Fully relativistic three-dimensional Cauchy-characteristic matching
for physical degrees of freedom
Sizheng Ma ,
1
,*
Jordan Moxon ,
1
Mark A. Scheel,
1
Kyle C. Nelli ,
1
Nils Deppe ,
2,3,1
Marceline S. Bonilla ,
4
Lawrence E. Kidder ,
3
Prayush Kumar,
5
Geoffrey Lovelace,
4
William Throwe,
3
and Nils L. Vu
1
1
TAPIR 350-17,
California Institute of Technology
, 1200 East California Boulevard,
Pasadena, California 91125, USA
2
Department of Physics,
Cornell University
, Ithaca, New York 14853, USA
3
Cornell Center for Astrophysics and Planetary Science,
Cornell University
, Ithaca, New York 14853, USA
4
Nicholas and Lee Begovich Center for Gravitational-Wave Physics and Astronomy,
California State University Fullerton
, Fullerton, California 92834, USA
5
International Centre for Theoretical Sciences,
Tata Institute of Fundamental Research
,
Bangalore 560089, India
(Received 20 August 2023; accepted 13 May 2024; published 11 June 2024)
A fully relativistic three-dimensional Cauchy-characteristic matching (CCM) algorithm is implemented
for physical degrees of freedom in a numerical relativity code
S
p
ECTRE
. The method is free of
approximations and can be applied to any physical system. We test the algorithm with various scenarios
involving smooth data, including the propagation of Teukolsky waves within a flat background, the
perturbation of a Kerr black hole with a Teukolsky wave, and the injection of a gravitational-wave pulse
from the characteristic grid. Our investigations reveal no numerical instabilities in the simulations. In
addition, the tests indicate that the CCM algorithm effectively directs characteristic information into the
inner Cauchy system, yielding higher precision in waveforms and smaller violations of Bondi-gauge
constraints, especially when the outer boundary of the Cauchy evolution is at a smaller radius.
DOI:
10.1103/PhysRevD.109.124027
I. INTRODUCTION
Since the detection of GW150914
[1]
, gravitational wave
(GW) astronomy has become a flourishing field. Accurate
modeling of GW signals is a key ingredient in extracting
signals from detector noise and understanding the proper-
ties of sources. To date, numerical relativity (NR) remains
the only
ab initio
method to simulate the major sources
of the GW signals: the coalescence of binary black hole
(BBH) systems.
Generally speaking, the formulations of NR can be
classified into two groups: Cauchy
[2,3]
and characteristic
[4
–
11]
formalism, depending on how spacetime is foli-
ated.
1
For the Cauchy approach, a spacelike foliation is
adopted, and Einstein
’
s equations are split into evolution
and constraint sets. This formalism has successfully led to
high-accuracy simulations of BBH systems
[3]
.
On the other hand, in the characteristic approach,
spacetime is sliced into a sequence of null hypersurfaces
that extend to future null infinity. Einstein
’
s equations are
formulated in terms of the unambiguous geometric treat-
ment of gravitational radiation in curved spacetimes due to
Bondi
et al.
[17]
, Sachs
[18]
, and Penrose
[19]
. Meanwhile,
future null infinity is rigorously encompassed on the
characteristic grid via a compactified coordinate system
and treated as a perfect absorbing outer boundary. In this
way, one is able to extract faithful GWs with the character-
istic formalism at future null infinity without any ambiguity
[20
–
24]
. However, the characteristic method cannot evolve
the near-field region of BBHs when caustics of null rays are
present
[25
–
30]
, because for this method coordinates are
chosen to follow null rays so caustics result in coordinate
singularities. Therefore, in practice, one can use the Cauchy
evolution to simulate the near-zone of the systems and
construct metric data on a timelike worldtube
[9
–
11,31]
.
Then the characteristic system propagates the world-
tube data nonlinearly to future null infinity, which in
turn yields GW information there. This procedure of
extracting GWs is known as
Cauchy-characteristic evolu-
tion
(CCE)
[4
–
11,32]
. Studies of characteristic evolution
and CCE date back to the 1980s. Isaacson
et al.
[33]
and
Winicour
[34,35]
considered a prototype of CCE by
shrinking the worldtube to a timelike geodesic and inves-
tigated the GWs emitted by an axially symmetric ideal
fluid. More complete and complicated CCE systems were
developed later
[6,7,36
–
40
–
42]
. Early applications of the
characteristic evolution were focused on simulating generic
*
sma@caltech.edu
1
The third group adopts hyperboloidal slicing
[12
–
16]
. Its
discussion is beyond the scope of this paper.
PHYSICAL REVIEW D
109,
124027 (2024)
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=
2024
=
109(12)
=
124027(28)
124027-1
© 2024 American Physical Society
three-dimensional (3D) single-black-hole spacetimes
[43]
,
Einstein-perfect fluid systems
[44
–
48]
, Einstein-Klein-
Gordon systems
[47,49
–
52]
, (nonlinear) perturbations of
BHs
[53
–
56]
, event horizons
[57
–
59]
, fissioning white
holes
[60]
, extreme mass ratio inspirals
[61]
, stellar core
collapse
[62]
, as well as linearized systems
[63]
. By using
finite-difference methods, PITT null
[6
–
8]
was the first
code to implement CCE and characteristic evolution, which
led to the first CCE simulation of BBH systems
[64
–
67]
.
The code was also used to extract GWs emitted by rotating
stellar core collapse
[68]
. Later, a spectral algorithm for
CCE was built as a module in
S
p
EC
[9,69
–
71]
and
S
p
ECTRE
[10,11]
, developed by the SXS collaboration
[3,72
–
75]
.
Bhagwat
et al.
[29]
used
S
p
EC
CCE to investigate the
start time of BBH ringdown. And
S
p
ECTRE
CCE has
been applied to computing memory effects
[76,77]
, fixing
the Bondi-Metzner-Sachs frame of GWs
[78,79]
, extracting
GWs emitted by black hole-neutron star binaries
[80]
,
computing GW echoes
[81]
, and constructing a NR
surrogate model
[82]
.
Although CCE has led to high-accuracy and unambigu-
ous GWs at future null infinity, CCE
’
s data flow is one-
way, meaning that the Cauchy evolution does not depend
at all on the characteristic evolution. This is inaccurate
because for a nonlinear set of equations like general
relativity, outgoing radiation at arbitrarily far distances
can backscatter off the spacetime curvature and eventually
affect the source; the Cauchy evolution (with or without
CCE) fails to capture this backscatter. To explain this in
more detail, note that to perform a Cauchy simulation, the
spatial Cauchy domain is typically truncated at a finite
distance from the source, with suitable boundary conditions
provided at the artificial outer boundary.
2
Ideally speaking,
perfect boundary conditions would make the artificial
boundary as transparent as possible so that the numerical
solution is identical to one that would be evolved on an
infinite domain, and these boundary conditions would
ideally include nonlinear backscatter. On the contrary, if
poor boundary conditions are prescribed, not only will the
backscatter be incorrect, but also spurious reflection can be
introduced at the boundary and contaminate the whole
simulation. In
S
p
EC
[73]
and
S
p
ECTRE
[74,75]
, the gener-
alized harmonic (GH) evolution system
[83]
is adopted for
the Cauchy simulation, whose boundary conditions can be
divided into three subsets: constraint-preserving, physical,
and gauge boundary conditions
[84]
. Effort has been made
to improve the accuracy of these boundary conditions, such
as Refs.
