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Implications of the statement Acceleration is not relative 
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#55
Feb1213, 11:37 AM

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PF Gold
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#56
Feb1213, 12:09 PM

PF Gold
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I have now realised that the OP was bothered because there is no treatment of the twins case with the travelling twin being inertial. As you and others have already pointed out, that is impossible. 


#57
Feb1213, 06:00 PM

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#58
Feb1213, 06:10 PM

PF Gold
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The Lorentz transformation works even if the β parameter depends on time, so we have a transformation from inertial to noninertial coordinates. 


#59
Feb1213, 06:15 PM

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 The mathematics of SR is simplest in inertial frames, but all phenomena may be analyzed in such frames, including noninertial motion.  There is no such thing as a global noninertial frame; noninertial frames are local.  It is possible, in many ways, to set up coordinates in which a noninertial world line has constant spatial coordinates of 0. For any such coordinates, you have to transform the Minkowski metric. This transformed metric leads to different formulas for time dilation, light paths, and geodesics. Different choices for such coordinates will produce different answers for coordinate dependent properties, but will produce the same answers as inertial frames for any observations or measurements. 


#60
Feb1213, 06:17 PM

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If you deal with noninertial frames within the confines of special relativity, then you have the same problem that Newton had: There is an absolute quality to acceleration; there is a preferred frame. I maintain my position that this does damage to the principle of relativity. 


#61
Feb1213, 06:21 PM

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#62
Feb1213, 06:27 PM

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In general relativity as well, acceleration is distinguishable, and there is a precise mathematical difference between a local inertial frame and a local noninertial frame in GR: in the former, the connection coefficients vanish, in the latter they do not. As for laws taking the same form, this is just a matter of the mathematical way you write them (stevendaryl has explained this before on this thread, I believe). If, in SR, you write laws explicitly using the metric and vector/tensor quantities, as you do in GR, then the laws will take the same form in nonintertial coordinates as they do in inertial coordinates. This is still not GR, because there is no gravity involved, nor is the EFE (the equation defining GR) used. 


#63
Feb1213, 06:30 PM

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PF Gold
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Also, arguing about definitions is not the same as arguing about physics. Can you state a *physical* objection that doesn't depend on a particular definition for what "the principle of relativity" says? 


#64
Feb1213, 06:52 PM

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However, the discussion about inertial vs noninertial frames is not relevant to the statement "acceleration is not relative". The statement "acceleration is not relative", as we have mentioned, refers to proper acceleration. Proper acceleration is a property of a worldline, not a property of a reference frame. It doesn't matter what reference frame you use, inertial or not, the proper acceleration is the same in all of them. So, the statement "acceleration is not relative" is about worldlines, not reference frames. I think that you are getting distracted by irrelevancies. The travelling twin has nonzero proper acceleration regardless of what reference frame is used. 


#65
Feb1213, 06:56 PM

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Once you know how the physics works in inertial frames, then figuring out the physics in any other frame is simply a matter of performing a change of variables to the coordinates (aka coordinate transform). All of the usual math for doing a chang of variables still applies. Thus, even though the postulates only describe physics in inertial frames, you can use them indirectly to derive the physics in noninertial frames. 


#66
Feb1213, 07:56 PM

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So let's consider an example, Newton's second law of motion. The relativistic 4D version of this, for a particle of constant mass, is[tex] F^\lambda = m \frac{d^2x^\lambda}{d\tau^2} [/tex]when measured in any inertial (Minkowski) coordinate system. This is pretty simple and almost the same as the nonrelativistic version. However in noninertial coordinates, the equation becomes[tex] F_\lambda = m \sum_{\mu=0}^3 g_{\lambda \mu} \frac{d^2x^\mu}{d\tau^2} + \frac{m}{2} \sum_{\mu=0}^3 \sum_{\nu=0}^3 \left( \frac{\partial g_{\lambda \mu}}{\partial x^\nu} + \frac{\partial g_{\lambda \nu}}{\partial x^\mu}  \frac{\partial g_{\mu \nu}}{\partial x^\lambda} \right) \frac{dx^\mu}{d\tau} \frac{dx^\nu}{d\tau} [/tex]You don't need to understand the meaning of this, just observe that it's very complicated. So, yes the laws of physics can be expressed in a form that is the same in all frames, inertial or noninertial, but such expression is much more complicated than the inertialframeversions of the laws. 


#67
Feb1313, 01:45 AM

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#68
Feb1313, 02:17 AM

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 http://web.archive.org/web/200608290...ry/gtext3.html It would be good if physics textbooks discussed this topic, but I don't know any that does. 


#69
Feb1313, 03:03 AM

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PF Gold
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It is a trivial mathematical fact that in flat spacetime, a geodesic=inertial path is an absolute maximum of clock time. 


#70
Feb1313, 03:16 AM

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#71
Feb1313, 06:06 AM

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#72
Feb1313, 07:16 AM

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