Einstein's Quadrapole Model for Gravity Waves

[yogi]

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.

 

[pervect]

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.

 

[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

 

[JesseM]

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:

Peter Ammon (Peter_Am...@rocketmail.com) wrote:

: Consider a system (of stars and whatnot) that's at rest relative to me.
: Because momentum is conserved, the center of mass of the system must
: always be at the same distance (and of course the system has constant
: mass). How, then, can this system generate a gravity wave?

Others have given good partial answers to this question, but there's
a piece that deserves more attention---it actually implies something
profound about gravitational waves.

Consider an isolated system that's acting as a source of gravitational
waves. If the system is really isolated, then as you say, its center
of mass cannot accelerate, and we may as well take it to be at rest.
Further, to a good approximation, its total mass will be constant
(except for energy carried off by gravitational waves, but that's
a higher order effect).

Now, whenever you're dealing with radiation, you can carry out a
multipole expansion. The monopole component, which you can think
of as spherical waves, depends on derivatives of the "monopole
moment," which for gravity is the mass. Since the mass is constant,
there can be no such spherical radiation. (The same is actually
true in electromagnetism, where the monopole moment is the charge.)

The next moment up is the dipole moment. Dipole radiation has (more
or less) two "lobes"; I can't draw a picture in this medium, but you
can find one in any good book about E&M. Now, the dipole moment
for gravity is basically the location of the center of mass, which
as you say doesn't change. So there can be no gravitational
dipole radiation. (In electromagnetism, in contrast, the "center
of charge" of a system certainly can vary in time, so dipole
radiation is allowed.)

This means that the lowest order gravitational radiation allowed
is quadrupole radiation. This is an important prediction, that
leads to many of the differences between gravitational and E&M
radiation. It also means, unfortunately, that gravitational
radiation is hard to generate---a symmetrically collapsing star,
for instance, won't emit quadrupole radiation, so any gravitational
radiation would have to come from assymetries in the collapse.

Steve Carlip

 

[pervect]

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).

 

[yogi]

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?

 

[pervect]

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.

 

[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.

 

[pervect]

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 statement - 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.

 

[Rob Woodside]

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.

 

[pervect]

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

 

[Rob Woodside]

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".

 

[yogi]

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?

 

[pervect]

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.

[edit-add]
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 traveling at 'c' are going to work in any theory that's Lorentz invariant.

 

[yogi]

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.

 

[pervect]

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 non-static 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 non-rotating 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.

 

[yogi]

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.

 

[Garth]

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?

 

[Creator]

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?

I would rephrase the question: At what point do we finally admit that the method of laser interferometry is unable to detect them?.

 

[pervect]

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.

As a practical matter, the first step would be to get a better understanding of classical relativistic electrodynamics. On the less-mathematical level, here are a lot of pictures out there that show how if you suddenly stop a rapidly moving charge, there must be a huge discontinuity in the field lines. This translates in that a rapidly deaccelerating charge must emit radiation.

I think I've posted some links to some Java applets that do this in the past.
You should be able to find them either b looking on the forum, or by doing a direct google for "field of a moving charge".

These same general arguments suggest that a mass that's rapidly moving and then stops suddenly should also emit gravitational radiation. A buzzword that might help is "larmor radiation".

 

[yogi]

I am familiar with the classical derivation of the em wave spawned by acceleration - my question related to the properties of space that determine the speed and nature of the hypothesized quadrupole wave. In electromagnetic theory we have a permeability and permittivity that define the free space velocity and the free space impedance - What are the factors in G wave propagation that correspond to the electrical properties of space that determine EM wave propagation?

In short - to get a G wave according to the theory, one needs a forth time rate of change of reduced quadruple moment. In contrast, the electric field is kinked by the acceleration of the charge - there is an existing field which is modified by the addition of energy to the charge - if we rapidly accelerate a mass we have an analogous situation - additional energy is added to the particle - moreover, if the particle is accelerated rapidly to a relativistic velocity relative to some test mass, there should be an abrupt increase in the inertia and consequently an abrupt increase in the G force (a la equivalence) acting between the accelerated particle and the test mass. This leads back to my original question which was the basis for the thread - namely is it possible to explain the energy relationship that was the basis for the quadruple hypothesis - in terms of a longitudinal wave that will impact spacetime curvature?

 

[Garth]

I would rephrase the question: At what point do we finally admit that the method of laser interferometry is unable to detect them?.

Point taken, however if no method of detection is positive then we might conclude that they do not actually exist, except as a solution to the field equation of a theory.

I am not suggesting that this might be the case but should we be worried yet? In other words should we have already made a detection?
Garth

 

[Chronos]

We do have indirect evidence of gravity waves. The orbital decay of binary neutron stars comes to mind. There is a fair amount of optimism LIGO has the sensitivity necessary to detect them.

 

[Garth]

I agree, but as LIGO has not positively detected gravity waves when can we expect it to detect a binary pulsar, or other known sources of gravitational radiation? Should it have done so already?

The indirect detection via the precise orbital decay of the binary pulsar is impressive, but the interpretation of that loss as gravitational wave energy loss is stilll theory dependent, it would be good to have an independent confirmation.