[85
–
88]
. In particular, the boundary conditions on
the physical degrees of freedom are expected to encode the
information of the backscattered (incoming) GWs that enter
the Cauchy domain. Accurate modeling of the backscat-
tered radiation is not a trivial task. Although there were
some attempts
[85,86]
to improve the physical boundary
conditions, in most
S
p
EC
production simulations
[3]
the
incoming GWs at the boundary are treated by freezing the
Weyl scalar
ψ
0
to zero
[84]
, which effectively eliminates all
backscatter from beyond the outer boundary.
It was pointed out that the characteristic evolution is a
natural system to compute the value of the backscattered
radiation in an exact and efficient way, e.g., see Ref.
[4]
and
references therein. A matching of the internal Cauchy
system and the exterior characteristic system is expected
to provide accurate physical boundary conditions for the
Cauchy module. In this way, the interface between the two
grids is transparent and GWs can pass cleanly off of and
onto the Cauchy grid. This algorithm is known as
Cauchy-
characteristic matching
(CCM). Historically, the idea of
CCM was outlined in Refs.
[89]
and
[90]
. Then the
algorithm was applied to the evolution of a scalar field
on a flat background
[91,92]
, and around a Schwarzschild
BH
[93]
(with metric being fixed). The CCM simulation of
gravitational systems was also visited by a series of papers
[94
–
98]
that assumed cylindrical symmetry
[94,95,98]
and
axial symmetry
[96,97]
. Meanwhile, CCM was used to
study an Einstein-perfect fluid system
[99]
and an Einstein-
Klein-Gordon system
[100]
with spherical symmetry.
Going to the 3D regime, Bishop
et al.
investigated a scalar
wave
[101]
. Szilagyi
et al.
[102]
performed the matching
in linearized harmonic coordinates. An alternative to
CCM is Cauchy-perturbative matching
[103
–
105]
, where
the exterior region is not evolved fully nonlinearly with a
characteristic code but instead is treated as a linearized
Schwarzschild BH. This algorithm led to a simulation
of a 3D Teukolsky wave
[106]
propagating on a flat back-
ground
[103]
. Later, this topic was revisited
[107]
in 2005
after years of progress in numerical relativity. However,
until now, all the existing matching algorithms for the
gravitational sector are based on either assumptions (sym-
metries) or approximations (perturbative matching, linear-
ized equations); a full matching in three spatial dimensions
is still missing. Further, although the existence and unique-
ness of CCE solutions have been established in
[108,109]
,
and a linearized characteristic system was found to be sym-
metric hyperbolic
3
[111]
, the full CCE system is only weakly
hyperbolic
[112
–
114]
, rendering CCM not well-posed.
As a step toward addressing those questions, in this
paper, we perform fully relativistic 3D CCM for physical
degrees of freedom of gravitational fields without any
approximation. The code is implemented in
S
p
ECTRE
[74,75]
. Unlike CCE, the data in CCM flows in both
directions, meaning that the Cauchy and characteristic
systems need to be evolved simultaneously. The commu-
nication from the Cauchy to the characteristic system has
been discussed extensively
[9
–
11]
. In this paper, we will be
2
In this paper we restrict our discussions to the outer boundary.
3
See also
[110]
for the well-posedness of the characteristic
formulation for the Maxwell equations.
SIZHENG MA
et al.
PHYS. REV. D
109,
124027 (2024)
124027-2
explaining how to feed the information of the characteristic
module back to the Cauchy system.
This paper is organized as follows. In Sec.
II
, we review
the Cauchy evolution in
S
p
ECTRE
, with particular attention
given to its physical boundary conditions. Next in Sec.
III
we discuss some basic information about the characteristic
module in
S
p
ECTRE
. Then a thorough algorithm to complete
the matching procedure is introduced in Sec.
IV
. Our code
is tested with several types of physical systems in Sec.
V
.
Finally, we summarize the results in Sec.
VI
.
Throughout this paper we use Latin indices
i; j; k;
...
to
denote 3D spatial components; and Greek indices
μ
;
ν
;
...
for 4D spacetime components. We generally avoid using
abstract indices, denoted by Latin letters from the first part
of the alphabet
a; b;
...
, to keep the text concise, unless
stated otherwise.
II. SUMMARY OF THE CAUCHY EVOLUTION
AND ITS BOUNDARY CONDITIONS
The detailed communication (matching) algorithm
depends on the formulation of the Cauchy evolution. For
instance, the perturbative matching in Ref.
[103]
was
performed through Dirichlet and Sommerfeld boundary
conditions. In
S
p
ECTRE
, the Cauchy data are evolved with
the GH formalism
[83]
. Outer boundary conditions are
imposed via the Bjørhus method
[84,115]
: the time deriv-
atives of the incoming characteristic fields are replaced on
the boundary. In this section, we provide a brief overview of
the Cauchy evolution and refer the reader to Ref.
[83]
for
more details. We specifically give more attention to the
physical subset of the boundary conditions
[84]
.
The Cauchy evolution relies on the
3
þ
1
decomposition
of a metric tensor
g
μ
0
ν
0
ds
2
¼
g
μ
0
ν
0
dx
0
μ
0
dx
0
ν
0
¼
−
α
2
þ
β
i
0
β
i
0
γ
i
0
j
0
dt
0
2
þ
2
β
i
0
γ
i
0
j
0
dx
0
j
0
dt
0
þ
γ
i
0
j
0
dx
0
i
0
dx
0
j
0
;
ð
2
:
1
Þ
with
α
the lapse function,
β
i
0
the shift function, and
γ
i
0
j
0
the spatial metric.
4
We use primes on the coordinates to
distinguish them from different coordinate systems that will
be introduced later; see Fig.
1
. Then the vacuum Einstein
equations,
R
μ
0
ν
0
¼
0
, can be cast into a first-order sym-
metric hyperbolic (FOSH) evolution system
FIG. 1. Coordinate systems used in the
S
p
ECTRE
CCE and CCM modules. The interior Cauchy evolution uses the Cauchy coordinates,
whereas the exterior characteristic system adopts the partially flat Bondi-like coordinates. To achieve their communication, two
intermediate coordinate systems are introduced.
4
In Ref.
[83]
, the authors used
ψ
a
0
b
0
and
g
i
0
j
0
to refer to the
spacetime metric and the spatial metric, respectively.
FULLY RELATIVISTIC THREE-DIMENSIONAL CAUCHY-
...
PHYS. REV. D
109,
124027 (2024)
124027-3
∂
t
0
u
α
0
þ
A
k
0
α
0
β
0
∂
k
0
u
β
0
¼
F
α
0
;
ð
2
:
2
Þ
where
u
α
0
¼f
g
μ
0
ν
0
;
Π
μ
0
ν
0
;
Φ
i
0
μ
0
ν
0
g
is a collection of dynami-
cal variables,
Π
μ
0
ν
0
¼
α
−
1
ð
β
i
0
∂
i
0
g
μ
0
ν
0
−
∂
t
0
g
μ
0
ν
0
Þ
and
Φ
i
0
μ
0
ν
0
¼
∂
i
0
g
μ
0
ν
0
are related to the time and spatial derivatives of the
metric.
The FOSH system in Eq.
