The institute started operations on 1 April , with Ehlers as its founding director and as the leader of its department for the foundations and mathematics of general relativity. On 31 December , Ehlers retired to become founding director emeritus.
Ehlers continued to work at the institute until his death on 20 May Ehlers' research was in the field of general relativity. In particular, he made contributions to cosmology , the theory of gravitational lenses and gravitational waves. His principal concern was to clarify general relativity's mathematical structure and its consequences, separating rigorous proofs from heuristic conjectures. For his doctoral thesis, Ehlers turned to a question that was to shape his lifetime research.
He sought exact solutions of Einstein's equations : model universes consistent with the laws of general relativity that are simple enough to allow for an explicit description in terms of basic mathematical expressions. These exact solutions play a key role when it comes to building general-relativistic models of physical situations. However, general relativity is a fully covariant theory — its laws are the same, independent of which coordinates are chosen to describe a given situation. One direct consequence is that two apparently different exact solutions could correspond to the same model universe, and differ only in their coordinates.
Ehlers began to look for serviceable ways of characterizing exact solutions invariantly , that is, in ways that do not depend on coordinate choice. In order to do so, he examined ways of describing the intrinsic geometric properties of the known exact solutions. During the s, following up on his doctoral thesis, Ehlers published a series of papers, all but one in collaboration with colleagues from the Hamburg group, which later became known as the "Hamburg Bible".
The analysis presented there uses tools from differential geometry such as the Petrov classification of Weyl tensors that is, those parts of the Riemann tensor describing the curvature of space-time that are not constrained by Einstein's equations , isometry groups and conformal transformations. This work also includes the first definition and classification of pp-waves , a class of simple gravitational waves. The work with Sachs studies, among other things, vacuum solutions with special algebraic properties, using the 2-component spinor formalism.
It also gives a systematic exposition of the geometric properties of bundles in mathematical terms: congruences of light beams. Spacetime geometry can influence the propagation of light, making them converge on or diverge from each other, or deforming the bundle's cross section without changing its area. One result is the Ehlers-Sachs theorem describing the properties of the shadow produced by a narrow beam of light encountering an opaque object.
The tools developed in that work would prove essential for the discovery by Roy Kerr of his Kerr solution , describing a rotating black hole — one of the most important exact solutions. The last of these seminal papers addressed the general-relativistic treatment of the mechanics of continuous media. However useful the notion of a point mass may be in classical physics; in general relativity, such an idealized mass concentration into a single point of space is not even well-defined. That is why relativistic hydrodynamics , that is, the study of continuous media, is an essential part of model-building in general relativity.
The paper systematically describes the basic concepts and models in what the editor of the journal General Relativity and Gravitation , on the occasion of publishing an English translation 32 years after the original publication date, called "one of the best reviews in this area". Another part of Ehlers' exploration of exact solutions in his thesis led to a result that proved important later. At the time Ehlers started his research on his doctoral thesis, the Golden age of general relativity had not yet begun and the basic properties and concepts of black holes were not yet understood.
In the work that led to his doctoral thesis, Ehlers proved important properties of the surface around a black hole that would later be identified as its horizon , in particular that the gravitational field inside cannot be static, but must change over time.
The simplest example of this is the "Einstein-Rosen bridge", or Schwarzschild wormhole that is part of the Schwarzschild solution describing an idealized, spherically symmetric black hole: the interior of the horizon houses a bridge-like connection that changes over time, collapsing sufficiently quickly to keep any space-traveler from traveling through the wormhole.
In physics, duality means that two equivalent descriptions of a particular physical situation exist, using different physical concepts.
This is a special case of a physical symmetry , that is, a change that preserves key features of a physical system. Whenever a particular pair of expressions for B and E conform to the laws of electrodynamics, switching the two expressions around and adding a minus sign to the new B is also valid. In his doctoral thesis, Ehlers pointed out a duality symmetry between different components of the metric of a stationary vacuum spacetime , which maps solutions of Einstein's field equations to other solutions.
This symmetry between the tt-component of the metric, which describes time as measured by clocks whose spatial coordinates do not change, and a term known as the twist potential is analogous to the aforementioned duality between E and B. This larger symmetry group has since become known as the Ehlers group. Its discovery led to further generalizations, notably the infinite-dimensional Geroch group the Geroch group is generated by two non-commuting subgroups , one of which is the Ehlers group.
