Heun equation, Teukolsky equation, and type-D metrics

0.24 MB·English

by  D. Batic

APS/123-QED
Heun equation, Teukolsky equation, and type-D metrics
D.Batic∗
Institute for Theoretical Physics
Swiss Federal Institute of Technology
CH-8093 Zu¨rich, Switzerland
H.Schmid
7
0 UBH Software & Engineering GmbH
0
2 D-92263 Amberg, Germany †

n
a (Dated: February 7, 2008)
J

5 Abstract
1

Startingwiththewholeclassoftype-Dvacuumbackgroundswithcosmological constantweshow
2
v
4 that the separated Teukolsky equation for zero rest-mass fields with spin s = 2 (gravitational
±
6
0 waves), s = 1 (electromagnetic waves) and s = 1/2 (neutrinos) is an Heun equation in disguise.
± ±
1
0
7 PACS numbers: 02.30Gp, 02.30.Hq, 0420.Jb,0.462.+v, 0.470.Bw
0
/
c Keywords: Heun equation, Teukolsky equation, type-D metrics, QFT in curved spacetimes
q
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r
g
:
v
i
X
r
a
∗Electronic address: [email protected]
†Electronic address: [email protected]
1
I. INTRODUCTION
According to Ronveaux (1995) the Heun equation (HE) is the most general second order
linear ODE of the form
d2y 1+2α 1+2α 1+2α dy αβz q
1 2 3 A
+ + + + − y = 0 (1)
dz2 z z 1 z z dz z(z 1)(z z )
(cid:18) − − S (cid:19) − − S
where a C 0,1 , α , α , α , α, β, q are complex arbitrary parameters, and 0, 1, a, and
1 2 3 A
∈ \{ }
are regular singularities with exponents 0, 2α , 0, 2α , 0, 2α , and α,β ,
1 2 3
∞ { − } { − } { − } { }
b b
respectively. Equation (1) has been originally constructed by the German mathematician
Karl Heun (1889) as a generalization of the hypergeometric equation (HYE). In order to see
how HE degenerates to the HYE we can first multiply (1) by z(z 1)(z a), then we set
− −
a = 1, and q = αβ, and finally we take out a factor (z 1), leaving the HYE in its standard
A
− b
form. Hence, we can always think to a HYE as a degenerated equation of Heun’s type. To
b
underline theimportance of (1) we recall thatit contains thegeneralized spheroidal equation
(GSWE), the Coulomb spheroidal equation, Lam´e, Mathieau, and Ince equations as special
cases. The fields of applications of the HE in physics are so large that it is not possible
to describe them here in detail. However, a review of many general situations relevant to
physics, chemistry, and engineering where the HE occurs can be found in Ronveaux (1995)
(pp 341). Here, we will show which role plays the HE in quantum field theory in curved
spacetimes.
To understand the motivation underlying the present work we give a short review on studies
concerning exact solutions of the Teukolsky equation in some black hole geometries. In
the 70’s, and 80’s we find a large number of publications regarding the angular equation
obtained after separation of variables from the Teukolsky wave equation on Kerr manifolds.
See, forinstance, Press, andTeukolsky (1973), Breuer et al. (1977), Fackerell, andCrossman
(1977), Leahy, and Unruh (1979), Chakrabarti (1984), and Seidel (1989). According to these
references we will also name the solution of the angular equation as spin-weighted spheroidal
function (SWSF). The radial equation has been investigated by Bardeen, and Press (1973),
Page (1973), Lee (1976), Arenstorf et al. (1978). The common picture emerging from
all previous studies is that the radial equation cannot be in general related to any known
differential equation of mathematical physics. This view changed with the work of Blandin
et al. (1983). They showed that the SWSF’s may be obtained by means of an elementary
transformation from Heun confluent functions. Three years later Leaver (1983) proved that
2
the radial, and angular parts of the Teukolsky master equation (TME) in the Kerr geometry
are generalized spheroidal wave equations. Finally, Suzuki et al. (1998) showed that the
radial, and angular part equations arising from the TME in the Kerr-Newman-deSitter
metric (KNdS) after separation of variables are Heun equations. It is interesting to observe
that if we let the cosmological constant go to zero (i.e. the KNdS geometry goes over to
the Kerr-Newman metric) their HEs become a confluent HE which, in turn, coincides with
the GSWE given by Leaver in 1983. Hence, the following question arises quite naturally,
namely: is it possible to reduce the TME in any physical relevant type D metric to a HE?
