Mon. Not. R. Astron. Soc. 000, 1–6 (0000)

Printed 8 November 2012

(MN LATEX style file v2.2)

On de-Sitter Geometry in Crater Statistics
G.W.
Gibbons1 and M. C. Werner2?
1

Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Wilberforce Road, Cambridge, CB3 0WA, United Kingdom

2 Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo, 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan,

Department of Mathematics, Duke University, Durham, NC 27708, USA

This draft 8 November 2012

ABSTRACTThe cumulative size-frequency distributions of impact craters on planetary bodies in the solar
system appear to approximate a universal inverse square power-law for small crater radii. In
this article, we show how this distribution can be understood easily in terms of geometricalstatistics, using a de-Sitter geometry of the configuration space of circles on the Euclidean
plane and on the unit sphere. The effect of crater overlap is also considered.
Key words: planets and satellites: general–cosmology:miscellaneous–methods:statistical

1

INTRODUCTION

When observations by the spacecraft Dawn enabled the first detailed survey of impact craters on the astroid 4 Vesta recently,
it was noted that the cumulative size-frequency distribution canbe approximated by a power-law with (double logarithmic) slope−1.9±0.1 for small craters, close to the geometric saturation slopeof −2 (Marchi et al. (2012)). This approximate inverse square
power-law has also been observed on other planetary bodies in thesolar system, for example on Mars (see, e.g., Werner & Tanaka(2011) and references therein). Since size-frequency distributionsof impact craters are a powerful tool in planetology to date surfaces
and reconstruct planetary evolution (see, e.g., Michael & Neukum(2010), and the review by Hartmann & Neukum (2001) for Mars),
it is important to understand their origin. The first detailed statistical model of impact cratering was given by Marcus (1964) inthe context of lunar exploration. Assuming that craters are circular
but may overlap, are Poisson-distributed over the lunar surface subject to a time-dependent production function with probability density p(r, t) for a crater radius to lie in the interval (r, r + dr), heshowed in a subsequent work that for any p(r, t) ∝ r−γ−1 , γ > 2,
the cumulative frequency of craters larger than radius r is indeed
approximately proportional to r−2 in a steady-state at late times(Marcus (1966)). Therefore, when this state is reached, the distribution of craters is the same for a class of underlying production
functions. Moreover, if a population of craters of radius r reaches
geometric saturation, then its frequency is exactly proportional tor−2 because of circle packing on the surface (cf. Gault (1970)).
It may also be recalled that power-law distributions are self-similar(for craters see, e.g., Takayasu (1990), pp. 18f and 34ff), but thisfeature alone does not explain the fairly consistent power of −2
itself.
The universality of this property, then, seems to lend itself naturally to a geometrical explanation. In this article, we present a new?

E-mail:marcus.werner@ipmu.jp

c 0000 RAS

geometrical approach to study crater distribution functions, whichis inspired by applications of measures to cosmological models of
inflation (e.g., Gibbons & Turok (2008)). While any size-frequency
distribution of craters is, in the absence of other resurfacing, the result of the production function which depends on properties of the
impactors creating the craters, it is possible that the size-frequency
distribution evolves to a steady-state at least partially independentof its creation history, as shown by Marcus’ result. In such a case,one might expect a priori a uniform distribution over the configuration space of craters. In order to investigate this point, we shallstart in this paper with the assumption of a uniform distribution ofcraters over their configuration space and show that this implies infact an inverse square power-law for the size-frequency distribution of small craters. Of course, by contraposition, this means thatany deviation from this relationship implies a non-uniformity of thecrater distribution over their configuration space, which in turn may
have physically interesting implications for the underlying class of
production functions.
The structure of this paper is as follows. In Sec. 2, we consider the 3-dimensional configuration spaces of circles on the Euclidean plane and on the unit sphere, and it turns out that thesecan be given a de-Sitter geometry. By interpreting the configuration space of craters as that of circles and computing its canonicalvolume measure, we show in Sec. 3 how an inverse square powerlaw for the cumulative size-frequency distribution of small craters
emerges. This model also shows that the global crater distribution
is no longer self-similar once crater size becomes comparable to thesize of the planetary body. Initially, we shall assume that craters aresparse with negligible overlap in the sense that their area filling factor F  1, corresponding to a small√ fraction of geometric saturation which has filling factor π/(2 3) ≈ 0.907 (cf. Gault (1970)).
This appears to be a reasonable assumption, as the observed distribution of craters is typically below 10% of geometric saturation(e.g. for the Moon, see again Gault (1970) who compares it to a
”Mare Exemplum” created in laboratory experiments). The effectof crater overlap on the distribution function is then considered as

