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Boundaries of Space-Times and Singularities in General Relativity.

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Annelen der Phyeik. 7. Folge, Band 32, Heft 2,1976, S. 146-168
J. A. Berth, Leipzig
Boundaries of Space-times and Singularities in General
Relativity
By H.-H. v. BORZESZKOWSKI
and U. KASPER
Zentralinstitut fur Astrophysik der Akdemie der Wissenschaften der DDR, Potsdam-Babelsberg
Abstract
I n general relativity, a cbtalogof bounda definitions is considered. Arguments are presented
which show that, in general, the complete escription of a physical situation does not admit
“singular coordinate transformations” and other extension procedures which change the physical
character of space-times; according to REICHENBACH,
a complete description is equivalent to the
fixation of the combination: topology (T)plus metric (M).
Therefore, any boundary which cannot
be ‘‘removed” by procedures which maintain topology haa to be considered a genuine boundary
(singularity). Some bounderies of the vacuum theory of general relativity are discussed.
7
Inhaltsil bersicb t
Es wird ein Katalog von Definitionen von Riindern allgemeinrelativistischer Raum-Zeiten
diskutiert. Es werden Ar mente angegeben, die zeigen, daD die vollstiindige Beschreibung einer
hysikalischen Situation E i n e ,,singuliiren Koordinetentransformationen“ und andere den physiist
galischen Charakter von bum-Zeiten iindernden Ausdehnungen zuliiDt ; nach REICHENBAOH
die Beschreibung einer physikalischen Situation einer Festlegun der Kombination: Topologie (T)
plus Metnk (M)iiquivalent. Daher mu0 jeder Rand einer Raum-kit, der nicht durch T-erhaltende
Ausdehnungsverfahren,,bemiti “ werden kann, ale eohter Rand (Singularitiit) angeaehen werden.
Ea werden einige Riinder der mmsmschen Vakuumtheorie diekutiert.
lf
1. lntroduction
A field theory is not completely determined by the formulation of field equations
in the form of partial differential equations; these equations must be supplemented
by suitable boundary conditions. This fact is closely related to the problem of singupoint of view [l] according to which
larities. This follows especially from EINSTEIN’S
one has to distinguish between two types of boundary conditions in every field theory,
namely between boundary conditions at infinity and those ones on a finite surface.
EINSTEIN
pointed out that singularities, i.e., “points (and lines etc.) where the field
equations are not valid”, are equivalent to the specification of boundary conditions
on closed surfaces which are arbitrary with regard to the field equations. These supfaces enclose the singularities and are, accordingly, an expression of the existence
of limits of the validity of the considered field theory.
In this paper, space-times which are solutions of the vacuum theory of general
relativity (vacuum GR) are discussed near the “points and lines etc.” where the
vacuum GR is not valid. Of course, these points etc. cannot be represented aa points
of the vacuum space-time but only as boundary points. (These boundary points are
often avoided by inclusion of the energy tensor of non-gravitational matter fields (see
below).) We shall not restrict our considerations to the above-mentioned boundaries
which can be enclosed by 2-dimensional surfaces.
Boundaries of Space-Times in General Relativity
147
To avoid misunderstandings it should be emphasized that we restrict ourselves
here to boundaries of vacuum GR only for methodical and not for physical reasons (in
particular, not to defend a geometrodynamic interpretation of the vacuum GR).
We simply wish to study aspects of the singularity (or boundary) problem by means
of a comparatively simple example. Of course, boundaries of the solutions of the
vacuum GR where
R,, = 0
(1 a)
are often no boundaries of the physical space-time; the incorporation of non-gravitational matter, i.e., the consideration of solutions of the equations
instead of eqs. (la), “removes” some of these boundaries. However, this does not
oppose the statement that they are boundaries in the sense of the vacuum GR. The
eqs.(lb) can be interpreted as modification of the theory that is determined by
eqs. (1a) and here considered a model. Just the removal of boundaries which, possibly,
can be carried out by a transition from eqs. (1a) to eqs. (1b) shows that the occurrence of boundaries can be an expression of the existence of limits of the validity of
the considered theory, here, of the vacuum CIR. (That is, these boundaries are genuine
definition.)’)
boundaries or singularities in the sense of EINSTEIN’S
When starting with eqs. (1b), one has to discuss the same problems; but then one
does not know the modification of the theory that could remove boundaries (cf. section 5).
