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How does GR handle metric transition for a spherical mass shell? 
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#181
Oct3011, 05:03 PM

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I'll look at what you posted in the other thread and make further comments on that if need be. 


#182
Oct3011, 05:13 PM

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(1) my answer is (b) (2) my answer could be (a), but only in the vacuous sense; in the presence of gravity there are no flat regions of spacetime, there is some curvature everywhere. (b) is the strictly correct answer, but as I said in an earlier post, for a given accuracy of measurement there will be some finite region where the deviations from flatness are not observable, and within that region a Lorentz transformation on a local coordinate patch of "flat" Minkowski coordinates will work (3) my answer is (b), but "similar" has to be interpreted carefully; Fredrik's comments in an earlier post in that thread on the properties a coordinate chart has to have to admit these transformations are worth reading (4) my answer is "it depends"; (a) is correct *only* if the coordinate chart meets the requirements for having a rotation transformation in the first place, per Fredrik's post; otherwise the answer is undefined because there is no welldefined assignment of coordinates to objects beyond a small local coordinate patch; (b) is never correct, if the transformation is valid at all when extended to distant objects it will change their coordinate positions 


#183
Oct3011, 05:29 PM

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One can have many objects in free fall all with lower and higher local velocities wrt a free falling object at escape velocity. 


#184
Oct3011, 08:18 PM

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An object in free fall is changing its velocity because it is acted upon by an external force. The force of gravity. Inertial movement is movement in a straight line. Freefall is not inertial movement. It's accelerated movement. This is a basic realitycheck. You mean to tell me that if an object is in orbit, you actually consider that to be a straight line? 


#185
Oct3011, 08:35 PM

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The object is "accelerated" with respect to the Earth, but that is coordinate acceleration, not proper acceleration; similarly, the "force" of gravity in Newtonian terms is not a "force" in GR, because it does not cause any 4acceleration of the object. That's how I was using the terms, and how they are standardly used in GR. 


#186
Oct3011, 08:36 PM

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From one perspective one certainly could consider it a straight line as a test object in orbit differs only from a radially free falling test object by having a nonzero angular momentum. For instance do you feel yourself turned often when the seasons pass? The same for an astronaut who travels in orbit, as far as he is concerned he travels in a straight line as no forces act on him. 


#187
Oct3011, 08:48 PM

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Whether a free falling object's path in space is straight depends solely on the chosen coordinates. In simple terms, the Sun just as much goes around the Earth as the Earth goes around the Sun it simply depends on the point of view. 


#188
Oct3011, 08:51 PM

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In fact, I would say that points to the sanity of Special Relativity. In fact, if Special Relativity somehow claimed that geodesic paths in spacetime were straight, I would find that to be much more alarming, and troubling. In the very next instant, the body will be comoving with an entirely different GLOBAL Lorentz Frame. It's like this. If I have a function whose domain is the real numbers, and its range is the real numbers, then I would say that function is valid globally. If I have a function whose domain is 0 to 1 and range is 0 to 1, then I would say the function is valid only locally. The Lorentz Transformations take as input and output global inertial reference frames, representing every event that ever has and ever will happen in the entire universe. 


#189
Oct3011, 08:57 PM

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The local phenomena within the area of a geodesic would be small, perhaps too small for your most sensitive equipment to detect, but traveling along a geodesic is different from traveling in a straight line. 


#190
Oct3011, 09:29 PM

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Of course it's easy to find geodesic paths in a curved spacetime that are not "straight" in the sense you're using the term, but the laws of SR don't apply in those spacetimes, precisely because they are curved. I've already given a specific example of such a law: SR requires initially parallel geodesics to remain parallel. (Or, in more ordinary language, SR requires that two objects, both in free fall and weightless, feeling no force, which are at rest relative to each other at one instant of time, must remain at rest relative to each other at all times.) In a curved spacetime, this law is violated, as my example of bodies falling towards Earth made clear. Btw, it's also worth noting that you speak of "changing velocity" without defining what that means. The 4velocity of a freely falling body does *not* change in the coordinateindependent sense I gave in my last post: its covariant derivative with respect to the body's proper time is zero. So if you think its velocity is changing, what is it changing relative to? Any such definition of velocity "changing" will be a coordinatedependent definition. The GR definition is not; it's a genuine physical observable (whether or not the body feels acceleration). 


