Gravitomagnetic tidal currents
References
Gravitomagnetic tidal currents
Eric Poisson
Department of Physics, University of Guelph
Atlantic GR Workshop, St. John’s, May 29–31, 2017
Eric Poisson
Gravitomagnetic tidal currents
Gravitomagnetic tidal currents
References
Setting and assumptions
We consider a self-gravitating body of mass M and radius R; the body
rotates rigidly with angular velocity Ω.
We assume that the rotation is slow, we work to first order in Ω, and
ignore centrifugal deformations.
The body is placed in a gravitomagnetic tidal field created by remote
bodies.
The tidal field is assumed to vary on long spatial and time scales.
We wish to determine the perturbation created by the coupling between
the gravitomagnetic tidal field and the body’s rotation.
Eric Poisson
Gravitomagnetic tidal currents
Gravitomagnetic tidal currents
References
Governing equations
Post-Newtonian gravity:
Euler’s equation:
Continuity equation:
Equation of state:
∇2 U = −4πGρ,
∇2 Ua = −4πGρ va
dva
= −∂a p + ρ ∂a U
ρ
dt
i
4ρ h
+ 2 ∂t Ua + (∂b Ua − ∂a Ub )v b
c
dva
= ∂t va + v b ∂b va
dt
∂t ρ + ∂a (ρv a ) = 0
p = p(ρ)
The unperturbed body is spherical, in hydrostatic equilibrium.
Eric Poisson
Gravitomagnetic tidal currents
Gravitomagnetic tidal currents
References
Gravitomagnetic tidal field
An orbiting companion of mass M 0 , position r 0 , and velocity v 0 creates a
gravitomagnetic potential Ua = GM 0 va0 /r0 .
Taylor expansion about the reference body’s centre of mass gives
1
Uatidal = − abc B cd xb xd ,
6
Bab =
6GM 0 0
(n × v 0 )(a n0b)
r03
where n0 = r 0 /r0 .
For a circular orbit in the x-y plane, the nonvanishing components of the
tidal moment Bab are
Bxz =
r
ω0 =
3GM 0 v 0
cos ω 0 t,
r03
G(M + M 0 )
,
r03
Eric Poisson
3GM 0 v 0
sin ω 0 t
r03
r
G(M + M 0 )
0
0 0
v =ωr =
r0
Byz =
Gravitomagnetic tidal currents
Gravitomagnetic tidal currents
References
Perturbation equation
Introduction of the gravitomagnetic tidal field, Ua = δUa = Uatidal ,
perturbs the body.
All other variables become perturbed, ρ → ρ + δρ, U → U + δU , and
va → va + δva = (Ω × x)a + δva
The Euler equation
i
1
4h
∂t va + v b ∂b va = − ∂a p + ∂a U + 2 ∂t Ua + (∂b Ua − ∂a Ub )v b
ρ
c
becomes
1
δρ
∂t δva + v b ∂b δva + (∂b va )δv b = − ∂a δp + ∂a U + ∂a δU
ρ
ρ
i
4h
b
+ 2 ∂t Ua + (∂b Ua − ∂a Ub )v
c
after perturbation.
Eric Poisson
Gravitomagnetic tidal currents
Gravitomagnetic tidal currents
References
Gravitomagnetic induction
Suppose first that the body is nonrotating, Ω = 0, so that va = 0.
Then all scalar perturbations vanish, δρ = δp = δU = 0, because scalars
formed from Bab have the wrong parity (they are pseudoscalars).
The only nonvanishing perturbation is δva , and the perturbation equation
reduces to ∂t δva = (4/c2 )∂t Ua .
It implies that δva − (4/c2 )Ua is independent of time.
Assuming that the tidal field vanishes at t = −∞, the solution is
δva =
4
Ua
c2
The tidal interaction creates a velocity field within the body:
gravitomagnetic induction. This is an inescapable consequence of the
relativistic circulation theorem.
Eric Poisson
Gravitomagnetic tidal currents
Gravitomagnetic tidal currents
References
Coupling with rotation
Now switch on the body’s rotation, so that v = Ω × x.
This triggers the remaining terms in the perturbation equation, including
the Lorentz-like term v × (∇ × U ).
This creates perturbations δρ, δp, δU , as well as an additional velocity
perturbation
δva =
4
Ua + wa ,
c2
wa = O(Ω)
The resulting system of partial differential equations is fairly complicated,
and can be dealt with by decomposing all variables in spherical harmonics.
