Bach-Einstein Gravitational Field Equations as a Perturbation of Einstein Gravitational Field Equations

Full text

Applied Mathematics, 2022, 13, 1022-1032
https://www.scirp.org/journal/am
ISSN Online: 2152-7393
ISSN Print: 2152-7385
DOI:
10.4236/am.2022.1312063 Dec. 30, 2022 1022
Applied Mathematics
Bach-Einstein Gravitational Field Equations as
a Perturbation of Einstein Gravitational Field
Equations
Fathy Ibrahim Abdel-Bassier1, Ahmed Fouad Abdel-Wahab1, Fayrouz Mostafa Abdel-Maboud2
1Mathematics Department, Faculty of Science, Minia University, El-Minia, Egypt
2Higher Institute for Engineering and Technology, El-Minia, Egypt
Abstract
The Bach equations are a version of higher-order gravitational field equa-
tions, exactly they are of fourth-order. In 4-dimensions the Bach-
Einstein
gravitational field equations are treated here as a perturbation of Einstein’s
gravity. An approximate inversion formula is derived which admits a com-
parison of the two field theories. An application to these theories is given
where the gravitational Lagrangian is expressed linearly in terms of
2
2
,,R R Ric
,
where the Ricci tensor
ddRic R x x
αβ
αβ
=
is inserted in some formulas which
are of geometrical or physical importance,
such as; Raychaudhuri equation
and Tolman’s formula.
Keywords
Gravitational Theory, Higher Order Gravity, Buchdahl’s Formula,
Bach-Einstein Gravitational Field Equations, Raychaudhuri Equation,
Tolman’s Formula
1. Introduction
In this paper we study the purely metrical fourth-order theories of gravitation in
4-dimensions which follow from a Lagrangian
: 2,
grav mat
LL L
χ
= +
(1)
which is the sum of a gravitational Lagrangian of the form [1] [2] [3]:
()
2
2
01
:2 ,
grav
L R a R a Ric=− Λ+ + +
(2)
and an appropriate matter Lagrangian
mat
L
.
How to cite this paper:
Abdel-Bassier,
F.I.,
A
bdel-Wahab, A.F. and Abdel-Maboud, F.M.
(20
22) Bach-Einstein Gravita
tional Field
Equ
ations as a Perturbation of Einstein Gra-
vit
ational Field Equations.
Applied Math
e-
ma
tics
,
13
, 1022-1032.
https://doi.org/10.4236/am.2022.1312063
Received:
October 24, 2022
Accepted:
December 27, 2022
Published:
December 30, 2022
Copyright © 20
22 by author(s) 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. I. Abdel-Bassier et al.
DOI:
10.4236/am.2022.1312063 1023 Applied
Mathematics
In fact, the most general quadratic gravitational Lagrangian:
22
2
10 1 2
:,L c R c Ric c Riem=++
(3)
effectively reduces to:
2
2
20 1
:,L a R a Ric= +
(4)
with
0 0 21 1 2
,4a c ca c c=−=+
, because of the fact that the Gauss-Bonnet expres-
sion
22
2
:4 ,
B R Ric Riem=−+
(5)
has vanishing variational derivatives with respect to the metric in 4-dimensions
[4]-[18]. Here
01
,,aa
χ
are real coupling constants,
Λ
is a “cosmological con-
stant” and we abbreviate
2:Riem R R
αβµν
αβµν
=
,
2
:Ric R R
αβ
αβ
=
, where the Ricci
tensor
Ric
has the components
:RR
µ
αβ µαβ
=
, and the scalar curvature reads
:R tr Ric g R
αβ
αβ
= ≡
,
tr
denotes the trace with respect to the metric:
2
d dd.s g xx
αβ
αβ
=
We adopt the usual conventions of tensor calculus: Greek letters
,,,
αβγ