2

G.W. Gibbons and M. C. Wernerwell. The conclusions are in Sec. 4, where an interpretation of thisresult and further applications are suggested.

2

DE-SITTER CIRCLE GEOMETRY

Consider n-dimensional unoriented spheres Sn
r in the n + 1dimensional Euclidean space En+1 . Any such sphere is uniquely
defined by its centre at a ∈ En+1 and its radius r > 0. Since
these are independent of each other, the configuration space of such
n+1
= {Sn
} is n + 2-dimensional. The spespheres Mn+2r ⊂ EEcial case n = 1 corresponds, of course, to circles on the Euclideancanplane. We will now see how the configuration space Mn+2
E
be interpreted as a n + 2-dimensional de-Sitter spacetime, and thisspacetime of constant curvature can be described in the usual way
as a quadric hypersurface in n + 3-dimensional Minkowski spacetime E1,n+2 with metric ηab = diag(−1, 1, ..., 1).
Firstly, one may regard the Euclidean space En+1 as the intersection of the null cone through the origin in an n + 3-dimensional
Minkowski spacetime E1,n+2 with a null hypersurface, as follows.
Given an arbitrary x ∈ En+1 , define
 2

x +1
x2 − 1
Xa =, x,∈ E1,n+2 ,22where the square of vectors in En+1 is with respect to the corresponding Euclidean metric, and let
N a = (1, 0, ..., 0, 1) ∈ E1,n+2 .
Then each x is identified with a point which is both on the null cone
through the origin,ηab X a X b = 0,

ηab Y a Y b = 1,

(3)aand use (3) to express Y in terms of the variables defining thecorresponding sphere,
1 a2 + 1
0
−r ,
(4)
Y
= −
2
r
a
Y = − ,(5)r
 2
1 a −1−r .
(6)
Y n+2 = −2
r

Mn+2
= {Y a ∈ E1,n+2 : ηab Y a Y b = 1} = dS n+2 ,E
as required. The metric on this de-Sitter spacetime is induced from
the Minkowski metric of E1,n+2 ,ds2bηab X N = −1,
which is null since ηab N a N b = 0.
Secondly, consider a spacelike vector Y a=(Y 0 , Y, Y n+2 ) ∈ E1,n+2 , so that ηab Y a Y b > 0, which isperpendicular to X a ,(1)

Writing Y · N = ηab Y a N b = −Y 0 + Y n+2 for short, one canrecast condition (1) as2Yηab Y a Y b
x+
=
.
(2)
Y ·N
(Y · N )2
So for any k 6= 0, a given kY a uniquely defines an n-sphere in
En+1 with centre and radius==−(dY 0 )2 + dY2 + (dY n+2 )2
1(−dr2 + da2 ),
r2

(7)

E
as the metric of the de-Sitter spacetime, one
using (4–6). With gab
can rewrite (7) asE
ds2 = gabdy a dy b ,y a = (r, a) ∈ dS n+2 ,to obtain
dVEn+2=q

E− det gab
drda1 . . . dan+1

1
drda1 . . . dan+1(8)
rn+2
as the volume measure of the de-Sitter and hence the configuration
space. Finally, as a special case of Eq. (8) one finds=

dVE3 =

1
drda1 da2 ,r3(9)

the volume measure of the 3-dimensional configuration space M3Eof circles on the Euclidean plane.

