ABSTRACT (original article): The gravitational waveform emitted by a particle moving in a Schwarzschild geometry can be computed, in perturbation theory, as a double series expansion in the limit where distances are large with respect to the size of the horizon,
z=2M/r≪1
(Post-Minkowskian approximation), but small with respect to the gravitational wavelength,
y=2iωr≪1
(Post-Newtonian approximation). While in the case of bounded systems the two expansions are linked, for scattering processes they describe complementary regions of spacetime. In this paper, we derive all order formulae for those two expansions of the waveform, where the entire dependence on the y or z variables is resummed under the assumption
ωM≪1
. The results are based on a novel hypergeometric representation of the confluent Heun functions and provide a new derivation of the recently discovered Heun connection formulae. Similar results are found for the non-confluent Heun case. Finally, in the case of circular orbits we compute our formulae at order 30PN and show excellent agreement against solutions obtained via numerical integrations and the MST method. CITATION (original article): Andrea Cipriani, Giorgio Di Russo, Francesco Fucito, José Francisco Morales, Hasmik Poghosyan, Rubik Poghossian, Resumming Post-Minkowskian and Post-Newtonian gravitational waveform expansions, arXiv:2501.19257. https://doi.org/10.48550/arXiv.2501.19257
In this notebook, the first section concern the computation of the solutions R_in, R_up and then of the energy per unit time of the gravitational radiation coming from the Schwarzchild black hole towards infinity, passing through the computation of the modes Z_{l,m} for the last purpose (all these quantities are defined inside the paper). The results in the second section concern the PN expansions (meaning expansions for small velocities) of some quantities always defined inside the paper, i.e. R_+(a), γ,
γ
κ
and a.

R_in, R_up, and energy per unit time

Download needed quantities files here

Definitions

We import the following quantities at order k=30:​1) from the files “P30.m”,“Pt30.m” the polynomials
P
0
(y)
,

P
0
(y)
(formula (4.4)). The coefficients
c
i,j
and

c
i,j
have been computed using the recursion relations (4.5-4.6)2) from the files “gg30.m”,“gk30.m” the quantities γ =
2a
x
-
∂
a
ℱ
inst
e
(formula (4.3)),
γ
κ
=1+
k
∑
i=1

c
i,i-1
i
x
(formula (4.2))​​3) from the file “a30.m” the a cycle computed using formula (4.8) for ℓ going from 1 to 30
SetDirectory[NotebookDirectory[]];​​Pf0=Import["P30.m"]/.η->1;​​Pf0t=Import["Pt30.m"]/.η->1;​​aa=Import["a30.m"]/.η->1;​​γγ=Import["gg30.m"]/.η->1;​​γκ=Import["gk30.m"]/.η->1;

Rin & Rup

Here we define the following quantities:1) formula (4.1): RRin =
ℛ
in
, RRup =
ℛ
up
​2) formula (4.2):
ccin=
C
in
,
ccup=
C
up
, HH0 =
0
H
+
, Rp=
R
+
, Rm=
R
-
3) formula (4.3):
γ
tot
In[]:=
γtot=
-2aπ

2
Gamma[1-2a]
Gamma-
3
2
+a-
x
2
Gamma
1
2
+a-
x
2
Gamma
5
2
+a-
x
2

2
Gamma[1+2a]
Gamma
1
2
(-3-2a-x)Gamma
1
2
-a-
x
2
Gamma
5
2
-a-
x
2

γγ;​​ccin=
-
1
2
π(5+2a+x)

x/2
x
Gamma-
3
2
+a-
x
2

(1-γtot)Gamma[1+2a]
;ccup=
3
(2M)
Gamma
1
2
(5-2a+x)
3+
x
2
x
γκGamma[1-2a]1-
-2aπ

Cos
1
2
π(x+2a)
Cos
1
2
π(x-2a)
;​​HH0[xi_]:=
-(a-5/2-x/2)
xi
Hypergeometric1F1
5
2
-a+
x
2
,1-2a,xi;​​Rp=
-
y
2

2-
x
2
1-
x
y
-1-
x
2
y
(Pf0HH0[y]+Pf0tyHH0'[y]);Rm=Rp/.a->-a;​​RRin1=ccinRm;RRin=RRin1+(RRin1/.a->-a);​​RRup1=ccupRp;RRup=RRup1+(RRup1/.a->-a);​​Rin[ll_,ωω_,rr_]:=Block[{l=ll,x=4Iωω,y=2Iωωrr,a=SetPrecision[aa[[ll]],100]},RRin]​​Rup[ll_,ωω_,rr_]:=Block[{l=ll,x=4Iωω,M=1,y=2Iωωrr,a=SetPrecision[aa[[ll]],100]},RRup]

