Considering background quantization of gravitational fields, it is generally assumed that the classical background satisfies Einstein's gravitational equations. However, there exist arguments showing that, for high frequency (quantum) fluctuations, this assumption has to be replaced by a condition describing the back reaction of fluctuations on the background. It is shown that such an approach leads to limitations for the quantum procedure which occur at distances larger than Planck's elementary lengthl=(Gh/c 3)1/2.

The analysis of the measurement of gravitational fields leads to the Rosenfeld inequalities. They say that, as an implication of the equivalence of the inertial and passive gravitational masses of the test body, the metric cannot be attributed to an operator that is defined in the frame of a local canonical quantum field theory. This is true for any theory containing a metric, independently of the geometric framework under consideration and the way one introduces the metric in it. Thus, to (...) establish a local quantum field theory of gravity one has to transit to non-Riemann geometry that contains (beside or instead of the metric) other geometric quantities. From this view, we discuss a Riemann–Cartan and an affine model of gravity and show them to be promising candidates of a theory of canonical quantum gravity. (shrink)

In discussing fundamentals of general-relativistic irreversible continuum thermodynamics, this theory is shown to be characterized by the feature that no thermodynamical degrees of freedom are ascribed to gravitation. However, accepting that black hole thermodynamics seems to oppose this harmlessness of gravitation one is called on to consider other approaches. Therefore, in brief some gravitational and thermodynamical alternatives are reviewed.

In discussing fundamentals of general-relativistic irreversible continuum thermodynamics, this theory is shown to be characterized by the feature that no thermodynamical degrees of freedom are ascribed to gravitation. However, accepting that black hole thermodynamics seems to oppose this harmlessness of gravitation one is called on to consider other approaches. Therefore, in brief some gravitational and thermodynamical alternatives are reviewed.

Empirical and theoretical evidence show that the astrophysical problem of dark matter might be solved by a theory of Einstein-Mayer type. In this theory, up to global Lorentz rotations, the reference system is determined by the motion of cosmic matter. Thus, one is led to a “Riemannian space with teleparallelism” realizing a geometric version of the Mach-Einstein doctrine. The field equations of this gravitational theory contain hidden matter terms, where the existence of hidden matter is inferred solely from its gravitational (...) effects. It is argued that, in the nonrelativistic mechanical approximation, they provide an inertia-free mechanics, where the inertial mass of a body is induced by the gravitational action of the comic masses. Interpreted from the Newtonian point of view, this mechanics shows that the effective gravitational mass of astrophysical objects depends on r such that one expects the existence of dark matter. (shrink)

The Einstein equations can be written as Fierz-Pauli equations with self-interaction, $W\gamma _{ik} = - G_{ik} + \tfrac{1}{2}g_{ik} g^{mn} G_{mn} - k(T_{ik} - \tfrac{1}{2}g_{ik} g^{mn} T_{mn} )$ together with the covariant Hilbert-gauge condition, $(\gamma _i^h - \tfrac{1}{2}\delta _i^k g^{mn} \gamma _{mn} )_{;k} = 0$ where W denotes the covariant wave operator and G ik the Einstein tensor of the metric g ik collecting all nonlinear terms of Einstein's equations. As is known, there do not, however, exist plane-wave solutions γ ik(z)with (...) g ik Z,i Z,k=0of these equations such that what is essential to the introduction of gravitons is not satisfied in general relativity. This nonexistence corresponds with the uncertainty relation,Δp(Δg*)2(Δx)3≥h hG/ c 3 concerning the total nonlinear gravitational field g *ik =g k +γ k. (shrink)

The analysis of the measurement of gravitational fields leads to the Rosenfeld inequalities. They say that, as an implication of the equivalence of the inertial and passive gravitational masses of the test body, the metric cannot be attributed to an operator that is defined in the frame of a local canonical quantum field theory. This is true for any theory containing a metric, independently of the geometric framework under consideration and the way one introduces the metric in it. Thus, to (...) establish a local quantum field theory of gravity one has to transit to non-Riemann geometry that contains (beside or instead of the metric) other geometric quantities. From this view, we discuss a Riemann–Cartan and an affine model of gravity and show them to be promising candidates of a theory of canonical quantum gravity. (shrink)

According to the Einstein-Mayer theory of the Riemanniann space-time with Einstein-Cartan teleparallelism, the local Lorentz invariance is broken by the gravitational field defining Machian reference systems. This breaking of symmetry implies the occurrence of “hidden matter” in the Einstein equations of gravity. The hidden matter is described by the non-Lorentz-invariant energy-momentum tensor $\hat \Theta _{ik}$ satisfying the relation $\hat \Theta _{i;k}^k = 0$ . The tensor $\hat \Theta _{ik}$ is formed from the Einstein-Cartan torsion field given by the anholonomy objects, (...) FAik=2hA[i,k], and appears together with Hilbert’s energy-momentum tensor T* ik and Poincaré’s pressure λgik on the right-hand side of Einstein’s equations so that one has $$R_{ik} - (1/2)g_{ik} R = - \kappa T*_{ik} - \lambda g_{ik} - \hat \Theta _{ik}$$ According to this theory, in the universe and in cosmic systems one must excep “invisible masses” described by the Poincaré and Einstein-Cartan terms to exist. The torsion field FAik makes the space-time a Machian universe; it is of the same nature as the “weak interacting matter” discussed in astrophysics. (shrink)

We discuss the meaning and prove the accordance of general relativity, wave mechanics, and the quantization of Einstein's gravitation equations themselves. Firstly, we have the problem of the influence of gravitational fields on the de Broglie waves, which influence is in accordance with Eeinstein's weak principle of equivalence and the limitation of measurements given by Heisenberg's uncertainty relations. Secondly, the quantization of the gravitational fields is a “quantization of geometry.” However, classical and quantum gravitation have the same physical meaning according (...) to limitations of measurements given by Einstein's strong principle of equivalence and the Heisenberg uncertainties for the mechanics of test bodies. (shrink)

A discussion of the diffraction and scattering of particles by a grating shows that the experiment discussed by H. Hönl and by L. Rosenfeld in 1965 and again in 1981 does not reveal any contradiction between general relativity and quantum theory. Moreover, these theories, in principle, cannot refute one another because the (weak) principle of equivalence, underlying general relativity theory, entails that gravitation does not alter the laws of microphysics.

It is shown that the covariance together with the quantum principle speak for an affinely connected structure which, for distances greater than Planck’s length, goes over in a metrically connected structure of space-time.