2.2

and on the hypersurface

ηab X a Y b = 0.

ηab Y a Y b,(Y · N )2

Hence, this configuration space of spheres in Euclidean space isseen to be an n + 2-dimensional de-Sitter spacetime dS n+2 ,On the planear=respectively. Because of the freedom to rescale Y a we may set,
without loss of generality,

We begin by showing that the geometry of the configuration spacesof circles on the Euclidean plane and on the unit sphere in Euclidean space can be regarded as de-Sitter spacetimes. The firstcase will be seen in Sec. 2.1 as special case of the configuration
spaces of n-spheres in n + 1-dimensional Euclidean spaces. The
discussion of this more general case will also help to establish thesecond case in Sec. 2.2. Then, in both cases, one can easily derive
the canonical volume measures for these configuration spaces. The
connection with more general sphere geometry is outlined in the
Appendix.

2.1

s

Y
,
a=−
Y ·N

On the sphere

In view of the application to the distribution of craters on a planetary surface, we shall proceed by extending the previous case to theconfiguration space of circles S1ρ on the 2-dimensional unit sphereS2 centered at the origin of E3 . Each circle is fully characterized
by its angular radius ρ ∈ (0, π/2) and by its centre n on the unit
2-sphere in Euclidean 3-space, so that n2 = 1 and the two spherical coordinates of the centre can be computed in the usual way.
The corresponding configuration space M3S is, as before, clearly
3-dimensional and can also be considered a de-Sitter spacetime.
To see this easily, one can utilize a trick to discuss this problem
in terms of the previous one. Each circle on our unit 2-sphere S2
may be regarded as the intersection of another 2-sphere S2r of radiusr centred at a ∈ E3 with S2 . As noted in the previous section,
the configuration space M4E of 2-spheres in Euclidean 3-space is
a 4-dimensional de-Sitter spacetime defined by the quadric (3) in
5-dimensional Minkowski spacetime E1,4 with coordinates (4–6).
c 0000 RAS, MNRAS 000, 1–6

3

On de-Sitter Geometry in Crater Statistics
Now, for a given circle of angular radius ρ, there is a family of
spheres S2r whose intersection with the unit 2-sphere produces thiscircle. Hence, their radius r depends on the choice of a which, bysymmetry, must be parallel to n. As shown in Fig. 1 (left panel),
we can choose a such that r = tan ρ and a = n sec ρ. Then from
(4–6), one finds Y 4 = 0 andY0=− cot ρ,

(10)

Y

=

−n csc ρ.

(11)With this choice, we have coordinates Z a = (Y 0 , Y) ∈ E1,3 on
the 4-dimensional Minkowski subspace Y 4 = 0 which determine
uniquely the centre and angular radius of a circle on the unit 2sphere and, since ηab Z a Z b = −(Y 0 )2 + Y2 = 1, define a 3dimensional de-Sitter spacetime as the corresponding configuration
space,

ΡSparse craters

as promised. The metric on this de-Sitter spacetime is again induced by the ambient Minkowski space E1,3 ,3.1.1

On the plane

dn2 = dθ2 + sin2 θdφ2 ,z a = (ρ, θ, φ) ∈ dS 3 ,with metricSgab= diag − csc2 ρ, csc2 ρ, csc2 ρ sin2 θ .

