Solving the Twin Paradox with General Relativity

[quangtu123]

I have read some basics knowledge about General Relativity and I see that it deal perfectly with gravity. But what about accelerated frames? Is there something similar to Lorentz Transformation for accelerated frame in General Relativity? (so that i can solve, maybe, the general twin paradox)

Thank you!

 

[WannabeNewton]

But what about accelerated frames? Is there something similar to Lorentz Transformation for accelerated frame in General Relativity? (so that i can solve, maybe, the general twin paradox)

You don't need Lorentz transformations to solve the generalized twin paradox but regardless, given an accelerating world-line $\gamma$ and another world-line (accelerating or freely falling) $\tilde{\gamma}$ and an event $p$ at which $\gamma,\tilde{\gamma}$ intersect, one can perform a Lorentz transformation from the instantaneous rest frame of $\gamma$ at $p$ to the instantaneous rest frame of $\tilde{\gamma}$ at $p$ in the exact same way one Lorentz boosts from one inertial frame to another in SR. In the context of accelerated frames and curved space-times these are called local Lorentz transformations. Local Lorentz transformations act on tetrad indices of frames transported along $\gamma$; one can also perform coordinate transformations of various kinds (e.g. non-rotating to rotating coordinates) and these act on coordinate indices.

Do you have something more specific in mind?

 

[Matterwave]

The twin paradox can be solved more simply, in my opinion, by simply integrating the differential proper time $d\tau_\gamma$ along each world line $\gamma$ and $\gamma'$. The result is simply:

$$\tau_\gamma = \int_\gamma \sqrt{g_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{d x^\nu}{d\tau}}d\tau$$

 

[pervect]

I have read some basics knowledge about General Relativity and I see that it deal perfectly with gravity. But what about accelerated frames? Is there something similar to Lorentz Transformation for accelerated frame in General Relativity? (so that i can solve, maybe, the general twin paradox)

Thank you!

Misner, Thorne, Wheeler, in "Gravitation", chapter six, go through and compute a coordinate system for an accelerated observer, which they occasionally call a frame (as in, for instance "Constraints on the Frame of an Accelerated Observer"). Note that as the above title indicates, said coordinate system is only local, it doesn't cover all of space time. MTW discusses in detail why this is a general limitation on any accelerated coordinate system. Knowing that this limit exists is an important first step in understanding accelerated coordinates and/or frames.

There are MUCH easier ways to "solve" the twin paradox - however, the section in MTW is worthwhile if you have the necessary background and are interested in accelerated frames and/or coordinates. It requires a basic familiarity with 4-velocity and the tensor notation, however. (The topic itself could be handled without the notation, in my opinion, but MTW"s treatment of the topic uses tensor notation).

The coordinate system that MTW calculates for a uniformly accelerated observer is basically equivalent to what's known as Fermi Normal coordinates. There do exist (complex) ways of transforming Fermi Normal coordinates from observer to observer, see for instance H Nikolic, "Notes on covariant quantities in noninertial frames and invariance of radiation in classical and quantum field theory" http://arxiv.org/abs/gr-qc/9909035, section 2 of which is "Coordinate transformation between two Fermi frames". Nikolic's paper will require much more than a basic knowledge to read, however.

One more comment. There are fairly simple formula for transforming between frame fields in GR, because frame fields are linear. However, the point of frame fields is that they only depend on velocity , so the nonlinear effects of acceleration aren't handled.

 

[quangtu123]

Thank you so much!
Before when I read science-popular books I thought in GR we will deal with accelerated frames (by instantaneous inertial frames and SR, as explaining the twin paradox); put it into equivalence principle and then we have a theory of gravitation.
Now I think gravitation and accelerated frames are 2 separated problems in GR.

 

[haushofer]

Locally there should be no physical distinction ;)

 

[WannabeNewton]

Now I think gravitation and accelerated frames are 2 separated problems in GR.

Accelerated frames are just facets of SR. They carry over unchanged into GR. The only novel thing (granted a profoundly novel thing) in GR is the identification of gravity with space-time curvature and the use of the equivalence principle to locally identify non-rotating freely falling frames with the inertial frames of SR.