Geometry of the quasi-hyperbolic Szekeres models

Andrzej Krasiński
N. Copernicus Astronomical Centre,
Polish Academy of Sciences,
Bartycka 18, 00 716 Warszawa, Poland∗

Krzysztof Bolejko
Sydney Institute for Astronomy, School of Physics A28,
The University of Sydney, NSW 2006, Australia†
(Dated:)
arXiv:1208.2604v3 [gr-qc] 19 Dec 2012

Geometric properties of the quasi-hyperbolic Szekeres models are discussed and related to the
quasi-spherical Szekeres models. Typical examples of shapes of various classes of 2-dimensional
coordinate surfaces are shown in graphs; for the hyperbolically symmetric subcase and for the
general quasi-hyperbolic case. An analysis of the mass function M (z) is carried out in parallel
to an analogous analysis for the quasi-spherical models. This leads to the conclusion that M (z)
determines the density of rest mass averaged over the whole space of constant time.

PACS numbers:
Keywords:

I. MOTIVATION of the quasi-hyperbolic case. This last task is carried out

in Secs. XII and XIII. The purpose of this was to iden-
tify the volume in a space of constant t, which could be
Continuing the research started in Refs. [1] and [2], related to the mass M (z). This goal was not achieved
the geometry of the quasi-hyperbolic Szekeres models is as intended, but it was shown that M (z) determines the
investigated. Unlike the quasi-spherical Szekeres mod- density of rest mass averaged over the space of constant
els that have been extensively investigated [3] – [20] and time. Section XIV is a summary of the results.
are rather well understood by now, the quasi-plane and The aim of this paper is to advance the insight into the
quasi-hyperbolic models are still poorly explored. This geometry of this class of spacetimes. This is supposed to
situation has somewhat improved recently: in Ref. [1] a be the next step after the exploratory investigation done
preliminary investigation of the geometry of both these in Ref. [1].
classes was carried out, and in Ref. [2] it was shown that
the physical interpretation of the plane symmetric mod-
els becomes clearer when a torus topology is assumed for II. INTRODUCING THE SZEKERES
the orbits of their symmetry. SOLUTIONS
The present paper is an attempt to understand the ge-
ometry of the quasi-hyperbolic model. In Sec. II, the This section is mostly copied from Ref. [2], mainly in
full set of the β,z 6= 0 Szekeres solutions is presented. order to define the notation.
In Sec. III limitations for the arbitrary functions in the The metric of the Szekeres solutions is
quasi-hyperbolic models are discussed that result from

the spacetime signature and from the evolution equa- ds2 = dt2 − e2α dz 2 − e2β dx2 + dy 2 , (2.1)
tion. It is also shown that a set where the mass func-
tion is zero is allowed to exist. In Sec. IV, it is repeated where α and β are functions of (t, x, y, z) to be deter-
after Ref. [2] that the quasi-hyperbolic Szekeres mani- mined from the Einstein equations with a dust source.
fold is all contained within an apparent horizon, i.e., is The coordinates of (2.1) are comoving, so the velocity
globally trapped. In Sec. VI, the geometry of various field of the dust is uµ = δ µ 0 , and u̇µ = 0.
2-dimensional surfaces in the hyperbolically symmetric There are in fact two families of Szekeres solutions,
subcase is investigated and illustrated with graphs. In depending on whether β,z = 0 or β,z 6= 0. The first family
Sec. VII, it is shown what deformations to the surfaces is a simultaneous generalisation of the Friedmann and
of constant t and ϕ ensue in the general quasi-hyperbolic Kantowski – Sachs [21] models. Since so far it has found
case. In Secs. VIII – XI various properties of the mass no useful application in astrophysical cosmology, we shall
function in the quasi-spherical models are discussed, in not discuss it here (see Ref. [17]), and we shall deal only
order to prepare the ground for an analogous discussion with the second family.
After the Einstein equations are solved, the metric
functions in (2.1) become

∗ Electronic address: akr@camk.edu.pl

eβ = Φ(t, z)eν(z,x,y) ,
† Electronic address: bolejko@physics.usyd.edu.au eα = h(z)Φ(t, z)β,z ≡ h(z) (Φ,z +Φν,z ) , (2.2)
 2


e−ν = A(z) x2 + y 2 + 2B1 (z)x + 2B2 (z)y + C(z), in which the spheres of constant mass are non-concentric.
The functions A(z), B1 (z) and B2 (z) determine how the
where Φ(t, z) is a solution of the equation center of a sphere changes its position in a space t =
const when the radius of the sphere is increased [16].
f(z) 1
2M
Φ,t 2 = −k(z) + + ΛΦ2 , (2.3) Often, it is practical to reparametrise the arbitrary
Φ 3 functions in the Szekeres metric as follows [22]. Even
f(z), A(z), B1 (z), B2 (z) and C(z) are if A = 0 initially, a transformation of the (x, y) coordi-
while h(z), k(z), M nates can restore A 6= 0, so we may assume A 6= 0 with
arbitrary functions obeying no loss of generality [17]. Then let g 6= 0. Writing
def

g(z) = 4 AC − B1 2 − B2 2 = 1/h2 (z) + k(z). (2.4) p
|g| def
(A, B1 , B2 ) = (1, −P, −Q), ε = g/|g|, (2.6)
The mass density ρ is 2S p
  k = −|g| × 2E, M f = |g|3/2 M, Φ = |g|R,
fe3ν ,z
2M
κρc2 = 2β β ; κ = 8πG/c4 . (2.5) we can represent the metric (2.1) as
e (e ) ,z
" 2  2 #
This family of solutions has in general no symmetry, e−ν def def S x−P y−Q
and acquires a 3-dimensional symmetry group with 2- p = E = + + ε , (2.7)
|g| 2 S S
dimensional orbits when A, B1 , B2 and C are constant
2
(then ν,z = 0). The sign of g(z) determines the geometry (R,z −RE,z /E) R2

of the surfaces of constant t and z and the symmetry of ds2 = dt2 − dz 2 − 2 dx2 + dy 2 .
ε + 2E(z) E
the ν,z = 0 subcase. The geometry is spherical, plane or (2.8)
hyperbolic when g > 0, g = 0 or g < 0, respectively.
With A, B1 , B2 and C being functions of z, the surfaces When g = 0, the transition from (2.1) to (2.7) – (2.8) is
z = const within a single space t = const may have dif- A = 1/(2S), B1 = −P/(2S), B2 = −Q/(2S), k = −2E,
ferent geometries, i.e., they can be spheres in one part f = M and Φ = R. Then (2.7) – (2.8) applies with
of the space and surfaces of constant negative curvature M
elsewhere, the curvature being zero at the boundary – see ε = 0, and the resulting model is quasi-plane.
a simple example of this situation in Ref. [1].1 The sign of Equation (2.3), in the variables of (2.8), becomes
k(z) determines the type of evolution when Λ = 0: with
k > 0 the model expands away from an initial singularity 2M (z) 1
R,t 2 = 2E(z) + + ΛR2 . (2.9)
and then recollapses to a final singularity; with k < 0 the R 3
model is ever-expanding or ever-collapsing, depending on
From now on, we will use this representation. The for-
the initial conditions; k = 0 is the intermediate case with
mula for density in these variables is
expansion velocity tending to zero asymptotically.
The Szekeres models are subdivided according to the
2 (M,z −3M E,z /E)
sign of g(z) into quasi-spherical (with g > 0), quasi-plane κρc2 = . (2.10)
(g = 0) and quasi-hyperbolic (g < 0). The geometry of R2 (R,z −RE,z /E)
the last two classes has, until recently, not been investi-
gated and is not really understood; work on their inter- For ρ > 0, (M,z −3M E,z /E) and (R,z −RE,z /E) must
pretation was only begun by Hellaby and Krasiński [1], have the same sign. Note that the sign of both these
and somewhat advanced for the quasi-plane models by expressions may be flipped by the transformation z →
the present author [2]. The sign of g(z) imposes limita- −z, so we may assume that
tions on the sign of k(z). For the signature to be the
physical (+ − −−), the function h2 must be non-negative R,z −RE,z /E > 0 (2.11)
(possibly zero at isolated points, but not in open subsets),
which, via (2.4), means that g(z) − k(z) ≥ 0 everywhere. at least somewhere. In this preliminary investigation we
Thus, with g > 0 all three possibilities for k are allowed; assume that we are in that part of the manifold, where
with g = 0 only the two k ≤ 0 evolutions are admissible (2.11) holds.
(k = 0 only at isolated values of z), and with g < 0, only In (2.7) – (2.8) the arbitrary functions are indepen-
the k < 0 evolution is allowed. dent.2 However, (2.7) – (2.8) creates the illusion that
The quasi-spherical models may be imagined as such the values ε = +1, 0, −1 characterise the whole space-
generalisations of the Lemaı̂ıtre – Tolman (L–T) model time, while in truth all three cases can occur in the same
spacetime.

