In this paper we study the purely metrical fourth-order theories of gravitation in 4-dimensions which follow from a Lagrangian

L : = L g r a v + 2 χ L m a t , (1)

which is the sum of a gravitational Lagrangian of the form [1] [2] [3]:

L g r a v : = − 2 Λ + R + ( a 0 R 2 + a 1 | R i c | 2 ) , (2)

and an appropriate matter Lagrangian L m a t .

In fact, the most general quadratic gravitational Lagrangian:

L 1 : = c 0 R 2 + c 1 | R i c | 2 + c 2 | R i e m | 2 , (3)

effectively reduces to:

L 2 : = a 0 R 2 + a 1 | R i c | 2 , (4)

with a 0 = c 0 − c 2 ,   a 1 = c 1 + 4 c 2 , because of the fact that the Gauss-Bonnet expression

B : = R 2 − 4 | R i c | 2 + | R i e m | 2 , (5)

has vanishing variational derivatives with respect to the metric in 4-dimensions [4] - [18]. Here χ , a 0 , a 1 are real coupling constants, Λ is a “cosmological constant” and we abbreviate | R i e m | 2 : = R α β μ ν R α β μ ν , | R i c | 2 : = R α β R α β , where the Ricci tensor R i c has the components R α β : = R μ α β μ , and the scalar curvature reads R : = t r     R i c ≡ g α β R α β , t r denotes the trace with respect to the metric:

d s 2 = g α β d x α d x β .

We adopt the usual conventions of tensor calculus: Greek letters α , β , γ , ⋯ take the values 0,1,2,3 . The signature of the metric g is assumed to be ( + − − − ) , R i e m = R α β μ ν ( d x α ∧ d x β ) ( d x μ ∧ d x ν ) denotes the Riemann curvature, their components R α β μ ν are introduced through the Ricci identity for a one-form u = u α d x α in terms of the Levi-Civita covariant derivatives ∇ α to g [19] [20] [21] [22] as:

( ∇ α ∇ β − ∇ β ∇ α ) u μ = R α β μ ν u ν .

Equivalently, there holds

R α β μ ν = ∂ β Γ α μ ν − ∂ α Γ β μ ν + Γ α μ λ Γ λ β ν − Γ β μ λ Γ λ α ν ,

in terms of the Christoffel symbols

Γ α β μ = 1 2 g μ ν ( ∂ α g β ν + ∂ β g α ν − ∂ ν g α β ) ,

while the Weyl conformal tensor, denoted by Weyl, is defined through its components [21] [22]:

W α β μ ν : = R α β μ ν + g α [ μ R ν ] β + g β [ ν R μ ] α − R 6 g α β μ ν .

Here and in the following ( ) or [ ] indicate the symmetrization or antisymmetrization respectively of indices and we abbreviate:

g α β μ ν = g α μ g β ν − g α ν g β μ .

Again in 4-dimensions, one can easily deduce the special quadratic expression [3] - [13]:

| W e y l | 2 : = W α β μ ν W α β μ ν = 1 3 R 2 − 2 | R i c | 2 + | R i e m | 2 (6)

is conformably invariant of weight −2, that means, a conformal transformation g → e ϕ g , ϕ variable, implies | W e y l | 2 → e − 2 ϕ | W e y l | 2 . Accordingly, that the Gauss-Bonnet expression (5) has vanishing variational derivatives with respect to the metric in 4-dimensions, thus (6) is equivalent to:

| W e y l | 2 : = W α β μ ν W α β μ ν = 2 | R i c | 2 − 2 3 R 2 . (7)

The most general Einstein’s equations [21] are given as:

G α β + Λ g α β = χ T α β , (8)

where

G α β : = R α β − 1 2 R g α β ,

is the Einstein tensor G = G α β d x α d x β , and T = T α β d x α d x β is the energy-momentum tensor.

It is obviously, that the most general Einstein’s Equations (8) have the alternative formula [20]:

R α β = χ ( T α β − t r T 2 g α β ) + Λ g α β . (9)

A spacetime for which

R α β = 1 4 R g α β ,   R = c o n s t . , (10)

is called an Einstein spacetime [22]. Inserting Equation (10) into the identity (7) one obtains

| W e y l | 2 = − 1 6 R 2 .

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 a 1 = − 3 a 0 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 α β to g can be represented by a covariant linear differential operator applied to a linear combination of T μ ν , g μ ν t r T , Λ g μ ν plus an error term with the factor ε 2 , where ε is a real parameter such that | ε | is so small, that is | ε | ≪ 1 . In Section 4; the Bach-Einstein gravitational field equations in 4-dimensions 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 R , R 2 , | R i c | 2 (28), where the Ricci tensor R i c = R α β d x α d x β is inserted in some formulas which are of geometrical or physical importance, such as; Raychaudhuri equation and 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 R , R 2 .

