BRS: The Weyl Vacuums. I. Definition, Geometrical Properties, Symmetries

[Chris Hillman]

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

• 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.

One basic question which often arises in PF is how phenomena known from Newtonian gravitation relate to analogous situations treated in gtr. For example, if we have two massive objects held apart somehow (perhaps by strong strings whose mass we neglect), we expect from sketching what we think the potential of the system should look like that we should have concentric ovals outside a "figure eight" sepatrix, and smaller concentric ovals inside each "lobe" of the sepatrix, and at the saddle point on the sepatrix, we should have an unstable equilibrium where a test particle can sit--- until it's position is perturbed. We should expect that at least in weak fields, the same basic picture should still be valid in gtr--- but how can we be sure?

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:
$$\begin{array}{rcl} ds^2 &=& -\exp(2u) \; dt^2 + \exp(2\,(v-u)) \; (dz^2+r^2) + \exp(-2u) \; r^2 \; d\phi^2 \\ && u_{zz} + u_{rr} + \frac{u_r}{r} = 0 \\ && v_z = 2 r \, u_z \, u_r \\ && v_r = r \; ( u_r^2-u_z^2) \end{array}$$
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
$$-\infty < t, \; z < \infty, \; 0 < r < \infty, \; -\pi < \phi < \pi$$

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 vacuums" 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 $\partial_t$ (timelike) and $\partial_\phi$ (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 $\partial_t$ 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 $\partial_\phi$ 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
$$\begin{array}{rcl} \vec{e}_1 & = & \exp(-u) \; \partial_t \\ \vec{e}_2 & = & \exp(u-v) \; \partial_z \\ \vec{e}_3 & = & \exp(u-v) \; \partial_r \\ \vec{e}_4 & = & \exp(u) \; \frac{1}{r} \; \partial_\phi \end{array}$$
Using this frame field (aka orthonormal basis, aka anholonomic basis) we can compute that the static observers have acceleration vector
$$\nabla_{\vec{e}_1} \vec{e}_1 = \exp(u-v) \; (u_z \; \vec{e}_2 + u_r \; \vec{e}_3)$$
And, as just mentioned, the expansion tensor of $\vec{e}_1$ vanishes (rigid congruence) and the vorticity tensor of $\vec{e}_1$ 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 $\partial_x, \; \partial_y$, or if you prefer $(1,0), \; (0,1)$, although the latter notation is dangerously imprecise when we consider non-cartesian charts and non-euclidean geometries!,
• every timelike unit vector field $\vec{U}$ defines a family of integral curves, which are timelike curves, but of course not neccessarily timelike geodesics,
• the covariant derivative $\nabla_{\vec{U}} \vec{U}$, 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 $u_z \vec{e}_2 + u_r \vec{e}_3$, 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 $\exp(2u) = 1 + 2u + O(u^2)$. This gives
$$\begin{array}{rcl} ds^2 &=& -(1 + 2u) \; dt^2 + (1-2u) \; (dz^2+r^2 + r^2 \, d\phi^2) \\ && u_{zz} + u_{rr} + \frac{u_r}{r} = 0 \end{array}$$
The static observers have acceleration vector
$$\nabla_{\vec{e}_1} \vec{e}_1 = (1+u) \; (u_z \; \vec{e}_2 + u_r \; \vec{e}_3)$$
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 $R_{abcd} \; R^{abcd}$, 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 $u_z = 0$ on z=0), this scalar simplifies considerably
$$R_{abcd} \; R^{abcd} = 16 \, \left( u_{rr} \; \left( u_{rr} + \frac{u_r}{r} \right) + \frac{u_r^2}{r^2} \right)$$
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.)

 

[Chris Hillman]

BRS: The Weyl Vacuums. II. Examples and physical interpretation

It is convenient to represent specific Weyl vacuums as pairs of axisymmetric functions (u,v), where we think of u as an axisymmetric Newtonian gravitational potential. Some examples:

• Newtonian point mass potential (mass M)
  $$u = \frac{-M}{\sqrt{z^2+r^2}}, \; v = -\frac{-M^2 \, r^2/2}{(z^2+r^2)^2}$$
  this is called the Chazy-Curzon vacuum (1924)
• Newtonian potential of a uniform thin line (linear density m)
  $$u = m \, \log(r), \; v = m^2 \, \log(r)$$
  this is static cylindrically symmetric; the additional Killing vector field $\partial_z$, so this is a subfamily of solutions with three dimensional self-isometry groups,
• Newtonian potential of a uniform thin ray (linear mass density m)
  $$u = m \; \log(z + \sqrt{z^2+r^2}), \; v = 2 m^2 \; \log \left( 1 + \frac{z}{\sqrt{z^2+r^2}} \right) -2 m^2 \; \log(2)$$
• Plane symmetric Newtonian potential
  $$u = a \,z, \; v = -a \,r^2/2$$
• Newtonian potential of a uniform density thin rod (mass M, length 2a):
  $$\begin{array}{rcl} u & = & \frac{M}{2a} \; \log \left( \frac{ \sqrt{(z-a)^2+r^2} + \sqrt{(z+a)^2+r^2}-2a }{ \sqrt{(z-a)^2+r^2} + \sqrt{(z+a)^2+r^2} +2a} \right) \\ v & = & \frac{M^2}{2a} \; \log \left( \frac{\sqrt{(z-a)^2+r^2} + \sqrt{(z+a)^2+r^2}-4a^2}{4 \; \sqrt{(z-a)^2+r^2} \; \sqrt{(z+a)^2+r^2} } \right) \end{array}$$
  (the potential we need can be found by integrating point masses, or by using prolate spheroidal coordinates and demanding that the equipotentials agree with coordinate spheroids),
• Newtonian potential of a uniform thin ring (mass M, radius a)
• Newtonian potential of a uniform thin disk (mass M, radius a)
  (the potential we need can be found by using oblate spheroidal coordinates and demanding that the equipotentials agree with coordinate spheroids).

Note that, as expected, v scales quadratically with the mass density parameter m in the above list.

In the important case of asymptotically flat Weyl vacuums, u,v vanish as z or r (or both) grow without bound. Physically, this corresponds to the case of an isolated source (some static configuration of matter near the origin). The Newtonian potentials of a point mass, thin rod, thin ring, and thin disk all give rise to asymptotically flat Weyl vacuums, but the Newtonian potentials of a thin line, thin ray, or thin planar sheet, do not. In the above list, "mass density m" and "mass M" refers to the Newtonian potentials, but it turns out that there is a suitable notion of mass valid for asympotically flat solutions, the Komar mass, and in the examples of a point mass, thin rod, thin ring, and thin disk the Komar mass turns out to be M.

An important point concerning Weyl vacuums is that while they correspond mathematically to Newtonian axisymmetric potentials, the physical relationship is straightforward only in the weak field limit (see below). In particular:

• the Chazy-Curzon vacuum is not spherically symmetric; the source contains higher order multipoles, and in the case analogous to a black hole, in adapted coordinates the "point" correspons to a kind of ring, and we can extend through the ring to a locally flat region! A hint that something is funny about this "point mass" is that the Kretschmann scalar $R_{abcd} \, R^{abcd}$ blows up for almost all directions of approach, but goes to zero for some (those are the directions that lead to the locally flat region, roughly speaking).
• the Weyl vacuum defined by the Newtonian potentials of uniform density lines and rays are geometrically unlike what you expect, particularly for "extreme" parameter values,
• the case m=-1/2 of the Weyl vacuum defined by the Newtonian potential of a uniform linear density thin line recovers one of Levi-Civita's Petrov type D static vacuums, the type A3, aka the Taub plane symmetric vacuum (one of the most frequently rediscovered solutions)--- in the general case, Levi-Civita's cylindrically symmetric vacuum has a three dimensional isometry group, but algebraically general Weyl tensor (Petrov I),
• the case m=1/2 of the Weyl vacuum defined by the Newonian potential of a uniform linear density thin rod recovers the Schwarzschild vacuum (the event horizon appears in Weyl canonical chart as the rod),
• the source of the Weyl vacuum defined by the Newtonian potential of a uniform density thin disk is not in fact "uniform"

Among the examples I listed, several (the Weyl vacuums generated by the Rindler ray potential, by a plane symmetric potential, and the Levi-Civita A3 or Taub plane-symmetric vacuum, have been put forward as candidates for the elusive and probably mythical "uniform gravitational field in gtr". A fourth candidate, Petrov's homogeneous vacuum (1962), which has a four-dimensional self-isometry group which acts transitively, is not a Weyl vacuum. While each of these possesses various features corresponding to some of the characteristics we would expect, on the basis of the Newtonian analog, all of them rather violently violate our intuition in other respects. Most experts now seem to agree that gtr simply does not admit any vacuum solution which can serve as "the (unique) uniform gravitational field" in all senses in which we might interpret this term. What to make of this lack? I say: it's only to be expected. Gtr is a more realistic theory than Newtonian gravitation, which means that

• it is valid more generally,
• it is more stringent (it rules out more scenarios as incompatible with the theory),
• thus, some idealizations which are useful in Newtonian gravitation simply have no analog in gtr.

Because the master equation is a linear PDE, we can put $u = u_1 + u_2$ where $u_1,u_2$ are each axisymmetric harmonic. Then defining $v_1, v_2$ from $u_1,u_2$ as above and putting $v = v_1 + v_2 + v_{12}$, and requiring that v satisfies the two first order equations in terms of u noted above, we can find two first order equations giving the "nonlinear interaction term" $v_{12}$ by quadrature from $u_1,u_2$. This gives a nonlinear superposition law which allows us to combine Weyl vacuum solutions freely.

For example, we can superimpose two Chazy-Curzon objects to obtain the Bach-Weyl vacuum (1922)
$$u_1 = m_1/\sqrt{(z-a_1)^2+r^2}, \; u_2 = m_2/\sqrt{(z-a_2)^2+r^2}$$
Then we have two massive objects in a static configuration, and it is initially mysterious why they do not fall toward each other! The solution (sort of) lies in noticing that the nonlinear interaction term in v implies the existence of line segment $a_1 < z < a_2, r= 0$ on the axis of rotational symmetry, which plays the role of a "strut" which is under compression and which holds the two objects apart. (Or, if you choose a difference integration constant for v, there will be "wires" extending to $z = \pm \infty$ which are under tension and which pull the two objects apart.) But a closer look reveals still further mysteries: the strut should show up as a third superimposed rod potential, but does not! Furthermore, the strut corresponds, not to a scalar curvature singularity, but to something more subtle: every point on the line segment in question is a "conical singularity".

To understand the simple meaning of this term, consider small circles (integral curves of the spacelike cyclic Killing vector) around r=0. The circumference of such a circle is given by
$$C = \int_0^{2\pi} \epsilon \exp(-u) d\phi = 2 \pi \; \exp(-u(z,\epsilon))$$
But the proper radius is
$$\tilde{\epsilon} = \int_0^\epsilon \exp(v-u) \; dr$$
So, take the limit of the ratio of actual to expected circumference, we find that the condition for "regularity" on the axis is
$$\lim_{r \rightarrow 0^+} v(z,r) = f(z) = 0$$
Usually this condition will only hold on some portions $z_1 < z < z_2, r=0$ of the axis; on the other portions we will have a line segment in which each point is "conical singularity" (not part of the spacetime!), which is so called because it exhibits the same kind of angular deficit in small disks as the vertex of an ordinary cone. By choosing a new integration constant for v we can generally eliminate conical singularities on one segment of the symmetry axis at the expense of introducing them on another.

