The Symmetries of Kibble’s gauge theory of gravitational ... · PDF filewe studied the Noether theorem for the transformation of the ... the conclusion that a gravitational field

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The Symmetries of Kibble’s Gauge Theory of Gravitational Field,

Conservation Laws of Energy-Momentum Tensor Density and the

Problems about Origin of Matter Field

Fangpei Chen

School of Physics and Opto-electronic Technology，Dalian

University of Technology，Dalian

Email:[email protected]

Abstract

Based on the analysis of the Kibble gravitational gauge field theory,

we studied the Noether theorem for the transformation of the

Poincare’ local group in the physics systems, derived the energy-

momentum tensor density conservation laws, and proved the

equivalence of this conservation laws to the Lorentz and Levi-Civita

energy-momentum tensor density conservation laws. Moreover, we

discussed the problems about the origin of matter field.

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Keywords

Lagrangian; matter field; gravitational field; energy-momentum

tensor density; conservation law； origin of matter field

1. Introduction

Kibble gravitational gauge field theory [1] is a

standard gravitational field theory with Poincare’

group as its gauge group. The symmetry of this gauge

theory is primarily the symmetry related to the

Poincare’ group transformations, which is important in

at least the following two aspects:

1) To explain the emergence of the gravitational field

from the global to local change of the Poincare’ group

transformation. Its explanation of the emergence of

gravitational field [1, 2] is as follows: when a

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gravitational field does not exist, and the matter field

possesses the symmetry of the Poincare group global

transformation, then by further requiring the symmetry

of the global transformation of the Poincare’ group to

be the symmetry of the local transformation of

Poincare’ group (i.e. the parameters of the group are

the function of space-time coordinates), one arrives at

the conclusion that a gravitational field must exist.

This can be also described as follows, according to the

Kibble gravitational field gauge theory, the existence

of the symmetry of the local transformation of the

Poincare’ group implies the existence of the

gravitational field, and the existence of the

gravitational field is represented by the existence of

the symmetry of the local transformation of Poincare’

group.

2) To examine the relations of conservation current.

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According to the Noether theorem, which is generally

applicable in theoretical physics, if the action

variable is unchanged under the transformation of some

group, then this must lead to the corresponding

conservation current. By applying this theorem, the

local transformation of Poincare’ group in a physics

system has also a conserved current. But this current

contains some parameters of the local transformation of

the Poincare’ group. These parameters are the function

of space-time coordinates, which vary independently from

each other, and can be mutually separated. Therefore,

multiple identity relations can be derived from the

conserved current of the Noether theorem, and these

identity relations reflect the important dynamic

properties of the physical system.

With regard to the above two aspects of the

symmetry, there have been more numerous and complex

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studies of the first aspect, but fewer and less

comprehensive investigation of the second aspect. This

paper describes my recent study of the second aspect

with particular focus on the derivation of the

conservation law of the energy-momentum tensor density

from the Noether theorem applied to the local

transformation of the Poincare’ group.

2. The local transformation of the Poincare’ group and

the Noether theorem

For a physics system, if it only contains a matter

field )(x , then its Lagrangian can be expressed as:

)](,

);([)(00

xxM

xM LL

（1）

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Here for simplicity, we assume the matter field has only

one component, although in reality the matter field can

be a spinor, tensor, vector or scalar. With the

exception of scalar, all the other quantities are of

multiple components.

Under the local transformation of the Poincare’

group, the space-time coordinates are changed to

)()(' xx xxxxxx

（2）

the matter field is transformed to

)(

2

1)(

)()()'()( '

xx

xxxx

S

（3）

where the group parameters )(x

（or )(x ）、 )(x

are

all variables. So )(x

、 )(x

are difficult to

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distinguish. One can define )()()( xxx x

，then

Equation (2) can be written as:

)(' xxxxxx

（2’）

from now on we will use )(x

to represent )(xx

.

The Lagrangian, action integrals, field equations

and conservation laws are all determined for a certain

physics system. Therefore, one needs to select a

relevant physics system for studying these problems.

