close

Вход

Log in using OpenID

embedDownload
Cover Page
The handle http://hdl.handle.net/1887/24880 holds various files of this Leiden University
dissertation.
Author: Dalhuisen, Jan Willem
Title: The Robinson congruence in electrodynamics and general relativity
Issue Date: 2014-03-25
8
The Robinson congruence in
general relativity
8.1 introduction
In previous chapters the Robinson congruence or its projection on a time-slice,
the Hopf fibration, was seen to correspond to a non-null twistor, an exact
solution to source-free Maxwell equations and a solution to linearized Einstein
equations. The purpose of this chapter is to show that it is also related to
an exact solution in general relativity. To this end we summarize and, to suit
our purposes, supplement an article by Debney, Kerr and Schild [13] in which
a formalism was established that can be considered as a solution generating
technique "avant-la-lettre".
Consider a metric that can be cast in so called Kerr-Schild form:
gµν = ηµν + 2heµ eν ,
in which η is the metric of Minkowski space (M 4 with usual coordinates
{t,x,y,z} that will also be used (with different interpretation of course) as
coordinates in the curved manifold), h = h(xµ ) and e is a null vector:
g µν eν eµ = eµ eµ = 0.
The contravariant form of the metric tensor can easily be seen to be
91
8. THE ROBINSON CONGRUENCE IN GENERAL RELATIVITY
g µν = η µν − 2heµ eν .
It follows that η µν eµ eν = 0: e is also a null vector with respect to "auxiliary" Minkowski space. Raising and lowering of indices of vectors that are
orthogonal to e, including e itself, can be done with η.
In terms of the null coordinates {u, v, w, w}:
¯
u≡
√1 (t
2
+ z)
v≡
√1 (t
2
− z)
w≡
√1 (x
2
+ iy)
w
¯=
√1 (x
2
− iy)
we can write a general (covariant) vector with unit coefficient in front of
du as
e = du + Adv + Y¯ dw + Y dw
¯
(the case of vanishing coefficient in front of du can be treated by a suitable limiting process, but will be of no concern here)
and the Minkowski metric tensor as


0
1
0
0
1
0
0
0
.
(ηµν ) = (η µν ) = 
0
0
0
−1
0
0
−1
0
The condition for e to be null with respect to η now gives A = Y Y¯ :
e = du + Y Y¯ dv + Y¯ dw + Y dw,
¯ or




1
Y Y¯
 1 
Y Y¯ 
µ
µν



(eµ ) = 
 Y¯  and (e ) = (η eν ) = −Y .
−Y¯
Y
The line element in null coordinates is ds2 = 2dudv − 2dwdw
¯ + 2h(du +
Y Y¯ dv + Y¯ dw + Y dw)
¯ 2 , and therefore
0
1
(gµν ) = 
0
0

92
1
0
0
0
0
0
0
−1


0
1
Y Y¯
0
 + 2h 
 Y¯
−1
0
Y
Y Y¯
(Y Y¯ )2
Y Y¯ 2
Y 2 Y¯
Y¯
Y Y¯ 2
Y¯ 2
Y Y¯
Y

Y 2 Y¯ 
.
Y Y¯ 
Y2
8.2. DEBNEY, KERR AND SCHILD
The inverse can be calculated most directly from g µν = η µν − 2heµ eν :
0

1
(g µν ) = 
0
0

1
0
0
0
0
0
0
−1


(Y Y¯ )2
0
¯

0
 − 2h  Y Y2
−Y Y¯
−1
−Y Y¯ 2
0
Y Y¯
1
−Y
−Y¯
−Y 2 Y¯
−Y
Y2
Y Y¯

−Y Y¯ 2
−Y¯ 
.
Y Y¯ 
Y¯ 2
¯
Using the null vector e, we can form a Newman-Penrose tetrad {l, n, m, m},
¯ = (m
l = (lµ ) ≡ (eµ ), n = (nµ ), m = (mµ ), m
¯ µ ), with, in null coordinates








