Why we cannot expect gravitation to be weird

Why we cannot expect gravitation to have weird
properties
R. Mirman, sssbbg@gmail.com
November 16, 2006
Abstract
General relativity seems to have unphysical solutions, like closed
time-like curves. This does not follow and is quite unlikely: the Einstein
equation is a necessary condition for a gravitational field but not
sufficient. There are additional requirements and before we can conclude
that there are fields with strange properties we must show that
all conditions are satisfied. This is implausible for weird fields.
Are there closed timelike curves, wormholes, …, in gravitational theory?
Doesn’t Einstein’s equation gives these? Satisfaction of the equation is a
necessary condition for a gravitational field, but not sufficient. There are
further conditions. Abnormal solutions imply that not all conditions are
satisfied. We cannot conclude that there are these unless it is shown that
all are satisfied for the strange solution.
This does not imply anything wrong with general relativity —it is almost
certainly correct. It just means that it is applied incorrectly.
What are other conditions ([2]; [3]; [4])?
The field must be produced, else it does not exist. What produces a
gravitational field? A sphere, a star, dust? But there are no spheres, stars,
dust. These are merely collections of protons, neutrons, electrons and such
— which are what creates and is acted upon. Such a collection must give
a strange field. However these objects are governed by quantum mechanics.
The uncertainty principle applies. Can a collection of such objects produce
strangeness? Before it is claimed that there are closed timelike curves, wormholes,
…, it must be shown that there is a collection of quantum mechanical
objects capable of producing them.
Would we expect a single proton, a single electron, to give closed timelike
curves? If not why would we expect a collection to? This implies that
the formalism is being used incorrectly. This can be tricky because we often
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think in ways different than the ones nature thinks in, like using classical
physics as a formalism while nature uses quantum mechanics, or using
large objects while nature produces gravitational fields from collections of
quantum mechanical ones. Using the proper formalism is essential.
There is another condition which is especially interesting since it requires
that general relativity be the theory of gravity (thus the quantum theory
of gravity, as it so clearly is ([2])). All properties of gravitation come from
it. This has been discussed in depth, with all the mathematics shown and
proven ([2]; [3]). Here we summarize.
A physical object, like a gravitational field, must be a representation
basis state of the transformation group of geometry, the Poincar´e group.
(The Poincar´e group is the transformation group, not the symmetry group,
although it is interesting that it is the symmetry group also ([4], sec. VI.2.a.ii,
p. 113)).
To clarify consider the rotation group and an object with spin up. Its
statefunction (a better term than wavefunction since nothing waves) gives
the spin as up. A different observer sees the spin at some angle, thus a different
statefunction. The statefunction of the first must be transformed to give
that of the second. Thus for each set of coordinates there is a statefunction
and these are transformed into each other when the coordinates are. For
each rotation there is a transformation of the statefunction. Moreover the
product of two transformations must correspond to the product of the two
rotations that they go with. Also a rotation, being a group element, can be
written as a product of two, or ten, or 1000, or in any of an infinite number
of ways. Each such product has a product of transformations on the statefunction
going with it, with each term in the product of transformations
corresponding to a term in the product of rotations. Thus the transformations
on the statefunction form a representation of the rotation group, and
each statefunction generated from any one by such a transformation is a
basis state of the rotation group representation.
This does not require that space or physics be invariant under the group.
Rotations are a property of geometry whether space is invariant under them
or not. Thus a state can be written as a sum of rotation basis states (spherical
harmonics) and is taken into another such sum by a rotation. Each term
in the latter is a sum of terms of the former (with coefficients functions of
the angles). Each term is a sum only of terms from the same representation
(states of angular momentum 1 go only into states of angular momentum 1,
and so on). This is true whether space is invariant under rotations or not
(say there is a direction, simulated by the vertical, that is different). An up
state may go into a down one, but that is irrelevant since these (mathemat-
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ical) transformations are considered at a single time. Also no matter how
badly symmetry is broken there cannot be an object with spin- 1
3 . These
would not be true if we expanded in unitary group states. The rotation
group is a property of our (real) geometry.
It is only a subgroup. The transformation group of space thus of the
fields is the Poincar´e group. Statefunctions (including those of gravity, the
connections) must be basis states of it. The Poincar´e group is an inhomogeneous
group so very different from the simple rotation group. Gravitation
is massless. The entire analysis depends on this.
Massless and massive representations are much different. The little group
of massive representations is semisimple (the rotation group), while that of
massless ones is solvable. Thus massless objects have difficulty in coupling
to massive ones. There are only three that can. Scalars apparently can. Helicity
1 gives electromagnetism (with its properties completely determined).
