AstroNuclPhysics ® Nuclear Physics - Astrophysics - Cosmology - Philosophy | Gravity, black holes and physics |
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"
3.4. Schwarzschild geometry
As already mentioned in §2.5, the problem of solving Einstein's equations is greatly simplified, if we prescribe a high degree of symmetry; this reduces the number of unknown functions (metric coefficients) and equations. Indeed, the first exact (and at the same time non-trivial) solution of Einstein's gravitational equations was found already in 1916 by K.Schwarzschild for the spherically symmetric case. In §4.3 we will see that this solution describes a spherically symmetric black hole.
Centrally
symmetric Schwarzschild solution
For a centrally
symmetric distribution of matter, we assume that the excited
gravitational field will also be centrally symmetric. First, we
will consider the case where the centrally symmetric distribution
of matter does not change over time and the gravitational field
is static (however, we will see below that
this assumption is not necessary, the result will be the same
also for the non-static case while maintaining central symmetry). Then all physical quantities
will be a function of only the distance from the center of
symmetry. Furthermore, we will assume that spacetime is asymptotically planar - at sufficiently large distances it
gradually changes into planar Minkowski spacetime.
In this case, it is natural to connect the frame of reference to the center of symmetry and to use spherical spatial coordinates (r, J, j) with the origin r = 0 also at the center of symmetry. The spacetime interval can then be searched for in the form
ds ^{2} = - A (r) .dt ^{2} + B (r) .dr ^{2} + r ^{2} (d J ^{2} + sin ^{2} J d j ^{2}) , | (3.10) |
so metric coefficients
g _{tt} = -A (r) , g _{rr} = B (r) , g _{JJ} = r ^{2} , g _{jj} = r ^{2} sin ^{2} J | (3.11) |
are only a function of the distance r from the center of symmetry. Coordinates of this type are called Schwarzschild coordinates. These are essentially spherical coordinates (r, J, j) spatial, where the radial coordinate r is defined as the proper length of the respective circle (with the center at the center of symmetry r = 0) of radius r divided by 2p, or square root of the sphere's own surface divided by 4p. The coordinate time t is measured with respect to a distant, asymptotically planar region.
Assume further that the source body is spatially limited and extends only to the distance R ; at r> R there is already a vacuum (T^{ik} = 0). First we will look for the so-called external solution, ie the space-time metric for r> R outside the body - in a vacuum. It is therefore necessary to solve the vacuum Einstein equations R_{ik} = 0. The Christoffel connection coefficients in the metric (3.10) are
(3.12) |
Einstein's equations R_{ik} = 0 then after adjustment give only two independent differential equations
dg _{tt} / dr = ( 1 / r) g _{tt} (1 + g _{rr}) , dg _{rr} / dr = - (1 / r) g _{rr} (1 + g _{rr} ) ,
which have a solution
g _{tt} = C _{1} . (1 + C _{2} / r) , g _{rr} = (1 + C _{2} / r) -1 ,
where C_{1} and C_{2} are integration constants. We choose the constant C_{1 }by setting the scale of the time coordinate t (by transforming t = C_{1} .t we get rid of the constant C_{1}). From the limit transition to Newton's law (g_{tt} ~ (1 - M/r) for r®¥) we get that the constant C_{2} must be equal to -2M, where M is the mass of the source body (ie the total mass enclosed in a sphere of radius r at r ®¥ ). We thus obtain an exact solution of Einstein's equations in vacuum for a spherically symmetric gravitational field; in this so-called Schwarzschild geometry, the space-time interval element has a shape
