text: – 70 – Table 2—Continued (cid:52) (cid:52) Obj ID Designation a e i Ω ω D r α HR n m PTFR Period (hr) m U ± ± ± H4473 (174473) 2003 AX16 3.10 0.14 4.03 270.5 154.0 3.6 2.77 1.79 2.90 15.99 0.14 4 59 19.71 0.10 3.16 0.04 0.34 2 ± ± ± H4511 (174511) 2003 BH77 3.12 0.24 17.14 117.8 290.9 4.6 2.79 1.88 9.51 15.45 0.21 4 59 19.68 0.09 5.03 0.05 0.59 2 ± ± ± H4680 (174680) 2003 SC304 2.67 0.16 11.19 307.8 324.7 3.0w3.03 2.10 7.34 15.39 0.25 2 50 19.92 0.09 4.36 0.02 0.79 3 ± ± ± H4946 (174946) 2004 CF85 2.98 0.15 9.55 296.1 306.9 5.8w3.18 2.23 5.03 15.24 0.16 4 66 19.88 0.11 9.32 0.18 0.56 2 ± ± ± H5313 (175313) 2005 MF15 3.24 0.14 1.84 205.4 91.7 5.2w3.70 2.72 2.44 15.15 0.23 4 44 20.40 0.14 4.92 0.08 0.77 2 ± ± ± H5347 (175347) 2005 NV122 3.17 0.06 8.47 314.8 151.1 5.4w3.03 2.05 3.35 15.18 0.15 4 74 19.44 0.08 7.27 0.05 0.39 3 ± ± ± H6988b (176988) 2002 YR29 3.12 0.05 7.31 295.4 186.0 5.3 2.97 2.01 4.75 15.14 0.28 4 68 19.42 0.07 7.74 0.06 0.91 3 ± ± ± H7002 (177002) 2003 AG59 3.15 0.24 16.78 118.3 43.8 6.5 2.52 1.58 8.35 14.71 0.16 4 70 18.23 0.03 3.14 0.02 0.30 3 ± ± ± H7054 (177054) 2003 EY35 3.16 0.09 2.35 95.4 62.3 4.3 2.88 1.90 1.07 15.57 0.15 2 22 19.44 0.09 2.72 0.03 0.45 2 ± ± ± H7109 (177109) 2003 GZ2 3.17 0.13 6.08 348.9 283.3 6.1w3.31 2.32 1.13 14.93 0.15 4 80 19.57 0.07 11.85 0.14 0.47 2 ± ± ± H7726 (177726) 2005 HX4 2.21 0.11 4.76 116.2 176.2 1.1 2.46 1.48 2.41 16.81 0.16 3 58 19.83 0.10 4.97 0.08 0.44 2 ± ± ± H9138 (179138) 2001 SU316 3.05 0.10 8.70 316.6 44.5 4.4 3.19 2.25 6.63 15.55 0.29 4 49 20.29 0.12 3.72 0.03 0.96 3 ± ± ± I0203 (180203) 2003 SU259 2.17 0.13 0.99 93.9 291.6 0.9 2.11 1.13 2.72 17.35 0.12 3 62 19.57 0.08 5.25 0.08 0.26 2 ± ± ± I1570 (181570) 2006 VZ21 2.17 0.12 2.61 289.2 310.4 0.9 2.31 1.33 3.00 17.31 0.21 4 51 20.07 0.11 10.43 0.35 0.64 2 ± ± ± I1966 (181966) 1999 UA42 2.23 0.21 4.57 327.7 7.2 1.7w2.59 1.68 9.70 16.61 0.12 2 46 20.37 0.13 2.24 0.03 0.43 2 ± ± ± I3130 (183130) 2002 RR201 2.22 0.12 3.49 116.5 164.0 1.1 2.48 1.52 6.41 16.80 0.14 3 53 20.15 0.11 3.60 0.04 0.37 2 ± ± ± I3597 (183597) 2003 UT14 2.74 0.04 3.42 204.1 232.4 2.6 2.70 1.72 4.36 15.97 0.26 4 67 19.69 