# Gravity waves

by yogi
Tags: gravity, waves
 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.
 Emeritus Sci Advisor P: 7,433 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 rho 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 point particles don't really belong in GR, so we have to assign the particle some fininte density rho over some finite volume v The analogous expression for a photon, which can't be stationary, moving in the plus or minus x direction is  rho n*rho 0 0 n*rho rho 0 0 0 0 0 0 0 0 0 0 here n is 1 or -1, depending on whether the photon is moving in the +x or -x direction 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.
 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
P: 8,470

## 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:
Emeritus
P: 7,433
 Quote by yogi 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
I still don't see why you think the resulting field will have any sort of spherical symmetry. There is a cylindrical spatial symmetry to the problem, but no spherical symmetry.

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).
 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?
 Emeritus Sci Advisor P: 7,433 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.
 P: 1,444 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.
 Emeritus Sci Advisor P: 7,433 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, non-rotating spherical collapse won't generate gravitational waves, nor will a symmetrical spherical non-rotating 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.
P: 90
 Quote by yogi 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.
I think we are at the dawn of a new age of astronomy with gravitational radiation. We have indirect evidence with Einstein's quadrupole formula and the spin down of binary pulsars and appear very close to direct detection with Ligo. Beyond the weak field approximation, we have a lot of Ricci flat solutions and a few good theorems. Check out the peeling theorem and the News function.

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.
 Emeritus Sci Advisor P: 7,433 Where could I find out more about the peeling theorem and the News function?
 P: 90 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".
 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?