Einstein Equations and Hilbert Action: What is missing on page 8 of the proofs for Hilbert's First Communication on the Foundations of Physics?

The history of the publication of the gravitational field equations of general relativity in November 1915 by Einstein and Hilbert is briefly reviewed. An analysis of the internal structure and logic of Hilbert's theory as expounded in extant proofs and in the published version of his relevant paper is given with respect to the specific question what information would have been found on a missing piece of Hilbert's proofs. The existing texts suggest that the missing piece contained the explicit form of the Riemann curvature scalar in terms of the Ricci tensor as a specification of the axiomatically underdetermined Lagrangian in Hilbert's action integral. An alternative reading that the missing piece of the proofs already may have contained the Einstein tensor, i.e. an explicit calculation of the gravitational part of Hilbert's Lagrangian is argued to be highly implausible.     

Strong dipolar effects in a quantum ferrofluid

Symmetry-breaking interactions have a crucial role in many areas of physics, ranging from classical ferrofluids to superfluid (3)He and d-wave superconductivity. For superfluid quantum gases, a variety of new physical phenomena arising from the symmetry-breaking interaction between electric or magnetic dipoles are expected. Novel quantum phases in optical lattices, such as chequerboard or supersolid phases, are predicted for dipolar bosons. Dipolar interactions can also enrich considerably the physics of quantum gases with internal degrees of freedom. Arrays of dipolar particles could be used for efficient quantum information processing. Here we report the realization of a chromium Bose-Einstein condensate with strong dipolar interactions. By using a Feshbach resonance, we reduce the usual isotropic contact interaction, such that the anisotropic magnetic dipole-dipole interaction between 52Cr atoms becomes comparable in strength. This induces a change of the aspect ratio of the atom cloud; for strong dipolar interactions, the inversion of ellipticity during expansion (the usual 'smoking gun' evidence for a Bose-Einstein condensate) can be suppressed. These effects are accounted for by taking into account the dipolar interaction in the superfluid hydrodynamic equations governing the dynamics of the gas, in the same way as classical ferrofluids can be described by including dipolar terms in the classical hydrodynamic equations. Our results are a first step in the exploration of the unique properties of quantum ferrofluids.     

Agricultural sustainability and intensive production practices

A doubling in global food demand projected for the next 50 years poses huge challenges for the sustainability both of food production and of terrestrial and aquatic ecosystems and the services they provide to society. Agriculturalists are the principal managers of global usable lands and will shape, perhaps irreversibly, the surface of the Earth in the coming decades. New incentives and policies for ensuring the sustainability of agriculture and ecosystem services will be crucial if we are to meet the demands of improving yields without compromising environmental integrity or public health.     

Biodiversity as a barrier to ecological invasion

Biological invasions are a pervasive and costly environmental problem that has been the focus of intense management and research activities over the past half century. Yet accurate predictions of community susceptibility to invasion remain elusive. The diversity resistance hypothesis, which argues that diverse communities are highly competitive and readily resist invasion, is supported by both theory and experimental studies conducted at small spatial scales. However, there is also convincing evidence that the relationship between the diversity of native and invading species is positive when measured at regional scales. Although this latter relationship may arise from extrinsic factors, such as resource heterogeneity, that covary with diversity of native and invading species at large scales, the mechanisms conferring greater invasion resistance to diverse communities at local scales remain unknown. Using neighbourhood analyses, a technique from plant competition studies, we show here that species diversity in small experimental grassland plots enhances invasion resistance by increasing crowding and species richness in localized plant neighbourhoods. Both the establishment (number of invaders) and success (proportion of invaders that are large) of invading plants are reduced. These results suggest that local biodiversity represents an important line of defence against the spread of invaders.     

Nova Geminorum 1912 and the origin of the idea of gravitational lensing

Einstein’s early calculations of gravitational lensing, contained in a scratch notebook and dated to the spring of 1912, are reexamined. A hitherto unknown letter by Einstein suggests that he entertained the idea of explaining the phenomenon of new stars by gravitational lensing in the fall of 1915 much more seriously than was previously assumed. A reexamination of the relevant calculations by Einstein shows that, indeed, at least some of them most likely date from early October 1915. But in support of earlier historical interpretation of Einstein’s notes, it is argued that the appearance of Nova Geminorum 1912 (DN Gem) in March 1912 may, in fact, provide a relevant context and motivation for Einstein’s lensing calculations on the occasion of his first meeting with Erwin Freundlich during a visit in Berlin in April 1912. We also comment on the significance of Einstein’s consideration of gravitational lensing in the fall of 1915 for the reconstruction of Einstein’s final steps in his path towards general relativity.     

Creating, moving and merging Dirac points with a Fermi gas in a tunable honeycomb lattice

Dirac points are central to many phenomena in condensed-matter physics, from massless electrons in graphene to the emergence of conducting edge states in topological insulators. At a Dirac point, two energy bands intersect linearly and the electrons behave as relativistic Dirac fermions. In solids, the rigid structure of the material determines the mass and velocity of the electrons, as well as their interactions. A different, highly flexible means of studying condensed-matter phenomena is to create model systems using ultracold atoms trapped in the periodic potential of interfering laser beams. Here we report the creation of Dirac points with adjustable properties in a tunable honeycomb optical lattice. Using momentum-resolved interband transitions, we observe a minimum bandgap inside the Brillouin zone at the positions of the two Dirac points. We exploit the unique tunability of our lattice potential to adjust the effective mass of the Dirac fermions by breaking inversion symmetry. Moreover, changing the lattice anisotropy allows us to change the positions of the Dirac points inside the Brillouin zone. When the anisotropy exceeds a critical limit, the two Dirac points merge and annihilate each other-a situation that has recently attracted considerable theoretical interest but that is extremely challenging to observe in solids. We map out this topological transition in lattice parameter space and find excellent agreement with ab initio calculations. Our results not only pave the way to model materials in which the topology of the band structure is crucial, but also provide an avenue to exploring many-body phases resulting from the interplay of complex lattice geometries with interactions.     