To conclude this short review we cite what Wu, and Cai wrote in 2003: ”it is not clear until
now whether the generalized Teukolsky equation in the general type D vacuum backgrounds
with cosmological constant can be transformed into a Heun equation.”
Our paper is organized as follows: in Sec. II we shortly present some results due to Kamran,
and McLenaghan (1987) concerning the separation of the TME in any type-D background.
In Sec. III-VII we show that the TME can be transformed in any physical relevant type D
metric into a HE.
II. BACKGROUND
Let D denote the class of algebraically special Petrov type D vacuum metrics with
0
cosmological constant. According to Thm. 2.1 in Kamran, and McLenaghan (1987) there
exists a system of local coordinates (u,v,w,x) in which such metrics can be written as
ds2 = 2 θ1θ2 θ3θ4 (2)
−
with a symmetric null tetrad (θ1,θ2,θ3,θ4)(cid:0)given by (cid:1)
Z(w,x) fW(w) dw
θ1 = (ǫ du+m(x) dv)+ ,
√2T(w,x) Z(w,x) 1 g2W(w)
p (cid:20) (cid:21)
Z(w,x) W(w) fdw
θ2 = (ǫ du+m(x) dv) ,
√2T(w,x) Z(w,x) 1 − g2W(w)
p (cid:20) (cid:21)
Z(w,x) X(x) dx
θ3 = (ǫ du+p(w) dv)+i = θ4,
2
√2T(w,x) Z(w,x) X(x)
p (cid:20) (cid:21)
1+f2
Z(w,x) := ǫ p(w) ǫ m(x), g :=
1 2
− 2
r
where all functions are real-valued and ǫ , ǫ , and f are constants such that ǫ2 + ǫ2 = 0.
1 2 1 2 6
Depending on whether fW2(w) is positive, negative or zero the metric (2) possesses a two-
3
parameter abelian group of isometries whose orbits are timelike, spacelike or null at a given
point, respectively. By integration of the Einstein-Maxwell field equation it results that the
general solution A∗ in the class D can be specified as follows (Thm. 2.2 ibid.)
0
ǫ = bf2cosγ, ǫ = sinγ,
1 2
ℓ2(1 b2cos2γ)
m(x) = c2x2 +b2k2 + − ǫ 2cℓx,
− ǫ2 2 −
(cid:20) 2 (cid:21)
k2(b2 ǫ2)
p(w) = + b2c2w2 +ℓ2 + − 2 cosγ +2b2ckw,
cos2γ
(cid:20) (cid:21)
T(w,x) = a(cwcosγ +k)(cxsinγ +ℓ)+1,
fW2(w) = c2b4g w4cos2γ +cf w3cosγ +f w2 +f w+f ,
4 3 2 1 0
X2(x) = c2g x4sin2γ +cκ x3sinγ +κ x2 +g x+g ,
4 1 2 1 0
κ := acf cosγ 2akf +3ak2f +4(ℓ ab4k3)g ,
1 1 2 3 4
− −
κ := 3acℓf cosγ (1+6akℓ)f +3k(1+3akℓ)f +6[ℓ2 b4k2(1+2akℓ)]g
2 1 2 3 4
− −
where f , f , f , f , g , g , g , a, b, c, k, ℓ, and γ are real parameters satisfying the relations
0 1 2 3 0 1 4
acg sinγ 3a2cℓ2f cosγ +2aℓ(1+3akℓ)f [1+3akℓ(2+3akℓ)]f
1 1 2 3
− −
+4[b4k aℓ3 +3ab4k2ℓ(1+akℓ)]g = 0, (3)
4
−
c2(g sin2γ b4f cos2γ)+c[(2aℓ3 +b4k)f cosγ ℓg sinγ] (b4k2 +ℓ2 +4akℓ3)f
0 0 1 1 2
− − −
+(b4k3 +3kℓ2 +6ak2ℓ3)f +(3ℓ4 b8k4 6b4k2ℓ2 8ab4k3ℓ3)g = 0, (4)
3 4
− − −
and are restricted to a range such that (2) is non-singular with signature minus two. More-
over, the cosmological constant Λ is expressed in terms of the above parameters by
Λ = 3[a2c2f cos2γ a2ckf cosγ +a2k2f a3k3f +(1+a2b4k4)g ]. (5)
0 1 2 3 4
− − −
Let A∗ denote the subclass of solutions in A∗ obtained by setting b = 1, c = √2, k = ℓ = 0
