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#55
Mar1610, 06:39 PM

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PF Gold
P: 1,841

The property of "being a geodesic" is a geometrical property that doesn't depend on the choice of coordinates; it is an intrinsic property of a worldline. The 4acceleration [tex](where U is 4velocity) is a 4vector, i.e. a tensor that transforms between coordinate systems in the correct way. (And the symbol [itex]dU^\mu/d\tau[/itex] does not represent a 4vector.) Its magnitude, "proper acceleration", [itex] \sqrt{A_{\mu} A^{\mu} } [/itex] is therefore a scalar invariant, the same value in all coordinate systems. The geodesic equation  in any valid coordinate system you like  is just the condition that the proper acceleration is zero, or equivalently, that the (spacelike) 4acceleration is the zero 4vector. 


#56
Mar1710, 03:25 AM

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P: 1,883

Your statement that "3acceleration" is frame dependent, while right, has no relevance whatsoever, and does not back up your disagreement with the statement. I think there are some misunderstandings that should be clarified. 


#57
Mar1710, 11:34 AM

P: 665

AB 


#58
Mar1710, 01:40 PM

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P: 1,883




#59
Mar1710, 03:21 PM

P: 665

AB 


#60
Mar1710, 04:09 PM

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P: 8,470

Also, do you disagree that, as DrGreg said, an object on a geodesic path feels zero 4acceleration everywhere along it rather than zero 3acceleration? And that the magnitude of the 4acceleration for a given point on a worldline determines what would be measured by a physical accelerometer moving along that worldline? Finally, do you disagree that if we have a path in Minkowski coordinates on the Rindler wedge that has zero 4acceleration everywhere as calculated by the Minkowski metric (i.e. a path with constant velocity in Minkowski coordinates), then if we use the coordinate transformation to transform this path into Rindler coordinates, the path will still have zero 4acceleration everywhere along it as calculated using the Rindler metric? 


#61
Mar1710, 04:50 PM

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PF Gold
P: 1,841

What has caused confusion in this thread is your describing [itex]d^2x^i/d\tau^2[/itex] as "proper acceleration". That is not the terminology that everyone else uses and so everyone has been disagreeing with you. I'm guessing that maybe you thought of this because some authors describe the 3vector [itex]dx^i/d\tau[/itex] as "proper velocity". That is not a terminology I like; I prefer to call that by its alternative name "celerity". In relativity, most "proper" things are invariant e.g. proper time, proper length and the correct definition of proper acceleration (& I've seen some people describe (rest) mass as "proper mass"). The 3vector [itex]dx^i/d\tau[/itex] is, of course, coordinatedependent. The 4vector [itex]dx^{\nu}/d\tau[/itex] is described as "4velocity" (rather than "proper velocity"). 


#62
Mar1810, 06:13 AM

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AB 


#63
Mar1810, 12:25 PM

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#64
Mar1810, 05:38 PM

P: 665

[tex]v=(x/2) a[/tex] where [tex]a[/tex] represents the coordinate acceleration. Thus it is clear that if one seeks out a constant [tex]v[/tex], it is necessary to let the particle be hovering at a constant [tex]x.[/tex] or be moving along paths with constant [tex]x.[/tex] Along nongeodesic trajectories, a constant [tex]v[/tex] means that the coordinate [tex]x[/tex] must remain constant duing the motion and this tells us that the proper 4acceleration vanishes. But since we are talking about general paths, the constancy of [tex]v[/tex] cannot be always guaranteed in the Rindler metric even if the particle has a constant coordinate velocity in Minkowski spacetime. AB 


#65
Mar1810, 06:03 PM

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#66
Mar1910, 04:24 AM

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AB 


#67
Mar1910, 02:06 PM

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#68
Mar1910, 03:18 PM

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AB 


#69
Mar1910, 03:40 PM

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P: 8,470

Assuming we've settled this issue, perhaps we can revisit my earlier claim that the equivalence principle would hold in arbitrarily large regions of Rindler coordinates? Hopefully you'd agree that in any flat SR spacetime described in Minkowski coordinates, it should be possible to construct a physical network of rulers and synchronized clocks moving inertially relative to this coordinate system (a network of the type that Einstein used to define the concept of an 'inertial frame'), such that if we use the readings on the rulers and clocks to define a new coordinate system, the laws of physics as seen in this coordinate system (which is just a different Minkowski frame) will be those seen in any SR inertial frame. Now, if you agree that Minkowski coordinates (on the Rindler wedge) and Rindler coordinates are just different descriptions of the same physical spacetime, this same physical network of rulers and clocks could also be described in Rindler coordinates, agreed? Now suppose we have such a physical network, and we want to describe where events in an experiment happen relative to that network in the context of our Minkowski or Rindler coordinate systems. Suppose a particular event E (a collision, say) happens next to the x=12 lightseconds mark on the physical ruler representing the xaxis in this network, and that the physical clock at that mark reads t=8 seconds when E happens next to it. Then if we are using a separate Minkowski coordinate system with coordinates x',t', to describe events in this spacetime, the coordinates of the physical clock at the x=12 l.s. mark on this ruler reading t=8 s may happen at some completely different coordinates in this system, say x'=23 l.s. and t'=100 s. Likewise, in the Rindler coordinate system, the coordinates x'' and t'' of this event will be different as well. However, both the Minkowski coordinate system and the Rindler coordinate system will agree that whatever the coordinates x' and t' (or x'' and t'') of the event of the physical clock at the x=12 l.s. mark reading t=8 s, the same coordinates x' and t' (or x'' and t'') would be assigned to the event E of the collisionthe Minkowski coordinate system and the Rindler coordinate system cannot disagree about local facts like whether two events (in this case the event of that clock at that marking reading 8 s and the event E of the collision) coincide at the same point in spacetime or not! If both the Minkowski coordinate system and the Rindler coordinate system (along with the metric and laws of physics expressed in those coordinate systems) agree in their predictions about what rulermarkings and clockreadings on the physical network coincide with which events in any physical experiment that takes place within the region covered by that physical network, then both coordinate systems should agree in their predictions about what the equations of the laws of physics will be when expressed in the coordinate system defined by the network (not when expressed in terms of their own coordinates). And of course in flat SR spacetime it should always be possible to construct a physical network of rulers and clocks which define a coordinate system where the laws of physics work the same way as in any other inertial coordinate system, right? This idea is essentially no different than the idea of the equivalence principle, that even if you have a curved spacetime described in some coordinate system like Schwarzschild coordinates, in any local region you should always be able to construct a network of freefalling rulers and clocks such that that the laws of physics as expressed in the coordinates defined by that network (not when expressed in Schwarzschild coordinates in that region) reduce to the same laws seen in an inertial frame in SR (at least to the first order or something). The only difference is that in this case we need not confine the grid of rulers and clocks to a small local region, they can cover any arbitrarily large region of the Rindler wedge and it'll still be true that the laws of physics as expressed in the coordinates defined by the network (not when expressed in Rindler coordinates in that region) will be exactly those seen in any SR inertial frame. 


#70
Mar1910, 07:18 PM

P: 665

I'm in a hurry to go somewhere, so I don't have enough time to read the whole post. I'll be back soon. AB 


#71
Mar2010, 06:15 AM

P: 665

My own view on the EP not only on Rindler wedge but in any spacetime with a vanishing Riemann tensor but nonvanishing Christoffel symbols is briefly stated in Papaetrou's book "Lectures on General Relativity" at page 56:
AB 


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