In general relativity, the Weyl metrics (named after the German-American mathematician Hermann Weyl)[1] are a class of static and axisymmetric solutions to Einstein's field equation. Three members in the renowned Kerr–Newman family solutions, namely the Schwarzschild, nonextremal Reissner–Nordström and extremal Reissner–Nordström metrics, can be identified as Weyl-type metrics.
Standard Weyl metrics
The Weyl class of solutions has the generic form[2][3]
where
and
are two metric potentials dependent on Weyl's canonical coordinates
. The coordinate system
serves best for symmetries of Weyl's spacetime (with two Killing vector fields being
and
) and often acts like cylindrical coordinates,[2] but is incomplete when describing a black hole as
only cover the horizon and its exteriors.
Hence, to determine a static axisymmetric solution corresponding to a specific stress–energy tensor
, we just need to substitute the Weyl metric Eq(1) into Einstein's equation (with c=G=1):
and work out the two functions
and
.
Reduced field equations for electrovac Weyl solutions
One of the best investigated and most useful Weyl solutions is the electrovac case, where
comes from the existence of (Weyl-type) electromagnetic field (without matter and current flows). As we know, given the electromagnetic four-potential
, the anti-symmetric electromagnetic field
and the trace-free stress–energy tensor
will be respectively determined by
![{\displaystyle (3)\quad F_{ab}=A_{b\,;\,a}-A_{a\,;\,b}=A_{b\,,\,a}-A_{a\,,\,b}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/760ecc08f7c06390595ed6d9742bccde7978d66e)
which respects the source-free covariant Maxwell equations:
Eq(5.a) can be simplified to:
in the calculations as
. Also, since
for electrovacuum, Eq(2) reduces to
Now, suppose the Weyl-type axisymmetric electrostatic potential is
(the component
is actually the electromagnetic scalar potential), and together with the Weyl metric Eq(1), Eqs(3)(4)(5)(6) imply that
![{\displaystyle (7.a)\quad \nabla ^{2}\psi =\,(\nabla \psi )^{2}+\gamma _{,\,\rho \rho }+\gamma _{,\,zz}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8bc0d608cd45d7028b244150caf33db3bd0a119)
![{\displaystyle (7.b)\quad \nabla ^{2}\psi =\,e^{-2\psi }(\nabla \Phi )^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/384be2290141d0501734b26fc36cd406775d056d)
![{\displaystyle (7.c)\quad {\frac {1}{\rho }}\,\gamma _{,\,\rho }=\,\psi _{,\,\rho }^{2}-\psi _{,\,z}^{2}-e^{-2\psi }{\big (}\Phi _{,\,\rho }^{2}-\Phi _{,\,z}^{2}{\big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2ef371ce323da1305b672ff44a1f3d763c8346d8)
![{\displaystyle (7.d)\quad {\frac {1}{\rho }}\,\gamma _{,\,z}=\,2\psi _{,\,\rho }\psi _{,\,z}-2e^{-2\psi }\Phi _{,\,\rho }\Phi _{,\,z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f4896251f398f07ea67863680c2dc4229e8b2e6)
where
yields Eq(7.a),
or
yields Eq(7.b),
or
yields Eq(7.c),
yields Eq(7.d), and Eq(5.b) yields Eq(7.e). Here
and
are respectively the Laplace and gradient operators. Moreover, if we suppose
in the sense of matter-geometry interplay and assume asymptotic flatness, we will find that Eqs(7.a-e) implies a characteristic relation that
Specifically in the simplest vacuum case with
and
, Eqs(7.a-7.e) reduce to[4]
![{\displaystyle (8.a)\quad \gamma _{,\,\rho \rho }+\gamma _{,\,zz}=-(\nabla \psi )^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de38c5cbd3536ca26eb89d96a8d265bfb2b83c46)
![{\displaystyle (8.b)\quad \nabla ^{2}\psi =0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30cb6be4788ab85cb16377bde2e8cf9d68dea5d5)
![{\displaystyle (8.c)\quad \gamma _{,\,\rho }=\rho \,{\Big (}\psi _{,\,\rho }^{2}-\psi _{,\,z}^{2}{\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78ce732126fa40c5b8b5ef319707e27fab57c0a8)
We can firstly obtain
by solving Eq(8.b), and then integrate Eq(8.c) and Eq(8.d) for
. Practically, Eq(8.a) arising from
just works as a consistency relation or integrability condition.
Unlike the nonlinear Poisson's equation Eq(7.b), Eq(8.b) is the linear Laplace equation; that is to say, superposition of given vacuum solutions to Eq(8.b) is still a solution. This fact has a widely application, such as to analytically distort a Schwarzschild black hole.
