Leszek M. Sokolowski

Metric gravity theories and cosmology: II. Stability of a ground state in f(R) theories

In the second part of the investigation of metric nonlinear gravity theories, we study a fundamental criterion of viability of any gravity theory: the existence of a stable ground-state solution being either Minkowski, de Sitter or anti-de Sitter space. Stability of the ground state is independent of which frame is physical. In general, a given theory has multiple ground states and splits into independent physical sectors. The fact that all L = f(gαβ, Rμν) gravity theories (except some singular cases) are dynamically equivalent to Einstein gravity plus a massive spin-2 and a massive scalar field allows us to investigate the stability problem using methods developed in general relativity. These methods can be directly applied to L = f(R) theories wherein the spin-2 field is absent. Furthermore for these theories which have anti-de Sitter space as the ground state we prove a positive-energy theorem allowing to define the notion of conserved total gravitational energy in the Jordan frame (i.e., for the fourth-order equations of motion). As is shown in 13 examples of specific Lagrangians the stability criterion works effectively without long computations whenever the curvature of the ground state is determined. An infinite number of gravity theories have a stable ground state and further viability criteria are necessary.

On the twin paradox in static spacetimes: I. Schwarzschild metric

Motivated by a conjecture put forward by Abramowicz and Bajtlik we reconsider the twin paradox in static spacetimes. According to a well known theorem in Lorentzian geometry the longest timelike worldline between two given points is the unique geodesic line without points conjugate to the initial point on the segment joining the two points. We calculate the proper times for static twins, for twins moving on a circular orbit (if it is a geodesic) around a centre of symmetry and for twins travelling on outgoing and ingoing radial timelike geodesics. We show that the twins on the radial geodesic worldlines are always the oldest ones and we explicitly find the the conjugate points (if they exist) outside the relevant segments. As it is of its own mathematical interest, we find general Jacobi vector fields on the geodesic lines under consideration. In the first part of the work we investigate Schwarzschild geometry.

On the issue of gravitons

We investigate the problem of whether one can anticipate any features of the graviton without a detailed knowledge of a full quantum theory of gravity. Assuming that in linearized gravity the graviton is in a sense similar to the photon, we derive a curious large number coincidence between the number of gravitons emitted by a solar planet during its orbital period and the number of its constituent nucleons (the coincidence is less exact for extra solar planets since their sample is observationally biased). The coincidence raises a conceptual problem of quantum mechanism of graviton emission, and we show that the problem has no intuitive solution and there is no physical picture of quantum emission from a macroscopic body. In Einsteins general relativity the analogy between the graviton and the photon turns out to be ill founded. A generic relationship between quanta of a quantum field and plane waves of the corresponding classical field is broken in the case of full GR. The graviton cannot be classically approximated by a generic pp wave nor by its special case, the exact plane wave. Furthermore and most important, the ADM energy is a zero frequency characteristic of any asymptotically flat gravitational field, this means that any general relationship between energy and frequency is a priori impossible. In particular, the formula E = ω does not hold. The graviton must have features different from those of the photon and these cannot be predicted from classical general relativity.

Symmetry properties under arbitrary field redefinitions of the metric energy-momentum tensor in classical field theories and gravity

We derive a generic identity which holds for the metric (i.e. variational) energy–momentum tensor under any field transformation in any generally covariant classical Lagrangian field theory. The identity determines the conditions under which a symmetry of the Lagrangian is also a symmetry of the energy–momentum tensor. It turns out that the stress tensor acquires the symmetry if the Lagrangian has the symmetry in a generic curved spacetime. In this sense, a field theory in flat spacetime is not self-contained. When the identity is applied to the gauge invariant spin-2 field in Minkowski space, we obtain an alternative and direct derivation of a known no-go theorem: a linear gauge invariant spin-2 field, which is dynamically equivalent to linearized general relativity, cannot have a gauge invariant metric energy–momentum tensor. This implies that attempts to define the notion of gravitational energy density in terms of the metric energy–momentum tensor in a field-theoretical formulation of gravity must fail.

Can the Local Energy-Momentum Conservation Laws be Derived Solely from Field Equations?

The vanishing of the divergence of the matter stress-energy tensor for General Relativity is a particular case of a general identity, which follows from the covariance of the matter Lagrangian in much the same way as (generalized) Bianchi identities follow from the covariance of the purely gravitational Lagrangian. This identity, holding for any covariant theory of gravitating matter, relates the divergence of the stress tensor with a combination of the field equations and their derivatives. One could thus wonder if, according to a recent suggestion [1], the energy-momentum tensor for gravitating fields can be computed through a suitable rearrangement of the matter field equations, without relying on the variational definition. We show that this can be done only in particular cases, while in general it leads to ambiguities and possibly to wrong results. Moreover, in nontrivial cases the computations turn out to be more difficult than the standard variational technique.

Every timelike geodesic in anti-de Sitter spacetime is a circle of the same radius

We refine and analytically prove an old proposition due to Calabi and Markus on the shape of timelike geodesics of anti--de Sitter space in the ambient flat space. We prove that each timelike geodesic forms in the ambient space a circle of the radius determined by

Quantum Spacetime and the Problem of Time in Quantum Gravity

\Lambda

Multidimensional gravitational waves. I. Purely radiative spacetimes

, lying on a Euclidean two--plane. Then we outline an alternative proof for

The Twin Paradox in Static Spacetimes and Jacobi Fields

AdS_4

Metric gravity theories and cosmology. I: Physical interpretation and viability

. We also make a comment on the shape of timelike geodesics in de Sitter space.