The equation of the initial value appears to impose constraints on the choice of boundary surface for such a system. These conditions imply that the exterior can be described by a Weyl solution of the Einstein field equations.3·4.
INTRODUCTION
Do these restrictions in any way affect the size of the boundary surface, which the ring conjecture suggests should be "larger in all directions" than. The main conclusions of this analysis are: There are constraints on the location of the matter/vacuum boundary in the Weyl exterior for non-collapsing systems as described here.
Junction Conditions .Across a 2-Surface
The three-dimensional geometry of the system on this hyper-surface (hereafter denoted ~) is governed by the single initial-valued equation. C3) R = 16rre (2.1). R is the curvature scalar for the two-dimensional geometry of 5, TrS=Sa a is the trace of the outer curvature of 5, Tr(sa)=SapsPa is the trace of its square (the sums over a and {3 span the two dimensions of 5) , and al on is the derivative with respect to the proper normal distance with fT.
Exterior Coordinate System. Metric, and Field Equations
Since ?'E can be determined from 'if!E by integrating equations (3.4) and (3.5), the entire exterior geometry is specified as 'if!E, or fictitious. 3.7) The coordinate basis vectors touching the boundary are a; arp and. where here and below denote primes derivatives with respect to A. The normal for the surface bait is .
Interior Coordinate System, Metric, and Field Equations
Here, as above, the subscript E indicates quantities calculated on the outside of the boundary. Since the metric (3.12) has the same form as (3.2), the description of the interface from the inside is the same as that from the outside.
A Boundary Constraint Inequality
The calculation of the scalar curvature (s)R for the geometry described by the metric (3.12) is straightforward and yields results. The internal coordinates leading to the metric form (3.12) and hence to the inequality (3.22) are not unique, since the region of the plane filled by the coordinates (pJ,ZJ) can be chosen arbitrarily (taking into account only the requirement of simple connectivity and conditions PI=O on the axis of symmetry, PI>O elsewhere).
Double-Matched Coordinates
The internal coordinates, specifically coordinate derivatives ;;· for matched coordinates, must be specified in some way to evaluate the inequality (3.30); then surfaces that violate the inequality are forbidden. Surfaces that satisfy the inequality may or may not be acceptable boundaries, since they may bound interiors that satisfy (3.20) but violate the f:~O condition.
A SECOND BOUNDARY CONSTRAINT
A boundary constraint different from (3.30) or (3.43) can be derived using the same procedure as in Section III, assuming a slightly different interior description.
Alternative Interior Description
Derivation of the Alternate Boundary Constraint
Specifically, the background source for these geometries is a line monopole, with linear density r;2, extending from zE=-a to zE=+a on the symmetry axis in the background space. Equation (3.3), the Laplace equation for '¢'E• is easily solved for such a source in prolate spheroidal coordinates (u,v ,ip), related to the Weyl coordinates (pE,zE,IJO) by PE =asinhusinv, zE=acoshucosv, with uE:[0,+00), v E:[O,rr].
The Spherical-Topology Boundary Constraint
This result means that for the Schwarzschild exterior geometry none of the u;::.constant surfaces, with u~O. The Schwarzschild coordinate radius of the matter-vacuum boundary at the moment of standstill can be chosen freely. The internal Friedmann metric on the hypersurface of time symmetry can be cast in the form (3.12), and the coordinates (p1,z1) can be chosen such that on the interface PI coincides with PE of the Weyl coordinates for the Schwarzschild exterior.
The three simplest measures of the size of u = constant surfaces in r geometries, consistent with axial symmetry, are the polar circumference. The form of the boundary constraints of Section III suggests that these constraints may constitute lower bounds on Cp, i.e., that surfaces with Cp values less than a minimum may be prohibited. These results seem to rule out any interpretation of the constraint (3.43) as a lower bound on Cp for acceptable boundary surfaces.
Application of the Boundary Constraint
EXTENSION OF THESE RESULTS
Some of the results of Sections III and IV can be applied to surfaces in a wider class of extrageometric geometries than the Weyl metrics. The spatial metric form (3.2) can be obtained in any axisymmetric, static exterior; the corresponding four-metric takes the form (3.1) only in special cases of this, irJ.including vacuum, 16 while the Weyl equations and (3. 5) occur only for the vacuum case. The derivation of the spherical-topology boundary constraint, up to the inequality (3.24), depends only on the three-metric form (3.2); the Weyl vacuum field equations are only used to obtain (3.30) from (3.24).
Thus, the spherical topological constraint .!11W'J~O can be applied to surfaces in any asymmetric, static exterior geometry, provided that .!11W'J is given by. It is thus possible to identify surfaces in arbitrary asymmetric, static exterior geometries that are forbidden as boundaries of temporally static, non-singular matter systems by means of the inequalities (3.24) or (4.11), for spherical or toroidal topology, respectively. Consequently, these constraints are likely to prove most useful for special exteriors, such as the Weyl vacuum geometries or electrovacuum generalizations thereof.