Box A: Remarks on the electrovac field equation
We employed the axisymmetric Laplace and gradient operators to write Eqs(7.a-7.e) and Eqs(8.a-8.d) in a compact way, which is very useful in the derivation of the characteristic relation Eq(7.f). In the literature, Eqs(7.a-7.e) and Eqs(8.a-8.d) are often written in the following forms as well:
![{\displaystyle (A.1.a)\quad \psi _{,\,\rho \rho }+{\frac {1}{\rho }}\psi _{,\,\rho }+\psi _{,\,zz}=\,(\psi _{,\,\rho })^{2}+(\psi _{,\,z})^{2}+\gamma _{,\,\rho \rho }+\gamma _{,\,zz}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/296b3867c53098501232711d82f44293dc870fe2)
![{\displaystyle (A.1.b)\quad \psi _{,\,\rho \rho }+{\frac {1}{\rho }}\psi _{,\,\rho }+\psi _{,\,zz}=e^{-2\psi }{\big (}\Phi _{,\,\rho }^{2}+\Phi _{,\,z}^{2}{\big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ae3a3ca53286a6dc6818cff7b62ab7c3714697ad)
![{\displaystyle (A.1.c)\quad {\frac {1}{\rho }}\,\gamma _{,\,\rho }=\,\psi _{,\,\rho }^{2}-\psi _{,\,z}^{2}-e^{-2\psi }{\big (}\Phi _{,\,\rho }^{2}-\Phi _{,\,z}^{2}{\big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9ee1ab4423d478a64bfe90c50c3a47e1041a157b)
![{\displaystyle (A.1.d)\quad {\frac {1}{\rho }}\,\gamma _{,\,z}=\,2\psi _{,\,\rho }\psi _{,\,z}-2e^{-2\psi }\Phi _{,\,\rho }\Phi _{,\,z}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83827a4c4d94169f4a14fc25b12fd49e9ce755eb)
and
![{\displaystyle (A.2.a)\quad (\psi _{,\,\rho })^{2}+(\psi _{,\,z})^{2}=-\gamma _{,\,\rho \rho }-\gamma _{,\,zz}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c26157d4dcf92d5bbb77d4a34bd886595fc9551)
![{\displaystyle (A.2.b)\quad \psi _{,\,\rho \rho }+{\frac {1}{\rho }}\psi _{,\,\rho }+\psi _{,\,zz}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/646968e0b9580ccfb71d42bdb021e4e432476b5d)
![{\displaystyle (A.2.c)\quad \gamma _{,\,\rho }=\rho \,{\Big (}\psi _{,\,\rho }^{2}-\psi _{,\,z}^{2}{\Big )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d8959bf6bcefbf17462c870e0e69f1f09bce57d)
Box B: Derivation of the Weyl electrovac
![{\displaystyle \psi \sim \Phi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/532f084d40ab13258e930a5ebac901fb44a36f5e)
characteristic relation
Considering the interplay between spacetime geometry and energy-matter distributions, it is natural to assume that in Eqs(7.a-7.e) the metric function
relates with the electrostatic scalar potential
via a function
(which means geometry depends on energy), and it follows that
Eq(B.1) immediately turns Eq(7.b) and Eq(7.e) respectively into
![{\displaystyle (B.2)\quad \Psi _{,\,\Phi }\cdot \nabla ^{2}\Phi \,=\,{\big (}e^{-2\psi }-\psi _{,\,\Phi \Phi }{\big )}\cdot (\nabla \Phi )^{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ea7c137cf8a4b8d943c42faf821a4b587346e9f)
which give rise to
Now replace the variable
by
, and Eq(B.4) is simplified to
Direct quadrature of Eq(B.5) yields
, with
being integral constants. To resume asymptotic flatness at spatial infinity, we need
and
, so there should be
. Also, rewrite the constant
as
for mathematical convenience in subsequent calculations, and one finally obtains the characteristic relation implied by Eqs(7.a-7.e) that
This relation is important in linearize the Eqs(7.a-7.f) and superpose electrovac Weyl solutions.
