Unimodular Gravity and Averaging
ABSTRACT
The question of the averaging of inhomogeneous spacetimes in cosmology is important for the correct interpretation of cosmological data. In this paper a conceptually simpler approach to averaging in cosmology is suggested, based on the averaging of scalars within unimodular gravity. As an illustration, the example of an exact spherically symmetric dust model is considered, and it is shown that within this approach averaging introduces correlations (corrections) to the effective dynamical evolution equation in the form of a spatial curvature term.
The question of the averaging of inhomogeneous spacetimes in cosmology is important for the correct interpretation of cosmological data. In this paper a conceptually simpler approach to averaging in cosmology is suggested, based on the averaging of scalars within unimodular gravity. As an illustration, the example of an exact spherically symmetric dust model is considered, and it is shown that within this approach averaging introduces correlations (corrections) to the effective dynamical evolution equation in the form of a spatial curvature term.
Cite this paper
A. Coley, J. Brannlund and J. Latta, "Unimodular Gravity and Averaging," Journal of Modern Physics, Vol. 3 No. 3, 2012, pp. 266-270. doi: 10.4236/jmp.2012.33036.
A. Coley, J. Brannlund and J. Latta, "Unimodular Gravity and Averaging," Journal of Modern Physics, Vol. 3 No. 3, 2012, pp. 266-270. doi: 10.4236/jmp.2012.33036.
References
[1] G. F. R. Ellis, “Gelativistivc Cosmology: Its Nature Aims and Problems,” In: B. Bertotti, F. de Felici and A. Pascolini, Eds., General Relativity and Gravitation, Reidel, Dordrecht, 1984, pp. 215-588. [2] G. F. R. Ellis and W. Stoeger, “Perturbed Spherically Symmetric Dust Solution of the Field Equations in Ob- servational Coordinates with Cosmological Data Func- tions,” Classical Quantum Gravity, Vol. 4, No. 6, 1987, p. 1697. doi:10.1088/0264-9381/4/6/025 [3] G. F. R. Ellis and T. Buchert, “The Universe Seen at Dif- ferent Scales,” Physics Letters A, Vol. 347, No. 1-3, 2005, pp. 38-46. doi:10.1016/j.physleta.2005.06.087 [4] R. A. Isaacson, “Gravitational Radiation in the Limit of High Frequency. I. The Linear Approximation and Geo- metrical Optics,” Physical Reviews, Vol. 166, No. 5, 1968, pp. 1263-1272. doi:10.1103/PhysRev.166.1263 [5] R. A. Isaacson, “Gravitational Radiation in the Limit of High Frequency. II. Nonlinear Terms and the Effective Stress Tensor,” Physical Reviews, Vol. 166, No. 5, 1968, pp. 1272-1280. doi:10.1103/PhysRev.166.1272 [6] R. M. Zalaletdinov, “Averaging Problem in Cosmology and Macroscopic Gravity,” General Relativity Gravita- tion, Vol. 24, No. 10, 1992, pp. 1015-1031. doi:10.1007/BF00756944 [7] R. M. Zalaletdinov, “Towards a Theory of Macroscopic Gravity,” General Relativity Gravitation, Vol. 25, No. 7, 1993, pp. 673-695. doi:10.1007/BF00756937 [8] R. M. Zalaletdinov, “The Gravitational Polarization in General Relativity: Solution to Szekeres’ Model of Quad- rupole Polarization,” General Relativity Gravitation, Vol. 20, No. 19, 2003, pp. 4195-4212. [9] M. Mars and R. M. Zalaletdinov, “Space-Time Averages in Macroscopic Gravity and Volume-Preserving Coordi- nates,” Journal of Mathematical Physics, Vol. 38, No. 9, 1997, p. 4741. doi:10.1063/1.532119 [10] A. A. Coley, N. Pelavas and R. M. Zalaletdinov, “Cos- mological Solutions in Macroscopic Gravity,” Physical Review Letters, Vol. 95, No. 15, 2005, p. 151102. doi:10.1103/PhysRevLett.95.151102 [11] A. A. Coley and N. Pelavas, “Averaging in Spherically Symmetric Cosmology,” Physical Review D, Vol. 75, No. 4, 2006, p. 043506. doi:10.1103/PhysRevD.75.043506 [12] A. A. Coley and N. Pelavas, “Averaging in Spherically Symmetric Cosmology,” Physical Review D, Vol. 74, No. 8, 2006, p. 087301. doi:10.1103/PhysRevD.74.087301 [13] J. Brannlund, R. J. van den Hoogen and A. Coley, “Av- eraging Geometrical Objects on a Differentiable Mani- fold,” International Journal of Modern Physics D, Vol. 19, No. 12, 2010, pp. 1915-1923. doi:10.1142/S0218271810018062 [14] A. Coley, “Averaging