10508388
The Newtonian limit of fourth-order gravity
The weak-field slow-motion limit of fourth-order gravity will be discussed. Let us consider the gravitational theory defined by the Lagrangian Lg = (8πG) −1 ( R/2 + (αRijR ij + βR)l ) . (1) G is Newton’s constant, l a coupling length and α and β numerical parameters. Rij and R are the Ricci tensor and its trace. Introducing the matter Lagrangian Lm and varying Lg + Lm one obtains the field equation Eij + αHij + βGij = 8πGTij . (2) For α = β = 0 this reduces to General Relativity Theory. The explicit expressions Hij and Gij can be found in STELLE (1978). In a well-defined sense, the weak-field slow-motion limit of Einstein’s theory is just Newton’s theory, cf. DAUTCOURT (1964). In the following we consider the analogous problem for fourth order gravity eqs. (1), (2). For the special cases α = 0 (PECHLANER, SEXL(1966), POLIJEVKTOVNIKOLADZE (1967)), α + 2β = 0 (HAVAS (1977), JANKIEWICZ (1981)) 1 and α+ 3β = 0 (BORZESZKOWSKI, TREDER, YOURGRAU (1978)) this has already been done in the past. Cf. also ANANDAN (1983), where torsion has been taken into account. The slow-motion limit can be equivalently described as the limit c→ ∞, where c is the velocity of light. In this sense we have to take the limit G→ 0 while G · c and l remain constants. Then the energy-momentum tensor Tij reduces to the rest mass density ρ: Tij = δ 0 i δ 0 j ρ , (3) x = t being the time coordinate. The metric can be written as ds = (1 − 2φ)dt − (1 + 2ψ)(dx + dy + dz) . (4) Now eqs. (3) and (4) will be inserted into eq. (2). In our approach, products and time derivatives of φ and ψ can be neglected, i.e., R = 4∆ψ − 2∆φ , where ∆f = f,xx + f,yy + f,zz . Further R00 = −∆φ, H00 = −2∆R00 − ∆R and G00 = −4∆R, where l = 1. Then it holds: The validity of the 00-component and of the trace of eq. (2), R00 −R/2 + αH00 + βG00 = 8πGρ (5) and − R− 4(α+ 3β)∆R = 8πGρ , (6) imply the validity of the full eq. (2). Now, let us discuss eqs. (5) and (6) in more details: Eq. (5) reads − ∆φ−R/2 + α(2∆∆φ− ∆R) − 4β∆R = 8πGρ . (7)
115391729
COMMENT: Fourth-order gravity and conformal transformations
15961894
Comparing selfinteracting scalar fields and R + R**3 cosmological models
We generalize the well-known analogies between m2φ2 and R + R2 theories to include the selfinteraction λφ4-term for the scalar field. It turns out to be the R + R3 Lagrangian which gives an appropiate model for it. Considering a spatially flat Friedman cosmological model, common and different properties of these models are discussed, e. g., by linearizing around a ground state the masses of the resp. spin 0-parts concide. Finally, we prove a general conformal equivalence theorem between a Lagrangian L = L(R), L′L″ ≠ 0, and a minimally coupled scalar field in a general potential.
120807507
Solutions of the linearized Bach‐Einstein equation in the static spherically symmetric case
The Bach-Einstein equation linearized around Minkowski space-time is completely solved. The set of solutions depends on three parameters; a two-parameter subset of it becomes asymptotically flat. In that region the gravitational potential is of the type φ = -m/r + e exp (-r/l) m and e being small parameters. 
 
 
 
Because of the different asymptotic behaviour of both terms, it became necessary to linearize also around the Schwarzschild solution φ = -m/r. The linearized equation resulting in this case is discussed using qualitative methods. The result is that for m ≤ 2l the asymptotic behaviour is φ = -m/r + er−m/l exp (-r/l) u, and for m ≧ 2l φ = -m/r + er−2 exp (-r/l) u, where u is some bounded function; m is arbitray and e again small. Further, the relation between the solution of the linearized and the full equation is discussed. 
 
 
 
Fur eine Linearisierung um den Minkowskiraum wird die im Titel angegebene Gleichung vollstandig gelost. Es handelt sich dabei um eine 3-parametrige Loungsmenge, wobei eine 2-parametrige Teilmenge zu asymptotisch flachen Losungen fuhrt. Dabei ergibt sich ein Gravitationspotential der Gestalt φ = -m/r + e exp (-r/l). 
 
 
 
Wegen des unterschiedlichen asymptotischen Verhaltens der beiden Summanden macht sich auch eine Linearisierung um die Schwarzschild-Losung (φ = -m/r) erforderlich. Die zugehorige Gleichung wird mittels qualitativer Methoden diskutiert mit dem Ergebnis, das das asymptotische Verhalten bei m < 2l φ = -m/r + er-m/l exp (-r/l) u und bei < φ = -m/r + er−2 exp (-r/l) u ist; hierbei ist u eine gewisse beschrankte Funktion. Schlieslich wird die Beziehung zwischen Loungen der linearisierten und der voll standigen Gleichung diskutiert.
973042
New exact solutions for power‐law inflation Friedmann models
We consider the spatially flat Friedmann model 
 
 
 
For a ≈ tp, especially, if p ≥ 1, this is called power-law inflation. For the Lagrangian L = Rm with p = − (m − 1) (2m − 1)/(m − 2) power-law inflation is an exact solution, as it is for Einstein gravity with a minimally coupled scalar field ϕ in an exponential potential V(ϕ) = exp (μϕ) and also for the higher-dimensional Einstein equation with a special Kaluza-Klein ansatz. The synchronized coordinates are not adapted to allow a closed-form solution, so we write 
 
 
 
The general solutions reads Q(a) = (ab + C)f/b with free integration constant C (C = 0 gives exact power-law inflation) and m-dependent values b and f: f = −2 + 1/p, b = (4m − 5)/(m − 1). Finally, special solutions for the closed and open Friedmann model are found.
31917251
On an algorithm solving two-level programming problems with nonunique lower level solutions
In the paper, an algorithm is presented for solving two-level programming problems. This algorithm combines a direction finding problem with a regularization of the lower level problem. The upper level objective function is included in the regularzation to yield uniqueness of the follower's solution set. This is possible if the problem functions are convex and the upper level objective function has a positive definite Hessian. The computation of a direction of descent and of the step size is discussed in more detail. Afterwards the convergence proof is given.Last but not least some remarks and examples describing the difficulty of the inclusion of upper-level constraints also depending on the variables of the lower level are added.