−Y¯
−Y
1 − hY Y¯
Y Y¯
 0 
 0 
 −h 
 1 
µ
µ






¯ µ) ≡ 
(lµ ) = 
 1 .
−Y  , (n ) ≡  hY  , (m ) ≡  0  , (m
0
1
hY¯
−Y¯
Note that in these coordinates m
¯ µ is not simply the complex conjugate of
µ
m , as would be the case in real coordinates. In terms of the more familiar
{t, x, y, z} these definitions would be more complicated. We will need only e in
these coordinates:

1 + Y Y¯
1 −(Y + Y¯ )

(eµ ) = √ 
2  i(Y − Y¯ ) 
−1 + Y Y¯

(8.1)
When the spinor field (ΠA ) = (Π1 , Π2 ) corresponds to the direction of e,
considered as a vector in auxiliary Minkowski space, M 4 , we find from
¯ X 0 and equations (8.1), (1.4) that
eµ ∝ σ µAX 0 ΠA Π
Y =−
Π1
Π2
=
.
2
Π
Π1
(8.2)
8.2 Debney, Kerr and Schild
The result of the article by Debney, Kerr and Schild that is of interest here can
be summarized as follows.
93
8. THE ROBINSON CONGRUENCE IN GENERAL RELATIVITY
For arbitrary (complex) analytic functions Φ and Ψ, constant q ∈ C and
constants p, c, m ∈ R we generate a solution of Einstein-Maxwell equations in
terms of null coordinates {u, v, w, w}:
¯
F ≡ Φ(Y ) + (qY + c)(w + Y v) − (pY + q¯)(u + Y w)
¯
generates, via F ≡ 0:
e.m. field:
φ0 = 0, φ1 = 12 Ψ(Y )P −2 ρ2 , φ2 = − 21 (∂Y Ψ)(∂Y F )−2 + 12 Ψ(Y )(∂Y F )−3 ∂Y ∂Y F
metric:
gµν = ηµν + 2heµ eν
where
P ≡ pY Y¯ + qY + q¯Y¯ + c
ρ ≡ −P (∂Y F )−1
¯ −4 ρ¯
ρ
h ≡ 12 mP −3 (ρ + ρ¯) − 12 ΨΨP
e ≡ du + Y Y¯ dv + Y dw
¯ + Y¯ dw
is null geodesic and shear-free
φ0 , φ1 , φ2 and ρ are defined as in the Newman-Penrose formalism with
¯ as in section 8.1.
{l, n, m, m}
Comparison of this function F with equations (8.2) and (1.10), which is
also valid in curved spaces of Kerr-Schild form, explains one part of the working
of this prescription to generate solutions to the combined system of Maxwell
and Einstein equations, the fact that e is geodesic and shear-free.
The defining equations of φ0 , φ1 and φ2 in terms of the tetrad components of the Faraday tensor in section 7.4.3 can be used to obtain the
inverse relations, expressing the tetrad components in terms of the φ’s:
∗
F(1)(3) = −F(3)(1) = F(1)(4)
= φo , F(1)(2) = −F(2)(1) = φ1 + φ∗1 , etc. Equation
(7.2) can now be used to obtain the F (k)(l) ’s. From these, the calculation of
F (k)(l) F(k)(l) leads to 4(φ0 φ2 + φ∗0 φ∗2 ) − 4(φ21 + φ∗2
1 ). This shows that the formalism
94
8.3. EXAMPLES
of Debney, Kerr and Schild does not lead to null electromagnetic fields, since,
from section 3.3, 2(E2 − B2 ) = F µν Fµν = F (k)(l) F(k)(l) = 0 only for Ψ(Y ) = 0:
when an electromagnetic field is present, at least one of the requirements for
this field to be null is not satisfied.
8.3 examples
In [13] the choice Φ = iaY, q = 0 and p = √12 = c was shown to lead to the
now famous Kerr metric for Ψ = 0 and to its electrically charged version, the
Kerr-Newman metric, for Ψ = e. Since the angular momentum of these solutions
is proportional to a it follows that we arrive at the Schwarzschild and the
charged Schwarzschild or Reissner-Nordström solutions by putting a = 0 in the
choices from above. Here we are interested in other choices of the parameters.
However, for comparison we first give the results of the present procedure for
the Schwarzschild and Kerr solutions.
Schwarzschild solution, Φ = 0 = Ψ, q = 0, p =
√
2
√1
2
=c:
2 2
m (x +y )
2
2 xz+r
≡ x2 + y 2 + z 2
2 +y 2 , h = 2 (r+z)2 r 3 , r
 
r
√


x
2(r+z)

