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Chapter 3
GEOMETRY AND
TOPOLOGY OF SPACE-TIME
3.1. Geometric-topological properties of
spacetime
3.2. Minkowski
planar spacetime and asymptotic structure
3.3. Cauchy's
problem, causality and horizons
3.4. Schwarzschild
geometry
3.5. Reissner-Nordström
geometry
3.6. Kerr
and Kerr-Newman geometry
3.7. Spatio-temporal singularities
3.8. Hawking's
and Penrose's theorems on singularities
3.9. Naked
singularities and the principle of "cosmic censorship"
3.7. Spatio - temporal
singularities
Regular and
singular solutions
In classical physics, bodies and fields have the usual and
expected physical, geometric and topological properties. They are
(relatively) smooth and continuous, have finite dimensions and
finite values of their physical quantities. We call such behavior
regular. And it can be mathematically modeled
using regular imagings , which are simple mutually
unambiguous imagings for which unambiguous inverse
imagings are available.
^{ }However, in mathematical modeling in
theoretical physics, we also encounter situations where the
relevant equations diverge and formally give infinite
or indeterminate values of physical quantities. A simple example
is the idealization of a point electric charge, where according
to Coulomb's law there is an infinitely large electric field
intensity in a place with zero distance (r = 0) from this charge.
Such anomalous behavior of a physical quantity is generally
referred to as singular (it is the opposite of
regular behavior). And the place or point of anomalous behavior
of a physical quantity is called singularity (Latin singularis = isolated, exceptional, unique
) .
^{ }In the analysis of some exact solutions of
Einstein's gravitational equations (eg Schwarzschild's solution)
we saw that in some places these solutions are
not regular.
We have seen that some of these "singularities" are
caused only by an inappropriate coordinate
system and the transition to other coordinates can remove the
relevant singular behavior (an example can be pseudosingularity of Schwarzschild
sphere). In such cases, therefore, these
are not the
singularities of spacetime itself (spacetime is regular here).
^{ }However, there are places where singular
behavior cannot be removed by switching to another coordinate
system, such as the point r = 0 in Schwarzschild geometry. In
such a case, it is a real, physical singularity of the spacetime
itself (where, for example, the invariants of the curvature
tensor reach infinite values). Under the singularities of
spacetime in the following we will understand only similar
"irremovable" singularities. Here we will clarify some
basic features of space-time singularities and try to give a general definition. We will talk about the conditions under
which singularities arise in the next §3.8 "Hawking's
and Penrose's theorems about singularities".
Physical
Unreality of Singularities
In this and other mathematically oriented chapters of our book
(§3.4-3.9, §5.3), we deal with singularities "seriously as
if they did exist". However, it should be borne in
mind that these are only mathematical
abstractions.
These singularities arise as a consequence of the
"straightforward" strict extrapolation of exact
solutions of the equations of the general theory of relativity to
certain special values of coordinates (eg to
t = 0, r = 0).
^{ }Real "physical" singularities,
in which real physical quantities would take on infinite values, cannot exist in nature ! We never see true infinity in
nature (the analysis of different types of infinity was performed
in §3.1, in the section "Infinity
in spacetime"). If a theory predicts them, it is
its serious shortcoming, indicating either its erroneousness or
excessive idealization, or that in a given situation it reaches
the limits of its validity. If infinity appears in a theory, we
try to remove it (the process of renormalization in quantum theory), at least
until the creation of a new more sophisticated theory.
^{ }It turns out that in those
"exotic" situations, play an important role the quantum laws of gravity as other kinds of interactions, wich emerge
immanently (as a consequence of quantum field fluctuations in
vacuum) together with "classical" gravity. The inclusion of these quantum interactions in the
general-relativistic gravitational model can remove
this singularity - spacetime does not have
to be singular here, it can have a very high but finite
curvature, at finite field concentration, finite density and
temperature of the gravitational substance. From a quantum point
of view, it can be expected that the singularity will "dissolve" in
the quantum foam. This is, yet indeterminate, a
view of quantum gravity and quantum geometrodynamics (§B.4 "Quantum
Geodynamics") .
^{ }Nevertheless, it is useful to investigate
mathematically under what circumstances in (non-quantum) theory
singularities arise and what properties they have, of which
something could "potentially" be applied in practice.
Let us first compare the
situation around singularities in gravity with the corresponding
situation in electrodynamics. Singularities also occur in
electrodynamics, eg in the place of a point electric charge (classical idealization!) the intensity
of the electric field reaches infinite values. However, the
metric of Minkowski spacetime, against which (and with the help
of) the electric field we monitor (and measure) remains regular
everywhere *). In contrast, in the general theory of relativity,
we do not have a "non-participating" metric with which
we monitor the behavior (and thus possible singularities) of some field.
Here, the space-time metric is the same field whose singularities
we investigate. Spacetime singularity therefore has much deeper
implications than the singularity eg. in clasical electrodynamics. In space-time
singularity, where regular space and time cease to exist, all
physical laws lose their validity, because all existing physical
laws are formulated on the basis of classical space and time.
