Rigidly rotating dust solutions depending upon harmonic functions
Rigidly rotating dust solutions depending upon
harmonic functions
Stefano Viaggiu
Dipartimento di Matematica,
Universitá di Roma “Tor Vergata”,
Via della Ricerca Scientifica, 1, I-00133 Roma, Italy
E-mail: viaggiu@mat.uniroma2.it
(or: stefano.viaggiu@ax0rm1.roma1.infn.it)
August 4, 2021
Abstract
We write down the relevant field equations for a stationary axially
symmetric rigidly rotating dust source in such a way that the general
solution depends upon the solution of an elliptic equation and upon
harmonic functions. Starting with the dipole Bonnor solution, we built
an asymptotically flat solution with two curvature singularities on the
rotational axis with diverging mass. Apart from the two point singu-
larities on the axis, the metric is regular everywhere. Finally, we study
a non-asymptotically flat solution with NUT charge and a massless
ring singularity, but with a well-defined mass-energy expression.
PACS numbers: 04.20.-q, 04.20.Jb, 04.40.Nr
Introduction
The problem of building a physically admissible metric for an isolated ro-
tating body is still long an unresolved problem [1, 9]. In fact, in order to
obtain a physically reasonable source, many restrictions must be imposed
(energy conditions, regularity, reasonable equation of state). In particular,
Einstein’s equations for a rotating body with a perfect fluid source seem
not to be integrable. A remarkable exception is given by dust pressureless
stationary axially symmetric spacetimes. In a remarkable paper [10], Wini-
cour showed that Einstein’s equations for a stationary axially symmetric
dust source with differential rotation can be reduced to quadratures. These
http://arxiv.org/abs/0704.1351v2
equations contain as a subclass the van Stockum one [11] of rigidly rotating
matter. The first asymptotically flat solution of the van Stockum class can
be found in [12]. The author in [12] shows that the van Stockum class of
solutions that are not cylindrically symmetric cannot exsist in the Newto-
nian theory. The solution in [12] has a curvature singularity with diverging
mass. Further, the technique named ”displace, cut, reflect” [13] to obtain
rotating discs immersed in rotating dust must be noted. Unfortunately, this
method generates distributional exotic matter on the z = 0 plane.
In this paper, starting from the Lewis [14, 15] form of the metric, we write
down the equations for stationary axially symmetric rigidly rotating space-
times in a co-moving reference frame in such a way that the general solution
depends upon the solution of an elliptic equation and upon harmonic func-
tions. The class of solutions contains the van Stockum line element for a
suitable choice of the harmonic function.
In this context, starting from the dipole Bonnor solution [12], we obtain an
asymptotically flat solution with two curvature singularities on the rotation
axis and showing simalar properties to the Bonnor solution.
In section 1 we derive the basic equations. In section 2 we present our
solution. Section 3 collects some final remarks and conclusions. In the ap-
pendix we derive a non asymtotically flat solution with a NUT charge and
a well-defined mass-energy expression.
1 Basic Equations
Our starting point is the Lewis [14] line element for a stationary axisym-
metric space-time:
ds2 = ev(ρ,z)
dρ2 + dz2
+ L(ρ, z)dφ2 + 2m(ρ, z)dtdφ − f(ρ, z)dt2, (1)
where x4 = t is the time coordinate, x1 = ρ is the radial coordinate in
a cylindrical system, x2 = z is the zenithal coordinate and x3 = φ is the
azimuthal angular coordinate on the plane z = 0. Also,
t ∈ (−∞,∞) , ρ ∈ (0,∞) , z ∈ (−∞,∞) , φ ∈ [0, 2π). (2)
Further, the root square of the determinant of the 2-metric spanned by the
Killing vectors ∂t, ∂φ is
|det g(2)| =
fL+m2 = W (ρ, z). (3)
Expression (3) characterizes the measure of the area of the orbits of the
isometry group. In the vacuum, the field equations for (1) imply thatW (ρ, z)
is harmonic, i.e. W,α,α = 0, where subindices denote partial derivative and
a summation with respect to α = ρ, z is implicit. Therefore, the function
W (ρ, z) can be chosen as a coordinate. Looking for regular solutions on the
axis, the simplest assumption can be made by setting W = ρ. In this way,
the van Stockum line element emerges by taking a dust source. However,
this is not the most general choice. Thanks to the gauge freedom, we can
fL+m2 = ρ2H(ρ, z), (4)
where H(ρ, z) is a sufficiently regular function to be specified by the field
equations.