Biodiversity and ecosystem stability in a decade-long grassland experiment

Human-driven ecosystem simplification has highlighted questions about how the number of species in an ecosystem influences its functioning. Although biodiversity is now known to affect ecosystem productivity, its effects on stability are debated. Here we present a long-term experimental field test of the diversity-stability hypothesis. During a decade of data collection in an experiment that directly controlled the number of perennial prairie species, growing-season climate varied considerably, causing year-to-year variation in abundances of plant species and in ecosystem productivity. We found that greater numbers of plant species led to greater temporal stability of ecosystem annual aboveground plant production. In particular, the decadal temporal stability of the ecosystem, whether measured with intervals of two, five or ten years, was significantly greater at higher plant diversity and tended to increase as plots matured. Ecosystem stability was also positively dependent on root mass, which is a measure of perenniating biomass. Temporal stability of the ecosystem increased with diversity, despite a lower temporal stability of individual species, because of both portfolio (statistical averaging) and overyielding effects. However, we found no evidence of a covariance effect. Our results indicate that the reliable, efficient and sustainable supply of some foods (for example, livestock fodder), biofuels and ecosystem services can be enhanced by the use of biodiversity.     

A Mott insulator of fermionic atoms in an optical lattice

Strong interactions between electrons in a solid material can lead to surprising properties. A prime example is the Mott insulator, in which suppression of conductivity occurs as a result of interactions rather than a filled Bloch band. Proximity to the Mott insulating phase in fermionic systems is the origin of many intriguing phenomena in condensed matter physics, most notably high-temperature superconductivity. The Hubbard model, which encompasses the essential physics of the Mott insulator, also applies to quantum gases trapped in an optical lattice. It is therefore now possible to access this regime with tools developed in atomic physics. However, an atomic Mott insulator has so far been realized only with a gas of bosons, which lack the rich and peculiar nature of fermions. Here we report the formation of a Mott insulator of a repulsively interacting two-component Fermi gas in an optical lattice. It is identified by three features: a drastic suppression of doubly occupied lattice sites, a strong reduction of the compressibility inferred from the response of double occupancy to an increase in atom number, and the appearance of a gapped mode in the excitation spectrum. Direct control of the interaction strength allows us to compare the Mott insulating regime and the non-interacting regime without changing tunnel-coupling or confinement. Our results pave the way for further studies of the Mott insulator, including spin-ordering and ultimately the question of d-wave superfluidity.     

Cavity QED with a Bose-Einstein condensate

Cavity quantum electrodynamics (cavity QED) describes the coherent interaction between matter and an electromagnetic field confined within a resonator structure, and is providing a useful platform for developing concepts in quantum information processing. By using high-quality resonators, a strong coupling regime can be reached experimentally in which atoms coherently exchange a photon with a single light-field mode many times before dissipation sets in. This has led to fundamental studies with both microwave and optical resonators. To meet the challenges posed by quantum state engineering and quantum information processing, recent experiments have focused on laser cooling and trapping of atoms inside an optical cavity. However, the tremendous degree of control over atomic gases achieved with Bose-Einstein condensation has so far not been used for cavity QED. Here we achieve the strong coupling of a Bose-Einstein condensate to the quantized field of an ultrahigh-finesse optical cavity and present a measurement of its eigenenergy spectrum. This is a conceptually new regime of cavity QED, in which all atoms occupy a single mode of a matter-wave field and couple identically to the light field, sharing a single excitation. This opens possibilities ranging from quantum communication to a wealth of new phenomena that can be expected in the many-body physics of quantum gases with cavity-mediated interactions.     

Nitrogen limitation constrains sustainability of ecosystem response to CO2

Enhanced plant biomass accumulation in response to elevated atmospheric CO2 concentration could dampen the future rate of increase in CO2 levels and associated climate warming. However, it is unknown whether CO2-induced stimulation of plant growth and biomass accumulation will be sustained or whether limited nitrogen (N) availability constrains greater plant growth in a CO2-enriched world. Here we show, after a six-year field study of perennial grassland species grown under ambient and elevated levels of CO2 and N, that low availability of N progressively suppresses the positive response of plant biomass to elevated CO2. Initially, the stimulation of total plant biomass by elevated CO2 was no greater at enriched than at ambient N supply. After four to six years, however, elevated CO2 stimulated plant biomass much less under ambient than enriched N supply. This response was consistent with the temporally divergent effects of elevated CO2 on soil and plant N dynamics at differing levels of N supply. Our results indicate that variability in availability of soil N and deposition of atmospheric N are both likely to influence the response of plant biomass accumulation to elevated atmospheric CO2. Given that limitations to productivity resulting from the insufficient availability of N are widespread in both unmanaged and managed vegetation, soil N supply is probably an important constraint on global terrestrial responses to elevated CO2.