0.09 9.60 0.10 0.95 3 ± ± ± I3609 (183609) 2003 UV160 2.13 0.10 1.84 239.1 114.0 0.8 2.32 1.34 4.28 17.52 0.19 4 58 20.38 0.11 5.39 0.03 0.70 3 ± ± ± I4384 (184384) 2005 LA18 2.20 0.09 6.03 107.0 331.7 1.1 2.04 1.13 13.67 16.90 0.19 3 56 19.46 0.07 6.00 0.07 0.56 3 ∗ ± ± ± I5086 (185086) 2006 RP92 3.09 0.07 1.15 300.2 66.1 3.7 3.25 2.27 2.80 15.90 0.25 3 36 20.51 0.13 7.68 0.19 0.97 2 ± ± ± I5199 (185199) 2006 TB25 3.13 0.10 0.83 341.5 37.2 4.1 3.34 2.35 1.06 15.70 0.12 4 52 20.45 0.14 7.74 0.52 0.44 2 ± ± ± I5225 (185225) 2006 TV74 3.20 0.15 0.70 20.2 25.2 3.5 3.28 2.29 0.42 16.06 0.22 4 60 20.48 0.12 9.06 0.17 0.74 2 ± ± ± I5506 (185506) 2007 TJ315 3.02 0.12 1.16 68.6 316.5 3.9 2.99 2.02 4.68 15.82 0.27 4 65 20.11 0.12 10.00 0.20 0.92 2 ± ± ± I5570 (185570) 2008 AA78 3.18 0.06 1.96 4.3 103.9 4.1 3.03 2.04 0.80 15.71 0.21 4 76 19.94 0.11 9.14 0.09 0.74 3 ± ± ± I5704 (185704) 1998 OD8 2.34 0.26 3.55 277.8 84.0 3.4w2.43 1.46 5.01 16.27 0.26 4 67 19.44 0.07 7.93 0.07 0.85 3 ± ± ± I5943 (185943) 2000 YB98 2.20 0.14 3.43 102.3 57.3 1.2 1.96 0.99 8.23 16.66 0.10 4 80 18.73 0.04 4.92 0.08 0.24 2 ± ± ± I5951 (185951) 2001 DE8 2.19 0.16 8.32 148.8 125.9 1.2 2.34 1.35 0.88 16.59 0.12 1 18 19.26 0.07 2.30 0.12 0.41 2 ± ± ± I6454b (186454) 2002 SU55 2.29 0.15 4.51 128.0 219.6 1.3 2.45 1.48 3.80 16.52 0.24 4 63 19.65 0.09 6.15 0.04 0.78 3 ± ± ± I6490 (186490) 2002 TB222 2.22 0.18 1.99 234.7 74.1 1.1 2.58 1.59 2.43 16.90 0.25 4 56 20.29 0.12 5.05 0.03 0.83 3 ± ± ± I6772 (186772) 2004 DQ20 2.28 0.08 6.36 113.6 295.7 1.3 2.19 1.21 3.47 16.39 0.19 4 75 18.82 0.05 5.52 0.03 0.52 3 ± ± ± I7926 (187926) 2001 BH31 2.19 0.10 4.17 153.4 347.9 1.3 1.98 0.99 1.12 16.46 0.12 4 72 18.09 0.03 2.69 0.01 0.17 3 ± ± ± I7931 (187931) 2001 DN107 2.21 0.02 7.02 118.6 90.2 1.2 2.22 1.25 7.23 16.68 0.19 3 58 19.42 0.07 4.95 0.03 0.64 3 ##LTLine## ##LTLine## ##LTLine##

text: Table