When there is no or a negligible gravitational force,

the Lagrangian only contains the matter field. But it is

noted that, apart from the gravitational force, all

other elementary interactions can be included in the

Lagrangian of matter field. When there is a non-

negligible gravitational field, the Lagrangian of a

physics system is usually written in two parts [1, 2]:

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)()()( xG

xM

x LLL （4）

Here, )(xML not only describes the pure matter field,

but also describes the gravitational force acted on the

matter. Therefore )(xML can be regarded as a

“generalized matter” Lagrangian. Whereas )(xGL only

describes pure gravitational force, and it can thus be

called as the Lagrangian of a pure gravitational field.

In the Kibble gravitational gauge field theory,

)(xML

and )(x

GL can be expressed by the following

generalized functional [1]:

)]();();(,

);([)( xixijxxM

xM hLL

（5）

)]();(,

);([)( xixijxijG

xG hLL (6)

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Where )(xih is a Vierbein field that can be used to

determine the metric of space-time, )(xij

is a tetrad

connection field that can be used for determining the

connection of space-time. For a flat space-time, there

is always ixih )( and 0

ij

. For a curved space-time,

there always be ixih )( and 0

ij

, then curvature and

torsion emerge in the space-time. According to the

gravitational gauge field theory, curvature and torsion

are all manifestations of the gravitational force, hence

the Vierbein field )(xih and the

tetrad connection field

)(xij

in the expression of (5) and (6) imply the

existence of gravitational force.

When discussing a physics system with gravitational

force phenomenon, in the action integral:

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xdLI xgx 4)()(

(7)

the Lagrangian )(xL must be the total Lagrangian

)()()( xG

xM

x LLL . Therefore,

ij

ij

ij

ij

i

i

LL

hh

LLLL

,0

,

0

0,0,

00

（8）

where 0 represents the variation of a function at

fixed value of x , and represents the variation of a

function under changing value of x . )(xg is in equation (7)

because of the need to consider the Jacobi matrix of coordinate

transformation in the existence of a gravitational force. It can

be proved [3] that:

)()()'(')'(' xxgxxg LL

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Under the local transformation of the Poincare’ group, the

variation of the action variable becomes[1]

xdxLx

xdL

xdLxdLI

xxgxxg

xgxxgx

4])()([4)]()([0

4)()(4)'(')'('

'

}{

'

（9）

where

ij

ij

xLxgij

ij

xLxg

i

i

xLxgi

i

xLxg

xLxgxLxgxLxg

hh

hh

,0

,

)]()([

0

)]()([

,0

,

)]()([

0

)]()([

,0,

)]()([

0

)]()([)]()([

0

(10)

Notice that there is an additional term hi

xLxg

,

)]()([

in

equation (10),

which is identical to 0 because )(xL is

not a function of hi

,. Therefore it does not affect

the result but makes it convenient for derivation

studies. After calculations, one can rewrite the terms

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inside the brackets {} of the integral in Equation (9)

as

xdxLgLg

hh

LgLg

x

hh

ij

ij

i

i

ijij

Lgii

LgLg

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,

0

,

0

,

)()()(

}])()()(

[

]{[ 0

,

0

,

0

,

(11)

where

0

,

)()()(

0

,

)()()(

0

,

)()()(

ij

Lg

ij

Lg

ij

Lg

i

Lg

i

Lg

i

Lg

LgLgLg

x

hxhh

x

（12）

are the Euler-Lagrange equations for matter field [4],Vierbein

field and tetrad connection field respectively. If these

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equations are all satisfied, and under the local transformation

of Poincare’ group the following condition holds

0)()'(' xx III ，

then the following conserved current may be derived from

Equations (9-12):

00

,

0

,

0

,

])()()(

[

xLgLg

hh

LgLg

x

ij

ij

i

i

(13)

which is the Noether theorem under the local transformation of

the Poincare group.

