Here we present two new FD methods for solving the types of PDEs we are presented with during the gauge transformation.
5.4.1 Partial annihilators and higher order EHS: general considerations
Consider a PDE of the form
W`ma ψ`m(t, r) =Sext`m(t, r), (5.4.1) where Wa
`m is an ath order partial differential operator in t and r, which is acting on a
scalar field ψ`m. The source Sext`m(t, r) is non-compact, and therefore not amenable to the EHS method. The annihilator method is a standard technique [60] for solving differential equations, wherein we search for a differential operator for which S`m
ext is a homogeneous solution. Then, we could act on both sides of Eq. (5.4.1) and produce a homogeneous differential equation, albeit of a higher order. Given the singular nature of the source in our problem, it is unlikely that we will be able to find such an operator. However, it turns out in practice with such sources to be possible to find an operator that nearly annihilates
S`m
ext, e.g.
Wb
`mSext`m(t, r) =Ssing`m (t, rp(t)). (5.4.2)
HereWb
`m is anbth order partial differential operator intand r, andSsing`m (t, rp(t)) only has support at the location of the particle. We refer to this as a partial annihilator. Therefore, acting withWb
`m on Eq. (5.4.1) we have
W`mb W`ma ψ`m(t, r) =Ssing`m (t, rp(t)). (5.4.3)
We now have an equation with a point-singular source, which we can solve using the EHS method, but at the price of having raised it from orderato order a+b.
Moving into the FD, we Fourier transform Eq. (5.4.3) to get
Lb`mnLa`mnψ˜`mn(r) =Zsing`mn(r). (5.4.4) The effect of the partial annihilator in the FD, Lb
`mn, is to make a non-compact source
Z`m
of this subsection we will suppress the mode indices. Recall that the tilde over a symbol indicates a quantity which has been Fourier transformed into the FD.
The ODE (5.4.4) in r will have a+blinearly independent homogeneous solutions. (We
have in mind systems whereaandbare even integers.) We can specify them by demanding
that half of them are purely down-going at the event horizon and the other half are purely out-going at spatial infinity. We denote the former by ˜ψj− and the latter by ˜ψ+
j , where j
runs from 1 to (a+b)/2. Now, the causally appropriate particular solution to Eq. (5.4.4)
will be a linear combination of the homogeneous solutions, ˜ ψp(r) =c−1(r) ˜ψ − 1(r) +· · ·c − (a+b)/2(r) ˜ψ − (a+b)/2(r) +c+1(r) ˜ψ+1(r) +· · ·c+(a+b)/2(r) ˜ψ(+a+b)/2(r). (5.4.5)
We get the various normalization functions c±j(r) by the general method of variation of
parameters [60]. This entails solving the equations
dc±j dr∗(
r) =Zsing(r)
Wj±(r)
W(r) (5.4.6)
whereW(r) is the Wronskian andWj±(r) is the “modified Wronskian,” which is the Wron-
skian with the column corresponding to the ψ±j homogeneous solution replaced by the
column vector (0,0, . . . ,1). Having solved Eq. (5.4.6) for the normalization functions, we
can return to the TD via the standard Fourier synthesis (recall that we have suppressed
`, m, nindices on ˜ψp) ψp(t, r) = X n ˜ ψp(r)e−iωmnt. (5.4.7)
This will yield a causally appropriate solution to Eq. (5.4.1).
In our system though, the source Ssing will have some degree of lack of differentiability, and the sum above will converge in the TD only algebraically (if at all) at the location of the particle, due to the Gibbs phenomenon. Therefore, we seek to use the EHS method to
find exponentially-convergent solutions. To that end, we define
Cj−≡c−j(rmin), Cj+ ≡c+j (rmax), (5.4.8) which are referred to as the normalization coefficients and are the result of integrating Eq. (5.4.6) through the entire source region. Then, we define the EHS in the FD to be
˜
ψ±(r)≡
(aX+b)/2
j
Cj±ψ˜±j (r), (5.4.9)
and the EHS in the TD are defined as
ψ±(t, r)≡X
n
˜
ψ±(r)e−iωmnt. (5.4.10) Then, as before with the original EHS method [1], the weak solution representation
ψEHS ≡ψ+(t, r)θ[r−rp(t)] +ψ−(t, r)θ[rp(t)−r] (5.4.11)
expresses the solution to Eq. (5.4.1).