(3.13) |
in geometrodynamic units; in common units it is
(3.13 ') |
The components of the metric tensor g_{ik} in Schwarszchild's geometry are thus (in geometrodynamic units) :
g_{ik} = | / | -(1-2M/r) | 0 | 0 | 0 | \ | . | |||
| | 0 | (1-2M/r)^{-1} | 0 | 0 | | | |||||
| | 0 | 0 | r^{ 2} | 0 | | | |||||
\ | 0 | 0 | 0 | r^{2 }sin^{2 }J | / |
So far, we have considered a static case and assumed that the gravitational field will be static. If we omit this assumption and accept the possibility of the dependence of metric coefficients (3.11) on time while maintaining spherical symmetry, ie g_{tt} = - g_{tt}(r, t), g_{rr} = g_{rr}(r, t), from Einstein's equations R_{ik} = 0 we get after adjusting that ¶g_{rr}/¶t = 0 (so g_{rr} does not depend on time) and g_{tt} = f(t)/g_{rr}, where f(t) is a function of time only. Because for r®¥ the investigated metric passes into the Minkowski metric, where g_{tt} = -1 and g_{rr} = 1, must be f(t) = const. and therefore neither g_{tt} depends on time. We come to the result that the centrally symmetric gravitational field in a vacuum must be automatically static, even if the excitation body shrinks, expands or pulsates radially (while maintaining the exact spherical symmetry). The obtained results can be summarized in the following statement :
Theorem 3.3 ( Schwarzschild-Birkhoff theorem ) |
In
asymptotically planar spacetime, the centrally symmetric
gravitational field in vacuum is described by
Schwarzschild geometry with metric form (3.13) in
Schwarzschild coordinates. This field is therefore static and is determined by only one parameter - the total mass M. |
This finding is very important in the study of gravitational collapse and the properties of black holes (Chapter 4) - this is actually a special case of Theorem 4.1 "black hole has no hair" (derived in §4.5 "Theorem" black hole has no hair"").
Internal metric of a
centrally symmetric static body
What is it inside a normal body (stars, planets) exciting a
gravitational field? In the simplified case of a non-rotating
body of mass M and radius R, consisting of an
incompressible substance with constant density
(ideal incompressible fluid) with zero surface pressure, the
geometry of spacetime is given by a somewhat more complex
so-called internal Schwarzschild solution, which
we present here without derivation:
ds^{2} = ^{1}/_{4} [3Ö(1-2GM/c^{2})/R) - Ö(1 - r^{2}.2GM/c^{2}/R^{3})]^{2} c^{2}dt^{2} - (1 - r^{2}.2GM/c^{2}/R^{3})^{-1}dr^{2} + r^{2}(dJ^{2} + sin^{2}J dj^{2}) . | (3.13b) |
This metric is regular everywhere. For r> R it passes smoothly into the external Scharzschild geometry (3.13).
Schwarzschild
geometry of spacetime - movement of bodies
Let's analyze properties (outer) Schwarzschild's
geometry. Since we will be interested in the whole
Schwarzschild's external solution, we will assume that the
excitation matter is concentrated at
the center r = 0 and there is a vacuum everywhere else. If
we look at the space-time element of Schwarzschild geometry
(3.13), it can be seen at first glance that it is not regular everywhere. At r®2M,
the time component of the metric tensor g_{tt}®0 and the spatial radial component g_{rr}®¥. Radius_{ }
r _{g} = 2 M º 2 G.M / c ^{2} | (3.14) |
is called the gravitational or Schwarzschild radius and the corresponding sphere r = r_{g} = 2 M the Schwarzschild sphere. Of course, the metric (3.13) is also singular for r = 0. To clarify the nature of these "singularities" of the Schwarzschild metric (and we will see that there is a fundamental difference between the two cases), let us first look at the motion properties of test particles in Schwarzschild spacetime.