(12)

3

Thinking of a sphere in E of radius R0 with a circle of radius
R measured on the spherical surface so that its angular radius isρ = R/R0 , then this metric (12) clearly reduces to the one in Eq.
(7) in the limit of small circles, ρ  1, as expected. The volume
measure corresponding to metric (12) becomesq
S
dVS3 =− det gabdρdθdφ=csc3 ρ sin θdρdθdφ = csc3 ρdρdω,Ρ¢Suppose that an infinitesimal area dA in the Euclidean plane contains the centres of dn craters of radii in the interval [r, r + dr].
Then the corresponding distribution function fE may be written
dn = fE drdA so that, given our assumptions and the result fromSec. 2.1, dn ∝ dVE3 and hence
CE
(14)
r3
from Eq. (9), where CE is a constant. If this constant is sufficiently
small, the assumption of sparse craters will hold. In order to determine this and estimate its value, let nE be the total number ofcraters with radius rmin 6 r 6 rmax wholly within a disk of radius R > rmax . Then the area filling factor of these craters is
Z rmaxrmax
FE =πr2 fE (r)dr = πCE ln
rmin
rmin

fE =

and thus express the line element of the configuration space asS
ds2 = gabdz a dz b ,aFigure 1. Left: A circle S1ρ of angular radius ρ at n on the unit 2-sphere S2(shown as solid circle) in Euclidean 3-space is represented as the intersection of a 2-sphere S2r of radius r = tan ρ at a = n sec ρ (shown as dashedcircle) with the unit 2-sphere. Right: The centre of a crater of angular radius ρ wholly overlapped by a crater of angular radius ρ0 may lie within theshaded area of solid angle ω(ρ0 − ρ).

3.1

from (10–11), subject to n2 = 1. One can write the line element on
the unit sphere in terms of the usual spherical coordinates (θ, φ),Ω HΡ¢ - ΡLn

M3S = {Z a ∈ E1,3 : ηab Z a Z b = 1} = dS 3 ,ds2 = −(dY 0 )2 + dY2 = csc2 ρ(−dρ2 + dn2 ),

Ρ

r

(13)

for the 3-dimensional configuration space M3S of circles on the unit
sphere, where dω = sin θdθdφ is the solid angle element on theunit sphere.

from (14), so that for sparse craters with FE  1 we must have

−1
rmax
CE  π ln
.
(15)
rmin
As R → ∞ and the condition (15) continues to hold, nE /πR2
tends to the total frequency of craters with this radial range per unit
area on the Euclidean plane,
Z rmax
νE =fE dr,rmin

3

APPLICATION TO CRATER DISTRIBUTIONSHaving determined the geometry of the configuration space M3E ofcircles on the Euclidean plane and M3S of circles on the unit sphere,
we shall now show how this may be applied to the distribution ofcraters on planetary surfaces. The assumptions of our model shallbe that, firstly, each crater can be represented by a circle and, secondly, the distribution of circles is uniform on their configuration
space as defined in Sec. 2. The latter assumption therefore serves
as a null hypothesis about the structure of the configuration spaceof craters. However, the configuration spaces of circles allow arbitrary overlap whereas, naturally, this is not the case for craters
erasing older and smaller ones within. In Sec. 3.1, we shall consider the case that craters are sparse so that their overlap may beignored. This condition is relaxed in Sec. 3.2, where we attempt toindicate the effect of overlap on the distribution function of craters.
c 0000 RAS, MNRAS 000, 1–6

whence, using again (14),

−1
r2
2
CE = 2νE rmin
1 − 2min
.
rmax

(16)

The cumulative frequency ν(r) of craters with radius greater thanr per unit area as a function of size can also be obtained from the
distribution function (14),
Z rmax
CE
1
1
.
(17)
ν(r) =fE (r0 )dr0 =
−
2
2
r2
rmax
r
Finally, note that the approximation rmin  rmax  R applied toEq.s (15–17), which is reasonable for a crater count, gives CE ≈2
2νE rmin 1 and an inverse square power-law for the cumulative
frequency,r2
ν(r) ≈ νE min
.
r2

(18)

4

G.W. Gibbons and M. C. Werner

3.1.2

On the spheren
100 000

When the maximum radius of the craters is not negligible comparedto the radius of the planetary body then, given our assumptions, theconfiguration space of circles on the unit sphere from Sec. 2.2 isapplicable. Let R0 be the radius of the planetary body and R be
the radius of a crater measured on the spherical surface so that its
angular radius is ρ = R/R0 ∈ (0, π/2) as before. Now suppose
dn is the number of the centres of craters with angular radius in theinterval [ρ, ρ + dρ] within the solid angle element dω = dA/R02 ,so dn = fS dρdω with distribution function fS ∝ dVS3 given byfS (ρ) = CS csc3 ρ,