3. The conservation law of the energy-

momentum tensor density under the local

transformation of Poincare’ group

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Based on the mathematics relationships, one can

obtain [5]:

,)()(

2

1

,)(

0x

mnxmnx s

hhhh ixixj

njim

xmni

,)()(

,)(

0

ijxj

nim

xmn

ijxiknk

j

mxmnkj

nkim

xmnij

,)()(

,

)(,

)()(0

By substituting ijihx

0,

0,

0, into Eq. (13), after

complicated calculations, one can divide Eq. (13) into

several identity equations because the parameters

)(,

),(,

),(),(,

),(,

),( xmnxmnxmnxxx

are mutually independent.These identity equations are

either conservation laws or other important relations.

In either case, they are important dynamic properties of

the physical systems. This paper cannot cover all of

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these properties, so we focus on how to derive the

conservation law of the energy-momentum tensor density

from the Noether theorem of the local transformation of

the Poincare’ group.

Due to the set of parameters )(,

),(,

),( xxx

with the set of parameters )(,

),(,

),( xmnxmnxmn are

independent of each other, the part containing the

parameters )(,

),(,

),( xxx

and the part containing the

parameters )(,

),(,

),( xmnxmnxmn in Eq.(13) should be

conserved respectively.

Hence from Eq. (13), one can obtain:

0}][)(

][)(

][)(

{

,,,

,,

,,

,

LgLg

hhh

LgLg

x

ijij

ij

ii

i

(14)

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Eq. (14) can be divided further into three identity

equations:

0

,

,

,

,

,

,

])()()(

[

LgLg

hh

LgLg

x

ij

ij

i

i

（15）

0

,,,

,,

,

,,

,,

])()(

[

])()()(

[

\

ij

ij

i

i

ij

ij

i

i

Lgh

h

Lg

x

LgLg

hh

LgLg

(16)

0

,,,

])()(

[

ij

ij

i

i

Lgh

h

Lg

(17)

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Since the parameters )(,

),( xx

,

,are also mutually

independent, hence Eqs (15),(16) and (17) can reduce to

other three equations without )(,

),( xx

,,

.

One can define:

LgLg

hh

LgLg

Tg

Mij

ij

Mi

i

MM

M

)(,

,

)(,

,

)(

,

,

)(

)(

)()()(

as the energy-momentum tensor density of the

“generalized matter” field, and define:

LgLg

hh

LgLg

Tg

Gij

ij

Gi

i

GG

G

)(,

,

)(,

,

)(

,

,

)(

)(

)()()(

as the energy-momentum tensor density of the pure

gravitational field. Because the parameter )(,

),( xx

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and ,

represents the translation of the space-time,

these definitions are appropriate. From Eq. (15), one

can obtain:

0)()( )(

TgTg

xGM

(18)

from Equation (16,17), one obtains:

0)()(

TgTg GM

(19)

Equation (18,19) are Lorentz and Levi-Civita energy-

momentum tensor density conservation laws.

In 1917-1918, Levi-Civita and other scientists had a

dispute with Einstein on the energy-momentum

conservation laws [6][7]. The readers are referred to

Reference [7] for the details of the dispute. The focus

of this paper is on the physical concept instead of the

dispute.

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4. The problems about“origin of matter field”

In the present studies of physics, there have been

insufficient investigations about the origin of matter

field. Just like in the investigation of the origin of

life, wherein one has to address the question of how

living things arise from non-living matter, in the

investigation of the origin of matter field, one also

needs to address the question of how the Universe

evolves from a matter-less state to a state with matter

field. But there has been a lack of study of how matter

field arise from matter-less state, and it is unclear

how to investigate this problem.

The basic property of matter field is the possession

of a positive energy ,which is determined by the

energy-momentum tensor density. To study the origin of

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matter field, there is a need to explain how the energy-

momentum density evolved from its non-existence state

(with zero energy ) to its existence state (with

positive energy ). In theoretical physics, there are

different definitions of the energy-momentum tensor

density, which can be used to derive different energy-

momentum tensor density conservation laws. A generally

applicable theory for origin of matter field must be

based on a generally applicable energy-momentum tensor

density conservation law. Under the local transformation

of the Poincare’ group, the energy-momentum tensor

density conservation law derived from the Noether

theorem is equivalent to the Lorentz and Levi-Civita

energy-momentum tensor density conservation laws. Because

Noether theorem is generally applicable to any physics

system, so do the Lorentz and Levi-Civita energy-

momentum tensor density conservation laws.

