Different aspects of relativity, mainly in a canonical formulation, relevant for the question “Is spacetime nothing more than a mathematical space (which describes the evolution in time of the ordinary three-dimensional world) or is it a mathematical model of a real 4D world with time entirely given as the fourth dimension?” are presented. The availability as well as clarity of the arguments depends on which framework is being used, for which currently special relativity, general relativity and some schemes of quantum gravity are available. Canonical gravity provides means to analyze the field equations as well as observable quantities, the latter even in coordinate independent form. This allows a unique perspective on the question of dimensionality since the space-time manifold does not play a prominent role. After reintroducing a Minkowski background into the formalism, one can see how distinguished coordinates of special relativity arise, where also the nature of time is different from that in the general perspective. Just as it is of advantage to extend special to general relativity, general relativity itself has to be extended to some theory of quantum gravity. This suggests that a final answer has to await a thorough formulation and understanding of a fundamental theory of space-time. Nevertheless, we argue that current insights into quantum gravity do not change the picture of the role of time obtained from general relativity.
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References
. R. M. Wald: General Relativity (The University of Chicago Press 1984)
R. Arnowitt, S. Deser, and C. W. Misner: The Dynamics of General Relativity (Wiley, New York 1962)
C. Barcel ó : Lorentzian Spacetimes from Parabolic and Elliptic Systems of PDEs (Springer, Fundamental Theories of Physics, Berlin 2006)
H. Friedrich and A. D. Rendall: The Cauchy Problem for the Einstein Equations, Lect. Notes Phys. 540, 127-224 (2000), gr-qc/0002074
G. F. R. Ellis: Physics in the Real Universe: Time and Spacetime (Springer, Fundamental Theories of Physics, Berlin 2006), gr-qc/0605049
P. G. Bergmann: Observables in General Relativity, Rev. Mod. Phys. 33, 510-514 (1961)
A. Komar: Construction of a Complete Set of Independent Observables in the General Theory of Relativity, Phys. Rev. 111, 1182-1187 (1958)
C. Rovelli: What is Observable in Classical and Quantum Gravity?, Class. Quantum Grav. 8, 297-316 (1991)
C. Rovelli: Quantum Reference Systems, Class. Quantum Grav. 8, 317-332 (1991)
A. Ashtekar, R. Tate, and C. Uggla: Minisuperspaces: Observables and Quantization, Int. J. Mod. Phys. D 2, 15-50 (1993), gr-qc/9302027
B. Dittrich: Partial and Complete Observables for Hamiltonian Constrained Systems, Class. Quantum Grav. 23, 6155-6184 (2006), gr-qc/0411013
. B. Dittrich: Aspects of Classical and Quantum Dynamics of Canonical General Relativity, PhD thesis, University of Potsdam (2005)
. D. Alba and L. Lusanna: Generalized Radar 4-Coordinates and Equal-Time Cauchy Surfaces for Arbitrary Accelerated Observers, gr-qc/0501090
S. B. Giddings, D. Marolf, and J. B. Hartle: Observables in effective gravity, Phys. Rev. D 74, 064018 (2006), hep-th/0512200
S. W. Hawking and R. Penrose: The Singularities of Gravitational Collapse and Cosmology, Proc. Roy. Soc. Lond. A 314, 529-548 (1970)
G. T. Horowitz and R. C. Myers: The Value of Singularities, Gen. Rel. Grav. 27, 915-919 (1995), gr-qc/9503062
. M. Dafermos: Black hole formation from a complete regular past, gr-qc/0310040
. M. Dafermos: The interior of charged black holes and the problem of uniqueness in general relativity, gr-qc/0307013
R. Haag: Local Quantum Physics (Springer, Berlin, Heidelberg, New York 1992)
M. Bojowald: Absence of a Singularity in Loop Quantum Cosmology, Phys. Rev. Lett. 86, 5227-5230 (2001), gr-qc/0102069
M. Bojowald: Homogeneous Loop Quantum Cosmology, Class. Quantum Grav. 20, 2595-2615 (2003), gr-qc/0303073
M. Bojowald, G. Date, and K. Vandersloot: Homogeneous Loop Quantum Cosmology: The Role of the Spin Connection, Class. Quantum Grav. 21, 1253-1278 (2004), gr-qc/0311004
A. Ashtekar and M. Bojowald: Quantum Geometry and the Schwarzschild Singularity, Class. Quantum Grav. 23, 391-411 (2006), gr-qc/0509075
L. Modesto: Loop Quantum Black Hole, Class. Quantum Grav. 23, 5587-5601 (2006), gr-qc/0509078
M. Bojowald: Non-singular Black Holes and Degrees of Freedom in Quantum Gravity, Phys. Rev. Lett. 95, 061301 (2005), gr-qc/0506128
C. Rovelli and L. Smolin: Spin Networks and Quantum Gravity, Phys. Rev. D 52, 5743-5759 (1995)
A. Ashtekar, J. Lewandowski, D. Marolf, J. Mour ão, and T. Thiemann: Quantization of Dif-feomorphism Invariant Theories of Connections with Local Degrees of Freedom, J. Math. Phys. 36, 6456-6493 (1995), gr-qc/9504018
M. Reisenberger and C. Rovelli: Sum over Surfaces form of Loop Quantum Gravity, Phys. Rev. D 56, 3490-3508 (1997), gr-qc/9612035
J. Ambjørn, J. Jurkiewicz, and R. Loll: Spectral Dimension of the Universe, Phys. Rev. Lett. 95,171301 (2005), hep-th/0505113
O. Lauscher and M. Reuter: Fractal Spacetime Structure in Asymptotically Safe Gravity, JHEP 0510, 050 (2005), hep-th/0508202
A. Vilenkin: Quantum creation of universes, Phys. Rev. D 30, 509-511 (1984)
J. B. Hartle and S. W. Hawking: Wave Function of the Universe, Phys. Rev. D 28, 2960-2975 (1983)
M. Bojowald: Dynamical Initial Conditions in Quantum Cosmology, Phys. Rev. Lett. 87, 121-301 (2001), gr-qc/0104072
M. Bojowald: Initial Conditions for a Universe, Gen. Rel. Grav. 35, 1877-1883 (2003), gr-qc/0305069
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Bojowald, M. (2007). Canonical Relativity and the Dimensionality of the World. In: Petkov, V. (eds) Relativity and the Dimensionality of the World. Fundamental Theories of Physics, vol 153. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-6318-3_8
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