(2.2)
is symmetric hyperbolic,
and its characteristic fields
u
ˆ
α
0
¼
e
ˆ
α
0
β
0
u
β
0
play an important
role in imposing boundary conditions. Here the left eigen-
vectors
e
b
α
0
β
0
are defined by
e
ˆ
α
0
μ
0
s
k
0
A
k
0
μ
0
β
0
¼
v
ð
ˆ
α
0
Þ
e
ˆ
α
0
β
0
;
ð
2
:
3
Þ
where
s
k
0
is the outward-directed unit normal to the
boundary of the computational domain:
s
t
0
¼
0
;s
k
0
¼
γ
i
0
k
0
∂
i
0
r
0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
γ
i
0
j
0
∂
i
0
r
0
∂
j
0
r
0
q
;
ð
2
:
4
Þ
and
v
ð
ˆ
α
0
Þ
are the eigenvalues. As pointed out by Kidder
et al.
[84]
, a convenient way to impose the Bjørhus
boundary conditions
[115]
can be achieved via
d
t
0
u
ˆ
α
0
¼
D
t
0
u
ˆ
α
0
þ
v
ð
ˆ
α
0
Þ
d
⊥
u
ˆ
α
0
−
d
⊥
u
ˆ
α
0
BC
;
ð
2
:
5
Þ
with
d
t
0
u
ˆ
α
0
≡
e
ˆ
α
0
β
0
∂
t
0
u
β
0
;d
⊥
u
ˆ
α
0
≡
e
ˆ
α
0
β
0
s
k
0
∂
k
0
u
β
0
;
ð
2
:
6
Þ
and
D
t
0
u
ˆ
α
0
≡
e
ˆ
α
0
β
0
−
A
k
0
β
0
α
0
∂
k
0
u
α
0
þ
F
β
0
:
ð
2
:
7
Þ
Here Eq.
(2.5)
replaces the normal derivative
d
⊥
u
ˆ
α
0
by its
desired value
d
⊥
u
ˆ
α
0
j
BC
on the boundary while leaving the
tangential derivative unchanged.
The boundary conditions in Eq.
(2.5)
must be imposed
on each incoming characteristic field
v
ð
ˆ
α
0
Þ
<
0
[116
–
118]
.
As discussed in Refs.
[83,84]
, for the fully first-order
generalized harmonic formulation there are fifty evolved
variables in the Cauchy domain (
g
μ
0
ν
0
;
Π
μ
0
ν
0
;
Φ
i
0
μ
0
ν
0
), and
there are (for typical values of the shift vector at the outer
boundary) forty incoming characteristic fields on the outer
boundary. Thirty-four of these incoming fields, namely
u
ˆ
0
μ
0
ν
0
,
u
ˆ
2
i
0
μ
0
ν
0
, and four components of
u
ˆ
1
−
μ
0
ν
0
[see Eqs. (32)
–
(34)
in Ref.
[83]
for their expressions], are directly related to the
influx of constraint violations. So the appropriate boundary
conditions for those forty fields are those that preserve the
constraints; we use Eqs. (63) through (65) of
[83]
, which
prevent influx of constraint violations without any approxi-
mation. Thus CCM matching is unnecessary for these
fields, and furthermore CCM matching for these fields is
not well-motivated because constraints are local and must
be preserved independent of the solution in the character-
istic domain.
This leaves six incoming characteristic fields, which are
the remaining six components of
u
ˆ
1
−
μ
0
ν
0
that are not related
to constraints. These fields require incorporating addi-
tional information into the Cauchy system. Four of these
correspond to incoming gauge modes, and two represent
incoming physical degrees of freedom, as described in
[83]
.
In this paper we use CCM to set the physical boundary
conditions, and we leave the gauge boundary conditions for
future work. We note that, after matching the physical and
gauge boundary conditions, our CCM system converges to
the exact infinite domain problem in the continuum limit.
The physical boundary condition for
u
ˆ
1
−
μ
0
ν
0
reads
d
t
0
u
ˆ
1
−
μ
0
ν
0
¼
P
P
ρ
0
τ
0
μ
0
ν
0
h
D
t
0
u
ˆ
1
−
ρ
0
τ
0
−
α
þ
s
j
0
β
j
0
×
w
−
ρ
0
τ
0
−
w
−
ρ
0
τ
0
BC
−
γ
2
s
i
0
c
3
i
0
ρ
0
τ
0
i
;
ð
2
:
8
Þ
where the constraint fields
c
3
i
0
ρ
0
τ
0
can be found in Eq. (57) of
Ref.
[83]
; and the physical projection operators
P
P
ρ
0
τ
0
μ
0
ν
0
are
given by
P
P
ρ
0
τ
0
μ
0
ν
0
≡
P
μ
0
ρ
0
P
ν
0
τ
0
−
1
2
P
μ
0
ν
0
P
ρ
0
τ
0
;
ð
2
:
9
Þ
with
P
μ
0
ν
0
¼
g
μ
0
ν
0
þ
n
μ
0
n
ν
0
−
s
μ
0
s
ν
0
;
ð
2
:
10
Þ
as well as the normal vector of the time slice
n
μ
0
.
Crucially,
w
−
ρ
0
τ
0
in Eq.
(2.8)
are the inward propagating
components of the Weyl tensor
C
μ
0
η
0
ν
0
α
0
,
w
−
ρ
0
τ
0
¼
P
P
μ
0
ν
0
ρ
0
τ
0
ð
n
η
0
þ
s
η
0
Þð
n
α
0
þ
s
α
0
Þ
C
μ
0
η
0
ν
0
α
0
;
ð
2
:
11
Þ
where
n
α
0
is the spacetime unit normal vector to the spatial
hypersurface and
w
−
ρ
0
τ
0
j
BC
are the desired values of
w
−
ρ
0
τ
0
at
the outer boundary. The effect of the boundary condition,
Eq.
(2.8)
, is to drive
w
−
ρ
0
τ
0
toward
w
−
ρ
0
τ
0
j
BC
. We find it is
convenient to write
w
−
ρ
0
τ
0
in terms of the Weyl scalar
ψ
0
0
:
w
−
ρ
0
τ
0
¼
2
ψ
0
0
̄
m
ρ
0
̄
m
τ
0
þ
̄
ψ
0
0
m
ρ
0
m
τ
0
;
ð
2
:
12
Þ
where we have used an identity [Eq.
(2.10)
]
P
ρ
0
τ
0
¼
m
ρ
0
̄
m
τ
0
þ
m
τ
0
̄
m
ρ
0
;
ð
2
:
13
Þ
and the definition of
ψ
0
0
:
ψ
0
0
¼
C
μ
0
ν
0
ρ
0
τ
0
l
μ
0
m
ν
0
l
ρ
0
m
τ
0
:
ð
2
:
14
Þ
SIZHENG MA
et al.
PHYS. REV. D
109,
124027 (2024)
124027-4
Here
f
l
μ
0
;k
μ
0
;m
μ
0
g
refer to the null tetrad within the
Newman-Penrose formalism. The choice of the null vectors
l
μ
0
(outgoing) and
k
μ
0
(ingoing) are determined
uniquely
by
Eqs.
(2.10)
and
(2.11)
(namely the Cauchy system):
l
μ
0
¼
1
ffiffiffi
2
p
ð
n
μ
0
þ
s
μ
0
Þ
;
ð
2
:
15a
Þ
k
μ
0
¼
1
ffiffiffi
2
p
ð
n
μ
0
−
s
μ
0
Þ
:
ð
2
:
15b
Þ
However, the choice of
m
μ
0
is not unique. The require-
ments on
m
μ
0
read:
m
μ
0
l
μ
0
¼
0
;m
μ
0
k
μ
0
¼
0
;m
μ
0
̄
m
μ
0
¼
1
:
ð
2
:
16
Þ
As we shall show later, the only allowed gauge freedom on
m
μ
0
is a rotation:
m
μ
0
→
m
μ
0
e
i
Θ
, but the values of
w
ρ
0
τ
0
in
Eq.