1 In most of the literature, these models have been considered sep- p 

2 |g| P 2 + Q2 /S + εS /2.


arately, but this was only for purposes of systematic research. Equation (2.4) defines C =
 3

Within each single {t = const, z = const} surface, in two coverings of the same surface, also in the general non-
the case ε = +1, the (x, y) coordinates of (2.1) can be symmetric case. Their spurious isolation is a property of
transformed to the spherical (ϑ, ϕ) coordinates by the stereographic coordinates used in (2.8).
Note that with ε = −1 (2.8) shows that for the signa-
(x − P, y − Q)/S = cot(ϑ/2)(cos ϕ, sin ϕ). (2.12)
ture to be the physical (+ − −−)
This transformation is called a stereographic projection.
For its geometric interpretation and for the correspond- E(z) ≥ 1/2 (3.1)
ing formulae in the ε ≤ 0 cases see Refs. [1] and [17].
The shear tensor for the Szekeres models is [17]
is necessary, with E = 1/2 being possible at isolated
1 values of z, but not on open subsets. We shall also assume
σ α β = Σ diag (0, 2, −1, −1), where
3
R,tz −R,t R,z /R M (z) ≥ 0 (3.2)
Σ= . (2.13)
R,z −RE,z /E

Since rotation and acceleration are zero, the limit σ α β → for all z, since with M < 0 (2.9) would imply R,tt > 0,
0 must be the Friedmann model [17, 23]. In this limit we i.e., decelerated collapse or accelerated expansion, which
have means gravitational repulsion.
In consequence of (3.1), only one class of solutions of
R(t, z) = r(z)S(t), (2.14) (2.9) is possible in the quasi-hyperbolic case:
and then (2.9) implies that
M
R = (cosh η − 1),
E/r2 , M/r3 and tB (z) are all constant. (2.15) 2E
M
However, with P (z), Q(z) and S(z) still being arbitrary, t − tB = (sinh η − η). (3.3)
(2E)3/2
the resulting coordinate representation of the Friedmann
model is very untypical. The more usual coordinates re-
sult when The second of the above determines η as a function of t,
with z being an arbitrary parameter, and then the first
P,z = Q,z = S,z = 0, (2.16) equation determines R(t, z).
Equations (3.1), (3.2) and (2.9) with Λ = 0 imply that
and r(z) is chosen as the z ′ coordinate. (We stress that
R,t 2 > 0 at all z, i.e., there can be no location in the
this is achieved simply by coordinate transformation, but manifold at which R,t = 0. In particular, there exists no
writing it out explicitly is an impossible task.) However,
location at which R = 0 permanently. The function R
(2.14) and (2.16) substituted in (2.8) give the standard attains the value 0 only at t = tB , i.e., at the Big Bang.
representation of the Friedmann model only when ε =
We have, at all points where M > 0,
+1. With ε = 0 and ε = −1, further transformations are
needed to obtain the familiar form [1, 24, 25].
The above is a minimal body of information about the lim R(t, z) = 0, lim R,t (t, z) = ∞, (3.4)
t→tB t→tB
Szekeres models needed to follow the remaining part of
this paper. More extended presentations of physical and
geometrical properties of these models can be found in but R > 0 at all values of z where t > tB . Thus, in
Refs. [1, 2, 16–19]. the quasi-hyperbolic model there exists no analogue of
the origin of the quasi-spherical model or of the center
of symmetry of the spherically symmetric model. (This
III. SPECIFIC PROPERTIES OF THE fact was demonstrated in Ref. [1] by a different method.)
QUASI-HYPERBOLIC MODEL However, a location z = zm0 at which M (zm0 ) = 0
is not prohibited, even though the parameter η in (3.3)
From now on we consider only the case Λ = 0, ε = −1 becomes undetermined when M = 0 6= E. Writing the
and only expanding models. The corresponding conclu- solution of (3.3) as
sions for collapsing models follow immediately.
It was stated in Ref. [1] that the surfaces H2 of con- hp
M
stant t and z in (2.8) in the quasi-hyperbolic case ε = −1 t − tB = 4E 2 R2 /M 2 + 4ER/M (3.5)
consist of two disjoint sheets. This was a conclusion from (2E)3/2
 p i
the fact that with ε = −1 the equation E = 0 has a so- − ln 2ER/M + 1 + 4E 2 R2 /M 2 + 4ER/M
lution for (x, y) at every value of z, and every curve that
goes into the set E = 0 has infinite length. However, it
will be shown in Sec. VI that the two sheets are in fact (where the log-term is the function inverse to cosh) we
 4

see that the limit of this as M → 0 is3 For an expanding model R,t > 0, and only past-
R trapped surfaces can possibly exist, for which k µ ;µ > 0.
lim (t − tB ) = √ , (3.6) Apart from shell crossings the first factor in (4.1) is pos-
z→zm0 2E z=zm0 itive everywhere. Hence, (4.1) implies
and the same result follows from (2.9) with
p M = 0 = Λ. R,t
Note that (3.6) implies R,t (t, zm0 ) = 2E (zm0 ) – an √ + e > 0. (4.2)
2E − 1
expansion rate independent of time. This agrees with
Newtonian intuition – expansion under the influence of For e = +1, and with R,t > 0 that we now consider, this
zero mass should proceed with zero acceleration. At all is fulfilled everywhere. For e = −1 we get
other locations, p where M (z) > 0, thepexpansion rate
is greater than 2E(z), and tends to 2E(z) only at R,t 2 > 2E − 1 ≥ 0 (4.3)
R(t, z) → ∞. However, E(z) at z 6= zm0 may be smaller
than E(zm0 ), so the expansion rate in the neighbourhood (the last inequality from (3.1)). With Λ = 0, (4.3) is also
p guaranteed to hold everywhere, by (2.9), since M ≥ 0
of the M = 0 set may in fact be smaller than 2E(zm0 ).
and R ≥ 0. This means that every surface of constant t
Conversely, at a location where R,t = constant (with
and z in an expanding model is past-trapped at all of its
Λ = 0), (2.9) implies that M = 0 (because R,t 6= 0 in
points. But then, every point of the Szekeres manifold
consequence of E ≥ 1/2, M ≥ 0 and R ≥ 0).
lies within one such surface. This, in turn, means that
The set where M = 0 may or may not exist in a given
every point of the Szekeres manifold is within a past-
quasi-hyperbolic Szekeres spacetime. It should be noted
trapped region. Therefore, the whole quasi-hyperbolic
that, if it exists, it is a 3-dimensional hypersurface in
expanding Szekeres manifold is within a past apparent
spacetime, unlike the origin in the quasi-spherical mod-
horizon.
els. The latter is a 2-dimensional surface in spacetime
The fact of being globally trapped is a serious limi-
and a single point in each space of constant t because in
tation on the possible astrophysical applications of the
the quasi-spherical case M = 0 implies R = 0 via the
quasi-hyperbolic model.
regularity conditions.

V. INTERPRETATION OF THE
IV. NO APPARENT HORIZONS
COORDINATES OF (2.8)

In Ref. [2] it was shown that a collapsing quasi-

In order to understand the geometry of (2.8), we begin
hyperbolic Szekeres manifold is all contained within the
with the hyperbolically symmetric subcase, P,z = Q,z =
future apparent horizon, i.e., that it represents the inte-
S,z = 0. It is most conveniently represented as
rior of a black hole. This is consistent with the fact that
the corresponding vacuum solution (the hyperbolically R,z 2 dz 2

symmetric counterpart of the Schwarzschild solution) has ds2 = dt2 − − R2 dϑ2 + sinh2 ϑdϕ2 . (5.1)
2E − 1
no event horizons and is globally nonstatic [1].
Here, we consider expanding models, and an adden- The two supposedly disjoint sheets of a constant-(t, z)
dum is needed to the result reported above. Consider a surface, in the coordinates of (2.8), are
surface of constant t and z in (2.8), and a family of null
 2  2
geodesics intersecting it orthogonally. As shown in Refs. x−P y−Q
[2] and [16], the expansion scalar for this family is sheet 1 : + > 1,
S S
   2  2
R,z E,z R, x−P y−Q
k µ ;µ = 2 − √ t +e , (4.1) sheet 2 : + < 1. (5.2)
R E 2E − 1 S S
where e = +1 for “outgoing” and e = −1 for “ingoing” The transformation from sheet 1 to (5.1) is
geodesics;4 eq. (4.1) was adapted to ε = −1.
(x, y) = (P, Q) + S coth(ϑ/2)(cos ϕ, sin ϕ), (5.3)

while the transformation from sheet 2 is

3 The result (3.6) shows that the argument used in deriving the
regularity conditions at the center for the Lemaı̂tre – Tolman (x, y) = (P, Q) + S tanh(ϑ/2)(cos ϕ, sin ϕ). (5.4)
model in Ref. [26], and repeated in Sec. 18.4 of Ref. [17], was
incorrect. The value of the parameter η need not be determined This shows that the two sheets are in truth two coordi-
at the center. However, the resulting regularity conditions are
correct because they can be derived in a different way. Once we
nate coverings of the same surface. The direct coordinate
know that R = 0, M = 0 and R ∝ M 1/3 at the center, the transformation between the two sheets is the inversion
behaviour of E at the center follows from eq. (2.9).
4 Since the surfaces of constant t and z are infinite, this labeling is S 2 (x′ − P, y ′ − Q)
(x − P, y − Q) = . (5.5)
purely conventional in this case, but the two families are distinct. (x′ − P )2 + (y ′ − Q)2
 5

