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In discussions of questions related to gtr, it is often useful to know that one can in fact "create solutions to order" in gtr, when one wishes to model specific physical scenarios. Sort of, not really--- and herein lies a tale which illustrates some of the many thorny technical and conceptual issues which arise as soon as we actually try to use gtr to compute some hard predictions about real astrophysics!

(Mechanical note: the PF server sometimes puts images generated from latex markup in the wrong places; I hope that won't happen here, but if my posts make no sense, try using "reply" with "quote" followed by "preview"; this often seems to put displayed equations in the right places again. I'll try to break this long post up into several smaller pieces, which may also help.)

In physics, we often wish to model situations featuring some configuration of matter which is static, or axisymmetric, or both. In Newtonian gravitation, it is well known (to anyone who has studied either potential theory in a math class, or electrostatics in a physics class!) that

When confronting such issues, it is useful to know that Weyl and Levi-Civita discovered (c. 1917) the family of

[tex]

\begin{array}{rcl}

ds^2 &=& -\exp(2u) \; dt^2 + \exp(2\,(v-u)) \; (dz^2+r^2) + \exp(-2u) \; r^2 \; d\phi^2 \\

&& u_{zz} + u_{rr} + \frac{u_r}{r} = 0 \\

&& v_z = 2 r \, u_z \, u_r \\

&& v_r = r \; ( u_r^2-u_z^2)

\end{array}

[/tex]

Here, subscripts denote partial derivatives, following a convention often used in the PDE literature, and also in the gtr literature. The intended range of the coordinates is some subset of

[tex]

-\infty < t, \; z < \infty, \; 0 < r < \infty, \; -\pi < \phi < \pi

[/tex]

In a plain English: a Weyl vacuum solution is obtained by choosing any axisymmetric harmonic function u and then finding v by quadrature using the two first order PDEs giving v in terms of u. It is useful to think of u as the "master metric function" and v as a "secondary function" which is entirely determined by u. Note that the equations giving v in terms of u are quadratic in u, so that if u is multiplied by a scale parameter m (often interpreted as a mass density), then v is multiplied by the factor m^2.

Note that I speak of "Weyl vacuum

The "master equation" defining the Weyl vacuums is the Laplace equation for axisymmetric functions on an unobservable flat background. But it turns out that

To see that a Weyl vacuum solution really is defined by each axisymmetric harmonic function, notice that

The Killing vector fields of a Weyl vacuum solution always include a two dimensional abelian Lie algebra of commuting and hypersurface orthogonal Killing vector fields [itex]\partial_t[/itex] (timelike) and [itex]\partial_\phi[/itex] (spacelike and cyclic); in exceptional cases there may be additional Killing vector fields which are not real linear combinations of these two. In other words, the self-isometry group of a Weyl vacuum is at least the two-dimensional abelian Lie group generated by time translations and rotations about the axis of symmetry. This is sometimes called boost-rotation symmetry and should be familiar to anyone who knows about loxodromic transformations, which arise as a special case when the Moebius group, aka the conformal group of the sphere, aka the Lorentz group acts on the Riemann sphere.

The integral curves of [itex]\partial_t[/itex] define the world lines of static observers. Because this congruence is vorticity free, and thus hypersurface orthogonal, it defines a family of orthogonal hyperslices which are all equivalent as Riemannian three-manifolds; hence "static solution". The integral curves of [itex]\partial_\phi[/itex] define nested circles, "latitude circles" on coordinate cylinders r=r0, t=t0. This Killing vector picks out a geometrically distinguished locus r=0, the axis of symmetry.

The congruence of world lines of static observers is not only vorticity-free (equivalent to hypersurface orthogonal) but expansionless (rigid). We can take the frame field of these observers to be

[tex]

\begin{array}{rcl}

\vec{e}_1 & = & \exp(-u) \; \partial_t \\

\vec{e}_2 & = & \exp(u-v) \; \partial_z \\

\vec{e}_3 & = & \exp(u-v) \; \partial_r \\

\vec{e}_4 & = & \exp(u) \; \frac{1}{r} \; \partial_\phi

\end{array}

[/tex]

Using this frame field (aka orthonormal basis, aka anholonomic basis) we can compute that the static observers have acceleration vector

[tex]

\nabla_{\vec{e}_1} \vec{e}_1 = \exp(u-v) \; (u_z \; \vec{e}_2 + u_r \; \vec{e}_3)

[/tex]

And, as just mentioned, the expansion tensor of [itex]\vec{e}_1[/itex] vanishes (rigid congruence) and the vorticity tensor of [itex]\vec{e}_1[/itex] also vanishes (hypersurface orthogonal congruence).