1000
100
10
1

(19)

from (13), where CS is a constant. A crater of angular radius ρ
subtends a solid angle
Z 2π Z ρ
ω(ρ) =sin θdθdφ = 2π(1 − cos ρ)(20)0

10 0000

on the planetary surface, so the area filling factor of craters withangular radius ρmin 6 r 6 π/2 is then
Z π
2
FS =
ω(ρ)fS (ρ)dρ

Ρ
0.001

0.01

0.1

1

Π
2

Figure 2. Double logarithmic plot of the total cumulative number n ofcraters larger than angular radius ρ under the assumption of sparse craters(solid curves), for ρmin = 10−3 and three values of the constant in Eq.
(24): from top to bottom CS = 10−2 , 10−3 , 10−4 , respectively. Note thatthe curves approach the inverse square power-law n ∝ ρ−2 for small ρ.
The dotted curve shows the effect of crater overlap for the top curve with
λ = −0.1 in Eq. (32).

ρmin


=

πCS

ρmin
cos ρmin+ ln cot
1 + cos ρmin2

.

For the assumption of sparse craters to hold, we require that FS 
1 as before. Now if ρmin  1, that is, the smallest craters counted
are much smaller than the radius of the planetary body, one must
therefore require that

−1
π
2
CS 
+ π ln
.
(21)
2
ρmin
The total number of craters with radius greater than ρmin is
Z 4π Z π
2nS =fS (ρ)dρdω
0

ρ

min
ρmin 
.
(22)= 2πCS cot ρmin csc ρmin + ln cot
2Again if ρmin  1, then we can approximate (22) by the leading
monomial to findnS ρ2min.
(23)
2π
The total cumulative number of craters with radius greater than ρ
is also obtained from (19),Z π2n(ρ) = 4π
fS (ρ0 )dρ0
CS ≈

ρ

ρ
= 2πCS cot ρ csc ρ + ln cot.
(24)2This function is shown in Fig. 2 for three values of CS , for which
condition (21) holds in the range shown. Also, if ρmin  1, one
can use the approximation (23) for the constant in Eq. (24) to findρn(ρ) ≈ nS ρ2min cot ρ csc ρ + ln cot.
(25)2This expression clearly recovers Eq. (18) in the limit of ρ  1
where 4πR02 ν = n and ρR0 = R ≈ r, as expected. Likewise, one
can easily convert (24) to the cumulative crater frequency per unit
area of the planetary body as a function of crater size. Note alsothat the global cumulative number (24) is given by trigonometricfunctions and not a power-law. Hence, the crater distribution is nolonger self-similar, as is the case for small craters or for craters on
the Euclidean plane in the limit rmax → ∞ of Eq. (17).


3.2

Overlapping craters

The spherical model discussed in Sec. 2.2 is, of course, the generalization of the planar model of Sec. 2.1 applicable to sparse craters
on a planetary surface. However, a more realistic model for the
distribution of craters should take into account the possibility ofoverlap. In this section, we shall consider how the more general
spherical model can thus be extended. For simplicity we shall assume that, firstly, there is a finite period of crater formation and,secondly, the crater distribution on the planetary surface is always
isotropic. Thirdly, we take (19) to be the intrinsic distribution ofcraters produced, which may differ from the distribution observedon the planetary surface due to overlap. Finally, we assume that thecentre and radius of a crater can be identified as long as a sector
of its perimeter survives, that is, only craters that are wholly overlapped by a later and larger crater are considered to be obliterated.
Hence, one may take the initial distribution to be fS (ρ) andthe final observed distribution fO (ρ) to be a function of ρ only.
Craters of radius ρ will have been wholly overlapped by larger
craters of radius ρ0 > ρ if their centres are within a solid angleofω(ρ0 − ρ) = 2π(1 − cos(ρ0 − ρ))(26)0from Eq. (20), concentric with a larger crater of radius ρ in the observed distribution, of which there are 4πfO (ρ0 )dρ0 . This is illustrated in Fig. 1 (right panel). In addition, there remain 4πfO (ρ)dρcraters of radius ρ in the distribution observed on the planetary surface. By assumption, we take the distribution of craters of radius ρthat are produced to be proportional to fS (ρ). Then this applies tothe overlapped craters as well as to the sum of the observed craters
plus the overlapped ones, of course with different constants of proportionality. We therefore expect to have
Z π
2
fO (ρ) +fO (ρ0 )fS (ρ)ω(ρ0 − ρ)dρ0 ∝ fS (ρ).
(27)
ρ

