AN EDUCATIONAL REVIEW OF GEOMETRODYNAMICS
Main Article Content
Abstract
This pedagogical review introduces geometrodynamics - the programme that treats gravity not as a force but as the dynamics of spacetime geometry itself - aimed at master’s students considering research in quantum gravity. Assuming only a working knowledge of general relativity at the level of the Einstein field equations, we trace how the geometric view of gravity naturally leads to a dynamical description of the gravitational field. The core of the review develops the ADM (Arnowitt–Deser–Misner) formalism, which recasts Einstein’s theory as a Hamiltonian system for the 3-metric of space and its conjugate momentum. This sets the stage for canonical quantization, culminating in the Wheeler–DeWitt equation and its conceptual puzzles: the problem of time, the role of observables in diffeomorphism-invariant theories, and the infinite-dimensional arena of superspace. We critically examine both the achievements and limitations of geometrodynamics, showing how its challenges motivated subsequent developments - particularly the Ashtekar variables and loop quantum gravity. Throughout, the emphasis is on the dynamics of spacetime: how geometry becomes a physical field that evolves, fluctuates, and ultimately demands quantization. We conclude with a summary of the open questions and an outlook on how modern approaches continue to address them
Article Details
This work is licensed under a Creative Commons Attribution 4.0 International License.
KNUMJ uses Creative Common (CC) Attribution (BY) 4.0 International License which guarantees people (readers) and organisations unlimited rights to share, use, and build upon works published in this journal; all they need to do is to acknowledge the sources (authors).
References
Alesci, E. and Rovelli, C. (2004). The complete lqg propagator. i. difficulties with the barrett–crane
vertex. Physical Review D, 70(4):044019.
Ambjørn, J., Görlich, A., Jurkiewicz, J., and Loll, R. (2014). Quantum gravity via causal dynamical
triangulations. In Springer handbook of spacetime. Springer.
Amelino-Camelia, G. (2013). Quantum gravity phenomenology. Living Reviews in Relativity, 16(1):5.
Arnowitt, R., Deser, S., and Misner, C. W. (1962). The dynamics of general relativity. In Witten, L., editor,
Gravitation: An introduction to current research, pages 227–265. Wiley.
Ashtekar, A. (1986). New variables for classical and quantum gravity. Physical Review Letters,
(18):2244–2247.
Ashtekar, A. (1987). New hamiltonian formulation of general relativity. Physical Review D, 36(6):1587–
Ashtekar, A., Lewandowski, J., Marolf, D., Mourão, J., and Thiemann, T. (1995). Quantization of
diffeomorphism invariant theories of connections with local degrees of freedom. Journal of
Mathematical Physics, 36(11):6456–6493.
Ashtekar, A. and Singh, P. (2011). Loop quantum cosmology: A status report. Classical and Quantum
Gravity, 28(21):213001.
Barbero G., J. F. (1995). Real ashtekar variables for lorentzian signature space-times. Physical Review D,
(10):5507–5510.
Baumgarte, T. W. and Shapiro, S. L. (2010). Numerical relativity: Solving Einstein’s equations on the
computer. Cambridge University Press.
Birrell, N. D. and Davies, P. C. W. (1982). Quantum fields in curved space. Cambridge University Press.
Bojowald, M. (2000). Loop quantum cosmology: I. kinematics. Classical and Quantum Gravity,
(6):1489–1508.
Bombelli, L., Lee, J., Meyer, D., and Sorkin, R. D. (1987). Space-time as a causal set. Physical Review
Letters, 59(5):521–524.
Brill, D. R. and Hartle, J. B. (1964). Method of the self-consistent field in general relativity and its
application to the gravitational geon. Physical Review, 135(2B):B271–B278.
DeWitt, B. S. (1967). Quantum theory of gravity. i. the canonical theory. Physical Review, 160(5):1113–
Einstein, A. (1916). Die grundlage der allgemeinen relativitätstheorie. Annalen der Physik, 354(7):769–
Fourès-Bruhat, Y. (1952). Théorème d’existence pour certains systèmes d’équations aux dérivées
partielles non linéaires. Acta Mathematica, 88(1):141–225.
Gell-Mann, M. and Hartle, J. B. (1990). Quantum mechanics in the light of quantum cosmology. In
Zurek, W. H., editor, Complexity, entropy, and the physics of information. Addison-Wesley.
Gibbons, G. W., Hawking, S. W., and Perry, M. J. (1978). Path integrals and the indefiniteness of the
gravitational action. Nuclear Physics B, 138(1):141–150.