Z
lm

Here we define the following quantities:1) formula (4.12):
zzR=
Z
lm
​2) formula (4.13): b0, b1,b2 =
(
012
b
)
lm
​2) formula (4.14): Y
=
sY
lm
3) formula (4.16): etalm
=
η
lm
Y[s_,l_,m_,θ_,ϕ_]:=
m
(-1)
Simplify
(l+m)!(l-m)!(2l+1)
(l+s)!(l-s)!4π
2l
Sin
θ
2

l-s
∑
r=0
Binomial[l-s,r]Binomial[l+s,r+s-m]
l-r-s
(-1)
mϕ

2r+s-m
Cot
θ
2

,Assumptions{ϕ∈Reals,θ∈Reals};​​b0[l_,m_]:=
π
2
1/2
((l-1)l(l+1)(l+2))
Y0,l,m,
π
2
,0;b1[l_,m_]:=π
1/2
((l-1)(l+2))
Y-1,l,m,
π
2
,0;​​b2[l_,m_]:=πY-2,l,m,
π
2
,0;​​zzR=-
2
J
R2
4
x
(x-y)
b2
l,m
16e
4
M
5
y
+R0
e
2
x
b0
l,m
2
2
M
(x-y)y
-
J
3
x
(4x-4y-
2
y
)
b1
l,m
8
3
M
(x-y)
3
y
+
2
J
4
x
(2x-4y-
2
y
)
b2
l,m
64e
4
M
(x-y)
3
y
+R1
J
3
x
b1
l,m
4
3
M
3
y
+
2
J
4
x
(2x-2y-
2
y
)
b2
l,m
16e
4
M
5
y
/.J->μ
M
v
1-3
2
v
,e->μ
(1-2
2
v
)
1-3
2
v
,ω->m
3
v
M
,y->2Imv;​​etalm[ll_,mm_,vv_]:=Block{l=ll,m=mm,v=vv,x=4Imm
3
vv
,M=1,μ=1,ω=m
3
v
,a=SetPrecision[aa[[ll]],100]},​​RR0=RRin;RR1=yD[RR0,y];RR2=
2
y
D[RR0,{y,2}];​​
1
32
5
2
M
2
μ
10
v
2
Abs[zzR]
2π
2
ω
/.{R0->RR0,R1->RR1,R2->RR2}/.y->2Immvv/.​​{
b0
l_,m_
->b0[l,m],
b1
l_,m_
->b1[l,m],
b2
l_,m_
->b2[l,m]}​​

Results
R
in
and
R
out
(Table 1)

We discard k>30 PM contributions, corresponding to terms of order
-11
10
forr=5and
-16
10
forr=100.
In[]:=
SetPrecisionRin2,
1
10
,5,10​​SetPrecisionRup2,
1
10
,5,10​​SetPrecisionRin2,
1
10
,100,15​​SetPrecisionRup2,
1
10
,100,15
Out[]=
-0.004086908784-0.000826769654
Out[]=
2632.048432-840.387207
Out[]=
252.949132499007-211.392731187846
Out[]=
-15205.329855167-989985.766975648

Results
Z
2,2
(Table 3)

We discard k>30 PM contributions, corresponding to terms of order
-13
10
forv=0.4and
-16
10
forv=0.1,0.3.
In[]:=
SetPrecisionetalm2,2,
1
10
,15​​SetPrecisionetalm2,2,
3
10
,15​​SetPrecisionetalm2,2,
4
10
,12
Out[]=
0.961483705502172
Out[]=
0.841176171325065
Out[]=
0.880705411714

Results dE/dt (Table 4)

We discard k>30 PM contributions, corresponding to terms of order
-13
10
forv=0.4and
-16
10
forv=0.1,0.3.

v=0.1

In[]:=
entemp01={};​​DoDoIfη01[l,m]=SetPrecisionetalml,m,
1
10
,15>=
-15
10
,entemp01=Join[entemp01,{{l,m,η01[l,m]}}],Break[],{m,l,1,-1};Ifη01[l,l]=SetPrecisionetalml,l,
1
10
,15<
-15
10
,Break[],{l,2,30}
In[]:=
dEdtv01=SetPrecision[Sum[entemp01[[i,3]],{i,1,Length[entemp01]}],15]
Out[]=
0.974719282291874

v=0.3

In[]:=
entemp03={};​​DoDoIfη03[l,m]=SetPrecisionetalml,m,
3
10
,15>=
-15
10
,entemp03=Join[entemp03,{{l,m,η03[l,m]}}],Break[],{m,l,1,-1};Ifη03[l,l]=SetPrecisionetalml,l,
3
10
,15<
-15
10
,Break[],{l,2,30}
In[]:=
dEdtv03=SetPrecision[Sum[entemp03[[i,3]],{i,1,Length[entemp03]}],15]

v=0.4

PN expansions

Download needed quantities files here

Definitions

CITE THIS NOTEBOOK

Resumming Post-Minkowskian and Post-Newtonian gravitational waveform expansions​
by Andrea Cipriani, Giorgio Di Russo, Francesco Fucito, José Francisco Morales, Hasmik Poghosyan, Rubik Poghossian
Wolfram Community, STAFF PICKS, April 22, 2025
​https://community.wolfram.com/groups/-/m/t/3448301