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Appendix B
UNITARY FIELD THEORY
AND QUANTUM GRAVITY
B.1. The process of unification in physics
B.2. Einstein's
visions of geometric unitary field theory
B.3. Wheeler's geometrodynamics. Gravity and topology.
B.4. Quantum
geometrodynamics
B.5. Gravitational
field quantization
B.6. Unification
of fundamental interactions. Supergravity.
Superstrings.
B.7. General
principles and perspectives of unitary field theory
B.3. Classical geometrodynamics. Gravity and topology.
In a sense, the successor of Einstein's efforts to create a unitary field theory, but in a slightly different way, became the American physicist John Archibald Wheeler, who, along with other collaborators, especially Ch.Misner, came from Einstein's original (verified and proven) general theory of relativity. He showed that to create a classic unitary field theory (while better than previous attempts) is not necessary in the general theory of relativity introduce any artificial and unjustified changes, it is enough to make full use of all geometric and topological possibility that general relativity provides.
In Chapter 2, we showed that the general theory of relativity overthrew space and time from the status of a kind of non-participating "scene" in which physical events take place, and made space-time a direct participant in physical events. According to GTR, the gravitational field is a manifestation of the curvature of empty spacetime - so we have a kind of "gravity without gravity"...
Electromagnetism
"without electromagnetism"
As shown in §2.5, Einstein's gravitational field equation R_{ik} - ^{1}/_{2} g_{ik}R = 8pT_{ik} have the important property that they not
only describe the behavior of the gravitational field, but
indirectly - via the laws of conservation of energy and momentum
T^{ik}_{;k} = 0 - even its sources. Therefore, if we take the
electromagnetic field in a vacuum, then from the Einstein
equations the gravitational field excited by it
R_{ik} - ^{1}/_{2} g_{ik} R = 2 F_{il} F^{l}_{k} - ^{1}/_{2} g_{ik} F_{lm} F^{lm} | (B.7) |
Maxwell's equations of this electromagnetic field F^{ik
}_{; k}
= 0 follow. If the curvature of spacetime is caused by an
electromagnetic field, then the trace of the Einstein tensor on
the left side (B.7) must be equal to zero, which gives R = 0 and
the square of the Ricci tensor R^{m}_{i} R^{k}_{m} = d^{k}_{i}.(^{1}/_{2} R_{lm}R^{lm}) is a multiple of the unit
matrix.
^{ }Einstein's and Maxwell's
equations (which are systems of 2nd order equations) can
therefore be combined into one system of 4th order
equations - Einstein-Maxwell's
equations, which in geometric form contains both
Maxwell's electrodynamics (without charges) in curved spacetime and Einstein's
equations indicating the curvature of spacetime by this
electromagnetic field *). The electromagnetic field leaves
characteristic "traces" on the geometry of space-time,
from which it can be "recognized" and whose behavior is
determined. The electromagnetic field (which is thus determined
by the expression containing the roots of the Ricci tensor of the
curvature R_{ik}) can thus be fully described
using only gravitational quantities, ultimately using the
components of the metric tensor g_{ik}. Maxwell's equations are then
given by the relationship between Ricci's curvature and the rate
at which this curvature changes; the laws of electrodynamics thus
become pure geometric character.
We get a kind of "electromagnetism without
electromagnetism",
in which the electromagnetic field is a manifestation of empty
curved spacetime.
*) However, it turned out
[281] that when integrating these Einstein-Maxwell equations, the
respective boundary conditions (values of g_{ik} and their time partial derivatives g_{ik, 0}) on
the initial Cauchy hypersurface x^{0} = 0 can correspond to more than one
Maxwell field at the same time, more values of F_{ik}. The
mentioned method of geometric description of electrodynamics thus
becomes ambiguous in the general case ...
Misner and Wheeler (using Rainich's earlier results for the electromagnetic field in GTR [213]) therefore concluded that Einstein's original general theory of relativity already fulfilled what Einstein had tried unsuccessfully for the rest of his life: a unified description of gravitational and electromagnetic field. A description yet completely natural and spontaneous, in which no existing theories do not change in any way.