The circle separating the two sheets, (x − P )2 + (y − This can be embedded in a flat 3-dimensional
Q)2 = S 2 , on which E = 0, corresponds to ϑ → ±∞ Minkowskian space with the metric
in the coordinates of (5.1). The center of this circle,
(x, y) = (P, Q), which is in sheet 2, is mapped by (5.4) dsM 2 = dT 2 − dX 2 − dY 2 (6.4)
to ϑ = 0. The infinity, (x − P )2 + (y − Q)2 → ∞, which
is in sheet 1, is mapped by (5.3) also to ϑ = 0. These by
relations are illustrated in Fig. 1. s  2
Z
There is no reason to allow negative values of ϑ in (5.1) dt
T = 1+ dR,
because, as both (5.3) and (5.4) show, the point of co- dR
ordinates (−ϑ, ϕ) coincides with the point of coordinates X = R cos ϑ, Y = R sin ϑ. (6.5)
(ϑ, ϕ + π), so the ranges ϑ ∈ [0, +∞) and ϕ ∈ [0, 2π)
cover the whole (ϑ, ϕ) surface. The embedding (6.5) projects a point of coordinates
Curves that go through ϑ = 0 are seen from (5.1) to (R, ϑ) and points of coordinates (R, ϑ + 2πn), where n
have finite length. At ϑ = 0 we have det (gαβ ) = 0, but is any integer, onto the same point of the Minkowskian
the curvature scalars given in Appendix A do not depend space (6.4). However, these points do not coincide in
on ϑ, so ϑ = 0 is only a coordinate singularity. the spacetime (5.1) – the identification of (R, ϑ) with
The geometry of the (x, y) surfaces in (2.8) is the same (R, ϑ + 2π) is not allowed because the transformation
in the hyperbolically symmetric case and in the full non- ϑ → ϑ + 2π is not an isometry in (5.1). Thus, the surface
symmetric case with E,z 6= 0. In the (x, y) coordinates with the metric (6.3) is covered by the mapping (6.5) an
of (2.8), ϑ = 0 corresponds to E = −1 in sheet 2, which infinite number of times. This shows that a hyperboli-
is clearly not a singularity, and to E → ∞ in sheet 1. cally symmetric geometry is a rather exotic and compli-
This seems to be a singularity in (2.8), but the curvature cated entity. We shall see this feature further on, while
scalars are not singular there, as shown in Appendix B. considering other surfaces.
Also the set E = 0 seems to be singular in (2.8), but the Using (2.9) with Λ = 0 we can write
same formulae in Appendix B show that it is nonsingular.
Hence, also in the general case there is no reason to treat  2
dt R
these two sheets as disjoint – they are two coordinate = , (6.6)
dR 2ER + 2M
coverings of the same surface.
and then the integral in (6.5) can be calculated explicitly:
VI. GEOMETRY OF SUBSPACES IN THE FG M √ √ 
HYPERBOLICALLY SYMMETRIC LIMIT T = − p ln 2EF + 2E + 1G +D,
2E E 2E(2E + 1)
(6.7)
A. Hypersurfaces of constant z where D is a constant and
def p def √
A hypersurface z = z1 = constant has the curvature F = (2E + 1)R + 2M , G = 2ER + 2M . (6.8)
tensor
The constant D can be chosen so that T = 0 at R = 0.
3
R0202 = 3 R0303 / sinh2 ϑ = RR,tt , Figure 2 shows the graph of the surface given by the para-


3
R2323 = R2 sinh2 ϑ 1 − R,t 2 , (6.1) metric equations (6.5) as embedded in the 3-dimensional
space with the metric (6.4). It is not exactly a cone, the
where (x0 , x2 , x3 ) = (t, ϑ, ϕ). Consequently, it is flat curves T (R) do have nonzero curvature
when R,t = ±1 and curved in every other case (also when
R,t = constant 6= ±1). However, (2.9) implies that with d2 T M
= , (6.9)
R,t = ±1 we have M = 0 and E = 1/2 (recall: we con- dR2 F G3
sider only the case Λ = 0). Such a subset in spacetime (if
but it is so small everywhere that it would not show up
it exists) is a special case of a neck – see the explanation
in a graph. Note that the vertex angle of this conical
to Fig. 10 later in this section.
surface is everywhere larger than π/4, since dT /dR > 1
The metric of a general hypersurface of constant z is
from (6.5).

Suppose that we made T unique by choosing D as in-
dsz1 2 = dt2 − R2 (t, z1 ) dϑ2 + sinh2 ϑdϕ2 . (6.2)
dicated under (6.8). The vertex of the conical surface in
To gain insight into its geometry, we first consider its sub- Fig. 2 corresponds to the Big Bang. If we want the image
space given by ϕ = ϕ0 = constant. The z1 is a constant in this figure to correspond to the history of the Universe
parameter within R and will be omitted in the formulae from the Big Bang up to now, then the upper edge of
below. The corresponding 2-dimensional metric is the funnel should be at T (Rp ), where Rp corresponds to
 2 the present moment. But this Rp depends on the value
2 2 2 2 dt of z = z1 . Consequently, the height of the funnel will be
dsz1 ,ϕ0 = dt −R (t)dϑ ≡ dR2 −R2 dϑ2 . (6.3)
dR different at different values of z.
 6
(ϑ, ϕ) (x,y)

ε=∞

I
ϑ=∞ ε=0
ϑ=0 II

III
Sheet 2

(x,y) = (P,Q) Sheet 1

FIG. 1: Relations between the (ϑ, ϕ) and (x, y) maps of a constant-(t, z) surface in (2.8) and (5.1). The arrow marked by I corresponds
to the transformation (5.4) that maps the set ϑ = 0 to (x, y) = (P, Q). Arrow II shows that both (5.3) and (5.4) map ϑ → ∞ to the circle
E = 0. Arrow III corresponds to (5.3) that maps ϑ = 0 to E → ∞.

Z s
R
≡ C0 1+ dR,
2EC0 R + 2M C0 2
2

X = C0 R cos ϕ, Y = C0 R sin ϕ. (6.11)

T
Now there is no multiple covering because ϕ is a cyclic
coordinate also in spacetime, and the surface given by
(6.11) looks qualitatively similar to that in Fig. 2, except
that the presence of C0 introduces some flexibility. The
second line of (6.11) shows that the explicit expression
for T is (6.7) multiplied by C0 , with (M, E) replaced by
Y C0 2 (M, E). The radius of a circle of constant R is now
(C0 R). The value of C0 is any in (−∞, +∞). When
X C0 → 0, the surface degenerates to the straight line X =
Y = 0. In order that C0 T in (6.7) allows a well-defined
FIG. 2: The surface of constant z = z1 and constant ϕ = ϕ0 limit C0 → 0, the constant D must have the form
in a spacetime with the metric (5.1). The embedding is in a
Minkowskian 3-space with the metric (6.4). The vertex at R = 0 M ln C0 D1
lies at the Big Bang. The circles represent the surfaces of constant t D= q
+ C0 , (6.12)
2
and z in (5.1). This embedding is not a one-to-one representation, C0 E 2E 2EC0 + 1
the surface in the figure is covered with that of (6.3) an infinite
number of times – see explanation in the text. where D1 is another constant. Again, it may be chosen
so that C0 T = 0 at R = 0.
From (6.11) we find
Now we go back to (6.2) and consider a surface of con- dT
stant ϑ = ϑ0 . Writing C0 = sinh ϑ0 we can write the lim = 1, (6.13)
2-metric as C0 →∞ d(C0 R)
"  2 #
dt so in the limit C0 → ∞ the surface (6.11) becomes ex-
2
dsz1 ,ϑ0 = C0 +2
dR2 −d (C0 R)2 −(C0 R)2 dϕ2 , actly a cone with the vertex angle π/4. However, with
dR C0 → ∞ the whole cone recedes to infinity, as can be
(6.10) seen from (6.7) and (6.11): the vertex of the cone, which
and then the embedding equations are is at R = 0, has the property limC0 →∞ (C0 T )|R=0 = ∞,
s  2 even with the value of D corrected as in (6.12).
Z
dt Note that the image in Fig. 2 will not change qualita-
T = C0 2 + dR
dR tively when we go over from the hyperbolically symmet-
 7

ric subcase (5.1) to the general (nonsymmetric) quasi-

hyperbolic case (2.7) – (2.8). Each hypersurface of con-
stant z in it is axially symmetric, and its metric can be
transformed to the form (6.2). Moreover, any surface of
constant z and y can have its metric transformed to the
form (6.3). The only change with respect to Fig. 2 is R
that the cone-like surfaces, while still being axially sym-
metric, can have their vertex angles different at different
values of z. τ
For completeness, we now consider the special flat hy-
persurface with R,t = 1, M = 0 and E = 1/2 men-
tioned below (6.1). It can be all transformed to the 3-
X
dimensional Minkowski form. The transformation to the
Minkowski coordinates (τ, X, Y ) is

τ = R cosh ϑ, X = R sinh ϑ cos ϕ,

Y = R sinh ϑ sin ϕ. (6.14) FIG. 4: The subspace {z = constant, ϕ = 0} of the spacetime
(5.1) with R = t. The curves of constant X are the hyperbolae
The surfaces of constant R are given by the equation τ 2 − R2 = X 2 (the one with X = 0 is the straight line τ = R).
The lines of constant τ are the circles X 2 + R2 = τ 2 .
τ 2 − X 2 − Y 2 = R2 . (6.15)

These are two-sheeted hyperboloids when R > 0 and a

and their initial points at t = tB are determined by tB (z),
cone when R = 0. They intersect the τ axis horizontally,
so both can vary arbitrarily when we proceed from one
and all tend asymptotically to the cone R = 0 as X 2 +
value of z to another. Fig. 5 shows a 3-d graph of an
Y 2 → ∞ (see Fig. 3).
example of a family of R(t, z) curves corresponding to
different values of z.
τ

M=0

R=0

X t = tB
R
z
FIG. 3: An axial cross-section through the family of hyperboloids
given by (6.15). t

FIG. 5: An exemplary collection of the R(t, z) curves for vari-

The surface ϕ = 0 of (6.14) is depicted in Fig. 4. Note, ous fixed values of z. The bang time curve t = tB (z) must be a
decreasing function of z to avoid shell crossings [1]. The rightmost
however, that Figs. 3 and 4 are graphs of a Lorentzian line is straight, corresponding to M = 0. The other t(R) functions
space mapped into a Euclidean space, so geometrical re- in this figure are given by (3.5) with tB (z) = 2−0.5z 2 , M = z 3 and
lations of (6.2) are not faithfully represented. E = 0.5 + z 3/2 . The values of z change by equal increments from
0 at the rightmost curve to 1 at the leftmost curve. The curves
have M increasing with z (so |R,tt | is increasing as a function of
z) and E increasing with z (so R,t is increasing). Note that all
B. The R(t, z) curves
curves except the M = 0 one hit the t = tB set with R,t → ∞.
The horizontal curves are those of constant R; the values of R on
Where M > 0, we have R,tt < 0 from (2.9) with Λ = 0. them change by equal increments from 0 on the lowest curve to 0.8
on the highest curve.
Consequently, R as a function of t must be concave. The
slopes of the curves R(t, z) at various z depend on E(z),
 8