Such line segments consisting of conical singularities frequently arise in studying specific Weyl vacuum solutions. Physically they are associated with rather dubious "struts" or "wires" which do not contribute to the source of the potential u, and thus lack active gravitational mass, but which nonetheless are strong enough to hold apart or pull on massive objects. In the simplest example, we superimpose two Chazy-Curzon objects at $r=0, z=\pm a$ and find we must either have a massless strut between the particles (under compression and hold them apart) or else two massless wires pulling them off to $r=0, z=\pm \infty$. By superimposing again with a plane potential Weyl vacuum, we can sometimes eliminate all conical singularities at the expense of introducing an "applied gravitational field" with mysterious (and dubious) sources. Or we can superimpose with a line or ray mass potential in hope of making the strut or wire look less dubious by giving it some actual mass.

 

[Chris Hillman]

BRS: The Weyl Vacuums. III. Relations with Some Other Families of Solutions

Weyl vacuums generally represent the static exterior regions of more extensive spacetimes which can be extended through event horizons and/or Rindler horizons. In a sense more than one weak field limit is of interest, since in addition to (u,v) = (0,0) which gives Minkowski vacuum in cylindrical chart, we have the m=1/2 case of the ray potential, which gives a locally flat vacuum with a Rindler horizon. (Which shows, incidently, that speaking of "the" canonical chart is potentially misleading.)

Bonnor introduced a neat exploitation of this phenomenon, in which we obtain Weyl vacuums from "accelerated potentials". The result is a new Weyl vacuum which we can write in Bonnor's version of the Weyl canonical chart:
$$\begin{array}{rcl} ds^2 &=& -\exp(2u) \; Z^2 \; dT^2 + \exp(2(w+v-u) \; (dZ^2+dR^2) + \exp(-2u) \; R^2 \; d\Phi^2 \\ && u_{ZZ} + \frac{u}{Z} + u_{RR} + \frac{u_R}{R} = 0 \end{array}$$
where the intended range of the coordinates is some subset of
$$-\infty < T < \infty, \; 0 < Z, \; R < \infty, \; -\pi < \Phi < \pi$$
There are at least two ways to find explicit examples:

• Given some Newtonian potential u, written in cylindrical coordinates on euclidean space (or if you prefer, in the original Weyl canonical chart), we can find the interaction w with the Rindler ray, and then apply Spiegel's transformation
  $$z = \frac{\left( Z^2- R^2 \right)}{2}, \; r = Z \, R$$
  whose inverse transformation is
  $$Z = \sqrt{z + \sqrt{z^2+r^2}}, \; R = \frac{r}{\sqrt{z + \sqrt{z^2+r^2}}}$$
  to obtain the metric functions (u,v,w).
• We can solve the master equation in the Spiegel-Bonnor chart, as shown above, and then find v+w by quadrature (I'll leave the task of finding the required first order equatoins as an exercise).

The results should agree up to integration constants, of course!

Note that in the case (u,v) = (0,0) (which gives the Rindler chart on Minkowski vacuum), the timelike Killing vector field $\partial_T = z \, \partial_t + t \, \partial_z$ corresponds to boosts, not time translations, so the integral curves are timelike "pseudocircles" (hyperbolas, curves of constant path curvature in Minkowski spacetime). This suggests that, in a general "Bonnor accelerated Weyl vacuum", the "static" observers should correspond to Rindler observers, who experience a constant acceleration away from Z=0, in addition to any additional accelerations required to maintain their position with respect to the source (and to maintain the rigidity of the timelike congruence of static observers). One way to verify this is to note that we can take the frame field of the static observers to be
$$\begin{array}{rcl} \vec{e}_1 & = & 1/Z \; \exp(-u) \; \partial_T \\ \vec{e}_2 & = & \exp(u-v-w) \; \partial_Z \\ \vec{e}_3 & = & \exp{u-v-w) \; \partial_R \\ \vec{e}_4 & = & 1/R \; \exp(u) \; \partial_\Phi \end{array}$$
and then the acceleration vector of these observers takes the form
$$\nabla_{\vec{e}_1} \vec{e}_1 = \exp(u-v-w) \left( (1/Z + u_Z) \; \vec{e}_2 + u_R \; \vec{e}_3$$

When we "accelerate" a Weyl vacuum by Bonnor's procedure, do we obtain a spacetime which is locally nonisometric to the original? In general the answer is yes. For a simple example, it is instructive to study some scalar curvature invariants on the Weyl vacuum from accelerated and unaccelerated plane symmetric potential. Recall that the locus $r=0$ has geometrical meaning, and the accelerated one has a Rindler horizon at $z=0$. But as the example of a Rindler horizon in Minkowski vacuum shows, that is not definitive proof that the spacetimes are distinct as Lorentzian manifolds! Note too that the radial coordinate does not permit simple comparision of functions on two different manifolds given a Weyl canonical chart! But we can look for relationships between two scalar invariants which may hold on one manifold but not the other. In particular, let us compare
$$\frac{R_{abcd;e} \; R^{abcd;e}}{R_{abcd} \; R^{abcd}}$$
on $r=0$ and $R=0, Z> 0$ respectively. In the unaccelerated potential solution this ratio has constant value 25/27, but in the accelerated potential solution the ratio rises from zero at Z=0 to 25/27, asymptotically as Z goes to infinity. Thus, these two Lorentzian manifolds (the Weyl vacuum generated by the Newtonian plane symmetric potential, and the Weyl vacuum generated by the "accelerated" plane potential) cannot be locally isometric.

Bonnor's acceleration procedure will generally yield a vacuum solution which can be extended through the Rindler horizon to past and future dynamical regions, in which the timelike Killing vector becomes a spacelike Killing vector. So in these regions we have an abelian Lie algebra of vorticity-free Killing vector fields, both spacelike, one cyclic, so these regions must be locally isometric to the Beck family of certain cylindrically symmetric, time varying vacuum solutions, given by
$$\begin{array}{rcl} ds^2 & = & \exp(2(u-v) ( -dt^2 + dr^2 ) + \exp(2u) dz^2 + r^2 \exp(-2u) d\phi^2 \\ && u_{tt} = u_{rr} + \frac{u_r}{r} \\ && v_t = 2 \, r \, u_t \, u_r \\ && v_r = r \; (u_t^2 + u_r^2) \end{array}$$
where the intended range of the coordinates is some subset of
$$-\infty < t,z < \infty, \; 0 < r < \infty, \; -\pi < \phi < \pi$$
The master function for these Beck vacuums is a cylindrically symmetric solution of the wave equation (once again, it turns out that this is true not only on the flat spacetime background, but on the Beck vacuum itself); in this form, the wave equation is sometimes called the Euler-Poisson-Darboux equation. This family was rediscovered by Einstein and Rosen and is sometimes known as Einstein-Rosen "waves", although Beck vacuums always include nonradiative components and may not include any radiative components, so this name is something of a misnomer.

In general, we should expect that the extended solutions should include past and future dynamical regions, and that there will be another static region paired with the one with which we began. This qualitatively somewhat resembles the Kruskal-Szekeres extension of the static region of the Schwarzschild vacuum solution, but is axisymmetric, not spherically symmetric. Carrying out this procedure for Weyl vacuum which arises from the accelerated point potential (the Chazy-Curzon vacuum) gives the Bonnor-Swaminarayn vacuum, in which two Chazy-Curzon objects separated by a Rindler horizon are pulled part by two massless strings. Similarly, from the accelerated rod potential with m=1/2 (the Schwarzschild vacuum, written in the Weyl canonical chart), we obtain the C vacuum[/i[, or "C metric", in which two Schwarzschild objects are pulled apart by two massless strings. We know from the theory of the power emitted by sources of linearized gravitational waves that when we have two masses and pull them away from each other, we change the quadrupole moment of the system and thus should produce radiation. While this theory does not apply to exact solutions, it does suggest (correctly) that we should find gravitational radiation somewhere (certainly not in the static regions covered by the two Rindler wedges, however).

In the dynamical regions of the extended solutions, we should perhaps more properly speak of a boost-rotation symmetric vacuum, in the sense of Bicak. It is possible, with sufficient care and effort, to employ the Bondi radiation formalism to compute how much mass-energy is carried away from the pair of accelerating sources by gravitational radiation.

Returning to the Weyl vacuums, it is possible to drop the assumption that the timelike Killing vector $\partial_t$ is vorticity-free (hypersurface orthogonal). This leads to the Ernst family of all stationary axisymmetric vacuum solutions of the EFE, which includes a third metric function w which is nonzero when the source has nonzero Komar angular momentum. Now u,w form a nonlinear system of master equations, and v is given by quadrature from these. We can again compute the point symmetry group in the sense of Lie's theory of symmetries, and this time this provides one of the few known approaches to finding explicit solutions. Another approach involves rewriting the system replacing w with a potential function, such that the two master equations become the real and imaginary parts of the Ernst equation, the axisymmetric case of the nonlinear wave equation
$$\begin{array}{rcl} p \; \Delta p & = & \| \nabla p \|^2 - \| \nabla q \|^2 \\ p \; \Delta q & = & 2 \; \nabla p \cdot \nabla q$$
which arises from the memorable Langrangian
$$L = \frac{\|\nabla p \|^2 +\| \nabla q \|^2}{p^2}$$
(Where p,q are the real and imaginary parts of the so-called Ernst function. Compare the ordinary wave equation, which arises from the Lagrangian $L = \| \nabla p \|^2$.) If we transform this formalism into prolate spheroidal coordinates, and seek rational functions solving the Ernst equation, we come up--- the Kerr vacuum! (All known derivations of this seem to be either nonelementary, or else to require a trick inspired by an astonishing insight.) Still another approach involves ideas from the theories of solitons to nonlinearly superimpose two previously known Ernst vacuum solutions, but seems to yield only solutions (such as the "double Kerr vacuum") which include those damnable massless struts!

In the case of the Beck vacuums, when we drop the requirement that $\partial_\zeta$ be hypersurface orthogonal, we obtain the Ehlers family of all cylindrically symmetric vacuum solutions, which includes a third metric function. As you would guess, "accelerated" Ernst vacuums (defined on stationary axisymmetric regions) can sometimes be extended through Rindler horizons to dynamical cylindrically symmetric regions (locally isometric to some Ehlers vacuum).

Both the Weyl vacuums and their generalization to the Ernst vacuums bear a very interesting relation to another important large family of solutions defined in terms of a system of PDEs (in which the metric functions break up into master functions and secondary functions): the colliding plane waves. These particularly important because they are one of the few known classes of solutions to the EFE which represent an actual physical interaction. One of Chanrasekhar's most famous discoveries (late in his career) was that the portion just inside the event horizon of the future interior region of certain black hole solutions is locally isometric to the interaction zone of certain colliding plane wave solutions. In particular, this is true for the Schwarzschild and Kerr vacuums. Even more important, perhaps, it is true for some attempts to model what might happen inside a generic black hole.

In addition to generalizing the symmetries (from static axisymmetric to stationary axisymmetric), it is possible to generalize the Weyl vacuums to allow for

• a source-free EM field (including both magnetostatic and electrostatic fields, as well as more elaborate fields in the case of rotating sources, e.g. the Kerr-Newman electrovacuum),
• a massless minimally coupled scalar field,

among other possibilities. The latter includes a large class of solutions in which the Riemann tensor is composed of "purely spatial curvature" (the hyperslices orthogonal to the static observers are curved), so that the tidal tensor and even any spin-spin effects vanish everywhere! In these generalizations, it is useful to compute the symmetry groups in sense of Lie of the defining systems of PDEs.

