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Journal of Modern Physics, 2017, 8, 1723-1736

http://www.scirp.org/journal/jmp

ISSN Online: 2153-120X

ISSN Print: 2153-1196

DOI:

10.4236/jmp.2017.810101 Sep. 12, 2017 1723 Journal of Modern Physics

Curvature Energy and Their Spectrum in the

Spinor-Twistor Framework: Torsion as

Indicium of Gravitational Waves

Francisco Bulnes

, Yuri Stropovsvky

, Igor Rabinovich

Research Department in Mathematics and Engineering, TESCHA, IINAMEI, Chalco, Mexico

Mathematics Department, Lomonosov Moscow State University, Moscow, Russia

Abstract

The twistor kinematic-energy model of the space-time and the kinema

ic-energy tensor as the energy-matter tensor in relativity are considered

demonstrate the

possible behavior of gravity as gravitational waves that derive

of mass-energy source in the space-

time and whose contorted image is the

spectrum of the torsion field acting in the space-time.

The energy of this field

is the energy of their second curvature. Likewise, it is assumed that the curv

ture energy as spectral curvature in the twistor kinematic frame is the curv

ture in twistor-

spinor framework, which is the mean result of this work. This

demonstrates the lawfulness of the torsion

as the indicium of the gravitational

waves in the space-time. A

censorship to detect gravitational waves in the

space-time is designed using the curvature energy.

Keywords

Censorship Condition, Contorted Surface, Curvature Energy, Gravitational

Waves, Matter-Energy Tensor, 3-Dimensional Sphere, Spinor Fields, Twistor

Kinematic-Energy Model, Weyl Curvature

1. Introduction

The twistor kinematic energy model could establish to the future-null-infinity in

the space-time, a quasi-local matter model represented through gravitational

waves of cylindrical type considering the condition on the spinor fields respec-

tive, in the null-infinity. Here, is obtained the asymptotical flat space-time far

away of the mass-energy source.

We consider the Penrose’s definition of the kinematic twistor associated to a

How to cite this paper:

Bulnes, F., Stro-

povsvky, Y

. and Rabinovich, I. (2017) Cur-

vature Energy and Their Spectrum in the

Spinor

Twistor Framework: Torsion as

Indicium of Gravitational Waves

Journal

of Modern Physics

, 1723-1736.

https://doi.org/10.4236/jmp.2017.810101

Received:

August 3, 2017

Accepted:

September 9, 2017

Published:

September 12, 2017

7 by authors and

Scientific

Research Publishing Inc.

This work is licensed under the Creative

Commons Attribution International

License (CC BY

4.0).

http://creativecommons.org/licenses/by/4.0/

Open Access

F. Bulnes et al.

DOI:

10.4236/jmp.2017.810101 1724 Journal of Modern Physics

2-surface in a general curved space when the total momentum of energy and

angular momentum to a system in special relativity and in linearized general re-

lativity can be characterized geometrically. Of fact, the geometrical evidence of

torsion through a contorted surface is wanted.

We consider a source as total charge depending of

(Killing vector) of the

Minkowski space background, modeled this as

,≅⊗⊗MC

which has an

important analytic system of twistor solution in the same space-time

+−

≅⊕MS S

Then their system has a complex 4-dimensional solutions family

(

≅ C

) and the family defines the 2-surface twistor space

( )

T S

But, in a surface of arbitrary genus

and of index

( )

− g

the solution

is a general twistor solution, which can be given though a model problem be-

tween bosonic fields deduced of the dual problem given by the relation:

βγ β γ

αβ γα α γ γ α

Σ= Σ

'' '

AI AI

(1)

to the energy-matter tensors

and the integral solution given to the kine-

matic tensor

αβ

through the energy-matter tensor,

αβ

σσ

= =

∫∫

ab cd a b

1 2 abcd ab

A Z Z R f d T kd

(2)

The exhibition of the kinematic tensor happens when the special surface in-

side space-time background

results to be the product

+−

⊗SS

of the

twistor 2-surface

( )

and also (2) defines a kinematic twistor tensor

αβ

as element of this symmetrized product of two 2-surfaces

( ) ( )

( )

αβ

∈⊗TTA S S*

which is a twistor space of (valence-2) and symmetric

dual twistor.

Proposition 1.1. The twistor kinematic tensor

αβ

is an element in duality

of the energy-mass tensor

We observe the following figure establishing the duality signed in the proposi-

tion 1.1. (see

Figure 1).

Proof

. Their demonstration is given considering the relation

βγ β γ

αβ γα α γ γ α

Σ= Σ

'' '

AI AI

where the second member can be had as a spinor using

the integral (2):

αβ

ωω σ

∫

A cd

1 2 ABcd 1 2

A ZZ R d

(3)

Figure 1. Duality between tensors

αβ

and

F. Bulnes et al.

DOI:

10.4236/jmp.2017.810101 1725 Journal of Modern Physics

which, using the spinor framework [1] [2] inside the integral (3) we have:

( )

(

)

(

)

( )

{ }

01 10 11

1 1 11 2 1 2 1 2 21 3 1 2

αβ

φ ψ ω ω φ ψ ωω ωω φ ψ ωω

= − + +Λ− + + −

∫

1 2 01 2

A Z Z dS

(4)

which is simplified using the spinor frame equations

( )

01 2

1 0 3 2 11

π ρπ ψ φ ω ψ φ ω

′′

℘ + = − −Λ + −

′′

℘ + = − + − −Λ

11 1 01

(5)

to the integral

( )

2 12

1 10

αβ

ππ ππ

′ ′′

−

= +

∫

12 0

A Z Z dS

(6)

which establishes the required duality.

◆

Of the integral involved in (6), we note that the twistor kinematic tensor

αβ

depends of

which has more mean, that is to say, depends on the first

and second fundamental forms of

This means the presence of curvature inside spinor terms in the integrating of

(6). This explains only the dependence of the energy due to curvature. Then to

spinor fields of the form

( )

ωπ

′

, we have the quantity [1]:

ωπ ωπ

′

′′

Σ= +

(7)

which is constant to constant curvature space. However, for a 2-surface in a

general space-time

there is no reason to that (7) could be constant. Like-

wise, we have the following proposition:

Proposition 1.2. (7) is constant for every 2-surface twistor if and only if the

2-surface with their field

( )

ωπ

′

is embedded in a conformally flat space-

time modulo certain genericity conditions.

Then in little words, the proposition 1.2. prepares a detection condition from

a contorted property of the 2-surface when is affected by the presence of a field

source. This in the conformally conditions detects curvature which is measured

and modeled in the spinor waves as is showed in

Figure 1 in the 2-surface twis-

tor of the twistor kinematic tensor

αβ

. In our study of spectral curvature we

can define this measure as curvature energy obtained through twistor frame of

the energy-mass tensor, as in the integrals (3) and (6). These have involved a

curvature tensor, which has curvature energy as spinor field energy or spinor

wave, this last understood as energy manifestation in the kinematic tensor space

(

) ( )

( )

⊗TTS S*

Likewise, the curvature energy as spectral curvature in the twistor kinematic

frame is the curvature in twistor-spinor framework.

The twistor equations to valence-2 on symmetric spinor

can be written as:

′′

∇ =−∈

A BC A(B C)

which has a 10-

dimensional complex solutions space. Their solution space is spanned by fields

of the form

ωω

(A B)

(such and is showed in

Figure 1

), where each

satisfies the twistor

equation

ωπ

∇ =−∈

A B AB

A' A'

whose solutions defines a 4-dimensional complex vector space which is the twistor space

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