(eµ ) = x2 +y2 
y 
z
ρ=
Note that the combination heµ eν that appears in the metric does not contain the common factor in (eµ ), in agreement with the sperical symmetry of the
solution. The form of the expansion, -Re(ρ) = ρ here, seems to contradict this
symmetry. However, the spin
√ coefficients depend on the chosen tetrad. A type
3) transformation with A= 2 r(xz+r
2 +y 2 ) leads to a manifest spherical symmetric
expansion.
Kerr solution, Φ = iaY, Ψ = 0, q = 0, p =
√
2
2
2 2
r˜
m (x +y )
r
2 xz+˜
2 +y 2 r
˜2 +iaz , h = 2 (˜
r +z)2


r˜
x˜
r
+ay
√
˜ 2 2 
r +z) r
r˜ +a 
(eµ ) = x2(˜
2 +y 2  y˜
r˜ r2 −ax2 
ρ=
r˜
r˜4 +a2 z 2
√1
2
=c:
, r˜4 + (a2 − x2 − y 2 − z 2 )˜
r2 − a2 z 2 = 0
r˜ +a
z
Note the difference between r˜ (an ellipsoidal radial coordinate) and r. In
the limit a → 0 we have r˜ → r, resulting in the Schwarzschild solution. Apart
95
8. THE ROBINSON CONGRUENCE IN GENERAL RELATIVITY
3
r
from the flat space-time part, the metric now contains the factor r˜42m˜
+a2 z 2 , shared
by all coefficients. It it clear that the angular momentum is directed in the
z-direction.
The coordinate transformation u = t + r˜, eiφ sinθ = (x + iy)(˜
r − ia)−1 , z = r˜cosθ
leads to the form of the Kerr metric as it appeared for the first time in the
literature [80]:
2
2m˜
r
2
− 2(du + asin2 θdφ)(d˜
r+
ds2 (= gµν dxµ dxν ) = [1 − r˜2 +a
2 cos2 θ ](du + asin θdφ)
2
2
2
2
2
2
2
asin θdφ) − (˜
r + a cos θ)(dθ + sin θdφ ).
The limit a → 0 of this expression gives the Schwarzschild metric in the
familiar advanced Eddington-Finkelstein coordinates. For r˜ = 0 and θ = π2
there is a curvature singularity (not just a coordinate singularity) that has the
geometry of a ring in auxiliary Minkowski space.
8.3.1 Robinson congruence
Φ=
√i Y,
2
Ψ = 0, p = 0 = q, c = 1
These choices lead to P = 1, F = ( √i2 + v)Y + w, thus F ≡ 0 ⇐⇒ Y =
√
− √2w
.
i+ 2v
In terms of the auxiliary coordinates {t, x, y, z} we have:
Y =
x+iy
−t+z−i ,
ρ=
√
2
1+(t−z)2 (−t
+ z + i),
√
−t+z
h = m 2( 1+(t−z)
2 ) and


1 + x2 + y 2 + (t − z)2


1
2(x(t − z) + y)