take the values
0,1,2,3
. The signature of the metric
g
is assumed to be
( )
+−−−
,
()( )
ddddRiemR xxxx
α β µν
αβµν
= ∧∧
denotes the Riemann curvature, their
components
R
ν
αβµ
are introduced through the Ricci identity for a one-form
duux
α
α
=
in terms of the Levi-Civita covariant derivatives
α
∇
to
g
[19] [20]
[21] [22] as:
( )
.u Ru
ν
α β β α µ αβµ ν
∇ ∇ −∇ ∇ =
Equivalently, there holds
,R
ν ν ν λν λν
αβµ β αµ α βµ αµ λβ βµ λα
=∂ Γ −∂ Γ +Γ Γ −Γ Γ
in terms of the Christoffel symbols
()
1,
2gg g g
µ µν
αβ α βν β αν ν αβ
Γ= ∂+∂−∂
while the Weyl conformal tensor, denoted by Weyl, is defined through its com-
ponents [21] [22]:
[] []
:.
6
R
W R gR gR g
αβµν αβµν α µ ν β β ν µ α αβµν
=++−
Here and in the following ( ) or [ ] indicate the symmetrization or antisym-
metrization respectively of indices and we abbreviate:
.g gg gg
αβµν αµ βν αν βµ
= −
Again in 4-dimensions, one can easily deduce the special quadratic expression
[3]-[13]:
2 22
2
1
:2
3
Weyl W W R Ric Riem
αβµν
αβµν
= =−+
(6)
is conformably invariant of weight −2, that means, a conformal transformation

F. I. Abdel-Bassier et al.
DOI:
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g eg
φ
→
,
φ
variable, implies
22
2
Weyl e Weyl
φ
−
→
. Accordingly, that the
Gauss-Bonnet expression (5) has vanishing variational derivatives with respect
to the metric in 4-dimensions, thus (6) is equivalent to:
22
2
2
: 2.
3
Weyl W W Ric R
αβµν
αβµν
= = −
(7)
The most general Einstein’s equations [21] are given as:
,GgT
αβ αβ αβ
χ
+Λ =
(8)
where
1
:,
2
G R Rg
αβ αβ αβ
= −
is the Einstein tensor
ddGG xx
αβ
αβ
=
, and
ddTT xx
αβ
αβ
=
is the ener-
gy-momentum tensor.
It is obviously, that the most general Einstein’s Equations (8) have the alterna-
tive formula [20]:
.
2
trT
RT gg
αβ αβ αβ αβ
χ

= − +Λ


(9)
A spacetime for which
1, .,
4
R Rg R const
αβ αβ
= =
(10)
is called an Einstein spacetime [22]. Inserting Equation (10) into the identity (7)
one obtains
22
1.
6
Weyl R= −
In Section 2; we introduce the variation derivatives of the Lagrangian (1) with
respect to
g
which produces the fourth-order gravitational field Equations (14).
It well known that the choice
10
3aa= −
of the gravitational Lagrangian (2),
yields the so-called Bach-Einstein gravitational field Equations (21). In Section 3;
a general algebraic structure is discussed, where we show that the Ricci tensor
components
R
αβ
to
g
can be represented by a covariant linear differential op-
erator applied to a linear combination of
,,
T g trT g
µν µν µν
Λ
plus an error term
with the factor
2
ε
, where
ε
is a real parameter such that
ε
is so small, that
is
1
ε

. In Section 4; the Bach-Einstein gravitational field equations in 4-di-
mensions are treated as a perturbation of Einstein’s gravity, where we derive an
approximate inversion Formula (32) which admits a comparison of the two field
theories. Exactly, we obtain an approximate inversion formula corresponding of
the Bach-Einstein gravitational field equations similar to the alternative Formula
(9). Finally, in the last section, an application to both the Einstein gravitational
field equations and Bach-Einstein gravitational field equations is given where the
gravitational Lagrangian is expressed linearly in terms of
2
2
,,R R Ric
(28),
where the Ricci tensor
ddRic R x x
αβ
αβ
=
is inserted in some formulas which are
of geometrical or physical importance, such as; Raychaudhuri equation and

