
Hans-Jürgen SchmidtUniversität Potsdam · Institute of Mathematics
Hans-Jürgen Schmidt
Privatdozent Dr. habil.
retired since August 2021.
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133
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Introduction
Additional affiliations
January 1986 - June 1986
Education
September 1974 - December 1978
Publications
Publications (133)
preprint, unpublished, contains formulas for solving equations up to fourth order, in German language.
The equation of motion announced in the title was already deduced for the
cases the inner metric being flat and the shell being negligibly small (test
matter), using surface layers and geodesic trajectories resp. Here we derive
the general equation of motion and solve it in closed form for the case of
parabolic motion. Especially the motion near th...
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) =...
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...
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...
A review of inflationary cosmological models from a geometrical point of view is made. Equivalence theorems relate different models and may be used to reduce them to a canonical form: the Einstein gravity minimally interacting with a number of coupled scalar fields (generally, coupling appears in kinetic terms, too). Double inflation appearing in t...
We deduce a new formula for the perihelion advance Θ of a test particle in the Schwarzschild black hole by applying a newly developed nonlinear 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-...
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 Friedmann models using a state-finder parametrization. Further we show that among all Lagrangians F(G) this L is the only one not having the form Gr with a real constant r but possessing a scale-invariant fiel...
We calculate the perihelion precession delta for nearly circular orbits in a central potential V(r). Differently from other approaches to this problem, we do not assume that the potential is close to the Newtonian one. The main idea in the deduction is to apply the underlying symmetries of the system to show that delta must be a function of r V''(r...
There exist two different dark-matter problems in cosmology: a) the discrepancy between the dynamical and luminous mass of astronomical systems and b) the discrepancy between the observed mean mass density of the universe and the critical one predicted by inflation. It will be shown that a "fifth force" ((Newtonian + Yukawa)-type potential) coming...
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 alt...
A new 5D thick brane solution is presented. We conjecture that the deduced thick brane is a plane defect in a bulk gauge condensate.
A scalar-tensor theory is a theory, where the dynamical degrees of freedom are described by one scalar field Φ and one second-rank tensor field g ij , thus giving rise to Spin 0 and Spin 2 contributions. Historically, Brans-Dicke theory is one of the most important scalar-tensor theories of gravitation. It contains, besides the gravitational consta...
We comment on the paper [1] by Albert Einstein from 1918 to Willem De Sitter's solution [2] of the Einstein field equation from today's point of view. To this end, we start by describing the geometry of the De Sitter space-time and present its importance for the inflationary cosmological model.
We complete the historical overview about the geometry of a Schwarzschild black hole at its horizon by emphasizing the contribution made by Synge in [6] to its clarification.
We present the history of fourth order metric theories of gravitation from its beginning in 1918 until 1988.
We present mathematical details of several cosmological models, whereby the topological and the geometrical background will be emphasized.
We present gravitoelectromagnetism and other decompositions of the Riemann
tensor from the differential-geometrical point of view.
Superthin and superlong solutions in 5D Kaluza-Klein gravity are considered. It is shown that they can be cosidered as a hybrid between Einstein's and string paradigmes.
We show [as a comment to H.-H. von Borzeszkowski and H.-J. Treder, Gen. Relativ. Gravitation 34, 1909-1918 (2002; Zbl 1014.83030)] that the square of the Weyl tensor can be negative by giving an example: ds 2 =-dt 2 +2yzdtdx+dx 2 +dy 2 +dz 2 · This metric has the property that in a neighbourhood of the origin, C ijkl C ijkl <0 . Note added: Possibl...
The vacuum solution ds2 = dx2 + x2 dy2 + 2 dz dt + x dt2 of the Einstein gravitational field equation follows from the general ansatz ds2 = dx2 + (x) dx
dx but fails to follow from it if the symmetric matrix g
(x) is assumed to be in diagonal form.
