@article{Schmidt2013,
  author    = {Schmidt, Hans-J{\"u}rgen},
  title     = {The tetralogy of Birkhoff theorems},
  series = {General relativity and gravitation},
  volume    = {45},
  journal   = {General relativity and gravitation},
  number    = {2},
  publisher = {Springer},
  address   = {New York},
  issn      = {0001-7701},
  doi       = {10.1007/s10714-012-1478-5},
  pages     = {395 -- 410},
  year      = {2013},
  abstract  = {We classify the existent Birkhoff-type theorems into four classes: first, in field theory, the theorem states the absence of helicity 0- and spin 0-parts of the gravitational field. Second, in relativistic astrophysics, it is the statement that the gravitational far-field of a spherically symmetric star carries, apart from its mass, no information about the star; therefore, a radially oscillating star has a static gravitational far-field. Third, in mathematical physics, Birkhoff's theorem reads: up to singular exceptions of measure zero, the spherically symmetric solutions of Einstein's vacuum field equation with can be expressed by the Schwarzschild metric; for , it is the Schwarzschild-de Sitter metric instead. Fourth, in differential geometry, any statement of the type: every member of a family of pseudo-Riemannian space-times has more isometries than expected from the original metric ansatz, carries the name Birkhoff-type theorem. Within the fourth of these classes we present some new results with further values of dimension and signature of the related spaces; including them are some counterexamples: families of space-times where no Birkhoff-type theorem is valid. These counterexamples further confirm the conjecture, that the Birkhoff-type theorems have their origin in the property, that the two eigenvalues of the Ricci tensor of 2-D pseudo-Riemannian spaces always coincide, a property not having an analogy in higher dimensions. Hence, Birkhoff-type theorems exist only for those physical situations which are reducible to 2-D.},
  language  = {en}
}
@article{SchmidtSingleton2013,
  author    = {Schmidt, Hans-J{\"u}rgen and Singleton, Douglas},
  title     = {Exact radial solution in 2+1 gravity with a real scalar field},
  series = {Physics letters : B},
  volume    = {721},
  journal   = {Physics letters : B},
  number    = {4-5},
  publisher = {Elsevier},
  address   = {Amsterdam},
  issn      = {0370-2693},
  doi       = {10.1016/j.physletb.2013.03.007},
  pages     = {294 -- 298},
  year      = {2013},
  abstract  = {In this Letter we give some general considerations about circularly symmetric, static space-times in 2 + 1 dimensions, focusing first on the surprising (at the time) existence of the BTZ black hole solution. We show that BTZ black holes and Schwarzschild black holes in 3 + 1 dimensions originate from different definitions of a black hole. There are two by-products of this general discussion: (i) we give a new and simple derivation of (2 + 1)-dimensional Anti-de Sitter (AdS) space-time; (ii) we present an exact solution to (2 + 1)-dimensional gravity coupled to a self-interacting real scalar field. The spatial part of the metric of this solution is flat but the temporal part behaves asymptotically like AdS space-time. The scalar field has logarithmic behavior as one would expect for a massless scalar field in flat space-time. The solution can be compared to gravitating scalar field solutions in 3 + 1 dimensions but with certain oddities connected with the (2 + 1)-dimensional character of the space-time. The solution is unique to 2 + 1 dimensions; it does not carry over to 3 + 1 dimensions.},
  language  = {en}
}
@article{SchmidtSingleton2013,
  author    = {Schmidt, Hans-J{\"u}rgen and Singleton, Douglas},
  title     = {Isotropic universe with almost scale-invariant fourth-order gravity},
  series = {Journal of mathematical physics},
  volume    = {54},
  journal   = {Journal of mathematical physics},
  number    = {6},
  publisher = {American Institute of Physics},
  address   = {Melville},
  issn      = {0022-2488},
  doi       = {10.1063/1.4808255},
  pages     = {14},
  year      = {2013},