C. Hypersurfaces of constant t

Formulae for the curvature of the spaces of constant t

in (2.7) – (2.8) are (from Ref. [1], in notation adapted to
that used here):
3 3
R1212 = R1313
R (R,z −RE,z /E) (E,z −2EE,z /E)
= − ,
(2E − 1)E 2
3 2ER2 Z
R2323 = − 4 , (6.16)
E
where the coordinates are labeled as (x1 , x2 , x3 ) =
(z, x, y). Equations (6.16) show that a space of constant t
becomes flat when E = 0, but then it has the Lorentzian
signature (+ − −). Consequently, with the Euclidean
signature, these spaces can never be flat.
Now let us consider the surfaces H2 of constant t = t0
and ϕ = ϕ0 in (5.1). For the beginning we will assume
that R,z > 0 for all values of z in the region under inves-
tigation. Then we can write the metric of H2 as follows: Y
[dR(t0 , z)]2
X
ds2 2 = + R2 dϑ2 . (6.17)
2E − 1
FIG. 6: Upper graph: A surface H2 of constant t and ϕ in the
When E ≡ 1, this is the metric of the Euclidean plane metric (5.1) in the case when E ≡ 1. It is locally isometric to the
in polar coordinates (R, ϑ). With some other constant Euclidean plane, but points having the coordinates (R, ϑ + 2πn) do
values of E, this will be the metric of a cone (see be- not coincide with the point of coordinates (R, ϑ), so the projection
low). With other functional forms of E, it is the metric of H2 covers the Euclidean plane multiply. The multiple covering
is depicted schematically. Lower graph: When E is not constant,
of a rotationally symmetric curved surface on which ϑ
the embedding of a surface of constant t and ϕ in the Euclidean
is the polar angular coordinate. We encounter here the space is locally isometric to a curved surface of revolution, with a
same phenomenon that was described in connection with similar multiple covering, also shown schematically. The surface in
Fig. 2: in each case, a point of coordinates (R, ϑ) and the figure is the paraboloid Z = R2 that results when E(z(R)) =
points of coordinates (R, ϑ + 2πn), where n is any inte- (2R2 + 1)/(4R2 + 1), where z(R) is the inverse function to R(t0 , z).
ger, are projected onto the same point of the plane, cone
or curved surface, respectively. However, as before, these
points do not coincide in the spacetime (5.1). Examples Now it is seen that with 1/2 < E < 1 we can embed this
of embeddings of H2 in the Euclidean E 3 are illustrated surface in the Euclidean space with the metric ds3 2 =
in Fig. 6. dX 2 + dY 2 + dZ 2 by
Note that with E = constant 6= 1 other interesting Z r
geometries come up. If we interpret ϑ as a polar coor- 2(1 − E)
X = R cos ϑ, Y = R sin ϑ, Z=± dR,
dinate, then the ratio of a circumference
√ of a circle R = 2E − 1
constant to its radius is 2π 2E − 1, which means that (6.19)
the surfaces (6.17) are ordinary cones when 1/2 < E < 1, while with E > 1 we can embed it in the Minkowskian
and cone-like surfaces that cannot be embedded in a Eu- space with the metric ds3 2 = −dT 2 + dX 2 + dY 2 by
clidean space when E > 1. With E being a function
Z r
of z, the cones and/or cone-like surfaces are tangent to 2(E − 1)
the (z, ϑ) surfaces at the appropriate values of z. Thus, X = R cos ϑ, Y = R sin ϑ, T =± dR,
2E − 1
with E > 1, the (z, ϑ) surfaces cannot be embedded in (6.20)
a Euclidean space. Whether this surface looks like the For later reference let us note, from (6.19), that
smooth surface of revolution in the lower panel of Fig. 6, dZ/dR → 0 when E → 1 and |dZ/dR| → ∞ when
or like a cone, depends on the behaviour of E(z) in the E → 1/2. This observation will be useful in drawing
neighbourhood of the axis R = 0. But attention: if the graphs and interpreting them.
value of t0 under consideration is such that t0 > tB (z) The surfaces on which t and ϑ are constant look similar
for all z, then the set R = 0 is not contained in the space to the surfaces described above, with two differences:
t = t0 . We will come back to this below. We can discuss 1. The coordinate ϕ changes from 0 to 2π also in the
the embedding when we write the metric (6.17) as spacetime (5.1), so there is no multiple covering of the
2(1 − E) 2 surfaces in the Euclidean space.
ds2 2 = dR + dR2 + R2 dϑ2 . (6.18) 2. The circumference to radius ratio is this time
2E − 1
 9
√
2π sinh ϑ 2E − 1, so the transition √ from cones to cone-
like surfaces occurs at sinh ϑ = 1/ 2E − 1.
Now let us recall what was said in the paragraph con-
taining (3.4): R(t, z) becomes zero only at t = tB . At
any t > tB , R > 0 for all z, even at M = 0 as (3.6)
shows. Thus, the surfaces in Fig. 6 can extend down to
the axis R = 0 only if, at the given instant t = t1 , the
function tB (z) attains the value t1 at some z = z1 : then
t = tB at z = z1 , so R(t1 , z1 ) = 0. This is illustrated in
Fig. 7. If t2 > tB (z) at all z, then R(t2 , z) is nowhere
zero. Let R0 > 0 be the smallest lower bound of R(t2 , z);
then R(t2 , z) ≥ R0 > 0 at all z, and the surface shown
in Fig. 6 has a hole of radius R0 around the axis. Since
R,t > 0, R0 is an increasing function of t, and the radius
of the hole increases with t. This is illustrated in Fig. 8.

t t = t2 FIG. 8: The surface of constant t and ϕ of (5.1), from the bottom

graph in Fig. 6, depicted at two instants t2 > tB (bottom graph)
and t3 > t2 (top graph). The multiple covering of the paraboloid
tB(z)
t = t1 is no longer taken into account. The hole around the axis expands
along with the whole surface.

9. The functions used for this picture are M = 10|z|3 ,

E = 0.6+0.5e−|z|, tB = −103 |z|+100. The time instants
z
z1 are (t1 , . . . , t6 ) = (1, 50, 100, 300, 500, 700).

Z
t1 t2 t3 t4 t5 t6
FIG. 7: The hypersurface t = t1 has a nonempty intersection
with the Big Bang set t = tB (z). The function R(t1 , z) attains the
value 0 at z = z1 , and the surface from Fig. 6 extends down to
the axis R = 0. At t = t2 we have t > tB at all values of z, so
R(t2 , z) is nowhere zero and has a smallest lower bound R0 > 0.
This means that the corresponding surface from Fig. 6 will have a
hole of radius R0 around the axis. Since R,t > 0, the radius of the
hole increases with time, as shown in Fig. 8.
expansion

So far, we have considered R as an independent vari-

able within the space t = t0 . Since it is a function of z,
the parameter along the radial direction in Figs. 6 and
8 is in fact the coordinate z. Now let us recall that R is
also a function of t and that at every z there exists such
a t (t = tB ), at which R = 0. Thus, as we consider the t1 t2 t3 t4 t5 t6

spaces t = t0 at consecutive values of t0 , the surfaces de- Y

picted in those figures get gradually “unglued” from the
Big Bang set (which is represented by the axis of sym- FIG. 9: Evolution of the surface from the lower panel of Fig. 6.
metry R = 0), and expand sideways. At the moment, The figure shows the axial cross-section of the surface at several
time instants, t1 < · · · < t6 . The Big Bang goes off along the
at which t0 begins to obey t0 > tB for all z, the surface Z axis, beginning at the top and at the bottom, and progressing
becomes completely detached from the axis and contin- toward the middle. The instant t3 corresponds to the last moment
ues to expand sideways. This is when the hole mentioned when the surface has no hole. Multiple covering not shown.
above first appears.
Let us also note the double sign in the definition of
Z, (6.19), which was not taken into account in Figs. The nondifferentiable cusp at the plane of symmetry
6 and 8. It means that each of those surfaces has its is a consequence of the assumption R,z > 0: to avoid a
mirror-image attached at the bottom. In summary, the singularity in the metric (5.1), E > 1/2 must hold every-
evolution of those surfaces progresses as shown in Fig. where, and then (6.19) implies |dZ/dR| < ∞ everywhere.
 10

This means that the upper half of the surface cannot go

over smoothly into the lower half.
Let us now consider the case when R,z = 0 at some z =
zn . To prevent a shell crossing at √ zn , E(z
n ) = 1/2 must
also hold, so that limz→zn R,z / 2E − 1 is finite. This
implies that R,z |zn = 0 for all t (i.e., that the extremum
of R is comoving), and then M,z = E,z = 0 at z = zn
from (2.9). This is an analogue of a neck – an entity well
known from studies of the Lemaı̂tre – Tolman model [17].
But, as remarked under (6.20), we have dZ/dR → ±∞
where E → 1/2. The evolution then looks like in Fig.
10. The functions used for drawing it are M = 102 |z|3 ,
E = 0.5 + |z|3/2 , tB = −103 |z|2 + 100, and the time
instants are (t1 , . . . , t6 ) = (1, 50, 100, 200, 300, 400).