F. I. Abdel-Bassier et al.
DOI:
10.4236/am.2022.1312063 1025 Applied
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Tolman’s formula. [1] [23]. D. Barraco and V. H. Hamity [1] mention Tolman’s
expression as a possible application of approximate inversion formulas, where
the gravitational Lagrangian is expressed linearly in terms of
2
,RR
.
2. The Fourth-Order Gravity
Variation derivatives of the Lagrangian (1) with respect to
g
produce the field
equations
,ET
αβ αβ
χ
=
where the variational derivative tensor
E
αβ
and the energy-momentum tensor
T
αβ
are defined by
11
22
:,
grav
det g E det g L
g
αβ αβ
δ
δ

=

11
22
:2 ,
mat
det g T det g L
g
αβ αβ
δ
δ

= − 

( )
:.det g det g
αβ
=
Here the symbol
δ
expresses variational derivatives (cf., e.g. [13] [24]).
Let us now calculate the variational derivative tensor
dd
EE xx
αβ
αβ
=
in the
general. Using Buchdahl’s formula: according to [13]-[18] there holds:
21
,
32
grav
E Z R Z gL
µ ν νµρ
αβ αβµν αρµν β αβ
=∇∇ − −
where
( )( )
[ ][ ]
, 2, .
grav
L
ZY Y X X R
αβµν αβµν αβµν
αβ µν αν βµ αβµν
∂
= = = ∂
Consequently, the fourth-order gravitational field equations of the Lagrangian
(1) read
( ) ( )
01
01
,E g G aE aE T
αβ αβ αβ αβ αβ αβ
χ
≡Λ + + + =
(11)
where
( )
( )
02
2
1
22
2
222 ,
2
E g R RR R g
R
Rg RRR g
µν
αβ µαβν αβ αβ
α β αβ αβ αβ
= ∇∇ + −
≡ ∇∇ − + −
(12)
()
2
1
1
2,
22
g
E R R R R R Ric g
αβ µν
αβ α β αβ µαβν αβ
=∇∇ − − + −
(13)
where
:g
αβ
αβ
= ∇∇
is the covariant d’Alembertian operator.
Thus, the fourth-order field Equation (11) takes the form:
2
0
2
1
11
222
22
1
2.
22
g R Rg a R g R RR R g
g
a R R R R R Ric g T
αβ αβ αβ α β αβ αβ αβ
αβ µν
α β αβ µαβν αβ αβ
χ

Λ + − + ∇∇ − + −



+ ∇∇ − − + − =




(14)

F. I. Abdel-Bassier et al.
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Inserting Equation (10) into the fourth-order tensors (12), (13), one easily
obtains:
( ) ( )
01
0,EE
αβ αβ
= =
anyway, in 4-dimensions, the variational derivative tensor
ddE E xx
αβ
αβ
=
that
corresponding to the most general quadratic Lagrangian (3) on an Einstein
spacetime (10) identically vanishes [3]. Consequently, the fourth-order field Eq-
uations (14) on an Einstein spacetime and the most general Einstein’s Equations
(8) are equivalent, where:
1,.
4R g T R const
αβ αβ
χ

Λ− = =


It is obvious that the choice
01
0aa= =
of the gravitational Lagrangian (2),
leads to the Einstein -Hilbert gravitational Lagrangian, that is:
:2 ,
EH
LR=− Λ+
which yields the most general Einstein’s Equations (8). On the other hand, the
choice
10
3aa= −
of the gravitational Lagrangian (4), leads to
( )
2
2
20
: 3.L a R Ric= −
(15)
Comparing (7), (15) we obtain the Bach gravitational Lagrangian, that is:
2
20
3
:,
2
Bach
L L a Weyl
−
≡=
(16)
which, leads, possibly supplemented by an appropriate choice of
mat
L
, to con-
formably invariant fourth-order field equations, namely the equations intro-
duced by R. Bach in 1921 [12]:
0
3,aB T
αβ αβ
χ
=
(17)
that called the Bach field equations, where the Bach tensor
ddBach B x x
αβ
αβ
=
[7]-[13] is given by:
( )
2
1.
63
B W WR
R
R g R W R Rg R
µ ν µν
αβ µαβν µαβν
µν
αβ αβ α β µαβν µαβν αµ βν
= ∇∇ −