SUMMARY
I rp=rpo is a constant i;o slice of the entire hypersurface of time symmetry (inside I and outside E). If a unit disk is chosen as the range of the internal coordinates, the coordinate map is displayed. Al) which, using the major branch of the square root, maps the unit disk bijectively to the right half of the unit disk.
The complete image depends on the interior geometry, but the first coordinates of the image points are given by the boundary conditions on p1. Everything on the right-hand side of this equation is gauge invariant, that is, invariant under conformal transformations of the interior coordinates, hence the integral of. The orientation of the endpoint values of e(A.) is determined by the stipulation that (3.15) gives the outward-directed normal.
INTRODUCTION
The model solution for a tidally distorted black hole studied in this paper is a special case of Geroch-Hartle solutions. The procedure used here is to model a simple physical situation representing the tidal distortion of a black hole: that of a Schwarzschild black hole perturbed by bodies fixed on the polar axis of the hole beyond the horizon. The internal geometry of the distorted horizon is explored in Section III; Section IV contains a discussion of various definitions of hole masses and perturbing bodies and a discussion of the role of binding energy in the solution.
In part V, the Riemann curvature tensor for the spacetime geometry near the distorted horizon is calculated for comparison with other tidal curvature calculations. The solution obtained here succeeds in illustrating some aspects of the effects of tidal perturbations on a black hole, in the static limit. This solution provides a consistent account of the masses involved in the configuration, and of the relationships between the locally measured masses.
GEOMETRY OF A SCHWARZSCHILD BLACK HOLE WITH TIDAL DISTORTIONS
General Formalism
When 1/J is known, the function 'l is calculated by integrating equations (2.3) and (2.4), with the boundary condition that 1=0 on the axis of symmetry in the absence of a conical singularity there. The Schwarzschild solution, which is static, axisymmetric, and vacuum, can be described by the Weyl formalism.12 The background space source for 1/J in this description is not a point, and not spherically symmetric: this is a manifestation of the distortion. inherent in the Weyl coordinates. The source occupies the segment -M~z~M, p=O on the symmetry axis in the background space (see figure 1).
Equation (2.2) for 1/J is best solved in the presence of such a source by transforming into elongated spheroidal coordinates (u,v), related to cylindrical Weyl coordinates by. This is the Schwarzschild metric in the usual Schwarzschild coordinates, which shows that (2.10) is indeed the desired solution. It describes a black hole of mass M, with the event horizon at the location u=O of the "line mass source".
A Schwarzschild Black Hole with Suspended "Moons"
INTRINSIC GEOMETRY OF THE DISI'ORTIID HORIZON
Two regimes are of interest in this problem: the weak-perturbation regime, where the change in horizon shape from its undisturbed sphericity is small. For configurations satisfying (3.13), the embedding surface defined by (3.8) represents the intrinsic geometry of the horizon section. Equations (3.31) and (3.32) then define an embedding surface of the form z=Z(Jf) which reproduces the horizon section geometry near the pole in the strong perturbation case.
Of course, the horizon section shape itself remains almost spherical everywhere, consistent with the original definition of the weak perturbation regime. This somewhat unwieldy result is consistent with the results of the previous sections on the horizon-section geometry. According to the Smarr formula, with the horizon cross-sectional area conserved in the process, the mass of the black hole is reduced proportionally by the perturbation.
The Masses of the Moons and the Contributions of the Ropes
RIEMANN TENSOR COMPONENTS IN THE VICINITY OF THE DISfORTED HORIZON
To facilitate such comparisons, the components here are calculated in the orthonormalized frame of a static near-horizon observer. The components of the Riemann tensor in this orthonormal frame are related to those in the coordinate frame in a simple way. The main goal of this calculation of the components of the Riemann tensor is to provide a standard of comparison in the static limit for dynamical calculations of tidal effects on black holes.
Other components of the Riemann tensor in this geometry can be calculated in the same way. Equations (5.8) through (5.16) then give the components of the Riemann curvature tensor measured in the orthonormal frame by a static observer near the pole of the horizon, in the case of strong perturbation. This symmetry of the Riemann tensor components implies that the components with two time indices and two indices in directions tangent to the horizon are the same as measured in the orthonormal frame by any observer who is static or moving normal to the horizon.
SUMMARY
This solution reveals many features of the tidal deformation of such a black hole in the static limit. This rotation is shown pictorially in Figure 4. The changes in the components of the transported vector Aa are given by. The form of the Riemann tensor on the rope follows from these properties of parallel transport.
The components of the Einstein tensor follow immediately from the components of the Riemann tensor32; the stress and energy tensor for the rope is then given by the Einstein field equations. Let the true mass-energy density of the rope be f: and let 77 be the tension in the rope. Embedding diagram profiles of tidal horizon bulges defined with respect to cylindrical coordinates in embedding space.