In our discussion of genuine boundaries or singularities, we shall give reasons for
the standpoint that the complele description of a concrete physical situation has to
take into consideration global (topological) aspects of gravity. J u s t the consequent
regard of this point of view destroys the “selection rule” which is often formulated for
singularity definitions (cf. section 2). This aspect also shows up in the connection
existing between the occurrence of boundaries and the boundary conditions a t infinity. 2)
‘As far as the aspect of limits of the validity of a theory is concerned, there is, in
principle, no difference between boundaries of non-relativistic and general relativistic field theories. But, for the theory of general relativity, a complication of the
discussion of the boundary problem follows from the fact that here the background
space-time is not given a priori; the field equations (plus boundary conditions) themselves determine the space-time metric. Accordingly, the occurrence of boundaries
means that points “on” or “behind” boundaries, by definition, do not belong to the
space-time : Where the field equations are not fulfilled there it is not sensible to speak
of a solution and, therefore, of a space-time in the sense of the considered theory. (For
the space-time definition cf. section 2.) Thus, one has to consider the properties of
space-time8 in the neighbuorhood of boundaries and to define a structure of boundaries
S by means of these properties of the neighbourhood (cf. also HAWKING
and ELLIS[3]).
Later on we take the view that the regular space-time regions a,, i.e., regions with
GIn S = 0 , are known and deduce properties of S by means of a limiting process
{xi> -+ S, or we consider EINSTEIN’S
vacuum equations to get properties of boundaries
which are compatible with the vacuum GR. Here the starting point will be the integral
1) Our vacuum GR-model has the advantage that we know that and how the theory has to
be modified to “remove” some singulerities.
2) For a discussion of the connection between the boundary conditions at infinity and the
behaviour of gravitational fields near singularities cf. also ”F~EDER[2].
10*
€I.-H. v. BORZESZKOWSKI
and U. KASPER
148
form
1RikdV4 = 0
GI
(2)
of the vacuum equations (la). Besides general physical arguments for this form of
the field equations (cf., for instance [4,5]) the need of the consideration of this form
results from our intention also to discuss violations of the validity of the vacuum
equations which are produced by the fact that, on a hypersurface S, the RICCItensor
is not defined by classic functions. In [5], TREDERdiscussed some integral formulations of (la) pointing out that boundaries of the vacuum space-times may be interpreted as existence of a &like matter distribution.
In GR, one does not only require structures of space-times which enable a definition of R, in terms of classic functions; on different (physical) grounds which are
not to be discussed here one demands that physical space-times have to fulfil a whole
catalog of conditions. Accordingly there exist different possibilities to define and
characterize boundaries, according to the condition that is violated somewhere. The
existence of these different mathematical definitions of boundaries is the source of
many misunderstandings in the field of the singularity discussion.
I n section 2, we shall list the conditions which define a space-time and discuss the
corresponding definitions of boundaries under consideration of physical aspects. It
will be shown that PENROSE,
HAWKING,
SCHMIDT
and others (1965 and later), in their
discussion of b- and g-boundaries (cf. for instance [3]), considered one of the various
possibilities of characterizing boundaries ; in general, the interpretation and fixation
of the physical situation (always necessary) remove a fictitious advantage of the gand b-boundary-definitions, namely the advantage to be “more invariant” than
other definitions.
Starting with other boundary definitions, TREDERdiscussed possibilities of a
realization of the particle programme within GR [6] in 1962. Later, TREDRE[7, 81
hm also discussed various other physical situations which are called boundaries in
our terminology. In this connection, also weighting methods for boundaries were considered [9, lo]. Under consideration of the aspects discussed in section 2, we will give
in section 3 some of the vacuum GR-boundaries investigated in [6- lo]. Finally, in
section 4 boundaries of static space-times are discueaed in more detail. I n particular,
it is shown that these space-times provide examples of the coincidence of different
types of boundaries.
2. Boundaries o! space-times in general relativity
Definition: A space-time is a set of events which satisfies the following conditions :3)
I. The set M is a four-dimensional HAUSDORFF
manifold with a countable basis.
11. M is a differentiable manifold of class CY.
111. a) M possesses a pseudo-RIEMANNian metric gik of h R E N T 7 , signature
(+ - - -). (By this, in general, poles of gtkand poles and zeros of the determinant
det gik of the metric gik are excluded.)
b) gik is differentiable of class C8 (s 5 r - 1). (Later on we assume s = 1; then
so-called surface distributions of sources may be interpreted as boundaries of spacetimes.)
IV. gik is a solution of EINSTEIN’S
vacuum equations.
