
#1
Jan2305, 06:47 AM

P: 1,444

Einstein proposed a quadrapole model for gravity waves  but why? If we consider two particles of finite mass like an electron and positron that combine to convert all their mass energy to a pair of photons   the distortion of space due the conversion of the gravitational mass to radiation should proceed at the velocity c in every direction  this implies a compressional wave. In other words, the change in the local curvature of space as measured by the radius excess [G/3c^2]M (where M is the combined mass of the positron and electron) is going to spherically
propagate (diverge) at velocity c so that a gravitational detector at a distance d (e.g., another mass M2 that is pulled toward the position of the electron and positron immediately prior to their combining) will experience a gravitational force decrease at the time t = d/c. But this force change is along the line of action connecting the center of the mass M2 with M  which implies a compressional wave. 



#2
Jan2305, 01:21 PM

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The photons that arise after the electron and positron annhilate are going to contribute to the stress energy tensor too  I think you are expecting that they'll just disappear gravitationally. This is not the case.
The stress energy tensor for a stationary particle looks like
The analogous expression for a photon, which can't be stationary, moving in the plus or minus x direction is
Again, we can't deal with point particles, so we approximate the photon as having some finite volume, just like we did with the particles. 



#3
Jan2305, 01:45 PM

P: 1,444

pervect  yes  I know the photons have an effective mass energy that contributes to the gravitational field  but if we construct for example a Gaussian surface that is centered on the combination event with a radius d that just touches the second test mass M2... then as the photons escape this surface there is no longer any energy or energy density w/i the Gaussian enclosure  so M2 at this point in time (t = d/c) will experience an abrupt change in force.. to avoid any interaction with the photons we can suppose that the two newly created Gamma ray photons travel orthogonal to the line of action joining their creation event with the center of M2
Also  the notion of point particles is useful in some ways  but in may not represent reality  in this caper you could substitute any number of antimatter particles that combine in an instant to create the radiation ...if you demand that there be a realistic volume over which to compute the initial energy density 



#4
Jan2305, 02:25 PM

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Gravity waves
Einstein didn't propose the quadrupole dependence of gravitational waves as an independent hypothesis, did he? I thought that this was only discovered after the equations for GR had already been written down. Steve Carlip gives a simple justification for why electromagnetic waves depend on the dipole moment while gravitational waves depend on the quadrupole moment in this post:




#5
Jan2305, 09:14 PM

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Jesse presents a good explanation of why gravity waves are generated by the quadropole moment (the dipole moment can't generate any waves, given that momentum is conserved the rate of change of the dipole moment must be zero). 



#6
Jan2405, 02:15 AM

P: 1,444

You have a spherical gravitational field that has been abruptly extinquished  that G field modifies the effective radius (the excess radius). The change in the spherically symmetrical distortion of space is felt by every mass at every distance d at the time d/c  I would think the same notion could be applied to the Sun and the earth  as matter is converted to radiation there should be a continuous change in the gravitational mass as the radiation passes the earth  the pressure wave results from the conversion process would be felt 8 minutes later  it would be a continuous diminution in the case of an ongoing process like the Sun  but if there were an abrupt interruption followed by a resumption of the hydrogen to helium reaction  it should be observed as a longitudinal wavefront.
Einstein seesawed back and forth for a period of time debating in his own mind whether gravitational waves existed. He finally decided they did  but the rationale that these waves be quadrapole may be the reason why they are not detected  in other words  as Carlip states: a symmetrical collapsing star does not generate quadrapole radiation.  But since it changes the spherical distortion of space  why can't the energy imbalance that is the basis for proposing G waves be considered as being satisfied by a longitudinal effect? 



#7
Jan2405, 02:25 AM

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I'm not at all convinced that your quasi Newtonian approach is going to give correct answers in such an extreme relativistic situation.
The probability of your answer being incorrect goes up to near 100% if your answer is inconsistent with gravity waves being generated by a quadropole moment. 



#8
Jan2405, 03:27 AM

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If they (quadrupole waves) are confirmed  that would lessen the need to explain anything further  but even then there is no reason why two different wave phenomena would/could not be generated  earthquakes for example generate both transverse and longitudinal waves  admittedly the medium is quite different.




#9
Jan2405, 04:10 AM

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I was thinking about this some more, and if you have many particles annihilating, not just one, then you would have spherical symmetry, like you said. (I got hung up on the notion of one particle anhillating one other, which was the original problem statment  that would have a cylindrical symmetry, the axis of the cylinderical symmetry being the direction in which the photons were emitted).
But if you do have spherical symmetry, you won't generate gravitational waves, by Birkhoff's theorem. A symmetrical, nonrotating spherical collapse won't generate gravitational waves, nor will a symmetrical spherical nonrotating explosion. Certainly there will be a change in gravity at a fixed point as the sphere expands (if one can somehow avoid being fried by the explosion), but I don't think there can be any gravitational waves. 