and γ = π/4. If in addition a = f = 1, such solutions recover the vacuum case with
f
cosmological constant of the seven-parameter family of Plebanski and Demianski (1976)
containing the Kerr-Newman-de Sitter metric as a special case.
Let B0 be the subclass of solutions in A∗ such that a = 0, b = c = 1, ℓ = 0 and γ = π/2.
−
If in addition f = 1, B0 reduces to the vacuum case with cosmological constant of Carter’s
− f
B (1968). By B0 we will denote the subclass of solutions in A∗ obtained by setting a = 0,
− +
e 4 f
b = c = 1, and k = γ = 0. If we let f = 1, such a solution becomes the vacuum case with
cosmological constant of Carter’s B . The Carter’s B describe all non-accelerating type
+ ±
D metrics in a coordinate system in which the components of the metric and the Maxwell
e e
field are rational functions.
Let C∗ denote the subclass of solutions in A∗ obtained by setting a = 1, b = 0, c = √2,
k = ℓ = 0 and γ = π/4. If in addition f = 1, C∗ reduces to the accelerating C-metric of
f
Levi-Civita (1918).
Finally, let C00 denote the subclass of solutions in A∗ obtained by setting a = 0, b = 1,
c = 0, k = 0, ℓ = 1 and γ = 0. If in addition f = 1, C00 becomes the Robinson-Bertotti
f
solution (1959).
Following Kamran, and McLenaghan (1987) the Teukolsky equation can be written in a
compact form by introducing a spin parameter s which can assume the values 2, 1, and
± ±
1/2. For s = 2, 1 and 1/2 we have
±
[(D (2s 1)ǫ+ǫ 2sρ ρ)(∆ 2sγ+µ) (δ+π α (2s 1)β 2sτ)(δ+π 2sα)
− − − − − − − − − − −
1
2(s )(s 1)Ψ ]Φ = 0, (6)
2 s
− − 2 −
and for s = 2, 1 and 1/2
− − −
[(∆ (2s+1)γ γ 2sµ+µ)(D 2sǫ ρ) (δ τ +β (2s+1)α 2sπ)(δ τ 2sβ)
− − − − − − − − − − −
1
2(s+ )(s+1)Ψ ]Φ = 0 (7)
2 s
− 2
where Ψ is the non-zero Newman-Penrose component of the Weyl tensor and the Φ ’s are
2 s
defined in terms of the field components as given in Table 1 (pp 286 ibid.). According to
Thm. 3.1 (ibid.) for all solutions in the class D , and for all s = 2, 1, 1/2 equations
0
± ± ±
(6), and (7) possess a separable solution of the form
T|s|+1
Φ = ei(ru+qv) ei|s|B/2Θ (w,x)
s Z|s|/2 s
where r, q are arbitrary real constants, dB = Z−1(ǫ m′dw + ǫ p′dx), and Θ (w,x) =
1 2 s
G (w)H (x). Moreover, (6), and (7) separate into the pair of decoupled ODE’s
s s
D L G (w)+f (w)G (w) = λ G (w), (8)
ws ws s s s s s
D L H (x) g (x)H (x) = λ H (x) (9)
xs xs s s s s s
−
5
where λ is a separation constant, the functions f , and g are given in Table 1 (pp 290-291
s s s
ibid.), and
d i 1 pr ǫ q dW
2
D = W + f (1+ǫ(s)) (1 ǫ(s)) − +(1 s ) ,
ws dw 1+f2 − f − W −| | dw
(cid:20) (cid:21)
d ǫ q mr dX
1
D = iX +iǫ(s) − +i( s 1) ,
xs
− dx X | |− dx
d i pr ǫ q dW
L = fW + (1+ǫ(s)) f2(1 ǫ(s)) − 2 s f ,
ws
− dw 1+f2 − − W −| | dw
d ǫ q (cid:2)mr dX (cid:3)
1
L = iX +iǫ(s) − +i s ,
xs
dx X | | dx
where ǫ(s) is a sign function such that ǫ(s) = +1 for s > 0, and ǫ(s) = 1 for s < 0.