*) Sometimes other definitions of space-time are considered. Using other definitions it is easy
to modify the following considerations accordingly.
Boundaries of Space-Timesin General Relativity
149
V. There are no curves in M such that, moving along these curves, physically
pathological situations arise. (For instance, many authors consider incomplete
geodesics and time-like curves of bounded acceleration and finite proper length as
such curves.)
Now we can define different types of boundaries, in dependence on the condition
that is violated; we shall call them I-boundaries, 11-boundariesetc. For example, the
violation of condition V leads to the occurrence of a V-boundary. Since the conditions
I-V defining a space-time are not independent of each other the boundary types defined in this manner can coincide, b.,the occurrence of a boundary of a certain type
can induce the occurrence of the boundary of another typ (see sections 3 and 4).
One could be tempted to call the boundary of a space-time G, (in the sense of a
violation of any of the conditions I-V), a genuine boundary or a singularity if there
is no extension G; of a space-time Q,. (An extension is a space-time, i.e. a set C; of
events which satisfies the conditions I-V, which contains a proper subset 5,that is
isometric to G,.) This definition of singularity is a generalization of a definition given
by some authors in connection with discussions of V-boundaries (cf. (31).
Before criticizingthisdefinition of singularities we yet want to make mine remarks
about boundaries of types 111and IV.
First, the JACOBIdeterminant D =
I :z-I
.of coordinate transformations zi + Zi
allowable in the framework of differential geometry must be regular everywhere. Therefore, e coordinate transformation d+ 9 can never remove a zero of the determinant
det (ga) (one has det (&) = D1 det (g*)). In this sense, the occurrence of a zero of
det (ga) (IIIa-boundary) is a coordinate invariant property of space-times. Thus the
operation xi + zi with DI, .= 00 which removes zeros of det (gfr) without producing
other types of boundaries IS an extension procedure ; sometimes it is called an “imegular coordinate transformation”.
Second, according to our definition of singularities, one has to take care of the fact
that the extended manifold has to be a space-time, i.e., in Q; all the conditions I-V
have to be satisfied. However, in general, irregular transformations produce poles in
some of the 9,-components, i.e. a new 111-boundary.
I t follows from these remarks that the often-discuased argument -det ( g f k ) = 0boundaries are physically irrelevant because they can be removed by coordinate
transformations - is not applicable, even if one admits so-called irregular coordinate
transformations.
There is a third argument against the a priori-exclusion of any of the listed boundarics. This argument implies a criticism of the singularity definition given above. I n
our opinion, it is not reasonable to admit every extension which fulfils the conditions
I-V. Indeed, what does the transition to an extended space-time Q& help if Q; descritm another physical situation than the examined one described by Q,? The rerequirement that G, and a subset 5,of G; are isometric to each other is only a necessary
but by no means a sufficient condition for GI and G; to describe the same situation.
Fixation and maintenance of the physical situetion rule out extension procedures
that change the physical character of space-time. Accordingly, in general, singular
coordinate transformations may not be admitted in discussing boundaries because they
generally change the topology of space-time and procedures changing topology
can change the physical character.
This point of view was proposed by HILBERT(1915116) [ll] and others and, in
the years of foundation and the first elaboration of GR,it was the generally accepted
in his “Philosophie
one. For instance, it was explicitely formulated by REICHENBACH
der Raum-Zeit-Lehre” [ 121.
a&
€I.-H. v. BOEZESZKOWSKI
and U. KASPER
160
According to REICHENBACH
the proper effect of the GAUSS-RIEMANN
geo,metry
consists in the fact that it divides the function which the usual rectangular coordinate
system in the Eucuman geometry has into two completely different functions; i t
leaves the coordinate system only the topological function of cataloguing (“numbering”) and transfers the metric function of length measurements to the system of the
geometry splits the description of space
metric coefficients. Thus the GAUSS-RIEMANN
REICHENBACH:
“Trotz der Willkur
into 8 topological part (T)and a metric part (M).
der Numerierung ist mit ihr bereits etwas sehr Wesentliches gegeben, denn die gegensind damit festgelegt.” ), According to REICHENseitigen Nachbarschaftsverha~tn~~
BACH, the metric plays a secondary role; it represents only the superstructure which
does not change the topological fundamentals.
Accordingly a complete characterization of space and space-time, resp., i.e., a
complete characterization of the physical situation, is only given when we know T
besides M ;only the combination ff = T M has a physical meaning. Therefore, a
change of T changes the physical content, too. (In GR, this combination is only
determined by the field equations plus initial, boundary and coordinate conditions.
