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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
13
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.