We consider a perfect fluid Tµν = (E + P )uµuν + Pgµν , with E being the
mass-energy density , P the hydrostatic pressure and uµ the 4-velocity of
the fluid. We consider a co-moving reference frame:
, uφ = uρ = uz = 0. (5)
Denoting with Rµν the Ricci tensor, the relevant field equations are
Rzz −Rρρ = 0, (6)
Rρρ +Rzz = (P − E)ev , (7)
Rρz = 0, (8)
Rφφ =
T − Tφφ
, (9)
Rtφ =
T − Ttφ
, (10)
Rtt = −
T + Ttt
, (11)
where T = 3P − E. Equation (9) involves a second-order partial equation
for L(ρ, z), while (10) and (11) give second order equations for m(ρ, z) and
f(ρ, z) respectively. Thanks to (4), equations (9)-(11) are not independent.
Therefore, from (4), we can express L(ρ, z) in terms of (f,m,H). Putting
this expression in (9) and using equations (10) and (11), we obtain the
following compatibility equation:
4HPev = H,α,α −
H,ρ. (12)
In what follows we study dust solutions for which P = 0. Setting ρ2H = F 2,
equation (12) becomes F,α,α = 0. Thus the compatibility condition for (9)-
(11) requires that F (ρ, z) be a harmonic function. Conversely, with F (ρ, z)
no more harmonic, the line element (1) is appropriate to describe spacetimes
with non-vanishing pressure P . For F = ρ the van Stockum line element is
regained together with the Papapetrou form of the metric [15]. Equations (6)
and (8) are linear first-order equations involving v,ρ and v,z and they permit
us to calculate v,ρ, v,z in terms of (f,m,F ). By applying the integrability
condition (v,ρ,z = v,z,ρ) for the equations so obtained, we read
f,zF,ρ = F,zf,ρ. (13)
Finally, when expressions for v,ρ, v,z are put in (7), we obtain
F,zf,z = −f,ρF,ρ. (14)
Excluding the case F = const (it can be see that this leads to the trivial
solution E = 0), we have f = const. We naturally choose f = 1. Therefore
our system of equations is
F,α,α = 0, (15)
m,α,α −
m,αF,α
= 0, (16)
e−vm2,α
, (17)
v,ρ =
m2zF,ρ −m2,ρF,ρ + 4FF,zF,z,ρ − 4FF,ρF,z,z − 2m,ρm,zF,z
2FF 2,α
,(18)
v,z =
4FF,zF,z,z + 4FF,ρF,z,ρ − F,zm2,z + F,zm2,ρ − 2m,ρm,zFρ
2FF 2,α
,(19)
L+m2 = F 2. (20)
First of all, for non-expanding spacetimes, the shear qik =
[ui;k + uk;i]
vanishes identically for (15)-(20), and therefore our system of equations de-
scribes rigidly rotating sources in a co-moving reference frame. When F = ρ,
equation (16) is invariant under the transformation z → z+a (a a constant)
and a solution can be expanded as
i m(ρ, z + ai). Setting F 6= ρ, if
F (ρ, z),m(ρ, z) are solutions, then also F (ρ, z+a),m(ρ, z+a) are, and thus
the solutions cannot be expanded.
Note that we have identified (ρ, z) with the radial and the zenithal coordinate
respectively in a cylindrical coordinate system. According to this assump-
tion, some conditions must be imposed. Firstly, by setting E = 0, the metric
must reduce to the standard flat expression ds2 = dρ2 + dz2 + ρ2dφ2 − dt2.