−
III. THE METRIC A∗ : b = 1, c= √2, k = ℓ = 0 , γ = π/4
In this case we have
√2
ǫ = ǫ = , m(x) = √2 x2, p(w) = √2 w2, T(w,x) = awx+1,
1 2
2 −
3 4
fW2(w) = g w4 + f wn = g (w w ),
4 n 4 i
−
n=0 i=1
X Y
4
X2(x) = g x4 +af x3 f x2 +g x+g = g (x x ).
4 1 2 1 0 4 i
− −
i=1
Y
Moreover, for i,j = 1, ,4 let w and x denote the i-th root of the polynomial equations
i i
···
fW2(w) = 0, and X2(x) = 0, respectively. Throughout this section we shall assume that
w = w , and x = x for every i = j. Furthermore, in the present case (5) reads
i j i j
6 6 6
Λ
g = a2f + .
4 0
− 3
(cid:18) (cid:19)
Making use of the expression for Ψ given by (2.7g) in Kamran, and McLenaghan (1984),
2
the functions f (w), and g (x) entering, respectively in (8), and (9) are computed to be
s s
f (w) = (1 2 s ) 2g (1 s )w2 + 2i√2rǫ(s)+ag (1 s ) w ,
s 4 1
− − | | −| | −| |
h (cid:16) (cid:17) i
g (x) = (1 2 s ) 2g (1 s )x2 2√2rǫ(s) af (1 s ) x .
s 4 1
− − | | −| | − − −| |
h (cid:16) (cid:17) i
Hence, the equation for G (w) becomes
s
d dG dG
fW2 s +Γ(w) s +Q (w)G = 0 (10)
s s
dw dw dw
(cid:18) (cid:19)
6
with
f2 1
Γ(w) := i√2 − (2rw2 q),
f2 +1 −
f2 1
Q (w) := λ +2g (1 s )(1 2 s )w2+ 2i√2r 2s+ − +ag (1 s )(1 2 s ) w+
s s 4 −| | − | | − f2 +1 1 −| | − | |
(cid:20) (cid:18) (cid:19) (cid:21)
W′ 2f (2rw2 q)2
f s WW′′ +(1 s )(W′)2 +i√2s(2rw2 q) + − .
| | −| | − W (1+f2)2 W2
(cid:16) (cid:17)
Equation (10) can be further simplified. To this aim let us make the substitution
G (w) = eh(w)ϕ (w).
s s
If we require that h′ = Γ/(2fW2) we obtain the following equation for ϕ (w)
s
−
d dϕ
fW2 s +R (w) ϕ = 0 (11)
s s
dw dw
(cid:18) (cid:19)
where
R (w) := λ +2g (1 s )(1 2 s )w2 w+
s s 4 s
−| | − | | −C
W′ (2rw2 q)2
f s WW′′ +(1 s )(W′)2 +i√2s(2rw2 q) + −
| | −| | − W 2fW2
(cid:16) (cid:17)
with
:= 4i√2sr ag (1 s )(1 2 s ).
