Using General Relativity to analyze the twin paradox

[Dale]

Builder refers to several follow-ups in the literature that corroborate Einstein's calculation.

I don't know how it is possible to corroborate something that doesn't exist.

 

[Jimster41]

How is the mass/energy component of the GR description of the "virtual field", accounted for in the SR flat s.t. non-inertial coordinates description? I got locked onto "proper acceleration" and the idea that in the flat SR description, the connection term had to step-wise distribute/account for the same amount of mass/energy as the source of a field in which twins were seen to be stationary, as it "became curved" - that was the part I thought was intriguing about looking at the paradox from that latter perspective. Playing with the velocity of two masses in flat s.t over some set of steps has to account for the same mass energy of two stationary objects in a field that is curved due to mass energy. Take that black hole (or whatever mass) away, s.t goes flat, now you put that mass back piece by piece, but in a noticeably different "form" by creating a path length difference between the twins.

Thanks for the earlier corrections by the way. Focusing on path length helped. If it sounds like I'm trying to contradict stevendaryls or Ben niehoff's recent posts then I'm not getting my question across. I buy the argument in post 139.

Hopefully I'm at least back on topic, even if wrong. Sorry guys but I'm learning something here.

 

[harrylin]

I don't know how it is possible to corroborate something that doesn't exist.

It's certainly possible to verify if a calculation method that is claimed to work indeed works. I provided two references that verified that Einstein wasn't bluffing, for those who hold that Einstein's description is too vague to verify.

 

[harrylin]

[..] The question of "which twin ages slower" is determined by the proper acceleration. The traveling twin experiences proper acceleration (because he's on the rocket and is physically subject to the non-gravitational force applied by it). Proper acceleration causes length contraction and time dilation and that's why he ages slower. [..]

In addition to the other answers: it's not generally relevant what the person feels, what matters are velocity and gravitational potential. In the first space travel "twin example" by Langevin, the traveler didn't even feel any force at turnaround as he was traveling in a sling shot around a star.

 

[PeterDonis]

Neither Einstein, nor Moller, nor Builder brought them up in this context.

Einstein did bring them up, just not explicitly. As has been pointed out to you multiple times now, when Einstein uses the term "gravitational field", he means "Christoffel symbols". If you look at his actual technical papers (DaleSpam linked to a good one), this is obvious. He just didn't use the term "Christoffel symbols" (or the mathematical equivalent in the terminology of the time) in the non-technical article you linked to because it was for a general audience that might not understand what the mathematical term meant.

 

[PeterDonis]

for those who hold that Einstein's description is too vague to verify

Einstein's description in the non-technical article is too vague to verify. His description in technical articles (like the one DaleSpam linked to) is not. The paper you linked to has to make assumptions about what Einstein meant in the non-technical article in order to "verify" it.

(I don't think that paper is a very good source anyway; it seems to me to exhibit a number of confusions and to make some statements that are just false.)

 

[stevendaryl]

How is the mass/energy component of the GR description of the "virtual field", accounted for in the SR flat s.t. non-inertial coordinates description?

GR doesn't say that "gravitational fields" have mass/energy as the source. It says that the curvature tensor has mass/energy as the source. The "induced gravitational field" that results from switching to a noninertial reference frame has zero curvature, so it has no mass/energy associated with it.

 

[Jimster41]

GR doesn't say that "gravitational fields" have mass/energy as the source. It says that the curvature tensor has mass/energy as the source. The "induced gravitational field" that results from switching to a noninertial reference frame has zero curvature, so it has no mass/energy associated with it.

I take your first sentence to be an encouragement to be precise (per the wiki below), and I think I understand. The second sentence I'm still trying to get.
There is some energy required to curve the world line of the "traveling twin" through flat space-time?The Einstein field equations (EFE) or Einstein's equations are a set of 10 equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy.[6] First published by Einstein in 1915[7] as a tensor equation, the EFE equate local spacetime curvature (expressed by the Einstein tensor) with the local energy and momentum within that spacetime (expressed by the stress–energy tensor).[8]

The Einstein Field Equations can be written as

where

is the Einstein tensor and

is the stress–energy tensor.