Variation derivatives of the Lagrangian (1) with respect to g produce the field equations

E α β = χ T α β ,

where the variational derivative tensor E α β and the energy-momentum tensor T α β are defined by

| d e t   g | 1 2 E α β : = δ δ g α β ( | d e t   g | 1 2 L g r a v ) ,

| d e t   g | 1 2 T α β : = − 2 δ δ g α β ( | d e t   g | 1 2 L m a t ) ,

d e t   g : = d e t ( g α β ) .

Here the symbol δ expresses variational derivatives (cf., e.g. [13] [24]).

Let us now calculate the variational derivative tensor E = E α β d x α d x β in the general. Using Buchdahl’s formula: according to [13] - [18] there holds:

E α β = ∇ μ ∇ ν Z α β μ ν − 2 3 R α ρ μ ν Z β ν μ ρ − 1 2 g α β L g r a v ,

where

Z α β μ ν = Y ( α β ) ( μ ν ) ,   Y α β μ ν = 2 X [ α ν ] [ β μ ] ,   X α β μ ν = ∂ L g r a v ∂ R α β μ ν .

Consequently, the fourth-order gravitational field equations of the Lagrangian (1) read

E α β ≡ Λ g α β + G α β + a 0 E α β ( 0 ) + a 1 E α β ( 1 ) = χ T α β , (11)

where

E α β ( 0 ) = 2 ∇ μ ∇ ν ( g μ α β ν R ) + 2 R R α β − 1 2 R 2 g α β ≡ 2 ∇ α ∇ β R − 2 g α β □ R + 2 R R α β − R 2 2 g α β , (12)

E α β ( 1 ) = ∇ α ∇ β R −   □ R α β − g α β 2 □ R + 2 R μ α β ν R μ ν − 1 2 | R i c | 2 g α β , (13)

where □   : = g α β ∇ α ∇ β is the covariant d’Alembertian operator.

Thus, the fourth-order field Equation (11) takes the form:

Λ g α β + R α β − 1 2 R g α β + a 0 ( 2 ∇ α ∇ β R − 2 g α β □ R + 2 R R α β − 1 2 R 2 g α β )   + a 1 ( ∇ α ∇ β R −   □ R α β − g α β 2 □ R + 2 R μ α β ν R μ ν − 1 2 | R i c | 2 g α β ) = χ T α β . (14)

Inserting Equation (10) into the fourth-order tensors (12), (13), one easily obtains:

E α β ( 0 ) = 0 = E α β ( 1 ) ,

anyway, in 4-dimensions, the variational derivative tensor E = E α β d x α d x β that corresponding to the most general quadratic Lagrangian (3) on an Einstein spacetime (10) identically vanishes [3]. Consequently, the fourth-order field Equations (14) on an Einstein spacetime and the most general Einstein’s Equations (8) are equivalent, where:

( Λ − 1 4 R ) g α β = χ T α β ,   R = c o n s t .

It is obvious that the choice a 0 = a 1 = 0 of the gravitational Lagrangian (2), leads to the Einstein -Hilbert gravitational Lagrangian, that is:

L E H : = − 2 Λ + R ,

which yields the most general Einstein’s Equations (8). On the other hand, the choice a 1 = − 3 a 0 of the gravitational Lagrangian (4), leads to

L 2 : = a 0 ( R 2 − 3 | R i c | 2 ) . (15)

Comparing (7), (15) we obtain the Bach gravitational Lagrangian, that is:

L 2 ≡ L B a c h : = − 3 2 a 0 | W e y l | 2 , (16)

which, leads, possibly supplemented by an appropriate choice of L m a t , to conformably invariant fourth-order field equations, namely the equations introduced by R. Bach in 1921 [12]:

3 a 0 B α β = χ T α β , (17)

that called the Bach field equations, where the Bach tensor B a c h = B α β d x α d x β [7] - [13] is given by:

B α β = 2 ∇ μ ∇ ν W μ α β ν − W μ α β ν R μ ν =   □ ( R α β − R 6 g α β ) − 1 3 ∇ α ∇ β R − ( W μ α β ν + R μ α β ν − R α μ g β ν ) R μ ν . (18)

One can easily show that the Bach tensor (18) is symmetric, trace-free; that is, g α β B α β = 0 , divergence-free; that is, ∇ α B α β = 0 , and is conformably invariant of weight −1 [3] [8].