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text: – 77 – Table 2—Continued (cid:52) (cid:52) Obj ID Designation a e i Ω ω D r α HR n m PTFR Period (hr) m U ± ± ± M2658 (222658) 2001 XJ194 2.52 0.22 2.84 127.1 281.5 2.3 2.22 1.26 7.92 16.21 0.16 4 71 19.01 0.05 5.89 0.11 0.21 2 ± ± ± M2965 (222965) 2002 QJ45 2.19 0.16 1.79 174.7 111.5 1.0 2.43 1.44 3.41 16.99 0.18 4 71 20.02 0.11 8.21 0.22 0.65 2 ± ± ± M3120 (223120) 2002 VZ32 2.34 0.22 0.99 246.6 121.2 0.9 2.35 1.37 3.34 17.32 0.15 4 54 20.16 0.13 2.44 0.02 0.39 2 ± ± ± M3170 (223170) 2002 YP7 2.38 0.28 5.81 109.3 322.6 1.0 1.83 0.86 8.35 17.03 0.21 4 95 18.55 0.04 8.35 0.07 0.60 3 ± ± ± M3193 (223193) 2003 AX71 2.34 0.21 1.59 239.4 167.7 0.9 2.36 1.37 2.09 17.34 0.25 2 40 20.06 0.09 4.68 0.05 0.91 2 ± ± ± M3512 (223512) 2004 CA43 2.27 0.05 1.94 69.5 137.1 1.0 2.26 1.31 8.03 16.97 0.24 4 65 19.85 0.09 4.97 0.05 0.63 3 ± ± ± M3521 (223521) 2004 CZ100 2.25 0.05 7.18 340.7 134.8 1.3 2.15 1.17 2.67 16.47 0.15 4 76 18.74 0.04 3.17 0.01 0.35 2 ± ± ± M4074 (224074) 2005 NJ53 2.26 0.11 6.54 145.0 311.3 0.9 2.10 1.11 1.18 17.19 0.23 4 73 19.16 0.06 6.44 0.04 0.70 3 ± ± ± M5415 (225415) 1999 XA261 2.28 0.14 2.35 87.9 51.9 0.9 1.97 1.02 9.77 17.32 0.13 4 72 19.27 0.07 2.96 0.03 0.31 2 ± ± ± M6193 (226193) 2002 TY285 2.26 0.23 3.36 330.5 24.1 1.3 2.74 1.75 0.76 16.39 0.22 4 59 20.07 0.12 6.19 0.08 0.69 2 ± ± ± M6311 (226311) 2003 CX7 2.41 0.17 3.20 103.6 357.0 1.1 2.00 1.03 7.12 16.82 0.24 4 78 18.95 0.05 6.49 0.04 0.71 3 ± ± ± M6604 (226604) 2004 CC86 2.27 0.13 2.16 136.0 121.0 1.0 2.49 1.53 5.80 17.03 0.15 4 54 20.37 0.14 3.56 0.05 0.49 2 ± ± ± M6682 (226682) 2004 HP43 2.29 0.15 0.19 243.6 301.9 0.9 2.12 1.17 8.72 17.34 0.23 4 61 19.99 0.10 4.47 0.02 0.77 3 ± ± ± M7189 (227189) 2005 QS67 2.35 0.17 2.82 18.3 67.9 0.7 2.15 1.16 2.45 17.65 0.18 4 68 19.69 0.09 4.17 0.04 0.47 2 ± ± ± M7234 (227234) 2005 SR5 2.35 0.18 1.73 252.0 69.6 1.1 2.71 1.73 3.64 16.75 0.14 2 36 20.43 0.14 2.27 0.03 0.57 2 ± ± ± M7837 (227837) 2007 CX56 2.36 0.17 3.14 113.8 30.8 1.2 2.01 1.04 5.32 16.62 0.23 4 77 18.68 0.04 5.71 0.03 0.68 3 ± ± ± M7839 (227839) 2007 CC73 2.29 0.20 2.00 347.4 60.6 0.6 2.31 1.32 2.07 17.99 0.16 2 31 20.63 0.15 2.41 0.02 0.56 2 ± ± ± M8821 (228821) 2003 BV57 2.38 0.15 1.55 321.4 166.5 0.9 2.03 1.08 9.60 17.14 0.27 4 68 19.43 0.07 8.21 0.07 0.90 3 ± ± ± M8830 (228830) 2003 DP 2.38 0.15 1.82 268.8 177.0 1.1 2.17 1.19 3.73 