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Does a free falling charge radiate ? 
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#37
Feb1913, 07:45 PM

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It seems like the answer to "does the sun radiate" should be obvioiusly "yes"  but without a timelike Killing vector, or an asymptotically flat spacetime, how do we justify something as simple as saying "the sun radiates"? On one hand, I feel like this may be nitpicking. On the other hand, it's a "nit" that has always bothered me. 


#38
Feb1913, 07:51 PM

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#39
Feb1913, 07:54 PM

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#40
Feb2013, 12:54 AM

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I would like to explain my ideas proposed a few months ago. First we should make clear whether we talk about pointlike test particles or whether we want to study extended objects. For the latter one it's clear that we expect radiation. For pointlike particles we don't expect radiation b/c of the modified geodesic equation. If we set the field to zero (neglecting selfinteraction of the test particle with it's own field) then we find the usual geodesic equation and we expect free fall w/o radiation. The main question is then whether the approximatin of pointlike test particles w/o selfinteraction makes sense. In reality we expect deviations from this idealized setup and therefore we expect radiation.
What I still don't like is the definition of radiation using 'energyloss' or the '1/r behaviour'. The problem is that we can neither measure nor define this energy; b/c it's a Coulomb field an integral over the energy density diverges both at r=0 and for r→∞; b/c we may have arbitrarily curved (expanding) spacetime we cannot define E = ∫d³x T^{00} even for wellbehaved T^{ab}; w/o a timelike Killing vector k_{a} field (e.g. in an expanding universe) we cannot define the 4vector J^{a} = T^{ab} k_{b} and again we do not have a reasonable definition of the el.mag. field energy E = ∫ d³x J^{0} That's why I would like to get rid of the concept of energy and 1/r behavour. My proposal is to study local, coordinatefree effects, i.e. the deviation from geodesic motion. Of course this means that we ask a different question: instead of Does a free falling charge radiate? we ask Are charged particles  pointlike test particles or exended objects  in freefall, i.e. do they follow geodesics? 


#41
Feb2013, 02:49 AM

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In my definition, motivated by the principle of general covariance, the lack of staticity is not a part of the definition of radiation. Or let me quote from page 8: " Now we turn back to the attempt to give an operational definition of radiation at large distances. In our opinion, the only reason why radiating fields deserve special attention in physics, is the fact that they fall off much slower than other fields, so their effect is much stronger at large distances. Actually, the distinction between “radiating” and “nonradiating” fields is quite artificial; there is only one field, which can be written as a sum of components that fall off differently at large distances. ... In this sense, we can say that radiation does not depend on the observer. " But if you wish, you can use any coordinates you want. The point is that the only physical quantity is F which transforms as a local tensor. So if, at a certain point far from the source of F, all components of F are of the order of r^1, then, at this SAME point, the components of F' will also be of the order of r^1. It is a trivial consequence of the fact that F transforms to F' as a tensor. 


#42
Feb2013, 05:16 AM

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But you seem to be mixing scenarios in a funny way for your own benefit. Take the HulseTaylor binary, a clear example of gravitational radiation, you have there two "extended" masses orbiting each other that are obviously not following exact geodesics. But it is clear that this is a different example from the earthsun example you were using because in this case it is alrigight to consider the earth as a test particle that is obviously moving in a geodesic, but you need to consider it an extended object to say it radiated and then you cannot claim it is followin a geodesic because in GR there is no defined center of gravity that you can pinpoint as the one that is following a geodesic . It should be straightforward that in the HulseTaylor binary you cannot model one body as a test body orbitting a source of gravitation since their masses are of similar order of magnitude. So you must make up your mind if in your example you really want to consider the orbitting bodies as extended objects or as test particles. 


#43
Feb2013, 05:27 AM

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#44
Feb2013, 05:40 AM

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#45
Feb2013, 06:04 AM

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#46
Feb2013, 08:45 AM

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@TrickyDicky regarding #43: fine, thanks



#47
Feb2013, 09:35 AM

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#48
Feb2013, 03:34 PM

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To decide whether what they are computing are geodesic paths or not it would be wise to look at the geodesic definitions, see for intance Wikipedia page on "Geodesics in General relativity": "True geodesic motion is an idealization where one assumes the existence of test particles. Although in many cases real matter and energy can be approximated as test particles, situations arise where their appreciable mass (or equivalent thereof) can affect the background gravitational field in which they reside. This creates problems when performing an exact theoretical description of a gravitational system (for example, in accurately describing the motion of two stars in a binary star system). This leads one to consider the problem of determining to what extent any situation approximates true geodesic motion. In qualitative terms, the problem is solved: the smaller the gravitational field produced by an object compared to the gravitational field it lives in (for example, the Earth's field is tiny in comparison with the Sun's), the closer this object's motion will be geodesic." If one then considers that "in metric theories of gravitation, particularly general relativity, a test particle is an idealized model of a small object whose mass is so small that it does not appreciably disturb the ambient gravitational field." And remembers that gravitational radiation is a gravitational field disturbance that in the case of binary neutron stars or black holes should be not negligible I think there's not really much more to discuss. 


#49
Feb2013, 03:44 PM

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The fact remains that (AFAIK) the trajectories computed for each neutron star turn out to be geodesics of the metric that is computed based on their masses and configurations, and this remains true as the system emits gravitational waves (meaning the metric changes with time). And this, as I said, is good evidence that gravitational waves can be generated by the geodesic motion of objects. 


#50
Feb2013, 04:04 PM

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For the general topic, a paper that points to much of the history of results, along with yet more rigorous derivation of geodesic motion for test particles (only): http://arxiv.org/abs/1002.5045 [see esp. section II, where the mass as well as size are required to approach zero to derive exact geodesic motion] Also: http://arxiv.org/abs/0907.0414 For the slight generalization only (and not exactly) true for extreme mass ratio, see the discussion of the The Detweiler–Whiting Axiom in section 24.1 of: http://relativity.livingreviews.org/.../fulltext.html [edit: See also this thread: http://physicsforums.com/showthread.php?t=498039] 


#51
Feb2013, 05:06 PM

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It's possible that the neutron stars in the binary pulsar are not moving on *exact* geodesics; of course we can't tell for sure because we can't make precise direct measurements of the system. But it does seem like they move on curves that are close enough to geodesics that we can't tell the difference with our current measurements, even though the two neutron stars are certainly not test objects. (At least, that's my understanding of the current models.) 


#52
Feb2013, 05:19 PM

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However, for a symmetrical two body problem, even with no GW (let alone with), I don't see how to even ask the question of moving on geodesics; no one on the thread I linked thought there was any hope of doing this, esp. Sam Gralla for whom this is a specialty of his. 


#53
Feb2013, 05:23 PM

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#54
Feb2013, 05:25 PM

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