## Why we cannot expect gravitation to be weird

Posted by impunv on June 10, 2007

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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