Selfforce in Radiation Reaction Formula

1
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
Index 1 Introduction LISA project 2
MiSaTaQuWa Self-force? 3 Adiabatic Metric
Perturbation 4 Radiation Reaction Formula 5
Gauge and Validity 6 Conclusion
Yasushi Mino Theoretical AstroPhysics Including
Relativity (TAPIR), CalTech E-mail
mino_at_tapir.caltech.edu
2
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
1 Introduction LISA Project
Everyday we experience gravity. Why should we
discuss the gravitational physics?
Newtonian Gravity Gravity as a potential No
dynamical Freedom
Einstein Gravity Gravity as a
geometry Dynamical Freedom in Gravity
We want to know the dynamics of the gravitational
physics!
3
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
The dynamical degree of freedom in Einsteins
Gravity propagates the space-time as gravity
waves. When linearized, we have a wave equation
of the metric perturbation.
Detection of gravitational waves is a strong
evidence of the dynamical nature of the
gravitational theory.
4
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
Einstein Gravity predicts a very strongly
self-gravitating object, called Black hole.
Newtonian Gravity It makes a singularity
if gravitationally collapsed.
Einstein Gravity The singularity is
hidden by the horizon.
We want to know the nature of the strong gravity!
5
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
Projects of Gravity Wave detection
Experimental test of the relativistic
gravitational theory New observational
window to distant astrophysical objects
Observation of highly relativistic gravitational
phenomena Promising Target NS/BH Binary
system, SuperNova, Primordial GWs, Pulser,
GammaRayBurst, ..
6
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
Space Project LISA (joint project by NASA ESA)
See LISA project homepage http//lisa.jpl.nasa.gov
/
7
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
Space Project LISA (joint project by NASA ESA)
See LISA project homepage http//lisa.jpl.nasa.gov
/
8
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
Space Project LISA (joint project by NASA ESA)
See LISA project homepage http//lisa.jpl.nasa.gov
/
9
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
LISA primary target A compact object (10M) is
inspiralling into a super-massive blackhole
(106M). extreme mass ratio eccentric
orbit relativistic motion We need to know what
gravitational waves are expected to be detected.
10
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
Theory is challenged by Experiment. Unlike other
theoretical physics, we do not (did not?) have a
theory to predict the observation until recently!
11
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
2 MiSaTaQuWa Self-force?
It is a good approximation to consider it as a
two-body problem in GR. The central black hole
is considered to be a Kerr black hole. For its
extreme mass-ratio, a linear perturbation might
be a good approximation.
m
M
12
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
Suppose we could use a linear perturbation We
approximate the supermassive black hole by a Kerr
black hole, and consider the linear metric
perturbation induced by an inspiralling compact
object.
One can calculate the gravitational waveform by a
linear perturbation, being given a orbit. We
need to solve the orbital equation.
13
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
A binary system is known to emit gravitational
waves, and the gravitational waves carry away the
binding energy of the system. As a result, the
orbit deviates from the background geodesic.
MiSaTaQuWa self-force was derived by a linear
perturbation. It is considered to include the
radiation reaction effect to the orbit.
(MiSaTaQuWaMino,Sasaki,TanakaQuinn,Wald)
14
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
We come to have a regularization method to
evaluate MiSaTaQuWa self-force. Radiation
Reaction Formula (Mino)
Mode-decomposition method Barack,Ori
Mino,Nakano,Sasaki Detweiler,Messaritaki,Whiting
,Kim Power-expansion method
(Mino,Nakano) but something weird
15
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
Do we have to consider the self-force in a
certain gauge? Is the orbital evolution
gauge-dependent?
16
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
One may choose a gauge condition such that the
self-force vanishes, and it is consistent with
the linear perturbation!
1) The linear perturbation is derived by solving
the linearized Einstein equation.
2) The linearized Einstein equation requires the
conservation law in the background, i.e. a
geodesic.
17
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
t
3) The linear approximation is valid only when
the orbital deviation from a geodesic is small.
4) Because the orbital deviation from a geodesic
is small, one can bring the orbits coordinates
back to those of the geodesic.
x
Note Gauge is a freedom to assign the
coordinates to a perturbed geometry. It has
nothing to do with the causality or hyperbolicity
of the Einstein equation.
18
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
Conclusion MiSaTaQuWa self-force makes no
physically meaningful prediction of the orbital
evolution by itself.
Conclusion There is no self-force.
Problem We have to extend the linear
perturbation formalism so that it can describe
the metric perturbation induced by a non-geodesic
orbit.
19
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
3 Adiabatic Metric Perturbation
We consider a quasi non-linear extension of the
linear metric perturbation so that one can
describe the metric perturbation induced by a
non-geodesic orbit. For this, we use 1) a
physically reasonable class of gauge
conditions, 2) the picture of adiabatic
approximation.
The adiabatic approximation is well known in
classical mechanics, but, the application to a
classical gauge field is not well known.
20
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
Kerr black hole is a stationary solution of the
Einstein equation, thus, the linearized Einstein
equation of a Kerr background is time-independent.

