COMMENT: Fourth-order gravity and conformal transformations

title: On Static Asymptotically Flat Solutions of Fourth Order Gravitational Field Equations
abstract: It is shown that each non-flat regular static asymptotically flat solution of the gravitational field equation following from the Lagrangian 
 
 
 
 
 
 
has in a certain sense positive energy. Further, for a set of parameters including the BACH-EINSTEIN theory 
 
 
 
 
 
 
some results concerning the full nonlinear behaviour of the solutions of the field equation will be given. 
 
 
 
Uber statisch asymptotisch flache Losungen von Gravitationsfeldgleichungen vierter Ordnung 
 
 
 
Es wird gezeigt, das jede nicht-flache regulare statische asymptotisch flache Losung der Gravitationsfeldgleichung, die aus einem Lagrangeausdruck 
 
 
 
 
 
 
folgt, in einem gewissen Sinne positive Energie hat. Ferner wird fur eine bestimmte Parametermenge, die die BACH-EINSTEIN-Theorie 
 
 
 
 
 
 
enthalt, einiges uber das volle nichtlineare Verhalten der Losungen der Feldgleichung mitgeteilt.

title: Solutions of the linearized Bach‐Einstein equation in the static spherically symmetric case
abstract: 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.

title: New exact solutions for power‐law inflation Friedmann models
abstract: 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.

title: The massive scalar field in a closed Friedmann universe model - new rigorous results
abstract: For the minimally coupled scalar field in Einstein's theory of gravitation we look for the space of solutions within the class of closed Friedmann universe models. We prove D ≥ 1, where D ≥ is the dimension of the set of solutions which can be integrated up to t ∞ (D > 0 was conjectured by PAGE (1984)). We discuss concepts like “the probability of the appearance of a sufficiently long inflationary phase” and argue that it is primarily a probability measure μ in the space V of solutions (and not in the space of initial conditions) which has to be applied. μ is naturally defined for Bianchi-type I cosmological models because V is a compact cube. The problems with the closed Friedmann model (which led to controversial claims in the literature) will be shown to originate from the fact that V has a complicated non-compact non-Hausdorff Geroch topology: no natural definition of μ can be given. 
 
 
 
We conclude: the present state of our universe can be explained by models of the type discussed, but thereby the anthropic principle cannot be fully circumvented. 
 
 
 
Fur das minimal gekoppelte Skalarfeld in Einsteinscher Gravitationstheorie betrachten wir den Losungsraum in der Klasse der geschlossenen Friedmannmodelle des Universums. Wir beweisen, das D ≥ 1 gilt, wobei D die Dimension der Menge derjenigen Losungen ist, die sich bis t ∞ integrieren lassen. (D > 0 war von PAGE (1984) vermutet worden.) Wir diskutieren Begriffe wie, ”die Wahrscheinlichkeit des Auftretens einer hinreichend langen inflationaren Phase„ und argumentieren, das man primar ein Wahrscheinlichkeitsmas μ im Raum der Losungen V (und nicht im Raum der Anfangswerte) anwenden mus. Fur kosmologische Modelle vom Bianchi-Typ I gibt es eine naturliche Definition fur μ, da dort V ein kompakter Wurfel ist. Die Probleme mit dem geschlossenen Friedmannmodell (die zu Kontroversen in der Literatur fuhrten) ergeben sich aus der Tatsache, das V dort eine komplizierte Geroch-Topologie hat, die weder kompakt noch hausdorffsch ist, so das sich keine naturliche Definition fur μ angeben last. 
 
 
 
Wir schliesen: Zwar last sich der heutige Zustand unseres Universums durch Modelle des hier diskutierten Typs erklaren, jedoch last sich dabei das anthropische Prinzip nicht vollig umgehen.

title: Comparing selfinteracting scalar fields and R + R**3 cosmological models
abstract: 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.

title: Models for Constrained Motion and d'Alembert's Principle
abstract: A theorem of Arnol'd and Gallavotti states that by a suitable potential of increasing rigidity a mechanical system with conservative forces can be constrained to a hypersurface of the configuration space and its motion approaches a solution of the reduced Lagrangian equation. In this paper the theorem will be generalized to time- and velocity-dependent forces including time-dependent constraints and proved with the aid of the method of averaging. This technique also allows us to identify the constraint force as the averaged negative gradient of the constraint potential.