= − − ∇∇ − + −



(18)
One can easily show that the Bach tensor (18) is symmetric, trace-free; that is,
0gB
αβ
αβ
=
, divergence-free; that is,
0B
α
αβ
∇=
, and is conformably invariant
of weight −1 [3] [8].
We can rewrite the gravitational Lagrangian (2) in terms of (16) as
2
0
3
:2 ,
2
grav
L R a Weyl
−
=− Λ+ +
(19)
which leads to the Bach-Einstein field equations
0
3.g G aB T
αβ αβ αβ αβ
χ
Λ+ + =
(20)
Using Equations (14)-(17) the Bach-Einstein field Equations (20) can be re-
written as:

F. I. Abdel-Bassier et al.
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Mathematics
0
2
11
3
22
26 .
2
g R Rg a R g R R
L
RR R R g T
αβ αβ αβ αβ αβ α β
µν
αβ µαβν αβ αβ
χ

Λ + − + − −∇ ∇



+− − =



(21)
3. Algebraic Structure
Generally, let us consider fourth-order gravitational field equations take the
form:
1,
2
R Rg g D R T
µν
αβ αβ αβ αβ µν αβ
εχ
− +Λ + =
(22)
where
ε
is a real parameter such that
ε
is so small, that is
1
ε

. The ten-
sor field
T
is assumed to be divergence-free:
0.T
β
αβ
∇=
According to that we require the identity
()
0.DR
β µν
αβ µν
∇=
We assume, without restriction of generality that,
D
µν
αβ
is symmetric in
α
and
β
as well as in
µ
and
ν
( ) ( )
,DD D
µν
µν µν
αβ αβ
αβ
= =
It is easy to see that (22) is a singular perturbation of (8) since the small para-
meter
ε
appears as a factor of the higher-order term
DR
µν
αβ µν
. Now, we show
that the Ricci tensor components
R
αβ
to
g
can be represented by a covariant
linear differential operator applied to a linear combination of
,,
T g trT g
µν µν µν
Λ
plus an error term with the factor
2
ε
.
Contraction of (22) with
g
αβ
yields
4.R D R trT
µν
µν
εχ
= Λ+ −
Inserting this value for
R
in (22), we get
( )
,DR T g
µ ν µν
α β αβ µν αβ αβ
δδ ε χ
+ = +Λ

(23)
where
:,
2
trT
TT g
αβ αβ αβ
= −

(24)
1
: ,: .
2
D D Dg D gD
µν µν µν µν αβ µν
αβ αβ αβ αβ
=−=

(25)
The linear tensor-operator with the components
D
µ ν µν
α β αβ
δδ ε
+
on the left-
hand side in (23) has an approximate inverse with the components
,D
µ ν µν
α β αβ
δδ ε
−
in analogy to the formula
( )
12
11 ,q qr
ε εε
−
+ =−+
where the remainder term
( )
12
1,r qq
ε
−
= +

F. I. Abdel-Bassier et al.
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is continuous in
ε
if
q
continuously depends on
ε
and
ε
is so small such
that
1q
ε
<
. Thus, in general, the Ricci tensor components
R
αβ
to
g
can be
represented approximately by a covariant linear differential operator applied to a
linear combination of
,,T g trT g
µν µν µν
Λ
as:
()
()
,
R DTg
µ ν µν
αβ α β αβ µν µν
δδ ε χ
≅ − +Λ