We can rewrite the gravitational Lagrangian (2) in terms of (16) as

L g r a v : = − 2 Λ + R + − 3 2 a 0 | W e y l | 2 , (19)

which leads to the Bach-Einstein field equations

Λ g α β + G α β + 3 a 0 B α β = χ T α β . (20)

Using Equations (14)-(17) the Bach-Einstein field Equations (20) can be rewritten as:

Λ g α β + R α β − 1 2 R g α β + a 0 ( 3 □ R α β − 1 2 g α β □ R − ∇ α ∇ β R   + 2 R R α β − 6 R μ α β ν R μ ν − L 2 2 g α β ) = χ T α β . (21)

Generally, let us consider fourth-order gravitational field equations take the form:

R α β − 1 2 R g α β + Λ g α β + ε D α β μ ν R μ ν = χ T α β , (22)

where ε is a real parameter such that | ε | is so small, that is | ε | ≪ 1 . The tensor field T is assumed to be divergence-free:

∇ β T α β = 0.

According to that we require the identity

∇ β ( D α β μ ν R μ ν ) = 0.

We assume, without restriction of generality that, D α β μ ν is symmetric in α and β as well as in μ and ν

D α β μ ν = D ( α β ) μ ν = D α β ( μ ν ) ,

It is easy to see that (22) is a singular perturbation of (8) since the small parameter ε appears as a factor of the higher-order term D α β μ ν R μ ν . Now, we show that the Ricci tensor components R α β to g can be represented by a covariant linear differential operator applied to a linear combination of T μ ν , g μ ν t r T , Λ g μ ν plus an error term with the factor ε 2 .

Contraction of (22) with g α β yields

R = 4 Λ + ε D μ ν R μ ν − χ t r T .

Inserting this value for R in (22), we get

( δ α μ δ β ν + ε D ˜ α β μ ν ) R μ ν = χ T ˜ α β + Λ g α β , (23)

where

T ˜ α β : = T α β − t r T 2 g α β , (24)

D ˜ α β μ ν : = D α β μ ν − 1 2 D μ ν g α β ,   D μ ν : = g α β D α β μ ν . (25)

The linear tensor-operator with the components δ α μ δ β ν + ε D ˜ α β μ ν on the left-hand side in (23) has an approximate inverse with the components δ α μ δ β ν − ε D ˜ α β μ ν , in analogy to the formula

( 1 + ε q ) − 1 = 1 − ε q + ε 2 r ,

where the remainder term

r = ( 1 + ε q ) − 1 q 2 ,

is continuous in ε if q continuously depends on ε and | ε | is so small such that | ε q | < 1 . Thus, in general, the Ricci tensor components R α β to g can be represented approximately by a covariant linear differential operator applied to a linear combination of T μ ν , g μ ν t r T , Λ g μ ν as:

R α β ≅ ( δ α μ δ β ν − ε D ˜ α β μ ν ) ( χ T ˜ μ ν + Λ g μ ν ) , (26)

where ≅ means equality up to terms with the factor ε 2 . It is obvious that for ε = 0 both (22) and (26) reduce to the most general Einstein’s Equations (8).

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

L : = L g r a v + 2 χ L m a t , (27)

here the gravitational Lagrangian has the form

L g r a v : = − 2 Λ + R + ε ( a 0 R 2 + a 1 | R i c | 2 ) , (28)

ε is a small parameter.

Thus, the fourth-order field equations take, simply, the symbol form:

E α β ≡ Λ g α β + G α β + ε ( a 0 E α β ( 0 ) + a 1 E α β ( 1 ) ) = χ T α β , (29)

where E α β ( 0 ) and E α β ( 1 ) 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 α β μ ν in the form

D α β μ ν = a 0 ( 2 ∇ α ∇ β − 2 g α β □   + 2 R α β − 1 2 R g α β ) g μ ν       − a 1 ( δ α μ δ β ν □   + 1 2 g α β g μ ν □   − 2 δ ( α ν ∇ μ ∇ β ) − 2 δ ( α ν R β ) μ + 1 2 R μ ν g α β ) .

