The small-scale structure of spacetime

Steven Carlip

doi:10.1017/CBO9780511920998.004

4 - The small-scale structure of spacetime

Published online by Cambridge University Press: 05 August 2012

Edited by

Amanda Weltman and

Steven Carlip: Affiliation:
University of California
Jeff Murugan: Affiliation:
University of Cape Town
Amanda Weltman: Affiliation:
University of Cape Town
George F. R. Ellis: Affiliation:
University of Cape Town

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Summary

Several lines of evidence hint that quantum gravity at very small distances may be effectively two-dimensional. I summarize the evidence for such “spontaneous dimensional reduction,” and suggest an additional argument coming from the strong-coupling limit of the Wheeler-DeWitt equation. If this description proves to be correct, it suggests a fascinating relationship between small-scale quantum spacetime and the behavior of cosmologies near an asymptotically silent singularity.

Introduction

Stephen Hawking and George Ellis prefaced their seminal book, The Large Scale Structure of Space-Time, with the explanation that their aim was to understand spacetime “on length-scales from 10−13 cm, the radius of an elementary particle, up to 1028 cm, the radius of the universe” [24]. While many deep questions remain, ranging from cosmic censorship to the actual topology of our universe, we now understand the basic structure of spacetime at these scales: to the best of our ability to measure such a thing, it behaves as a smooth (3+1)-dimensional Riemannian manifold.

At much smaller scales, on the other hand, the proper description is far less obvious. While clever experimentalists have managed to probe some features down to distances close to the Planck scale [43], for the most part we have neither direct observations nor a generally accepted theoretical framework for describing the very small-scale structure of spacetime. Indeed, it is not completely clear that “space” and “time” are even the appropriate categories for such a description.

Type: Chapter
Information: Foundations of Space and Time
Reflections on Quantum Gravity
, pp. 69 - 84

DOI: https://doi.org/10.1017/CBO9780511920998.004 [Opens in a new window]

Publisher: Cambridge University Press

Print publication year: 2012

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References

[1] J., Ambjørn, J., Jurkiewicz, and R., Loll, Phys. Rev. Lett. 85, 924 (2000), eprint hep-th/0002050.

[2] J., Ambjørn, J., Jurkiewicz, and R., Loll, Phys. Rev. Lett. 93, 131301 (2004), eprint hep-th/0404156.

[3] J., Ambjørn, J., Jurkiewicz, and R., Loll, Phys. Rev. D72, 0640141 (2005), eprint hep-th/0505154.

[4] J., Ambjørn, J., Jurkiewicz, and R., Loll, Phys. Lett. B607, 205 (2005), eprint hep-th/0411152.

[5] J., Ambjørn, J., Jurkiewicz, and R., Loll, Phys. Rev. Lett. 95, 171301 (2005), eprint hep-th/0505113.

[6] J. J., Atick and E., Witten, Nucl. Phys. B310, 291 (1988).

[7] S., Basu and D., Mattingly, Class. Quant. Grav. 22, 3029 (2005), eprint astro-ph/0501425.

[8] V. A., Belinskii, I.M., Khalatnikov, and E.M., Lifshitz, Adv. Phys. 19, 525 (1970)

[9] V. A., Belinskii, I.M., Khalatnikov, and E.M., Lifshitz, Adv. Phys. 31, 639 (1982).

[10] D., Benedetti and J., Henson, “Spectral geometry as a probe of quantum spacetime,” eprint arXiv:0911.0401v1 [hep-th].

[11] A. L., Berkin, Phys. Rev. D46, 1551 (1992).

[12] D., Blas, O., Pujolas, and S., Sibiryakov, JHEP 0910, 029 (2009), eprint arXiv:0906.3046[hep-th].

[13] P., Candelas and D.W., Sciama, Phys. Rev. Lett. 38, 1372 (1977).

[14] S., Carlip, Rept. Prog. Phys. 64, 885 (2001), eprint gr-qc/0108040.

[15] D. F., Chernoff and J. D., Barrow, Phys. Rev. Lett. 50, 134 (1983).

[16] A., Connes, “Noncommutative geometry year 2000,” in Highlights of Mathematical Physics, edited by A., Fokas et al., American Mathematical Society, 2002, eprint math.QA/0011193.

[17] L., Crane and L., Smolin, Nucl. Phys. B267, 714 (1986).

[18] B. S., DeWitt, Phys. Rev. 160, 1113 (1967).

[19] B. S., DeWitt, “Dynamical theory of groups and fields,” in Relativity, Groups and Topology, edited by C., DeWitt and B., DeWitt, Gordon and Breach, New York, 1964, pp. 587–820.

[20] R., Epp, “The symplectic structure of general relativity in the double null (2+2) formalism,” eprint gr-qc/9511060.

[21] G., Francisco and M., Pilati, Phys. Rev. D31, 241 (1985).