The motion of a free test particle in a Schwarzschild field is given by the equation of geodetic (2.5b)
where the connection coefficients G^{i}_{kl }are calculated from the components of the metric tensor in (3.13) :
(3.15) |
the other components are either symmetrical with them or equal to zero. Spherical symmetry allows us to set J = p/2 for the plane of motion (follow the motion in the equatorial plane), thus eliminating the equation for the J coordinate. The remaining equations are
(3.16) |
Dividing the first equation dt/d t gives
or
d t = K ( 1 - 2 M / r ) dt , | (3.17) |
where the integration constant K is related to the velocity of the particle (see below). An analogous modification of the second equation gives ln (dj/dt) + 2.ln r = const., from which we get the equation
r ^{2} d j / d t = const. = ^{def} ` L | (3.18) |
expressing the law of conservation of angular momentum` L. From the third equation (3.16) we then get
(dr / d t ) ^{2} = 1 / K ^{2} - (1 - 2M / r) (1 + ` L ^{2} / r ^{2} ) . | (3.19) |
Radial
motion of a particle
Let us now have a particle falling radially in the Schwarzschild
field in the direction of the center r = 0. In equation (3.19)
there will now be ¯L = r^{2}dj/dt = 0, so (dr/dt)^{2} = 1/K^{2} - 1 + 2M/r. In the limit r®¥ we get (dr/dt)^{2}
= 1/K^{2} - 1, so 1/K^{2} - 1 is equal to the square of
its own velocity v_{¥} , which the particle would have
at an infinitely large distance from the center. So
(dr / d t ) ^{2} = v _{¥ }^{2} + 2.M / r , | (3.20) |
which coincidentally is the same result as in Newton's theory. From this equation we can calculate the interval of the proper time t, that the particle needs to get from some (final) distance r = r_{2} to a distance r = r_{1} from the center :
(3.21) |
This interval of the proper time is always finite, even for r_{1} = 0. Thus, the free-falling test particle reaches both the "critical" radius r = 2M and even the point r = 0, in a finite intrinsic (proper) time.
To determine the corresponding interval of the coordinate time t (which is the observer's own time at infinity), we use equations (3.19) and (3.17), which gives :
(dr / dt) ^{2} = (1 - 2M / r) ^{2} - K ^{2} (1 - 2M / r) ^{3} . | (3.22) |
The limit transition r®¥ shows that 1-K^{2} has the meaning of the coordinate velocity v_{¥} at infinity. The interval of the coordinate time required to move from distance r_{2} to distance r_{1} is then equal to
(3.23) |
The interval of the
coordinate time required to reach the point r = r_{1}
is close to infinity at r_{1}®2M *).
*) Really infinitely
long time?^{ }
It should be noted that from a physical point of view, this
result of infinitely long time is only "mathematical
fiction" - it applies to the idealized case of
"intangible" test particle of point
dimensions (which does not affect the
studied spacetime). The motion of such
particles is suitable be analyzed to study the
geometric-topological properties of a given spacetime and the
coordinate systems used.
^{ }For real material bodies
falling on a black hole the situation will be somewhat different.
When such a body approaches a black hole, the horizon
will be deformed by mutual gravity, which will (due to the "bump" or protrusion) of the incident body "come out to meet".
The extreme case of such a process in the fusion of black holes
is shown in enlarged sections in Fig.4.13-GW in §4.8, passage
"Binary
gravitationally coupled systems of black holes. Precipitation and
fusion of black holes".
The real effective time of absorption of a body
by a black hole, or the merging of two black holes into one, is
therefore finite.
Relationships (3.21) and (3.24) can be
summarized more simply :
Particles moving from a distance r_{0 }>_{ }r_{g}
radially in Schwarzschild spacetime towards the center (to the horizon, to a black hole)
exceed the event horizon r_{ }=_{ }r_{g} for a finite
proper time interval t = (2r_{g}/3c). [(r_{0}/r_{g})^{3/2
}-1]; then it will
continue to move until the central singularity r=0, and it will
also travel this trajectory in finite proper time. From the point
of view of measurement by an external observer, each
interval of the proper time Dt is extended to the interval Dt = Dt/(1-r_{g}/r)^{1/2 }by dilation of
time; the closer the particle approaches the horizon, its
movement slows down from the point of view of an
external observer and on the horizon r_{
}=_{ }r_{g}
it stops completely, it "freezes" for an
infinitely long time. This different temporal dynamics of radial
motion of particles in external and internal view is caused by
the effect of gravitational dilation of time in the general
theory of relativity (we discussed it in §2.4
"Physical laws in curved spacetime", passage "Space and time in the gravitational
field").