For helicity 2 the indices do not match. Fortunately the formalism gives a
nonlinear condition, the Bianchi identities, that allow gravitation to interact
with massive objects. Gravitation must be nonlinear else it could not couple,
so could not exist. Einstein’s equation then follows from the formalism,
but is not all of it.
A supposed gravitational field must be shown to form a representation
basis state of a massless helicity 2 representation of the Poincar´e group or
it is not a gravitational field. Unless ones with strange properties are shown
to be that then they are results of the wrong or incomplete formalism, so
nonexistent.
Since the Poincar´e group is inhomogeneous the momentum operators
(the Hamiltonian is one) must commute. There would be many problems if
not ([2],sec. 6.3.8, p. 110). It must be checked for a proposed field that the
momenta commute on it.
The proper way to find fields is thus to find functions satisfying these
properties — extremely difficult. To see if a field can be produced we must
find if the momentum operators of the entire system commute. These consist
of three sets of terms, for the field, for massive matter and for the interactions.
Thus we have to find a (quantum mechanical) distribution of matter
which, with the fields it produces, gives these operators, and such that they
commute.
It is likely to be very rare that we can do this. Great caution is required;
we cannot jump to conclusions about the existence of strange solutions.
Appendix:
To illustrate the importance of proper formalism, properly applied, we
consider other related topics ([2]).
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Are there ”graviton”’s ([2],sec. 11.2.2, p. 187)? We are used to taking
electromagnetic fields as sets of photons so try to apply it to gravity. But
electromagnetism is linear, gravitation nonlinear. What is a photon? It is
not a little ball, a ridiculous idea. If we Fourier expand an electromagnetic
potential (a solution of the equations) each term is a solution. Each term
is then a photon. A solution is a sum of solutions. If we do the same for
a field that is a solution of the gravitational equations the terms are not
solutions. A gravitational field is a collection of ”graviton”s each producing
a collection of ”graviton”s, each … . Obviously the concept is useless.
Consider a gravitational wave extending over a large part of the universe.
That single wave is a ”graviton”. The concept is not likely useful.
Are there magnetic monopoles ([2],sec. 7.3, p. 131)? Maxwell’s equation
have an asymmetry. But these are classical, so irrelevant. Quantum electrodynamics
does not have such an asymmetry. There is no hole to be filled
and, using the correct formalism, there is no way a magnetic monopole can
act on a charge. There are no magnetic monopoles.
What is the value of the cosmological constant? In Einstein’s equation
one side is a function of space, the other a constant (obvious nonsense), that
is one side is a function of a massless representation, the other a momentumzero
representation. This is like equating a scalar and a vector. The cosmological
constant is trivially 0, unfortunately else gravitation would have
a fascinating property: a wave would be detected not only an infinitely long
time before arrival but before emission ([2], sec. 8.1.4, p. 139).
Are there Higg’s bosons? Gauge transformations are the form Poincar´e
transformations take for massless objects, and these only ([2], sec. 3.4, p. 43).
This is explained in one paragraph ([4], sec. E.2.1, p. 445). They cannot be
applied to massive objects because of the mathematics, not because of some
new field. People are entranced by gauge invariance and decided to apply
it to objects where it cannot hold. This is like deciding that orbital angular
momentum is integral so spins must be. They are not so there must be some
new field that makes them half-integral. But the mathematics gives both
types of spin, does not allow spin- 1
3 , gives gauge invariance for massless
objects, and does not allow it for massive ones. This is a result of the
mathematics, not of some new field. There are no Higgs bosons.
References
[1] Borstnik, Norma Mankoc, Holger Bech Nielsen, Colin D. Froggatt, Dragan
Lukman (2004), ”Proceedings to the 7th Workshop ’What comes
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beyond the Standard models’, July 19 – July 30. 2004, Bled, Slovenia”,
Bled Workshops in Physics, Volume 5, #2, December.
[2] Mirman, R. (1995c), Massless Representations of the Poincar´e Group,
electromagnetism, gravitation, quantum mechanics, geometry (Commack,
NY: Nova Science Publishers, Inc.; republished by Backinprint.
com).
[3] Mirman, R. (2004a), Geometry Decides Gravity, Demanding General
Relativity — It Is Thus The Quantum Theory Of Gravity, in Borstnik,
Nielsen, Froggatt and Lukman (2004), p. 84-93.
[4] Mirman, R. (2006), Our Almost Impossible Universe: Why the laws of
nature make the existence of humans extraordinarily unlikely (Lincoln,
NE: iUniverse, Inc.)
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