s 1
C − −| | − | |
Let us introduce the following functions
σ (w) := 2rw q, f (w) := 2g (1 s )(1 2 s )w2 w +λ ,
±q s 4 s s
± −| | − | | −C
and let us define constants
3
c−1 := g (w w ), i = 1,2,3.
i 4 i − j
j=1
Yj6=i
By means of the homographic substitution
w w w w
1 2 4
z = − − (12)
w w w w
4 2 1
− −
mapping the points w ,w ,w ,w , to 0,1,z , ,z with
1 2 3 4 S ∞
∞ ∞
w w w w
2 4 3 1
z := − , z := − z (13)
∞ S ∞
w w w w
2 1 3 4
− −
7
equation (11) becomes
d2ϕ dϕ
s s
+P(z) +Q (z)ϕ = 0 (14)
dz2 dz s s
with e
1 1 1 2
P(z) = + + ,
z z 1 z z − z z
S ∞
− − −
B B B 2 A A A A
1 2 3 1 2 3 ∞
Q (z) = + + + + + + +
s z2 (z 1)2 (z z )2 (z z )2 z z 1 z z z z
S ∞ S ∞
− − − − − −
e
where for i = 1,2,3
3
1 3ag
A = (2 s 2 3 s +2) w +(1 s )(1 2 s ) 1 (2 s )(1+2 s )w ,
∞ i 4
z (w w ) | | − | | −| | − | | g − −| | | |
∞ 4 1 4 !
− i=1
X
s c σ (w2) 2
B = i i −q i ,
i
− 2 − √2(w w )
(cid:18) i − 4 (cid:19)
c w w w w
1 2 1 3 1
A = g (w ), A = c − g (w ), A = c − g (w ),
1 s 1 2 2 s 2 3 3 s 3
−z w w z (w w )
∞ 4 1 S 4 1
− −
and
g √2s
g (w ) = 4 s p(w ) f (w )+i σ (w (w 2w ))+
s i i s i +q i 4 i
2 | | − w w −
i 4
−
g c2 3 3
4 i σ (w2) σ (w2) w 2w σ w ,
w w −q i  +q i j − i +q j 
i − 4 j=1 j=1
 Xj6=i Yj6=i 
  
3 3 3
p(w ) := 2(2 s 3)w2+(4 3 s )w w +( s 2)w w 2w +2( s 1) w .
i | |− i − | | i j | |− 4 j − i | |− j
j=1 j=1 j=1
Xj6=i Xj6=i  Yj6=i
  (15)
If we make the F-homotopic transformation
ϕ (z) = zα1(z 1)α2(z z )α3(z z )ϕ (z)
s S ∞ s
− − −
and require that α2 = B , then (14) becomes e
i − i
d2ϕ dϕ
s s
+P (z) +Q (z)ϕ = 0 (16)
dz2 s dz s s
e e
with b b e
1+2α 1+2α 1+2α A A A A
1 2 3 1 2 3 ∞
P (z) = + + , Q (z) = + + + .
s s
z z 1 z z z z 1 z z z z
S S ∞
− − − − −
b b b b
b b
8
Now, a direct computation gives
4
(1 s )(1 2 s ) ag
1
A = −| | − | | w + , (17)
∞ i
z (w w ) g
∞ 4 1 4 !