  abstract  = {We study a class of isotropic cosmologies in the fourth-order gravity with Lagrangians of the form L = f(R) + k(G) where R and G are the Ricci and Gauss-Bonnet scalars, respectively. A general discussion is given on the conditions under which this gravitational Lagrangian is scale-invariant or almost scale-invariant. We then apply this general background to the specific case L = alpha R-2 + beta Gln G with constants alpha, beta. We find closed form cosmological solutions for this case. One interesting feature of this choice of f(R) and k(G) is that for very small negative value of the parameter beta, the Lagrangian L = R-2/3 + beta Gln G leads to the replacement of the exact de Sitter solution coming from L = R-2 (which is a local attractor) to an exact, power-law inflation solution a(t) = t(p) = t(-3/beta) which is also a local attractor. This shows how one can modify the dynamics from de Sitter to power-law inflation by the addition of a Gln G-term.},
  language  = {en}
}
@article{Schmidt2013,
  author    = {Schmidt, Hans-J{\"u}rgen},
  title     = {The tetralogy of Birkhoff theorems},
  issn      = {0001-7701},
  year      = {2013},
  abstract  = {We classify the existent Birkhoff-type theorems into four classes: First, in field theory, the theorem states the absence of helicity 0- and spin 0-parts of the gravitational field. Second, in relativistic astrophysics, it is the statement that the gravitational far-field of a spherically symmetric star carries, apart from its mass, no information about the star; therefore, a radially oscillating star has a static gravitational far-field. Third, in mathematical physics, Birkhoff's theorem reads: up to singular exceptions of measure zero, the spherically symmetric solutions of Einstein's vacuum field equation with Lambda = 0 can be expressed by the Schwarzschild metric; for Lambda unequal 0, it is the Schwarzschild-de Sitter metric instead. Fourth, in differential geometry, any statement of the type: every member of a family of pseudo-Riemannian space-times has more isometries than expected from the original metric ansatz, carries the name Birkhoff-type theorem. Within the fourth of these classes we present some new results with further values of dimension and signature of the related spaces; including them are some counterexamples: families of space-times where no Birkhoff-type theorem is valid. These counterexamples further confirm the conjecture, that the Birkhoff-type theorems have their origin in the property, that the two eigenvalues of the Ricci tensor of two- dimensional pseudo-Riemannian spaces always coincide, a property not having an analogy in higher dimensions. Hence, Birkhoff-type theorems exist only for those physical situations which are reducible to two dimensions.},
  language  = {en}
}
@article{SchmidtSingleton2013,
  author    = {Schmidt, Hans-J{\"u}rgen and Singleton, Douglas},
  title     = {Isotropic universe with almost scale-invariant fourth-order gravity},
  year      = {2013},
  abstract  = {We study a broad class of isotropic vacuum cosmologies in fourth-order gravity under the condition that the gravitational Lagrangian be scale-invariant or almost scale-invariant. The gravitational Lagrangians considered will be of the form L = f(R) + k(G) where R and G are the Ricci and Gauss-Bonnet scalars respectively. Specifically we take f(R) = R^2n and k(G) = G^n or k(G) = G ln G. We find solutions in closed form for a spatially flat Friedmann space-time and interpret their asymptotic early-time and late-time behaviour as well as their inflationary stages. One unique example which we discuss is the case of a very small negative value of the parameter b in the Lagrangian L = R^2 + b G ln G which leads to the replacement of the exact de Sitter solution from L = R^2 (being a local attractor) to a power-law inflation exact solution also representing a local attractor. This shows how one can modify the dynamics from de Sitter to power-law inflation by the addition of the G ln G-term.},
  language  = {en}
}
@article{SchmidtSingleton2013,
  author    = {Schmidt, Hans-J{\"u}rgen and Singleton, Douglas},
  title     = {Exact radial solution in 2+1 gravity with a real scalar field},
  issn      = {0370-2693},
  year      = {2013},