Newtonian analogue of metric potential Ψ(ρ,z)
In Weyl's metric Eq(1),
; thus in the approximation for weak field limit
, one has
and therefore
This is pretty analogous to the well-known approximate metric for static and weak gravitational fields generated by low-mass celestial bodies like the Sun and Earth,[5]
where
is the usual Newtonian potential satisfying Poisson's equation
, just like Eq(3.a) or Eq(4.a) for the Weyl metric potential
. The similarities between
and
inspire people to find out the Newtonian analogue of
when studying Weyl class of solutions; that is, to reproduce
nonrelativistically by certain type of Newtonian sources. The Newtonian analogue of
proves quite helpful in specifying particular Weyl-type solutions and extending existing Weyl-type solutions.[2]
Schwarzschild solution
The Weyl potentials generating Schwarzschild's metric as solutions to the vacuum equations Eq(8) are given by[2][3][4]
where
From the perspective of Newtonian analogue,
equals the gravitational potential produced by a rod of mass
and length
placed symmetrically on the
-axis; that is, by a line mass of uniform density
embedded the interval
. (Note: Based on this analogue, important extensions of the Schwarzschild metric have been developed, as discussed in ref.[2])
Given
and
, Weyl's metric Eq(\ref{Weyl metric in canonical coordinates}) becomes
and after substituting the following mutually consistent relations
![{\displaystyle (15)\quad L+M=r\,,\quad l_{+}-l_{-}=2M\cos \theta \,,\quad z=(r-M)\cos \theta \,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7761e68bb9a4b70b8e3326d2c326a1e35b386a55)
one can obtain the common form of Schwarzschild metric in the usual
coordinates,
The metric Eq(14) cannot be directly transformed into Eq(16) by performing the standard cylindrical-spherical transformation
, because
is complete while
is incomplete. This is why we call
in Eq(1) as Weyl's canonical coordinates rather than cylindrical coordinates, although they have a lot in common; for example, the Laplacian
in Eq(7) is exactly the two-dimensional geometric Laplacian in cylindrical coordinates.
Nonextremal Reissner–Nordström solution
The Weyl potentials generating the nonextremal Reissner–Nordström solution (
) as solutions to Eqs(7} are given by[2][3][4]
where
Thus, given
and
, Weyl's metric becomes
and employing the following transformations
![{\displaystyle (20)\quad L+M=r\,,\quad l_{+}-l_{-}=2{\sqrt {M^{2}-Q^{2}}}\,\cos \theta \,,\quad z=(r-M)\cos \theta \,,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2996cbea2e1256c17ad3e68900b2ee276ff38da1)
one can obtain the common form of non-extremal Reissner–Nordström metric in the usual
coordinates,
Extremal Reissner–Nordström solution
The potentials generating the extremal Reissner–Nordström solution (
) as solutions to Eqs(7} are given by[4] (Note: We treat the extremal solution separately because it is much more than the degenerate state of the nonextremal counterpart.)
Thus, the extremal Reissner–Nordström metric reads
and by substituting
we obtain the extremal Reissner–Nordström metric in the usual
coordinates,
Mathematically, the extremal Reissner–Nordström can be obtained by taking the limit
of the corresponding nonextremal equation, and in the meantime we need to use the L'Hospital rule sometimes.
Remarks: Weyl's metrics Eq(1) with the vanishing potential
(like the extremal Reissner–Nordström metric) constitute a special subclass which have only one metric potential
to be identified. Extending this subclass by canceling the restriction of axisymmetry, one obtains another useful class of solutions (still using Weyl's coordinates), namely the conformastatic metrics,[6][7]
where we use
in Eq(22) as the single metric function in place of
in Eq(1) to emphasize that they are different by axial symmetry (
-dependence).
Weyl vacuum solutions in spherical coordinates
Weyl's metric can also be expressed in spherical coordinates that
which equals Eq(1) via the coordinate transformation
(Note: As shown by Eqs(15)(21)(24), this transformation is not always applicable.) In the vacuum case, Eq(8.b) for
becomes
The asymptotically flat solutions to Eq(28) is[2]
where
represent Legendre polynomials, and
are multipole coefficients. The other metric potential
is given by[2]
See also
References
- ^ Weyl, H., "Zur Gravitationstheorie," Ann. der Physik 54 (1917), 117–145.
- ^ a b c d e f g h Jeremy Bransom Griffiths, Jiri Podolsky. Exact Space-Times in Einstein's General Relativity. Cambridge: Cambridge University Press, 2009. Chapter 10.
- ^ a b c Hans Stephani, Dietrich Kramer, Malcolm MacCallum, Cornelius Hoenselaers, Eduard Herlt. Exact Solutions of Einstein's Field Equations. Cambridge: Cambridge University Press, 2003. Chapter 20.
- ^ a b c d R Gautreau, R B Hoffman, A Armenti. Static multiparticle systems in general relativity. IL NUOVO CIMENTO B, 1972, 7(1): 71-98.
- ^ James B Hartle. Gravity: An Introduction To Einstein's General Relativity. San Francisco: Addison Wesley, 2003. Eq(6.20) transformed into Lorentzian cylindrical coordinates
- ^ Guillermo A Gonzalez, Antonio C Gutierrez-Pineres, Paolo A Ospina. Finite axisymmetric charged dust disks in conformastatic spacetimes. Physical Review D, 2008, 78(6): 064058. arXiv:0806.4285v1
- ^ Antonio C Gutierrez-Pineres, Guillermo A Gonzalez, Hernando Quevedo. Conformastatic disk-haloes in Einstein-Maxwell gravity. Physical Review D, 2013, 87(4): 044010. [1]