- #1

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

an asymmetric star collision radiates quadrupole gravitational radiation

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

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

an asymmetric star collision radiates quadrupole gravitational radiation

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