16.79 0.23 4 65 19.21 0.06 4.30 0.02 0.64 3 ± ± ± M9062 (229062) 2004 GB24 2.33 0.12 5.42 99.8 161.8 1.0 2.53 1.59 7.99 16.95 0.15 4 43 20.54 0.15 2.46 0.03 0.52 2 ± ± ± M9468 (229468) 2005 UE253 2.44 0.16 1.07 359.6 82.9 0.9 2.11 1.16 9.66 17.32 0.22 4 55 19.88 0.10 4.97 0.03 0.76 2 ± ± ± N0162 (230162) 2001 QE244 2.38 0.13 6.33 167.2 191.6 1.2 2.64 1.65 2.38 16.55 0.17 4 63 19.94 0.10 5.78 0.07 0.47 2 ± ± ± N0224 (230224) 2001 TS160 2.41 0.13 0.67 321.2 216.7 0.9 2.12 1.14 1.19 17.33 0.21 4 65 19.62 0.08 4.82 0.02 0.57 2 ± ± ± N1336b (231336) 2006 DL197 2.58 0.22 9.98 290.4 239.9 1.6w2.20 1.22 3.42 16.49 0.27 4 69 18.93 0.05 9.23 0.09 0.78 3 ± ± ± N1865 (231865) 2000 SY318 2.60 0.29 11.44 315.1 104.2 3.5 2.34 1.36 5.82 15.29 0.14 4 77 18.22 0.03 3.02 0.01 0.34 3 ∗ ± ± ± N2206 (232206) 2002 GN107 2.56 0.20 12.14 160.8 74.1 1.7w2.44 1.47 4.71 16.45 0.21 4 61 19.64 0.08 11.29 0.13 0.77 3 ± ± ± N2486 (232486) 2003 NN 2.60 0.21 7.93 158.5 195.9 2.8 3.09 2.10 0.56 15.76 0.19 4 68 19.87 0.09 3.86 0.03 0.62 2 ± ± ± N2775 (232775) 2004 PA75 2.35 0.21 4.92 344.1 284.8 1.0 2.44 1.46 1.98 16.97 0.11 4 63 19.99 0.11 2.31 0.01 0.39 2 ± ± ± N3214 (233214) 2005 XT110 2.64 0.14 1.92 2.0 16.1 1.7 2.67 1.74 8.52 16.88 0.18 3 27 20.78 0.17 4.19 0.06 0.59 2 ± ± ± N3350b (233350) 2006 DT61 2.64 0.22 13.24 158.3 73.8 2.3w2.47 1.49 4.42 15.42 0.25 4 75 18.73 0.04 5.33 0.03 0.71 3 ± ± ± N3378 (233378) 2006 DP199 2.55 0.12 15.84 149.4 14.4 2.3 2.26 1.27 0.98 16.18 0.19 4 80 18.60 0.04 5.52 0.03 0.54 3 ± ± ± N3486 (233486) 2006 UF1 3.11 0.01 8.24 292.0 67.7 6.4w3.12 2.16 4.07 15.26 0.16 4 68 19.73 0.09 12.63 0.34 0.50 2 ##LTLine## ##LTLine## ##LTLine##

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text: – 89 – Table 3—Continued (cid:52) (cid:52) Obj ID Designation a e i Ω ω D r α HR n m PTFR Period (hr) m U ± ± ± 83034 (83034) 2001 QT183 2.99 0.13 3.61 138.2 196.9 7.1 3.38 2.40 1.56 14.52 0.11 4 79 19.22 0.07 8.73 0.23 0.36 2 ± ± ± 83962 (83962) 2001 XW123 3.17 0.09 2.61 99.1 176.8 4.7w3.44 2.46 1.58 14.76 0.20 4 65 19.68 0.09 5.19 0.09 0.66 2 ± ± ± 84847 (84847) 2003 AW28 3.11 0.10 12.93 120.3 299.6 6.3 2.93 1.99 6.49 14.76 