.
(eµ ) = √


2
2(y(t − z) − x)
2(1 + (t − z) )
2
2
2
x + y − 1 − (t − z)
(8.3)
We recognise the Robinson congruence or Hopf fibration in this form, see
equations (4.2), (4.9) and (4.10). Alternatively, the corresponding spinor field
Π2
(Π1 , Π2 ) with Π
= Y is (Π1 , Π2 ) = f (xµ )(−t + z − i, x + iy) which is exactly
1
equation (4.6), which led to the Robinson congruence.
In all the previous appearances of the Hopf fibration we were able to give
a physical interpretation of the integral curves of the vector field: as electric
or magnetic field lines, lines related to energy flow ((super-)Poynting vector),
96
8.3. EXAMPLES
tendex or vortex lines. But now the vector field appears twice in the metric
without obvious interpretation. In addition, we do not have a clear and global
distinction between time and space here, as was the case in past examples,
where fields could be considered to be defined on flat space-time. The vector
field in (8.3) is still to be considered as a Robinson congruence, but a projection
on a time-slice in order to arrive at the structure of a Hopf fibration cannot be
done, except in auxiliary space.
Of course, the congruence is built from null geodesics and these are possible
photon paths or light rays. But in every point there are possible photon paths in
all directions.
If we change Ψ from zero to constant e, the difference appears only in the
expressions for h and the electromagnetic field. There is an extra term in
h due to the contribution of the electromagnetic energy density to the cur−e2
The electromagnetic field is represented as
vature of space-time: 1+(z−t)
2.
2
(z−t+i)
φ0 = 0 = φ2 , φ1 = e (1+(z−t)
2 )2 . This expression is not very illuminating, but
again a Robinson congruence pops up, since the corresponding electromagnetic
field can also be obtained from a one form α: Fµν dxµ ∧ dxν = dα, in which
√
2(z−t)
α = e 21+(z−t)
2 e.
8.3.2 degenerate Robinson congruence
Φ = 0 = Ψ, p = 0 = q, c = 1
This leads to Y =
x+iy
z−t ,
ρ=
√
2
z−t ,
h=
√
m 2
z−t
and

x2 + y 2 + (t − z)2


2x(t − z)
,



2y(t − z)
x2 + y 2 − (t − z)2

(eµ ) =
√
1
2(t−z)2
in which we recognise a degenerate Robinson congruence, see (4.22).
2
e
Putting Ψ = e, a constant, leads to an additional term in h: − (z−t)
2 , and
an electromagnetic
field that can be represented as in example (8.3.1), with
√
α = e 2z−t2 e.
97
8. THE ROBINSON CONGRUENCE IN GENERAL RELATIVITY
8.4 conclusion and final remarks
In this chapter we used a method developed by Debney, Kerr and Schild to
show that the Robinson congruence also appears in exact solutions of the full
Einstein equations. However, within the confines of classical general relativity
it is fair to say that in contrast with previous cases we do not know whether
a physical interpretation is possible. An attempt at an interpretation along the
lines of [35, 36, 81, 82] could be worthwhile.
An interesting possibility suggests itself when comparing the solution of section
8.3.1 with the solution in chapter 6. Could the latter be the linearized version
of the former? If yes, can the interpretation of the curves of the Robinson
congruence in the solution of the linearized theory in some sense be taken over
to the exact solution in the full theory?
It has not yet been investigated whether the solution in section 8.3.1 belongs to the class of solutions for which no general solution is known [83] or
perhaps to the class of which only a few solutions are known [84], or neither of
these possibilities.
It may further be noted that from [85] we may conclude the Petrov typeD character of at least the vacuum solutions in sections 8.3.1 and 8.3.2, as well
as the fact that these solutions must contain singularities not confined to a
bounded region. This makes it unlikely that they will bring any changes to
the observation in [86]: ’But the hope of finding metrics amongst the solutions
[. . . ] which describe the radiation field of a physically meaningful matter
distribution has not been realized.’
As in chapter 4, there is a complex shift related to the transition degenerate Robinson congruence → Robinson congruence. The relation between the two
shifts has not been investigated. Again, there is a possible connection with work
done by E. Newman [31, 33].
Finally, it is of interest to note that in the case of rotation-free Kerr-Schild
metrics there is a geometrical interpretation for (eµ ) and other quantities appearing in the present procedure [87] (also described in [88]). For this interpretation
to work for the degenerate Robinson congruence (rotation-free, since ρ is a real
function) we need the unphysical assumption of a massive particle travelling
along the z-axis with the speed of light in order to have the correct retarded
distance [88]. In addition, other quantities do not fit into the scheme presented
in said references.
98
1/--pages
Пожаловаться на содержимое документа