#191
Oct3011, 10:04 PM

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Geodesics are of course described by the geodesic equation. [tex] \frac{d^2x^\lambda }{dt^2} + \Gamma^{\lambda}{}_{\mu \nu }\frac{dx^\mu }{dt}\frac{dx^\nu }{dt} = 0 [/tex] Futhermore, if we multiply through by m, and replace the right hand side by a force, this is about as close as GR comes to Newton's equations of motion. Specifically, if we have a test particle moving under the action of an external nongravitational force, (for instance, an electric field), we can write in GR [tex] m\,\frac{d^2x^\lambda }{ds^2} + m\,\Gamma^{\lambda}{}_{\mu \nu }\frac{dx^\mu }{ds}\frac{dx^\nu }{ds} = F [/tex] So, it shouldn't be too surprising that the Christoffel symbols act pretty much like forces. In particular, we can identify some of them as being equal to what we used called the "force" of gravity in Newtonian theory. And in GR, we can replace solving F=ma with solving the geodesic equations (well, there are occasions where this doesn't work, and we have to worry about the Paperpatrou equations, but this is rare) BUT As I remarked earlier, the components of the Christoffel symbols transform in a complex way. From wiki http://en.wikipedia.org/w/index.php?...ldid=455650120 they transform like: [tex] \overline{\Gamma^k_{ij}} = \frac{\partial x^p}{\partial y^i}\, \frac{\partial x^q}{\partial y^j}\, \Gamma^r_{pq}\, \frac{\partial y^k}{\partial x^r} + \frac{\partial y^k}{\partial x^m}\, \frac{\partial^2 x^m}{\partial y^i \partial y^j} [/tex] which is not the tensor transformation law. So it's convenient to think of Christoffel symbols as "forces" in any one particular coordinate system that you want to work in, but it's a mistake to think they'll transform in the same manner as the forces in flat spacetime that you may be thinking of them as being analogous to. 


#192
Oct3111, 08:16 AM

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A couple of notes that I wanted to throw out before I go to work:
I think if I redo it with equipartition of rapidity, we'll have distribution that looks the same (densitywise) in the center after Lorentz Transformation. (2) I want to make an analogy. Let's say I take a cone, lay it out flat, and draw a straight line on it, and then wrap it back up in the cone shape again. We live in a cartesian coordinate system. Has that Cartesian Coordinate system failed us because that line that we have defined as straight is now curved? No. The cartesian coordinate system is alive and well, but it has a cone in it, and it has curved lines in it. I'm not trying to criticize the application of geometry to take a cone and wrap it, and say, YES, you can draw "straight" lines on a curved object. I have no problem with that. But when you go on to say that the overlying cartesian geometry has somehow been invalidated because you can shape a paper into a cone, that's where I have a disagreement. (3) I'm not saying that straight lines are always easy to detect. Of course if I look at the moon on the horizon, it looks distorted, because the light has NOT moved in a straight line (mostly because of atmospheric rather than gravitational effects). But the fact that the light doesn't move in a straight line does NOT mean that straight lines don't exist. But on the whole, how often does that happen? The great majority of things in the universe are not significantly affected by this sort of phenomenon. You point at them, and that's the direction they (not are) WERE, when the light left them. You can't see where the object IS, but you can see where the object WAS when the light left it. 


#193
Oct3111, 11:15 AM

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Now try the same thought experiment with a sphere. You can't do it. There is no way to "lay a sphere out flat" and draw straight lines on the flat version, and then wrap it all back up into a sphere again. It's impossibleany such operation will distort the surface and invalidate its geometric invariants. That's one way of expressing the fact that a sphere has nonzero *intrinsic* curvature. The connection coefficients in pervect's formulas are nonzero on a sphere (i.e., there is *no* coordinate chart on the sphere that makes them all zero). And that's the kind of curvature we're talking about when we say that gravity is curvature of spacetime. The spacetime we are living in has intrinsic curvature because of gravity; it is a spacetime analogue of something like a sphere (actually more like a saddle, but the same argument I gave for a sphere would apply to a saddle too). It is *not* the spacetime analogue of something like a cone or a cylinder that can be laid out flat and wrapped up again without changing its intrinsic geometry. And in the presence of intrinsic curvature, all your intuitions about how things work in flat Euclidean space, or flat Minkowskian spacetime, are simply wrong on any large scale; they only approximately work over very small patches. You can only treat the Earth's surface as flat over a small area, and you can only treat spacetime as flat over a small region. 


#194
Nov111, 06:33 PM

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I have made several further posts in the other thread.
http://www.physicsforums.com/showthread.php?t=545002 I think I have explained myself better there. I'm trying to find the most concise description of our disagreement: You seem to take the attitude that infinite straight lines do not exist. And I take the attitude that infinite straight lines do exist, or at least are definable. In my thinking, regardless of the curvature of paths caused by gravity, it is always possible to imagine what would happen if matter wasn't there. In your thinking, regardless of how we try to imagine what would happen if matter wasn't there, there would always be more matter, screwing things up. Now I won't argue with that, but the question is not whether we can really map the straight lines with perfect accuracy. The question is whether those straight lines exist at all. I think they do. You think they don't. I explain why I think they do in the other thread. 


#195
Nov111, 11:57 PM

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I'll comment in the other thread.



#196
Feb2512, 07:17 AM

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The bottom result is that I was not able to conclusively prove or disprove either position. I still think that I am probably correct, but not so strongly as before. 


#197
Feb2512, 08:37 AM

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#198
Feb2512, 08:47 AM

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