0.13 4 77 19.05 0.06 8.28 0.22 0.35 2 ± ± ± 86474 (86474) 2000 CT79 2.80 0.04 4.06 163.9 341.9 2.9w2.70 1.71 0.47 15.16 0.09 2 43 18.59 0.03 9.50 0.19 0.21 2 ± ± ± 90138 (90138) Diehl 2.29 0.06 0.52 218.3 124.6 1.2 2.43 1.45 2.45 16.64 0.13 4 59 19.61 0.08 8.07 0.20 0.43 2 ± ± ± 93000 (93000) 2000 RJ83 2.55 0.12 12.03 309.1 142.1 3.5 2.26 1.32 9.18 15.31 0.14 3 65 18.26 0.03 6.04 0.08 0.27 2 ± ± ± 96990 (96990) 1999 TQ215 2.69 0.21 2.99 117.2 339.4 2.5w2.33 1.34 3.16 15.59 0.24 4 72 18.29 0.03 8.57 0.08 0.71 2 ± ± ± 97158 (97158) 1999 VV166 2.69 0.04 2.82 354.3 225.7 2.5w2.70 1.80 10.50 14.98 0.12 4 67 19.14 0.06 5.82 0.10 0.50 2 ± ± ± 98617 (98617) 2000 WQ81 2.55 0.08 2.95 311.4 34.2 3.1 2.75 1.76 0.82 15.55 0.15 4 77 19.01 0.05 11.71 0.29 0.46 2 ± ± ± A5742 (105742) 2000 SH90 3.14 0.05 21.12 301.7 340.6 8.6 3.27 2.34 6.23 14.09 0.32 4 74 18.95 0.05 16.27 0.57 0.97 2 ± ± ± A6275 (106275) 2000 UY69 2.60 0.23 3.21 296.5 175.4 2.4w2.00 1.04 6.83 15.21 0.15 4 81 17.06 0.01 19.59 0.77 0.26 2 ± ± ± A9018 (109018) 2001 QA6 2.37 0.20 6.52 146.5 222.5 1.7 2.69 1.71 1.84 15.92 0.24 4 90 19.47 0.07 11.29 0.26 0.71 2 ± ± ± A9512 (109512) 2001 QA236 2.39 0.18 5.38 141.5 345.9 5.0w1.99 1.00 1.29 15.70 0.21 3 44 17.36 0.02 5.27 0.03 0.56 2 ± ± ± B4122 (114122) 2002 VW49 2.29 0.18 6.11 107.7 219.3 1.4 2.66 1.72 8.44 16.36 0.23 4 79 20.20 0.12 5.75 0.13 0.87 2 ± ± ± B4348b (114348) 2002 XY74 2.36 0.23 3.94 299.1 108.5 3.9w2.04 1.07 6.10 16.03 0.29 4 78 18.18 0.03 14.33 0.21 0.90 2 ± ± ± B4561 (114561) 2003 BP52 2.40 0.18 1.00 133.9 321.3 1.5w1.98 1.00 3.85 16.24 0.19 4 84 18.05 0.03 22.33 1.46 0.58 2 ± ± ± B7401 (117401) 2005 AL8 2.75 0.12 9.89 110.6 31.8 4.8w2.45 1.48 5.97 15.29 0.08 4 77 18.56 0.04 8.21 0.21 0.34 2 ± ± ± C1192 (121192) 1999 NC22 2.57 0.13 6.90 108.1 178.0 3.3 2.90 1.93 3.88 15.43 0.15 3 60 19.53 0.07 11.16 0.40 0.38 2 ± ± ± C2735 (122735) 2000 SW49 2.53 0.17 7.89 329.1 83.3 2.2 2.52 1.53 1.84 16.29 0.12 4 91 19.40 0.07 10.00 0.43 0.32 2 ± ± ± C2783 (122783) 2000 SY85 2.59 0.20 3.91 105.1 300.9 1.8 2.34 1.39 8.49 16.69 0.26 4 57 19.78 0.10 6.23 0.12 0.90 2 ± ± ± C8209 (128209) 2003 SS65 4.02 0.15 3.13 115.2 