^{ }Thus, we come to the conclusion
that from the point of view of an external observer, each test particle
slows down when approaching the radius r = 2M and needs an infinitely
long coordinate time to achieve it (taking
into account the above reservation). On the other hand, from the point of
view of the observer falling together with the particle, the
radius r = 2M is reached for a finite interval of its own time,
the particle passes freely through this point and then reaches
the center r = 0 for a finite own time. And in the same way will
behave the surface of a gravitationally
collapsing star into a black hole, as will be analyzed in §4.2, passage
"Two different views of
gravitational collapse - external and internal".
Coordinate and physical singularity
If we calculate the components of the Riemann curvature tensor in
Schwarzsehild spacetime on the basis of the connection
coefficients (3.15) and transform them into the reference system
of a falling observer, (those of which are non-zero) will be
proportional to M/r^{3}, for
example
R_{trtr} = -2M/r^{3} , R_{t}_{J}_{t}_{J} = R_{t}_{j}_{t}_{j} = M/r^{3 }, R_{JjJj} = 2M/r^{3} , R_{r}_{J}_{r}_{J} = R_{r}_{j}_{r}_{j} = -M/r^{3} , | (3.24) |
so in the Schwarzschild sphere they reach values of the order of 1/M^{2}; similarly scalar invariants (eg R= R_{iklm}R^{iklm}= 48M^{2}/r^{6}) of the curvature tensor, which do not depend on the coordinate system. The curvature of spacetime, and thus the gradients of gravitational forces (tidal forces), are finite in the Schwarzschild sphere - and the smaller, the greater the mass parameter M.
We conclude that the singular behavior of the Schwarzschild spacetime element (3.13) for r = 2M does not have its origin in the singular character of the geometry of spacetime on the Schwarzschild sphere, but is caused by the Schwarzschild coordinates used, which are not suitable here. By switching to another suitable reference system, such as a system associated with free-falling test particles, this pseudosingularity in the Schwarzschild sphere disappears.
How an unsuitable coordinate system can cause an seeming singularity can be illustrated by a simple example according to Fig.3.15. If we follow the spherical surface using coordinates indicating the "geographical" width and length of the J and j (Fig.3.15a), all "meridians" will converge at the points of the "northern" and "southern" pole, so these points are not geographical length defined, the metric tensor g_{jj} is equal to zero here. Or similarly, if we make a map of the globe using a cylindrical projection according to Fig.3.15b, the images P' of the pole P will be at infinity and the metric tensor g_{JJ}®¥. However, the geometry of the spherical surface in these poles is completely normal - just turn the globe by a certain angle and the points that appeared singular in the previous coordinate system will be completely regular and other points, the new "poles", will appear singular.
Fig.3.15.
An example of pseudosingularity caused by an
inappropriate coordinate system. a ) The point P on a spherical surface (pole) appears to be singular because no "longitude" is defined in the coordinates used for it. b ) If we make a map of the globe using a cylindrical projection, the pole P will appear singular, because its image P' will be at infinity. |
The geometry of spacetime itself is completely regular in the Schwarzschild sphere, the observer can freely pass through the Schwarzschild sphere during the finite interval of his own time, he will not find anything special here locally and will continue his movement. The peculiarity of space-time geometry in the Schwarzschild sphere, therefore, does not lie in some abnormal local properties, but as we will see below, it has a significant global property - it is a horizon of events.
Otherwise it is with the second singular m place in the metric (3.13) - the point r = 0. Here both the components of the curvature tensor (3.24) and its scalar invariants reach infinite values, so there are infinitely large gradients of gravitational forces. A particle that reaches the point r = 0 can no longer continue its motion, is crushed by these infinite tidal forces, it literally ceases to exist within a given spacetime. This is a real, physical singularity of the space-time geometry, which cannot be removed by any choice of the frame of reference. The properties of spacetime singularities and the assessment of the possibilities of their existence are devoted to §3.7 and 3.8.