#10
Jan2405, 11:37 AM

P: 90

Still most of our ideas come from the weak field approximation and the analogy to electromagnetism in flat space. Wigner's little group is of use here. The little group is a subgroup of the Poincare group with 4 translations and 6 rotations (three spatial rotations and three boost "rotations"). Fix the momentum of the particle and ask what subgoup of the Poincare group leaves the momentum invariant. For a massive particle, go to its rest frame and notice that the rotations SO(3) leave the 4 momentum fixed. Thus any property of the particle must transform as a representation of SO(3). This group is compact so the property will have a discrete spectrum. This quantizes the angular momentum. Look at something with a null momentum. The luxury of a rest frame is unavailable and the full little group is E(2), the 2 dimensional Euclidean group with 2 translations perpendicular to the null momentum and SO(2) for the rotations in that plane spanned by the translations. Thus with E(2) the angular momentum is continuous and not quantized, but it is quantized if only the SO(2) component is available. The former happens when a component of the angular momentum is in the spatial plane of the translations and the latter when all the angular momentum is parallel to the direction of motion. The continuous spin case are the longitudinal waves and the discrete spin case are the transverse waves. In the transverse case the spin is a Lorentz invariant, either parallel or antiparallel to the direction of motion. Such an object may have its energy red shifted arbitrarily close to zero, but that red shifting will leave the spin invariant!!! If you accelerate a massive particle arbitrarily close to the speed of light, you get SO(2) as the little group. I suspect, but have never shown that the continuous spin case arises by slowing a tachyon down to arbitrarily close to the speed of light. Yes earthquakes in the sold crust have both longitudinal ( P waves or Principle waves) and transverse ( S waves or secondary waves or shear waves) sound waves. The P waves are faster and the difference in arrival times allows the location of the epicentre. We also infer a liquid region inside the earth as a liquid will not support shear waves. So if the path from the epicentre is through solid crust you will first receive the P waves and a little later the S waves, but if the path encounters a liquid region you will only get P waves. This allows you to map the liquid core. The media are very different as there are only transverse electromagnetic waves. 



#11
Jan2405, 10:50 PM

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Where could I find out more about the peeling theorem and the News function?




#12
Jan2505, 12:24 AM

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It should be in Wald and more modern texts. Wald talks about the "peeling prperty" on pg 285, referencing Geroch and Penrose. MTW has a few other references on pg 1165. I'm going to get some more modern texts and somewhere recently I've seen a nice discussion pointing out that the peeling theorem which tells you how curvature arrives at null infinity is not as powerful as a multipole expansion in electromagnetism. Try googling "Bondi's news function" and "peeling theorem" or "peeling property".




#13
Jan2505, 04:37 AM

P: 1,444

pervect  a symmetrical spherical explosion will generate a longitudinal pressure wave  the premise of the discussion is based upon the notion that the energy of the mass conversion event is carried by a longitudinal phenomena  or perhaps by more than one type of wave.
When we think in terms of transverse wave phenomena  our experience with common examples such as a violin string always conjures up the need for a restoring force  in the case of em waves we either need some sort of medium in tension or we require interacting magnetic and electical fields. Where or what is the physical restoring medium for gravitational waves? 



#14
Jan2505, 02:16 PM

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A spherically symmetic explosion will not generate gravity waves in standard General relativity.
Formally, all spherically symmetric solutions of Einsteins equations with no matter density (G_{ab}=0) are the the Schwarzschild solution [Birkhoff's theorem]. The good news is I think this probably does make your "Gaussian sphere" idea work correctly (initially I was uncertain about it). Birkhoff's theorem can be interpreted as saying that there is no spherically symmetric radiation (monople radation) in GR. This should not be surprising, there isn't any such beast in electromagnetism, either. Unless you contemplate throwing out the conservation of momentum, I think you are going to have a hard time finding ANY reasonable theory of gravity that has dipole radiation, as per the arguments originally posted by Jesse M, because the conservation of momentum implies that the dipole moment is a conserved quantity. [editadd] But what about monople radiation? (The conservation of momentum argument gets rid of dipole radiation  but the original question was more clesly related to monople radiation). What prevents monople radiation is the conservation of charge in EM theory, and the conservation of mass in GR. This will likely also extend to any other reasonable theory of gravitation. The remaning issue is longitudinal waves. I know we don't see them, but offhand I can't think of a mathematical reason why not, though I suspect there is one. I'm not familiar enough with the E(2) group to see if Rob Woodside's group theoretical arguments would forbid longitudinal waves that move at the speed of light. Perhaps Rob or someone else can comment on this point. [add^2] I've got a rather hazy recollection that the two possible forms of photon polarization are left circular and right circular, and that this comes from group theoretical arguments. (Other forms of photon polarization can be exrpessed as superpositions of left circular & right circular photons). Also, since a photon or anything else moving at the speed of light has no valid concept of time, I don't think that pressure waves travelling at 'c' are going to work in any theory that's Lorentz invariant. 



