In
theoretical physicsTheoretical physics is a branch of physics which employs mathematical models and abstractions of physics to rationalize, explain and predict natural phenomena...
,
geometrodynamics generally denotes a program of reformulation and unification which was enthusiastically promoted by
John Archibald WheelerJohn Archibald Wheeler was an American theoretical physicist who was largely responsible for reviving interest in general relativity in the United States after World War II. Wheeler also worked with Niels Bohr in explaining the basic principles behind nuclear fission...
in the 1960s.
Einstein's geometrodynamics
The term geometrodynamics is rather loosely used as a synonym for
general relativityGeneral relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
. More properly, some authors use the phrase
Einstein's geometrodynamics to denote the
initial value formulationThe initial value formulation of general relativity is a reformulation of Albert Einstein's theory of general relativity that describes a universe evolving over time....
of general relativity, introduced by Arnowitt, Deser, and Misner (
ADM formalismThe ADM Formalism developed in 1959 by Richard Arnowitt, Stanley Deser and Charles W. Misner is a Hamiltonian formulation of general relativity...
) around 1960. In this reformulation,
spacetimeIn physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
s are sliced up into
spatial hyperslices in a rather arbitrary fashion, and the vacuum Einstein field equation is reformulated as an
evolution equation describing how, given the geometry of an initial hyperslice (the "initial value"), the geometry evolves over "time". This requires giving
constraint equations which must be satisfied by the original hyperslice. It also involves some "choice of gauge"; specifically, choices about how the
coordinate system used to describe the hyperslice geometry evolves.
Wheeler's geometrodynamics
As described by Wheeler in the early 1960s, geometrodynamics attempts to realize three catchy slogans
- mass without mass,
- charge without charge,
- field without field.
These slogans (due to Wheeler himself), which are discussed in more detail below, capture the general hope that geometrodynamics would "do more with less".
Before the
International Congress for Logic, Methodology, and Philosophy of Science in 1960 Wheeler began by quoting
William Kingdon CliffordWilliam Kingdon Clifford FRS was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his honour, with interesting applications in contemporary mathematical physics...
's "Space theory of Matter" of 1870. He continues, "The vision of Clifford and Einstein can be summarized in a single phrase, 'a geometrodynamical universe': a world whose properties are described by geometry, and a geometry whose curvature changes with time – a dynamical geometry."
Another way of summarizing the goals of Wheeler's original formulation of geometrodynamics is that Wheeler wished to lay the proper conceptual and mathematical foundation for
quantum gravityQuantum gravity is the field of theoretical physics which attempts to develop scientific models that unify quantum mechanics with general relativity...
, and also to unify gravitation with electromagnetism (the strong and weak interactions were not yet sufficiently well understood in 1960 to be included in the program). Wheeler's vision for accomplishing these goals can be summarized as a program of
reducing physics to geometry in an even more fundamental way than had been accomplished by the ADM reformulation of general relativity.
Wheeler introduced the notion of
geonIn theoretical general relativity, a geon is an electromagnetic or gravitational wave which is held together in a confined region by the gravitational attraction of its own field energy. They were first investigated theoretically in 1955 by J. A...
s, gravitational wave packets confined to a compact region of spacetime and held together by the gravitational attraction of the (gravititational) field energy of the wave itself. Wheeler was intrigued by the possibility that geons could affect test particles much like a massive object, hence the slogan
mass without mass.
Wheeler was also much intrigued by the fact that the (nonspinning) point-mass solution of general relativity, the
Schwarzschild vacuumIn Einstein's theory of general relativity, the Schwarzschild solution describes the gravitational field outside a spherical, uncharged, non-rotating mass such as a star, planet, or black hole. It is also a good approximation to the gravitational field of a slowly rotating body like the Earth or...
, has the nature of a
wormholeIn physics, a wormhole is a hypothetical topological feature of spacetime that would be, fundamentally, a "shortcut" through spacetime. For a simple visual explanation of a wormhole, consider spacetime visualized as a two-dimensional surface. If this surface is folded along a third dimension, it...
. Similarly, in the case of a charged particle, the geometry of the
Reissner-Nordström electrovacuumIn physics and astronomy, the Reissner–Nordström metric is a static solution to the Einstein-Maxwell field equations, which corresponds to the gravitational field of a charged, non-rotating, spherically symmetric body of mass M.-The metric:...
solution suggests that the symmetry between electric (which "end" in charges) and magnetic field lines (which never end) could be restored if the electric field lines do not actually end but only go through a wormhole to some distant location or even another branch of the universe.
George RainichGeorge Yuri Rainich was a leading mathematical physicist in the early twentieth century.-Career:Rainich studied mathematics in Odessa and Munich, eventually obtaining his doctorate in 1913 from the University of Kazan...
had shown decades earlier that one can obtain the electromagnetic field tensor from the electromagnetic contribution to the
stress-energy tensorThe stress–energy tensor is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields...
, which in general relativity is directly coupled to spacetime curvature; Wheeler and Misner developed this into the so-called
already unified field theory which partially "unifies" gravitation and electromagnetism. This is very roughly the idea behind the slogan
charge without charge.
Finally, in the ADM reformulation of general relativity, Wheeler argued that the full Einstein field equation can be recovered once the
momentum constraint can be derived, and suggested that this might follow from geometrical considerations alone, making general relativity something like a logical necessity. Specifically, curvature (that is, the gravitational field, as treated in general relativity) might arise as a kind of "averaging" over very complicated topological phenomena at very small scales, the so-called
spacetime foam, which would realize geometrical intuition suggested by quantum gravity. This is roughly the idea behind the slogan
field without field.
These ideas were very imaginative, and they captured the imagination of many physicists, even though Wheeler himself quickly dashed some of the early hopes for his program. In particular, spin 1/2
fermionIn particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....
s proved difficult to handle.
Geometrodynamics also attracted attention from philosophers intrigued by the suggestion that geometrodynamics might eventually realize mathematically some of the ideas of Descartes and Spinoza concerning the nature of space.
Modern notions of geometrodynamics
More recently,
Christopher IshamChristopher Isham is a theoretical physicist at Imperial College London. His main research interests are quantum gravity and foundational studies in quantum theory. He was the inventor of an approach to temporal quantum logic called the HPO formalism, and has worked on loop quantum gravity and...
,
Jeremy ButterfieldJeremy Butterfield is a philosopher at the University of Cambridge, noted particularly for his work on philosophical aspects of quantum theory, relativity theory and classical mechanics....
, and their students have continued to develop
quantum geometrodynamics to take account of recent work toward a quantum theory of gravity and further developments in the very extensive mathematical theory of initial value formulations of general relativity. Some of Wheeler's original goals remain important for this work, particularly the hope of laying a solid foundation for quantum gravity. The philosophical program also continues to motivate several prominent contributors.