These so-called hidden symmetries play an important role in the Kaluza—Klein reduction of both general relativity and its generalizations, such as eleven-dimensional supergravity. Other applications include their use as a tool in the discovery of previously unknown solutions and their role in a proof that solutions in the stationary axi-symmetric case form an integrable system. In the s, Ehlers collaborated with Felix Pirani and Alfred Schild on a constructive-axiomatic approach to general relativity: a way of deriving the theory from a minimal set of elementary objects and a set of axioms specifying these objects' properties.
The basic ingredients of their approach are primitive concepts such as event , light ray, particle and freely falling particle. At the outset, spacetime is a mere set of events, without any further structure. They postulated the basic properties of light and freely falling particles as axioms, and with their help constructed the differential topology , conformal structure and, finally, the metric structure of spacetime, that is: the notion of when two events are close to each other, the role of light rays in linking up events, and a notion of distance between events.
Key steps of the construction correspond to idealized measurements, such the standard range finding used in radar. The final step derived Einstein's equations from the weakest possible set of additional axioms. The result is a formulation that clearly identifies the assumptions underlying general relativity. In the s, in collaboration with Ekkart Rudolph, Ehlers addressed the problem of rigid bodies in general relativity. Rigid bodies are a fundamental concept in classical physics. However, the fact that by definition their different parts move simultaneously is incompatible with the relativistic concept of the speed of light as a limiting speed for the propagation of signals and other influences.
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While, as early as , Max Born had given a definition of rigidity that was compatible with relativistic physics, his definition depends on assumptions that are not satisfied in a general space-time, and are thus overly restrictive. Ehlers and Rudolph generalized Born's definition to a more readily applicable definition they called "pseudo-rigidity", which represents a more satisfactory approximation to the rigidity of classical physics.
With Peter Schneider, Ehlers embarked on an in-depth study of the foundations of gravitational lensing. One result of this work was a monograph co-authored with Schneider and Emilio Falco. It was the first systematic exposition of the topic that included both the theoretical foundations and the observational results. From the viewpoint of astronomy, gravitational lensing is often described using a quasi-Newtonian approximation—assuming the gravitational field to be small and the deflection angles to be minute—which is perfectly sufficient for most situations of astrophysical relevance.
In contrast, the monograph developed a thorough and complete description of gravitational lensing from a fully relativistic space-time perspective. This feature of the book played a major part in its long-term positive reception. A basic derivation of the Newtonian limit of general relativity is as old as the theory itself. Einstein used it to derive predictions such as the anomalous perihelion precession of the planet Mercury. Ehlers extended this work by developing a frame theory that allowed for constructing the Newton—Cartan limit, and in a mathematically precise way, not only for the physical laws, but for any spacetime obeying those laws that is, solutions of Einstein's equations.
This allowed physicists to explore what the Newtonian limit meant in specific physical situations. For example, the frame theory can be used to show that the Newtonian limit of a Schwarzschild black hole is a simple point particle.
General relativity is non-linear : the gravitational influence of two masses is not simply the sum of those masses' individual gravitational influences, as had been the case in Newtonian gravity. Ehlers participated in the discussion of how the back-reaction from gravitational radiation onto a radiating system could be systematically described in a non-linear theory such as general relativity, pointing out that the standard quadrupole formula for the energy flux for systems like the binary pulsar had not yet been rigorously derived: a priori, a derivation demanded the inclusion of higher-order terms than was commonly assumed, higher than were computed until then.
His work on the Newtonian limit, particularly in relation to cosmological solutions, led Ehlers, together with his former doctoral student Thomas Buchert, to a systematic study of perturbations and inhomogeneities in a Newtonian cosmos. This laid the groundwork for Buchert's later generalization of this treatment of inhomogeneities. This generalization was the basis of his attempt to explain what is currently seen as the cosmic effects of a cosmological constant or, in modern parlance, dark energy , as a non-linear consequence of inhomogeneities in general-relativistic cosmology.
Complementing his interest in the foundations of general relativity and, more generally, of physics, Ehlers researched the history of physics. Up until his death, he collaborated in a project on the history of quantum theory at the Max Planck Institute for the History of Science in Berlin.
Ehlers showed a keen interest in reaching a general audience. He was a frequent public lecturer, at universities as well as at venues such as the Urania in Berlin. He authored popular-science articles, including contributions to general-audience journals such as Bild der Wissenschaft. He edited a compilation of articles on gravity for the German edition of Scientific American.
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