Here it should be emphasized that T and M are not completely independent of each
other.)
To illustrate this point of view let us discuss here an example which was important for the foundation of GR6). Let G, be the space-time that is globally covered by
za, 23) and whose metric is given by the line element
a coordinate system (t, 9,
U P = (a)adta - (dz’)a - (dz%)a - (d23)2,
(38)
where the relations
-oo<<<+oo,
O<Z’<+oo,
-oo<z%,z3<oo
(3b)
give the global coordinate cart. Here d = 0 is a 111-boundaryof ff4. Of course, mathematically it is possible to go to the extension Q; which is given by
+
= &a
- (d21)*- (d9)’ - (d5?)*
with
< t < +oo,
< ea < +oo
(u = 1,2,3).
(4 b)
Intending now a description of the gravitation free pace, then, of course, it is physically sensible to go from 0,to the extension G i (possessink no boundary). Then ff, is
the description of the gravitation free space with regard to a reference system that is
uniformely accelerated with regard to the inertial system
2”).
However, this transition is not a physically trivial one since it would not be possible
if another physical interpretation would be intended than that of the gravitation free
MINKOWSKIspace. For instance, such other physical interpretation (or: situation) is
given in the cme of a homogeneous gravitational field discussed by EINSTEIN[16]; in
this c m , Q, is the space-time which describes this situation. According to EINSTEIN
[16], the global coordinate system (t, 2”)covering this space-time is equivalent to an
inertial system.
Despite the local isometryof the relations (3) and (4) for d # 0, these space-times
differ from each other by their global behaviour, i.e., by their topology T.Going from
ff, to ff; by means of a singular transformation z’+ Si (with DIZnrnO
=oo),T is changed.
--oo
-00
6,
H. REICHENBACH,
Philosopie der hum-%it-Lehre, Berlin u. Leipzig 1938, p. 280.
SCHWAF~ZSOHILD
spacle-time, the
reader is referred, particularly, to the article of C~~PEESTOCK
and JIJNWICUS
(131. In particular,
the authors discusa argumehts given by ROSEN(141 and the question whether or not singular and
complex transformations are admissible. - For the meaning of coordinate conditions cf. also
FWK [15].
4)
*) For a detailed discussion of the r = 2n-boundary of the
Boundaries of ’space-Timesin General Relativity
151
To summarize, not restricting oneself arbitrarily to the consideration of invariant
structures (i.e., to scalars and, thus, to structures which are also invariant under
singular coordinate transformations) or to the geodesic structuree), without considering the problem of the solution of the field equations and the physical interpretation
of t h e e solutions, one haa to require: An allowable extemion a: has to describe the
same physical situation as a,. That is, in general G, must have the mme topology T
aa a;.
Therefore, the arguments against coordinate-dependent descriptions (often given)
are too strong. They overemphaeize the covariance of the field equations and take no
notice of the fact that a complete dynamics only follows “when one goes from infinitesimal to finita space-time regions, i.e., when the differential equations of field and
niotion are integrated in a specified coordinate system’’ [18].
Thus the question of extension arising in connection with the definition of singularities is not a purely mathematical question. On the other hand, this does not mean
that there me no sufficient mathematical conditions for the inextensibility of a spacetime; by no means, a space-time is extensible if the scalar invariants of the curvature
tensor tend to infinite for {xi> +. S.
3. Boundaries and matter which is concentrated on hypersurlaces
Let us consider the integral form (2) of EINSTEIN’S
vacuum field equations. There
is a IV-boundary if the relation
holds for some a,. Here we are going to discuss such cases in which the IV-boundary
E (or 0 < z 5 E )
is caused by R, not being defined in a neighbourhood - E 5 z 2
of z = 0 by classical functions:
+
R j k = --jkd(z)
(6)
or is induced by or coincides with other types of boundaries. In this section, we suppose
that xo is not a null coordinate so that we can write the metric ae
as2 = goo ( 2 0 ) (dXO)2
g,,” dz’ dd
(p, r = 1, 2, 3).
(7)
Let us consider the following situations:
by a hypersurface
a) A domain G4 is split into two space-time domains (2,- and
S : xo = 0 in such a way that S is a IIIb-boundary of both space-time domains and
Q, and a$ are isometric to each other. The I11b-boundary is caused by discontinuities
of the first derivatives of gik. - Examples were discussed in [83. r t is singular (i.e.,
det (t;)lt-,,= 0).
b) A domain G, is split into two space-time domains G , and G t by a hypersurface
S : x o = 0 in such a way that S is a IIIb-boundary. This boundary is caused by discontinuities of the first derivatives of the g,,.. goo and its derivatives are continuous.