Movement
of light in the Schwarzschild field
We will not continue the analysis of the movement of test
particles in the Schwarzschild field, we will move it to §4.3,
where this analysis will have a direct physical connection with
the properties of black holes. Here we focus more on the
geometric properties of Schwarzschild's solution. For this
purpose, we will show some properties of light
propagation
in the Schwarzschild field (for the
geometric-topological properties of the Schwarzschild
gravitational field, the propagation of light in the radial
direction is important). The coordinate velocity c_{r }of light in the radial direction is
obtained if we set ds^{2} = 0 in the
expression for space-time interval (3.13) _{ }at
dj = dJ = 0
c _{r} = dr / dt | _{(ds = 0)} = 1 - 2M / r . | (3.25) |
In the Schwarzschild sphere, the coordinate speed of light in the radial direction is zero. The time it takes light to get from a point with the coordinate r = r_{1} to a point r = r_{2}, will be according to (3.25)
(3.26) |
For r_{1}® r_{g} = 2M, this time approaches
infinity for any value of the target r_{2} > r_{1}. The light emitted from the
Schwarzschild sphere r = r_{g} takes an infinitely
long time
to reach any place farther from the center. From places with
coordinates r < r_{g} - from under the Schwarzschild sphere -
therefore, no object can get out, because even light cannot get
out of there (and no object can move faster than light). Thus, no
event that takes place inside the Schwarzschild sphere can
manifest itself from the outside and
it cannot be observed in
any way from the outside. Schwarzschild's sphere is therefore the horizon of events, which in the sense of causality separates the inner area from the rest of
spacetime (see §3.3).
Note:^{ }This
event horizon is also sometimes referred to as the Killing
horizon, because the Killing vector field x_{o} º ¶/¶t (introduced in §2.4), its time component, changes the
space-time character here: outside the horizon it is of the time
type, while inside the horizon acquires a spatial character.
^{ }For the gravitational spectral shift (red shift) of light in the radial direction in the
Schwarzschild metric follows the simple formula w_{A} = w_{r }.(1 - 2M/r)^{1/2}, where w_{r} is the frequency of the signal
sent by the static observer from the distance r
from the center of the black hole and w_{A} is the frequency of this signal
detected by the remote observer (at infinity). This remote
observer will detect this signal with a smaller value of
frequency w_{A} due to the fact that this signal loses
energy as it overcomes the gravitational field of the black hole.
If a signal of any frequency w_{r}
were sent from the Schwarzschild sphere r=2M, then w_{A} = 0 - no signal would be detected... From a
general relativistic point of view, the issue of gravitational
spectral shift is discussed in §2.4, passage "Gravitational spectral shift".
In the region of the Schwarzschild horizon the components of the metric tensor g_{rr}=(1 - 2M/r)-1 and g_{tt}= -(1 - 2M/r) change their signs: for r> 2M is g_{rr} > 0 and g_{tt}< 0, while at r <2M is g_{rr}< 0 and g_{tt} > 0. It can be said that the time t and radial spatial r coordinates, in a sense, they exchange roles with each other. Inside the horizon, the role of the flow of time into the future is taken over by the continuous shrinking of r. The space-time light cones are completely turned inwards here, so that every real body will move here so that r is constantly decreasing - it must therefore "fall" towards the center r=0 (Fig.3.16). The catastrophic consequences of this phenomenon are shown in Chapter 4 on the gravitational collapse and black holes.
Fig.3.16. Spacetime diagram of radial motion of test particles and photons in Schwarzschild spacetime using ordinary Schwarzschild coordinates. |
Curvature
of light rays - the effect of a gravitational lens
For the analysis of the
geometrical-topological properties of the Schwarzschild
gravitational field, which we are dealing with here, the
propagation of light in the radial direction is
important. From astrophysical standpoint,
however, important properties of the electromagnetic radiation
propagation (and therefore light) in the azimuthal direction
- gravitational bending of photon trajectories
and curvature of light rays when passing around
gravitational bodies and systems. For the propagation of light in a
gravitational field and the effect of "lensing" will be discussed in more detail in §4.3, section "Gravitational lenses. Optics of black holes".