The coupling between Ωa and Bab implies that the equations decouple
into dipole (` = 1), quadrupole (` = 2), and octupole (` = 3) sectors.
For a simple polytropic model, p = Kρ2 , the quadrupole and octupole
sectors can be handled analytically. The dipole sector requires numerical
integration.
Eric Poisson
Gravitomagnetic tidal currents
Gravitomagnetic tidal currents
References
Approximation
To illustrate the outcome of this detailed analysis, we neglect the scalar
perturbations δρ, δp, and δU .
With δva = (4/c2 )Ua + wa , the perturbation equation
∂t δva + v b ∂b δva + (∂b va )δv b =
i
4h
b
∂
U
+
(∂
U
−
∂
U
)v
t
a
b
a
a
b
c2
becomes
i
4h
b
b
(∂
v
)U
+
v
∂
U
b
a
a
b
c2
i
2 h
= 2 Ωa xb xc Bbc − 2Ωb xb xc Bac + r2 Ωb Bab
3c
∂t wa = −
The solution is
"
#
Z
Z
Z
2
b c
b c
2 b
wa = 2 Ω a x x
Bbc dt − 2Ωb x x
Bac dt + r Ω
Bab dt
3c
Eric Poisson
Gravitomagnetic tidal currents
Gravitomagnetic tidal currents
References
Gravitomagnetic tidal currents
When Bab is idealized as stationary, wa grows linearly with time.
in circular orbit, Bab ∝ cos ω 0 t and
RFor a binary system
0−1
Bab dt ∝ ω
sin ω 0 t.
The velocity field acquires a factor of 1/ω 0 . It scales as
w∼
0 0
GM 0 ΩR2
1
2 1 GM v
ΩR
=
c2
ω 0 r03
c2 r02
For a binary neutron star of relevance to LIGO,
w∼
M0
1.4 M
2.8 M
M + M0
2/3 Eric Poisson
R
12 km
2 100 ms
P
Gravitomagnetic tidal currents
f
100 Hz
4/3
km
s
Gravitomagnetic tidal currents
References
Velocity field
Eric Poisson
Gravitomagnetic tidal currents
Gravitomagnetic tidal currents
References
Recapitulation
We have a body of mass M , radius R, angular velocity Ω, immersed in a
gravitomagnetic tidal field
B tidal = ∇ × U tidal ∼ GM 0 v 0
x
r03
The tidal field exerts a force density
f=
1
ρ v × B tidal ,
c2
v =Ω×x
Euler’s equation ρ ∂t δv = f implies
∂t δv ∼
1
R
(ΩR)(GM 0 v 0 ) 03
c2
r
When the tidal field oscillates with frequency ω 0 ,
δv ∼
0 0
1
2 1 GM v
(ΩR
)
c2
ω 0 r03
Eric Poisson
Gravitomagnetic tidal currents
Gravitomagnetic tidal currents
References
Mode analysis
The response of a fluid body to an applied force can be calculated by
expanding the fluid perturbation in normal modes.
Each mode behaves as a driven harmonic oscillator,
ξ̈ + ω 2 ξ = f
ξ = displacement of a fluid element from its unperturbed position
ω = mode’s natural frequency
The natural frequency of each mode corresponds to the restoring forces
exerted by the fluid.
The fluid’s complete response is obtained by summing over all modes.
Eric Poisson
Gravitomagnetic tidal currents
Gravitomagnetic tidal currents
References
Zero-frequency modes
When ω 6= 0 and the external force f varies slowly, the acceleration term
can be neglected in the mode equation, and the perturbed equilibrium is
described by ξ = f /ω 2 .
But in the absence of a restoring force, ω = 0, and an equilibrium state is
not possible.
Instead the mode equation gives
Z
δv = ξ̇ =
f dt
in agreement with the previous analysis.
The gravitomagnetic tidal currents correspond to a class of
zero-frequency modes (g-modes and r-modes) for the fluid.
Eric Poisson
Gravitomagnetic tidal currents
Gravitomagnetic tidal currents
References
References
1
P. Landry and E. Poisson, Dynamical response to a stationary tidal
field, Phys. Rev. D 92, 124041 (2015).
2
E. Poisson and J. Douçot, Gravitational tidal currents in rotating
neutron stars, Phys. Rev. D 95, 044023 (2017)
3
P. Landry, Tidal deformation of a slowly rotating material body:
Interior metric and Love numbers, arXiv:1703.08168 (accepted for
publication in Phys. Rev. D).
Eric Poisson
Gravitomagnetic tidal currents