Centrally symmetric gravitational field

 AstroNuclPhysics ® Nuclear Physics - Astrophysics - Cosmology - Philosophy Gravity, black holes and physics

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 (Tik = 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 Rik = 0. The Christoffel connection coefficients in the metric (3.10) are

 (3.12)

Einstein's equations Rik = 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 C1 and C2 are integration constants. We choose the constant C1 by setting the scale of the time coordinate t (by transforming t = C1 .t we get rid of the constant C1). From the limit transition to Newton's law (gtt ~ (1 - M/r) for r®¥) we get that the constant C2 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 gik in Schwarszchild's geometry are thus (in geometrodynamic units) :

 gik   = / -(1-2M/r) 0 0 0 \ . | 0 (1-2M/r)-1 0 0 | | 0 0 r 2 0 | \ 0 0 0 r2 sin2 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 gtt = - gtt(r, t), grr = grr(r, t), from Einstein's equations Rik = 0 we get after adjusting that grr/t = 0 (so grr does not depend on time) and gtt = f(t)/grr, where f(t) is a function of time only. Because for r®¥ the investigated metric passes into the Minkowski metric, where gtt = -1 and grr = 1, must be f(t) = const. and therefore neither gtt 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:

 ds2  =  1/4 [3Ö(1-2GM/c2)/R) - Ö(1 - r2.2GM/c2/R3)]2 c2dt2 - (1 - r2.2GM/c2/R3)-1dr2 + r2(dJ2 + sin2J dj2)  . (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 gtt®0 and the spatial radial component grr®¥. 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 = rg = 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 Gikl 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)

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 = r2dj/dt = 0, so (dr/dt)2 = 1/K2 - 1 + 2M/r. In the limit r®¥ we get (dr/dt)2 = 1/K2 - 1, so 1/K2 - 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 = r2 to a distance r = r1 from the center :

 (3.21)

This interval of the proper time is always finite, even for r1 = 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-K2 has the meaning of the coordinate velocity v¥ at infinity. The interval of the coordinate time required to move from distance r2 to distance r1 is then equal to

 (3.23)

The interval of the coordinate time required to reach the point r = r1 is close to infinity at r1®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 > rg radially in Schwarzschild spacetime towards the center (to the horizon, to a black hole) exceed the event horizon r = rg for a finite proper time interval t = (2rg/3c). [(r0/rg)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-rg/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 = rg 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

 Rtrtr = -2M/r3 ,  RtJtJ = Rtjtj = M/r3 ,  RJjJj = 2M/r3 ,  RrJrJ = Rrjrj = -M/r3 , (3.24)

so in the Schwarzschild sphere they reach values of the order of 1/M2; similarly scalar invariants (eg R= RiklmRiklm= 48M2/r6) 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 gjj 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 gJJ®¥. 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 cr of light in the radial direction is obtained if we set ds2 = 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 = r1 to a point r = r2, will be according to (3.25)

 (3.26)

For r1® rg = 2M, this time approaches infinity for any value of the target r2 > r1. The light emitted from the Schwarzschild sphere r = rg takes an infinitely long time to reach any place farther from the center. From places with coordinates r < rg - 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 xo 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 wA = wr .(1 - 2M/r)1/2, where wr is the frequency of the signal sent by the static observer from the distance r from the center of the black hole and wA 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 wA 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 wr were sent from the Schwarzschild sphere r=2M, then wA = 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 grr=(1 - 2M/r)-1 and gtt= -(1 - 2M/r) change their signs: for r> 2M is grr > 0 and gtt< 0, while at r <2M is grr< 0 and gtt > 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 v2 - u2 = 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 u2 > v2, 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 M2, 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".

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Vojtech Ullmann