287.5 9.0 3.81 2.86 4.61 14.00 0.21 4 69 19.56 0.09 18.46 1.62 0.49 2 ± ± ± D1048 (131048) 2000 YM38 2.62 0.06 2.83 171.0 308.5 2.3 2.48 1.50 4.56 16.24 0.26 4 96 19.50 0.07 11.43 0.27 0.83 2 ± ± ± D2327 (132327) 2002 GA24 2.59 0.10 2.17 4.9 281.6 2.4 2.72 1.74 2.28 16.09 0.15 1 22 19.71 0.07 3.90 0.24 0.52 2 ± ± ± D2435 (132435) 2002 GL164 2.63 0.03 3.53 104.4 50.8 2.5 2.56 1.59 4.50 16.04 0.16 4 68 19.48 0.08 8.28 0.21 0.48 2 ± ± ± D5274 (135274) 2001 SH126 3.06 0.09 3.71 315.3 135.1 6.3 2.80 1.86 7.17 14.76 0.11 4 74 18.82 0.05 5.68 0.07 0.25 2 ± ± ± D5316 (135316) 2001 SN277 3.08 0.14 10.17 323.3 56.1 7.3 3.31 2.32 2.01 14.46 0.11 4 75 19.17 0.06 6.36 0.08 0.28 2 ± ± ± D7712 (137712) 1999 XU96 2.27 0.21 1.73 24.8 39.0 1.1 2.18 1.19 1.80 16.72 0.13 4 81 19.05 0.05 5.93 0.07 0.48 2 ± ± ± D8108 (138108) 2000 DC102 2.81 0.07 17.75 147.2 55.9 3.9w2.67 1.69 0.80 15.08 0.16 1 18 18.46 0.03 8.97 1.64 0.48 2 ± ± ± D8503 (138503) 2000 KC58 2.36 0.15 7.90 112.3 69.5 2.0 2.15 1.21 11.13 15.52 0.13 4 70 18.40 0.03 6.86 0.10 0.20 2 ± ± ± E7270 (147270) 2002 YT16 2.35 0.18 1.65 302.3 178.9 1.5w1.92 0.96 9.93 16.18 0.12 4 78 17.94 0.02 18.46 1.13 0.21 2 ± ± ± F1957 (151957) 2004 GE14 2.34 0.01 6.55 126.7 1.2 1.7 2.31 1.34 4.13 15.83 0.14 4 65 18.82 0.05 20.43 1.23 0.45 2 ± ± ± F3672 (153672) 2001 TO189 2.43 0.12 5.63 271.6 33.4 1.6 2.72 1.75 3.34 16.04 0.21 4 69 19.72 0.06 28.24 1.57 0.66 2 ##LTLine## ##LTLine## ##LTLine##

text: Table

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text: PTFR

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text: Diehl

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text: – 93 – Table 4—Continued (cid:52) (cid:52) Obj ID Designation a e i Ω ω D r α HR n m PTFR m ± ± 42297 (42297) 2001 UL73 3.16 0.17 1.52 137.1 354.8 6.5 2.62 1.66 5.52 14.71 0.22 3 60 18.31 0.03 0.76 ± ± 45610 (45610) 2000 DJ48 2.86 0.11 1.50 352.9 14.3 7.5 2.94 1.98 5.43 14.38 0.13 4 73 18.62 0.05 0.52 ± ± 45904 (45904) 2000 YV29 2.68 0.13 13.31 116.9 340.2 4.2w2.37 1.45 10.83 13.75 0.21 4 76 17.06 0.01 0.64 ± ± 46714 (46714) 1997 HF7 2.33 0.18 1.46 54.7 10.2 2.1w2.02 1.06 7.80 15.07 0.17 4 62 17.23 