It is noticeable that, the Riemann curvature tensor has been eliminated by means of the Ricci identity

2 R μ α β ν R μ ν = 2 ∇ μ ∇ β R α μ − ∇ α ∇ β R + 2 R α μ R β μ .

Applying the results of (24)-(26) to the present situation yields

R α β ≅ ( δ α μ δ β ν − ε D ˜ α β μ ν ) ( χ T ˜ μ ν + Λ g μ ν ) ,

where, in this case

T ˜ α β : = T α β − t r T 2 g α β ,

D ˜ α β μ ν = a 0 ( 2 ∇ α ∇ β + g α β □   + 2 R α β − 1 2 R g α β ) g μ ν       − a 1 ( δ α μ δ β ν □   − g α β g μ ν □   + g α β ∇ μ ∇ ν − 2 δ ( α ν ∇ μ ∇ β ) − 2 δ ( α ν R β ) μ + 1 2 R μ ν g α β ) .

We arrive at

R α β ≅ χ T ˜ α β + Λ g α β + ε a 0 ( 2 ∇ α ∇ β + g α β □   + 2 R α β − 1 2 R g α β ) ( χ t r T − 4 Λ )   + ε a 1 [ δ α μ δ β ν □ − 2 δ ( α ν ∇ μ ∇ β ) − 2 δ ( α ν R β ) μ       − g α β ( g μ ν □ − ∇ μ ∇ ν − 1 2 R μ ν ) ] ( χ T ˜ μ ν + Λ g μ ν ) . (30)

Since we neglect the terms of order ε 2 , then we substitute by the following expressions for R α β and R:

R α β ≅ χ T ˜ α β + Λ g α β ,   R ≅ − χ t r T + 4 Λ ,

in (30), so we get the perturbation of (29) as:

R α β ≅ χ ( T α β − t r T 2 g α β ) + Λ g α β + ε a 0 χ [ 2 ∇ α ∇ β t r T + 2 ( χ t r T − 4 Λ ) T α β       + g α β ( □ t r T − 1 2 χ ( t r T ) 2 + 2 Λ t r T ) + ε a 1 χ [ □ T α β + ∇ α ∇ β t r T       − 2 ∇ μ ∇ ( α T β ) μ − 2 χ T μ α T β μ + 2 ( χ t r T − 2 Λ ) T α β   + g α β ( 1 2 χ | T | 2 − 1 2 χ ( t r T ) 2 + Λ t r T ) ] , (31)

up to terms with the factor ε 2 . The trace part of (31) reads

R ≅ χ ( 2 ε ( 3 a 0 + a 1 ) □   − 1 ) t r T + 4 Λ .

Accordingly, (15)-(20), (27)-(29) and (31), we can easily deduce:

R α β ≅ χ ( T α β − t r T 2 g α β ) + Λ g α β − ε a 0 χ [ ∇ α ∇ β t r T − 6 ∇ μ ∇ ( α T β ) μ   + 3 □ T α β − g α β ( □ t r T − 3 2 χ | T | 2 + χ ( t r T ) 2 − Λ t r T ) − 6 χ T μ α T β μ   + 4 ( χ t r T − Λ ) T α β ] , (32)

which are a perturbation on the Bach-Einstein field equations.

Simply, the choice a 1 = − 3 a 0 of the perturbation Equation (31), leads to a perturbation on the Bach-Einstein field Equations (32). On the other hand, the choice ε = 0 or a 0 = a 1 = 0 of the perturbation Equation (31), leads to the most general Einstein’s Equations (9). Of course, the choice ε = 0 or a 0 = 0 of a perturbation on the Bach-Einstein field Equations (32) leads, also, to the most general Einstein’s Equations (9).

There is a well-established theory and a broad literature on singular perturbations of differential equations [3]. We circumvent here this theory by assuming the existence of solutions regular in the perturbation parameter ε , and we deduce 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 consideration. Let us recall that the Ricci tensor R i c = R α β d x α d x β 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 R i c 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 u = u α ∂ α reads

R α β u α u β = ∇ α u ˙ α + ω 2 − σ 2 − θ ˙ − 1 3 θ 2 ,

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 ∇ α u β + u ˙ α u β 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

M = 2 χ ∫ S     R α β n α u β d σ ,

where n = n α ∂ α denotes the unit normal to the spacelike hypersurface, u = u 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 expression as a possible application of approximate inversion formulas.

The Formulas (31) and (32) express the Ricci tensor in terms of the energy-momentum tensor T = T α β d x α d x β . 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.

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.

The authors declare no conflicts of interest regarding the publication of this paper.

Abdel-Bassier, F.I., Abdel-Wahab, A.F. and Abdel-Maboud, F.M. (2022) Bach-Einstein Gravitational Field Equations as a Perturbation of Einstein Gravitational Field Equations. Applied Mathematics, 13, 1022-1032. https://doi.org/10.4236/am.2022.1312063