Analytical extension of Schwarzschild
geometry
Because the apparent singularity on the horizon of Schwarzschild
geometry is caused only by the nature of the reference system
used (Schwarzschild coordinates), it is offered to be
analyzed the Schwarzschild
spacetime using coordinates that do not have this unpleasant
property. The simplest in terms of "practical"
implementation would
be a frame of reference associated with the radially falling test
particles. Although such a system does not have a coordinate
singularity, it is not suitable for the study of global geometric
properties (too complex and confusing
transformation relations). For monitoring the geometric
properties of Schwarzschild spacetime, the so-called Kruskal-Szekeres coordinate system [160] is very advantegeus, which was created
by coupling the previously
introduced Eddington-Finkelstein shrinking and expanding
coordinate system [76], [84]. For this purpose, the
modified r* coordinate is first introduced
(3.27) |
[ rÎ(0, +A) T r*Î(-A, +A) ] and further isotropic coordinates p and q :
p = t + r * , q = t - r * . | (3.28) |
The significance of these coordinates is that, the outward light geodesics are given by the equation q = const. and inward-facing geodesics by the equation p = const. The Schwarzschild metric in coordinates (q, r, J, j) then has the form
ds ^{2} = - ( 1-2M / r) dq ^{2} + 2dq dr + r ^{2} (d J ^{2} + sin ^{2} J d j ^{2} ) ;
this so-called shrinking Eddington-Finkelstein coordinate system well (without pseudosingularity) describes the fall of particles below the gravitational radius, but not in the opposite direction. The expanding Eddington-Finkelstein coordinate system uses the coordinates r and p instead of r and t , so the expression for the metric here is
ds ^{2} = - ( 1-2M / r) dp ^{2} + 2dp dr + r ^{2} (d J ^{2} + sin ^{2} J d j ^{2}) ;
here, on the contrary, the outward movement of the particles is well described, while when the particles fall below the gravitational radius, pseudosingularity appears. To remove pseudosingularities and obtain a complete extension of the Schwarzschild geometry, it is proposed to combine both of these systems, ie to exclude r and use the coordinates (p, q, J, j). The space-time element of Schwarzschild geometry then has a shape
ds ^{2} = - (1-2M / r) dp dq + r ^{2} (d J ^{2} + sin ^{2} J d j ^{2}) .
To remove the singular coefficient 1-2M/r, it is also necessary to perform a suitable transformation p' = p'(p), q' = q'(q). Kruskal found such a transformation in the form
p' = e ^{p / 4M} , q' = e ^{-q / 4M} .
In these new coordinates, the metric has a shape
ds ^{2} = - (16.M ^{2} /r).e ^{-r / 2M} dp' dq' + r ^{2} (d J ^{2} + sin ^{2} J d j ^{2}) .
To get this metric in the usual form with separated temporal and spatial members, we finally introduce instead of the isotropic coordinates p' and q' the temporal u and the spatial v coordinate :
u = (q ' - p') / 2 , v = (q ' + p') / 2 .
In the Kruskal system, therefore, the coordinates t and r are replaced by dimensionless temporal and radial spatial coordinates u and v by means of a transformation
u = |r - 2M|^{1/2} e^{ r/4M} cosh(t/4M) , v = |r - 2M|^{1/2} e^{ r/4M} sinh(t/4M) for r > 2M | (3.29) |
and analogously for r < 2M by exchanging "cosh" for "sinh" and vice versa. So, in fact, they are two consecutive coordinate maps (similar to, for example, the planar representation of the globe). The space-time element of Schwarzschild geometry then has a shape in Kruskal coordinates
ds ^{2} = - (16.M ^{2} /r).e ^{-r / 2M} (du ^{2} - dv ^{2}) + r ^{2} (d J ^{2} + sin ^{2} J d j ^{2}) , | (3.30) |
where r as a function of u and v is given by the equation
(r - 2M) e ^{r / 2M} = u ^{2} - v ^{2} . | (3.30 ') |
It can be seen from (3.30) that the metric of Schwarzschild geometry in Kruskal coordinates is regular everywhere except for the center r = 0 (where it is a real physical singularity that cannot be removed by any transformation of the coordinate system).