− i=1
X
b3 (1 s )(1 2 s ) 4 ag
1
A = −| | − | | w + . (18)
i i
− z (w w ) g
∞ 4 1 4 !
i=1 − i=1
X X
b
Taking into account that for the metric A∗ equation (3) reduces to ag f = 0, and
1 3
−
replacing f by ag in the equation fW2 = 0 it can be checked that the following relation
3 1
holds
4
ag
1
w = .
i
− g
4
i=1
X
Hence, the coefficients (17), and (18) are zero, and (16) reduces to an Heun equation with
terms αβ and accessory parameter q (according to the notation in (1)) given by
A
αβ = (z +1)A z A A = z A +A , q = z A (19)
S 1 S 2 3 S 3 2 A S 1
− − − −
b b b b b b
where
α +α (1+2α ) 1
1 3 1
A = A α α (1+2α ) , (20)
1 1 1 2 1
− − − z − z
S ∞
α +α (1+2α ) 1
Ab = A +α +α (1+2α ) 2 3 2 , (21)
2 2 1 2 1
− z 1 − z 1
S ∞
− −
α +α (1+2α ) α +α (1+2α ) 1
Ab = A + 2 3 2 + 1 3 1 . (22)
3 3
z 1 z − z z
S S ∞ s
− −
b
Concerning (9) we find the following equation for H (x), namely
s
d dH
X2 s + λ +2g (1 s )(1 2 s )x2 + x+
s 4 s
dx dx − −| | − | | C
(cid:18) (cid:19)
h
X′ (2rx2 +q)2
s XX′′ ( s 1)X′2 √2s(2rxb2 +q) H = 0 (23)
| | − | |− − X − 2X2 s
(cid:21)
(cid:16) (cid:17)
with
:= 4√2rs+(1 s )(1 2 s )af .
s 1
C −| | − | |
Let us introduce the followinbg notation
f (x) = 2g (1 s )(1 2 s )x2 + x λ ,
s 4 s s
−| | − | | C −
3
bc −1 = g (x x ), i = 1,2,3b.
i 4 i j
−
j=1
Yj6=i
b
9
By means of the homographic substitution
x x x x
1 2 4
z = − − (24)
x x x x
4 2 1
− −
mapping the points x ,x ,x ,x , to 0,1,x , ,x with
1 2 3 4 S ∞
∞ ∞
x x x x
2 4 3 1
x = − , x = − x (25)
∞ S ∞
x x x x
2 1 3 4
− −
our equation (23) becomes
d2H dH
s s
+P (z) +Q (z)H = 0 (26)
dz2 s dz s s
with
1 1 1 2
P (z) = + + ,
s
z z 1 z x − z x
S ∞
− − −
B B B 2 A A A A
1 2 3 1 2 3 ∞
Q (z) = + + + + + + +
s z2 (z 1)2 (z x )2 (z x )2 z z 1 z x z x
S ∞ S ∞
− − − − − −
e e e e e e e
where for i = 1,2,3
3
1 3ag
A = (2 s 2 3 s +2) x +(1 s )(1 2 s ) 1 (2 s )(1+2 s )x ,
∞ i 4
x (x x ) | | − | | −| | − | | g − −| | | |
∞ 4 1 4 !
− i=1
X
e s c σ (x2) 2
B = + i +q i ,
i
− 2 √2(x x )
(cid:18) i − 4 (cid:19)
c b x x x x
e 1 2 1 3 1
A = g (x ), A = c − g (x ), A = c − g (x ),
1 s 1 2 2 s 2 3 3 s 3
−x x x x (x x )
∞ 4 1 S 4 1
− −
b
ande b e b b e b b
g √2s
4
g (x ) = s p(x ) f (x ) σ (x (x 2x ))
s i i s i −q i 4 i
2 | | − − x x − −
i 4
−
b e b g c2 3 3
4 i σ (x2) σ (x2) x 2x σ x ,
x x +q i  −q i j − i −q j 
i − 4 j=1 j=1
b  Xj6=i Yj6=i 
  
3 3 3
p(x ) := 2(2 s 3)x2+(4 3 s )x x +( s 2)x x 2x +2( s 1) x . (27)
i | |− i − | | i j | |− 4 j − i | |− j
j=1 j=1 j=1
Xj6=i Xj6=i  Yj6=i
e  
If we make the F-homotopic transformation
H (z) = zαb1(z 1)αb2(z x )αb3(z x )H (z)
s S ∞ s
− − −
e
10