(2.11)
do not depend on the gauge variable
Θ
. There-
fore, in our following calculations, we will take advantage
of this fact and choose
m
μ
0
as close as possible to that of the
characteristic system, in order to simplify calculations.
As mentioned earlier, production
S
p
EC
simulations set
w
−
ρ
0
τ
0
j
BC
in Eq.
(2.8)
to zero. But within the CCM frame-
work, we shall use the characteristic system to determine
w
−
ρ
0
τ
0
j
BC
from Eq.
(2.12)
, where the
ψ
0
0
in Eq.
(2.12)
will be
computed from the characteristic evolution and interpolated
back to the Cauchy grid. We will explain more details in
Sec.
IV
below.
III. SUMMARY OF THE CHARACTERISTIC
EVOLUTION
In this section, we briefly summarize the
S
p
ECTRE
char-
acteristic system as described in Refs.
[10,11]
. The proce-
dures for extracting the Cauchy quantities on the worldtube,
evolving the characteristic variables in the exterior region,
and computing waveform quantities at future null infinity are
identical for CCE versus CCM. For more details of the
characteristic algorithm, see Refs.
[10,11]
.
The
S
p
ECTRE
characteristic system is based on Bondi-
Sachs metric in partially flat Bondi-like coordinates
f
ˆ
r;
ˆ
x
ˆ
A
;
ˆ
u
g
[10,11]
ds
2
¼
−
e
2
ˆ
β
ˆ
V
ˆ
r
−
ˆ
r
2
ˆ
h
ˆ
A
ˆ
B
ˆ
U
ˆ
A
ˆ
U
ˆ
B
d
ˆ
u
2
−
2
e
2
ˆ
β
d
ˆ
ud
ˆ
r
−
2
ˆ
r
2
ˆ
h
ˆ
A
ˆ
B
ˆ
U
ˆ
B
d
ˆ
ud
ˆ
x
ˆ
A
þ
ˆ
r
2
ˆ
h
ˆ
A
ˆ
B
d
ˆ
x
ˆ
A
d
ˆ
x
ˆ
B
;
ð
3
:
1
Þ
where
ˆ
x
ˆ
A
stands for the pair of angular coordinates
f
ˆ
θ
;
ˆ
φ
g
.
With this coordinate system, a few gauge conditions have
been imposed:
g
ˆ
r
ˆ
r
¼
0
,
g
ˆ
r
ˆ
A
¼
0
, and the determinant of the
angular components
ˆ
h
ˆ
A
ˆ
B
is set to that of the unit sphere
metric
q
ˆ
A
ˆ
B
det
ð
ˆ
h
ˆ
A
ˆ
B
Þ¼
det
ð
q
ˆ
A
ˆ
B
Þ¼
sin
2
ˆ
θ
:
ð
3
:
2
Þ
Consequently, the system is characterized by 6 degrees of
freedom (4 quantities):
ˆ
W
¼ð
ˆ
V
−
ˆ
r
Þ
=
ˆ
r
2
,
ˆ
h
ˆ
A
ˆ
B
,
ˆ
U
ˆ
B
, and
ˆ
β
.
Near future null infinity, the metric components need to
follow falloff rates
[10,11]
:
lim
ˆ
r
→
∞
ˆ
W
¼
O
ð
ˆ
r
−
2
Þ
;
ð
3
:
3a
Þ
lim
ˆ
r
→
∞
ˆ
U
ˆ
A
¼
O
ð
ˆ
r
−
2
Þ
;
ð
3
:
3b
Þ
lim
ˆ
r
→
∞
ˆ
h
ˆ
A
ˆ
B
¼
q
ˆ
A
ˆ
B
þ
O
ð
ˆ
r
−
1
Þ
:
ð
3
:
3c
Þ
Note that the conditions in Eq.
(3.3)
are not sufficient for
the metric to asymptotically approach the Minkowski
metric, as true Bondi-Sachs coordinates do. To bring the
partially flat Bondi-like coordinates to a true Bondi-Sachs
system (up to BMS transformations), one needs to further
impose
lim
ˆ
r
→
∞
ˆ
β
¼
O
ð
ˆ
r
−
1
Þ
:
ð
3
:
4
Þ
In practice, it was found that most computations are more
straightforward in partially flat Bondi-like coordinates
f
ˆ
r;
ˆ
x
ˆ
A
;
ˆ
u
g
where Eq.
(3.4)
is not satisfied. We transform
into true Bondi-Sachs coordinates only when necessary
for computing waveform quantities at future null infinity
[10,11]
. See Fig.
1
(and also Table I of
[10]
) for the various
coordinate systems used in CCE and CCM.
Following the algorithm outlined in Refs.
[10,11]
, the
characteristic system needs to take boundary data on a
timelike worldtube from the inner Cauchy system. There-
fore, one has to perform gauge transformations to convert
the Cauchy
3
þ
1
metric in Eq.
(2.1)
to the Bondi-Sachs
metric in Eq.
(3.1)
. The procedure involves three steps, and
we summarize them in Fig.
1
. First, the spacelike foliation
of Eq.
(2.1)
is converted to a null foliation. To achieve this
goal, one needs to construct a class of null vectors
∂
λ
at the
worldtube surface.
ð
∂
λ
Þ
a
¼
δ
a
a
0
n
a
0
þ
s
a
0
α
−
γ
i
0
j
0
β
i
0
s
j
0
;
ð
3
:
5
Þ
where
λ
is an affine parameter,
a
0
and
a
are abstract indices,
the unit vector
s
a
0
is defined in Eq.
(2.4)
, and
n
a
0
still stands
for the normal vector of the time slice. A new null
coordinate system
f
u;
λ
;
x
A
g
is introduced, and quantities
are transformed into this coordinate system. This coordi-
nate system is discussed in more detail in Sec.
IVA 1
.
The second step is to transform the null-radius coor-
dinates to so-called Bondi-like coordinates
f
u; r; x
A
g
by
FULLY RELATIVISTIC THREE-DIMENSIONAL CAUCHY-
...
PHYS. REV. D
109,
124027 (2024)
124027-5
imposing the gauge condition in Eq.
(3.2)
. At this stage, the
metric is brought into Bondi-Sachs form
ds
2
¼
−
e
2
β
V
r
−
r
2
h
AB
U
A
U
B
du
2
−
2
e
2
β
dudr
−
2
r
2
h
AB
U
B
dudx
A
þ
r
2
h
AB
dx
A
dx
B
:
ð
3
:
6
Þ
The coordinates still differ from the partially flat Bondi-like
coordinates because the falloff rates in Eq.
(3.3)
are now
relaxed to
lim
r
→
∞
W
¼
O
ð
r
0
Þ
;
ð
3
:
7a
Þ
lim
r
→
∞
U
A
¼
O
ð
r
0
Þ
;
ð
3
:
7b
Þ
lim
r
→
∞
h
AB
¼
O
ð
r
0
Þ
:
ð
3
:
7c
Þ
The transformation to this coordinate system is discussed
in detail in Sec.
IVA 2
.
