Geometrodynamics
Encyclopedia
In theoretical physics
, geometrodynamics generally denotes a program of reformulation and unification which was enthusiastically promoted by John Archibald Wheeler
in the 1960s.
. More properly, some authors use the phrase Einstein's geometrodynamics to denote the initial value formulation
of general relativity, introduced by Arnowitt, Deser, and Misner (ADM formalism
) around 1960. In this reformulation, spacetime
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.
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 Clifford
'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 gravity
, 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 geon
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 vacuum
, has the nature of a wormhole
. Similarly, in the case of a charged particle, the geometry of the Reissner-Nordström electrovacuum
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 Rainich
had shown decades earlier that one can obtain the electromagnetic field tensor from the electromagnetic contribution to the stress-energy tensor
, 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 fermion
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.
, Jeremy Butterfield
, 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.
Theoretical physics
Theoretical 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 Wheeler
John Archibald Wheeler
John 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
General 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 formulation
Initial value formulation (general relativity)
The 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 formalism
ADM formalism
The 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, spacetime
Spacetime
In 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 Clifford
William Kingdon Clifford
William 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 gravity
Quantum gravity
Quantum 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 geon
Geon (physics)
In 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 vacuum
Schwarzschild metric
In 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 wormhole
Wormhole
In 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 electrovacuum
Reissner-Nordström metric
In 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 Rainich
George Yuri Rainich
George 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 tensor
Stress-energy tensor
The 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 fermion
Fermion
In 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
Christopher 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 Butterfield
Jeremy Butterfield
Jeremy 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.