At the beginning of
§B.1, we emphasized the unsatisfaction of the concept that the
field is excited by a source
different from the field. For the electromagnetic field as a
source of gravity in a vacuum, the situation here has been
successfully solved, at least in principle. The
"material" sources of the electromagnetic field are electric charges. Here, too, geometrodynamics proposes an
elegant solution: the topological
interpretation of electric charges by means of some "tunnels" in a
curved space, which capture electric lines of force and
effectively behave like electric charges (see
below "Topological
interpretation of electric charge").
^{ }The source of gravity in
classical physics is mainly general, unspecified and unstructured
matter - objects (bodies, particles) having mass. In previous
unitary theories the particles as trying to interpret
some particularities (singularities) in the field, which leads to many difficulties (see
§3.7), or as a some continuous densified structures having their laws of internal motion; however, these laws of
internal motion were introduced from the outside, and it was not
clear how to derive them within a closed theory.
^{ }It is different in geometrodynamics. If we
take Schwarzschild's geometry in empty asymptotically planar
spacetime, the ordinary law of gravity will apply at great
distances from the center as if there were real matter in the
center. It is thus an empty curved space in which gravitational
attraction acts, so Schwarzschild's geometry is the simplest
geometrodynamic model of "matter
without matter".
However, it is a artifical model with a topology different
from the Euclidean topology and contains a singularity (§3.4).
Matter "without
matter"
Geons
However, the laws of general theory of relativity allow
the existence of objects with the usual Euclidean topology and
without singularities, behaving like real matter (exciting the
gravitational field and responding to this field), and these
objects are composed purely of the field itself. When
electromagnetic waves propagate through space, they excite a
gravitational field around them - they curve the space-time in
which they propagate, and this does not remain without affecting
their motion. According to the general theory of relativity, very
powerful electromagnetic waves can create such
a strong
gravitational field around them, that they will be forced to permanently move
along the closed
paths. The
electromagnetic waves thus create a kind of gravitational "waveguide" curved around this geometry
of spacetime (from the gravitational field) in which they circulate permanently - Fig.B.2a. Such a formation
of electromagnetic waves, held together by the gravity of its own
field energy, is called a (electromagnetic) geon
[284].
Fig.B.2. Massive electromagnetic or gravitational waves can
create such a strong gravitational field (curvature space-time)
around them, that they will
be permanently forced to circulate in a closed
"gravitational waveguide" - a metastable material
formation geon is created. a) Average field distribution in the geon.
b) Due to its
gravitational effects, the geon behaves like any other matter
(even a planet) - we can, for example, put a satellite into orbit
around a geon.
The geon can be reached,
for example, by the following imaginary experiment: Let us have a
black hole of mass M, to whose photon sphere r = 3M
(§4.3) we will send massive doses of electromagnetic waves. With
the increasing amount of energy of electromagnetic waves moving
on photon sphere, these waves will increasingly contribute to the
overall gravitational field that keeps them there, so we can reduce the weight of the black hole by the appropriate value (in reality, however, it is prohibited by
the 2.law of dynamics of black holes - see §4.6). With a sufficiently large
accumulation of electromagnetic waves, we can completely
"remove" the black hole *), because the energy ~ mass
of the waves themselves is enough to create a sufficiently strong
gravitational field to keep itself on the "photon
sphere" - to maintain the circulation
of waves in the geon.
*) However, no mechanism is known for
removing (or gradually "controlled
removal") the initiating black hole
inside the genon (apart from the
hypothetical or sci-fi topological tunnel
"wormholes" - cf. §4.4, passage "Wormholes - bridges
to other universes? Time machines?"). This is just an
artificial imaginary experiment. In a "natural" way,
perhaps geons could arise in massive primordial fluctuations in
Planck's time at the beginning of the universe..?.. - cf. §5.5
"Microphysics and cosmology.
Inflationary universe.".
^{ }If the geon of total mass M is
spherically symmetric, it will evoke a spherically symmetric
gravitational field and the space-time metric will be (compare
with §3.4)^{ }
ds ^{2} = - g _{tt} dt ^{2} + g _{rr} dr ^{2} + r ^{2} (d J ^{2} + sin ^{2} J d j ^{2} ).