Fig.3.17. Kruskal's spacetime diagram of Schwarzschild geometry.
a ) Coordinate network
u, v in relation to
the Schwarzschild coordinates r and t .
b ) Overall structure of
spacetime and motion of test particles and photons. A is the outer asymptotically planar region,
B is the inner region
below the horizon.
Some basic features of Schwarzschild geometry in Kruskal-Szekeres coordinates are schematically shown in the Kruskal diagram (which is a space-time diagram in Kruskal coordinates u and v) in Fig.3.17b, which can be compared with the corresponding diagram in Schwarzschild coordinates in Fig.3.16. Above all, the singularity r = 0 is given here by the relation v^{2} - u^{2} = 1, which describes two separate singularities as shown by the hyperbolas in Fig.3.17. The horizon r = 2M is formed here by two lines u = ± v. The outer region r> 2M is expressed by unequality u^{2} > v^{2}, which again describes the two outer areas. For radial zero (isotropic) geodesics ds = 0 we get du = ± dv; these radial light geodesics are therefore straight lines at an angle of 45° to the axes of the Kruskal system. This property is very advantageous because the light cones look exactly the same as in the Minkowski planar spacetime diagram STR. Thus, in this Kruskal diagram, real masses can only move at an angle of less than 45° from the vertical axis v (inside space-time light cones). The analysis of the geometric properties of Schwarzschild spacetime and the observation of causal relations and motion of material objects is therefore much easier and more illustrative than in ordinary Schwarzschild coordinates (if we compare Fig.3.16 and 3.17b, we see that the movement of test particles and photons looks much clearer on the Kruskal diagram).
It is very strange that from the original Schwarzschild solution containing only one asymptotically planar outer region r®¥ , one horizon r = 2M and one singularity r = 0, we got spacetime with two outer regions, two horizons and two singularities by switching to Kruskal coordinates. The explanation is that the space-time obtained by the transition to Kruskal coordinates is a complete analytical extension of the original Schwarzschild geometry, ie an example of the procedure mentioned in the conclusion of §3.1. Real spacetime M, which is the solution of Einstein's equations for the island spherically symmetric case, is a more extensive manifold, than would be expected from the original solution (3.13) in Schwarzschild coordinates. Schwarzschild's coordinates are able to contain only a part of this complete manifold, while Kruskal's coordinates cover it in its entirety.
Fig.3.18. a) Illustrative
drawing of the geometric
structure of the section (spatial hyperplanes) v = t = 0,
J = p/2 of
Schwarzschild spacetime in
the form of nesting into auxiliary three-dimensional
Euclidean space. This auxiliary three-dimensional space
has no physical significance (it is only a means of drawing); only the internal geometry of
the nested surface, which shows the two asymptotically
planar regions A and A' connected by
the Einstein-Rosen bridge, is relevant. b) A topological tunnel between two places in the same universe. c) "Wormhole" between two places in the same universe. |
If
area A in Fig.3.17a is the original outer asymptotically planar area of the
Schwarzschild geometry ("our universe"), another mirror-inverted
asymptotically planar region A' apears on the other side of the diagram - a kind of "second
universe". If we take a section
through the Kruskal diagram along the axis u (ie the section t = 0 according
to Fig.3.16), we see that the spherical surfaces r = const. with
decreasing r they first normally decrease their
areas, but not to zero, but to a minimum value of 16p M^{2}, and then grow again, as if
they widened by a second asymptotically
plane space. This
situation is illustrated in Fig.3.18a, which is by nesting a
section t = 0 with one omitted dimension into the auxiliary
three-dimensional space. The complete geometry contains in time
section a certain "bridge", called
the Einstein-Rosen
bridge, connecting two different asymptotically planar
universes *). As can be clearly seen from Kruskal's diagram, no
one can penetrate the second (mirror-inverted) universe through
the Einstein-Rosen bridge, because he would have to move at
superlight speed to escape the singularity.
*) Assuming the usual Euclied global
topology; with a more complex topology, such a bridge could
connect two different places of one multiple-connected universe -
Fig.3.18b, c.