Finally, the Bondi-like coordinates are transformed to the
partially-flat Bondi-like coordinates
f
ˆ
r;
ˆ
x
ˆ
A
;
ˆ
u
g
by removing
the asymptotic value of
U
A
at null infinity,
U
ð
0
Þ
A
. Here we
define
U
ð
0
Þ
A
by
U
A
¼
U
ð
0
Þ
A
þ
O
ð
r
−
1
Þ
:
ð
3
:
8
Þ
Details can be found in Sec.
IVA 3
.
Once the worldtube quantities have been computed in
partially flat Bondi-like coordinates, they serve as inner
boundary conditions to evolve the characteristic system.
This evolution step is described in detail in Refs.
[10,11]
and is identical for CCM versus CCE. Here we emphasize
again that the characteristic evolution with the partially-flat
Bondi-like coordinates is only weakly hyperbolic, which in
turn makes CCM not well-posed
[114]
.
After determining all the metric components with the
characteristic algorithm, we can now compute Weyl scalars.
For CCM, we need the Weyl scalar
ψ
0
in the exterior
characteristic region, which will be used in the outer-
boundary condition for the interior Cauchy system. To
assemble
ψ
0
from the metric components, we adopt the
tetrad provided by Ref.
[10]
m
μ
¼
−
1
ffiffiffi
2
p
r
ffiffiffiffiffiffiffiffiffiffiffiffi
K
þ
1
2
r
q
μ
−
J
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
ð
1
þ
K
Þ
p
̄
q
μ
!
;
ð
3
:
9a
Þ
k
μ
¼
ffiffiffi
2
p
e
−
2
β
δ
μ
u
−
V
2
r
δ
μ
r
þ
1
2
̄
Uq
μ
þ
1
2
U
̄
q
μ
;
ð
3
:
9b
Þ
l
μ
¼
1
ffiffiffi
2
p
δ
μ
r
;
ð
3
:
9c
Þ
where
J
,
K
, and
U
are spin-weighted scalars, defined by
5
U
≡
U
A
q
A
;J
≡
1
2
q
A
q
B
h
AB
;
ð
3
:
10a
Þ
K
≡
1
2
q
A
̄
q
B
h
AB
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
þ
J
̄
J
p
;
ð
3
:
10b
Þ
The complex dyads
q
A
and
q
A
read
q
A
∂
A
¼
−
∂
θ
−
i
sin
θ
∂
φ
;
ð
3
:
11a
Þ
q
A
dx
A
¼
−
d
θ
−
i
sin
θ
d
φ
:
ð
3
:
11b
Þ
They obey the identity
q
A
q
A
¼
0
;q
A
̄
q
A
¼
2
:
ð
3
:
12
Þ
Note that the tetrad vectors in Eq.
(3.9)
are constructed with
the Bondi-like coordinates, of which the partially flat
Bondi-like coordinates are subclasses. Therefore, Eq.
(3.9)
can be applied directly to the partially flat Bondi-like co-
ordinates as long as all variables are replaced by their
partially flat Bondi-like counterparts. With the tetrad
vectors at hand, we can now derive a full expression of
the Bondi-like
ψ
0
in relation to Bondi quantities
[10]
ψ
0
¼
r
∂
r
β
−
1
4
Kr
ð
1
þ
K
Þ
∂
r
J
−
J
2
∂
r
̄
J
1
þ
K
þ
J
ð
1
þ
K
2
Þ
∂
r
J
∂
r
̄
J
8
K
3
þ
1
8
K
J
2
∂
2
r
̄
J
1
þ
K
−
ð
1
þ
K
Þ
∂
2
r
J
−
J
̄
J
2
ð
∂
r
J
Þ
2
þ
J
3
ð
∂
r
̄
J
Þ
2
16
K
3
:
ð
3
:
13
Þ
Similarly, Eq.
(3.13)
is also applicable to the partially-flat-
Bondi-like
ˆ
ψ
0
when the Bondi quantities are evaluated with
the partially flat Bondi-like coordinates.
IV. MATCHING CHARACTERISTIC
AND CAUCHY SYSTEMS
We are now in a position to accomplish Cauchy-
characteristic matching for the physical degrees of freedom.
The goal is to use the Weyl scalar
ψ
0
obtained with the
characteristic system to compute the boundary value
w
−
ρ
0
τ
0
j
BC
that goes into the physical boundary condition of
the Cauchy system [Eq.
(2.8)
]. This is done by evaluating
w
−
ρ
0
τ
0
j
BC
by inserting the characteristic system
’
s
ψ
0
into
Eq.
(2.12)
. Notice that the tetrad adopted by the character-
istic system in Eq.
(3.9)
differs from the one used by
5
We note that there is a typo in Eq. (10e) of Ref.
[10]
; the
correct expression is given in Eq.
(3.10b)
.
SIZHENG MA
et al.
PHYS. REV. D
109,
124027 (2024)
124027-6
Cauchy evolution in Eq.
(2.15)
, so we need to perform
Lorentz transformations to obtain (a) the Cauchy Weyl
scalar
ψ
0
0
[defined in Eq.
(2.14)
] and (b) the null covariant
vector
m
μ
0
in Eq.
(2.12)
. Necessary ingredients for the
Lorentz transformations involve a set of Jacobian matrices
across different coordinate systems. So in Sec.
IVA
we first
work out the explicit expressions for these Jacobians,
and then in Secs.
IV B
and
IV C
we carry out the trans-
formations. Notice that the evaluation of
ψ
0
with the
characteristic system [Eq.
(3.13)
] could be done in either
the partially flat Bondi-like coordinates [Eq.
(3.1)
] or the
Bondi-like coordinates [Eq.
(3.6)
], and different choices
lead to different Lorentz transformations. In order to keep
our discussions as general as possible, we consider both
choices in Secs.
IV B
and
IV C
, respectively. We also
illustrate these two options in Fig.
1
(see two arrows labeled
by
“
CCM
”
), serving as a roadmap for the CCM algorithm.
The final step toward finishing the matching is to inter-
polate the values of
ψ
0
0
and
m
μ
0
from the characteristic grid
to the Cauchy grid. This is done in Sec.
IV D
.
For ease of future reference, we summarize the two
primary matching steps below:
(1) Construct
ψ
0
0
and
m
μ
0
from either the partially flat
Bondi-like coordinates (Sec.
IV B
) or the Bondi-like
coordinates (Sec.
IV C
).
(2) Interpolate
ψ
0
0
and
m
μ
0
from the characteristic grid to
the Cauchy grid. See Sec.
IV D
.
A. Jacobians for CCM
As outlined in Sec.
III
and summarized in Fig.
1
,
two intermediate coordinate systems are introduced to
convert the worldtube data from the Cauchy coordinates
to the partially flat Bondi-like coordinates. Below, we
provide the definition of these transformations and their
Jacobians.
1. Cauchy and null-radius coordinates
The null-radius coordinates consist of
f
u;
λ
;
x
A
g
, where
λ
is the affine parameter of the null vector in Eq.
(3.5)
.