It might help to recall here that

Except in special cases, the Weyl tensor will be algebraically general at each point.

If we interpret u as a Newtonian potential, the Newtonian gravitational acceleration would be [itex] u_z \vec{e}_2 + u_r \vec{e}_3[/itex], but we must be careful, because the Weyl coordinates z,r do not, in general, have any simple relation to euclidean distances! We must also be careful not to assume that, merely because every Weyl vacuum can be written in a Weyl canonical chart, that things like the asymptotic behavior of tensor components wrt the coordinates z,r can be easily compared between two different Weyl vacuum solutions!

The weak-field limit is obtained when we expand to first order in the maximal value of |u|, so by the remark about the quadratic scaling of |v| wrt |u|, we can obtain the weak field limit very simply by neglecting v and expanding [itex]\exp(2u) = 1 + 2u + O(u^2)[/itex]. This gives

[tex]

\begin{array}{rcl}

ds^2 &=& -(1 + 2u) \; dt^2 + (1-2u) \; (dz^2+r^2 + r^2 \, d\phi^2) \\

&& u_{zz} + u_{rr} + \frac{u_r}{r} = 0

\end{array}

[/tex]

The static observers have acceleration vector

[tex]

\nabla_{\vec{e}_1} \vec{e}_1 = (1+u) \; (u_z \; \vec{e}_2 + u_r \; \vec{e}_3)

[/tex]

which is is close the "gravitational acceleration" in Newtonian gravitostatics, since u is assumed small, as expected. So in the weak field limit, there is a very simple connection between the Weyl vacuums and the corresponding Newtonian models. However, this is not at all true for the strong field situation!

To verify that Weyl vacuums are in general curved Lorentzian manifolds (not that there is much doubt of this!), perhaps the easiest approach is to compute Kretschmann scalar [itex]R_{abcd} \; R^{abcd}[/itex], the kind of computation best entrusted to a symbolic computational program. In the weak field limit, in the equatorial plane z=0, assuming a discrete reflection symmetry across the this plane (so that [itex]u_z = 0[/itex] on z=0), this scalar simplifies considerably

[tex]

R_{abcd} \; R^{abcd} = 16 \, \left( u_{rr} \; \left( u_{rr} + \frac{u_r}{r} \right) + \frac{u_r^2}{r^2} \right)

[/tex]

which vanishes wherever u does not depend upon r--- as we would expect, bearing in mind our interpretation of u, in the weak field limit, as a Newtonian gravitational potential.

It is possible to study the

(I originally wrote "I won't way more here", which makes for an inadvertent pun.)

(Mechanical note: the PF server sometimes puts images generated from latex markup in the wrong places; I hope that won't happen here, but if my posts make no sense, try using "reply" with "quote" followed by "preview"; this often seems to put displayed equations in the right places again. I'll try to break this long post up into several smaller pieces, which may also help.)

In physics, we often wish to model situations featuring some configuration of matter which is static, or axisymmetric, or both. In Newtonian gravitation, it is well known (to anyone who has studied either potential theory in a math class, or electrostatics in a physics class!) that

- a vacuum gravitational field is given (in Newtonian gravitation) by a potential which is a harmonic function,
- we can employ various mathematical tricks to construct such Newtonian gravitational potentials, by taking advantage of any symmetries; this works particularly well for axisymmetric potentials,
- in particular, a common exercise asks the student to find the potential of a uniform density rod, ring, or disk.

When confronting such issues, it is useful to know that Weyl and Levi-Civita discovered (c. 1917) the family of

*all*static axisymmetric vacuum solutions of the EFE, generally known as "the Weyl vacuums". These can be defined, in terms of the so-called*Weyl canonical coordinate chart*, as follows:[tex]

\begin{array}{rcl}

ds^2 &=& -\exp(2u) \; dt^2 + \exp(2\,(v-u)) \; (dz^2+r^2) + \exp(-2u) \; r^2 \; d\phi^2 \\

&& u_{zz} + u_{rr} + \frac{u_r}{r} = 0 \\

&& v_z = 2 r \, u_z \, u_r \\

&& v_r = r \; ( u_r^2-u_z^2)

\end{array}

[/tex]

Here, subscripts denote partial derivatives, following a convention often used in the PDE literature, and also in the gtr literature. The intended range of the coordinates is some subset of

[tex]

-\infty < t, \; z < \infty, \; 0 < r < \infty, \; -\pi < \phi < \pi

[/tex]

In a plain English: a Weyl vacuum solution is obtained by choosing any axisymmetric harmonic function u and then finding v by quadrature using the two first order PDEs giving v in terms of u. It is useful to think of u as the "master metric function" and v as a "secondary function" which is entirely determined by u. Note that the equations giving v in terms of u are quadratic in u, so that if u is multiplied by a scale parameter m (often interpreted as a mass density), then v is multiplied by the factor m^2.

