Slicing Black Hole Spacetimes
A general framework is developed to investigate the properties of useful choices of stationary spacelike slicings of stationary spacetimes whose congruences of timelike orthogonal trajectories are interpreted as the world lines of an associated family of observers, the kinematical properties of which in turn may be used to geometrically characterize the original slicings. On the other hand properties of the slicings themselves can directly characterize their utility motivated instead by other considerations like the initial value and evolution problems in the 3-plus-1 approach to general relativity. An attempt is made to categorize the various slicing conditions or “time gauges” used in the literature for the most familiar stationary spacetimes: black holes and their flat spacetime limit.
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Keywords: spacelike slicings; black holes
A century after the birth of general relativity, we now take for granted the existence of various stationary spacelike slicings of stationary spacetimes which have certain special geometrical properties useful in studying the astrophysical consequences of say, black hole spacetimes. Many of these slicings arise from geometrical properties of their irrotational congruences of orthogonal timelike trajectories, interpreted as the world lines of an associated family of observers which may be either geodesic or accelerated. Other slicings are instead characterized by the intrinsic or extrinsic geometry of the slicing itself. We here survey the various categories of such useful slicings for nonrotating and rotating black hole spacetimes, but starting with the limiting flat Minkowski spacetime which allows the greatest variety of examples of special slicings.
Among the observer-defined slicings of black hole spacetimes is the Boyer-Lindquist time coordinate slicing associated with the usual stationary accelerated observers referred to equivalently as fiducial or locally non-rotating or zero angular momentum observers, abbreviated as FIDOs, LNOs or ZAMOs. Geodesic observer families instead characterize the rain, drip and hail coordinate systems \@cite?,? which include the Painlevé-Gullstrand coordinates \@cite?,? and their generalizations \@cite?. For nonrotating black holes the latter are also characterized by the intrinsic curvature properties of their associated slicing, whose induced geometry is flat. Smarr and York \@cite? pioneered linking the preservation of kinematical properties of the slicing to the choice of time lines in an evolving spacetime, while constant mean curvature slicings were seen as privileged from the point of view of the initial value problem even earlier \@cite?,?,?. Special spherically symmetric slicings of the nonrotating Schwarzschild black hole spacetime were first considered by Estabrook et al \@cite?,?. Recent work has investigated the geometry associated with analogue black holes \@cite?,? and shown that neither intrinsically flat nor conformally flat slicings of the Kerr spacetime exist \@cite?,?,?,?.
On the other hand, the analysis of the Cauchy problem of general relativity has also led to the introduction of spacetime slicings useful in simplifying the evolution equations, like harmonic slicing \@cite? (and closely related time gauges \@cite?), defined by requiring the time coordinate of the spacetime associated with the slicing to be a harmonic function. Furthermore, the numerical relativity study of multi-black hole dynamics \@cite? takes advantage of the use of “hyperbolic slicings,” requiring spatial compactification techniques at infinity \@cite?,?,?, as well as horizon penetrating coordinates like Painlevé-Gullstrand coordinates. We refer to these as “analytic slicings,” belonging to the Cauchy problem literature, in contrast with those of a “geometrical” nature.
Motivated primarily by a desire to better understand the underlying geometrical structure of these spacetimes, we systematically review, develop and discuss special slicings of black hole spacetimes together with their flat Minkowski limit. We use geometric units with . Greek indices run from to and refer to spacetime quantities, while Latin indices from to and refer to spatial quantities.
2 Spacetime, general coordinates and spacelike -surfaces
Let be a generic set of coordinates adapted to a slicing of spacetime by spacelike hypersurfaces of constant values of the time coordinate and write the metric as
The convenient Wheeler lapse-shift notation \@cite? re-expresses the metric in the form
defining the lapse function and shift vector field which satisfy
while the 1-forms are orthogonal to the unit timelike 1-form aaa The symbol denotes here the completely covariant form of a tensor . associated with the unit normal vector field to the slicing
Here we use the equivalent shortened notations for the the spatial coordinate frame . The induced Riemannian -metric on the time slices is simply
with spacetime volume element . Note that is the dual frame to the frame reflecting the orthogonal decomposition of each tangent space adapted to the slicing and its normal direction.
For a stationary spacetime, one can choose these coordinates so that the stationary symmetry corresponds to translation in the time coordinate , with associated Killing vector field . To transform to another stationary slicing, without loss of generality one can consider restricted choices of the new spacelike time coordinate of the form
which retain the time lines of the original coordinate system if the spatial coordinates are not changed. One is still free to choose new time lines by changing the spatial coordinates as well, but unless the time lines are associated with a Killing vector field, the metric will become explicitly time-dependent.