 

[Chris Hillman]

BRS: The Weyl Vacuums. IV. References

There is a huge literature dealing with various aspects of the Weyl vacuums and related families of solutions, or with specfic examples of interest. Here I'll only list some review papers and some general gtr resources:

• Ehlers and Kundt, Exact Solutions of the Gravitational Field Equations, in Gravitation: an Introduction to Current Research, edited by Louis Witten, Wiley, 1962.
• Bonnor, Physical Interpretation of Vacuum Solutions of Einstein's Equations I, Gen. Rel. Grav. 24 (1992): 551-574
• Bonnor, Griffiths, and MacCallum, Physical Interpretation of Vacuum Solutions of Einstein's Equations II, Gen. Rel. Grav. 26 (1994): 687-729
• Bicak, Selected Solutions of Einstein's Field Equations, http://arxiv.org/abs/gr-qc/0004016
• Stephani et al., Exact Solutions of Einstein's Field Equations, 2nd edition, Cambridge University Press, 2003, (see section 20.2)
• Scott, A Survey of the Weyl Metrics, in Conference on Mathematical Relativity (Canberra, 1988): 175-195.
• V. Pravda and V. Pravdova, Boost-rotation symmetric spacetimes - review,
  http://arxiv.org/abs/gr-qc/0003067
• E. Poisson, A Relativist's Toolkit, University of Cambridge Press, 2004.
• S. Carroll, Spacetime and Geometry, Addison-Wesley, 2004.

Ideally, the place to start is with the classic review paper of Ehlers and Kundt (a landmark review which proved influential during the Golden Age of Relativity, c. 1960-1975), which is unfortunately not available on-line. The review by Bonnor et al. and the review by Scott is particularly relevant to grappling with the difficulty of interpreting specific Weyl vacuum solutions. The review by Pravda and Pravdova (former students of Bicak) is relevant to Bonnor's acceleration procedure. The monograph by Stephani et al. contains much useful background information on frames and other matters. The gtr textbook by Carroll contains a good discussion of Komar mass and angular momentum. The book by Poisson contains a good discussion of the kinematic decomposition of a timelike congruence (acceleration vector, expansion tensor, vorticity tensor).

 

[Chris Hillman]

BRS: The Weyl Vacuums. V. Geodesics, Light Bending, Precession

Returning to the general Weyl vacuum solution: by the magic of Noether's theorem on variational symmetries (referring to Lie's theory of symmetries of PDEs; Lie was a friend of Klein, a friend and colleague of Hilbert, who was Noether's supervisor when she was a post-doc at Goettingen), our two Killing vectors immediately yield two quantities which must be conserved along any geodesic. From $\partial_t$ we get
$$\small E = \left[ \begin{array}{cccc} 1&0&0&0 \end{array} \right] \; \left[ \begin{array}{cccc} -\exp(2u) & 0 & 0 & 0 \\ 0 & \exp(2\,(v-u)) & 0 & 0 \\ 0 & 0 & \exp(2\, (v-u)) & 0 \\ 0 & 0 & 0 & \exp(-2u) \, r^2 \end{array} \right] \; \left[ \begin{array}{c} \dot{t} \\ \dot{z} \\ \dot{r} \\ \dot{\phi} \end{array} \right] = \dot{t} \; \exp(2 u)$$
while from $\partial_\phi$ we get
$$\small L = \left[ \begin{array}{cccc} 0&0&0&1 \end{array} \right] \; \left[ \begin{array}{cccc} -\exp(2u) & 0 & 0 & 0 \\ 0 & \exp(2\,(v-u)) & 0 & 0 \\ 0 & 0 & \exp(2\, (v-u)) & 0 \\ 0 & 0 & 0 & \exp(-2u) \, r^2 \end{array} \right] \; \left[ \begin{array}{c} \dot{t} \\ \dot{z} \\ \dot{r} \\ \dot{\phi} \end{array} \right] = r^2 \; \dot{\phi} \; \exp(-2 u)$$
(This is one occasion when it is useful to write vectors expanded with respect to the coordinate basis in matrix form!) That is,
$$\dot{t} = E \, \exp(-2u), \; \dot{\phi} = \frac{L}{r^2} \; \exp(2u)$$
Plugging these into
$$\epsilon = -\exp(2u) \; \dot{t}^2 + \exp(2 \, (v-u)) \; \left( \dot{z}^2 + \dot{r}^2 \right) + \exp(-2u) \; r^2 \; \dot{\phi}^2$$
where [itex]\epsilon = -1, \; 0, \; 1[/tex] for timelike, null, and spacelike geodesics respectively, we obtain the first integrals
$$\begin{array}{rcl} \dot{t} & = & E \; \exp(-2u) \\ \dot{\phi} & = & \frac{L}{r^2} \; \exp(2u) \\ \dot{z}^2 + \dot{r}^2 & = & E^2 \, \exp(-2v) + \epsilon \; \exp(2 \, (u-v)) - \frac{L^2}{r^2} \, \exp(4u-2v) \end{array}$$

Now assume that $\dot{z} = z = 0$; that is, consider geodesics in the equatorial plane, and assume that our spacetime is symmetric across the coordinate plane z=0 (an additional, "discrete" symmetry which won't show up in the Lie algebra of Killing vector fields!). Then the null geodesics in the equatorial plane are given by
$$\begin{array}{rcl} \dot{t} & = & E \; \exp(-2u) \\ \dot{\phi} & = & \frac{L}{r^2} \; \exp(2u) \\ \dot{r}^2 & = & E^2 \, \exp(-2v) - \frac{L^2}{r^2} \, \exp(4u-2v) \end{array}$$
This gives a null congruence which in the z=0 plane consists of null geodesics:
$$E \; \exp(-2u) \; \partial_t \pm \exp(-v) \; \sqrt{E^2 - \frac{L^2}{r^2} \; \exp(4u)} \; \partial_r + \frac{L}{r^2} \exp(2u) \; \partial_\phi$$
The integral curves of this vector field (with z=0) are null geodesics, the world lines of "photons" (better: radar pips) whose motion is confined to the equatorial plane. The choice of sign is determined by whether we are consider ingoing or outgoing null geodesics.

We can use this to derive a light bending formula, valid for null geodesics in the equatorial plane $z=0$. Notice first that in our first integrals for null geodesics in the equatorial plane, recalling that we are assuming an additional discrete symmetry under reflection across this plane, so that $u_z, \; v_z = 0$ on z=0, the minimal canonical radial coordinate r is given by solving the somewhat awkard functional equation
$$r_{\operatorname{min}} = \frac{L}{E} \; \exp(2 \, u(0, r_{min})$$
Assuming we can solve this at least numerically, consider
$$\frac{d \phi}{d r} = \frac{\dot{\phi}}{\dot{r}} = \frac{ \exp(3u)}{ r \; \sqrt{ r \, E/L - \exp(2u)} \; \sqrt{ r \, E/L + \exp(2u)} }$$
(The right hand side depends only on the ratio E/L, which is consistent with the general fact that in gtr, light bending is not frequency dependent.) Integrating this over $r_{\operatorname{min}} < r < \infty$, doubling, and subtracting $\pi$ gives the light bending angle
$$\delta \phi = -\pi + 2 \; \int_{r_{\operatorname{min}}}^\infty \frac{ \exp(3u) \; dr}{ r \; \sqrt{ r \, E/L - \exp(2u)} \; \sqrt{ r \, E/L + \exp(2u)} }$$
This is an exact result, provided we can find the minimal radius and carry out the integration. In practice, we may find it easier to consider asymptotic expansions in r or to use other tricks.

As a crude check, putting u=v=0 in this formula gives zero, as it should (no light bending in flat spacetime!).

Careful readers will note that I slipped in another assumption, which implies asymptotic flatness: I tacitly assumed that u is monotonic decreasing to zero, so that we can think of null geodesics as coming in from $r=\infty$, reaching a minimal "radius" at $r=r_{\operatorname{min}}$, and zooming back out to $r=\infty$.

As noted earlier, we can take the frame field of static observers (whose world lines are integral curves of the timelike Killing vector--- these are timelike curves but certainly not, in general, timelike geodesics!) to be
$$\begin{array}{rcl} \vec{e}_1 & = & \exp(-u) \; \partial_t \approx (1-u) \; \partial_t \\ \vec{e}_2 & = & \exp(u-v) \; \partial_z \approx (1+u) \; \partial_z \\ \vec{e}_3 & = & \exp(u-v) \; \partial_r \approx (1+u) \; \partial_r \\ \vec{e}_4 & = & \frac{1}{r} \; \exp(u) \; \partial_\phi \approx \frac{1}{r} \; (1+u) \; \partial_\phi \end{array}$$
where the approximations are suitable for expanding everything to first order in |u| (the weak-field approximation). The electro-Riemann or tidal tensor (which agrees with the electro-Weyl tensor in a vacuum!) with respect to a timelike unit vector $\vec{U}$ is:
$$E\left[ \vec{U} \right]_{ab} = R_{ambm} \; U^m \; U^n$$
where of course we should put $\vec{U} = \vec{e}_1$ in order to compute the tidal accelerations measured by static observers. In the case of static observers in a Weyl vacuum, in the weak field limit, in the equatorial plane (again assuming a reflection symmetry across this plane), the tidal tensor reduces to
$$E\left[ \vec{e}_1 \right]_{ab} = \left[ \begin{array}{ccc} -u_{rr} - \frac{u_r}{r} & 0 & 0 \\ 0 & u_{rr} & 0 \\ 0 & 0 & \frac{u_r}{r} \end{array} \right]$$
which agrees with our identification, in this limit, of the metric function u with the Newtonian gravitational potential. We earlier noted something similar for the acceleration vector of the static observers.

The magneto-Riemann tensor (agrees with magneto-Weyl tensor in a vacuum!) with respect to a timelike unit vector $\vec{U}$ is:
$$B\left[ \vec{U} \right]_{ab} = {^{\ast}R}_{ambm} \; U^m \; U^n$$
where we take the left dual (equals right dual in a vacuum), and this always vanishes for static observers in a static vacuum. (It might help to recall that the electro-Riemann tensor controls tidal accelerations, while the magneto-Riemann tensor controls spin-spin forces, a subtle topic usually neglected in textbooks because these forces--- which give rise to nonzero accelerations when a small spinning object moves through the gravitational field of a massive spinning object--- are too small to measure. And it might help to note that this Bel decomposition of the Riemann tensor is mathematically analogous to the familiar decomposition of the electromagnetic field tensor into electric and magnetic field vectors. There is a third piece of the Bel decomposition, however, the topo-Riemann tensor which describes "spatial" curvatures associated with hyperplane elements.)