Another interesting feature of Schwarzschild's solution, which can be seen from the Kruskal diagram, is that in the region r <2M the geometry has a dynamic character. If we observe the time sections (parallel to the u- axis ) by the Kruskal diagram gradually from large negative values v to large positive values v (ie in the process of time evolution), we will initially see two unrelated asymptotically planar universes, each of which has its singularity r=0. As soon as the incision stops passing through the lower (past) singularity, an Einstein-Rosen bridge appears between the two outer regions. This bridge will widen (the maximum will be t = 0 in the section) and then narrow again, until finally when it touches the section with the upper singularity it disappears and again two unrelated universes with their singularities remain. It can be said, that the bridge disappears so fast, that no real object can penetrate it into the second universe (it always ends in a singularity). The geometry of the Schwarzschild solution has a dynamic character in the region r <2M due to the fact, that here due to the exchange of roles of temporal and spatial metric components, the role of temporal evolution is taken over by "spatial evolution".
A thorough analysis of Schwarzschild's geometry reveals a completely different global topology than would be expected from Schwarzschild's "innocent-looking" expression (3.13). It can be seen at first glance that the topology will not be Euclidean at the point r = 0 where is the singularity; however, this might seem to be the only difference from ordinary topological properties. However, the full extension shows a completely different global topology - two different asymptotically planar "universes" connected by a "bridge" that has a dynamic character.
Some properties of Schwarzschild geometry in Kruskal coordinates are the same as in Minkowski spacetime - (same shapes of light cones). Therefore, for the same reasons as we did for §3.2 for Minkowski spacetime, it is advantageous to "improve" the Kruskal diagram even further by introducing a conformal transformation converting regions of infinity to finite coordinates. A suitable transformation relationship will again be similar to (3.5):
c = arctg (v + u) - arctg (v - u) , h = arctg (v + u) + arctg (v- u) .
The resulting metric then has the form
(3.31) |
The corresponding conformal Penrose space-time diagram of the complete analytical extension of the Schwarzschild solution is shown in Fig.3.19. The properties of geometry, the motion of particles and the causal relationships between the individual parts of Schwarzschild spacetime can be seen even more clearly here than in the Kruskal diagram :
Fig.3.19. Penrose-Kruskal conformal space-time diagram of the
complete extension of Schwarzschild geometry.
a ) Coordinate lines
in relation to Schwarzschild coordinates (hypersurfaces r
= const. and t = const.).
b ) Shapes of light cones and
radial geodesics of time and light type.
Critical
Note: Double Solution in
Analytic Extension - Reality or Fiction ?
The unusual characteristics of analytical
extension Schwarzschild and
especially Reissner-Nordström and Kerr-Newman geometry (§3.5 and 3.6) cause frequent
discussion and differing opinions among experts and broader public. Do they
reflect physical reality ? - or is it just an artificial
mathematical structures that
are in nature and the universe has never and nowhere do
not apply ?
On the one hand, these are exact solutions to
Einstein's proven gravitational
equations, so we should
take them very seriously, at least on a theoretical level. On the
other hand, we also have experience in other areas of applications
of mathematics in physics and
science that if the (correct) equations used have two or more
solutions, some of these solutions may not be applicable - they
do not correspond to the assignment, do not make sense, eg are
negative or imaginary.
A simple example might be the
use of quadratic equations in geometric tasks to
determine the dimensions of geometric shapes (eg, relationships
of perimeters, areas, and land diagonals). These equations
usually have two roots, while only one of them satisfies
the task assignment and is usable, while the other is often
negative or imaginary. We cannot attach technical or scientific
significance to this second result. In the theory of equations
itself, however, it is necessary to consider all solutions ...
Similarly, some extended
solutions of Einstein's equations, although mathematically
correct, probably do not solve any real situation in nature (in space).
From an (astro) physical point of view, these questions are
discussed in more detail in §4.4 "Rotating and
Electrically Charged Kerr-Newman Black Holes", section "Black
Holes - Bridges to Other Universes? Wormholes".
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 | ||
AstroNuclPhysics ® Nuclear Physics - Astrophysics - Cosmology - Philosophy |