in Cosmological Models Using Scalars,” Classical Quantum Gravity, Vol. 27, No. 24, 2010, p. 245017. doi:10.1088/0264-9381/27/24/245017 [15] T. Buchert, “On Average Properties of Inhomogeneous Fluids in General Relativity: Dust Cosmologies,” General Relativity Gravitation, Vol. 32, No. 1, 2000, pp. 105-125. doi:10.1023/A:1001800617177 [16] T. Buchert, “On Average Properties of Inhomogeneous Fluids in General Relativity: Perfect Fluid Cosmologies,” General Relativity Gravitation, Vol. 33, No. 8, 2001, pp. 1381-1405. doi:10.1023/A:1012061725841 [17] S. Weinberg, “The Cosmological Constant Problem,” Re- views of Modern Physics, Vol. 61, No. 1, 1989, pp. 1-23. doi:10.1103/RevModPhys.61.1 [18] Y. J. Ng and H. van Dam, “A Small but Nonzero Cos- mological Constant,” International Journal of Modern Physics D, Vol. 10, No. 1, 2001, pp. 49-55. doi:10.1142/S0218271801000627 [19] D. R. Finkelstein, A. A. Galiautdinov and J. E. Baugh, “Clif- ford Algebra as Quantum Language,” Journal of Mathe- matical Physics, Vol. 42, No. 1, 2001, p. 340. doi:10.1063/1.1328077 [20] W. G. Unruh, “Time and the Interpretation of Canonical Quantum Gravity,” Physical Review D, Vol. 40, No. 4, 1989, pp. 1048-1052. doi:10.1103/PhysRevD.40.1048 [21] A. Einstein, “The Field Equations of Gravitation,” Preus- sische Akademie der Wissenschaften Berlin (Mathematical Physics), Vol. 1915, 1915, pp. 844-847. [22] A. Einstein, “Cosmological Considerations in the General Theory of Relativity,” Preussische Akademie der Wissenschaften Berlin (Mathematical Physics), Vol. 1917, 1917, p. 142. [23] A. Einstein, “Do Gravitational Fields Play an Essential Role in the Structure Of Elementary Particles of Matter,” Preussische Akademie der Wissenschaften Berlin (Mathe- matical Physics), Vol. 1919, 1919, p. 349. [24] L. Smolin, “Quantization of Unimodular Loop Quantum Gravity,” Physical Review D, Vol. 80, No. 8, 2009, p. 084003. [25] G. F. R. Ellis, J. Murugun and H. van Elst, “On the Trace- Free Einstein Equations as a Viable Alternative to Gen- eral Relativity,” Classical Quantum Gravity, Vol. 28, No. 22, 2011, p. 225007. doi:10.1088/0264-9381/28/22/225007 [26] D. J. Shaw and J. D. Barrow, “Testable Solution of the Cosmological Constant and Coincidence Problems,” Physi- cal Review D, Vol. 83, No. 4, 2011, p. 043518. doi:10.1103/PhysRevD.83.043518 [27] B. Li, T. P. Sotirou and J. D. Barrow, “(f)T Gravity and Local Loretz Invariance,” Physical Review D, Vol. 83, No. 6, 2011, p. 064035. doi:10.1103/PhysRevD.83.064035 [28] S. R. Green and R. M. Wald, “A New Framework for Treating Small Scale Inhomogeneities in Cosmology,” Physical Review D, Vol. 83, No. 8, 2011, p. 084020. doi:10.1103/PhysRevD.83.084020 [29] A. P. Billyard and A. A. Coley, “Interactions in Scalar Field Cosmology,” Physical Review D, Vol. 61, No. 8, 2000, p. 083503. doi:10.1103/PhysRevD.61.083503 [30] K. Bolejko, M. N. Celerier, C. Hellaby and A. Krasinski, “Structures in the Universe by Exact Methods; Formation, Evolution, Interactions,” Cambridge University Press, Cam- bridge, 2009. doi:10.1017/CBO9780511657405 [31] A. G. Riess, et al., “Observational Evidence from Super- novae for an Accelerating Universe and a Cosmological Constant,” The Astronomical Journal, Vol. 116, No. 3, 1998, p. 1009. doi:10.1086/300499 [32] S. Perlmutter, et al., “Measuring Cosmology with Supernovae,” The Astronomical Journal, Vol. 517, No. 2, 1999, p. 565. doi:10.1086/307221 [33] D. N. Spergel, et al., “First-Year Wilkinson Microwave Anisotropy Probe (WMAP, Observations: Determination of Cosmological Parameters,” The Astrophysical Journal Supplement Series, Vol. 148, No. 1, 2003, p. 175. [34] D. L. Wiltshire, “Exact Solution to the Averaging Prob- lem in Cosmology,” Physical Review Letters, Vol. 99, No. 25, 2007, p. 251101. doi:10.1103/PhysRevLett.99.251101 [35] I. A. Brown, J. Behrend and K. A. Malik, “Gauges and Cosmological Backreaction,” Journal of Cosmology and Astroparticle Physics, Vol. 2009, No. 11, 2009, p. 027. [36] A. Paranjape, “The Averaging Problem in Cosmology,” Ph.D. Thesis, Cornell University, Ithaca, 2009.
