Weyl metrics

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In general relativity, the Weyl metrics (named after the German-American mathematician Hermann Weyl) refer to the 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[edit]

The Weyl class of solutions has the generic form[1][2]


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,[1] 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[edit]

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


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





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[3]





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.

Newtonian analogue of metric potential Ψ(ρ,z)[edit]

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,[4]


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.[1]

Schwarzschild solution[edit]

The Weyl potentials generating Schwarzschild's metric as solutions to the vacuum equations Eq(8) are given by[1][2][3]


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.[1])

Given and , Weyl's metric Eq(\ref{Weyl metric in canonical coordinates}) becomes


and after substituting the following mutually consistent relations



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[edit]

The Weyl potentials generating the nonextremal Reissner–Nordström solution () as solutions to Eqs(7} are given by[1][2][3]


where


Thus, given and , Weyl's metric becomes


and employing the following transformations



one can obtain the common form of non-extremal Reissner–Nordström metric in the usual coordinates,


Extremal Reissner–Nordström solution[edit]

The potentials generating the extremal Reissner–Nordström solution () as solutions to Eqs(7} are given by[3] (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,[5][6]


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[edit]

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[1]


where represent Legendre polynomials, and are multipole coefficients. The other metric potential is given by[1]


See also[edit]

References[edit]

  1. ^ 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.
  2. ^ 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.
  3. ^ 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.
  4. ^ James B Hartle. Gravity: An Introduction To Einstein's General Relativity. San Francisco: Addison Wesley, 2003. Eq(6.20) transformed into Lorentzian cylindrical coordinates
  5. ^ 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
  6. ^ Antonio C Gutierrez-Pineres, Guillermo A Gonzalez, Hernando Quevedo. Conformastatic disk-haloes in Einstein-Maxwell gravity. Physical Review D, 2013, 87(4): 044010. [1]