*) This is the case in STR
when we take the electromagnetic field in isolation. In reality,
however, the electromagnetic
field also evokes a
gravitational field (curves spacetime), so that in the place of infinite
intensity of electromagnetic field,
an infinitely strong gravitational field (infinite curvature of spacetime), and thus spacetime singularity would be
created.
^{ }The singular point cannot be considered as
a part of the observed space-time manifold, in principle no physical
measurements can be performed in it. In order for spacetime to
remain a manifold, all singular points of it are necessary exclude
("cut out"). The remaining spacetime will then again be
a manifold, where the usual physical laws will apply
at each point and where physical measurements can be made.
However, the remaining spacetime cannot be declared regular, and we
cannot think that we have completely got rid of
the singularity. As we will see below, this is remaging spacetime has certain "pathological" properties
(related to the fact that some geodesics ends its existence in
places "carved" singular points), preventing consider him regulars.
Let us try to capture
the common characteristics of spacetime singularities. In other
words: what fundamental "dangers" can they wait for a
real observer (moving along world lines of the time or light type)
when moving through space-time? For the time being, we will limit
ourselves to moving around geodetics for simplicity. We have seen
an example of one such danger in the analysis of particle motion
in the Schwarzschild geometry of a centrally symmetric
gravitational field. If the observer comes under the
Schwarzschild sphere, his destiny is sealed - in a short interval
of his own time he will necessarily reach the point r = 0, where
he will be destroyed by infinite tidal forces.
^{ }Thus, a clear sign of the singularity
could be the singularity of the curvature^{ }-
unlimited large curvature of spacetime in
the vicinity of singular point. However, in the exact analysis,
we encounter the problem that the spacetime around the singular
point is not regular (the singular point itself is even excluded
from the observed spacetime), so it may not be entirely clear
what is "close surroundings" and "infinitely large
curvature". Since the numerical values of the curvature
tensor components depend on the reference frame used (vector base
that can degenerate near the singularity), it is necessary to
observe either the behavior of scalar polynomials (independent of
the base used) formed from the metric tensor components g_{ik} and curvature tensor R^{i }_{klm}, or measure the components of the
curvature tensor R^{i }_{klm} at the base transmitted in
parallel along the investigated worldline from a safely distant
regular region to the investigated points near the singularity.
Fig.3.26. Incomplete geodtic G, which ends after the finite proper time (or at the finite value of the affine parameter) and can no longer be extended within M, is a sign of spacetime singularity. |
However, singularities
have another common feature, which is simpler and more
characteristic. It is important that the observer's worldline,
when encountering a singular point (eg r = 0 in Schwarzschild's
geometry) ends discontinuously here for a finite interval
of its own time. This geodetic cannot be further extended
within the existing manifold, the object describing the geodetic simply
ceased to exist for it and can therefore no longer be described
by any world point from the given manifold (Fig.3.26). We can say that, for
example in the Schwarzschild spacetime geodesic
there are geodesics (especially, all those geodesics that
intersect the Schwarzschild sphere), which have only a finite total
length and discontinuously ends after the finite
interval of their own time (or at the final value of the
affine parameter in the case of isotropic geodesics). Such
geodesics are "incomplete". The concept of completeness and incompleteness can be extended from geodesics
to general worldlines, while the affine parameter is replaced by
a generalized affine parameter *).
*) The generalized affine
parameter is introduced as follows [225], [127] :
1. At the point p Î M , through which the curve C passes: x = x
(t), the three-dimensional vector base ( e 1, e 2, e 3) in the tangent space is chosen.
2. Parallel transmission along curve C a vector base can be obtained at each
point of the curve (for each value of t).
3. The tangent vector V(t) = ( ¶/¶t)_{x(t)} at
each point of the curve can be expressed using the transferred
base: V = V^{a}(t). e_{a} .
4. The generalized affine parameter is then defined by the
integral l = _{p} ò^{x(t)} (Vi V^{i} )^{1/2} dt; depends on the point p and on the vector base e_{a} at the point p .
If x(t) is geodetic, then l
is
affine parameter on C: x = x(t). Generalized affine parameter l but
can be implemented on any curve in M .
Definition 3.7 |
World lines G is called
complete (full) in
space (M , g), if in the M is defined for each value of
the affine parameter (own time to time trajectory type) lÎ (-¥, +¥). Otherwise, a worldline that has a start or end point (beyond which it can no longer be extended within M) at the final value of the affine parameter l (proper time for time-type curves) is called incomplete . |
Definition 3.8 |
Spacetime (M , g) is
called complete (or
geodetically complete) if every worldline (or every
geodesy) in M is complete. Otherwise, M is called incomplete (or geodetically incomplete). |
In other words, in a complete spacetime M each world lines G(l) of final "length" (as
measured by the generalized parameter affine l) has an endpoint in M - is extensible.