  abstract  = {In this paper we give some general considerations about circularly symmetric, static space-times in 2+1 dimensions, focusing first on the surprising (at the time) existence of the BTZ black hole solution. We show that BTZ black holes and Schwarzschild black holes in 3+1 dimensions originate from different definitions of a black hole. There are two by-products of this general discussion: (i) we give a new and simple derivation of 2+1 dimensional Anti-de Sitter (AdS) space-time; (ii) we present an exact solution to 2+1 dimensional gravity coupled to a self-interacting real scalar field. The spatial part of the metric of this solution is flat but the temporal part behaves asymptotically like AdS space-time. The scalar field has logarithmic behavior as one would expect for a massless scalar field in flat space- time. The solution can be compared to gravitating scalar field solutions in 3+1 dimensions but with certain oddities connected with the 2+1 dimensional character of the space-time. The solution is unique to 2+1 dimensions; it does not carry over to 3+1 dimensions.},
  language  = {en}
}
@article{Schmidt2011,
  author    = {Schmidt, Hans-J{\"u}rgen},
  title     = {Perihelion advance for orbits with large eccentricities in the Schwarzschild black hole},
  issn      = {1550-7998},
  year      = {2011},
  abstract  = {We deduce a new formula for the perihelion advance \$Theta\$ of a test particle in the Schwarzschild black hole by applying a newly developed non-linear transformation within the Schwarzschild space-time. By this transformation we are able to apply the well-known formula valid in the weak-field approximation near infinity also to trajectories in the strong-field regime near the horizon of the black hole. The resulting formula has the structure \$Theta = c_1 - c_2 ln(c^2_3 - e^2) \$ with positive constants \$c_{1,2,3}\$ depending on the angular momentum of the test particle. It is especially useful for orbits with large eccentricities \$e < c_3 < 1\$ showing that \$Theta o infty\$ as \$e o c_3\$.},
  language  = {en}
}
@article{Schmidt2011,
  author    = {Schmidt, Hans-J{\"u}rgen},
  title     = {Gauss-Bonnet Lagrangian G ln G and cosmological exact solutions},
  issn      = {1550-7998},
  year      = {2011},
  abstract  = {For the Lagrangian L = G ln G where G is the Gauss-Bonnet curvature scalar we deduce the field equation and solve it in closed form for 3-flat Friedman models using a statefinder parametrization. Further we show, that among all lagrangians F(G) this L is the only one not having the form G^r with a real constant r but possessing a scale-invariant field equation. This turns out to be one of its analogies to f(R)-theories in 2-dimensional space-time. In the appendix, we systematically list several formulas for the decomposition of the Riemann tensor in arbitrary dimensions n, which are applied in the main deduction for n=4.},
  language  = {en}
}
@article{Schmidt2007,
  author    = {Schmidt, Hans-J{\"u}rgen},
  title     = {Fourth order gravity : equations, history, and application to cosmology},
  year      = {2007},
  abstract  = {The field equations following from a Lagrangian L(R) will be deduced and solved for special cases. If L is a non-linear function of the curvature scalar, then these equations are of fourth order in the metric. In the introduction we present the history of these equations beginning with the paper of H. Weyl from 1918, who first discussed them as alternative to Einstein's theory. In the third part, we give details about the cosmic no hair theorem, i.e., the details how within fourth order gravity with L= R + R^2 the inflationary phase of cosmic evolution turns out to be a transient attractor. Finally, the Bicknell theorem, i.e. the conformal relation from fourth order gravity to scalar- tensor theory, will be shortly presented.},
  language  = {en}
}
@article{Schmidt2005,
  author    = {Schmidt, Hans-J{\"u}rgen},
  title     = {Einsteins Arbeiten in Bezug auf die moderne Kosmologie : de Sitters L{\"o}sung der Einsteinschen Feldgleichung mit positivem kosmologischen Glied als Geometrie des inflationaeren Weltmodells},
  year      = {2005},
  abstract  = {Die Arbeit von Albert Einstein von 1918 zu Willem De Sitters Loesung der Einsteinschen Feldgleichung wird unter heutigem Gesichtspunkt kommentiert. Dazu wird zunaechst die Geometrie der De Sitterschen Raum-Zeit beschrieben, sowie ihre Bedeutung fuer das inflationaere Weltmodell erlaeutert.},
  language  = {de}
}