Therefore, limE→0 F = ρ. Further, looking for regular spacetimes on the
rotation axis , the norm of the space-like Killing vector ∂φ must be van-
ishing (except at isolated points) at ρ = 0, i.e. limρ→0 L = 0. Finally, for
asymptotically flat spacetimes, at spatial infinity F (ρ, z) looks as follows:
F = ρ+ o(1).
2 Generating an asymptotically flat solution
Our starting point is the Bonnor dipole solution [12] F = ρ,m =
(ρ2+z2)
We can obtain a solution of (16) by taking the map ρ → F (ρ, z), z → G(ρ, z),
where F = ρ
1 + bc
(ρ2+z2)
, G = z
1− bc
(ρ2+z2)
with c ≥ 0, being b a
constant. Therefore we get the solution
F = ρ
(ρ2 + z2)
, (21)
cρ2[ρ2 + z2 + bc]
ρ2 + z2[(ρ2 + z2)
+ 2bcρ2 + b2c2 − 2bcz2]
v = ln(α) +
γ(ρ2 + z2 + bc)
[(ρ2 + z2)
+ 2ρ2bc+ b2c2 − 2z2bc]4
c2e−vβ∆
[(ρ2 + z2)
+ 2ρ2bc+ b2c2 − 2z2bc]4
β = α(ρ2 + z2)
= (ρ2 + z2)
+ c2b2 − 2ρ2cb+ 2z2cb,
γ = (ρ2 − 8z2)(ρ2 + z2)2 + 2ρ4cb+ 18ρ2z2bc+ ρ2c2b2 + 16z4bc− 8z2c2b2,
∆ = (ρ2 + 4z2)(ρ2 + z2)
2 − 8z4cb+ 4z2c2b2 − 6ρ2z2cb+ ρ2c2b2 + 2ρ4cb.
Solution (21) is asymptotically flat. Note that the map ρ → F (ρ, z) , z →
G(ρ, z) is not bijective, i.e. is not a diffeomorphism. Concerning the features
of (21), they depend on the sign of the constant b.
For b > 0, apart from ρ = 0, z = ±
bc, our solution is regular everywhere.
At these two points, we have curvature singularities with properties close
to the ρ = 0, z = 0 singularity of the dipole Bonnor solution (see [12]). In
particular, the mass-energy diverges at these points. Otherwise, the energy
density E(ρ, z) is integrable. For b = 0, we regain the Bonnor solution.
Finally, for b < 0, the two point singularities disappear, but emerges a
curvature ring singularity for z = 0, ρ =
|b|c with diverging mass. Inde-
pendently on the parameter b, at spatial infinity the metric reduces to the
standard expression in asymptotical cylindrical coordinates, and so also by
setting c = 0 (E = 0). Note that, because of the non-invertibility of the map
between the Bonnor solution and solution (21), the curvature singularity at
the origin of [12] is shifted in the two curvature singularities of (21) (for
b > 0) on the rotation axis. As a final consideration, it must be noted that
there exists for (21) a finite non singular region about the origin. Thus,
our solution could be matched, in principle, with some asymptotically flat
vacuum solution. We do not enter in this discussion, but only mention this
possibility.
3 Conclusions and final remarks
We have studied stationary axially symmetric rigidly rotating dust space-
times in terms of harmonic functions. In [16] Bonnor found the general
solution for charged dust with zero Lorentz force in terms of harmonic func-
tions. However, the use of such kind of functions in [16] is different from
the one in our paper. In fact, in our paper harmonic functions appear in
equation (4) thanks to the gauge freedom, while in [16] the function F (ρ, z)
is chosen to be the cylindrical polar coordinate ρ. In the Bonnor paper,
harmonic functions arise in order to obtain the most general solution for
equation (16) with F = ρ. In this case, all the solutions of the equation
m,α,α − 1ρm,ρ = 0 are given by taking a generic harmonic function η(ρ, z),
with m = ρηρ. In a similar way, another harmonic function is introduced
when charged dust comes in action. Therefore, a direct relation between
does not exist between the harmonic function F (ρ, z) and η(ρ, z) of [16].