It may be useful to express the distribution function asfO (ρ) = CO f (ρ)
where CO is the scaling factor proportional to the total number ofc 0000 RAS, MNRAS 000, 1–6

On de-Sitter Geometry in Crater Statistics
observed craters nO greater than ρmin , which reduces to CS in thelimit of sparse craters,
Z π
2f (ρ)dρ.
(28)
nO = 4πCOρminHence, using (19) and (26), one can rewrite (27) asZ π
2f (ρ) = g(ρ) + λf (ρ0 )K(ρ, ρ0 )dρ0 ,

(29)

ρ

5

Integrating (32) in Eq. (28), the leading monomial in ρmin yields
the approximation



2πCO
λ
2
.
nO ≈ 21−
3 − 2 ln
ρmin
4
ρmin
Hence, the constant λ indicates the importance of crater overlap.
In the current formulation, it is a free parameter that will dependon the total number of craters produced and on the number nO ofcraters that remain observable.

whereg(ρ) = csc3 ρ,4andK(ρ, ρ0 ) = csc3 ρ(1 − cos(ρ0 − ρ)),and λ < 0 is a (by definition negative) constant. The problem ofoverlapping craters, at least in this simplified form, can therefore be
expressed as the integral equation (29), which is the standard formof a linear, inhomogeneous Volterra equation of the second kind.
Note that its kernel is of separable form since it may be writtenX
K(ρ, ρ0 ) =ki (ρ)ki0 (ρ0 ),iwhere the ki and ki0 are functions only of ρ and ρ0 , respectively. Alternatively, one can turn the integral equation (29) into a differential
equation. Lettingf (ρ)f˜(ρ) =,g(ρ)
we getλg f˜ +df˜ d3 f˜
+
= 0,
dρ
dρ3

(30)

a linear, homogeneous ordinary differential equation of third orderwith non-constant coefficients. Furthermore, one can defined ln f˜
fˆ(ρ) =dρ
to reduce the order of (30) and convert it todfˆ d2 fˆ
+= 0,
λg + fˆ + fˆ3 + 3fˆ
dρ
dρ2

(31)

a non-linear, inhomogeneous ordinary differential equation of second order with constant coefficients. While a detailed discussion of
the properties of Eq.s (29)–(31) and the numerical comparison withobserved crater distributions is beyond the scope of the present article, we shall use the Volterra equation (29) to illustrate the effectof little overlap for craters with small lower limit of the angular radius, ρmin  1. By this we mean a small deviation of fO from thesparse crater distribution so that the constant in the Volterra equation is small, |λ|  1. Then an approximate solution of Eq. (29)for f can be expressed naturally as the leading terms of a Neumann
series expansion of the full solution,f (ρ) =∞
X