It is interesting to compare these results with the physical experience of observers in the equatorial plane (assuming reflection symmetry) who fall in freely and radially. To study them, we should introduce an appropriate frame field. To find it, boost the static frame radially (focusing attention on z=0)
$$\begin{array}{rcl} \vec{e^\prime}_1 & = & \sqrt{f^2+1} \; \vec{e}_1 - f \; \vec{e}_3 \\ \vec{e^\prime}_2 & = & \vec{e}_2 \\ \vec{e^\prime}_3 & = & -f \; \vec{e}_1 + \sqrt{f^2+1} \; \vec{e}_3 \\ \vec{e^\prime}_4 & = & \vec{e}_4 \end{array}$$
where f is an undetermined function of r. In the plane z=0 we now demand that $\nabla_{\vec{e^\prime}_1} \vec{e^\prime}_1 = 0$, which gives an equation which we solve for f. The desired frame is:
$$\begin{array}{rcl} \vec{e^\prime}_1 & = & \exp(-2u) \; \partial_t - \exp(-v) \; \sqrt{1-\exp(2u)} \; \partial_r \\ \vec{e^\prime}_2 & = & \exp(u-v) \; \partial_z \\ \vec{e^\prime}_3 & = & -\exp(-2u) \; \sqrt{1-\exp(2u)} \; \partial_t + \exp(-v) \; \partial_r \\ \vec{e^\prime}_4 & = & \frac{1}{r} \; \exp(2u) \; \partial_\phi \end{array}$$
Alert readers will notice that we require $u(0,r) < 0$ here!

The vorticity tensor of the new frame still vanishes, so the congruence is nonrotating (and geodesic in the equatorial plane), but the expansion tensor of the new congruence is nonvanishing in the equatorial plane, as we should expect. In the weak field limit, in the equatorial plane, the electro-Riemann or tidal tensor and the magneto-Riemann tensor have the same form as in the case of static observers (generalizing a phenomenon well known in the special case of Schwarzschild vacuum), and the expansion tensor is
$$H\left[ \vec{e}_1 \right]_{ab} = \left[ \begin{array}{ccc} \sqrt{-2 \, u} \; u_r & 0 & 0 \\ 0 & \frac{1}{\sqrt{-2 \, u}} \; u_r & 0 \\ 0 & 0 & \frac{\sqrt{-2u}}{r} \end{array} \right]$$
where (as we should expect from Newtonian intuition) we must assume that $u < 0$.

In the equatorial plane z=0, recalling our assumption that (by the assumed reflection symmetry) $u_z = 0, v_z=0$ on this plane, the acceleration vector of static observers becomes $\exp(u-v) \; u_r \vec{e}_3$, which shows that if u_r vanishes anywhere (perhaps in a "saddle point" lying between two massive objects, a static observer may be inertial (although such a timelike geodesic is likely to be unstable under small perturbations).

More interesting is the case of observers in stable circular orbits in the equatorial plane, often called Hagihara observers (because Hagihara considers them in his celestial mechanics textbook). Once again, we study their physical experience by introducing a suitable frame field and computing tensorial components of interest wrt the new frame. To find the Hagihara frame, we can this elementary procedure (easier to carry out than to describe, perhaps!):

• starting with the static frame, boost at each event in the $\vec{e}_4$ direction, where the boost is given by terms like $\cosh(f)$, where f is an undetermined function of r,
• require that the acceleration vector of the new congruence vanish, leading to a differential equation which we can solve for the undetermined function,
• at each event, rotate the new frame by the appropriate amount so that one spatial frame vector always points at a certain distant star,
• compute Fermi derivatives to show that the new frame $\vec{e^\prime}_1, \ldots \vec{e^\prime}_4$ is now inertial but not gryostabilized (the frame vectors are still rotating as our observers orbit),
• to fix this, rotate around $\vec{e^\prime}_2$, with the rotation given by terms like $\cos(g \,t)$, where g is an undetermined function of r,
• require that the Fermi derivatives vanish in the equatorial plane, leading to an equation which can be solved for the undetermined function g,
• the new frame $\vec{e^{\prime\prime}}_1, \ldots \vec{e^{\prime\prime}}_4$ is the desired Hagihara frame.

The final result shows that the gyrostabilized Hagihara frame "rotates with respect to the fixed stars" at a rate which is constant for each Hagihara observer. This is of course the geodetic precession, and we can state the grand result like this: the geodetic precession per orbit is given by
$$\delta \phi = 2 \pi \; (1 - \sqrt{1 + r\, u_r} \; \sqrt{1 + 2 r \, u_r} \exp(-v) )$$
a result due to Rindler and Perlick. This is an exact result, although in concrete examples, an asymptotic expansion wrt r may give more insight than the exact expression.

 

[Chris Hillman]

BRS: The Weyl Vacuums. VI. Three classes of observers

First, I forgot to mention in Post I that the volume form of a Weyl vacuum written in standard canonical chart is
$$r \; \exp(v-u) \; dt \wedge dz \wedge dr \wedge d\phi$$
which shows that when we plot a "Weyl picture" in cylindrical coordinates on flat spacetime (probably with some coordinates suppressed!), our picture will not in general preserve volume, because of the factor $\exp(v-u)$.

I should also have written out the curved space wave operator (aka generalized Laplacian) for a Weyl vacuum written in standard canonical chart:
$$\Box h = -\exp(-2u) \; h_{tt} \; + \; \exp(2\,(u-v)) \; \left( h_{zz} + h_{rr} \; + \; \frac{h_r}{r} \right) \; + \; \exp(2u) \; \frac{h_{\phi \phi}}{r^2}$$
which shows that, as I said, the metric function u is harmonic in the smooth manifold underlying the Weyl vacuum solution (as well as the fictitious flat spacetime background).

In Post V, when I introduced "the equatorial plane", I should have said that any coordinate plane $z=z_0$ where $u_z = 0, \; v_z = 0$ will work; for convenience we may as well assume this happens at z=0. When computing geometric objects (vectors and tensors) in this plane, don't forget to plug in this assumption, which usually greatly simplifies everything!

Now let's see some more detail on how to use the frame fields I have introduced to study the physical experience of (respectively)

• static observers,
• Lemaitre observers (radially infalling inertial observers in the equatorial plane),
• Hagihara observers (inertial observers in circular orbits in the equatorial plane)

If we compute the components of the electroriemann tensor (tidal tensor) and magnetoriemann tensor wrt the Lemaitre frame $\vec{e^\prime}_1, \ldots \vec{e^\prime}_4$, we get expressions of the form
$$E\left[\vec{e}_1\right]_{ab} & = & \left[ \begin{array}{ccc} -f & 0 & 0 \\ 0 & h & 0 \\ 0 & 0 & f-h \end{array} \right]$$
(which is diagonal and traceless) and
$$B\left[\vec{e}_1\right]_{ab} & = & \left[ \begin{array}{ccc} 0 & 0 & \ell \\ 0 & 0 & 0 \\ \ell & 0 & 0 \end{array} \right]$$
respectively. The typical appearance of a nonzero term in the magnetoriemann tensor when we boost from a static observer to a new observer in a static spacetime is analogous to the typical appearance of a magnetic field when we boost to from a static observer to a new observer in an electrostatic field. In that case, the EM field invariants
$$F_{ab} \; F^{ab}, \; F_{ab} \; {^{\ast}F}^{ab}$$
are respectively nonzero and zero, indicating "no intrinsic magnetism". In a nonrotating gravitational field, the principal curvature invariants
$$R_{abcd} \; R^{abcd}, \; R_{abcd} \; {^{\ast}R}^{abcd}$$
are respectively nonzero and zero, indicating "no intrinsic gravitomagnetism". And I should probably say that, just as the electric and magnetic vectors "live" in spatial hyperplane elements orthogonal to the world line of an observer, the electroriemann and magnetoriemann tensors live in these spatial hyperplane elements. I didn't write "orthogonal hyperslice" because if the timelike congruence of world lines of observers is not vorticity-free, we cannot "knit together" these hyperplane elements to form hypersurfaces, but the formalism of the Bel decomposition (and vectorial decomposition of the EM field tensor) is still valid. This issue applies to observers with vorticity in flat spacetime as well as curved spacetimes, so this is a matter of the underlying mathematics of manifold theory (specifically, the Frobenius theorem), not an issue specific to gtr.

I should probably insert another warning here: I like to write the expansion tensor as $H[\vec{U}]_{ab}$ (and likewise for vorticity tensor, vorticity vector, electric field vector, magnetic field vector, electroriemann tensor, magnetoriemann tensor) in order to emphasize that not only are these quantities defined with respect to a specific $\vec{U}$, the expansion tensors wrt to two different timelike unit vector fields $\vec{U}, \; \vec{V}$ are two different tensor fields living on two different sets of hyperplane elements (and likewise for vorticity tensor, vorticity vector, electric field vector, magnetic field vector, electroriemann tensor, magnetoriemann tensor). Once again, this is a matter of the mathematics of Lorentzian manifolds, not an issue involving the physics of gtr.

In the weak field limit (neglect v and expand everything to first order in the maximal value of |u|) the electroriemann tensor for the Lemaitre observers agrees with that for the static observers, and the magnetoriemann tensor vanishes for both families of observers. But the strong field components do distinguish between these observers.

I should add a remark about computing the expansion tensor of our Lemaitre congruence in the equatorial plane: the congruence as I defined it (for mathematical convenience) is well-defined (if u < 0), but only geodesic in the equatorial plane. But applying the conditions $u_z = 0, \; v_z = 0$ ensures that our expansion tensor components, if we kill the first row and first column, refer to the relative motion of inertial infalling observers. Also, the weak-field limit
$$H_{33} = \frac{u_r}{\sqrt{-2u}}, \; H_{44} = \frac{\sqrt{-2u}}{r}$$
is very close to the Newtonian result.

Now for some more detail about the Hagihara observers. I explained how to construct the desired nonspinning inertial frame in two stages: first, boost each frame in the static frame field in the [itex[\vec{e}_4[/itex] direction by just the right amount to ensure that the acceleration vector of the new family of observers should vanish (in the equatorial plane); second, rotate each frame in the new frame field by just the right angle $t \, h$ where h is some function of r, such that the Fermi derivatives (in the equatorial plane) all vanish. When you carry out this procedure, the first step yields the frame field
$$\begin{array}{rcl} \vec{e^\prime}_1 & = & \frac{\sqrt{1-r \, u_r}}{\sqrt{1-2 r \, u_r}} \; \exp(-u) \; \partial_t + \frac{\sqrt{r \, u_r}}{\sqrt{1-2 r\, u_r}} \; \partial_\phi \\ \vec{e^\prime}_2 & = & \vec{e}_2 = \exp(u-v) \; \partial_z \\ \vec{e^\prime}_3 & = & \vec{e}_2 = \exp(u-v) \; \partial_r \\ \vec{e^\prime}_4 & = & \frac{\sqrt{r \, u_r}}{\sqrt{1-2 r \, u_r}} \; \exp(-u) \; \partial_t + \frac{\sqrt{1 - r \, u_r}}{\sqrt{1-2 r\, u_r}} \; \partial_\phi \\ \end{array}$$
where we must assume (granted that u < 0) that |u| is decreasing with r, but not too fast, so that $0 < u_r < 1/2/r$. The condition that u_r not decay too fast should not be suprising if you recall that in the special case of Hagihara observers in the equatorial plane of the Schwarzschild vacuum, these observers only exist outside $r=6m$. Inside that radius (but outside the event horizon), u_r decreases too quickly to admit the existence of inertial observers in circular orbits.

Note that the third frame vector field (a spacelike unit vector field) rotates to maintain alignment with "the radial direction" as our observers move around in their circular orbits, so while this frame is inertial, the spatial frame vectors must be rotating with respect to a suitably oriented ideal gryoscope carried by our Hagihara observer. The second step corrects for this by introducing a rotation of each frame about $\vec{e^\prime}_2$, given by terms like
$$\sin \left( t \; \frac{\sqrt{r \, u_r} \; \sqrt{1-2 r \, u_r}}{r} \; \exp(2u-v) \right)$$

Both of these two Hagihara frames are useful. The first one, in which the spatial vectors rotate to keep aligned with "the radial direction", more clearly shows the positional dependence of tensorial quantities.