Also, the paper [17] must be noticed in which charged dust solutions are
given in terms of Bessel functions of first and second kind and hyperbolic
functions. Also in the paper [17] the condition F = ρ is retained.
In this paper, section two, starting with the dipole Bonnor solution, we
build a class of asymptotically flat solutions containing the Bonnor one as a
subclass by a suitable choice of the functions F (ρ, z), G(ρ, z). Obviously, it
must be noted that not all the harmonic functions generate physically sen-
sible solutions. For a physically sensible solution we mean a regular (apart
from isolated singularities) asymptotically flat solution. Generally, it is a
simple matter to verify that, if F (ρ, z) = ρ,m(ρ, z) is a regular differen-
tiable solution for (16), then also m(F (ρ, z), G(ρ, z)) is a solution for (16)
with F (ρ, z) harmonic being G(ρ, z) the harmonic conjugate to F (ρ, z) i.e.
Fρ = Gz, Fz = −Gρ. Further, in order to generate a new asymptotically
flat and regular solution on the axis starting with a seed solution with these
two properties, we must build a non-bijective (not a diffeomorphism) map
ρ → F (ρ, z), z → G(ρ, z) such that limE→0 F = ρ , limE→0G = z, and
limρ→0 L = limρ→0 F = 0 (apart from isolated points) and such that at spa-
tial infinity the functions F (ρ, z), G(ρ, z) look as follows: F = ρ+o(1) , G =
z + o(1). Concerning isolated singularities, no general conclusions can be
made. The functions F (ρ, z), G(ρ, z) of section two satisfy all the conditions
mentioned above.
Finally, in the appendix we present a non-asymptotically flat solution not
obtained from a seed solution with the technique discussed above and there-
fore it represents an ad hoc solution. In particular, it is possible to build ad
hoc solutions for (16) by setting F (ρ, z) = ρ(1 + c
ρ2+z2
) (with c a constant)
and m(ρ, z) a homogeneous function such that m,α,α − m,ρρ = 0.
Appendix
We consider the following solution:
F = ρ
ρ2 + z2
, m = c
ρ2 + z2
, (22)
ln[ρ4 − 2cρ2 + 2ρ2z2 + c2 + z4 + 2cz2]
ln[c+ ρ2 + z2]− 2 ln[ρ2 + z2] , E =
c2e−v
[c+ ρ2 + z2]
with c a constant. When c > 0, the solution (22) has a curvature ring sin-
gularity when (ρ, z) = (
c, 0) The axis is regular for z > 0, while it shows a
conical (no curvature) singularity when z ≤ 0. Further, for z < 0 there is a
region where closed time-like curves (CTC) appear resulting in a vioaltion of
causality. However, it is possible to take a simple coordinate transformation
found in [18], i.e. τ = t+ 2cφ giving the whole rotational axis free of coni-
cal singularities. Unfortunately, we are forced to introduce a periodic time
coordinate τ and therefore once again CTC appear. Therefore, the problem
of violation of causality cannot be avoided with an opportune coordinate
transformation.
The spacetime asymptotically reads the expression appropriate for asymp-
totic NUT metrics [19, 20] with NUT charge q given by c = 2q. We can
estimate the mass inside an infinite cylinder of radius R by means of the
integral
M(R) =
EFevdφ. (23)
The integral (23) is well defined everywhere. Thus, for the mass we get the
formula
M(R) = 2π2c2
1 +R−
R2 + 1
. (24)
Because of the non-asymptotical flatness, the solution (22) is not interesting
in an astrophysical context. However, the natural arena for this solution is in
the extra relativistic context given by non-Abelian gauge theories or in the
low energy string theory [21, 22] where, in order to obtain supersymmetries,
NUT charge comes in action.
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	Basic Equations
	Generating an asymptotically flat solution
	Conclusions and final remarks