Z
t1 t2 t3 t4 t5 t6
FIG. 11: Left: The view from above of the surface from the
lower panel in Fig. 6. The circles are images of the curves of
constant R (and thus of constant z). Right: An example of a
corresponding image in the general quasi-hyperbolic case. Now the
geodesic distance between the circles depends on the position along
the circle.

expansion VII. SPACES OF CONSTANT t IN THE

GENERAL QUASI-HYPERBOLIC CASE

The main difference between the hyperbolically sym-

metric case, where E,z = 0, and the full quasi-hyperbolic
case, where E,z 6= 0, is seen in (2.8). Consider two sur-
faces S1 and S2 such that t = t0 = constant on both,
t1 t2 t3 t4 t5 t6
z = z1 on S1 and z = z2 on S2 . When E,z = 0, the
Y geodesic distance between S1 and S2 along a curve of
constant (x, y) is the same for any (x, y). When E,z 6= 0,
FIG. 10: The analogue of Fig. 9 for the situation when R,z = 0 at this distance depends on (x, y) and varies as the func-
some z = zn (in the middle horizontal plane). Then the upper half
of each constant-(t, ϕ) surface goes over smoothly into the lower
tions P (z), Q(z) and S(z) dictate. Figure 11, left panel,
half. shows an exemplary family of constant-R curves in a sin-
gle surface of constant t and ϕ with E,z = 0. This is a
surface of constant t and (y/x) in the coordinates of (2.8)
– a contour map of the surface from the lower panel in
The minimum of R with respect to z need not exist in Fig. 6. With a general E(x, y, z) the geodesic distance
any space of constant t. (By minimum we mean not only between the constant-R curves will depend on the posi-
a differentiable minimum similar to the one in Fig. 10, tion along each curve, and the whole family would look
but also a cusp at the minimal value like the one in Fig. like in the lower panel of Fig. 11. 5
9.) It will not exist when the function tB (z) has no upper
bound, i.e., when the Big Bang keeps going off forever,
moving to ever new locations. In that case, the surfaces VIII. INTERPRETATION OF THE MASS
shown in the upper half of Fig. 9 will never get detached FUNCTIONS M (z) AND M(z) IN THE
from the Big Bang set, only the vertex of each conical QUASI-SPHERICAL CASE
surface will keep proceeding along the axis. In the special
case tB = constant, the whole surface of constant t and ϕ In the quasi-spherical case, the function M (z) of (2.9),
gets “unglued” from the axis R = 0 at the same instant. by analogy with the Newtonian and the Lemaı̂tre – Tol-
The image would look similar to Fig. 10, but there would man cases, is understood as the active gravitational mass
be no conical surfaces with the vertices progressing along
R = 0. The generators of the surface in the picture are in
general not vertical. They become vertical when R,z = 0
everywhere, i.e., when R = R(t), which can happen only 5 Fig. 11 is in fact deceiving. The curves shown there as circles
in the β,z = 0 family of Szekeres solutions that we do not are images of infinite curves, as explained under (6.5), and each
discuss here. image is covered an infinite number of times.
 11

inside the sphere of coordinate radius z. In fact, it is puz- In the quasi-spherical case we are able to calculate the
zling why it depends only on z when the mass density integral with respect to x and y over the whole (x, y)
(2.10) so prominently depends also on t, x and y. Some- surface because its surface area is finite. Such a calcula-
what miraculously, as shown √ below, the denominator in tion cannot be repeated for the ε ≤ 0 cases because both
(2.10)
R pis canceled by the −g11 term inside the inte- integrals analogous to (8.2) are infinite. Let us then con-
gral ρ |g3 |d3 x that determines the mass in a sphere.6 sider what happens with M and M when we calculate
The term containing E,z in the numerator gives a zero the integrals in (8.2) over a part of the sphere.
contribution to the integral. This is consistent with the
fact, known from electrodynamics, that the total charge
of a dipole is zero (see Refs. [11, 17, 20] for the split- IX. THE MASS IN THE SPHERICALLY
ting of (2.10) into the monopole and the dipole part in SYMMETRIC CASE
the quasi-spherical case). It is also consistent with the
result of Bonnor [8, 9] that the Szekeres solution can be In nearly all the papers concerning the LT model and
matched to the Schwarzschild solution. the quasi-spherical Szekeres model it was assumed that
The considerations of this and the next three sections each space of constant t has its center of symmetry (in the
are intended to prepare the ground for an analogous in- LT case) or origin (in the quasi-spherical Szekeres case),
vestigation in the quasi-hyperbolic case further on. The where M = 0 and R = 0 at all times. We will assume the
questions we seek to answer are: Can M still be inter- same here, but this is an assumption. It is possible that
preted as mass, and where does the mass M (z) reside the center of symmetry is not within the spacetime in the
when a surface of constant z has infinite surface area? LT case, and the corresponding quasi-spherical Szekeres
Let us calculate, in the quasi-spherical case, the generalization will then have no “origin”.8
amount of rest mass within the sphere of coordinate ra- For the beginning we will consider the spherically sym-
dius z at coordinate time t, assuming that z = z0 is metric (Lemaı̂tre–Tolman) subcase, in which P,z = Q,z =
the center, where the sphere has zero geometrical
R p radius S,z = 0, so P = Q = 0 can be achieved by coordinate
(see Ref. [16]). This amount equals M = V ρ |g3 |d3 x, transformations. Then E,z ≡ 0 in (8.2). Now suppose
where V is the volume of the sphere and ρ is the mass that we calculate the integral in the first of (8.2) over
density given by (2.10). Substituting for ρ and g3 we get a circular patch C of the sphere (circular in order that
Z +∞ Z +∞ Z z no (x, y) dependence appears from the boundary shape).
1 The boundary of C is an intersection of the sphere with
M = dx dy du a cone whose vertex is at the center of the sphere. Let
4π −∞ −∞ z0
  the vertex angle θ of the cone be π/n. This translates
M,u (u) 3M E,u
√ − √ , (8.1) to the radius of C in the original (x, y) coordinates being
1 + 2EE 2 1 + 2EE 3 def
u0 = S tan(π/2n) = Sβ. Then we have in place of the
where u is the running value of z under the integral. Note first of (8.2)
that E is the only quantity that depends on x and y, it is Z
an explicitly given function, and so the integration over 1 u0 2
d2 xy 2 = 4π 2 ≡ 2π [1 − cos(π/n)]
x and y can be carried out: C E S + u0 2
Z +∞ Z +∞ Z +∞ Z +∞ β2
1 E,z ≡ 4π . (9.1)
dx dy 2 = 4π, dx dy 3 = 0. 1 + β2
−∞ −∞ E −∞ −∞ E
(8.2) This tends to 4π when n → 1 (u0 → ∞). Note that
(The first of these just confirms that this is the surface the final result does not depend on S – this happened
area of a unit sphere.) Using this in (8.1) we get becausepwe have chosen the coordinate radius in each
Z z circle, x2 + y 2 = u0 , to be a fixed multiple of S.
M,u In the spherically symmetric case now considered, we
M= √ (u)du, (8.3)
z0 1 + 2E choose the same cone to define the circles of integration
in (9.1) in all surfaces of constant z. Instead of (8.3) we
which is√the same relation as in the LT model,7 and shows get for the amount of rest mass within the cone, MC :
that 1/ 1 + 2E is the relativistic energy defect/excess Z Z z
function (when 2E < 0 and 2E > 0 respectively). 1 M,u
MC = d2 xy du √
4π C z0 1 + 2EE 2

6 g3 is the determinant of the metric of the 3-space t = constant

in (2.1). 8 The authors are aware of just one paper, in which LT models
7 For the quasi-spherical case also other integral relations are sim- without a center of symmetry were considered. These are the
ilar to the ones in the L–T model, which is a consequence of the “in one ear and out the other” and the “string of beads” models
fact that the dipole contribution vanishes after averaging over of Hellaby [28], described also in Ref. [17]. In both of them,
x-y surface [27]. M 6= 0 throughout the space.
 12
Z z hp i
β2 M, x′ = C 2 − 1 sin(2λ + D) /U1 , (10.5)
= √ u (u)du. (9.2)
1 + β2 z0 1 + 2E
The term of (8.1) that contained E,u disappeared here in where C and D are arbitrary constants of integration.
consequence of the assumed spherical symmetry, but it These have to obey the initial conditions (x′ , y ′ )|λ=0 =
will not disappear when we go over to the general case, (x, y). After solving for C and D this leads to
and its contribution will have to be interpreted. 