On a generic slice of the new slicing, the 1-form
vanishes identically, i.e.,
If we retain the spatial coordinates and only introduce this new time coordinate,
then one must distinguish the spatial coordinate frame vector fields tangent to the old and new time coordinate hypersurfaces
Re-expressing the spacetime metric then leads to
where (the induced metric on ) and and (the new lapse and shift) are given by
Similarly to Eq. (3), we then have
One can also evaluate the contravariant metric
while the spacetime and spatial metric determinants satisfy
Note that is the dual frame to the frame adapted to the orthogonal slicing decomposition of the tangent space and that on one has . Any “spatial tensor” has only Latin indexed components allowed to be nonzero in this frame.
The unit timelike 1-form normal to the slicing is given by
with associated unit timelike normal vector field
In turn can be expressed in terms of as
where is the relative velocity of with respect to and the associated gamma factor, explicitly
as follows from Eq. (16) after re-expressing and in terms on and . A straightforward calculation shows that the new lapse function and shift vector field for the same time coordinate lines are given by
The above decomposition gives a more transparent kinematical meaning to the various quantities, as we will show below.
Starting from the spacetime unit volume 4-form , one can associate with any timelike unit vector field , whether or , a spatial volume 3-form which can be used to define the cross product and the curl operator in the local rest space of , as well as a spatial duality operation for antisymmetric spatial tensor fields. In a spatial frame adapted to that subspace, for spatial vector fields and in that subspace, one has
where the spatial covariant derivative of any tensor field (including ) is obtained by projecting all indices of the spacetime covariant derivative of that tensor into the local rest space using the associated projection tensor whose fully covariant form is the spatial metric.
We conclude this section by introducing the relevant tensor quantities needed to evaluate both the intrinsic and extrinsic curvature of a typical slice as well as provide a geometrical characterization of the kinematical properties of its normal congruence .
Intrinsic curvature of .
This is obtained evaluating the Riemann tensor components of the -metric induced on , i.e.,
Note that in the three-dimensional case the Riemann tensor is completely determined by the associated Ricci tensor.
Conformal flatness of .
The Cotton tensor associated with the induced 3-metric (22) is given by
where all operations including the covariant derivative refer to the 3-metric. A vanishing Cotton tensor characterizes the conformal flatness of the spatial metric. Taking the spatial dual of the two covariant indices gives the equivalent Cotton-York tensor
which is symmetric because of the twice contracted Bianchi identities of the second kind
and the symmetric tensor curl “Scurl” operation \@cite?,?,? is defined by
The spatial metric is conformally flat if the spatial Ricci tensor has vanishing Scurl.
Extrinsic curvature of .
This is obtained evaluating the Lie derivative of the spacetime metric along the unit normal to and projecting the result orthogonally to and raising an index to make a mixed tensor, with an extra numerical factor
Its trace is the mean curvature of the slice. The constant mean curvature (CMC) time gauge is a slicing with constant mean curvature on each slice, though it may vary from slice to slice. A maximal slicing instead has vanishing mean curvature on every slice. When the tensor itself vanishes, the slicing is called totally geodesic, or extrinsically flat. An invariant characterization of the extrinsic curvature can be obtained by studying its eigenvalues, namely those of the matrix of components in an adapted frame since it is a spatial tensor. The three eigenvalues in turn can be expressed in terms of the three scalar trace invariants of the powers of the extrinsic curvature , and .
Kinematical fields associated with : acceleration, vorticity, expansion and shear.
These are obtained by decomposing the covariant derivative of into its irreducible parts under a change of frame
namely the acceleration of the congruence, its vanishing vorticity tensor since the congruence is hypersurface-forming and the expansion tensor . The scalar expansion of the congruence is just the sign-reversed mean curvature of the slicing, while the tracefree part of the expansion tensor is the shear tensor.
Finally, there exist slicings motivated more by analytic considerations than geometrical ones. One example is represented by the so-called harmonic slicings. In this case the foliation is characterized by a harmonic condition
A similar condition imposed on all spacetime coordinates specifies the so-called de Donder gauge choice of coordinates. In the literature, there are also variations of this condition (“” slicing, etc.), which will not be explored here (see, e.g., Ref. \@cite?). Eq. (30) is equivalent to
which becomes in a coordinate system with time coordinate
Recalling Eq. (13) the previous equation can be also written as
which in turn can be expressed as an evolution equation for the new lapse function .
