- #1
Jonathan Thornburg -- remove -animal to reply
[[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