x′ = 2x cos(2λ) + 1 − x2 − y 2 sin(2λ) /U3
def

U3 = 1 + x2 + y 2 + 1 − x2 − y 2 cos(2λ)
X. SYMMETRY TRANSFORMATIONS OF A
SPHERE IN THE COORDINATES OF (2.12) −2x sin(2λ),
′
y = 2y/U3 . (10.6)
In the spherically symmetric case rotations around a
point are symmetries of the space. In this case, if we It is instructive to calculate the effect of the transfor-
rotate the whole cone around its vertex to any other po- mation (10.6) in the (ϑ, ϕ) coordinates of (10.2). Let
sition, eq. (9.2) will not change. Each z = const circle us then take a point of coordinates (x, y) = (x, 0), i.e.,
will then be rotated by the same angles to its new po- (ϑ, ϕ) = (ϑ, 0), and let us apply (10.6) to it. After a little
sition, and the result of such a rotation will be a cone trigonometry we get
isometric to the original one.
1 − cos ϑ cos(2λ) − sin ϑ sin(2λ)
However, in the general, nonsymmetric case the tan(ϑ′ /2) =
spheres z = constant are not concentric. Suppose we sin ϑ cos(2λ) − cos ϑ sin(2λ)
build, in the general case, a surface composed of circles, ≡ tan(ϑ/2 − λ) =⇒ ϑ′ = ϑ − 2λ, (10.7)
each circle taken from a different sphere. If we rotate
each sphere by the same angles, whatever surface existed i.e., (10.6) is equivalent to rotating the sphere around the
initially, will be deformed into a shape non-isometric to (ϑ, ϕ) = (π/2, π/2) axis by the angle (−2λ).
the original one. We want to calculate the effect of such a It can now be verified that the quantity:
transformation, and for this purpose we need the formu- 2 2
lae for the O(3) rotations in the (x, y) coordinates. We def 1 + x2 + y 2 1 + x′ + y ′
I(x, y) = ≡ (10.8)
calculate them now. 2y 2y ′
The generators of spherical symmetry, in the ordinary
spherical coordinates, are [17] is an invariant of the transformation (10.6). The set I =
C is the circle x2 + (y − C)2 = C 2 − 1.
∂ ∂ ∂ An arbitrary circle of radius A and center at x = y = 0,
J1 = , J2 = sin ϕ + cos ϕ cot ϑ ,
∂ϕ ∂ϑ ∂ϕ x2 + y 2 = A2 , is transformed by (10.6) into the circle
∂ ∂
J3 = cos ϕ − sin ϕ cot ϑ (10.1) "
#2
∂ϑ ∂ϕ ′ 1 + A2 sin(2λ) 2
x − + y′
We transform these to the (x, y) coordinates of (2.12), (1 + A2 ) cos(2λ) + 1 − A2
for the beginning with P = Q = 0, S = 1, by 4A2
= . (10.9)
x = cot(ϑ/2) cos(ϕ), y = cot(ϑ/2) sin(ϕ) (10.2) [(1 + A2 ) cos(2λ) + 1 − A2 ]2
and obtain The coordinate radius of the (x′ , y ′ ) circle is different
∂ ∂ from the original radius A except when λ = 0, which is
J1 = x −y ,
∂y ∂x the identity transformation. But the coordinate radius
∂
∂ is not an invariantly defined quantity. An invariant mea-
J2 = 2xy + 1 − x2 + y 2 , sure of the circle, its surface area, does not change under
∂x ∂y

∂ the transformation (10.6), and neither does the invariant
∂
J3 = 1 + x2 − y 2 + 2xy . (10.3) distance between any two points, as we show below.
∂x ∂y The Jacobian of the transformation (10.6) is
The transformations generated by J1 are rotations in the
(x, y) plane. To find the transformations generated by J3 ∂(x′ , y ′ ) 4
= . (10.10)
we have to solve (see Ref. [17] for explanations): ∂(x, y) U3 2
dx′ 2 2 dy ′ Together with (10.8) and (10.6) this shows that the sur-
= 1 + x′ − y ′ , = 2x′ y ′ , (10.4)
dλ dλ face element under the integral (9.2), 4dxdy/E 2 , does not
where λ is the parameter of the group generated by J3 . change in form after the transformation. (This must be
The general solution of this is so, since (10.6) is just a change of variables that does not
p change the value of the integral.) Thus, (10.6) preserves
def
y ′ = 1/U1 , U1 = C + C 2 − 1 cos(2λ + D), the area of any circle, as is appropriate for a symmetry.
 13

The invariant distance between the points (x, y) = The transformations generated by J2 result from those
(0, 0) and (x, y) = (A, 0) (i.e., the invariant radius of for J3 by interchanging x′ with y ′ and x with y; then
the original circle referred to in (10.9)) is, from (2.7) – all the conclusions about invariant properties follow also
(2.8) with ε = +1, P = Q = 0, S = 1: for these transformations, and, in consequence, for any
Z composition of (10.6) with them.
A
dx
= 2 arctan A. (10.11)
0 1 + x2

The image under (10.6) of any point (x, 0) is (x1 (x), 0), XI. THE MASS IN THE GENERAL
where, from (10.6): QUASI-SPHERICAL CASE


2x cos(2λ) + 1 − x2 sin(2λ) Now let us consider the general case, and integrals
x1 (x) = . analogous to (8.2), where the (x, y) integration extends
1 + x2 + (1 − x2 ) cos(2λ) − 2x sin(2λ)
(10.12) only over a circular subset of each sphere, the radius of
Thus, the image of (0, 0) is (x0 , 0), where each circle being a fixed multiple of S. In the general
case, each sphere has a geometrically preferred center at
sin(2λ) (x, y) = (P (z), Q(z)), and, for the beginning, we choose
x0 = . (10.13)
1 + cos(2λ) the center of the disc of integration C at that point. As
before, the radius of each circle will be a fixed multiple
p def
The invariant distance between the images of (0, 0) and of S: x2 + y 2 = u0 = Sβ. This means, this time
of (A, 0) is then the volume of integration will not be a simple cone, but
Z x1 (A)
a ‘wiggly cone’ – the circles in the different z = const
dx1 x (A) surfaces will have their centers not on a straight line or-
2
= 2 arctan(x1 )|x10 ≡ 2 arctan A,
x1 (0) 1 + x1 thogonal to their planes, but on the curve given by the
(10.14) parametric equations x = P (z), y = Q(z) that is not
by employing the identity arctan α − arctan β = orthogonal to the planes of the circles. The result (9.1)
arctan [(α − β)/(1 + αβ)]. This certifies that the invari- still holds within each z = const surface, but the ana-
ant distance between the center of a circle and a point on logue of the second integral in (8.1), calculated over the
the circle is the same as the invariant distance between interior of the ‘wiggly cone’ here, will no longer be zero.
their images (but the image of the center is no longer the Instead, introducing in each z = const surface the polar
center of the image-circle, compare (10.9) and (10.13)). coordinates x = P + u cos ϕ, y = Q + u sin ϕ, we get

Z Z 2π Z u0

E,z (S,z /2) 1 − u2 /S 2 − S1 (P,z u cos ϕ + Q,z u sin ϕ)
d2 xy 3 = dϕ udu 3
C E 0 0 (S 3 /8) (1 + u2 /S 2 )
Z u0

uS,z S 2 − u2 u0 2 S,z β2
≡ 8πSS,z du = 4πSS, z = 4π . (11.1)
0 (S 2 + u2 )3 (S 2 + u0 2 )2 S (1 + β 2 )2

In agreement with (8.2) this goes to zero when u0 → ∞. Consequently, from (8.1), (9.2) and (11.1), the total mass
within the wiggly cone is
Z z Z z
β2 M,u β2 M S,u
M= 2
√ (u)du − 3 2
√ (u)du. (11.2)
1 + β z0 1 + 2E (1 + β ) z0 S 1 + 2E
2

It contains a contribution from S,z that is decreasing with interpretation – but it becomes proportional to the mass
increasing β, i.e., the greater volume we take, the less within the cone in the spherically symmetric limit.
significant the contribution from S,z gets. It will vanish As an example, consider the axially symmetric family
when the integrals extend over the whole infinite range of spheres whose axial cross-section is shown in Fig. 12.
of x and y (in the limit β → ∞). This can be interpreted The circles are given by the equation
so that in a wiggly cone the dipole components of mass  p 2
distribution do contribute to M – but less and less as the x − b2 + u 2 + y 2 = u 2 , (11.3)
volume of the cone increases. Thus, with such choice of
the integration volume M does not have an immediate where b is a constant that determines the center of the
limiting circle of zero radius, while u is the radius of the
 14

circles. (The same family of spheres was used in Ref. [1] We derived the transformation (10.6) in the coordi-
to construct Szekeres coordinates for a flat space.) Figure nates in which the constants (P, Q, S) were set to (0, 0, 1)
13, left graph, shows the initial wiggly cone constructed by coordinate transformations. With general values of
for these spheres – the one referred to in (11.1) and (11.2). (P, Q, S), the result would be

   
x′ − P x−P (x − P )2 + (y − Q)2
= 2 cos(2λ) + 1 − sin(2λ) /U4
S S S2
y−Q
y′ − Q = 2 ,
U4
 
def (x − P )2 + (y − Q)2 (x − P )2 + (y − Q)2 x−P
U4 = 1+ + 1 − cos(2λ) − 2 sin(2λ). (11.4)
S2 S2 S

Now let z = z1 correspond to the base of the wig-

gly cone, where the values of the arbitrary functions are 10
P1 = P (z1 ), Q1 = Q(z1 ) and S1 = S(z1 ). Apply the
transformation (11.4) with (P, Q, S) = (P1 , Q1 , S1 ) to
each sphere intersecting the wiggly cone. In the base
z = z1 this will be a symmetry, in other spheres this will
5
not be a symmetry. One should in principle calculate
the effect of this transformation in other spheres on the
integrands in (9.2) and (11.1) to see what happens. But
then, (11.4) is merely a change of variables under the in-
tegral that does not change the value of the calculated 0
integral. Thus, eq. (11.2) applies independently of where
we choose the base of the wiggly cone, and its position
that we chose initially (each circle had its center in the
geometrically distinguished center of the (x, y) surface)
is only necessary to fix the relation between circles cor- -5
responding to different values of z.
The result of the transformation (11.4) applied to the
wiggly cone of the left graph in Fig. 13 is shown in the
same figure in the right graph.
-10
-10 -5 0 5 10

XII. THE SYMMETRY TRANSFORMATIONS

FIG. 12: An axial cross-section through the family of spheres
IN THE HYPERBOLICALLY SYMMETRIC CASE
given by (11.3).