[22] T., Futamase, Phys. Rev. D29, 2783 (1984).

[23] G. W., Gibbons, “Quantum field theory in curved spacetime,” in General Relativity: An Einstein Centenary Survey, edited by S. W., Hawking and W., Israel, Cambridge University Press, Cambridge, 1979, pp. 639–79.

[24] S.W., Hawking and G. F. R., Ellis, The Large Scale Structure of Space-Time, Cambridge University Press, Cambridge, 1973.

[25] J. M., Heinzle, C., Uggla, and N., Röhr, Adv. Theor. Math. Phys. 13, 293 (2009), eprint gr-qc/0702141.

[26] A., Helfer et al., Gen. Rel. Grav. 20, 875 (1988).

[27] M., Henneaux, Bull. Math. Soc. Belg. 31, 47 (1979).

[28] M., Henneaux, M., Pilati, and C., Teitelboim, Phys. Lett. 110B, 123 (1982).

[29] P., Hořava, Phys. Rev. D79, 084008 (2009), eprint arXiv:0901.3775 [hep-th].

[30] P., Hořava, Phys. Rev. Lett. 102, 161301 (2009), eprint arXiv:0902.3657 [hep-th].

[31] B. L., Hu and D. J., O'Connor, Phys. Rev. D34, 2535 (1986).

[32] V., Husain, Class. Quant. Grav. 5, 575 (1988).

[33] C. J., Isham, Proc. R. Soc. London A351, 209 (1976).

[34] D., Kabat and M., Ortiz, Nucl. Phys. B388, 570 (1992), eprint hep-th/9203082.

[35] A. A., Kirillov, Int. J. Mod. Phys D3, 431 (1994).

[36] A. A., Kirillov and G., Montani, Phys. Rev. D56, 6225 (1997).

[37] R., Kommu, in preparation.

[38] O., Lauscher and M., Reuter, JHEP 0510, 050 (2005), eprint hep-th/0508202.

[39] D. F., Litim, Phys. Rev. Lett. 92, 201301 (2004), eprint hep-th/0312114.

[40] D. F., Litim, AIP Conf. Proc. 841, 322 (2006), eprint hep-th/0606044.

[41] R., Loll, Living Rev. Relativity 1, 13 (1998), eprint gr-qc/9805049.

[42] K., Maeda and M., Sakamoto, Phys. Rev. D54, 1500 (1996), eprint hep-th/9604150.

[43] D., Mattingly, Living Rev. Relativity 8, 5 (2005), eprint gr-qc/0502097.

[44] C.W., Misner, Phys. Rev. Lett. 22, 1071 (1969).

[45] L., Modesto, “Fractal structure of loop quantum gravity,” eprint arXiv:0812.221 [gr-qc].

[46] G., Montani, Class. Quant. Grav. 12, 2505 (1990).

[47] M., Niedermaier, Class. Quant. Grav. 24, R171 (2007), eprint gr-qc/0610018.

[48] M., Pilati, “Strong coupling quantum gravity: an introduction,” in Quantum Structure of Space and Time, edited by M. J., Duff and C. J., Isham, Cambridge University Press, Cambridge, 1982, pp. 53–69.

[49] M., Pilati, Phys. Rev. D26, 2645 (1982).

[50] M., Pilati, Phys. Rev. D28, 729 (1983).

[51] T., Regge, Nuovo Cimento A 19, 558 (1961).

[52] M., Reuter and F., Saueressig, Phys. Rev. D65, 065016 (2002), eprint hep-th/0110054.

[53] D., Rideout, “Dynamics of causal sets,” eprint arXiv:gr-qc/0212064.

[54] M., Rocek and R. M., Williams, Phys. Lett. B104, 31 (1981).

[55] C., Rovelli, Phys. Rev. D35, 2987 (1987).

[56] D. S., Salopek, Class. Quant. Grav. 15, 1185 (1998), eprint gr-qc/9802025.

[57] R., Sorkin, personal communication.

[58] J. L., Synge, Relativity: The General Theory, North-Holland, Amsterdam, 1960.

[59] G., 't Hooft, Phys. Lett. 198B, 61 (1987).

[60] C., Teitelboim, Phys. Rev. D25, 3159 (1982).

[61] D. V., Vassilevich, Phys. Rept. 388, 279 (2003), eprint hep-th/0306138.

[62] H. L., Verlinde and E. P., Verlinde, Nucl. Phys. B371, 246 (1992), eprint hep-th/9110017.

[63] A., Wall, “Proving the achronal averaged null energy condition from the generalized second law,” eprint arXiv:0910.5751.

[64] S., Weinberg, “Ultraviolet divergences in quantum theories of gravitation,” in General Relativity: An Einstein Centenary Survey, edited by S. W., Hawking and W., Israel, Cambridge University Press, Cambridge, 1979, pp. 790–831.

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