The second one, the "gyrostabilized" Hagihara frame, shows clearly that a nonspinning inertial observer in a circular orbit around a massive object, who carefully measures tidal accelerations inside his spaceship, without looking out the porthole, will find that he can deduce the a projective direction from his spaceship to the massive object, and that this distinguished projective direction is slowly rotating with respect to his gryostabilized spatial frame vectors. I said "projective direction" (think of RP^2 viz. S^2) because in general the observer will deduce an axis aligned with the direction to the massive object, but won't be able to decide between the two possible directions without looking out the porthole. The reason is that "Coulomb contribution" to the tidal tensor (in gtr as in Newtonian gravitation) has the form, in an aligned frame $p \; \operatorname{diag}(-2,1,1)$, so if we "drop" an initially spherical cloud of test particles, this will deform into a prolate spheroid whose long axis distinguishes the projective direction, but due to the reflection symmetry, does not say in which of the two possible directions the massive object lies!

You are probably thinking that this projective direction should be time delayed, but we can't really look for unambiguous time delays in a static spacetime! There is a very interesting issue here which we could study if our source was time varying: the EM field (as treated in Maxwell's theory) is updated to take account of changes in the source of an EM field via "spin-one" radiation, while the gravitational field (as treated in gtr) is updated to take acount of changes in the source of a gravitational field via "spin-two" radiation. It turns out that EM uses a linear predictor, whereas gtr uses a quadratic predictor, so in approximately circular orbits around a time varying source, we should expect some interesting differences between the behavior of the electric and magnetic vectors viz. the electroriemann and magnetoriemann tensors. (In a later thread, incidently, I plan to generalize the discussion in this thread to Weyl-Maxwell electrovacuums, and in particular plan to study Hagihara observers in circular orbits around an isolated massive charged object, and to compare the behavior of the electric and magnetic field vectors with that of the electroriemann and magnetoriemann tensors in the "gyrostabilized" Hagihara frame.)

In more detail: in the radially aligned Hagihara frame, in the equatorial plane we obtain components of form:

• the expansion tensor
  $$H\left[\vec{e^\prime}_1\right]_{ab} & = & \left[ \begin{array}{ccc} 0&0&0\\ 0&0&w\\ 0&w&0 \end{array} \right]$$
  shows our congruence is shearing, as we should expect from Newtonian intuition (because inner observers are rotating faster, they "slide past" nearby observers rotating a bit further out),
• the vorticity vector (equivalent to the vorticity tensor--- this is the same idea as representing an antisymmetric two-form as a vector in the standard vector calculus on euclidean three-space) has form
  $$\vec{\Omega} = j \; \vec{e^\prime}_2$$
  showing nonzero vorticity orthogonal to the equatorial plane,
• the electroriemann or tidal tensor
  $$E\left[\vec{e^\prime}_1\right]_{ab} & = & \left[ \begin{array}{ccc} -f&0&0\\ 0&h&0\\ 0&0&f-h \end{array} \right]$$
  is diagonal traceless (and slightly different from the tidal tensor of the static or Lemaitre observers),
• the magnetoriemann tensor
  $$B\left[\vec{e^\prime}_1\right]_{ab} & = & \left[ \begin{array}{ccc} 0 & \ell &0\\ \ell &0 &0\\ 0 &0 &0 \end{array} \right]$$
  (compare which component is nonzero here with the case of the Lemaitre observers) shows that in principle, our observers could measure tiny-tiny spin-spin forces on a suitably oriented gyroscope (but in practice, these are unfortunately much too tiny to be measured).

In the nonspinning inertial Hagihara frame, the tensor components reflect the fact that we have introduced a steady rotation of each primed frame about $\vec{e^\prime}_2$.

From the components of the timelike (geodesic) unit vector field $\vec{e^\prime}_1$ we obtain that the orbital coordinate speed (in the canonical chart) of the Hagihara observers is
$$\frac{d\phi}{dt} = \frac{\sqrt{u_r/r}}{\sqrt{1-r \, u_r}} \; \exp(-2u)$$
In other words, the world lines of the Hagihara observers appear in the canonical chart as coordinate helices with this coordinate angular velocity, which is constant for each Hagihara observer. Recalling that we tacitly assumed asymptotic flatness in this discussion, this is in fact the orbital angular velocity measured by a static observer "at spatial infinity" (the point is that static observers near the origin will experience "gravitational time dilation"). And drat, I can no longer edit Post V to correct the typo I just noticed: the Rindler-Perlick expression for the geodetic precession per orbit should of course read
$$\Delta \phi = 2 \pi \; \left(1 - \sqrt{1-r \, u_r} \; \sqrt{1-2r \, u_r} \; \exp(v) \right)$$
See http://arxiv.org/abs/gr-qc/0010003

Let me stress again a general point: one advantage of using frame fields is that the components of vectors and tensors computed with respect to a frame have geometrical meaning; assuming we have properly defined our frame, the frame vectors can be expanded in any chart wrt the coordinate basis, and the value of the frame components (but certainly not the coordinate components) of vectors and tensors is therefore coordinate independent. It would be difficult to overstate the usefulness of studying components in ordinary vector calculus when interpreting solutions (for example, the electric and magnetic field vectors in some solution of Maxwell field equations, or the vorticity vector in hydrodynamics), so it is good to know that by using frames we can do the same thing in any spacetime

I should also mention a general point about nonspinning inertial frames: these are as close as we can come, in a curved Lorentzian manifold, to the special case of a cartesian frame in flat spacetime, i.e. a Lorentz frame (where the distinction between the frame field and the frame at some event, and the frame and coordinate basis vector fields/vectors, is blurred because for this case, and only for this case, these notions agree). For an inertial frame (i.e. one in which $\vec{e}_1$ is a timelike geodesic vector field), the Fermi derivatives wrt $\vec{e}_1$ reduce to covariant derivatives, so the general condition that a frame field be nonspinning inertial is
$$\nabla_{\vec{e}_1} \vec{e}_1 = \nabla_{\vec{e}_1} \vec{e}_2 = \nabla_{\vec{e}_1} \vec{e}_3 = \nabla_{\vec{e}_1} \vec{e}_4 = 0$$

 

[Chronos]

I warned you about bone headed questions - I am having trouble seeing how Weyl vacuum preserves Lorentz invariance. Is this unnecessary or have I missed the point?

 

[Chris Hillman]

I warned you about bone headed questions - I am having trouble seeing how Weyl vacuum preserves Lorentz invariance. Is this unnecessary or have I missed the point?

That is in fact a very good question!

Let me rephrase it: where is the Lorentz invariance in a general Lorentzian manifold? This question is analogous to: where is the rotation invariance in a general Riemannian two-manifold?

And we here's another: when is a Lorentz manifold invariant under some self-isometries? What does it mean to say that some such Manifold is axially symmetric? This is analogous to: what does it mean to say that a Riemannian two-manifold is rotationally symmetric?

I think these questions are best addressed in another thread containing some further background for manifolds generally. So I'm going to start a new BRS thread on the simplest examples I can think of, in the case of Riemannian two-manifolds, including some Ctensor files so everyone with Maxima installed can verify my claims (modulo correctness of Ctensor, of course), which I hope will address your question, and then I'll edit this post to link to the new thread.

Edit: the BRS thread where I will attempt to answer the question asked by Chronos is here:
https://www.physicsforums.com/showthread.php?t=379533

But please do ask any questions you might have about a BRS thread in a post in that thread, because a future SA/M who might happen along might have the very same question!

We are all exploring new ground here, but one of the goals I have for BRS is to try to ensure that the threads, which might be inspired by some "topical discussion" (e.g. sophisticated background for a current PF thread in the public areas), are useful at a later date.

A general remark about what I have in mind for BRS threads: I certainly hope to address background which I have been assuming (without believing I can really get away with this!) in the Weyl vacuums thread, including:

• Monge patches for a euclidean surface treated as a Riemannian two-manifold
  • obtaining the line element from the height function of the surface,
  • obtaining a coframe giving the correct metric tensor,
  • visualizing frames as small disks, plus rotating an arbitrary frame arbitrarily at each event,
  • computing connection one-forms and curvature two-forms and obtaining Gaussian curvature,
  • obtaining the Laplacian from the coframe,
  • comparision with any possible self-isometries generated by Killing vectors,
• background on vector fields, including integral curves, Lie algebra vs. Lie groups of transformations,
• background on Lie's theory of symmetry of DEs with examples (unfortunately Maxima can't handle these, but there are several excellent textbooks to refer to)

I started with Weyl vacuums and am vaguely planning to next write about Bondi radiation formalism, not because these are the logical place to start, but because in organizing complex material it is easier to not try to do too much at once. So I am just writing about whatever interests me at the moment, and currently plan to try to address questions in new threads when the question is related to a topic I have had in mind for a future BRS thread anyway.

 

[Chris Hillman]

BRS: The Weyl Vacuums. VII. Alternative charts

It is often useful to have at hand some alternative coordinate charts for the Weyl vacuums. There are a half dozen of these which are often encountered in reading arXiv eprints dealing with the Weyl vacuums, all inspired by similar alternative charts for euclidean three-space, such as can be found elsewhere in mathematical physics. In this post, I'll give a brief overview of the most common charts are their uses. In each case, I'll discuss

• transformation to and from the Weyl canonical chart
• axisymmetric solutions of Laplace equation by separation of variables

The charts I plan to discuss all keep the coordinates $t, \; \phi$ but replace $z, \, r$.
To save typing I'll abuse notation by using $x, \, y$ for the new coordinates for each alternative chart.

[size=+1]Rational Spherical Chart[/size]

The most important alternative chart is particularly convenient for studying asymptotically flat Weyl vacuums, and is closely related to ordinary polar spherical coordinates. It can be obtained from the canonical chart using the coordinate transformation
$$x = \sqrt{z^2+r^2}, \; y = \frac{z}{\sqrt{z^2+r^2}}$$
which has inverse transformation
$$z = x \, y, \; r = x \, \sqrt{1-y^2}$$
(Needless to say, the relationship with polar spherical coordinates is $x = \rho, \; y = \cos(\theta)$.) In rational spherical coordinates, the line element for the Weyl vacuums reads
$$\begin{array}{rcl} ds^2 & = & -\exp(2u) \, dt^2 + \exp(2 \,(v-u)) \left( dx^2 + \frac{x^2}{1-y^2} \, dy^2 \right) + \exp(-2u) \; x^2 \; (1-y^2) \; d\phi^2, \\ &&-\infty < t < \infty, \; 0 < x < \infty, \; -1 < y < 1, \; -\pi < \phi < \pi \end{array}$$
Note that in terms of the canonical coordinate chart,

• x = x_0 is a sphere centered at z=0,r=0
• y = y_0 is a coordinate cone with vertex at z=0,r=0, with three degenerate cases:
  • y = 0 corresponds to the equatorial plane
  • y=1 corresponds to the ray r=0, z > 0
  • y = -1 corresponds to the ray r=0, z < 0

(See the figure which should appear at bottom, with shading corresponding to y, and with contours of constant x also indicated.)

