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This paper has been selected by the Editors of General Relativity and Gravitation for re-publication in the Golden Oldies series. It is one of the papers cited in N. D. Birrell and P. C. W. Davies’ book Quantum Fields in Curved Space (Cambridge University Press, Cambridge, 1982) on the generalization of the Dirac Equation to a curved spacetime and is notable as containing the first occurrence of the well-known formula $$g^{\mu \nu }\nabla _\mu \nabla _\nu +m^2+R/4$$ for the ‘square’ of the Dirac operator. The original is in German and in a poorly accessible journal: it is presented here in English. The paper is accompanied by an editorial note by B. S. Kay which clarifies the historical development of the topic, and by a brief biography written by M. A. H. MacCallum.

We investigate geodesic orbits and manifolds for metrics associated with Schwarzschild geometry, considering space and time curvatures separately. For `a-temporal' space, we solve a central geodesic orbit equation in terms of elliptic integrals. The intrinsic geometry of a two-sided equatorial plane corresponds to that of a full Flamm's paraboloid. Two kinds of geodesics emerge. Both kinds may or may not encircle the hole region any number of times, crossing themselves correspondingly. Regular geodesics reach a periastron greater than the $r_S$ Schwarzschild radius, thus remaining confined to a half of Flamm's paraboloid. Singular or $s$-geodesics tangentially reach the $r_S$ circle. These $s$-geodesics must then be regarded as funneling through the `belt' of the full Flamm's paraboloid. Infinitely many geodesics can possibly be drawn between any two points, but they must be of specific regular or singular types. A precise classification can be made in terms of impact parameters. Geodesic structure and completeness is conveyed by computer-generated figures depicting either Schwarzschild equatorial plane or Flamm's paraboloid. For the `curved-time' metric, devoid of any spatial curvature, geodesic orbits have the same apsides as in Schwarzschild space-time. We focus on null geodesics in particular. For the limit of light grazing the sun, asymptotic `spatial bending' and `time bending' become essentially equal, adding up to the total light deflection of 1.75 arc-seconds predicted by general relativity. However, for a much closer approach of the periastron to $r_S$, `time bending' largely exceeds `spatial bending' of light, while their sum remains substantially below that of Schwarzschild space-time., 14 pages, 19 figures

The paper focuses on finding out several physical quantities to exert an influence on the spatial parameters of complex-octonion curved space, including the metric coefficient, connection coefficient, and curvature tensor. In the flat space described with the complex octonions, the radius vector is combined with the integrating function of field potential to become a composite radius vector. And the latter can be considered as the radius vector in a flat composite-space (a function space). Further it is able to deduce some formulae between the physical quantity and spatial parameter, in the complex-octonion curved composite-space. Under the condition of weak field approximation, these formulae infer a few results accordant with the General Theory of Relativity. The study reveals that it is capable of ascertaining which physical quantities are able to result in the warping of space, in terms of the curved composite-space described with the complex octonions. Moreover, the method may be expanded into some curved function spaces, seeking out more possible physical quantities to impact the bending degree of curved spaces.

The discovery of cosmic acceleration has raised the intriguing possibility that we are witnessing the first breakdown of General Relativity on cosmological scales. In this article I will briefly review current attempts to construct a theoretically consistent and observationally viable modification of gravity that is capable of describing the accelerating universe. I will discuss f(R) models, and their obvious extensions, and the DGP model as an example of extra-dimensional implementations. I will then briefly describe the Galileon models and their very recent multifield and curved space extensions - a class of four-dimensional effective field theories encoding extra dimensional modifications to gravity. This article is dedicated to the career of my friend and former colleague, Joshua Goldberg, and is written to appear in his festschrift., 17 pages, to appear in a festschrift for Joshua Goldberg

Using a WKB approximation of the Dirac equation in a curved spacetime, we obtain the expression for the phase shift between two neutrino mass eigenstates in a generic gravitational field. We apply this expression to two specific space-time geometries, namely the Kerr-Newman metric describing a rotating and charged black hole, and the Friedmann-Robertson-Walker metric., Comment: 15 pages, two figures

We study a scalar field in curved space in three dimensions. We obtain a static perturbative solution and show that this solution satisfies the exact equations in the asymptotic region at infinity. The new solution gives rise to a singularity in the curvature scalar at the origin. Our solution, however, necessitates the excising the region near the origin, thus avoiding the naked singularity.

It is shown that the motion of a string in the field of a massive object can be derived from the Bianchi identities using the method of Papapetrou. The time average equation of motion is that of point geodesic only at lowest order, and the method may be extended to obtain structure dependent, non-geodesic motion.

In Minkowskian spaces, the helicity of a massless fermion is a conserved quantity. In principle, this property may not hold when gravitational effects are not neglected. In this context, this work proves that the helicity of a massless neutrino is not conserved in curved spaces. In order to show this fact, the time variation of the helicity in the Heisenberg picture is calculated. Also, we verify that the differential cross-section due to helicity flip of a massless neutrino in a curved space does not vanish as a result of the coupling between the spin and the curvature of space-time.

The strong analogy between states defined in the context of quantum field theory in curved space-time (QFT-CST) and the ones defined in the thermo field dynamics (TFD) of Takahashi and Umezawa [1] is shown. This analogy is useful in order to introduce the entropy operator in CST in the same way as in TFD. When the extremum condition in the thermodynamical potential is imposed, a family of Bogoliubov transformations that give us a planckian spectrum is found, even in “pathological” cases such as the minimally coupled scalar field.