We report on the period-doubling bifurcation recently detected for strongly anisotropic Bianchi I quantum cosmology by M. Bachmann and H.-J. Schmidt and present further arguments related to the quantum boundary.
One of variants of the quantum scenario of creation of the Universe from "Nothing" assumes quantum tunneling of a wave function (WF) of the Universe Psi through a potential barrier. In the elementary statement, this problem was considered with a constant scalar field 1,2, when the one-dimensional Wheeler -DeWitt equation(WDE) is analyzed similarly...
We speculate that the universe is filled with stars composed of electromagnetic and dilaton fields which are the sources of the powerful gamma-ray bursts impinging upon us from all directions of the universe. We calculate soliton-like solutions of these fields and show that their energy can be converted into a relativistic plasma in an explosive wa...
We present an example that non-isometric space-times with non-vanishing curvature scalar cannot be distinguished by curvature invariants.
We answer the following question: Let l, m, n be arbitrary real numbers. Does there exist a 3-dimensional homogeneous Riemannian manifold whose eigenvalues of the Ricci tensor are just l, m and n ?
We consider the Newtonian limit of the theory based on the Lagrangian L = R + \sum a_k R \Box^k R. The gravitational potential of a point mass turns out to be a combination of Newtonian and Yukawa terms. For sixth-order gravity the coefficients are calculated explicitly. For the general case one gets as a result: The the potential is always unbound...
The space of all Riemannian metrics is infinite-dimensional. Nevertheless a great deal of usual Riemannian geometry can be carried over. The superspace of all Riemannian metrics shall be endowed with a class of Riemannian metrics; their curvature and invariance properties are discussed. Just one of this class has the property to bring the lagrangia...
Some mathematical errors of the paper commented upon [W.-M. Suen, Phys. Rev. D 40, (1989) 315] are corrected. Comment: 3 pages, LaTeX, reprinted from Phys. Rev. D 50 (1994) 5452
The Bach equation and the equation of geometrodynamics are based on two quite different physical motivations, but in both approaches, the conformal properties of gravitation plays the key role. In this paper we present an analysis of the relation between these two equations and show that the solutions of the equation of geometrodynamics are of a mo...
For a spatially flat Friedmann model with line element $ds^2=a^2 [ da^2/B(a)-dx^2-dy^2-dz^2 ] $, the 00-component of the Einstein field equation reads $8\pi G T_{00}=3/a^2$ containing no derivative. For a nonlinear Lagrangian ${\cal L}(R)$, we obtain a second--order differential equation for $B$ instead of the expected fourth-order equation. We dis...
For a spatially flat Friedmann model with line element $ds^2=a^2 [ da^2/B(a)-dx^2-dy^2-dz^2 ] $, the 00-component of the Einstein field equation reads $8\pi G T_{00}=3/a^2$ containing no derivative. For a nonlinear Lagrangian ${\cal L}(R)$, we obtain a second--order differential equation for $B$ instead of the expected fourth-order equation. We dis...
The evolution of the closed Friedmann Universe with a packet of short scalar waves is considered with the help of the Wheeler–DeWitt equation. The packet ensures conservation of homogeneity and isotropy of the metric on average. It is shown that during tunneling the amplitudes of short waves of a scalar field can increase catastrophically promptly...
We argue that the Lagrangian for gravity should remain bounded at large curvature, and interpolate between the weak-field tested Einstein-Hilbert Lagrangian ℒEH = R/16πG and a pure cosmological constant for large R with the \(\underline c {\text{urvature - }}\underline s {\text{aturated}}\) ansatz ℒcs = ℒEH/\(\sqrt {{\text{1 + }}l^4 R^2 } \), where...
We study extended theories of gravity where nonminimal derivative couplings of the form Rklϕ,kϕ,l are present in the Lagrangian. We show how and why the other couplings of similar structure may be ruled out and then deduce the field equations and the related cosmological models. Finally, we get inflationary solutions which do follow neither from an...