The radial component of
the metric has the usual Schwarzschild form g_{rr} = 1/[1 - 2m(r)/r], where m(r) is the mass~energy
contained inside a sphere of radius r. The time component of the
metric outside the geon also has the Schwarzschild form g_{tt} = 1 - 2M/r, inside the geon it has the
value g_{tt} = 1/9 (the time
inside the geon flows three times slower than far from the geon). The geon is
not completely stable,
but only metastable - part of the energy of the
waves penetrates through the centrifugal and gravitational
barriers, the geon slowly dissolves (the slower the greater the number of wavelengths
around the circumference), or vice versa, it could collapse to form a black hole. For a distant
observer, the geon will have
the same gravitational effects as any other matter (such as a planet) -
we can, for example, orbit a satellite into
orbit around a geon (Fig.B.2b).
^{ }Such a mass composed of
electromagnetic waves may seem strange to us, but the material
nature of electromagnetic waves is sufficiently established. We
get an even more suggestive picture when we replace
electromagnetic waves with
gravitational waves. Gravitational waves also transfer energy
(§2.7 and 2.8), curved space-time (universal
excitation of gravity) and according to the general
theory of relativity, they can also create a
"gravitational" geon, which will externally manifest
itself as real matter by its gravitational effects .
^{ }But what are gravitational waves?
- are waves of the gravitational field, ie fluctuations in the
geometry of empty spacetime. The external observer
thus witnesses an interesting thing: the rippling curvature of
empty space-time "without matter" will appear as a
material formation! The gravitational geon is thus an
illustrative model of a kind of "matter
without matter",
matter formed literally from the "emptiness" of space with rippling
curvature.
^{ }Here it must be said that the
whole concept of geometrodynamics meets certain philosophical- methodological
logical problems; however, this does not mean that
the above concepts contradict the basic postulates of materialist
philosophy *), which is a natural platform for physics and
science in general.
*) If we observe matter
either on ever smaller scales of the microworld or, conversely,
on ever larger scales of the megasworld, matter will gradually
lose some attributes to which we are accustomed from the common
experience of our macroworld,
and eventually new attributes will begin to appear. However, the
basic feature of matter always remains - to be an objective
reality.
However, the
hypothetical (model) geon is only a certain extreme example of
the construction of a material object from the geometry of
spacetime; above were mentioned some fundamental problems with
its formation and with the stability of the geone against the
radiation and scattering of gravitationally trapped orbiting waves-quantums
of electromagnetic or gravitational waves. The initial hopes that
the geon might be used as a classical model of stable
(elementary?) Particles were not fulfilled. It is a product of classical field theory, and until a fully satisfactory quantum
theory of gravity is built, the possible relationship between
geons and quantum-mechanical elementary particles cannot be
specified. There is no hope yet that there
might be some "micro-geons" stabilized by
quantum effects (omehow, like the orbits of
electrons in atomic shells, are stabilized by quantum
corpuscular-wave dualism?), which could
model elementary particles ...
^{ }From a theoretical point of view, we don't
even have to go to such complex constructions. In fact, every
gravitational wave described by its Isaacson
tensor of nonlocal energy-momentum (see §2.8) is such a "matter without matter", composed of a
"vacuum" (understood in the usual
sense). The fenomenon, as even in "empty" space without
the usual "real" material sources, appear kind of effective mass having global gravitational effects, is, after all, a similar situation in electrodynamics, where even in a
vacuum without the presence of electric charges (and currents)
for a non-stationary electromagnetic field, a Maxwell shear current appears, having magnetic effects
the same as the "real" current of electric charges ...
Topological
interpretation of electric charge
Charge "without
charge"
Now let's notice
the electric charges. Electric charges (and their
currents) are the sources of the electromagnetic field, but they
are also something "foreign" in the theory of the
electromagnetic field itself - a kind of substance different from the field. In
places of positive electric charges, electric lines of force
begin and exit on all sides, in places of negative electric
charges, lines of force enter and end there from all sides
(Fig.B.3a); Maxwell's field equations do not apply here.
According to Gauss's theorem, the total electric charge in any
part of space can be determined by surrounding the investigated
area with an (imaginary) closed surface S and measuring the intensity E electric
field in all places of this closed area - we determine the
"number of lines of force" that go in or out
(Fig.B.3a).