3 Minkowski spacetime
As a simple example of the problem of finding special slices in a given spacetime, let us consider the flat Minkowski spacetime geometry in an inertial time slicing, with its line element written in spherical coordinates to compare later with a black hole spacetime in Boyer-Lindquist coordinates
The hypersurfaces are both intrinsically and extrinsically flat. Because of their simplicity, the Minkowski slicing examples are useful in better understanding the corresponding general relativistic situations considered below.
Consider a new spherical symmetric slicing by a time function , retaining the spatial coordinates (and hence the orthogonal time lines). The induced metric on a typical slice is
with the new lapse and shift functions
One can define the new spatial frame
with dual frame
In this frame the extrinsic curvature tensor has components
The intrinsic curvature is characterized in three dimensions equivalently either by the Riemann or Ricci tensor, with nonvanishing components respectively
and the curvature scalar is
Finally, the Cotton-York tensor (24) is identically zero for this slicing, independent of the choice of , ensuring the conformal flatness of any spherical slicing.
The new time coordinate hypersurfaces have unit normal
with radial relative velocity (which must satisfy ) and associated Lorentz factor . The associated observers moving orthogonal to the slicing follow radial trajectories which are ingoing for and outgoing for in comparison with the usual static observers () following the original time lines. The simple transformation flips the radial motion of the new observers.
Let us consider some explicit examples. In order to deal with dimensionless quantities, we introduce a positive scaling constant with the dimensions of a length, say . For simplicity the solutions of the various conditions on the slicing function below will be given modulo an overall sign subject to the initial condition for the sake of easy comparison.
Let , with a dimensionless constant. Eq. (7) then gives
where is a dimensionless integration constant, implying that
The function is then given by
These slices are characterized by the vanishing of the trace of the extrinsic curvature tensor Eq. (7) which leads to
modulo a constant chosen to be 1 with no loss of generality. This equation can be integrated in terms of elliptic functions, i.e.,
where and are the incomplete and the complete elliptic integrals of first kind, respectively (such that ). Note that in the limit of we have .
Vanishing Ricci scalar:
Requiring that the spatial Ricci scalar Eq. (10) vanish leads to a parabola of revolution about an inertial time axis orthogonal to its symmetry axis
The induced metric is
with extrinsic curvature
and intrinsic curvature specified by
Smarr and York \@cite? introduced a special constant mean curvature slicing of Minkowski spacetime by translating a single sheet of a spacelike pseudosphere (hyperboloid) of a given fixed radius (and therefore fixed intrinsic and extrinsic curvature) along the time lines of an inertial coordinate system, in contrast with the geodesically parallel family of all such pseudospheres of varying radii centered on one spacetime point. In the metric (1) in spherical coordinates in an inertial coordinate system, consider the time slices where is given implicitly by
so that , where is the radius of the pseudosphere and we have chosen only one of the two possible signs, i.e., the plus sign. One finds then the corresponding radial velocity given by
The normal congruence to the slicing defines the associated Smarr-York observers. The intrinsic metric of the slices evaluates to
while the extrinsic curvature tensor
and Ricci curvature tensor
are both (spacetime) constant multiples of the spatial metric reflecting the constant intrinsic and extrinsic curvature conditions.
The lapse function and shift vector field are given by
respectively. The spatial metric then takes the form
or transforming the radial coordinate by ,
more familiar from cosmology. Note that when the slice tends to be both intrinsically and extrinsically flat. This shift vector field satisfies the minimal-distortion equation of Smarr and York.
The behavior of the slicing function as well as of the associated spatial velocity as a function of is shown in Fig. 1 in all cases discussed above.
4 Schwarzschild spacetime
Consider now the Schwarzschild spacetime representing a nonrotating black hole, whose line element written in standard coordinates is given by
Introduce an orthonormal frame adapted to the static observers following the time lines, i.e.,
with dual frame
The hypersurfaces form a slicing which is extrinsically flat (as the orthogonal hypersurfaces to the static Killing vector congruence ), but not intrinsically flat. The induced metric on these hypersufaces
has nonzero Ricci curvature
with vanishing Ricci scalar .
Let us look for general slicings of the Schwarzschild geometry which are compatible with the Killing symmetries of the spacetime, i.e., spherical slices . Their timelike unit normal is
with relative velocity (see Eq. (19))
and associated Lorentz factor . As in the Minkowski spacetime case, reversing the sign of interchanges the ingoing and outgoing radially moving observers associated with the new slicing.