We solve the Wheeler-DeWitt equation for the minisuperspace of a cosmological model of Bianchi type I with a minimally coupled massive scalar field $\phi$ as source by generalizing the calculation of Lukash and Schmidt [1]. Contrarily to other approaches we allow strong anisotropy. Combining analytical and numerical methods, we apply an adiabatic a...
Vacuum solutions for multidimensional gravity on the principal bundle with the SU(2) structural group as the extra dimensions are found and discussed. This generalizes the results of Ref. \cite{vds3} from U(1) to the SU(2) gauge group. The spherically symmetric solution with the off-diagonal components of the multidimensional metric is obtained. It...
A regular vacuum solution in 5D gravity on the principal bundle with the U(1) structural group is proposed as a 4D wormhole. This solution has two null hypersurfaces where an interchange of the sign of some 5D metric components happens. For a 4D observer living on the base of this principal bundle this is a wormhole with two asymptotically flat Lor...
The spherically symmetric solutions in Weyl gravity interacting with U(1) or SU(2) gauge fields are examined. It is shown that these solutions are conformally equivalent to an infinite flux tube with constant (color) electric and magnetic fields. This allows us to say that Weyl gravity has in some sense a classical confinement mechanism. We discuss...
The Bach equation, i.e., the vacuum field equation following from the Lagrangian L=C_{ijkl}C^{ijkl}, will be completely solved for the case that the metric is conformally related to the cartesian product of two 2-spaces; this covers the spherically and the plane symmetric space-times as special subcases. Contrary to other approaches, we make a cova...
A contribution linear in r to the gravitational potential can be created by a suitable conformal duality transformation: the conformal factor is 1/(1+r)^2 and r will be replaced by r/(1+r), where r is the Schwarzschild radial coordinate. Thus, every spherically symmetric solution of conformal Weyl gravity is conformally related to an Einstein space...
We study extended theories of gravity where nonminimal derivative couplings of the form $R^{kl}\phi_{, k}\phi_{, l}$ are present in the Lagrangian. We show how and why the other couplings of similar structure may be ruled out and then deduce the field equations and the related cosmological models. Finally, we get inflationary solutions which do fol...
We discuss the instabilities appearing in the cosmological model with a quasi de Sitter phase following from a fourth-order gravity theory. Both the classical equation as well as the quantization in form of a Wheeler - De Witt equation are conformally related to the analogous model with Einstein's theory of gravity with a minimally coupled scalar f...
The solutions of two-dimensional gravity following from a non-- linear Lagrangian L = f(R) p g are classified, and their symmetry and singularity properties are described. Then a conformal transformation is applied to rewrite these solutions as analogous solutions of twodimensional Einstein-dilaton gravity and vice versa. KEY : Dilaton gravity in 1...
Power-law inflation is shown to be a local attractor for t to infinity in the space of inhomogeneous solutions of the Einstein theory of gravity with a minimally coupled scalar field having a positive exponential potential. Applying the conformal equivalence theorem the authors show that a similar result holds true for fourth-order gravity with the...
From the Lagrangian R Square Operator Square Operator R one gets an eighth-order theory of gravitation. It has more promising properties than the previously discussed sixth-order ones. The de Sitter solution has the attractor property; we explicitly show how the modes decay. Further, exactly one power law and one pole-like solution exist. Adding th...
For the Lagrangian L=F(R, Square Operator R, . . ., Square Operator kR) square root -g the author deduces the field equation and discusses its relation to Einstein's theory with many scalar fields. It turns out that they are conformally equivalent. The masses of the scalar fields can be calculated from the linearised field equation of L. For F=R Sq...
The authors generalise the result derived for the Einstein theory with the cosmological term (the general asymptotic solution containing four arbitrary functions of three coordinates) to fourth-order gravity. For scale-invariant fourth-order gravity the expanding generalised de Sitter solution is found to be an attractor, i.e. for t to infinity an...
