Re: Is Intrinsic Spin Compatible with GR?
- From: tessel@xxxxxxx
- Date: Wed, 11 May 2005 07:01:14 +0000 (UTC)
Subject: Re: Is Intrinsic Spin Compatible with GR?
I remarked:
>> See again CW
---that is, chapter 6 of the interesting semitechnical book
author = {Ignazio Ciufolini and John Archibald Wheeler},
title = {Gravitation and Inertia},
publisher = {Princeton University Press},
year = 1995}
>> and note that if the elementary scalar invariant
>>
>> R_(abcd) *R^(abcd)
>>
>> is nonvanishing, at least some spinning particles will experience
>> spin-spin forces (spinning particles with "aligned motion" might not, but
>> generic freefall motion will not be aligned).
Oz asked:
> Er, would you mind an elementary question?
Not at all.
> I'm trying to grasp this in a general (non mathematical) way.
[...]
> It seems to me that for a null-interaction between two spinning
> particles has either to be with spin axes mutually parallel or in the
> plane entirely mutually perpendicular. Personally I would go for the
> latter, would that be right?
Not quite, but you are right to suspect that the relative orientation of
the spin axes plays a key role.
In principle, according to gtr, two small spinning objects will
experience a spin-spin force (except perhaps for special relative
orientations of their spin axes), but this will be very -very- tiny!
Indeed, even in the situation I mentioned--- a small rapidly spinning
object (such as a gyroscope carried on board an artificial satellite)
orbiting a much larger spinning object (such as the Earth)--- the
spin-spin force is quite tiny. Still, it is reasonable to estimate it to
see whether we can hope to verify the existence of spin-spin forces (as
predicted by gtr) in an Earth-orbiting satellite experiment. To do this,
we can develop a straightforward idealized model of this situation by
considering test particles in equatorial or polar orbits (say) around a
rotating object modeled by the Kerr exterior vacuum.
Recall that on many past occasions we have discussed here the
interpretation of the Bel decomposition of the Riemann tensor in terms
of tidal forces and spin-spin forces
dot dot (delta x^a) = -E^a_b (delta x^b),
delta x^a = "deviation vector field"
and
(dot v^a)_initial = B^a_b s^b
v^a = velocity of spinning particle,
s^b = intrinsic spin of spinning particle
This works out most neatly for nonspinning inertial frames, which
therefore play a special role in gtr (they are the closest we can get,
in a general spacetime, to Lorentz frames). I won't repeat the basic
facts about the Bel decomposition or its physical interpretation here,
since we have discussed this so often in this group.
Recall further that some years ago we discussed the Bel decomposition
wrt the Hagihara and Doran frames in the Kerr vacuum, which we studied
using the Boyer/Lindquist and Doran coordinate charts respectively.
Recall that Hagihara observers are nonspinning inertial observers in
circular orbits in the equatorial coordinate plane (either corotating or
counter rotating). OTH, Doran observers are inertial observers who fall
in, if you will, "radially from rest at infinity"; they are the most
natural generalization to the Kerr vacuum of the well-known
Painleve/LeMaitre observers in the Schwarzschild vacuum, and except for
polar or equatorial infall, they are spinning (i.e. their spatial local
frame is "spinning" in the sense that the Fermi derivatives of e_1, e_2,
e_3 wrt e_0 are nonzero). Since spinning frames disturb the
interpretation of the components of B_(ab) in terms of spin-spin forces,
I'll only recall our discussion of polar and equatorial infalling Doran
observers.
Finally, recall that the Boyer/Lindquist chart is the natural
generalization to Kerr of the familiar Schwarzschild chart for the
Schwarzschild vacuum, whereas the Doran chart is the natural
generalization to Kerr of the Painleve chart for the Schwarzschild
vacuum.
OK, enough preliminaries! To answer your question, we showed that wrt
the timelike geodesic congruence defined by the world lines of these
observers, the Bel decomposition of the Riemann tensor (or Weyl tensor,
since we are dealing with a vacuum solution) reads as follows:
Hagihara observers:
E_(ab) = m/r^2 diag(-2,1,1) + m^2/r^4 diag(-3,3,0) + O(1/r^(9/2))
B_(12) = -3 m a/r^4 + O(1/r^(9/2)) (the other components vanish)
where m = mass of Earth, a = specific angular momentum of Earth (angular
momentum divided by mass), r = Boyer/Lindquist radial coordinate, theta
= polar angle, and where we use the Hagihara frame to compute "physical
components" of the Riemann tensor.