We write the three-dimensional tensor g,,, in the neighbourhood of xo = 0 m (cf. [19])
+
’) By the way, in the case of g-boundaries the problem arises that geodesics generally cannot
be interpreted near the boundary [l?];accordingly this description of boundaries Beems to be
less physically sensible than one generally assumes. - Now, this definition has the advantage that
general singularity theorems can be established by means of this definition. But this is not the
advantage of this definition alone. At least for timelike incomplete geadeaics, the singularity
typ can be formulated by means of the boundary
theorems of the HA~INO-PENROSB-GEROCH
definition: “Occurrence of zeros of det (gir) in synchroneous coordinates”.
I&-€€. v. BOEZESZKOWSKI
and U.KASPEB
162
for P
< 0. Let apvand BpVbe holomorphic (or sufficiently smooth) functions of
k
xa in
k
the domain considered with the following properties :
It was proved by TBEDER
[7] that in this case relation (6) holds, where tk,is given by
by a hypersurface
c) A domain 0;is split into two space-time domains GF and
S :P = 0 in such a way that S is a III-boundary. The boundary is caused by a zero
of order 2m (m > 1) in goo = q,(zo) together with discontinuities of derivatives of
1 of qpv. - We can maintain eqs. (8),but eqs. (9) have to be replaced by
the order m
+
p
for 1 < m
v
(114
= I?.
+ 1 and
Supposing
900
= $!ol
(P)*"(,
one can see that in this C B ~ Bthe relation (6) holds [7], where tk,is given by
d) A III-boundary is caused by zeros and poles of g,,,. (In [lo] it was shown that,
because of the validity of EINSTEIN'S vacuum equations in the neighbourhood of S,
in this caw only zeros and no poles of det (Sir) are possible on S.) - We can interpret
this III-boundary 88 a N-boundary in the sen* of (6) (cf. [lo]). If we do 80, we obtain
functions 7: which have poles and the following relations hold [lo]:
too-
(2
2 - 1).
(14)
Generalizing notions of the theory of linear partial differential equations TBEDER[9]
has proposed to define the following notions and to use them for a classification or
weighting of boundaries.
(xO)-',
7;-
(XO)'
Boundaries of Gpace-Times in General Relativity
163
Definition: EINSTEIN’S
mixed-variant vacuum field equations are called
strongly fulfilled, if the relation
$ fR:d4x
QCD
=0
(15)
holds with any arbitrarily chosen function f . EINSTEIN’S
mixed-variant vacuum field
equations are called weakly fulfilled, if the relation
J FR,kd% = 0
QSD
(16)
holds with any arbitrarily chosen function F that is r e g u l a r in D. To obtain a claesification of boundaries in t h e above-mentioned sense we try to go from eq. (6) to weakly
fulfilled equations
cpRt = 0
(17)
by multiplying eq. (6) by a functional cp[gik] which strongly enough tends to nought
when 20 + 0; that is, cp must have a zero of such an order that
-0
lim
(cprt)- (20)” (n 2 1)
(18)
holds. Now one can classify space-time manifolds with boundaries according to the
following aspects: 1. Is there a functional F [ & k ] 80 that we obtain, by multiplying
EINSTEIN’Svacuum field equations with 9, weakly fulfilled equations on a domain
that is prolonged beyond the boundary? (Here, we understand the domain to be
given by the range of the coordinates, and “prolonged beyond the boundary” means
that the coordinates of the points of the boundary are inside that range.)
2. If the answer is “yes”, then the boundary is characterized by the system of
coordinates xi and the functional producing weakly fulfilled equations.