• We call this a physically reasonable class of
  gauge conditions.
• One can easily extract the physical information
  of gravitational waves.
• Technically feasible metric perturbation
  formalisms belong to this class.

21
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
Adiabatic extension of the linear perturbation
Step 1 We consider the spacelike foliation.
Step 2 We approximate the orbit by a geodesic on
each foliation hypersurface.
Step 3 We patch the linear metric perturbations
of geodesics on each foliation surface.
22
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
Linear perturbation
g a geodesic
Linear perturbation is valid as long as the orbit
does not deviate from a geodesic. (around a week)
Adiabatic metric perturbation
g(t) time-evolving geodesic
Adiabatic metric perturbation is valid as long as
the extra term L is sufficiently small. (around a
year)
23
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
4 Radiation Reaction Formula
Radiation Reaction Formula was originally
formulated by a linear perturbation in the
physically reasonable gauge. (Strictly speaking,
all the formula so far is based on the linear
perturbation, and we cannot make any physical
interpretation.) For a meaningful discussion of
the orbital evolution, we have to consider the
adiabatic extension of Radiation reaction formula
in a manner consistent with the adiabatic metric
perturbation.
24
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
4-a Original Radiation Reaction Formula
We consider a geodesic. Geodesics are
characterized by 6 parameters
Primary constants Secondary constants
25
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
r/q-motion
t/f-motion
Asymptotic gravitational field in the physically
reasonable class of gauge conditions
three principal frequencies, functions
of (E,L,C)
26
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
We consider the self-force acting on (E,L,C).
In the physically reasonable class of gauge
conditions, the linear metric perturbation gives
the force as
It is proven that the zero-mode is
gauge-invariant and can be obtained by the
radiative part of the field.
27
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
Perturbative evolution of the orbital constants
becomes
One can see that the perturbative evolution of
the orbit and the linear metric perturbation
becomes in valid at the dephasing time scale
(around a week)
28
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
4-b Adiabatic Radiation Reaction Formula
Now the constants could evolve
non-perturbatively
In the physically reasonable class of gauge
conditions, the adiabatic metric perturbation
gives the force as
(We ignore O(m2) terms here.)
29
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
We have the adiabatic evolution equation of the
orbit. (See the upcoming paper in detail.)
The behavior of these evolution equations
Einstein Equation for the adiabatic metric
perturbation
30
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
Validity of the adiabatic metric perturbation
Beyond this time scale, we do not solve the
Einstein equation to the accuracy O(m).
Validity of the adiabatic orbital evolution
Beyond this time scale, the second order effect
will change the orbital evolution.
This time scale is around several months for LISA.
31
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
5 Gauge and Validity
is proven to be gauge invariant in the physically
reasonable class. What about, non-zero components?
We found that the non-zero modes are totally
gauge dependent. By a special choice of gauge,
one can eliminate the non-zero mode.
32
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
We call this gauge condition the radiation
reaction gauge.
In this gauge, the self-force has only the
dissipative term. This DOES NOT mean that the
self-force does not have conservative terms. The
conservative effect of the self-force is
renormalized into the initial conditions.
33
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
General gauge in the physically reasonable class
Radiation reaction gauge
(v) is a typical velocity of the system, and is
0.3 at most.
34
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
Validity of the adiabatic metric perturbation
Beyond this time scale, we do not solve the
Einstein equation to the accuracy O(m).
Validity of the adiabatic orbital evolution
Beyond this time scale, the second order effect
will change the orbital evolution.
This time scale is around several years for LISA.
35
Astrophysics Seminar at University of Florida,
Gainesville, Sept. 24, 2004 Self-Force in
Radiation Reaction Formula
6 Conclusion
A method for an orbital prediction using a linear
metric perturbation is established for the first
time. The method can predict the orbital
evolution for long enough for LISA project. The
calculation technique of the method proposed here
is already established and the required
computational power is minimum. Coding to
calculate the waveform is in progress.