Note that I speak of "Weyl vacuum

**s**" but "Schwarzschild vacuum". The distinction is basically that the Weyl vacuums comprise a family defined using the solutions of a system of PDEs (thus, defined by one or more*functions*), while the Schwarzschild vacuum is defined using the solutions of a simple ODE (thus, defined by finitely many*numbers*, the physical parameters, in this case "mass"). Thus, Weyl vacuums are capable of modeling any static axisymmetric gravitational field admissible in gtr (if no nongravitational fields or matter are present), while the Schwarzschild vacuum models a much more specific (but quite important!) situation.The "master equation" defining the Weyl vacuums is the Laplace equation for axisymmetric functions on an unobservable flat background. But it turns out that

*u is also axisymmetric harmonic on the Lorentzian manifold which it defines, as well as on the fictitious flat background*, so in fact we are perfectly justified of speaking of u as being a harmonic function without mentioning any fictitious background metric!To see that a Weyl vacuum solution really is defined by each axisymmetric harmonic function, notice that

*the condition that the two first order equations giving v in terms of u must be self-consistent is precisely the master equation*. To see this, take the partial derivative of the RHS of the second equation by r, and take the partial of the RHS of the third equation wrt z. The two mixed partials must agree, which gives a second order equation which u must satisfy in order for the system consisting of the second and third equations to be self consistent, and this equation is precisely the Laplace equation. This suggests (correctly) a connection with Baecklund transformations and certain ideas in the theory of solitons. For us the important point is that the system is always self-consistent.The Killing vector fields of a Weyl vacuum solution always include a two dimensional abelian Lie algebra of commuting and hypersurface orthogonal Killing vector fields [itex]\partial_t[/itex] (timelike) and [itex]\partial_\phi[/itex] (spacelike and cyclic); in exceptional cases there may be additional Killing vector fields which are not real linear combinations of these two. In other words, the self-isometry group of a Weyl vacuum is at least the two-dimensional abelian Lie group generated by time translations and rotations about the axis of symmetry. This is sometimes called boost-rotation symmetry and should be familiar to anyone who knows about loxodromic transformations, which arise as a special case when the Moebius group, aka the conformal group of the sphere, aka the Lorentz group acts on the Riemann sphere.

The integral curves of [itex]\partial_t[/itex] define the world lines of static observers. Because this congruence is vorticity free, and thus hypersurface orthogonal, it defines a family of orthogonal hyperslices which are all equivalent as Riemannian three-manifolds; hence "static solution". The integral curves of [itex]\partial_\phi[/itex] define nested circles, "latitude circles" on coordinate cylinders r=r0, t=t0. This Killing vector picks out a geometrically distinguished locus r=0, the axis of symmetry.

The congruence of world lines of static observers is not only vorticity-free (equivalent to hypersurface orthogonal) but expansionless (rigid). We can take the frame field of these observers to be

[tex]

\begin{array}{rcl}

\vec{e}_1 & = & \exp(-u) \; \partial_t \\

\vec{e}_2 & = & \exp(u-v) \; \partial_z \\

\vec{e}_3 & = & \exp(u-v) \; \partial_r \\

\vec{e}_4 & = & \exp(u) \; \frac{1}{r} \; \partial_\phi

\end{array}

[/tex]

Using this frame field (aka orthonormal basis, aka anholonomic basis) we can compute that the static observers have acceleration vector

[tex]

\nabla_{\vec{e}_1} \vec{e}_1 = \exp(u-v) \; (u_z \; \vec{e}_2 + u_r \; \vec{e}_3)

[/tex]

And, as just mentioned, the expansion tensor of [itex]\vec{e}_1[/itex] vanishes (rigid congruence) and the vorticity tensor of [itex]\vec{e}_1[/itex] also vanishes (hypersurface orthogonal congruence).