λj uj (ρ),

whence the first approximation will be given by≈

u0 + λu1=

csc3 ρ + λZ π2

csc3 ρ0 csc3 ρ(1 − cos(ρ0 − ρ))dρ0ρ=Starting from the assumption that impact craters are uniformly distributed over their de-Sitter configuration space, it is shown by Eq.s
(18) and (25) that this can indeed explain the approximate inverse
square power-law for the cumulative size-frequency distribution ofsmall craters, which is widely observed as noted in the Introduction.
Moreover, while Eq. (24) shows that the global crater distribution is
not self-similar, Fig. 2 illustrates that the deviation from the inverse
square power-law due to the finite size of the planetary body is infact mostly negligible, so that the crater distribution is effectively
self-similar. Also, since smaller craters are relatively more affectedby overlap, we expect that they are systematically undercounted
relative to the distribution of sparse craters with negligible overlap.
This is the case, as indicated by Eq. (32) and the dotted curve in
Fig. 2 for small ρ.
Hence, the approach promoted in this article suggests that thede-Sitter configuration space may be a useful prior for the studyof cratering histories. Deviations from the inverse square powerlaw correspond to non-uniform distributions of small craters on thisconfiguration space. We suggest that such deviations from uniformity will indicate physically interesting processes like resurfacing
due to igneous activity and erosion, as well as properties of the
impactors before a steady-state of the crater population, as in theformalism of Marcus mentioned in the Introduction, is reached. A
first step in this direction would be to classify production functions
and resurfacing processes in terms of the deviation of the resulting
crater configuration space from this de-Sitter geometry, which will
depend on time. Hence, comparing these templates with observed
maps of crater configuration spaces may yield new insights into
cratering histories.
Finally, this article notes a perhaps surprising formal connection between the geometry of a cosmological spacetime and a problem in planetology. Recall that the 3-dimensional configuration
space of circles, and hence of craters as discussed, was interpreted
as the 3-dimensional de-Sitter spacetime with the metric given by(12) and hence the line element,
ds2 = − csc2 ρdρ2 + csc2 ρ(dθ2 + sin2 θdφ2 ).
Now in order to connect this with the de-Sitter spacetime familiarfrom cosmology, one can use the coordinate transformation csc ρ =cosh t so thatds2 = −dt2 + cosh2 t(dθ2 + sin2 θdφ2 ),

j=0

f (ρ)

CONCLUSION



λ
ρ
csc3 ρ 1 +− cos ρ + ln cot.
22

c 0000 RAS, MNRAS 000, 1–6(32)

where t takes its usual interpretation as coordinate time. Extending
our argument from two to three spatial dimensions, one obtains ameasure on the space of spheres in 3-dimensional Euclidean spacefrom Eq. (8) for n = 2. This may be useful to study the statistics
of bubbles in the universe, for instance supernova remnants within
the interstellar medium or voids in the large-scale structure. Also,
eternal inflation treats of bubble universes which are expected to6G.W. Gibbons and M. C. Wernercollide (see, e.g., Garriga, Guth & Vilenkin (2007)), so finding a
version of the overlap formula (29) applicable to this problem may
further elucidate the collision process of bubble universes.

and this is exactly the reflexive and symmetric contact relation studied in sphere geometry. Moreover, V can be equipped with a product defined asc1 c2 = −r1 r2 + a1 a2 .
Letting dc = (dr, da), we therefore find that

ACKNOWLEDGMENTS
MCW gratefully acknowledges support by World Premier International Research Center Initiative (WPI Initiative), MEXT, Japan,
and thanks Dr Steven Gratton for discussions at the 2012 Kavli Astrophysics Symposium, Chicheley Hall, UK.

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Timerding H. E., 1912, in: Jahresber. Deut. Math.-Ver., 21, 274
Werner S. C., Tanaka K. L., 2011, Icar., 215, 603