Continues the investigation by Starobinsky and Schmidt (1987) on a general vacuum solution of fourth-order gravity. Now the author includes the Bach tensor. For L2=1/3( mu R2)+1/2( alpha C2) the expanding de Sitter spacetime is an attractor in the set of axially symmetric Bianchi type-I models if and only if alpha mu <or=0 or alpha >4 mu holds. (Th...
The authors derive field equations for sixth-order gravity with the Lagrangian containing the terms R Square Operator R and R2 and show its conformal equivalence to the Einstein theory with two interacting scalar fields. Their masses are calculated in the weak-field limit. This equivalence is generalised to higher order gravity in the linearised ca...
The authors consider sixth-order theories of gravity with Lagrangians containing the term RN Square Operator R, where N>1 is an integer. A conformal transformation gives a picture with two scalar fields in Einstein's theory. They find de Sitter and power law inflationary solutions. However, the theory has an unstable weak-field behaviour, because t...
We report on the cited papers refs. 1 - 18 from the following points of view: What do we exactly know about solutions when no exact solution (in the sense of "solution in closed form") is available? In which sense do these solutions possess a singularity? In which cases do conformal relations and/or dimensional reductions simplify the deduction? Fu...
Introduction to differential geometry" at Potsdam university. If no metric exists at all, then covariant vectors and contravariant vectors are different types of objects. If a metric exists, then there is a canonical isomorphism between them; so we introduce vectors, and after fixing a coordinate system, we speak about their covariant and their con...
The conformal equivalence of fourth-order gravity following from a non-linear Lagrangian L(R) to theories of other types is widely known, here we report on a new conformal equivalence of these theories to theories of the same type but with different Lagrangian. For a quantization of fourth-order theories one needs a Hamiltonian formulation of them....
The Einstein equation in D dimensions, if restricted to the class of space-times possessing n = D - 2 commuting hypersurface-orthogonal Killing vectors, can be equivalently written as metric-dilaton gravity in 2 dimensions with n scalar fields. For n = 2, this results reduces to the known reduction of certain 4-dimensional metrics which include gra...
Assuming SO(3)-spherical symmetry, the 4-dimensional Einstein equation reduces to an equation conformally related to the field equation for 2-dimensional gravity following from the Lagrangian L = R^(1/3). Solutions for 2-dimensional gravity always possess a local isometry because the traceless part of its Ricci tensor identically vanishes. Combinin...
We define under which circumstances two multi-warped product spacetimes can be considered equivalent and then we classify the spaces of constant curvature in the Euclidean and Lorentzian signature. For dimension D=2, we get essentially twelve representations, for D=3 exactly eighteen. More general, for every even D, 5D+2 cases exist, whereas for ev...
We prove that for non-linear L = L(R), the Lagrangians L and \hat L give
conformally equivalent fourth-order field equations being dual to each other.
The proof represents a new application of the fact that the operator
<D'Alembertian minus R/6> is conformally invariant.
We systematically investigate the possible transitions between classical homogeneous Riemannian 3-hypersurfaces of cosmological models on one hand, and between homogeneous Lorentzian minisuperspace 3-geometries on the other hand. For the Riemannian case, in contrast to our earlier approaches where we only evaluated the three scalar invariants from...
We present three reasons for rewriting the Einstein equation. The new version is physically equivalent but geometrically more clear. 1. We write $4 \pi$ instead of $8 \pi$ at the r.h.s, and we explain how this factor enters as surface area of the unit 2--sphere. 2. We define the Riemann curvature tensor and its contractions (including the Einstein...
We use gravitational Lagrangians R □ k √− g and linear combinations of them motivated from trials how to overcome the non‐ renormalizability of Einstein' s theory. We ask under which circumstances the de Sitter space‐ time represents an attractor solution in the set of spatially flat Friedman models. This property ensures the inflationary model to...
The following four statements have been proven decades ago already, but they continue to induce a strange feeling: - All curvature invariants of a gravitational wave vanish - in spite of the fact that it represents a nonflat spacetime. - The eigennullframe components of the curvature tensor (the Cartan ''scalars'') do not represent curvature scalar...