The induced metric on is
and the new lapse function and shift vector field are
A (nonorthonormal) basis on is given by
with dual frame
The acceleration is given by
and the extrinsic curvature of the slices then turns out to be
where , and the traces of its powers then have the following values
The intrinsic curvature is described by the following nonvanishing components of the spatial Riemann tensor
or equivalently by the spatial Ricci tensor, whose nonzero components are
Finally, the spatial Ricci scalar is
showing that corresponds to vanishing spatial scalar curvature, but to vanishing spatial curvature.
The Cotton-York tensor (24) vanishes identically, so that spherical slicings are automatically conformally flat.
Geodesic slicings correspond to a spatially constant lapse function , as from Eq. (12). In this case
and the only surviving component of the spatial Riemann tensor is
whereas the spatial Ricci tensor is fully specified by
with . The value of the parameter determines the sign of the intrinsic curvature of the slices, corresponding to either vanishing (), positive () or negative () curvature. Note that interpreting as the 4-velocity field of a family of (geodesic) test particles, the parameter coincides with the (conserved, Killing) energy per unit mass of the particles
Therefore, also represents the case of particles with vanishing radial velocity at spatial infinity; can be associated with particles starting at infinity with nonzero (inward) velocity; finally, corresponds to particles starting moving at a finite radial position. In the literature, adapted coordinates to these situations (in the special case of a Schwarzschild spacetime) are referred to as “hail” (), “rain” () and “drip” () coordinates (see, e.g., Ref. \@cite? and references therein).
Let us consider some explicit examples. We will let denote in each case the dimensionless integration constant appearing in the solution of the ordinary differential equation for the radial velocity , all quantities being rescaled by the characteristic length scale of the background curvature (i.e., the mass ). For simplicity the solutions for the slicing function will be given up to an overall sign and the constant of integration will be chosen so that if this function is finite at , or if it approaches a finite limit at , in order to compare the time slices from a common reference point at the horizon or at spatial infinity relative to the usual slicing. Typical behaviors of the slicing functions as well as of the corresponding linear velocities are shown in Fig. 2.
Let . Then using Eq. (14) with one finds
formally equivalent to Eq. (12) in the case of a flat spacetime, where is a dimensionless integration constant, implying that
The function is then given by
Let . Then the radial velocity approaches for large , independent of the value of . In contrast the behavior at the horizon depends on the sign of the quantity . In fact, in the limit one has , i.e., if , while if .
For the above relations simplify to
so that and
which can be expressed in terms of elliptic functions. Therefore the radial velocity behaves as for large and for , independent of the value of .
Vanishing Ricci curvature:
Setting in Eq. (17) gives , leading to
Therefore the radial velocity behaves as for large and for , independent of the value of .
The harmonic condition (30) gives
so that and
is a common choice in the literature. Therefore the radial velocity behaves as for large and for (for that choice of the integration constant).
Let us investigate now spherical slicings characterized by intrinsic flatness, i.e., by the vanishing of the spatial Riemann tensor. Examining Eq. (15) it is sufficient to impose the condition
which obviously yields a flat induced metric apparent from Eq. (8) since , while similarly the acceleration vanishes as well from Eq. (12), leading to a geodesically parallel slicing. Because of the above choice of the lapse function, the slicing is associated with the “rain” coordinates introduced above, whose adapted observers are known as Painlevé-Gullstrand (PG) observers \@cite?,?. They are geodesic and irrotational, and admit orthogonal time hypersurfaces which are intrinsically flat. Their radial velocity relative to the static observers is , having selected the minus sign corresponding to a free fall from rest at infinity (ingoing PG, for example). Integrating Eq. (7) then gives
However, PG slicings are not extrinsically flat since
PG observers are in a sense “complementary” to the static observers, which instead have extrinsically flat orthogonal time hypersurfaces.
5 Splitting of the curvature tensor by a generic family of observers
For rotating black holes, the situation is more complicated, so we examine first the splitting of the curvature tensor using the test observer congruence associated with a given stationary spacelike slicing. Let be the unit timelike -velocity vector tangent to the test observer world lines orthogonal to the slicing. The orthogonal decomposition of the covariant derivative of this irrotational congruence is
where is the acceleration and
is the expansion tensor with trace , the expansion scalar usually referred to as just the expansion of the congruence, while the tracefree part of the expansion tensor describes the shear of the congruence.
Details of the decomposition of the curvature tensor and corresponding Einstein equations may be found in many places, including Ref. \@cite?. Using an observer-adapted frame, the latter lead to the following set of equations in vacuum, setting the Ricci tensor to zero