^{ }From the example of Schwarzschild geometry
(which is obviously singular) it can be seen that geodetic
incompleteness - ie the existence of incomplete temporal and
isotropic geodesics - is a sufficient condition for the
singularity of spacetime. From the physical point of view, we
must consider as singular also such spacetime, which is geodetically
complete, but in which there exist
incomplete time-similar world lines (not-geodetics). The physical significance of
incompleteness with respect to spatial-type world lines, on which
no real object can move, is not clear, and therefore we will not
consider it here.
^{ }Thus, we can finally state the general
definition of spacetime singularity :^{
}
Definition 3.9 (singularity of spacetime) |
Non-extensible spacetime (M, g) is
called singular if it
contains incomplete worldlines of the time or isotropic
type. The points at which these worldlines end at the finite value of the affine parameter (proper time) and beyond which they can no longer be extended within M, are called spacetime singularities . |
The condition
"non-extensible" is in definition 3.9 because each
extensible spacetime automatically contains incomplete
worldlines. For example, if we take only a part of the planar
(and of course regular) Minkowski spacetime, there will be
incomplete geodesics that will end (or begin) at the
"edge" of our defined spacetime. Of course, this
phenomenon would not
be reasonable consider as a singularity. The first sentence in
definition 3.9 can also be equivalently stated as follows :
^{ }Spacetime (M, g) is called
singular, if each of its
extensions contains incomplete
worldlines of the time or isotropic type.
^{ }According to definition 3.9 we have therefore chosen as a sign
of the singularity of spacetime the presence of a worldline,
which at some point ends in its finite proper time and cannot
continue. In the next §3.7 it will be shown under what
assumptions such defined singularities in the spacetime of the general theory of relativity arise.
It remains to clarify the physical properties and structure of singularities. It is not easy to build a "physics of singularities", because singularities are in themselves something "non-physical" that must be excluded from the physical manifold M in order for it to remain a manifold. One of the methods of singularity analysis was proposed by Geroch [95] and Hawking [130] and further improved by Schmidt [225]. Each incomplete light line is assigned an appropriate "end point", and those endpoints which belong to such world lines G_{1} and G_{2} are considered to be identical so that the world line G_{2} enters the space-time region formed around the word line G_{1} its small variations, and it already remains there (Fig.3.27). Such identification creates classes of equivalence of endpoints of incomplete worldlines, and the set of all these classes of equivalence creates a certain boundary ¶M around the singularity. This boundary is uniquely determined by the structure (M,g), i.e. it can be determined by measuring the non-singular points of M .
Fig.3.27. Around the singularity is constructing the boundary ¶M, consisting of equivalence classes "endpoints" incomplete world lines in M. The endpoints of such worldlines G_{1} and G_{2} are considered to be identical (i.e. belonging to the same equivalence class), where G_{2} enters the dashed area formed around G_{1 }by small variations and no longer leaves it. |
That the question of the
definition and classification of spacetime singularities is by no means trivial,
we can show by a simple example. If we consider only relativistic
kinematics, we can imagine a rocket that accelerates so much in
ordinary Minkowski spacetime, that its worldline becomes
isotropic (the rocket reaches the speed of light) so fast that
the total interval in the observer's last time in the
rocket will be finite - the worldline will have a finite
"length". After the end time
expiry, it will no
longer be possible to display the rocket by any world point in
the given Minkowski spacetime. However, this situation is
physically unfeasible because a rocket moving after such a saint
would have to have an infinite amount of fuel. However, Geroch
[96] constructed spacetime, which is geodetically complete but
contains incomplete worldlines of the time type with finite
acceleration; from such spacetime it would be possible to
"fly away" on a rocket with a finite supply of
fuel.
^{ }Intuitively, we expect that the
singularity is related to the infinitely large curvature of
spacetime. Indeed, definition 3.9 includes this, because one of
the reasons why a world line cannot be extended beyond a certain point, there
may be an infinitely large curvature of spacetime near this
place. However, the incompleteness of spacetime may not always be
due to the singularity of the curvature. Examples have been
constructed, such as Taub, Newman, Tamburin, and Unti spacetime
[188], [127], which satisfies the definition of singularity 3.9
(it contains horizons beyond which some geodesics cannot be
extended), but has finite curvature everywhere. The
incompleteness of geodesics here has a metric rather than a
topological character. However, spacetime of this kind is a
purely artificial construction and cannot be used if there is any
matter in spacetime. In general, it can be expected that in
physically real situations, the singularities of spacetime will
actually be caused by infinitely large curvature, or its discontinuity (but let us still
keep in mind the above discussion "Physical
unreality of singularities"!).
Gravity, black holes and space-time physics : | ||
Gravity in physics | General theory of relativity | Geometry and topology |
Black holes | Relativistic cosmology | Unitary field theory |
Anthropic principle or cosmic God | ||
Nuclear physics and physics of ionizing radiation | ||
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