The master equation of the Weyl system (the Laplace function for axisymmetric functions) becomes
$$x^2 \, u_{xx} + (1-y^2) \, u_{yy} + 2 x \, u_x - 2 y \, u_y = 0$$
so we see again that the axisymmetric harmonic functions on the unobservable flat background are precisely the axisymmetric harmonic functions on our Weyl vacuum solution. I'll leave transforming the other two PDEs as an unimportant exercise.

If we solve the Laplace equation by separation of variables, choosing the separation constant with regard only for neatness of appearance, we find two families of solutions:

• solutions regular at the origin are given by
  $$u = \sum_{n=0}^\infty b_n \; P_n(y) \; x^{n}$$
• asympotically vanishing solutions are given by
  $$u = \sum_{n=0}^\infty a_n \; \frac{P_n(y)}{x^{n+1}}$$

where $P_n$ is the Legendre polynomial. (There are two other families involving $Q_n$, the Legendre function with logarithmic terms, which we discard because we are assuming that -1 < y < 1.) Looking ahead a bit, in the next post in this thread I plan to explain how to obtain axisymmetric Newtonian potentials from a given (axisymmetric) distribution of mass, and we'll see that this involves matching across some sphere one of the regular solutions to one of the asymptotically vanishign ones.

Here, P_n is of course an axially symmetric spherical harmonic (an eigenfunction of the Laplace operator on the ordinary round sphere). A useful theorem from potential theory states that if H(y) is a spherical harmonic which is a polynomial of degree n in y, then $H(y)/x^n$ is a solution of the Laplace equation on E^3. So, in the asympotically vanishing case, we are decomposing axisymmetric harmonic functions into rational functions of the form polynomial in y divided by a power of x; this is simply the multipole expansion of an axisymmetric harmonic function. If we put the origin at the center of mass of the source, and assume reflection symmetry across the equatorial plane, only the odd powers of x survive
$$u = \sum_{n=0}^\infty a_n \; \frac{P_{2n}(y)}{x^{2n+1}}$$
Let's write the first two terms as
$$u = \frac{-M}{x} + \frac{-Q \, \left( \frac{3 \,y^2-1}{2} \right)}{x^3} + O(1/x^5)$$
where in Newtonian theory, M would be the mass and Q the quadrupole moment of the source.

Incidentally, two useful reference books for background in the potential theory I will use in this thread are:
Kellogg, Foundations of Potential Theory, Dover reprint
Sneddon, Elements of Partial Differential Equations, Dover reprint
A handy reference for the special functions I will use is
Bell, Special Functions for Scientists and Engineers, Dover reprint

Let us verify that

• the Komar mass of an asympotically flat Weyl vacuum is M
• the Komar spin vanishes

In general, the ingredients for computing the Komar integrals are

• the tangent vectors to the stationary observers, $\vec{T}$, a timelike unit vector field, which in our chart is $\vec{T} = \exp(-u) \, \partial_t$
• the outward pointing spherical normals $\vec{N}$, a spacelike unit vector field, which is here $\vec{N} = \exp(2\, (u-v)) \; \partial_x$,
• a timelike Killing vector $\vec{\xi}$, which is here $\vec{\xi} = \partial_t$,
• a spacelike cyclic Killing vector $\vec{\chi}$, which is here $\vec{\chi} = \partial_\phi$

We must compute the Komar mass using the following steps:

• compute the covariant derivative $\nabla_{\vec{T}} \vec{\xi}$
• take the inner product of this with $\vec{N}$
• average this scalar over a sphere of constant radius; in rational spherical coordinates, taking account of the Jacobian factor, the formula we need is
  $$\left< F \right> = \frac{1}{4 \pi} \; \int_{\phi=-\pi}^\pi \; \int_{y=-1}^1 \; F(y, \phi) \; x^2 \, \exp(v-2u) \; dy \, d\phi$$
• take the limit as the radius of our sphere tends to infinity.

In our case, we are averaging an axisymmetric scalar over x=x_0, and the average reduces to
$$\left< \vec{N} \cdot \nabla_{\vec{T}} \vec{\xi} \right> = \int_{-1}^1 \; x_0^2 \; u_x(x_0,y) \, dy$$
where we assume $u < 0, \; u_x > 0$, so that u is tending to zero from below as x grows; then the sign of u will agree with the standard Newtonian potential. Now we plug in our expression for a general asympotically vanishing u (with center of mass at the origin and symmetric under reflection across the equatorial plane). Interchanging the limit and the integral over y gives simply M, which is the desired result.

The computation of the Komar spin is similar:

• compute the covariant derivative $\nabla_{\vec{T}} \vec{\chi}$
• take the inner product of this with $\vec{N}$
• average this scalar over a sphere of constant radius,
• take the limit as the radius tends to infinity

except that we need a multiplicative factor of -3/2. In our case, the scalar vanishes, so the Komar spin vanishes, as should happen for a static asympotically flat vacuum solution.

[size=+1] Rational Paraboloidal Chart[/size]

This chart can be obtained from the canonical chart using the transformation
$$x = \sqrt{z + \sqrt{z^2+r^2}}, \; y = \sqrt{-z + \sqrt{z^2+r^2}}$$
which has inverse transformation
$$z = \frac{x^2-y^2}{2}, \; r = xy$$
In rational paraboloidal coordinates, the line element for the Weyl vacuums reads
$$\begin{array}{rcl} ds^2 & = & -\exp(2u) \, dt^2 + \exp(2 \,(v-u)) \; (x^2+y^2) \; (dx^2 + dy^2 ) + \exp(-2u) \; x^2 \; y^2 \; d\phi^2, \\ &&-\infty < t < \infty, \; 0 < x, \, y < \infty, \; -\pi < \phi < \pi \end{array}$$
The master equation (the Laplace equation for axisymmetric functions) becomes
$$( x \, u_x)_x + (y \, u_y)_y = 0$$
which has a pleasantly symmetric form. We have already seen how this form arises in Bonnor's "accelerated potential" Weyl vacuums.

[size=+1] Rational Prolate Spheroidal Chart[/size]

This chart can be obtained from the canonical chart using the transformation
$$x = \frac{\sqrt{(r+A)^2+z^2}+\sqrt{(r-A)^2+z^2}}{2A}, \; \; y = \frac{\sqrt{(r+A)^2+z^2}-\sqrt{(r-A)^2+z^2}}{2A}$$
which has inverse transformation
$$z = A \, x \, y, \; \; r = A \, \sqrt{x^2-1} \, \sqrt{1-y^2}$$

 

[Chris Hillman]

BRS: The Weyl Vacuums. VIIb. Alternative charts, cont'd

Imprecations! I came back too late to continue to edit the previous post, which I left unfinished.
I mean to add a line labeling the figures: left to right: level surfaces for x,y in

• rational spherical coordinates,
• rational parabolic coordinates,
• rational prolate spheroidal coordinates,

all plotted in cylindrical coordinates z,r (with axial symmetry about the vertical axis r=0, so you can mentally rotate to see the level surfaces in E^3).

And since I left the last section unfinished, and even made a small slip in my haste, let me start that part over.

[size=+1]Rational Prolate Spheroidal Chart[/size]

This chart can be obtained from the canonical chart using the transformation
$$x = \frac{\sqrt{(z+A)^2+r^2}+\sqrt{(z-A)^2+r^2}}{2A}, \; \; y = \frac{\sqrt{(z+A)^2+r^2}-\sqrt{(z-A)^2+r^2}}{2A}$$
which has inverse transformation
$$z = A \, x \, y, \; \; r = A \, \sqrt{x^2-1} \, \sqrt{1-y^2}$$
where A is a positive constant which sets the scale of the dimensionless coordinates x,y. Here, if you attach a string of fixed length at the endpoints of the line segment $r=0, \; -A < z < A$ in euclidean three-space, hook the string with one end of a paper clip, and while keeping the string taut in a V shape, move the paper clip around as much as possible, the moving vertex of the "V" will trace out a prolate spheroid. IOW,

• the level surfaces x=x_0 are coordinate prolate spheroids, degenerating to the line segment $r=0, -A < z < A$ as $x \rightarrow 1^+$,
• the level surfaces y=y_0 are everywhere orthogonal to the family of prolate spheroids, with three degenerate cases:
  • y = 1 is the ray r=0, z > 0
  • y = 0 is the equatorial plane z=0
  • y = -1 is the ray r=0, z < 0

I should probably add that a prolate spheroid is an axially symmetric ellipsoid, with smaller "equatorial diameter" than "axial diameter"--- think of a sphere flattened equatorially.

The line element now takes the form
$$\begin{array}{rcl} ds^2 & = & -\exp(2u) \; dt^2 + A^2 \, \exp(2 \, (v-u)) \; \left( \frac{dx^2}{x^2-1} + \frac{dy^2}{1-y^2} \right) + A^2 \, (x^2-1) \; (1-y^2) \; \exp(-2u) \; d\phi^2, \\ && -\infty <t < \infty, \; 1 < x < \infty, \; -1 < y < 1, \; -\pi < \phi < \pi \end{array}$$

As with most alternative charts, the primary interest of this one is in separating variables in Laplace equations for axisymmetric functions, which reads (in either the flat background or in the Weyl vacuum itself)
$$((x^2-1) \, u_x)_x + ((1-y^2) \, u_y)_y = 0$$
Separating variables in the form u(x,y) = f(x) g(y), we find

• solutions regular near x=1 have the form
  $$u = \sum_{n=0}^\infty b_n \; P_n(x) \, P_n(y)$$
• asymptotically vanishing solutions have the form
  $$u = \sum_{n=0}^\infty a_n \; Q_n(x) \, P_n(y)$$

Different authors use different normalizations, so the Legendre functions used by Maple may differ by multiplicative factors from those used by some authors:
$$\begin{array}{rcl} P_0(x) & = & 1 \\ P_1(x) & = & \frac{3 x^2-1}{2} \\ P_2(x) & = & \frac{(5 x^2-3) \, x}{2}$$
and
$$\begin{array}{rcl} Q_0(x) & = & \log \sqrt{(x-1)/(x+1)} \\ Q_1(x) & = & x \; \log \sqrt{(x-1)/(x+1)} - 1 \\ Q_2(x) & = & \frac{3 x^2-1}{2} \; \log \sqrt{(x-1)/(x+1)} - \frac{3x}{2} \end{array}$$
and so forth. The graph of the Legendre polynomials $P_n(y)$ always fits inside a square $[-1, 1]\times[-1,1]$, with values $\pm 1$ at $y = \pm 1$, and they are odd or even according to whether n is odd or even. That is, the even index Legendre polynomials obey $P_{2n}(-y) = P_{2n}(y)$ and tend to 1 as |y| tends to 1, a factoid which we will need in the next post.

The importance of this chart probably won't become fully clear until, in a planned followup thread on the Ernst vacuums, I explain how a "rotating" version leads to a particularly simple solution of a certain PDE, which, when "unwound", leads to the Kerr vacuum! (All methods I know of obtaining this most important vacuum solution of the EFE employ either unmotivated tricks or systematic but nonobvious preparation.)