#15
Jan2505, 11:24 PM

P: 1,444

Rob W. Didn't mean to ignor your instructive post #10  when it comes to group theory I understand only about every tenth word. The only meaningful comment I can make is that from what I have interpreted from the condensed data, the jury is still out on quadrupole G waves  but the dipole theories are in deep trouble.
pervect  For me the idea of a longitudinal wave follows directly from the supposition that the energy density in a gravitational field must cease when the mass upon which the G field depends is converted into a form that is radiated away. For example, if we placed a spherical Gaussian surface of radius one light year to encompass a type 1a supernova, there should be no evidence of any change in either the G field or the light flux for one year  When the radially divergent light flux is detected by sensors on the Gaussian surface  conservation of energy should demand that the energy w/i the surface drop to zero at the same time (it would have been collapsing outwardly at all points w/i the instantaneous volume defined by the radius of the photon flux  but since the matter sensor cannot know this until the effect reaches the gaussian sphere where it is measured by its influence upon a test mass). From the standpoint of the test mass on the Gaussian surface  it is attracted toward the center of the sphere for one year until the collapsing G field energy reaches the gaussian surface  then zip  there is a change in the force along the line of action between the supernova and the test mass. 



#16
Jan2805, 10:06 PM

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I believe that your description of what a person would feel in the event of a hypothetical massive explosion is basically correct. The argument for it being correct is that a vacuum solution for any spherically symmetric body must be a Schwarzschild solution even for a nonstatic metric  the technical term for the explosion you describe. There is some messy math, but IIRC one will find a function talking about the mass enclosed within a radius r in the interior Schwarzschild solution, along with some rather long justifications of why this is possible in the spherically symmetric case, which basically boils down to your Gaussian sphere idea.
However, there would not be any actual "gravity wave" emitted by this process. The gravity would be high, then when the wavefront passed, the gravity would go away. But there still would be no actual gravitational radiation, just a change in the field. Any actual gravity waves would mess up the math, but for a nonrotating spherical explosion they won't exist. Probably the clearest example of this is an explosion of a large object where the debris is physical debris that travles slower than light. If gravity waves existed, they'd carry off some energy at the speed of light, and you'd see the first disturbance in gravity at a time d/c, d being the distance to the explosion. But since there aren't any gravity waves, you won't see any disturbance in gravity until a time d/v, where v is the outward velocity of the debris  i.e. you won't see anything happen until the debris actually pass your position. This is of course all in standard theory. As far as shock waves go, I still don't think they work for anything going at the speed of light. Basically, pressure waves are all about matter compressing and decompressing, which takes time  and moving at 'c', there is no time. I would need a more rigorous argument to make this really convincing. I've got to run, if I can think of a way to make this clearer or more rigorous at the same time not getting too mathematical, I will. [add] The best analogy I can come up with is the neutrion oscillations. Neutrino oscillations are not compatible with neutrino's being massless and hence moving at 'c'. One way of saying this is that oscillations take time, and nothing moving at 'c' experiences time. (This isn't perhaps the most rigorous way of explaining this point, but you'll find Baez in sci.physics, for instance, on record as using this explanation). Similarly, pressure waves need time to move, and again, moving at 'c', there is no time. 



#17
Jan2905, 04:14 AM

P: 1,444

pervect  I agree with your analysis  what I have failed to get across is that the rationale Einstein used to convince himself that, in order to conserve energy in the case of relatively accelerating masses that communicate their position to each other at the velocity of light, it was necessary to propose a wave phenomena that dissipated (at least carried away) energy  apparently to the spatial medium (whatever that is). But how can this spatial medium perform its role in sustaining the wave as it must if there is nothing that is waving. Do we not have to have a better description of how space works or what it is. Now in the case of a spherically symmetrical explosion you would not get an oscillatory wave as you pointed out  but the effect is more akin to a solitron or tsunami  a one shot deal from the standpoint of the time domain  but nontheless comprising many Fourier components in the frequency domain. Just how this type of wave does the job in the case of acceleration is not easy to see  if in fact it does.




#18
Jan3105, 06:19 AM

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On another tack  the sensitivity of gravitational wave detectors, LIGO etc. has been increasing all the time yet so far no detection has been confirmed. At what lower bound do we get worried and say that the predicted radiation does not exist?
In other words when do we expect to get a positive detection? 


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