local neighborhood of any point in curved spacetime we can define locally inertial frames, so can we say that any worldline that passes through that point must have either zero or nonzero instantaneous acceleration relative to these local inertial frames?

Yes.

 

[Jonnyb42]

This is related to the fact that metric field is not a first-rank tensor (i.e., a vector), but a second-rank tensor. This is just a rough, hopefully intuitive explanation, while a proper explanation requires more math.

Daaang. Just guessing, but would an acceleration be relative in a third-rank tensor??
I am ashamed to not know what tensors are and what these things mean. I want to know enough math. I have to finish up some projects for the end of the year (I am a senior right now.) and when summer starts I am going to study vector calculus and general relativity like mad.

Thanks for all the help.
This stuff is crazy.

 

[Demystifier]

I am ashamed to not know what tensors are and what these things mean. I want to know enough math. I have to finish up some projects for the end of the year (I am a senior right now.) and when summer starts I am going to study vector calculus and general relativity like mad.

Thanks for all the help.
This stuff is crazy.

There are many excellent textbooks on this stuff, but if you need a recommendation, I suggest
http://xxx.lanl.gov/abs/gr-qc/9712019

 

[Jonnyb42]

Thank you very much, I will definitely read that, whenever I do!

I have another question that comes to my mind,

Is the theory of General Relativity being expanded on/updated, or has it been the same since Einstein published it?
I wonder the same with quantum mechanics.

 

[Fredrik]

GR hasn't really been modified since 1915, but people are still working on ways to find new interesting solutions to the main equation of the theory. That equation (Einstein's equation) describes the relationship between how things are distributed in the universe and how things move. It's so hard to solve that it took 48 years to find the solution that describes a universe that's completely empty except for a single rotating star.

QM hasn't really changed since the early 1930's, but a lot of work has been done on theories of matter and interactions in the framework of QM (mostly quantum field theories), and people are still working on them.

The big thing right now is of course to figure out how these two fit together.

 

[Aaron_Shaw]

and who happened to be momentarily adjacent at the instant they are both born,

Does this make sense as they are moving relative to each other? I don't think you can say this and then continue to calculate with accelerating frames assuming they started with a certain simultaneity.

 

[JesseM]

Does this make sense as they are moving relative to each other? I don't think you can say this and then continue to calculate with accelerating frames assuming they started with a certain simultaneity.

Why would it make a difference? Any physical situation can be analyzed using any frame you like, a frame is just a coordinate system for analyzing events in spacetime. And all frames agree about local events, so if it's true in one frame that their mothers were passing right next to each other at the moment each baby was born, then this is true in all frames.