The closely related trigonometric prolate spheroidal chart can be obtained from the rational version using the transformation
$$x = \cosh(\chi), \; y = \cos(\theta)$$

[size=+1] Rational Oblate Spheroidal Chart[/size]

This chart can be obtained from the canonical chart using the transformation
$$x = \frac{\sqrt{(r+A)^2+z^2}+\sqrt{(r-A)^2+z^2}}{2A}, \; \; y = \frac{\sqrt{(r+A)^2+z^2}-\sqrt{(r-A)^2+z^2}}{2A}$$
which has inverse transformation
$$z = A \, \sqrt{x^2-1} \, \sqrt{1-y^2}, \; \; r = A \, x \, y$$
The line element now takes the form
$$\begin{array}{rcl} ds^2 & = & -\exp(2u) \; dt^2 + A^2 \, \exp(2 \, (v-u)) \; \left( \frac{dx^2}{x^2-1} + \frac{dy^2}{1-y^2} \right) + A^2 \, x^2 \, y^2 \; \exp(-2u) \; d\phi^2, \\ && -\infty <t < \infty, \; 1 < x < \infty, \; 0 < y < 1, \; -\pi < \phi < \pi \end{array}$$
where we should only try to cover the upper half space. This time

• the level surfaces x = x_0 are coordinate oblate spheroids, degenerating as $x \rightarrow 1^+$ to the disk z = 0, 0 < r < A,
• the level surfaces y=y_0 are certain surfaces everywhere orthogonal to these spheroids, with two degenerate cases:
  • y = 1 is the equatorial plane
  • y= 0 is the ray r=0, z > 0

I'll leave the separation of variables as an exercise.

[size=+1] Rational Toroidal Chart[/size]

This is obtained from the canonical chart using the transformation
$$x = \frac{z^2+r^2+A^2}{\sqrt{z^2+(r+A)^2} \; \sqrt{z^2+(r-A)^2}}, \; \; y = \frac{z^2+r^2-A^2}{\sqrt{z^2+(r+A)^2} \; \sqrt{z^2+(r-A)^2}}$$
which has inverse transformation
$$z = \frac{A \; \sqrt{1-y^2}}{x-y}, \; \; r= \frac{A \; \sqrt{x^2-1}}{x-y}$$
where again we should only try to cover the upper half space. The line element now takes the form
$$\begin{array}{rcl} ds^2 & = & -\exp(2u) \; dt^2 + \frac{A^2}{(x-y)^2} \; \left( \exp(2 \, (v-u)) \; \left( \frac{dx^2}{x^2-1} + \frac{dy^2}{1-y^2} \right) + (x^2-1) \; \exp(-2u) \; d\phi^2 \right), \\ && -\infty <t < \infty, \; 1 < x < \infty, \; -1 < y < 1, \; -\pi < \phi < \pi \end{array}$$
where

• the surfaces x=x_0 are coordinate tori (circular cross section) nested around (but not centered around) the ring z=0, r=A, with two degenerate cases:
  • x = 1 is the ray r=0, z > 0
  • as $x \rightarrow \infty$ we obtain the circle r=A, z=0,
• the surfaces y=y_0 are certain surfaces everywhere orthogonal to these tori, with two degenerate cases:
  • y =1 corresponds to the equatorial planar region z=0, r > A
  • y = 0 corresponds to the equatorial planar disk z=0, 0 < r < A

[size=+1] Rational Bipolar Spherical Chart[/size]

You can probably guess that this one is obtained from the canonical chart using the transformation
$$x = \frac{z^2+r^2+A^2}{\sqrt{(z+A)^2+r^2} \; \sqrt{(z-A)^2+r^2}}, \; \; y = \frac{z^2+r^2-A^2}{\sqrt{(z+A)^2+r^2} \; \sqrt{(z-A)^2+r^2}}$$
which has inverse transformation
$$z = \frac{A \; \sqrt{x^2-1}}{x-y}, \; \; r= \frac{A \; \sqrt{1-y^2}}{x-y}$$
where once again we should only try to cover the upper half space. The line element now takes the form
$$\begin{array}{rcl} ds^2 & = & -\exp(2u) \; dt^2 + \frac{A^2}{(x-y)^2} \; \left( \exp(2 \, (v-u)) \; \left( \frac{dx^2}{x^2-1} + \frac{dy^2}{1-y^2} \right) + (1-y^2) \; \exp(-2u) \; d\phi^2 \right), \\ && -\infty <t < \infty, \; 1 < x < \infty, \; -1 < y < 1, \; -\pi < \phi < \pi \end{array}$$

Figures: left to right: level surfaces for x,y in

• rational oblate spheroidal coordinates,
• rational toroidal coordinates,
• rational bipolar spherical coordinates,

all plotted in z,r; the surfaces are axially symmetric in each case, so you can mentally rotate about the axis r=0.

 

[Chris Hillman]

BRS: The Weyl Vacuums. VIII. Axisymmetric Newtonian potentials

To construct explicit Weyl vacuum solutions, we require a plentiful supply of axisymmetric harmonic functions. Since I have already given fairly general series expressions for asympototically vanishing axisymmetric harmonic functions, it is clear that such a supply exists. In this post, I will focus on the problem of finding the Newtonian potential outside a static axisymmetric configuration of matter in Newtonian gravitation. Each such potential immediately yields a corresponding Weyl vacuum solution; as I mentioned in Post II in this thread, it turns out that geometrically/physically speaking, the sources for this exact vacuum solution will not bear any simple relation with the Newtonian sources, except in the weak field limit, so this is not the same thing as constructing Weyl vacuum solutions "to order". Rather, we will construct weak field approximations "to order". The weak field limit is important, so this is worthwhile.

But first, a correction: in Post II, regarding the Newtonian potential of a uniform thin disk (mass M, radius a), I wrote:

the potential we need can be found by using oblate spheroidal coordinates and demanding that the equipotentials agree with coordinate spheroids.

I can't believe I said that! Particularly since explaining why this yields a Newtonian potential which is not the one we need was always part of my plan for this thread! :blush: And in Post VIII, a typo I can no longer correct: the first few Legendre polynomials are
$$P_0(y) = 1, \; P_1(y) = y, \; P_2(y) = \frac{3y^2-1}{2}, \; P_3(y) = \frac{5y^3-3y}{2}$$

Onwards. Let me begin by highlighting a few salient facts from potential theory. First and foremost: the theory of axisymmetric harmonic functions in E^3 is quite different from the theory of harmonic functions in E^2! The best way to understand this, perhaps, is to note that the symmetry group (in the sense of Lie) of $u_{zz} + u_{rr} + \frac{u_r}{r} = 0$ (u a function of z,r only) is quite different from that of $v_{xx} + v_{yy} = 0$ (v a function of x,y only). The latter includes the full euclidean planar conformal group; the former includes a subgroup of the finite dimensional group of conformal transformations on E^3. So you can't write down a holomorphic function, take its real or imaginary part, and expect to obtain an axisymmetric harmonic function in E^3!

That said, some important and useful ideas familar to anyone who has taken an advanced undergraduate level course in complex analysis do carry over to harmonic functions on E^n:

• Gauss's mean value theorem: if you average the values of any harmonic function u on E^n over any sphere in E^n, the resulting number is the value of u at the center of the sphere,
• Koebe proved a converse result: a continuous function which satisfies the mean value property is harmonic,
• As a consquence of the mean value property, no nonconstant harmonic function can assume its maximum or minimum, taken over any ball, in the interior of the ball, and a bounded harmonic function on E^3 must be constant.

Regarding the wealth of E^3 harmonic functions, the interior Dirichlet problem is the boundary value problem
$$\begin{array}{rcl} \Delta u = 0, & {\rm inside} \; \Omega \\ u =f & \partial \Omega \end{array}$$
and similarly for the exterior Dirichlet problem. It is shown in potential theory that (for rather general functions f and domains $\Omega$), these BVPs always have a unique solution. So we can describe the wealth of E^3 harmonic functions by saying that every (suitably nice) function on the unit sphere generates a unique harmonic function, so locally the wealth of harmonic functions is comparable to the wealth of suitably nice functions of two variables. Furthermore, a function u on a suitably nice domain which satisfies $u=f$ on $\partial \Omega$ and minimizes the magnitude of the divergence $\| \nabla u\|^2$ on the interior must be a solution of the Dirichlet problem; the converse is not true unless certain conditions are satisified (these conditions were found by Hilbert, who corrected an error by Riemann which was noticed by Weierstrass.)

(My first exposure to the Dirichlet problem was interesting: early in my undergraduate career, my faculty advisor was John Hubbard, at the very moment that he was working with Adrien Douady on his most famous result, the proof that the Mandlebrot set is connected. This work utilized the idea of thinking of M as a material object, charging it up, and studying the equipotentials and the family of orthogonal curves, the trajectories of charged test particles. These are smooth even though M has a fractal boundary. The same idea works for Julia sets too, and there is a symbolic dynamics which arises by coloring basins suitably and then noting the sequence of colors as a charged particle approaches the boundary. I vividly remember running into Hubbard in the hall one day. He was clutching a sheaf of enlarged color prints, which he laid down on a glass display case to show me how the symbolic dynamics works. I couldn't have known then that one day I'd write a diss on the symbolic dynamics of generalized Penrose tilings! Hubbard was one of the first to draw those marvelous pictures which are universally familiar today--- in those days, there was no color printer, so he set up a camera on a tripod and photographed the video display of the local supercomputer. Today such pictures can be easily drawn on any desktop computer. See the articles by Linda Keen and Bodil Branner in DeVaney and Keen, editors, Chaos and Fractals, Proceedings of Symposia in Applied Mathematics 39, AMS, for more information about the electrostatics of Julia sets, labeling the boundary of the Mandlebrot set, and the Hubbard-Douady theorem.)

Similarly, axisymmetric harmonic functions are locally specified by giving the values on the axis of symmetry, so the wealth of axisymmetric harmonic functions is comparable to the wealth of suitably nice functions of one variable.

The situation in which we have a thin shell on the boundary of some region $\Omega$ is important. Then the potential is continuous but the normal derivative "jumps" as we move across the shell. The potential is harmonic on the inner and outer regions and in the limit as we approach the shell from the interior or exterior, we have
$$\lim_{\rm outer} \nabla_{\vec{n}} u - \lim_{\rm inner} \nabla_{\vec{n}} u = 4 \pi \, \mu$$
where $\mu$ is the surface density in the shell and $\vec{n}$ is the outward pointing normal to the shell.

The name "harmonic" arises, incidently, from Euler's observation that if we have a spherical harmonic $f(\theta,\phi)$ which is a polynomial (homogeneous of degree n), then $f/\rho^n$ is a solution of the Laplace equation, i.e. a harmonic function. A spherical harmonic is an eigenfunction of the Laplace equation on S^2. And those are called "harmonics" because they can be used to model "vibrations" of a spherical membrane, and thus are two dimensional generalizations of sines and cosines, which can be used to model vibrations of a linear mass, such as a vibrating piano string or violin string. A major theme of the subject involves an issue I already mentioned: potential theory differs radically in one, two, and three or more dimensions; in particular, the higher dimensions are less "musical" in various senses. (C.f. Mark Kac's problem: can you hear the shape of a drum? Unfortunately explaining this would take us too far afield.)

It is worth pointing out that for asympotically vanishing harmonic functions on E^3, it is true generally that the equipotentials become more and more spherical as one moves away from the source. This agrees with Newtonian intuition because far from an isolated configuration of matter, we expect the field to closely resemble the field of a spherically symmetric object with the same mass. Furthermore, a family of surfaces $f(x,y,z) = c$ arises as the equipotentials of some harmonic function only if $\Delta f/\| \nabla f \|^2$ is a function of f only.