[1] G. F. R. Ellis, “Gelativistivc Cosmology: Its Nature Aims and Problems,” In: B. Bertotti, F. de Felici and A. Pascolini, Eds., General Relativity and Gravitation, Reidel, Dordrecht, 1984, pp. 215-588. [2] G. F. R. Ellis and W. Stoeger, “Perturbed Spherically Symmetric Dust Solution of the Field Equations in Ob- servational Coordinates with Cosmological Data Func- tions,” Classical Quantum Gravity, Vol. 4, No. 6, 1987, p. 1697. doi:10.1088/0264-9381/4/6/025 [3] G. F. R. Ellis and T. Buchert, “The Universe Seen at Dif- ferent Scales,” Physics Letters A, Vol. 347, No. 1-3, 2005, pp. 38-46. doi:10.1016/j.physleta.2005.06.087 [4] R. A. Isaacson, “Gravitational Radiation in the Limit of High Frequency. I. The Linear Approximation and Geo- metrical Optics,” Physical Reviews, Vol. 166, No. 5, 1968, pp. 1263-1272. doi:10.1103/PhysRev.166.1263 [5] R. A. Isaacson, “Gravitational Radiation in the Limit of High Frequency. II. Nonlinear Terms and the Effective Stress Tensor,” Physical Reviews, Vol. 166, No. 5, 1968, pp. 1272-1280. doi:10.1103/PhysRev.166.1272 [6] R. M. Zalaletdinov, “Averaging Problem in Cosmology and Macroscopic Gravity,” General Relativity Gravita- tion, Vol. 24, No. 10, 1992, pp. 1015-1031. doi:10.1007/BF00756944 [7] R. M. Zalaletdinov, “Towards a Theory of Macroscopic Gravity,” General Relativity Gravitation, Vol. 25, No. 7, 1993, pp. 673-695. doi:10.1007/BF00756937 [8] R. M. Zalaletdinov, “The Gravitational Polarization in General Relativity: Solution to Szekeres’ Model of Quad- rupole Polarization,” General Relativity Gravitation, Vol. 20, No. 19, 2003, pp. 4195-4212. [9] M. Mars and R. M. Zalaletdinov, “Space-Time Averages in Macroscopic Gravity and Volume-Preserving Coordi- nates,” Journal of Mathematical Physics, Vol. 38, No. 9, 1997, p. 4741. doi:10.1063/1.532119 [10] A. A. Coley, N. Pelavas and R. M. Zalaletdinov, “Cos- mological Solutions in Macroscopic Gravity,” Physical Review Letters, Vol. 95, No. 15, 2005, p. 151102. doi:10.1103/PhysRevLett.95.151102 [11] A. A. Coley and N. Pelavas, “Averaging in Spherically Symmetric Cosmology,” Physical Review D, Vol. 75, No. 4, 2006, p. 043506. doi:10.1103/PhysRevD.75.043506 [12] A. A. Coley and N. Pelavas, “Averaging in Spherically Symmetric Cosmology,” Physical Review D, Vol. 74, No. 8, 2006, p. 087301. doi:10.1103/PhysRevD.74.087301 [13] J. Brannlund, R. J. van den Hoogen and A. Coley, “Av- eraging Geometrical Objects on a Differentiable Mani- fold,” International Journal of Modern Physics D, Vol. 19, No. 12, 2010, pp. 1915-1923. doi:10.1142/S0218271810018062 [14] A. Coley, “Averaging in Cosmological Models Using Scalars,” Classical Quantum Gravity, Vol. 27, No. 24, 2010, p. 245017. doi:10.1088/0264-9381/27/24/245017 [15] T. Buchert, “On Average Properties of Inhomogeneous Fluids in General Relativity: Dust Cosmologies,” General Relativity Gravitation, Vol. 32, No. 1, 2000, pp. 105-125. doi:10.1023/A:1001800617177 [16] T. Buchert, “On Average Properties of Inhomogeneous Fluids in General Relativity: Perfect Fluid Cosmologies,” General Relativity Gravitation, Vol. 33, No. 8, 2001, pp. 1381-1405. doi:10.1023/A:1012061725841 [17] S. Weinberg, “The Cosmological Constant Problem,” Re- views of Modern Physics, Vol. 61, No. 1, 1989, pp. 1-23. doi:10.1103/RevModPhys.61.1 [18] Y. J. Ng and H. van Dam, “A Small but Nonzero