Before proceeding to the case of interest, let us con-

sider the hyperbolically symmetric subcase, in which
P,z = Q,z = S,z = 0, and so P = Q = 0 by a transfor-
mation of x and y. The symmetry group of the resulting where (x0 , x1 , x2 , x3 ) = (T, R, ϑ, ϕ). To find the symme-
metric is a subgroup of that for the corresponding vac- tries explicitly, we have to transform (12.2) to the coor-
uum solution [1]: dinates of (2.8).
  We are interested in the transformations within an
2 2m 1 (x, y) surface, and those are generated by k2 , k3 and k4 .
ds = − 1 + dT 2 + dR2
R 1 + 2m/R The transformation from the (ϑ, ϕ) coordinates of (12.1)


− R2 dϑ2 + sinh2 ϑdϕ2 . (12.1) to sheet 2 of the (x, y) coordinates of (2.8) is (5.4). The
inverse formulae are
The full set of Killing vectors for this metric is
k1 α = δ0 α , k4 α = δ3 α , ϕ = arctan(y/x),
k2 = cos ϕδ2 − coth ϑ sin ϕδ3 α ,
α α
1 + x2 + y 2
cosh ϑ = ,
k3 α = sin ϕδ2 α + coth ϑ cos ϕδ3 α , (12.2) 1 − (x2 + y 2 )
 15

FIG. 13: Left graph: a wiggly cone constructed for the family of spheres shown in Fig. 12. The vertex angle for this cone is π/32.
Right graph: the result of the transformation (11.4) applied to the cone from the left graph. The initial cone is shown in thin lines. The
rest mass contained in the new wiggly cone is the same as it was in the initial cone. Dotted lines show the image of the initial cone that
would result if each circle of intersection of the original cone with a sphere were rotated by the same angle around the center of its sphere.

p


def
2 x2 + y 2 W3 = 1 − x2 + y 2 + 1 + x2 + y 2 cosh(2λ)
sinh ϑ = . (12.3)
1 − (x2 + y 2 ) +2x sinh(2λ),
′
We now transform the Killing fields (12.2) by (12.3). In y = 2y/W3 . (12.10)
the (x, y) coordinates we get for the generators
The equations corresponding to (12.7) for the genera-
∂ ∂ tor J3 result from (12.7) simply by interchanging x′ with
J1 = x −y , (12.4)
∂y ∂x y ′ . The corresponding initial condition then results by

 ∂ ∂ interchanging x with y. Thus, from (12.10) we can read
J2 = 1 + y − x2
2
− 2xy , (12.5) off the transformation generated by J3 ; it is
∂x ∂y
∂
∂ x′ = 2x/W4 ,
J3 = −2xy + 1 + x2 − y 2 . (12.6)
∂x ∂y def


W4 = 1 − x2 + y 2 + 1 + x2 + y 2 cosh(2λ)
The J1 generates rotations in the (x, y) surface. To find +2y sinh(2λ), (12.11)
the transformations generated by J2 , we have to solve ′
 2 2


y = (1/W4 ) 2y cosh(2λ) + 1 + x + y sinh(2λ) .
the set
dx′ 2 2 dy ′ Note that the (x′ , y ′ ) given by (12.10) also obey the
= 1 + y ′ − x′ , = −2x′ y ′ , (12.7) third equation in (12.9), so the quantity
dλ dλ
with the initial condition that at λ = 0 we have (x′ , y ′ ) = def 1

I1 = = x2 + y 2 − 1 (12.12)
(x, y). The general solution of this set is 2y
def
p
y ′ = 1/W1 , W1 = − C + C 2 + 1 cosh(2λ + D), is an invariant of the transformations (12.10). The cor-
p responding invariant for (12.11) is
x′ = (1/W1 ) C 2 + 1 sinh(2λ + D), (12.8)
def 1

where C and D are arbitrary constants to be determined I2 = = x2 + y 2 − 1 (12.13)
2x
from x′ (0) = x, y ′ (0) = y. They are

These facts are helpful in calculations, and so is the fol-
sinh D = 2x/W2 , cosh D = x2 + y 2 + 1 /W2 , lowing identity that follows from (12.12)
q
def 2

W2 = (x2 + y 2 − 1) + 4y 2 , 2 2 2
x′ + y ′ − 1 ≡ x2 + y 2 − 1 . (12.14)
1
W3
C = x2 + y 2 − 1 . (12.9)
2y The fact that I1 is an invariant of (12.10) means that
So, finally the transformation (12.10) maps the set I1 = C = con-

 stant
√ into itself for every C. This set is a circle of radius
x′ = (1/W3 ) 2x cosh(2λ) + 1 + x2 + y 2 sinh(2λ) , C 2 + 1 and center in the point (x, y) = (0, C).
 16

The inverse transformation to (12.10) results from does not matter where the center of the circle is because
(12.10) by the substitution λ → (−λ). This can be veri- we can freely move the circle around the (x, y) surface by
fied using the above identities. symmetry transformations without changing its area.
These same identities can be used to verify that the
transformation (12.10) maps the circle x2 + y 2 = A2 into
the following circle in the (x′ , y ′ ) coordinates
"
#2 XIII. THE MASS FUNCTION IN THE
2
1 − A sinh(2λ) 2 QUASI-HYPERBOLIC CASE
x′ − + y′
1 + A2 + (1 − A2 ) cosh(2λ)
4A2 In the quasi-hyperbolic case, the (x, y) surfaces are in-
= 2. (12.15) finite, so they do not surround any finite volume. Thus,
[1 + A2 + (1 − A2 ) cosh(2λ)]
unlike in the quasi-spherical case, we should not expect
The radius of this new circle equals the original radius, the value of the mass function M (z) to correspond to
A, only in two cases: λ = 0, which is an identity trans- a mass contained in a well-defined volume. We should
formation, or A = 1. In both cases, also the center of rather observe the analogy to a solid cylinder of finite ra-
the circle remains unchanged. The radius meant here is dius and infinite length in Newton’s theory, in which the
a coordinate radius that has no invariant meaning. The mass density depends only on the radial coordinate. Its
meaningful quantity is the geometric radius, which is the exterior gravitational potential is determined by a func-
invariant distance between the center of the circle and a tion that has the dimension of mass, whose value is pro-
point on its circumference. It can be verified that the portional to mass contained in a unit of length of the
invariant distance between any pair of points is the same cylinder.
as the invariant distance between their images. We now proceed by the same plan as we did in the
Below we present some remarks about the transforma- quasi-spherical case. We can freely move a circle of inte-
tion (12.10). The same statements apply to (12.11). gration around each (x, y) surface. We first consider the
The Jacobian of the transformation (12.10) is hyperbolically symmetric case and we erect over a cho-
sen circle a solid column in the z direction that contains
∂(x′ , y ′ ) 4 a certain amount of rest mass. Then we go over to the
= , (12.16) quasi-hyperbolic nonsymmetric space and erect a wiggly
∂(x, y) W3 2
column that contains the same amount of rest mass.
which, together
R with (12.14), shows that the integrand in We will integrate over the interior of a circle in sheet 2
the integral d2 xy/E 2 is form-invariant under this trans- whose radius u0 is, for the beginning, unknown. We only
formation. In this integral, (12.10) is an ordinary change know that the radius must be smaller than S, so that
of variables, and the area of integration in the (x′ , y ′ ) the integration region does not intersect the circle where
variables will be an image of the original area under the E = 0 (since, we recall, E = 0 is the image of infinity, and
same transformation. This means that also the value of the integral over a region that includes E = 0 would be
this integral is an invariant of (12.10). Thus, if we choose infinite). Thus, instead of (9.1) and (11.1) we have this
the region of integration to be a circle around a point, it time

Z Z 2π Z u0
1 4uS 2 u0 2
d2 xy = dϕ 2 du = 4π , (13.1)
U E2 0 0 (u2 − S 2 ) S 2 − u0 2
Z Z 2π Z u0
E,z −8uS 2
4πSS,z u0 2
d2 xy 3 = dϕ 3 u cos ϕP,z +u sin ϕQ,z +u2 S,z /(2S) + SS,z /2 du = 2.
U E 0 0 (u2 − S2) (u0 2 − S 2 )

Z
The first integral in (13.1) will be independent of S when E,z 4π(S,z /S)β 2
d2 xy = 2 . (13.3)
u0 is a fixed multiple of S: U E3 (1 − β 2 )

u0 = βS, β < 1. (13.2) Instead of (11.2) we now have:

Z z
β2 M,u (u)
Then M = 2
du √
1 − β z0 2E − 1
Z 2 Z z
1 β2 β 3M S,u
d2 xy = 4π , − 2 du √ . (13.4)
U E2 1 − β2 (1 − β 2 ) z0 S 2E − 1
 17
Z z
The meaning of the limits of integration has to be 4πβ 2 R3 S,u
− 2
√ du. (13.7)
explained here. In the quasi-spherical case, and in the (1 − β2) z0 S 2E − 1
spherically symmetric LT subcase, one usually assumes
that each space of constant t has its center of symmetry, For the ratio M/V we now calculate two consecutive lim-
where M = 0 = R. As explained at the beginning of Sec. its: first β → 1, to cover the whole of each z = constant
IX, this is an additional assumption – the center of sym- hyperboloid, and then z → z∞ , where z∞ is the value
metry need not belong to the spacetime. But the center of z at which R → ∞, to cover the whole t = constant
of symmetry, or origin, is the natural reference point at space. After taking the first limit, we get
which the mass function has zero value. In the quasi- Z z   Z z 
hyperbolic case now considered, a similar role is played M 3M S,u R3 S,u
lim = √ du 4π √ du .
by the set M = 0, so we will assume that this set exists. β→1 V z0 S 2E − 1 z0 S 2E − 1
Again, as mentioned earlier, this set is a 2-dimensional (13.8)
surface in each space of constant t, and not a single point. Since in general (apart from special forms of the functions
With this assumption made, z0 in (13.4) will be the involved) both the numerator and the denominator above
value of z at√ which M (z0 ) = 0. In √ addition, we assume become infinite when z → z∞ , we apply the de l’Hôpital
that (M,u / 2E − 1) and [S,u /(S 2E − 1)] are finite at rule and obtain
z = z0 and in a neighbourhood of z0 , so that both inte-
grals in (13.4) tend to zero as z → z0 . M 3M
lim lim = lim . (13.9)
These equations are very similar to the corresponding z→z∞ β→1 V z→z∞ 4πR3
ones in the quasi-spherical case, (11.1) – (11.2), so one
is tempted to interpret M , by analogy with that case, as The l.h.s. of the above is the global average of rest mass
a quantity proportional to the active gravitational mass M per volume. The r.h.s. looks very much like the same
contained within a solid (wiggly) tube with circular sec- type of global average for the active gravitational mass
tions, the radius of a circular section at z = z1 being M , except that it is taken with respect to a flat 3-space.
βS(z1 ). The base of the tube is in the surface (t = t0 = A very similar result follows when we take z → z0
const, z), its coordinate height is (z − z0 ), and its top instead of z → z∞ in (13.8). Then both the numerator
is at (t, z) = (t0 , z0 ). There is no problem with this in- and denominator tend to zero and we obtain
terpretation in the hyperbolically symmetric case, where M 3M
S,u = 0 and the second integral in (13.4) vanishes. lim lim = lim 3
. (13.10)
z→z0 β→1 V z→z 0 4πR
However, there is a significant difficulty when going
over to the quasi-hyperbolic case. In the quasi-spherical On the l.h.s here we have a global average of M/V over
case we were free to take the limit of the integral extend- the (x, y) surface taken at the value of z at which M = 0.
ing over the whole sphere, which was β → ∞. Here, the Equation (13.10) results also when the integrals in
integral over the whole hyperboloid would correspond to (13.4) – (13.8) are taken over the interval [z1 , z2 ], where
β → 1, and in this limit all the integrals (13.1) – (13.4) z0 < z1 < z2 < z∞ , and then the limit z2 → z1 is taken.
become infinite. Worse still, the contribution from the All the calculations in this section were done in sheet
dipole – the second integral in (13.4) – tends to infinity 2 of the (x, y) map. The corresponding results for sheet
faster than the monopole component (the first integral), 1 are obtained by taking all integrals with respect to u
while in the spherical case increasing the area of integra- over the interval [u0 , ∞) (with u0 > S now) instead of
tion caused decreasing the influence of the dipole. [0, u0 ], substituting 1/u0 for u0 in (13.1), and 1/β for β
We can do another operation on (13.4) that will shed (with the new β obeying β > 1) in (13.3), (13.4) and
some light on the meaning of M . The volume of the (13.7). Equations (13.8) – (13.10) do not change.
region containing the rest mass M is The meaning of the limits on the r.h. sides of (13.9)
Z z Z p and (13.10) requires further investigation. Note that they
V= du d2 xy |g3 (t, u, x, y)|, (13.5) arise from the dipole contributions to mass and volume.
z0 U

where g3 is the determinant of the metric of the 3-

dimensional subspace t = constant of (2.1), thus XIV. SUMMARY
Z z Z
R2 (R,u −RE,u /E)
V= du d2 xy √ . (13.6) The aim of this paper was to clarify the geometrical
z0 U 2E − 1E 2 structure of the quasi-hyperbolic Szekeres models [3, 4]
This has the same structure as the integral representing given by (2.7) – (2.9), and of the associated hyperbol-
M. Since R and E do not depend on x and y, the inte- ically symmetric dust model given by (5.1). The main
gration with respect to (x, y) can be carried out, and by results achieved are the following:
(13.1) – (13.3) we get
Z z 1. Although there exists no origin, where R would be
4πβ 2 R2 R,u zero permanently, a set where M = 0 can exist. At
V = √ du
2
1 − β z0 2E − 1 this location, R,t is constant (section III).
 18

2. The whole spacetime is both future- and past- glob- the quasi-spherical models follows only in the (hy-
ally trapped (section IV). perbolically) symmetric case. In the general case,
terms arising from the dipole component of the
3. The geometrical interpretation of the (x, y) coor- mass distribution cause difficulties that were not
dinates in a constant-(t, z) surface was clarified in fully resolved. It has only been demonstrated that
Sec. V. Contrary to an earlier claim [1], this sur- the average value of M/V over the whole space
face consists of just one sheet, doubly covered by t = t0 is determined by the average value of M/V0 ,
the (x, y) map. where V0 is the flat space limit of V.
4. The geometries of the following surfaces for the This problem requires further investigation, but it
metric (5.1) were shown in illustrations, all of them is hoped that the results achieved here will be of
in Sec. VI: use for that purpose.
(a) z = constant, ϕ = 0 for (5.1) in Figs. 2 – 4.
(b) The collection of R(t, z) curves in Fig. 5.
Appendix A: The curvature tensor for the metric
(c) t = constant, ϕ = 0 for (5.1) in Figs. 6 and 8. (5.1)
It turned out that the surfaces listed under (a) are
locally isometric to ordinary surfaces of revolution The formulae given below are the tetrad components
in the Euclidean space (in special cases to a plane of the curvature tensor for the metric (5.1). The tetrad
and a cone) when E ≥ 1, but cover the latter an is the orthonormal one given by
infinite number of times. When 1/2 < E < 1,
they cannot be embedded in a Euclidean space even R,z
e0 = dt, e1 = √ dz,
locally. The values E ≤ 1/2 are prohibited by the 2E − 1
spacetime signature. e2 = Rdϑ, e3 = R sinh ϑdϕ, (A1)
The time evolution of the surfaces under 3(c) was
illustrated in Figs. 9 and 10. with the labeling of coordinates (t, r, ϑ, ϕ) =
(x0 , x1 , x2 , x3 ). The components given below are
5. For the general metric (2.7) – (2.8), the geometry of scalars, so any scalar polynomial in curvature compo-
the surfaces t = constant, ϕ = 0 was investigated in nents will be fully determined by them. Since they do
Sec. VII and shown in Fig. 11. The other surfaces not depend on ϑ, they have no singularity caused by any
listed above are the same as in the hyperbolically special value of ϑ. 
symmetric case (5.1).
6. In Secs. VIII – XI a detailed analysis was car- 2M M,z
R0101 = 3
− 2 , (A2)
ried out of the relation between the mass function R R F
M (z) and the sum of rest masses in a volume M(z) 1 M
R0202 = R0303 = R2323 = − 3 , (A3)
in the quasi-spherical Szekeres model. The func- 2 R
tion M (z) represents the active gravitational mass M M,z
within a sphere of coordinate radius z, while M(z) R1212 = R1313 = 3 − 2 . (A4)
R R F
is the sum of rest masses of particles contained in
the same volume: The formulae in both appendices were calculated by
Z p the algebraic program Ortocartan [29, 30].
M= |g3 |ρd3 x, (14.1)
V
Appendix B: The curvature tensor for the metric
where V is any volume in a space of constant t = t0 , (2.8)
g3 is the determinant of the metric in that space
and ρ is the mass density at t = t0 . The relation The formulae given below are the tetrad components
(8.3) follows in the limit when V is the volume of the of the curvature tensor for the metric (2.8) with ε = −1.
whole space t = t0 . The calculations in Secs. VIII The tetrad is the orthonormal one given by
– XI demonstrated how to calculate (14.1) within
various relevant volumes. F
e0 = dt, e1 = √ dz,
2E − 1
7. In Secs. XII and XIII calculations analogous to R R
those from Secs. VIII – XI were carried out for e2 = dx, e3 = dy, (B1)
E E
the quasi-hyperbolic Szekeres models. The aim
was to interpret the function M (z) in this case by with the labeling of coordinates (t, z, x, y) =
identifying the volume in which the active gravi- (x0 , x1 , x2 , x3 ), where E is given by (2.7) and
tational mass is contained. Integrals analogous to
def
(14.1) can be calculated, but the full analogy with F = R,z −RE,z /E. (B2)
 19

The components given below are scalars, so any scalar − coth(ϑ/2) (P,z cos ϕ + Q,z sin ϕ) . (B7)
polynomial in curvature components will be fully deter-
mined by them. The only quantity in (B3) – (B5) that depends on ϑ is
E,z /(EF ). Using (B6) – (B7) we easily find
2M 3M E,z M,z
R0101 = + 2 − 2 , (B3)
R3 R EF R F
1 M E,z 1
R0202 = R0303 = R2323 = − 3 , (B4) lim = , (B8)
2 R ϑ→∞ EF R
M 3M E,z M,z E,z 1
R1212 = R1313 = 3 + 2 − 2 . (B5) lim = . (B9)
R R EF R F ϑ→0 EF R,z +RS,z /S
Note that these reproduce (A2) – (A4) when E,z = 0.
We wish to find out whether the sets E = 0 and E → ∞ The loci where these can become infinite do not depend
are curvature singularities. For this purpose it is useful on ϑ. Hence, ϑ → ∞ and ϑ = 0 are not curvature singu-
to introduce the coordinates (ϑ, ϕ) by (5.3). Since the larities, and neither are E = 0 or E → ∞. 
quantities (B3) – (B5) are scalars, we only need to sub-
stitute (5.3) in them. The two suspected sets become Acknowledgements The research for this paper was
ϑ → ∞ and ϑ = 0, respectively. After the transforma- inspired by a collaboration with Charles Hellaby, initi-
tion we have ated in 2006 at the Department of Mathematics and Ap-
S plied Mathematics in Cape Town. It was supported by
E= , (B6) the Polish Ministry of Education and Science grant no N
2 sinh2 (ϑ/2) N202 104 838.
S,z  
E,z = 2 1 − 2 cosh2 (ϑ/2)
2 sinh (ϑ/2)

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