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The Lorentz and Levi-Civita energy-momentum tensor

density conservation laws dictate that when the energy-

momentum tensor density of the matter field of a physics

system increases, the energy-momentum tensor density of

the gravitational field will decrease, whereas when the

energy-momentum tensor density of the matter field of a

physical system decreases, the energy-momentum tensor

density of the gravitational field will increases, but

the sum remains constant. This implies that the energy-

momentum tensor density of the gravitational field can

be converted into the energy-momentum tensor density of

the matter field.

Under special conditions, the energy-momentum tensor

density may be zero, If the above mentioned

transformation of the energy-momentum tensor density

still exists, then correspondingly, the system in this

particular space-time location changes from the state of

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non-existence of energy-momentum tensor density of

matter field to that of the existence of energy-momentum

tensor density of matter field (with simultaneous

emergence of a negative energy-momentum tensor density

of gravitational field). Because the matter field is

always linked with its energy-momentum tensor density ,

the existence of the energy-momentum tensor density of

the matter field means the existence of matter field.

The non-existence of the energy-momentum tensor density

of the matter field means the non-existence of matter

field. The above analysis indicates that the Universe

can evolve from a state without matter field to a state

with matter field. The above opinion has been discussed

in a recent publication [8]. As this problem is closely

related to the problems discussed in this paper, it will

be further described below.

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There's a question of whether the energy-momentum

tensor density is identical to matter field? From the

following considerations, the two seem to be non-

identical but closely related. The energy and its

related energy-momentum tensor density must have a

bearer, which is likely the vacuum, namely Minkowski

space-time. When there is no matter field completely,the

space-time is a vacuum. A matter field emerges only when

the vacuum carries positive energy and the corresponding

energy-momentum tensor density. It is recognized that

there are different opinions about this question, and

further investigations are needed to fully resolve this

question.

Physics is an experiment-based science. Physics

theories must be based on the laboratory studies and

observations. Although there is no experiment to

definitively prove it, the concept of the creation of

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matter field from the state without matter field is not

inconsistent with existing experiments and observations.

For instance, the great energy phenomena of the quasar

and at the center of galaxies had been explained by the

existence of black holes, but there have been opinions

that black holes may not exist at all. An alternative

explanation is that these great energy phenomena are due

to the creation of matte field, which is definitely a

very promising explanation. While this is still a

hypothesis to be further verified by experimental

evidences, the current evidences seem to suggest that it

is not a totally impossible event for matter field to

emerge from the non-existence state, and this problem is

worth future investigations.

References

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[1] Kibble T.W.B.(1961), “Lorentz invariance and

the gravitational field”, J.Math.Phys.2：212.

[2] Chen F P (2014), “Space-time, the basic concept and

laws of physics” Scientific Publisher Beijing

[3] Carmeli M.(1982), “Classical fields:General

Relativity and Gauge Theory ”,John Wiley & Sons.New

York.

[4] Held F.W.,von der Heyde P.,Kerlick G.D.(1976),

“General relativity with spin and torsion:Foundations

and prospects”Rev.Mod.Phys.,48,393.

[5] Chen F P.(1990), International Journal of

Theoretical Physics, 29: 16

[6] Chen F P (2000), J Herbei Normal University, 24: 326

[7] Cattani C, De Maria M. (1993), Conservation Laws and

Gravitational Waves in General Relativity. // Earman J,

Janssen M, Norton J D. The Attraction of Gravitation,

Boston: Birkhauser.

[8] Chen F.P.(2015), “The Conservation Law of Energy-

Momentum Tensor Density and the Origin of

Matter”,Astronomy and Astrophysics , 3: 13.