Cos- mological Constant,” International Journal of Modern Physics D, Vol. 10, No. 1, 2001, pp. 49-55. doi:10.1142/S0218271801000627 [19] D. R. Finkelstein, A. A. Galiautdinov and J. E. Baugh, “Clif- ford Algebra as Quantum Language,” Journal of Mathe- matical Physics, Vol. 42, No. 1, 2001, p. 340. doi:10.1063/1.1328077 [20] W. G. Unruh, “Time and the Interpretation of Canonical Quantum Gravity,” Physical Review D, Vol. 40, No. 4, 1989, pp. 1048-1052. doi:10.1103/PhysRevD.40.1048 [21] A. Einstein, “The Field Equations of Gravitation,” Preus- sische Akademie der Wissenschaften Berlin (Mathematical Physics), Vol. 1915, 1915, pp. 844-847. [22] A. Einstein, “Cosmological Considerations in the General Theory of Relativity,” Preussische Akademie der Wissenschaften Berlin (Mathematical Physics), Vol. 1917, 1917, p. 142. [23] A. Einstein, “Do Gravitational Fields Play an Essential Role in the Structure Of Elementary Particles of Matter,” Preussische Akademie der Wissenschaften Berlin (Mathe- matical Physics), Vol. 1919, 1919, p. 349. [24] L. Smolin, “Quantization of Unimodular Loop Quantum Gravity,” Physical Review D, Vol. 80, No. 8, 2009, p. 084003. [25] G. F. R. Ellis, J. Murugun and H. van Elst, “On the Trace- Free Einstein Equations as a Viable Alternative to Gen- eral Relativity,” Classical Quantum Gravity, Vol. 28, No. 22, 2011, p. 225007. doi:10.1088/0264-9381/28/22/225007 [26] D. J. Shaw and J. D. Barrow, “Testable Solution of the Cosmological Constant and Coincidence Problems,” Physi- cal Review D, Vol. 83, No. 4, 2011, p. 043518. doi:10.1103/PhysRevD.83.043518 [27] B. Li, T. P. Sotirou and J. D. Barrow, “(f)T Gravity and Local Loretz Invariance,” Physical Review D, Vol. 83, No. 6, 2011, p. 064035. doi:10.1103/PhysRevD.83.064035 [28] S. R. Green and R. M. Wald, “A New Framework for Treating Small Scale Inhomogeneities in Cosmology,” Physical Review D, Vol. 83, No. 8, 2011, p. 084020. doi:10.1103/PhysRevD.83.084020 [29] A. P. Billyard and A. A. Coley, “Interactions in Scalar Field Cosmology,” Physical Review D, Vol. 61, No. 8, 2000, p. 083503. doi:10.1103/PhysRevD.61.083503 [30] K. Bolejko, M. N. Celerier, C. Hellaby and A. Krasinski, “Structures in the Universe by Exact Methods; Formation, Evolution, Interactions,” Cambridge University Press, Cam- bridge, 2009. doi:10.1017/CBO9780511657405 [31] A. G. Riess, et al., “Observational Evidence from Super- novae for an Accelerating Universe and a Cosmological Constant,” The Astronomical Journal, Vol. 116, No. 3, 1998, p. 1009. doi:10.1086/300499 [32] S. Perlmutter, et al., “Measuring Cosmology with Supernovae,” The Astronomical Journal, Vol. 517, No. 2, 1999, p. 565. doi:10.1086/307221 [33] D. N. Spergel, et al., “First-Year Wilkinson Microwave Anisotropy Probe (WMAP, Observations: Determination of Cosmological Parameters,” The Astrophysical Journal Supplement Series, Vol. 148, No. 1, 2003, p. 175. [34] D. L. Wiltshire, “Exact Solution to the Averaging Prob- lem in Cosmology,” Physical Review Letters, Vol. 99, No. 25, 2007, p. 251101. doi:10.1103/PhysRevLett.99.251101 [35] I. A. Brown, J. Behrend and K. A. Malik, “Gauges and Cosmological Backreaction,” Journal of Cosmology and Astroparticle Physics, Vol. 2009, No. 11, 2009, p. 027. [36] A. Paranjape, “The Averaging Problem in Cosmology,” Ph.D. Thesis, Cornell University, Ithaca, 2009.