Meanwhile, the time and angular coordinates are the same
as the Cauchy coordinates:
8
>
>
<
>
>
:
u
¼
t
0
;
x
A
¼
δ
A
A
0
x
0
A
0
;
λ
¼
λ
ð
t
0
;r
0
Þ
:
ð
4
:
1
Þ
Consequently, the metric components in the null-radius
coordinates are
[10]
g
λ
u
¼
−
1
;g
λ
λ
¼
0
;g
λ
A
¼
0
;g
u
u
¼
g
t
0
t
0
;
g
u
A
¼
δ
A
0
A
g
t
0
A
0
;g
A
B
¼
δ
A
0
A
δ
B
0
B
g
A
0
B
0
:
ð
4
:
2
Þ
Equation
(4.2)
lead to the Jacobian between two coordinate
systems:
∂
ð
t
0
;r
0
;x
0
A
0
Þ
∂
ð
u;
λ
;
x
A
Þ
¼
0
B
@
1
∂
λ
t
0
0
0
∂
λ
r
0
0
00
δ
A
0
A
1
C
A
ð
4
:
3
Þ
2. Null-radius and Bondi-like coordinates
To bring the null-radius coordinates to Bondi-like
coordinates, one needs to impose the gauge condition in
Eq.
(3.2)
and define the Bondi-like radius:
r
¼
det
ð
g
A
B
Þ
det
ð
q
A
B
Þ
1
=
4
:
ð
4
:
4
Þ
Then the Bondi-like coordinates
f
u; r; x
A
g
can be written as
8
>
<
>
:
u
¼
u;
x
A
¼
δ
A
A
x
A
;
r
¼
r
ð
u;
λ
;
x
A
Þ
:
ð
4
:
5
Þ
Equation
(4.5)
result in the Jocobian
∂
ð
u; r; x
A
Þ
∂
ð
u;
λ
;
x
A
Þ
¼
0
B
@
100
∂
u
r
∂
λ
r
∂
A
r
00
δ
A
A
1
C
A
;
ð
4
:
6
Þ
and its inverse
∂
ð
u;
λ
;
x
A
Þ
∂
ð
u; r; x
A
Þ
¼
0
B
@
10 0
−
∂
u
r=
∂
λ
r
ð
∂
λ
r
Þ
−
1
−
δ
A
A
∂
A
r=
∂
λ
r
00
δ
A
A
1
C
A
:
ð
4
:
7
Þ
3. Bondi-like and partially flat Bondi-like coordinates
One difference between these two coordinate systems is
that the quantity
U
A
is finite at future null infinity, but the
quantity
ˆ
U
ˆ
A
vanishes. To remove the asymptotically con-
stant part of
U
A
, the angular coordinates
ˆ
x
ˆ
A
must satisfy
∂
u
ˆ
x
ˆ
A
¼
−
∂
A
ˆ
x
ˆ
A
U
ð
0
Þ
A
;
ð
4
:
8
Þ
where
U
ð
0
Þ
A
is defined by Eq.
(3.8)
. In practice, the
quantities
ˆ
x
ˆ
A
are evolved numerically on the characteristic
grid along with the evolution of the characteristic metric
components. The Bondi-like radius
r
also needs to be
adjusted accordingly to meet the gauge condition in
Eq.
(3.2)
. Finally, the time coordinate
ˆ
u
¼
u
remains
unchanged. In summary, the transformation is given by
FULLY RELATIVISTIC THREE-DIMENSIONAL CAUCHY-
...
PHYS. REV. D
109,
124027 (2024)
124027-7
8
>
<
>
:
ˆ
u
¼
u;
ˆ
x
ˆ
A
¼
ˆ
x
ˆ
A
ð
u; x
A
Þ
;
ˆ
r
¼
r
ˆ
ω
ð
u; x
A
Þ
;
ð
4
:
9
Þ
where
ˆ
ω
ð
u; x
A
Þ
is a conformal factor
ˆ
ω
¼
1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ˆ
b
̄
ˆ
b
−
ˆ
a
̄
ˆ
a
q
;
ð
4
:
10
Þ
and two spin-weighted Jacobian factors
ˆ
a
and
ˆ
b
are given by
ˆ
a
¼
ˆ
q
ˆ
A
∂
ˆ
A
x
A
q
A
;
ð
spin-weight
2
Þð
4
:
11
Þ
ˆ
b
¼
̄
ˆ
q
ˆ
A
∂
ˆ
A
x
A
q
A
:
ð
spin-weight
0
Þð
4
:
12
Þ
Since
f
q
A
;
̄
q
A
g
(
f
ˆ
q
ˆ
A
;
̄
ˆ
q
ˆ
A
g
) form a complete basis for the
angular subspace spanned by
f
x
A
g
(
f
ˆ
x
ˆ
A
g
), we can expand
∂
ˆ
A
x
A
into
6
∂
ˆ
A
x
A
¼
1
4
ˆ
q
ˆ
A
;
̄
ˆ
q
ˆ
A
̄
ˆ
a
̄
ˆ
b
ˆ
b
ˆ
a
q
A
̄
q
A
!
;
ð
4
:
13
Þ
where the expression is written in terms of matrix products.
Note that the determinant of the middle
2
×
2
matrix
(together with the factor of
1
=
4
) is equal to
−
ˆ
ω
2
[see
Eq.
(4.10)
]. In practice, we find it is also convenient to
define spin-weighted factors that are related to the inverse
of the Jacobian:
a
¼
q
A
∂
A
ˆ
x
ˆ
A
ˆ
q
ˆ
A
;
ð
spin-weight
2
Þð
4
:
14
Þ
b
¼
̄
q
A
∂
A
ˆ
x
ˆ
A
ˆ
q
ˆ
A
;
ð
spin-weight
0
Þð
4
:
15
Þ
as well as the conformal factor
ω
ð
ˆ
u;
ˆ
x
ˆ
A
Þ
associated with
them
ω
¼
1
2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
b
̄
b
−
a
̄
a
p
:
ð
4
:
16
Þ
Similarly, the counterpart of Eq.
(4.13)
reads
∂
A
ˆ
x
ˆ
A
¼
1
4
q
A
;
̄
q
A
̄
a
̄
b
ba
ˆ
q
ˆ
A
̄
ˆ
q
ˆ
A
!
:
ð
4
:
17
Þ
At the same spacetime point, the identity
∂
ˆ
A
x
A
∂
A
ˆ
x
ˆ
B
¼
δ
ˆ
B
ˆ
A
leads to
a
¼
−
ˆ
a
ˆ
ω
2
;b
¼
̄
ˆ
b
ˆ
ω
2
:
ð
4
:
18
Þ
Plugging Eq.
(4.18)
into Eq.
(4.16)
we obtain another
identity
ω
ˆ
ω
¼
1
:
ð
4
:
19
Þ
We then use Eq.
(4.9)
to get the Jacobian between the
Bondi-like and the partially flat Bondi-like coordinates
∂
ð
ˆ
r;
ˆ
x
ˆ
A
;
ˆ
u
Þ
∂
ð
r; x
A
;u
Þ
¼
0
B
@
ˆ
ω
r
∂
A
ˆ
ω
r
∂
u
ˆ
ω
0
∂
A
ˆ
x
ˆ
A
∂
u
ˆ
x
ˆ
A
00 1
1
C
A
:
ð
4
:
20
Þ
Its inverse reads
∂
ð
r; x
A
;u
Þ
∂
ð
ˆ
r;
ˆ
x
ˆ
A
;
ˆ
u
Þ
¼
0
B
@
ω
r
δ
A
ˆ
A
∂
A
ln
ω
r
∂
u
ln
ω
þ
rU
ð
0
Þ
A
∂
A
ln
ω
0
∂
ˆ
A
x
A
U
ð
0
Þ
A
00
1
1
C
A
;
ð
4
:
21
Þ
where we have used Eq.
(4.8)
to simplify the result.
B. Choice 1: Transforming
m
ˆ
μ
and
ˆ
ψ
0
to the Cauchy tetrad
We first consider choice 1, as summarized in Fig.