From the discussion of the examples a) d) we know that different III-boundaries induce IV-boundaries in the sense of eq. (6). Especially, the boundaries discussed
in a) and b) are different from those ones in c) and d) because all the ttkin the examples
a) and b) are regular, whereas mme of the rikin c) and d) have got poles on S. Applying
TREDER’S
classification method we also can say: The boundaries in a) and b) differ
from those ones in c) and d ) in such a way that, in general, there is no functional
[det (qik)Imsuch so that EINSTEIN’S
vacuum field equations become weakly fulfilled;
in example c) and d) such a functional [det (sir)$exists. The exponent m can be used
for further classifications of this boundary type. Between m and order n of the zero of
det (qik),we have the relation [lo]:
2
m=-.
n
If it is correct to suppose that every delta-like matter distribution yields a pole-like
tensor, then EINSTEIN’S
equations
singularity of one of the invariants of the RIEMANN
with delta-like “sources” are equivalent to weakly fulfilled vacuum equations, and
tensor.
the functional cp is an invariant concomitant of the RIEMANN
4. Boundaries of static space-time8
In the following we consider gravitational fields with a KILLINGvector field the
trajectories of which are orthogonal to a family of hypersurfaces so that there is a
coordinate system in which we have (cf. [6])
qkl.0
= O,
(20)
90N
= 0.
(21)
H.-H. v. BORZESZKOWSKI
and U.KASPER
164
(Here, zois not a null coordinate.) Then we can write
det ( g d = 900 det (gpv),
where det (grv) is the determinant of the metric tensor induced on hypersurfaces
xo = const. Furthermore, we have
1
g#
= drQ, p = 0, p =
-.
900
Without changing eqs. (20),..., (23) it is possible to describe the hypersurface on
which det (gik) = 0 by 21 = 0. Additionally, we suppose that the metric on xo = const
0 holds and, consequently, the zero of
is negatively definite so that det (qJ
det (gkl) is only caused by a zero of goo.
Finally, without changing the obtained simplifications, we can transform gp to
+
0 9x4 933
I n the neighhourhood of z' = 0 we write goo88
where n and m are to be greater than zero; it is not supposed that n and m are
exnatural numbers. det (gik) gives a similar series. In caw that (25) is not a TAYLOR
pansion the expression (25) is only correct when a9 2 0. That means x1 = 0 has to be
considered a boundary of the space-time.
We write gpvin the following form
ya, the vacuum field equations (1)read:
Defining gooE Va and -det (gJ
30,
= 0,
R,, = Prv
+ p1
(V,pv
- r;. V * J *
In the neighbourhood of z1 = 0 we havo
(I/detg,,vgll)z~=Op(pl)(zl)-a+...=O,
n n
@=b
.- -
R,'=
$," (z')'+**.= 0
(28 b)
where s 2 -2. Thus it follows that we can interpret space-times with such a kind of
boundaries as solutions of weakly fulfilled EINSTEIN'S
vacuum equations. Again we
have
70"
( z y , 7;
(zI)Z,
(29)
where 12 1. The above-introduced functional 9 is
-
g = 900 det
-
(30)
Especially, n = 2 follows from equation (28a). Such space-times were discussed by
TBBDER
in connection with the particle problem in EINSTEIN'S
theory of gravitation.
I n our general discussion of space-times with boundaries, these static space-times
are especially interesting because they are an example for the possibility that III(srv)
Boundaries of Bpace-Times in General Relativity
155
boundaries or IV-boundaries can coincide with V-boundaries. Moreover, it is possible
to prove that generally those space-times are not extensible beyond the boundary
S :x1 = 0 because there is at least one invariant of the RIEMANN
tensor which is
[20]:
singular in general. The latter is a consequence of a theorem proved by ISRAEL
In a static space-time, let Z be any special hypersurface t = const, maximally
vector) and let 8, be
extended consistent with & . tk< 0 (tk= timelike KILLING
the class of static fields such that the following conditions are satisfied on 2:
is regular, empty, noncompact, and asymptotically E u c m a n . More prea)
cisely, the last term means that the metric has the asymptotic form
ga#9= ,
6
v
+
(qw)1/2
O@--l),
=1
ga#,y
= O(r-*),
- - + 7, m
m
= const,
(31)
11 = 0(r-2), 7,a = o(r-a), rl,,#9 = 0 ( ~ - 4 )
when r = (duo x%fi)1’2 -P 00.
b) The equipotential surfaces V = const > 0, t = const are regular, simply
connected closed 2-spaces.
formed from the four dimensional RIEMANNtenc) The invariant 4Ru,,,n 4Rk1mn
sor is bounded on C.
d) If V hm a vanishing lower bound on Z,the intrinsic geometry (characterized
by 2R) of the 2-spaces V = c approaches a limit as c --f +0, corresponding to a closed
regular 2-space of finite area.
Theorem :The only static space time satisfying a), b), c), and d) is SCHWARZSCHILDS’
spherically symmetric vacuum solution.