We discuss the consequences of the incorrectness [see the Erratum Phys. Rev. D 49, 1145(E) (1994)] of Cotsakis’ paper and add two related remarks. The scope of this Comment is to encourage further research on ‘‘which of the conformally equivalent metrics is the physical one.’’
Space-times which allow a slicing into homogeneous spatial hypersurfaces generalize the usual Bianchi models. One knows already that in these models the Bianchi type may change with time. Here we show which of the changes really appear. To this end we characterize the topological space whose points are the 3-dimensional oriented homogeneous Riemann...
The use of time-like geodesics to measure temporal distances is better justified than the use of space-like geodesics for a measurement of spatial distances. We give examples where a “spatial distance” cannot be appropriately determined by the length of a space-like geodesic.
Mechanics is developed over a differentiable manifold as space of possible positions. Time is considered to fill a one--dimensional Riemannian manifold, so having the metric as lapse. Then the system is quantized with covariant instead of partial derivatives in the Schr\"odinger operator. Consequences for quantum cosmology are shortly discussed. Co...
We use gravitational Lagrangians $R \Box \sp k R \sqrt{-g}$ and linear combinations of them; we ask under which circumstances the de Sitter space-time represents an attractor solution in the set of spatially flat Friedman models. Results are: For arbitrary $k$, i.e., for arbitrarily large order of the field equation, on can always find examples whe...
An alternate Hamiltonian H different from Ostrogradski's one is found for the Lagrangian L = L(q, \dot q, \ddot q). We add a suitable divergence to L and insert a=q and b=\ddot q. Contrary to other approaches no constraint is needed because \ddot a = b is one of the canonical equations. Another canonical equation becomes equivalent to the fourth-or...
We consider the Lagrangian L=F(R) in classical (i.e. non-quantized) two-dimensional fourth-order gravity and give new relations to Einstein's theory with a non-minimally coupled scalar field.
We distinguish between scale-invariant Lagrangians and scale-invariant field equations. L is scale invariant for and diverges for . The field equation is sca...
We analyze the presuppositions leading to instabilities in theories of order higher than second. The type of fourth-order gravity which leads to an inflationary (quasi-de Sitter) period of cosmic evolution by inclusion of one curvature-squared term (i.e., the Starobinsky model) is used as an example. The corresponding Hamiltonian formulation (which...
We prove the theorem valid for (Pseudo)-Riemannian manifolds $V_n$: "Let $x \in V_n$ be a fixed point of a homothetic motion which is not an isometry then all curvature invariants vanish at $x$." and get the Corollary: "All curvature invariants of the plane wave metric $$ds \sp 2 \quad = \quad 2 \, du \, dv \, + \, a\sp 2 (u) \, dw \sp 2 \, + \, b\...
We analyze the presumptions which lead to instabilities in theories of order higher than second. That type of fourth order gravity which leads to an inflationary (quasi de Sitter) period of cosmic evolution by inclusion of one curvature squared term (i.e. the Starobinsky model) is used as an example. The corresponding Hamiltonian formulation (which...
The space-times of plane thin domain walls are studied in the context of the Brans-Dicke (BD) theory of gravity by using distribution theory. In particular, the BD field equations are divided into two groups: one holding in the regions outside of the wall and the other holding on the wall. It is found that the equations on the wall take a very simp...
For the LagrangianL = R
2,the de Sitter space-time is known to be an attractor solution. Here, we classify for closed Friedmann models in which initial conditions lead asymptotically to a de Sitter phase and what the behaviour is for the other solutions. Four types of solutions form together a generic set, and three of them are asymptotically de Si...
We review on the main geometric properties of the space-time of constant curvature which is the foundation of the inflationary cosmological model. We show e.g.: The spatially flat Friedmann model with exponentially increasing scale factor is only a local description for the de Sitter space-time, whereas the closed Friedmann model can lead to a glob...