This implies that the curvature of space (represented by the Einstein tensor) is directly connected to the presence of matter and energy (represented by the stress–energy tensor).

 

[PeterDonis]

There is some energy required to curve the world line of the traveling twin through flat space-time?

Curvature of a worldline is not the same as curvature of spacetime. You can have a curved worldline in flat spacetime, and a straight (geodesic) worldline in curved spacetime. They're two different concepts.

It is true that the traveling twin must have some means of curving his worldline (accelerating) in order to change course, such as a rocket, and this requires energy, and energy can in principle curve spacetime. However, the standard twin paradox scenario assumes that the energy required for the traveling twin to change course is too small to have any significant effect on the geometry of spacetime, so spacetime can be assumed to be flat. As a practical assumption, this works very well for all ordinary objects; to significantly curve spacetime, you need a very large object, like a planet or a star.

 

[Jimster41]

Curvature of a worldline is not the same as curvature of spacetime. You can have a curved worldline in flat spacetime, and a straight (geodesic) worldline in curved spacetime. They're two different concepts.

It is true that the traveling twin must have some means of curving his worldline (accelerating) in order to change course, such as a rocket, and this requires energy, and energy can in principle curve spacetime. However, the standard twin paradox scenario assumes that the energy required for the traveling twin to change course is too small to have any significant effect on the geometry of spacetime, so spacetime can be assumed to be flat. As a practical assumption, this works very well for all ordinary objects; to significantly curve spacetime, you need a very large object, like a planet or a star.

Thanks for seeing the part I'm confused about. The above definitely highlights it for me. So that's good.

I know you can't possibly teach me this here, so I'll let it go, and keep studying my Relativity textbooks (just now getting to curvature so... maybe all will be made clear for me soon) The part I'm missing is what physical thing is happening to the "traveling twin" in a single acceleration step, which is dependent on energy, by which time dilation and length contraction occur, but by which no other local physics is affected, that is not a local space-time "bending" moment... and to what is it happening?

Anyway, sincere thanks for your efforts.

 

[PeterDonis]

what physical thing is happening to the "traveling twin" in a single acceleration step

He's accelerating. That is, he is "changing direction" in spacetime, so his worldline is curved, not straight. More precisely, his worldline has two straight segments with a curved segment in between; the curved segment is where he is accelerating. The stay-at-home twin's worldline is straight the whole time.

If you mean, what causes him to accelerate, anything that causes him to feel a force will work: he can fire rockets, he can be pushed by a giant laser, he can turn on an electromagnet in his ship and get deflected by an external magnetic field, etc.

which is dependent on energy, by which time dilation and length contraction occur

Time dilation and length contraction don't depend on "energy". The stay-at-home twin never expends any energy--his worldline is straight the whole time--but he still is time dilated and length contracted relative to the traveling twin.

that is not a local space-time "bending" moment

As long as the energy the traveling twin needs to expend to curve his worldline is small enough, it will have negligible effect on the spacetime geometry. But SR doesn't explain why this is true; SR just assumes it (and assumes that we are only dealing with situations where all the energies are small enough). To explain why energies that are small enough don't affect the spacetime geometry, you need GR; the Einstein Field Equation, which tells you how much spacetime curvature is produced by a given amount of energy, is part of GR, not SR.

 

[Dale]

It's certainly possible to verify if a calculation method that is claimed to work indeed works. I provided two references that verified that Einstein wasn't bluffing, for those who hold that Einstein's description is too vague to verify.

They made a lot of assumptions in the German-to-Math translation. I am not saying that their assumptions are bad ones, but I still hold that his description is too vague to verify.

 

[Dale]

Neither Einstein, nor Moller, nor Builder brought them up in this context.

Clearly Einstein did. That is what he was referring to when he said "gravitational field". I showed in his technical paper where he stated that explicitly, but even just his pop-sci paper makes it clear from the way he describes the properties of the gravitational field.

there remain fictional terms in his description with magical effects.

With this I think it is time to close this thread. This now primarily about Einstein's word-choice and not about physics, and this type of language is deliberately unhelpful. Let's keep future threads to actual physics and not semantics.