From that theorem we draw the following conclusion. If we demand that 4Rk1mn
x
4RuA has got no poles in side the manifold and that the conditions a), b), d) are
does not exist; for any statia vacuum space-time
satisfied, then lim ‘Rklmn4RHmn
v+o
that is not isometric to the SCHWARZSCHILD
metric.
The hypersurface V = 0 is identical with our hypersurface zl = 0, if we additionally require that d = 0, xo = const are two-dimensional closed surfaces. I n that case
we have the result that ,Rkl,,,,, ‘RHmnis not defined on the boundary of the spacetime given by det (qk,)= 0 in cam that the metric of the space-time is not isometric
to the SCIIWARZSCHILD
metric. So we are not able to extend the space-time beyond
the boundary in such a way that the invariant of the R I E U N i a n tensor have no
singularities’).
These space-times are not simply connected.
On condition that at least the first two terms in the relations (26) and (26) are
terms of a TAYLOEseries, the metric reads
’) This corresponds to the opinion given above that solutions of eq. (6) are solutions of the
weakly fulfilled vacuum field equations ( l a )where the functional introduoed above is a function
of invariants of the RIEMANN
teneor.
1.56
H.-H. v. BOBZESZXOWSKI
and U.KASPER
if XI is an element of a sufficiently small neighbourhood of x' = 0. u,Z, ad, ud are
2 9 0
2
determined by the vacuum field equations.
From an inspection of the equation of time-like geodesics in the neighbourhood
of x1 = 0 we conclude that the boundary under consideration is in general a V-boundary, too. Paying attention to (31) we obtain
ds
ax0
(2')2 = = const > 0
(33)
as a consequence of the 0-component of the equation of the geodesics, and
dx' d&
gik
= 1 yields
dsds
where N is defined by
Now we introduce x1 as a parameter and obtain
and
where xk is an element of a neighbourhood of x1 = 0. Eqs. (36) and (37) are integrals
of eqs. (33) and (34).
The answer to the question whether the manifold under consideration is timelike
incomplete depends on whether and how strongly N tends to naught when x1 + 0.
Necessary and sufficient condition that lim N = 0 is !
m [(2')*g&uaub]
= -p.
2'+0
5 +o
But that requires g& uaub < 0 in a sufficiently small neighbourhood of x' = 0.
Althou h we have supposed that grvAflAv is negatively definite, this only means
gabua u < -g11(u1)2 and not necessarily gd uaub < 0. Whether all these conditions are fulfilled can only be seen by a closer inspection of the vacuum field equations.
Here we only want to consider the consequencea in the case that the vacuum field
equations allow such solution.
Let us introduce x1 as a parameter into those components of the equation of geodesics which contain dualds. Then we obtain
e
Now we put u2 N (x')'", u3 N
and the condition lim [(a+)' gd uaub]= o'+O
allows the following possibilities:
1. a < 1, z = 1,
2. 0 = 1, z = 1,
3. u = 1, 7 < 1.
From eq. (38) we obtain N N "1 in a sufficiently small neighbourgoodof x1 = 0
for all the three listed poasibilitiese).As a consequence of eqs. (36) and (37) we have the
result that x" tends to infinite whereas 8 is bounded when 2' + 0, both in the case
8)
+
We m u m e gao,o 0.
157
Boundaries of Space-Times in General Relativity
that gild is of the order zero (which in general is the case in (32)) or of higher order
(cf. [22]) in x1 and in the case that the first term of glla is -($)I-'
(0 < s < 1).The
series in the
latter could be possible in case that gll is not expressible as a TAYLOR
neighbourhood of 21 = 0.
So we have proved that space-time8satisfying all the listed assumptions are timelike incomplete.
A necessary condition that a space-time with boundary considered in this section
(x1)-l-*
is a t least time-like complete, is that a t least one of the zP-componentsis
(s > 0). In that caae, N would be naught though d is not equal to zero (Here we have
in mind the case that x1 = 0 is a two-dimensional cloaed surface.) Perhaps the following two possibilities could arise. Firstly, the particles moving towards d = 0 are
deflected and, because of their divergent angular momentum, never reach x1 = a. Together with xo, the eigentime s could possibly be divergent. Secondly, the particles
moving towards x1 = 0 are deflected and, because of their enormous angular momentum, remove from z1 = 0. Possibly, 8 is divergent in that case, too.
-
6. Concluding remarks
Regarding boundaries of solutions of the equations (1a) there arjpes the question:
Are those boundaries removable by a transition to equations (lb), i. e., by an incorporation of phenomenological matter ? Boundaries removable in such a way are only
a hint that we have to complete the theory ( I a) by consideration of non-gravitational
matter [21].