(26)
where
≅
means equality up to terms with the factor
2
ε
. It is obvious that for
0
ε
=
both (22) and (26) reduce to the most general Einstein’s Equations (8).
4. Perturbation on the Bach-Einstein Field Equations
Let us apply the approximation procedure of section 3 to a class of fourth-order
gravitational field equations in 4-dimensions, whence, the Bach-Einstein field
Equations (21). Namely, let us consider a Lagrangian
: 2,
grav mat
LL L
χ
= +
(27)
here the gravitational Lagrangian has the form
( )
2
2
01
:2 ,
grav
L R a R a Ric
ε
=− Λ+ + +
(28)
ε
is a small parameter.
Thus, the fourth-order field equations take, simply, the symbol form:
( ) ( )
( )
01
01
,E g G aE aE T
αβ αβ αβ αβ αβ αβ
εχ
≡Λ + + + =
(29)
where
( )
0
E
αβ
and
( )
1
E
αβ
are given respectively in (12) and (13).
The field equations derived from the Lagrangian (27), with the gravitational
Lagrangian (28) have the form (22) with
D
µν
αβ
in the form
0
1 ( ) ()
1
2 22
2
11
22 .
22
D a g R Rg g
a gg R Rg
µν µν
αβ α β αβ αβ αβ
µ ν µν ν µ ν µ µν
α β αβ α β α β αβ
δδ δ δ

= ∇∇ − + −



− + − ∇∇ − +




It is noticeable that, the Riemann curvature tensor has been eliminated by
means of the Ricci identity
2 2 2.R R R R RR
µν µ µ
µαβν µ β α α β αµ β
= ∇∇ −∇∇ +
Applying the results of (24)-(26) to the present situation yields
( )
( )
,R DTg
µ ν µν
αβ α β αβ µν µν
δδ ε χ
≅ − +Λ

where, in this case
:,
2
trT
TT g
αβ αβ αβ
= −

0
1 ( ) ()
1
22
2
1
22 .
2
D a g R Rg g
a gg g R Rg
µν µν
αβ α β αβ αβ αβ
µ ν µν µ ν ν µ ν µ µν
α β αβ αβ α β α β αβ
δδ δ δ

= ∇∇ + + −



− − + ∇∇ − ∇∇ − +




We arrive at

F. I. Abdel-Bassier et al.
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Mathematics
( )
( )
0
1 ( ) ()
1
22 4
2
22
1.
2
R T g a g R Rg trT
aR
gg R T g
αβ αβ αβ α β αβ αβ αβ
µν ν µ ν µ
αβ α β αβ
µν µ ν µν
αβ µν µν
χε χ
ε δδ δ δ
χ

≅ +Λ + ∇ ∇ + + − − Λ



+ − ∇∇ −




− −∇ ∇ − + Λ









(30)
Since we neglect the terms of order
2
ε
, then we substitute by the following
expressions for
R
αβ
and
R
:
, 4,R T g R trT
αβ αβ αβ
χχ
≅ +Λ ≅− + Λ

in (30), so we get the perturbation of (29) as:
( )
( )
( )
( )
0
2
1
()
22
2 24
2
12
2
2 2 22
11 ,
22
trT
R T g g a trT trT T
g trT trT trT a T trT
T T T trT T
g T trT trT
αβ αβ αβ αβ α β αβ
αβ αβ α β
µµ
αβµ µαβ αβ
αβ
χ εχ χ
χ εχ
χχ
χχ


≅ − +Λ + ∇ ∇ + − Λ
 
 


+ − + Λ + +∇ ∇




− ∇∇ − + − Λ


+ − +Λ





(31)
up to terms with the factor
2
ε
. The trace part of (31) reads
( )
( )
01
2 3 1 4.R a a trT
χε
≅ + − +Λ
Accordingly, (15)-(20), (27)-(29) and (31), we can easily deduce:
( )
( )
0 ()
22
6
2
3
36
2
4,
trT
R T g g a trT T
T g trT T trT trT T T
trT T
µ
αβ αβ αβ αβ α β α β µ
µ
αβ αβ µα β
αβ
χ εχ
χχ χ
χ
 