In the following we give a necessary and sufficient criterion for the existence of theU(1) radiation gauge in a curved space-time with isometry. This criterion is purely geometric and leads to a (local) 3+1-split of space-time with vanishing extrinsic curvature. If the symmetry is timelike and in absence of charges the generalization of the Coulomb gauge leads to a time evolution of the Maxwell field which is analogous to that in flat space-time.

The presence of the action (1) leads to a “metrical elasticity” of space, i.e., to generalized forces which oppose the curving of space. Here we consider the hypothesis which identifies the action (1) with the change in the action of quantum fluctuations of the vacuum if space is curved. Thus, we consider the metrical elasticity of space as a sort of level displacement effect (cf. also [1a]). In present-day quantum field theory it is assumed that the energy momentum tensor of the quantum fluctuations of the vacuum T k(0) and the corresponding action S(0), formally proportional to a divergent integral

Two criteria to find the good vacuum in a curved space-time are studied. In a generic Robertson-Walker universe they are not coincident; yielding to the well-known ambiguity in the vacuum definition. In few cases they coincide and give well-established vacua. An approximate vacuum is introduced in intermediate situations.

The formulation of this limit given by Dautcourt [1] is slightly improved using the notions of Galilei manifold and Newtonian connection. It is then shown under what conditions the conservation equations ▽μ μα = 0 for an arbitrary relativistic continuum have the correct (also covariantly formulated) Newtonian limit. For electromagnetism one obtains a curved space generalization of the electric or magnetic Galileian theory of LeBellac and Levy-Leblond [4] depending on whether the contravariant or the covariant Maxwell tensor is required to have a regular Galileian limit.

The question of to what extent zeta function regularization respects the invariances of a quantum field theory in a background gravitational field is investigated. It is shown that zeta function regularization provides a generalization to curved space-time of analytic propagator regularization which is known not to respect gauge invariance. Furthermore, a study of the regularized stress tensor of a conformally invariant scalar field indicates that both conformai and general coordinate invariance are violated.

A prescription is given for point splitting in a curved space-time background which is a natural generalization of that familiar in quantum electrodynamics and Yang-Mills theory. It is applied (to establish its validity) to the verification of the gravitational anomaly in the divergence of a fermion axial current. Notable features of the prescription are that it defines a point-split current that can be differentiated straightforwardly, and that it involves a natural way of averaging (four-dimensionally) over the directions of point splitting. The method can extend directly from the spin-1/2 fermion case treated to other cases, e.g., to spin-3/2 Rarita-Schwinger fermions.

A simple kinetic model of dipole particles moving in gravitational and electromagnetic fields is considered. Conservation of phase volume is proved. For stationary fields approximate expressions for the macroscopic energy-momentum and spin tensor of the system are obtained.

A generalization is given of some previous work in which a momentum space representation for the Feynman propagator,G(x, y), of a scalar field in an arbitrary curved space-time was obtained. The pointsx andy are allowed to vary in a normal neighborhood of an arbitrary fixed pointz which is taken as an origin of normal coordinates and the representation is obtained by Fourier transformation in the coordinate differencexα-yα. The generality of this representation enables it to be applied to the evaluation of the divergences in any Feynman graph. As an example, the third-order (two-loop) corrections to the four-point function of λo4 field theory are shown to be renormalizable in curved space-time.

It is shown using a classical wave theory approach that photons and neutrinos have different propagation properties in curved space-time. It is also shown that neutrinos have the anomalous property that the sign of their energy density may change as they propagate, and they may even become ghost neutrinos in some regions.

Algebraic constraints are derived for higher-spin fields in a curved space-time manifold. Comparison is made with previously obtained results. A particular solution of the zero-restmass field equations is given for the plane wave Einstein-Maxwell space-times.

A one-electron atom is considered in a general curved space-time. The Hamiltonian of the Dirac equation is written in Fermi normal coordinates, including all interaction terms of first order in the Riemann tensor of the space-time. Expressions are obtained for the shifts in various atomic energy levels caused by the curvature. There is a possibility that these shifts would be observable in the spectrum of hydrogen falling into small black holes (radius about 10−3 cm) left over from the early universe.

Zero-rest-mass higher-spin fields in algebraically special vacuum background space-times are considered. It is shown that the algebraic speciality of the background metric strongly restricts the form of the solutions of these fields. These results are used to study perturbations of the Schwarzschild black hole.

Starting from the Klein-Gordon equation, the single-particle approximation for a reiativistic scalar particle in the presence of external electromagnetic and gravitational fields is performed. The nonrelativistic limit is obtained by a Foldy-Wouthuysen transformation on a Schrodinger-type equation. The results are then compared with those obtained in classical mechanics.

In general, a first order Lagrangian gives rise to second order Euler-Lagrange equations. However, there are important examples where the associated Euler-Lagrange equations are of first order only, the Weyl neutrino equations being of this type. In this paper we therefore consider first order spinor Lagrangians which give rise to firstorder Euler-Lagrange equations. Specifically, the most general first order spinor field equations of rank one in curved space-time which are derivable from a first order Lagrangian of the same type are explicitly constructed. Subject to a certain restriction, the Weyl neutrino equation is the only possibility. Furthermore, if the spinor field satisfies the Weyl neutrino equation, then the associated energy momentum tensor is the conventional neutrino energymomentum tensor.

It is demonstrated that (1) there exist infiniteG1 that satisfy Lichnerowicz's conditions (L conditions) in a globally hyperbolic manifold; and, (2) there is noG1 in an expanding universe that would satisfy those conditions and that would behave as the ordinary Δ1 of flat space whenx → x′. The author thinks that these results present a serious problem for finding a semiclassical theory of scalar field in curved space-time.