1
,
where the tetrad vector
m
ˆ
μ
and the Weyl scalar
ˆ
ψ
0
are
evaluated in the partially flat Bondi-like coordinates, using
Eqs.
(3.9a)
and
(3.13)
. Before transforming them into the
Cauchy tetrad, we first observe a useful and important fact:
The characteristic outgoing null tetrad vector
l
ˆ
a
at the
worldtube surface, as defined in Eq.
(3.9c)
, is by con-
struction
proportional to
that of the Cauchy system
l
a
0
,
defined in Eq.
(2.15a)
. Again, here
ˆ
a
and
a
0
stand for
abstract indices. To see this, we write
l
ˆ
a
¼
1
ffiffiffi
2
p
ð
∂
ˆ
r
Þ
ˆ
a
¼
1
ffiffiffi
2
p
ð
∂
λ
ˆ
r
Þ
−
1
δ
ˆ
a
a
ð
∂
λ
Þ
a
¼
1
ffiffiffi
2
p
e
2
ˆ
β
δ
ˆ
a
a
ð
∂
λ
Þ
a
;
ð
4
:
22
Þ
where the first equality comes from Eq.
(3.9c)
, the second
equality is due to the combination of the Jacobian matrices
in Eq.
(4.7)
and
(4.21)
, and the final equality is based on a
relationship [see Eqs. (19a) and (33a) of Ref.
[10]
]
ˆ
β
¼
−
1
2
ln
ð
∂
λ
ˆ
r
Þ
:
ð
4
:
23
Þ
On the other hand, the null vector
ð
∂
λ
Þ
a
in Eq.
(4.22)
is
proportional to the Cauchy outgoing null vector
l
a
0
needed
6
To obtain Eq.
(4.13)
, one can exhaust all the possible linear
combinations formed by the two bases
f
ˆ
q
ˆ
A
;
̄
ˆ
q
ˆ
A
g
and
f
q
A
;
̄
q
A
g
,
and then determine the coefficients uniquely via Eqs.
(3.12)
,
(4.11)
, and
(4.12)
.
SIZHENG MA
et al.
PHYS. REV. D
109,
124027 (2024)
124027-8
by the boundary condition [see Eqs.
(2.15a)
and
(3.5)
], but
with a different normalization. After combining Eq.
(4.22)
with
(2.15a)
and
(3.5)
, we obtain:
l
a
0
¼
α
−
γ
i
0
j
0
β
i
0
s
j
0
e
−
2
ˆ
β
l
ˆ
a
δ
a
0
ˆ
a
:
ð
4
:
24
Þ
Therefore, the statement
l
a
0
∝
l
ˆ
a
is proven. Under this
constraint, the allowed Lorentz transformation between
the characteristic and Cauchy tetrads can be split into two
categories
(i) Type I: (
l
unchanged)
l
→
l
;
k
→
k
þ
̄
κ
m
þ
κ
̄
m
þ
κ
̄
κ
l
;
m
→
m
þ
κ
l
;
̄
m
→
̄
m
þ
̄
κ
l
:
ð
4
:
25
Þ
(ii) Type II: (both
l
and
k
changed)
l
→
A
l
;
k
→
A
−
1
k
;
m
→
e
i
Θ
m
;
̄
m
→
e
−
i
Θ
̄
m
;
ð
4
:
26
Þ
where the complex scalar
κ
has a spin weight of 1,
A
and
Θ
are real scalars. The Weyl scalar
ˆ
ψ
0
transforms
correspondingly:
(i) Type I:
ˆ
ψ
0
→
ˆ
ψ
0
:
ð
4
:
27
Þ
(ii) Type II:
ˆ
ψ
0
→
A
2
e
2
i
Θ
ˆ
ψ
0
:
ð
4
:
28
Þ
Notice that
ˆ
ψ
0
is not mixed with other Weyl scalars. In
particular, it remains unchanged within the t I category.
Below we will take advantage of this observation to
simplify the calculation.
As summarized in Fig.
1
, for choice 1, we need to
transform both
m
ˆ
μ
and
ˆ
ψ
0
to the Cauchy tetrad in order to
evaluate the inward propagating components of the Weyl
tensor
w
−
ρ
0
τ
0
j
BC
[Eq.
(2.12)
] in the correct tetrad. We treat the
transformation of
m
ˆ
μ
and
ˆ
ψ
0
separately in the two follow-
ing sections.
1. Type I transformation of m
ˆ
μ
The characteristic system
’
s
m
ˆ
μ
[Eq.
(3.9a)
] is not aligned
with that of the Cauchy system [Eq.
(2.16)
]. This is because
our choice of the ingoing null vector
k
μ
for the character-
istic system [Eq.
(3.9b)
] is not the same as
k
μ
0
used in the
Bjørhus boundary condition, which is defined uniquely by
Eq.
(2.15b)
. To transform the characteristic vector
m
to the
corresponding choice in the Cauchy boundary condition, it
suffices to add some multiple of the outgoing null vector
l
to
m
; thus we need to perform a type I transformation. We
want to emphasize that the value of
ˆ
ψ
0
is not impacted by a
type I transformation, so when performing such a trans-
formation it is not necessary to keep track of the explicit
Lorentz parameter [namely
κ
in Eq.
(4.25)
] that was used in
the transformation. Accordingly, in the vector expressions
below, we will simply drop terms that are proportional
to the outgoing null vector
l
, since these terms can be
eliminated through a type I transformation. Whenever
this is done we will indicate that such terms have been
dropped by a type I transformation by using the symbol
≈
instead of
¼
.
By combining Jacobians in Eqs.
(4.3)
,
(4.7)
, and
(4.21)
,
we obtain the relationship
∂
ˆ
A
¼ð
∂
ˆ
A
x
A
Þ
δ
A
0
A
∂
A
0
þ
∂
λ
×
∂
ˆ
A
r
∂
λ
r
−
∂
A
r
∂
λ
r
ð
∂
ˆ
A
x
A
Þ
δ
A
A
ð
4
:
29
Þ
Since
∂
λ
is the outgoing null vector given in Eq.
(4.22)
, the
second term in Eq.
(4.29)
can be removed via a type I
Lorentz transformation. We then insert Eqs.
(4.29)
and
(4.13)
into Eq.
(3.11)
, which yields
q
ˆ
μ
≈
1
2
ˆ
a
δ
ˆ
μ
μ
0
̄
q
μ
0
þ
1
2
̄
ˆ
b
δ
ˆ
μ
μ
0
q
μ
0
;
ð
4
:
30
Þ
where
≈
implies that a type I Lorentz transformation has
been performed, as described above. Plugging Eq.
(4.30)
into Eq.
