## No weirdness in gravitation

Posted by impunv on June 3, 2007

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\author{R. Mirman

sssbbg@gmail.com}

\title{Why we cannot expect gravitation to have weird properties}

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\begin{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.

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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~(\cite{ml}; \cite{bna}; \cite{imp})?

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 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~(\cite{ml})). All properties of gravitation come from it. This has been discussed in depth, with all the mathematics shown and proven~(\cite{ml}; \cite{bna}). 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~(\cite{imp}, 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 (mathematical) transformations are considered at a single time. Also no matter how badly symmetry is broken there cannot be an object with spin-${1\over 3}$. 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~(\cite{ml},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~(\cite{ml}).

Are there “graviton”‘s~(\cite{ml},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~(\cite{ml},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 momentum-zero 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~(\cite{ml}, 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~(\cite{ml}, sec.~3.4, p.~43). This is explained in one paragraph~(\cite{imp}, 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-${1\over 3}$, 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.

\begin{thebibliography}{99}

\bibitem{nmb} Borstnik, Norma Mankoc, Holger Bech Nielsen, Colin D. Froggatt, Dragan Lukman (2004), “Proceedings to the 7th Workshop ‘What comes beyond the Standard models’, July 19 — July 30. 2004, Bled, Slovenia”, Bled Workshops in Physics, Volume 5, \#2, December.

\bibitem{ml} 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).

\bibitem{bna} 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.

\bibitem{imp} 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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