Giulini, D. (2009). The superspace of geometrodynamics. General Relativity and Gravitation, 41(4):785–
Goroff, M. H. and Sagnotti, A. (1986). The ultraviolet behavior of einstein gravity. Nuclear Physics B,
(3):709–736.
Gourgoulhon, É. (2012). 3+1 formalism in general relativity: Bases of numerical relativity, volume 846
of Lecture Notes in Physics. Springer.
Halliwell, J. J. (1989a). Decoherence in quantum cosmology. Physical Review D, 39(10):2912–2923.
Halliwell, J. J. (1989b). Many-worlds interpretation of quantum cosmology. In Blair, D. and Buckingham,
M., editors, Proceedings of the 5th Marcel Grossmann Meeting. World Scientific.
Halliwell, J. J. (1991). Introductory lectures on quantum cosmology. In Piran, T., editor, Proceedings of
the 7th Jerusalem Winter School on Quantum Cosmology and Baby Universes. World Scientific.
Halliwell, J. J. (2009). Introductory lectures on quantum cosmology. arXiv:0909.2566.
Hartle, J. B. (1983). Minisuperspace. In Gibbons, G., Hawking, S., and Siklos, S., editors, The very early
universe. Cambridge University Press.
Hartle, J. B. and Hawking, S. W. (1983). Wave function of the universe. Physical Review D, 28(12):2960–
Hawking, S. W. and Ellis, G. F. R. (1973). The large scale structure of space-time. Cambridge University
Press.
Immirzi, G. (1997). Real and complex connections for canonical gravity. Classical and Quantum Gravity,
(10):L177–L181.
Isham, C. J. (1993). Canonical quantum gravity and the problem of time. In Ibort, L. A. and Rodríguez,
M. A., editors, Integrable systems, quantum groups, and quantum field theories. Kluwer.
Kiefer, C. (2012). Quantum gravity. Oxford University Press, 3 edition.
Kuchař, K. V. (1992). Time and interpretations of quantum gravity. In Proceedings of the 4th Canadian
Conference on General Relativity and Relativistic Astrophysics. World Scientific.
Markopoulou, F. (2008). Quantum graphity. arXiv:0909.1861.
Marolf, D. (1995). Refined algebraic quantization: Systems with a single constraint. arXiv:gr-
qc/9508015.
Misner, C. W., Thorne, K. S., and Wheeler, J. A. (1973). Gravitation. W. H. Freeman.
Misner, C. W. and Wheeler, J. A. (1957). Classical physics as geometry: Gravitation, electromag- netism,
unquantized charge, and mass as properties of curved empty space. Annals of Physics, 2(6):525–
Page, D. N. and Wootters, W. K. (1983). Evolution without evolution: Dynamics described by stationary
observables. Physical Review D, 27(12):2885–2892.
Pinto-Neto, N. and Fabris, J. C. (2013). Quantum cosmology from the de broglie–bohm perspective.
Classical and Quantum Gravity, 30(14):143001.
Pretorius, F. (2005). Evolution of binary black-hole spacetimes. Physical Review Letters,
(12):121101.
Reuter, M. (1998). Nonperturbative evolution equation for quantum gravity. Physical Review D,
(2):971–985.
Rovelli, C. (2004). Quantum gravity. Cambridge University Press.
Rovelli, C. and Smolin, L. (1995). Spin networks and quantum gravity. Physical Review D, 52(10):5743–
’t Hooft, G. and Veltman, M. (1974). One-loop divergencies in the theory of gravitation. Annales de
l’Institut Henri Poincaré, 20(1):69–94.
Thiemann, T. (1998). Quantum spin dynamics (qsd). Classical and Quantum Gravity, 15(4):839–873.
Thiemann, T. (2007). Modern canonical quantum general relativity. Cambridge University Press.
Vilenkin, A. (1984). Quantum creation of universes. Physical Review D, 30(2):509–511.
Wald, R. M. (1984). General relativity. University of Chicago Press.
Wald, R. M. (1994). Quantum field theory in curved spacetime and black hole thermodynamics.
University of Chicago Press.
Wheeler, J. A. (1955). Geons. Physical Review, 97(2):511–536. Wheeler, J. A. (1962). Geometrodynamics.
Academic Press.
Wheeler, J. A. (1964). Geometrodynamics and the issue of the final state. In DeWitt, C. and DeWitt, B.,
editors, Relativity, groups and topology. Gordon and Breach.
Wheeler, J. A. (1968). Superspace and the nature of quantum geometrodynamics. In DeWitt, C. and
Wheeler, J. A., editors, Battelle rencontres. Benjamin.