- Geometrodynamics
In
theoretical physics , geometrodynamics generally denotes a program of reformulation and unification which was enthusiastically promoted byJohn Archibald Wheeler in the 1960s.Einstein's Geometrodynamics
As a search of the
arXiv with the keyword "geometrodynamics" will show, many authors-- sometimes including Wheeler himself-- rather loosely use this term as a synonym for general relativity.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) around 1960. In this reformulation, spacetimes are sliced up into "spatial hyperslices" in a rather arbitrary fashion, and the vacuumEinstein 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".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 geons, 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 theReissner-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 theelectromagnetic field tensor from the electromagnetic contribution to thestress-energy tensor , which in general relativity is directly coupled tospacetime 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 andSpinoza concerning the nature of space.Modern Notions of Geometrodynamics
More recently,
Christopher Isham ,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.References
*cite web | author=Anderson, E. | title=Geometrodynamics: Spacetime or Space? | work=arXiv eprint server | url=http://www.arxiv.org/abs/gr-qc/0409123 | accessmonthday=September 30 | accessyear=2004 This Ph.D. thesis offers a readable account of the long development of the notion of "geometrodynamics".
*cite book | author=Butterfield, Jeremy | title=The Arguments of Time | location=Oxford | publisher=Oxford University Press | year=1999 | id=ISBN 0-19-726207-4 This book focuses on the philosophical motivations and implications of the modern geometrodynamics program.
*cite book | author=Prastaro, Augustino | title=Geometrodynamics: Proceedings, 1985 | location=Philadelphia | publisher=World Scientific | year=1985 | id=ISBN 9971-978-63-6
*cite book | author=Misner, Charles W; Thorne, Kip S. & Wheeler, John Archibald | title=Gravitation | location=San Francisco | publisher=W. H. Freeman | year=1973 | id=ISBN 0716703440 See "chapter 43" for superspace and "chapter 44" for spacetime foam.
*cite book | author=Wheeler, John Archibald | title=Geometrodynamics | location=New York | publisher=Academic Press | year=1963 | id=LLCN 62013645
*cite journal | author=Misner, C.; and Wheeler, J. A. | title=Classical physics as geometry | journal=Ann. Phys. | year=1957 | volume=2 | issue=6 | pages=525 | doi=10.1016/0003-4916(57)90049-0 [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WB1-4DF56P8-2M4&_user=4421&_coverDate=12%2F31%2F1957&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000059598&_version=1&_urlVersion=0&_userid=4421&md5=a6fb8ee2cbcff70062f80f24a5328287 online version (subscription required)]
*cite journal | author=J. Wheeler | title=Geometrodynamics and the Problem of Motion | journal=Rev. Mod. Physics | year=1961 | volume=44 | number=1 | pages=63 | doi=10.1103/RevModPhys.33.63 [http://prola.aps.org/abstract/RMP/v33/i1/p63_1 online version (subscription required)]
*cite journal | author=J. Wheeler | title=On the nature of quantum geometrodynamics | journal=Ann. Phys. | year=1957 | volume=2 | issue=6 | pages=604–614 | doi=10.1016/0003-4916(57)90050-7 [http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6WB1-4DF56P8-2M5&_user=4421&_coverDate=12%2F31%2F1957&_rdoc=1&_fmt=&_orig=search&_sort=d&view=c&_acct=C000059598&_version=1&_urlVersion=0&_userid=4421&md5=7d7c66d2cfb70b77a0dfec6fa0834896 online version (subscription required)]
Wikimedia Foundation. 2010.