5.4.2 Extended particular solutions method
As an alternative to the partial annihilator method we consider solving Eq. (5.4.1) without promoting it to a higher-order equation. We start by moving Eq. (5.4.1) to the FD, yielding
La`mnψ˜`mn(r) =Zext`mn(r). (5.4.12) With its non-compact source, the EHS method is not immediately applicable to solve Eq. (5.4.12). As usual, we expect ˜ψ`mn to consist of both a particular solution and ho-
mogeneous solutions. We inspect the asymptotic nature of Z`mn
ext (r) at infinity and the event horizon. Between this and our differential operator La
`mn we should be able to find
the leading order nature of the causal particular solution. On the largerside, we denote this
the`, m, nindices, and will continue to do so for the remainder of this subsection.
We first take ˜ψ∞p (r), (though the opposite choice would work as well) as a boundary con-
dition at infinity to begin our ODE integration of Eq. (5.4.12). We integrate this differential equation inward, through the region of the source and on to the horizon. At this point, in addition to having obtained a particular solution, we will have excited all ahomogeneous
solutions, which will be evident in the behavior near the horizon. Half of these homogeneous terms will be causal waves traveling down into the black hole, and the other a/2 will be
acausal waves coming up from the black hole. We eliminate this acausal behavior by solving the homogeneous version of Eq. (5.4.12) for the a/2 acausal pieces and subtracting them
off. The homogeneous solutions on the infinity side are ˜ψ+h,j where j runs from 1 to a/2.
Likewise, there are a/2 homogeneous solutions on the horizon side, which we denote ˜ψh,j− .
We sum up the scaled homogeneous solutions and return to the TD via
ψh±(t, r) = ∞ X n=−∞ a/2 X j=1 ˜ ψ±h,j(r) e−iωmnt. (5.4.13) This is a fairly straightforward process if Eq. (5.4.1) has a source term which is differ- entiable everywhere. Unfortunately, the system we work with does not have such a source, and we must be careful. The source Sext will be a linear combination of singular pieces (δ, δ0, etc.) and the master function Ψ (either Ψeven or Ψodd) and its derivatives. Since we are working with linear equations, we can always solve for the singular parts with the EHS method, and we therefore consider only the extended source pieces which come from Ψ. When we Fourier transform Eq. (5.4.1) to get Eq. (5.4.12) there is an ambiguity that arises. Because the TD source of Eq. (5.4.1) contains Ψ, the FD source of Eq. (5.4.12) will contain R, which has two forms,
Rstd(r) =c+(r) ˆR+(r) +c−(r) ˆR−(r), and R±(r) =C±Rˆ±(r). (5.4.14)
The particular solution that we get from using Rstd as the source we call the standard particular solution and denote as ˜ψ∞/H. The superscript ∞/H is to distinguish between
whether the integration starts at infinity or the horizon. On the other hand, when using usingR± as the source we compute theextended particular solution (EPS) which we denote as ˜ψ±p. The superscript± is to distinguish between whether the integration starts at infinity
or the horizon.
BecauseRstdis the Fourier transform of Ψ, it must be used when solving for the correct homogeneous solutions, as described above. Returning ˜ψp∞/H to the TD produces ψ
∞/H p ,
which will exhibit the usual Gibbs phenomenon that is always present when the source is singular. The convergence will be algebraic at best. The way around this rests on generalizing the EHS method and using the extended particular solutions (EPS).
Having computed the EPS, we have in hand what it takes to form the true solution to Eq. (5.4.1). We use the Fourier synthesis to take the EPS to the TD,
ψp±(t, r) =X
n
˜
ψ±p(r)e−iωmnt. (5.4.15) By the same continuity arguments that apply to the EHS method, we claim that the causally appropriate solution to the inhomogeneous equation with non-compact source (5.4.1) is
ψ(t, r) =ψp+(t, r) +ψh+(t, r)θ[r−rp(t)] +
ψp−(t, r) +ψ−h(t, r)θ[rp(t)−r]. (5.4.16)
We have verified this claim by demonstrating numerically that this approach is entirely equivalent to the partial annihilator method. The new higher-order homogeneous solu- tions introduced by the annihilator are precisely the same as the particular solutions found here. Note that those homogeneous solutions come in standard and EHS form, just as the particular solutions here come in standard and EPS form.