≅ − +Λ − ∇∇ − ∇∇
 


+ − − + −Λ −



+ −Λ 

(32)
which are a perturbation on the Bach-Einstein field equations.
Simply, the choice
10
3aa= −
of the perturbation Equation (31), leads to a
perturbation on the Bach-Einstein field Equations (32). On the other hand, the
choice
0
ε
=
or
01
0aa= =
of the perturbation Equation (31), leads to the
most general Einstein’s Equations (9). Of course, the choice
0
ε
=
or
0
0a=
of a perturbation on the Bach-Einstein field Equations (32) leads, also, to the
most general Einstein’s Equations (9).
5. Conclusions and Discussions
There is a well-established theory and a broad literature on singular perturba-
tions of differential equations [3]. We circumvent here this theory by assuming
the existence of solutions regular in the perturbation parameter
ε
, and we de-
duce the result (32) on the latter.
The approximate inversion Formulas (31) and (32) derived here stress the role
of the Ricci tensor in the class of alternative gravitational theories under consid-

F. I. Abdel-Bassier et al.
DOI:
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eration. Let us recall that the Ricci tensor
ddRic R x x
αβ
αβ
=
appears in several
formulas of geometrical or physical importance:
 The volume of geodesic balls in Riemannian geometry can be expanded with
respect to the radius [25]; analogously the volume of truncated light cones in
Lorentzian geometry can be expanded with respect to the truncation time
parameter [7]. The leading terms of the deviations from the flat space or flat
spacetime values are linear in the Ricci tensor. Moreover, some estimate for
Ric
leads to estimates for the volume of geodesic balls [26] [27].
 The Raychaudhuri equation for the so-called geometrical expansion
θ
of a
family of timelike geodesics with tangent vector field
uu
α
α
= ∂
reads
22 2
1,
3
R uu u
αβ α
αβ α
ωσθ θ
=∇ + − −−


where the dot abbreviates the derivative
u
α
α
∇
and where the rotation
ω
, the
shear
σ
, and the expansion
θ
of
u
arise from the decomposition of
u uu
αβ αβ
∇+

into irreducible parts (cf. e.g., [28] [29] [30]).
 Singularity theorems of Hawking-Penrose type are based on assumptions on
the Ricci tensor [31] [32].
 Tolman’s formula expresses the total active mass of a static, asymptotically
flat spacetime as
2d,
S
M R nu
αβ
αβ
σ
χ
=
∫
where
nn
α
α
= ∂
denotes the unit normal to the spacelike hypersurface,
uu
α
α
= ∂
is the timelike Killing vector field, and
d
σ
is the natural volume element of the
hypersurfaces [1] [23]. D. Barraco and V. H. Hamity [1] mention Tolman’s ex-
pression as a possible application of approximate inversion formulas.
The Formulas (31) and (32) express the Ricci tensor in terms of the ener-
gy-momentum tensor
ddTT xx
αβ
αβ
=
. Such a result can be inserted into each of
the above-mentioned geometrical or physical formulas where the Ricci tensor
plays a dominant role. By this, the influence of the energy-momentum tensor
becomes transparent.
Acknowledgements
The first two authors would like to express their gratitude to their advisor Prof.
R. Schimming for his excellent teaching as well as kind support to them during
their stay in Greifswald University. The authors express gratitude to the referees
for their valuable comments and suggestions. Also, it is our pleasure to extend
our sincere thanks and appreciation for the constructive cooperation from the
editor of the journal for the important and technical modifications he made to
us, which make the work in a good form.
Conflicts of Interest
The authors declare no conflicts of interest regarding the publication of this
paper.

F. I. Abdel-Bassier et al.
DOI:
10.4236/am.2022.1312063 1031 Applied
Mathematics
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