# Explanation of why there can't be dipole gravitational radiation

Jonathan Thornburg -- remove -animal to reply

## Main Question or Discussion Point

[[disclaimer: this is *not* a homework assignment]]

In general relativity, the lowest non-vanishing multiple of gravitational
radiation is generically the quadrupole: the monopole is forbidden
by Birkhoff's theorem, and conservation of momentum is forbidden by
conservation of momentum. I thought I understood this latter argument
(set out in Misner, Thorne, and Wheeler section 36.1)... but in
discussing this point with a colleague, I've become less certain
that I understand it.

To focus the discussion, let's consider the asymmetric collision of
two stars (merging to form a bigger star), and let's suppose the system
is *not* relativistic, i.e. let's suppose that Newtonian gravity/mechanics
provide a good approximation to the dynamics. MTW's argument simply
says that any change in the mass dipole moment of the system would
violate conservation of linear momentum.

The problem is, gravitational radiation can carry linear momentum,
and the MTW formula only applies to the dipole moment of the *masses*
in the system. How do I know that the system can't emit dipole
gravitational radiation, with the final merged stars recoiling in
the opposite direction so that the total linear momentum of the
mass+gravitational-radiation system is conserved?

To look at the issue from a slightly different perspective, let's
look at standard quadrupolar gravitational radiation. We know that
which (in general) *does* carry a net linear momentum, with the merged
star recoiling in the opposite direction. How do I know that this
can't also apply to dipole gravitational radiation?

Just to make matters more interesting, suppose we now drop the
Newtonian-gravity approximation, and consider (say) the asymmetric
collision of two black holes. In this case it's known from numerical
simulations that asymmetric collisions generally radiate a net linear
momentum in (quadrupole) gravitational radiation (see, eg, Sperhake
et al, Physical Review Letters 98, 091101). How do I now that this
isn't also the case for dipole gravitational radiation?

Can anyone offer any insights here?

thanks, ciao,

--
-- "Jonathan Thornburg -- remove -animal to reply" <jthorn@aei.mpg-zebra.de>
School of Mathematics, U of Southampton, England
"Washing one's hands of the conflict between the powerful and the
powerless means to side with the powerful, not to be neutral."
-- quote by Freire / poster by Oxfam

## Answers and Replies

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Jonathan Scott
On 3 Apr, 19:21, "Jonathan Thornburg -- remove -animal to reply"
<jth...@aei.mpg-zebra.de> wrote:
> ...
> The problem is, gravitational radiation can carry linear momentum,
> and the MTW formula only applies to the dipole moment of the *masses*
> in the system. How do I know that the system can't emit dipole
> gravitational radiation, with the final merged stars recoiling in
> the opposite direction so that the total linear momentum of the
> mass+gravitational-radiation system is conserved?

The MTW expression for the dipole moment in terms of masses is a
simple Newtonian approximation; exact conservation of energy and
momentum would require it to include all other forms of energy and
momentum within the system, including gravitational. That means that
regardless of what is happening in the system (including any higher
order multipole gravitational radiation) it is not possible to move
the center of mass of the system as a whole, and hence it is not
possible to create dipole radiation.

Jonathan Scott (Chandlers Ford, near Southampton, England)

Hi this (old) topic interest me a lot. I am not sure to understand what you say correctly. Because it is not possible to move the center of mass, ok, but there exist a "frame" where radiation carry energy, thus it should be possible to see the same center of mass/energy no? I explain myself: In this reference frame for example, a small body falling onto a big body will acelerate a lot. If I take the analogy with electromagnetism, what we call dipolar radiation, is not only really the dipolar one, but also the "late part" of electromagnetic field, in 1/r that also comes from a sole accelerating charge. Why a small body falling on a big shouldn't radiate, or more accurately have the same 1/r part? We don't speak about point test particle here, and carrying on the analogy with electromagnetism, a small body accelerating a lot radiates (so loses) more energy than a big accelerating just a little. This is why some accelerators use protons instead of neutrons. So maybe the small body will lose more energy than the big one, and the retroaction should be more important. I guess this is the question, the center of mass doesn't look to change in a quadrupolar radiation, it doesn't seem to hange in a two body system orbiting, but bodies don't follow geodesics.
In electromagnetism the energy is slightly radiate in front of the charge, factor depending on the relative speed of the charge to the observer. So are you definitely totally sure there shouldn't be a radiation???

Second, let's imagine two equal masses, one charged, the other not. Maybe the charged one would radiate from a far perspective while falling on the body, or the equivalence principle prevent it from losing energy by electromagnetic radiation?

If it radiates electromagnetic radiations, then the repelling that comes from self interaction should slow down the charged particle, and the other could radiate gravitationally more because it would accelerate more, or then the center of mass/energy should change.

Sorry for the english, I am french.

I read more carefully what you said, and I think I understood, sorry... So there are gravitationnal dipolar radiations, IF we don't take it as a part of the stress energy tensor source? Am I right?