(3.9a)
, we obtain
m
ˆ
μ
≈−
δ
ˆ
μ
μ
0
ffiffiffi
2
p
ˆ
r
2
4
0
@
ffiffiffiffiffiffiffiffiffiffiffiffi
ˆ
K
þ
1
2
s
1
2
ˆ
a
−
ˆ
J
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
ð
1
þ
ˆ
K
Þ
q
1
2
ˆ
b
1
A
̄
q
μ
0
þ
0
@
ffiffiffiffiffiffiffiffiffiffiffiffi
ˆ
K
þ
1
2
s
1
2
̄
ˆ
b
−
ˆ
J
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
ð
1
þ
ˆ
K
Þ
q
1
2
̄
ˆ
a
1
A
q
μ
0
3
5
:
ð
4
:
31
Þ
Or equivalently
m
ˆ
μ
≈
δ
ˆ
μ
μ
0
m
μ
0
;
ð
4
:
32
Þ
with
m
μ
0
being the components of a new contravariant
vector
m
a
0
m
a
0
¼
ˆ
M
θ
0
ð
∂
θ
0
Þ
a
0
þ
ˆ
M
φ
0
i
sin
ˆ
θ
ð
θ
0
Þ
ð
∂
φ
0
Þ
a
0
;
ð
4
:
33
Þ
and
4
ˆ
r
ˆ
M
θ
0
¼ð
ˆ
a
þ
̄
ˆ
b
Þ
ffiffiffiffiffiffiffiffiffiffiffiffi
ˆ
K
þ
1
p
−
ð
̄
ˆ
a
þ
ˆ
b
Þ
ˆ
J
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð
1
þ
ˆ
K
Þ
q
;
ð
4
:
34
Þ
4
ˆ
r
ˆ
M
φ
0
¼ð
̄
ˆ
b
−
ˆ
a
Þ
ffiffiffiffiffiffiffiffiffiffiffiffi
ˆ
K
þ
1
p
−
ð
̄
ˆ
a
−
ˆ
b
Þ
ˆ
J
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð
1
þ
ˆ
K
Þ
q
:
ð
4
:
35
Þ
FULLY RELATIVISTIC THREE-DIMENSIONAL CAUCHY-
...
PHYS. REV. D
109,
124027 (2024)
124027-9
At this stage, we have constructed a Cauchy tetrad vector
m
a
0
in Eq.
(4.33)
that differs from the original characteristic
tetrad vector
m
ˆ
a
by only a type I transformation. Mean-
while, we can see
m
a
0
has components only within the
Cauchy angular subspace
f
θ
0
;
φ
0
g
. Therefore it meets all
the requirements in Eq.
(2.16)
. Consequently we can
convert it to its covariant form
m
a
0
and insert it into
Eq.
(2.12)
to evaluate
w
−
ρ
0
τ
0
j
BC
.
Since Cartesian coordinates are used to evolve the
Cauchy system, we write down the Cartesian components
of two angular bases
ð
∂
θ
0
Þ
a
0
and
ð
∂
φ
0
Þ
a
0
for completeness:
ð
∂
θ
0
Þ
a
0
¼
R
0
wt
ð
cos
φ
0
cos
θ
0
;
sin
φ
0
cos
θ
0
;
−
sin
θ
0
Þ
;
ð
4
:
36
Þ
ð
∂
φ
0
Þ
a
0
¼
R
0
wt
sin
θ
0
ð
−
sin
φ
0
;
cos
φ
0
;
0
Þ
:
ð
4
:
37
Þ
2. Type II transformation of
ˆ
ψ
0
Equation
(4.24)
indicates that two outgoing null vectors
l
a
0
and
l
ˆ
a
are related by a type II transformation
[Eq.
(4.26)
], with the Lorentz parameter
ˆ
A
given by
ˆ
A
¼ð
α
−
γ
i
0
j
0
β
i
0
s
j
0
Þ
e
−
2
ˆ
β
;
ð
4
:
38
Þ
which leads to
ψ
0
0
¼
ˆ
A
2
ˆ
ψ
0
:
ð
4
:
39
Þ
On the other hand, there is one more gauge freedom:
the rotation of
m
with a phase factor
e
i
Θ
. However, the
combination
ψ
0
0
̄
m
ρ
0
̄
m
τ
0
that appears in
w
−
ρ
0
τ
0
[Eq.
(2.12)
]is
invariant under such a phase rotation because
ψ
0
0
is also
transformed accordingly due to Eq.
(4.28)
. Physically
speaking, the incoming characteristics
w
−
ρ
0
τ
0
do not depend
on the choice of the angular tetrad vector. Therefore, we can
neglect this gauge freedom while performing the matching.
C. Choice 2: Transforming
m
μ
and
ψ
0
to the Cauchy tetrad
Then we consider choice 2, where the characteristic
quantities
m
μ
and
ψ
0
are evaluated in the Bondi-like
coordinates. Similar to Sec.
IV B
, below we treat the
transformation of
m
μ
and
ψ
0
separately.
1. Type I transformation of m
μ
By combining Jacobians in Eqs.
(4.3)
and
(4.7)
,we
obtain
∂
A
≈
δ
A
0
A
∂
A
0
;
ð
4
:
40
Þ
which leads to
q
μ
≈
δ
μ
μ
0
q
μ
0
:
ð
4
:
41
Þ
Here we have used the definition of
q
μ
in Eq.
(3.11a)
.
Inserting Eq.
(4.41)
into Eq.
(3.9a)
yields
m
μ
≈
δ
μ
μ
0
m
μ
0
;
ð
4
:
42
Þ
with
m
μ
0
being the components of the vector
m
a
0
m
a
0
¼
M
θ
0
ð
∂
θ
0
Þ
a
0
þ
M
φ
0
i
sin
θ
0
ð
∂
φ
0
Þ
a
0
;
ð
4
:
43
Þ
and
2
rM
θ
0
¼
ffiffiffiffiffiffiffiffiffiffiffiffi
K
þ
1
p
−
J
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð
1
þ
K
Þ
p
;
ð
4
:
44
Þ
2
rM
φ
0
¼
ffiffiffiffiffiffiffiffiffiffiffiffi
K
þ
1
p
þ
J
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ð
1
þ
K
Þ
p
:
ð
4
:
45
Þ
We remark that the null vector
m
μ
0
, which differs from
m
μ
by only a type I transformation, is now in the Cauchy
angular subspace
f
θ
0
;
φ
0
g
, as required by the Cauchy
boundary condition in Eq.
(2.16)
. Therefore, its covariant
form can be used to construct
w
−
ρ
0
τ
0
j
BC
in Eq.
(2.12)
.
In practice, the characteristic system is evolved with the
partially flat Bondi-like coordinates, as summarized in
Fig.
1
. Therefore, we need to transform the partially flat
Bondi quantities
ˆ
J
and
ˆ
K
[Eq.
(3.10)
] to the Bondi-like
coordinates via
[10]
J
¼
̄
b
2
ˆ
J
þ
a
2
̄
ˆ
J
þ
2
a
̄
b
ˆ
K
4
ω
2
;
ð
4
:
46
Þ
K
¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
þ
J
̄
J
p
;
ð
4
:
47
Þ
and then insert the results into Eq.
(4.43)
to construct the
tetrad vector
m
μ
0
for matching.
2. Type II transformation of
ψ
0
In the meantime, after obtaining
J
and
K
from Eqs.
(4.46)
and
(4.47)
, we can evaluate
ψ
0
with Eq.
(3.13)
. Similar to the
discussion in Sec.
IV B 2
, the two outgoing null vectors
l
μ
0
and
l
μ
are related by a Type II transformation, and the
corresponding Lorentz parameter
A
reads
A
¼ð
α
−
γ
i
0
j
0
β
i
0
s
j
0
Þ
e
−
2
β
:
ð
4
:
48
Þ
Consequently, the desired
ψ
0
0
is given by
ψ
0
0
¼
A
2
ψ
0
:
ð
4
:
49
Þ
D. Interpolating to the Cauchy coordinates
Now we have obtained the desired tetrad vector
m
μ
0
and
the Weyl scalar
ψ
0
0
. But they are still evaluated on the
SIZHENG MA
et al.
PHYS. REV. D
109,
124027 (2024)
124027-10