Starting with a term in EINSTEIN'S
equations describing nongravitational matter,
all the discussed definitions of boundaries do not lose their validity and are applicable
to this c m , too. (Only condition I V haa to be changed accordingly.) But then the
discussion of singularity problems is more complicated because the answer to many
questions depends on the properties of the non-gravitational matter. Partly, these
difficulties have been removed by certain physically plausible assumptions concerning
the behaviour of non-gravitational matter. With these assumptions it was possible,
for instance, to establish general theorems which give an answer to the question
[23],
whether a space-time has a III-boundary or a V-boundary (RAYCHAUDHURI
KHALATNIKOV,
1,ieSmrz [24], HAWKING,
PENROSE
[25] and others).
The authors wish to thank Professor Dr. H.-J. TBEDERfor helpful discussions and
for drawing their attention to REICHENBACH'S
arguments.
References
(13 A. EINSTEIN,Grundzhge der Relativitibtheorie, Braunschweig 1964, unveriinderte Aufl.,
Berlin, Oxford, Braunschweig. 1969.
[2] H.-J. TBEDER.
Ann. Physik. Leipzi (1974).
and G . F. ELLIS,T i e large scale structure of space-time, Cambridge UN(31 S. W. HAWKING
versity Press 1973.
(41 A. EINSTBIN,Sitzungsber. Preuse. Akad. Wise. 468 (1918).
[6] H.-J. TREDER,in: Pers ctives in Geometry and Relativity, Indiana University Press 1966.
Ann. P g s i k . Leipzig 9, 283 (1962).
[6] H.-J. TIZEDER,
[7] H.-J. TREDER,
Ann. Physik, Leipzig 24, 234 (1970).
[8] H.-J. TREDER,
Ann. Physik, Leipzig 25, 261 (1970).
(93 H.-J. TREDBR,Monatsber. Dtech. Akad. Wise. Berlin 12, 768 (1970).
[lo] H.-H. v. BOMESZKOWSKI,
Ann. Physik, Leipzig 80, 291 (1973).
[ll] D. HILBERT,Nachr. k 1. Ges. Wiss. Gattingen, Math.,-phys. Klasse (1915) 395. Vgl. Gesammelte Abhandlungen, i d . 111 Berlin, Heidelberg, New York 1970, 258.
Philosopie der hum-Zeit-Lehre, Berlin 1928.
(121 H. REICHENBACH,
(131 E.I. COOPERSTOOK
and G . J. G. JUNEVICUS,
Nuovo Cimento 16, 387 (1973).
168
H.-H. v. BORZESZKOWSKI
and U. KASPER
1141 N. ROSEN,“Relativity’~,F’roc. Midwest Relativity Cod.,1969, p. 229, Plenum, New York
1970.
1161 V. FOCK,Theorie von Raum, %it und Gravitation, Berlin 1960.
[161 A. EINSTEIN,Ober spzielle und allgemeine Relativitiitstheorie Berlin 1969.
Int. J. Theor. Phys. 8, 331 (1973).
[17] H.-H. v. BORZESZKOWSKI,
[18] H.-J. TREDER,
Wiss. Z. Humboldt Univ. $22, 17 (1973).
[19] H.-J. TREDBR,Gravitative StoSwellen, Berlin 1962.
[20] W. ISRAEL,
Phys. Rev. 164, 1776 (1967).
[21] J. A. WHEELER,Geometrodynamics, New York 1962.
LIEBSCEEB,D.-E.. and H.-J. TBIDER, Ann. Phyeik. Lei zig 12, 196 (1963).
[22] E. KFGEISEL,
Relativiatio Cosmology, Phys. Rev. 98, 1123 (19g6).
[23] A. RAYCEIAUDHURI,
[24] I. M. KHALATNIKOV
and E.M. L m m . Investigations in Relativiatic Cosmology, Adv.
Phys. 12, 186 (1963).
[25] S. HAWKING
and R.PBNBOSE,
The Singularities of Gravitational Collapee and Cosmology.
Proc. R. Soc. Lond. A 814, 626 (1970).
Bei der Redaktion eingegangen am 17. Juni 1974.
Anschr. d. Verf.: Dr. H.-H. V. BOBZESZKOWSKI
und Dr. U.KASPEB
Zentralinst. f. Astrophysik d. AdW der DDR
DDR-1602 Potadam-Babelsberg. Rosa-Luxemburg-Str. 17a
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