The vacuum field equations are solved in a V2 following from the scale-invariant gravitational Lagrangian £ = Rk + 1. For £ = R2, exactly six solutions exist. Just one of them breaks the scale-invariance and has indefinite signature. For k →0, nontrivial results also arise. Their relation to two-dimensional quantum gravity is discussed. For £ = F(R...
The conformal equivalence theorem between fourth order gravity and Einstein's theory with a scalar field is generalized to higher order and higher dimensions. This is applied to cosmological models with more than one inflationary phase. Especially it is discussed under which conditions double inflation is a typical solution.
The vacuum field equations are solved in a V2 following from the scale-invariant gravitational Lagrangian £=Rk+1. For £=R2, exactly six solutions exist. Just one of them breaks the scale-invariance and has indefinite signature. For k↠0, nontrivial results also arise. Their relation to two-dimensional quantum gravity is discussed. For £=F(R) a Birkh...
We consider the Newtonian limit of the theory based on the Lagrangian . The gravitational potential of a point mass turns out to be a combination of Newtonian and Yukawa terms. For sixth-order gravity (p = 1) the coefficients are calculated explicitly. For general p one gets . Therefore, the potential is always unbounded near. The origin.
Wir betra...
Homogeneous isotropic, anisotropic, and inhomogeneous cosmological models are studied using Einstein's general relativity with quntum corrections in field theoretical approximation. In particular we discuss coherent scalar fields and curvature squared terms in the gravitational Lagrangian. The conformal equivalence of the field equations of fourth...
Generalized inflationary stages as power-law solutions result from the addition of R3 and R□R terms to the quadratic gravitational Lagrangian. Both contributions lead to new attractor solutions in the high-curvature limit. We emphasize the properties of critical values of the curvature scalar in the R3 theory. Contrarily to models with two scalar f...
The conformal relation between scale- invariant fourth-order gravity and Kaluza-Klein models as derived is applied to Friedmann cosmological models. Especially, the results that power-law inflation is an attractor solution can be carried over, but the conformal transformation brings power-law inflation to de Sitter-like exponential inflation, or po...
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 P AGE (1984)). We discuss concepts like “the probability...
We consider the spatially flat Friedmann model. For a(t) = t^p, especially,
if p is larger or equal to 1, this is called power-law inflation. For the
Lagrangian L = R^m 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 Phi in an exponential potential V(P...
The authors consider the Wheeler-DeWitt equation for multidimensional
cosmological models including phenomenological matter. In this way they
find a generalized mechanical system in minisuperspace, which is
integrable for a special mixture of tracefree matter components,
generalizing the result of Ivashchuk, Melnikov, and Zhuk for the case of
pheno...
The higher-dimensional Einstein vacuum equation with Lambda-term is shown to be conformally equivalent to the four-dimensional field equation of scale-invariant fourth-order gravity. This holds for a general warped product between space-time and internal space of arbitrary dimension m which turns out to be an Einstein space. (The limit m-->∞ makes...
We review on new results for fourth-order gravity and for Einstein's theory plus a scalar field ϕ under the view point of cosmology. We apply theorems on a conformal rescaling between different types of gravitational Lagrangians under consideration and discuss its physical interpretation. The transformation between ϕ≈0 andФ
′≈∞ is analysed.
The author proves a general conformal equivalence between pairs of Lagrangians (L(R); L(R)) and applies it to the pair (Rm; Rm) with m=(3m-4)/(2m-3).
Double inflation is a typical solution of fourth-order gravity,
Lgrav = M2 R - R2 /6L_{grav} = M^2 R - R^2 /6
minimally coupled to a scalar field with mass m, and 0<mp = 1 - mM/Mpl2p = 1 - mM/M_{pl}^2
Fort and a scalar field or ideal fluid as source, the solutions for the scale factor oscillate around at2/3, thus theR
2 term gives effectively dust...