APPENDIX
Here we give a brief discussion of the geometry underlying theconstruction in Sec. 2. Circle geometry, beginning with the classical problem of Apollonius to find the circles touching three given
ones, is a special case of the sphere geometries developed by Lie(1872) and Laguerre (1881). Following partially the treatment by
Benz (2012), consider the Euclidean space En+1 with its standard Euclidean metric. Then a Laguerre cycle is either a point or
an oriented n-sphere of radius r 6= 0 centered at a ∈ En+1 , withorientation expressed by the sign of r. Laguerre cycles can thus
be regarded as elements of a real vector space V = R ⊕ En+1 .
Assigning coordinates c = (r, a) to a Laguerre cycle is called cyclographic projection, and addition of Laguerre cycle coordinates
is defined as for vectors. The zero Laguerre cycle is 0 = (0, 0).
Now given two Laguerre cycles c1 = (r1 , a1 ), c2 = (r2 , a2 ), thereal number
P (c1 , c2 ) = −(r2 − r1 )2 + (a2 − a1 )2 ,
n+1

(33)

where the product of vectors in E
is understood to be the innerproduct with respect to the Euclidean metric, is called power of
c1 and c2 . It turns out that c1 and c2 touch each other respecting
orientation if, and only if, (Benz (2012), proposition 3.43)

dc2 = −dr2 + da2 = P (c, c + dc)(34)

measures the infinitesimal deviation of Laguerre cycles with respect to the contact relation, and this is seen to define a Minkowski
metric on V . Note that (34) may be regarded as a line element on
V which is invariant under transformations of Laguerre cycles preserving power (33). Hence, there is a close connection between
the Minkowski spacetime of special relativity and the geometryof Laguerre cycles, which appears to have been pointed out firstby Timerding (1912). In addition to the set of Laguerre cycles Γ,
one can introduce the set of oriented Euclidean hyperplanes Σ in
En+1 and the object infinity ∞, and define corresponding contact relations. Then ∆ = Γ ∪ Σ ∪ {∞} is called the set of Liecycles, and bijections Λ : ∆ → ∆ that preserve contact relations form the group of Lie transformations of En+1 . It turns out(Benz (2012), proposition 3.56) that to every Lie cycle a homogeneous Lie cycle coordinate L = [L0 , L, Ln+2 , Ln+3 ], where[L0 , L, Ln+2 , Ln+3 ] = [kL0 , kL, kLn+2 , kLn+3 ] for any k 6= 0,
can be assigned bijectively, which satisfies the Lie quadric−(L0 )2 + L2 + (Ln+2 )2 − (Ln+3 )2 = 0.

(35)

For a Laguerre cycle c = (r, a), the Lie cycle coordinate is (c.f.
Benz (2012), p. 154, applying a sign change and reordering)
P (0, c) + 1
P (0, c) − 1
L(c) =, a,, −r .
2
2
By homogeneity of the Lie cycle coordinates, we can set Ln+3 = 1
to obtain



 

a
1 a2 − 1
1 a2 + 1− r ,− ,−− r ,1
L(c) =
−
2
r
r
2
r
 0

=
Y , Y, Y n+2 , 1 ,using (33) and the coordinates (4)-(6) defined in Sec. 2. Then theseare seen to satisfy−(Y 0 )2 + Y2 + (Y n+2 )2 = 1(36)

because of the Lie quadric (35). Suppose now that we consider theline element
ds2=−(dY 0 )2 + dY2 + (dY n+2 )2

(37)

2

=

1
dc
(−dr2 + da2 ) = 2
r2rusing (34). Clearly, any transformation (Y 0 , Y, Y n+2 ) 7→ (Y 0 +dY 0 , Y + dY, Y n+2 + dY n+2 ) that preserves our choice (36)
leaves this line element invariant, and in fact null ds2 = 0 so thatdc2 = 0. Hence, such transformations preserve the contact relations of the corresponding Laguerre cycles, as described above.
Under this assumption, then, (Y 0 , Y, Y n+2 ) may be regarded asa point on the de-Sitter configuration space of Sec. 2 defined by thequadric hypersurface (36) in the Minkowski spacetime E1,n+2 withline element (37). A connection between Laguerre sphere geometry
and de-Sitter spacetime seems to emerge first in Graf (1934), whouses the opposite metric signature.

P (c1 , c2 ) = 0,
c 0000 RAS, MNRAS 000, 1–6