It might help to recall here that

- every vector field on a smooth manifold M can be usefully viewed as a linear first order partial differential operator on smooth funcions on M; in the standard cartesian chart on the Euclidean plane, the coordinate vector fields can be written [itex]\partial_x, \; \partial_y[/itex], or if you prefer [itex](1,0), \; (0,1)[/itex], although the latter notation is dangerously imprecise when we consider non-cartesian charts and non-euclidean geometries!,
- every timelike unit vector field [itex]\vec{U}[/itex] defines a family of integral curves, which are timelike curves, but of course not neccessarily timelike
*geodesics*, - the covariant derivative [itex]\nabla_{\vec{U}} \vec{U}[/itex], evaluated at some event E on some integral curve C, is nothing other than the
*path curvature*of C at E, which is physical interpreted as the acceleration of the observer whose world line in C, - frame fields define both a family of observers (whose world lines are the congruence of curves defined by the timelike unit vector field in the frame field) and their spatial frame vectors at each event; the frame fields which are easiest to interpret are gyrostabilized frames, which have the property that the Fermi derivatives of the spatial frame vectors wrt the timelike frame vector vanishes, as happens here,
- the dual notion of coframe fields (orthonormal basis of covectors, or one-forms) is the foundation of Cartan's approach to computing useful things like connection one-forms and curvature two-forms, and ultimately, to classifying Lorentzian manifolds up to local isometry,
- for our purposes, the most important virtue of computing frame field components is that they are (after supplying sufficient explanation) physically significant; these are the components of vector and tensorial quanitities of physical interest which could, in principle, be measured by the family observers corresponding to the frame field.

Except in special cases, the Weyl tensor will be algebraically general at each point.

If we interpret u as a Newtonian potential, the Newtonian gravitational acceleration would be [itex] u_z \vec{e}_2 + u_r \vec{e}_3[/itex], but we must be careful, because the Weyl coordinates z,r do not, in general, have any simple relation to euclidean distances! We must also be careful not to assume that, merely because every Weyl vacuum can be written in a Weyl canonical chart, that things like the asymptotic behavior of tensor components wrt the coordinates z,r can be easily compared between two different Weyl vacuum solutions!

The weak-field limit is obtained when we expand to first order in the maximal value of |u|, so by the remark about the quadratic scaling of |v| wrt |u|, we can obtain the weak field limit very simply by neglecting v and expanding [itex]\exp(2u) = 1 + 2u + O(u^2)[/itex]. This gives

[tex]

\begin{array}{rcl}

ds^2 &=& -(1 + 2u) \; dt^2 + (1-2u) \; (dz^2+r^2 + r^2 \, d\phi^2) \\

&& u_{zz} + u_{rr} + \frac{u_r}{r} = 0

\end{array}

[/tex]

The static observers have acceleration vector

[tex]

\nabla_{\vec{e}_1} \vec{e}_1 = (1+u) \; (u_z \; \vec{e}_2 + u_r \; \vec{e}_3)

[/tex]

which is is close the "gravitational acceleration" in Newtonian gravitostatics, since u is assumed small, as expected. So in the weak field limit, there is a very simple connection between the Weyl vacuums and the corresponding Newtonian models. However, this is not at all true for the strong field situation!

To verify that Weyl vacuums are in general curved Lorentzian manifolds (not that there is much doubt of this!), perhaps the easiest approach is to compute Kretschmann scalar [itex]R_{abcd} \; R^{abcd}[/itex], the kind of computation best entrusted to a symbolic computational program. In the weak field limit, in the equatorial plane z=0, assuming a discrete reflection symmetry across the this plane (so that [itex]u_z = 0[/itex] on z=0), this scalar simplifies considerably

[tex]

R_{abcd} \; R^{abcd} = 16 \, \left( u_{rr} \; \left( u_{rr} + \frac{u_r}{r} \right) + \frac{u_r^2}{r^2} \right)

[/tex]

which vanishes wherever u does not depend upon r--- as we would expect, bearing in mind our interpretation of u, in the weak field limit, as a Newtonian gravitational potential.

It is possible to study the

*point symmetry group*in the sense of Lie of the above system of PDEs, and this exhibits some very interesting continuous symmetries, but I won't say more about this here. It is also possible to write down "gauge transformations" for the Weyl vacuums (coordinate transformations which preserve the form of the Weyl canonical chart, but possibly not the spacetime geometry), but I won't say more here.(I originally wrote "I won't way more here", which makes for an inadvertent pun.)

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