Fig.B.3. Classical and topological interpretation of electric
charges.
a) The usual understanding of electric charge Q
as "substance", from which the field lines of the excited
electric field originate (or enter).
b) Topological interpretation of electric
charge - there is no
"real" charge as a substance, the lines of force do not
begin or end anywhere, they are just captured and pass through a
topological tunnel, whose throats then appear as
"apparent" charges "Q".
But can't the lines of
force that go in, somehow "unnoticed" get out again
without noticing it on the enclosed area bounding this interior
(or similarly, the lines of force that go out to get back in)? At
first glance, such a question seems absurd. If we lock someone in
jail on all sides, acording to common sense
is unthinkable to get out, without having to go through the wall of
his prison - to break through the wall, open the door in
it.
^{ }Let's draw this situation in a
two-dimensional case on a piece of paper; instead of humans we
think of ants, which we will consider here as two-dimensional
beings (Fig.B.4). In Fig.B.4a, the two-dimensional world of
ants has the usual characteristics, and a prisoner inside a
closed curve cannot really get out without passing through this
boundary of his prison.
^{ }
Fig.B.4. Influence of topological properties of space on the
possibilities of movement.
a) A prisoner (an
ant) surrounded on all sides by a prison wall cannot get out in a
space (here two-dimensional) with the usual topological
characteristics without passing through the prison wall.
b) In an
area with multiple continuous topologies, it is possible to leave
a closed prison without having to go through its wall. The ant
can walk through the topological tunnel and look at the intact
wall of his prison from the outside.
But what if the two-dimensional world of ants looks topologically as shown in Figure B.4b? An ant trapped in an area surrounded on all sides by a closed curve (prison wall) can go through a "tunnel" and look at its prison from the outside without having to go through a closed wall of its prison at some point. From the point of view of the three-dimensional environment, in which this construction is embedded, there is nothing strange about it - the ant, although it still moves within its two-dimensional surface (its world), "creeps or overcomes" the wall of its prison, so to speak, through "another dimension". From the point of view of the two-dimensional ants themselves, for which no "third dimension" exists, however, a miracle happened: a prisoner surrounded on all sides by a wall suddenly came from somewhere outside to look at the intact wall of his prison! The reason is that said two-dimensional space has different topological properties than in Fig.B.4a. It is multiple continuous. The closed curve here no longer has to be the boundary of the area inside! The local geometric properties at each location while doing so can be quite common (only slight curvature).
Now we can return to electric charges. In Fig.B.3a, a positive and a negative electric charge are shown in the usual way in a two-dimensional drawing - the lines of force emerge from the positive charge (according to the agreement) and end at a negative charge. If we surround the charge with an imaginary closed surface S, we can "count" the force lines entering or leaving to determine the value of the charge Q inside. However, there may not be any "real" electric charge here! With a suitable topology of the space, as shown in Fig.B.3b, the force lines will enter inside the closed area S, but they will not end there, but they will pass through a topological "tunnel" to another place in the space, from where they will come out again and return. To the external observer, who will measure the electric field, one "tunnel mouth" will appear as a negative charge -"Q" (with the lines of the line going in) and the other throat of the tunnel as a positive charge +"Q" (the lines of force go out). The electric field, whose lines of force pass through a topological tunnel, satisfies Maxwell's equations everywhere. Consequently, the total flux of the electric field intensity trough the mouth of the tunnel cannot change over time, if the topology is not changed; it does not matter the variability of the electromagnetic field, the curvature of space, changes in the "cross-section" of the topological tunnel or the distance between its two mouths. Electric field flow through each closed surface S surrounding the tunnel
thus it complies with the law of conservation of electric charge and the Gauss theorem of electrostatics.
Such a topological interpretation of an electric charge is actually a "charge without charge": no "real" electric charges exist here, electric lines have no beginnings or ends, they are only captured and pass through a topological tunnel of space, the individual mouths of which then appear as positive and negative charges "Q". Thus, a free electromagnetic field in a vacuum without charges can generate (effective) electric charges due to the suitable topological structure of the space. In this theory, electric charge appears as a nonlocal property of electrodynamics (without sources) in multiple continuous spaces.
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