Here, m, a will correspond to the Newtonian values (rewritten in
relativistic units in which c = G = 1), as measured by observers very
far from the Earth (i.e. with large radial coordinate).
Then the form of E_(ab) tells us that the tidal forces on a small object
orbiting around the Earth in an equatorial circular orbit is the usual
Coulomb type radial stretching and orthogonal compression, scaling like
m/r^3, -plus- augmented radical stretching and augmented compression
parallel to the rotation axis of the Earth scaling like m^2/r^4 (in
these units, m << 1); the Coulomb tidal compression in the direction of
motion is unchanged. Note that this result doesn't depend upon a;
indeed, exactly the same thing happens for a nonrotating source
(Schwarzschild vacuum case). Note too that the expression for E_(ab) is
-traceless symmetric-; this must happen in any vacuum.
OTH, the form of B_(ab) tells us that if we carry a gyroscope inside our
satellite, it will experience a tiny spin-spin force, scaling like
ma/r^4, unless its spin axis is aligned with the direction of motion of
the satellite. Note that B_(ab) is -traceless symmetric-; this must
happen in any vacuum. (In any vacuum, as I said last time, E_(ab) and
B_(ab) each account for five of the ten linearly independent components
of the Weyl tensor. In the particularly symmetric case of the Kerr
vacuum, of course, there are far fewer linearly independent components.)
Notice that B_(ab) decays much faster than E_(ab); to wit, inverse
fourth power of radius, rather than inverse third power. In addition,
the magnitude of the spin-spin acceleration depends on the magnitude of
spin vector, which is for a gyroscope constructed with current
technology, e.g. using titanium, rather tiny. If you restore the usual
G, c numerical factors for ordinary physical units and plug in
reasonable values to compare the magnitude of the spin-spin and tidal
accelerations for reasonable pair of gyroscopes which might be carried
in a satellite in a low Earth orbit, the former is many orders of
magnitude smaller than the latter. (IIRC, last time we discussed this,
we did plug in some numbers, so I won't repeat the computation here.)
Doran observers:
E_(ab) = m/r^3 diag(-2,1,1) + O(1/r^5)
B_(ab) = 3 m a cos(theta)/r^4 diag(-2,1,1)
[ 0 1 0 ]
- 3 m a sin(theta)/r^4 [ 1 0 0 ]
[ 0 0 0 ]
where now r is the Doran radial coordinate (which at least at this level
of approximation is essentially the same as the Boyer/Lindquist
coordinate), and theta is the polar angle (zero or pi on the rotation
axis of the Earth), and where we use the Doran frame to compute the
"physical components" of the Riemann tensor.
Here, the form of E_(ab) tells us that the tidal forces are, to order
1/r^5, just the usual Coulomb forces (radial stretching plus orthogonal
compression).
The form of B_(ab) tells us that if one of these "radially" infalling
satellites (you probably want to think of them as "spiralling in", as
demanded by "frame dragging") should carry a gyroscope, we will measure
a tiny spin-spin force on the gyro which scales like m a/r^4, unless the
spin axis is orthogonal to the "gravitomagnetic field".
For example, consider a satellite falling in along the rotation axis of
the Earth (falling directly toward the North pole, say, so that theta =
0). If it carries a gyroscope, we will measure a tiny spin-spin force on
the gyro, which is largest in magnitude if the spin axis is parallel
(antiparallel) to that of the Earth, in which case we find a repulsive
(attractive) spin-spin force on the gyro.
Similarly, consider a satellite spirally in within the equatorial plane,
so that theta = pi/2. If it carries a gyroscope with spin axis which is
-not- parallel to e_3 (parallel to the @/@phi direction), we will
measure a tiny spin-spin force on the gyro, with direction and magnitude
depending on the orienation of the spin axis and on the location of the
gryoscope. Notice that B_(ab) is off-diagonal for Hagihara observers and
for equatorially infalling Doran observers, but diagonal for axially
infalling Doran observers.
Recall that to obtain these results, we took an exact solution (the Kerr
vacuum) and simply expanded the exact results in power series wrt 1/r,
in order to obtain simpler and approximately valid expressions. By a
curious coincidence, last week some new preprints appeared on the arxiv
which are relevant to our discussion. See in particular (26) in
gr-qc/0504146, where the same result as above is obtained for a
satellite falling radially inward along the rotation axis of the Earth.
The authors of this paper use the "gravitomagnetic formalism", which is
valid for weak fields.