In Post VII, when I discussed rational spherical coordinates, I mentioned the fact that separation of variables leads to general expressions for two types of axisymmetric harmonic functions:

• regular at the origin, $u = \sum_{n=0}^\infty b_n \, P_n(y) \, x^n = \sum_{n=0}^\infty b_n \, P_n(\cos(\theta)) \, \rho^n$
• asymptotically vanishing, $u = \sum_{n=0}^\infty a_n \, P_n(y)/x^{n+1} = \sum_{n=0}^\infty a_n \, P_n(\cos(\theta))/\rho^{n+1}$

The latter is of course the multipole expansion of an axisymmetric harmonic function.

We need to know a bit about the convergence properties of these "exterior" and "interior" series. Suppose that the source of the Newtonian gravitational field which we are representing using an axisymmetric harmonic function is just enclosed by a sphere $x=a$ and $x=b, \; b > a$, is a slightly larger sphere. Then $u = \sum_{n=0}^\infty a_n \, P_n(y)/x^{n+1}$ converges outside $x=a$ and converges uniformly outside $x=b$. Harnack showed this is also true for all the partials of u, so u is real analytic. A similar statement holds for for $u = \sum_{n=0}^\infty b_n \, P_n(y) \, x^n$ in the case of a spherical void partially or wholly enclosed by matter.

More generally, the idea is that should we require the Newtonian potential generated by some isolated configuration of matter, whenever we have a sphere centered at some point which stays just outside the matter filled regions, we can find a series which uniformly converges to our potential inside the sphere, and by matching values we can systematically "fill in the blanks" via analytic extension. So we can use one "exterior series" (valid outside a sphere enclosing the entire configuration) and possibly many "interior series". (See the figure below.)

With these preliminaries established, let's construct some Newtonian potentials.

The most elementary method is to start with the potential of a point mass at $z=\bar{z}, r=\bar{r}, \phi = \bar{\phi}$, namely
$$u_{\rm point} = \frac{\mu}{\sqrt{ (z-\bar{z})^2 + r^2 + \bar{r}^2 - 2 r \, \bar{r} \, \cos(\phi-\bar{\phi}) }}$$
and to integrate over the object which is the source of the Newtonian gravitational field, taking $\mu$ to be an integrable function giving the mass density at each point inside the object. Here it is crucial to remember to use the euclidean volume form written in cylindrical coordinates, namely $\omega = dz \, dr \, r \, d\phi$. (In other words, don't forget the Jacobian factor when you integrate).

Two examples are particularly useful:

• the potential of a uniform density thin rod (mass M, length 2L) lying in the axis of symmetry r=0 with centroid at the origin (see Post II above) is obtained by integrating over $-A < \bar{z} < A$ with $\bar{r} = 0$,
• the potential of a uniform density thin ring (mass M, radius A) lying in the plane z=0,
  $$u = \frac{ 2 M \; K \, \left( \frac{ \sqrt{4 A \, z \,r}}{\sqrt{(r^2+A)^2+z^2}} \right) } {\pi \, \sqrt{z^2+(r+A)^2}}$$
  where K is the elliptic integral of the second kind, is obtained by integrating over $-\pi < \bar{\phi} < \pi$ with $\bar{r} = A$.

Each of these examples can also be obtained by using one of the symmetry adapted charts described in Post VII. Namely:

• in the rational prolate spheroidal chart on E^3, assume u is a function of x only, solve the resulting ODE, and transform the result back to cylindrical coordinates,
• in the rational toroidal chart on E^3 (after solving the Laplace equation written in rational toroidal coordinates by separation of variables), assume u is a function of x only, solve the resulting the ODE, and transform the result back to cylindrical coordinates.

(In Post VII I discussed rational charts for Weyl vacuums, but by setting u=v=0 and ignoring the time coordinate one immediately obtains the corresponding rational charts on euclidean three space.) Notice that this method works only because the surfaces of constant x coordinate in these two coordinate systems satisfy the necessary condition mentioned above for a family of surfaces to arise as the equipotentials of some harmonic function! This is a special property which will not hold for arbitrary orthogonal coordinates on E^3.

At this point, it is natural to integrate over a family of rings in an effort to obtain the potential of a thin uniform density disk (mass M, radius A). Unfortunately, this integral is hard even for Maple to evaluate in closed form! Fortunately, there is another way which in practice is just as effective. The idea is to extend from the values on $r=0, \; z > A$ to the "exterior series", which will converge on the region $z^2+r^2>A^2$, and similarly to extend from the values on $r=0, \; 0 < z < A$ to the "interior series", which will converge on $z >0, \; r> 0, \; z^2 + r^2 < A^2$. (As noted above, the convergence will get a bit wobbly near the sphere of radius A, particularly near the poles and on the equator, but on slightly smaller regions we can obtain uniform convergence with known error terms, so by taking enough terms and using the symmetry under reflection across the equatorial plane, we can approximate our harmonic function quite passably on all of E^3.) This extension method will yield series valid in at least some regions for any axisymmetric distribution of matter for which we can find the values of the potential on the axis r=0.

I'll explain it by sketching how it works for the case of the uniform density thin disk:

• We must find the axial values; putting $r=0$ in the point potential, and integrating over $-\pi < \bar{\phi} < \pi$ and $0 < \bar{r} < A$ we obtain
  $$u(z,0) = \frac{2M}{A^2} \; \left( z - \sqrt{z^2+A^2} \right)$$
  which we write in rational spherical coordinates (on the locus y=1) as
  $$f(x) = \frac{2M}{A^2} \; \left( x - \sqrt{x^2+A^2} \right)$$
• We find the coefficients $a_n$ in the exterior series by matching these coefficients with the result of expanding f in powers of 1/x; in this example, only odd powers survive,
• We find the coefficients $b_n$ in the interior series by matching them with the result of expanding f in powers of x; in this example, only even powers survive with one exception ($b_1 \neq 0$).

The result is a piecewise expression in terms of two infinite series, which we can truncate if we are only interested in obtaining a good approximation. The first few coefficients turn out to be
$$\begin{array}{l} a_0 = \frac{-M}{A}, \; a_1 = 0, \; a_2 = \frac{M}{4\,A}, \; a_3 = 0, \; a_4 = \frac{-M}{8\.A}, \; a_5 = 0, \\ a_6 = \frac{5\,M}{64\,A}, \; a_7 = 0, \; a_8 = \frac{-7\,M}{128\,A}, \; a_9 = 0, \; a_{10} = \frac{21\,M}{512A}, \ldots \end{array}$$
for the exterior series, and
$$\begin{array}{l} b_0 = \frac{-2\,M}{A}, \; b_1 = \frac{2\,M}{A}, \; b_2 = \frac{-M}{A}, \; b_3 = 0, \; b_4 = \frac{M}{4\,A}, \; b_5 = 0, \\ b_6 = \frac{-M}{8\,A}, \; b_7 = 0, \; b_8 = \frac{5\,M}{64A}, \; b_9 = 0, \; b_{10} = \frac{-7\,M}{128A}, \ldots$$
for the interior series. Finally, if desired, we can transform to cylindrical coordinates to plot the equipotentials; see the figure below.

The final result can be expressed in rational spherical coordinates $x =\sqrt{z^2+r^2} = \rho, \; y = z/\sqrt{z^2+r^2} = \cos(\theta)$ as
$$u(x,y) = \left\{ \begin{array}{ll} \frac{2M}{A} \; \left( x \, P_1(y) \; + \; \sum_{n=0}^\infty \frac{(-1)^n \, \Gamma(n-1/2) \; (x/A)^{2n} \; P_{2n}(y)} {2 \sqrt{\pi} \; \Gamma(n+1)} \right) & 0 < x < A \\ \frac{-2M}{A} \; \sum_{n=0}^\infty \frac{ (-1)^n \, \Gamma(n+1/2) \; (A/x)^{2n+1} \; P_{2n}(y)} {2 \, \sqrt{\pi} \; \Gamma(n+2)} & A < x < \infty \end{array} \right$$
where $\Gamma$ is the Euler Gamma function. (Compare Sneddon, Elements of Partial Differential Equations, Dover reprint, which uses a different notation.)

Note that

• the equipotentials become more spherical at large distances, as promised,
• because of the reflection symmetry across the equatorial plane, the equipotentials are orthogonal to the equatorial plane outside the disk $z=0, \; r > A$,
• they are not orthogonal to the surface of the disk $z=0, \; 0 < r < A$.

Also, the problem of computing the potential of a thin uniform density rod superficially appears to be "dual" to the problem of computing the potential of a thin uniform density disk (interchange the role of z,r), but as we have seen, this is actually not the case at all the case (the problem is again the paucity of symmetries of the E^3 Laplace equation compared to the E^2 Laplace equation). If you use the oblate rational spheroidal coordinates to write a harmonic function of x only, solve the resulting ODE, and transform to cylindrical coordinates, you obtain a harmonic function (this claim is plausible, since spheroids $x=x_0$ satisfy the necessary condition for a family of surfaces to be the equipotentials of a harmonic function), but it turns out to arise from a distribution of matter in a thin disk of radius A which is not uniform.

If you apply the extension method to a thin uniform density tube (mass M, radius A, height 2L), you should use a large sphere of radius $\sqrt{A^2+L^2}$, and a small sphere of radius A. The interior series actually converges on a slightly larger domain, but as this example shows, in general one must do more work to obtain a piecewise expression valid on (almost) all of E^3, in the case of more complicated shaped bodies which do not lie entirely in a sphere of radius A with center at the origin.

Figures: from left to right,

• domains of convergence for exterior/interior series (rational spherical coordinates or polar spherical coordinates)
• equipotentials for a uniform density thin ring (seen in cross section as a small red dot), plotted in cylindrical coordinates z,r (mentally rotate about r=0 to see the equipotential surfaces),
• equipotentials for a uniform density thin disk (seen in cross section as a red line segment), plotted in cylindrical coordinates z,r (mentally rotate about r=0 to see the equipotential surfaces).

 

[Chris Hillman]

BRS: The Weyl Vacuums. VIII. Axisymmetric Newtonian potentials, contd

There are other methods of constructing harmonic functions in E^3, of course. A Green's function is great if you can find it. (See Zachmanoglou and Thoe, Introduction to Partial Differential Equations with Applications, Dover reprint.) Integral transforms are sometimes useful. (See Duffy, Transform Methods for Solving Partial Differential Equations, CRC Press.) Lie's theory of the symmetries of a PDE applies very nicely to $u_{zz} + u_{rr} + \frac{u_r}{r}$ and yields solutions which are invariant under specified flows. For example, the spatial dilation flow is $z \, \partial_z + r \, \partial_r$; it generates the one dimensional subgroup of spatial dilations
$$\begin{array}{rcl} z & \rightarrow & \exp(\lambda) \, z \\ r & \rightarrow & \exp(\lambda) \, r \\ u & \rightarrow & u \end{array}$$
The rational invariants of this group of transformations are
$$u = c_1, \; \frac{z}{r} = c_2$$
which leads to the symmetry Ansatz $u = F(z/r)$. Plugging this into the Laplace equation yields the solutions
$$u = A + B \, \operatorname{arctanh} \left( \frac{z}{\sqrt{z^2+r^2}} \right)$$
which are indeed invariant under spatial dilations.

Here, Lie's methods (or a solid intuition) are required only to find the flows under which the form of the equation $u_{zz} + u_{rr} + u_r/r =0$ is unchanged; after that, elementary reasoning yields the rational invariants of the flow, the appropriate symmetry Ansatz, and the corresponding solutions. (For Lie's theory of symmetry, see Olver, Applications of Lie Groups to Differential Equations, 2nd edition, Springer.)