This formalism is very well suited to an intuitive picture, because the
"gravitomagnetic field" behaves qualitatively, for an isolated rotating
object, much like the magnetic field produced by an isolated dipole
magnet. See the simple expression they derive for the spin-spin force on
a gryoscope carried by an orbiting satellite. Compare the simple
expression derived in Ciufolini & Wheeler (again using the
"gravitomagnetic formalism") for the Lense/Thirring precession of the
spin axis of a torque-free gryoscope carried by an orbiting satellite.
You can readily draw pictures which should help you confidently predict
the qualitative forces/precessions depending upon the direction of the
spin axis of our gyroscope.
You might recall that a remarkable fact about the Kerr vacuum is that
the geodesic equations are explicitly solvable; this happens only
because of the existence of a remarkable nontrivial Killing tensor,
which reflects a very special symmetry of the Kerr vacuum. OTH, AFAIK
there are essentially no nontrivial exact solutions for the problem of
-spinning- test particle motion (which as we have seen departs from the
motion of nonspinning freely falling test particles) in the Kerr vacuum.
As I said, the motion of spinning test particles in any spacetime is
given (according to gtr) by the Papapetrou/Dixon equations (so-called
because they were first found by Mathisson 1937, later rediscovered by
Papapetrou 1951 and then studied more carefully by Dixon 1973; strictly
speaking, there are various distinct formulations depending upon which
algebraic supplementary condition is imposed on the differential
equations in order to give a system with a unique solution). These
equations appear to be quite difficult to solve exactly, and IIRC no
nontrivial -exact- solutions are known even in such special cases as
Schwarzschild or Kerr! There are by now quite a few papers in the arxiv
reporting studies of the motion of test particles in the Kerr vacuum;
these employ either approximations or numerical simulations (or both).
Going back to my earlier remark as quoted above, in the Kerr vacuum,
written in either the Boyer/Lindquist chart or the Doran chart, the
principal curvature scalars are
R_(abcd) R^(abcd) = 48 m^2/r^6 + O(1/r^8)
R_(abcd) *R^(abcd) = -288 m^2 a cos(theta)/r^7 + O(1/r^9)
Notice that in the special case of the Schwarschild vacuum (a=0), the
latter vanishes, but otherwise (a =/= 0) is nonzero except in the
equatorial coordinate plane (theta = pi/2). OTH, the former does not
depend (to this order) on a.
When I mentioned "motion aligned with the gravitational field", I was
referring to the gravitational analog of a familar phenomenon in
Maxwell's theory of electromagnetism. Namely, the decomposition of the
EM tensor field into electric and magnetic vector fields is strongly
observer-dependent. Even in an electrostatic field, suitably moving
observers might measure a nonzero magnetic field; "aligned" observers
will measure only an electric field. OTH, in a "dynamic" EM field, -all-
observers will measure some magnetic field.
Similarly, in a -static- gravitational field, suitable observers can be
found who measure no gravitomagnetic field (i.e. in the Bel decompositon
adapted to their motion, B_(ab) vanishes), but in a -dynamic- field,
this will not be possible in general. In the Kerr vacuum, R_(abcd)
*R^(abcd) happens to vanish in the equatorial coordinate plane, so we
can find very special observers there for whom their B_(ab) vanishes.
BTW, it has been recognized for a very long time that Newtonian gravity
can eject the occasional star from a galaxy via an unlucky close
encounter with another star --- think of how NASA whips spaceprobes
around the solar system using judicious applications of force by the
field of Jupiter or Saturn. Some early work by Chandrasekhar and von
Neumann attempts to estimate the rate at which this occurs, as I think I
discussed in connection with our discussion last year of the solution of
PDEs by taking advantage of their symmetry groups.
GTR may provide at least two additional modes by which "gravitational
ejection" might occur. First, during the formation of a new black hole
by asymmetrical collapse of matter, "recoil" from a violent
gravitational wave emission mainly in some preferred direction might
literally "kick" the new hole right out of the galaxy. Second, during a
close encounter of two rapidly spinning black holes or neutron stars,
one of the pair might perhaps sometimes be ejected to spin-spin forces.
My understanding is that the second idea currently appears dubious, but
the first might be viable.
If they are effective, these various ejection processes should provide a
contribution to the famous "missing matter". However, AFAIK, all these
ejection mechanisms together are currently thought to be insufficient to
account for a substantial fraction of this missing matter.
Still, there could be quite a few isolated stars and other objects
wandering through intergalactic space, so science fiction authors can
contemplate the possible existence of some very lonely planetary
systems.
"T. Essel"
.
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