0.01 0.51 ± ± 49589 (49589) 1999 CQ149 2.95 0.09 2.05 171.4 315.7 6.0 2.72 1.73 0.75 14.86 0.21 4 77 18.34 0.04 0.69 ± ± 50886 (50886) 2000 GW39 2.97 0.13 0.82 65.1 276.7 6.4w3.18 2.20 1.69 14.57 0.24 3 26 18.87 0.08 0.64 ± ± 51202 (51202) 2000 JR6 2.97 0.12 13.70 119.2 296.7 7.5w2.75 1.77 2.87 13.76 0.12 4 82 17.69 0.02 0.52 ± ± 54158 (54158) 2000 HY48 2.37 0.12 2.24 66.2 108.3 1.5 2.19 1.23 7.59 16.14 0.23 2 45 18.80 0.05 0.85 ± ± 54587 (54587) 2000 QM189 3.13 0.04 11.17 110.3 157.7 4.4w3.23 2.29 6.44 14.13 0.20 3 43 18.85 0.04 0.51 ± ± 55652 (55652) 4048 P-L 2.98 0.08 2.44 217.8 139.3 6.9w3.19 2.20 1.93 14.21 0.18 4 73 18.68 0.04 0.57 ± ± 56702 (56702) 2000 LQ28 3.14 0.25 15.78 313.6 37.0 12.1w3.79 2.82 3.59 13.07 0.26 3 60 18.56 0.04 1.15 ± ± 57175 (57175) 2001 QD24 2.44 0.21 3.09 111.9 287.7 1.9w2.21 1.23 4.45 15.67 0.27 2 24 18.33 0.05 0.54 ± ± 59263 (59263) 1999 CK30 2.43 0.19 2.25 100.6 301.1 1.9 2.24 1.30 8.96 15.64 0.25 4 69 18.54 0.04 0.69 ± ± 59812 (59812) 1999 RA18 3.24 0.04 14.94 343.7 225.0 11.9w3.17 2.18 1.37 13.53 0.17 4 88 17.93 0.03 0.62 ± ± 61480 (61480) 2000 QY39 3.04 0.09 9.89 328.4 322.6 8.1 3.23 2.24 0.74 14.21 0.17 4 76 18.65 0.04 0.57 ± ± 62043 (62043) 2000 RH65 3.23 0.10 8.72 170.3 248.1 9.6w3.18 2.20 3.61 13.78 0.22 2 34 18.42 0.05 0.81 ± ± 64387 (64387) 2001 UW144 2.38 0.15 2.40 91.5 145.4 1.9 2.28 1.29 1.88 15.64 0.23 3 64 18.09 0.02 0.55 ± ± 67917 (67917) 2000 WU109 2.52 0.15 4.37 317.0 327.4 2.4w2.76 1.77 0.93 14.91 0.25 2 34 18.70 0.06 0.98 ± ± 69727 (69727) 1998 HD145 2.57 0.12 8.10 140.5 125.1 2.5w2.66 1.67 1.45 15.21 0.23 2 40 18.51 0.05 0.61 ± ± 70615 (70615) 1999 TZ204 2.71 0.06 2.75 302.8 164.3 3.0w2.56 1.58 3.29 14.99 0.26 4 82 18.29 0.03 0.86 ± ± 75653 (75653) 2000 AG64 2.33 0.04 9.38 122.1 129.5 5.2w2.40 1.42 3.60 14.52 0.25 1 14 17.48 0.02 0.85 ± ± 77831 (77831) 2001 QL226 3.00 0.07 7.73 310.4 178.5 9.2 2.79 1.81 3.54 13.93 0.11 4 81 17.72 0.02 0.56 ± ± 78479 (78479) 2002 RO52 2.84 0.07 3.07 128.4 240.1 3.5w3.00 2.01 1.00 14.71 0.19 1 19 18.71 0.05 0.66 ± ± 80448 (80448) 1999 YA17 2.31 0.11 9.72 293.9 99.1 4.1w2.25 1.28 5.19 15.31 0.20 3 51 18.04 0.04 0.65 ± ± 80451 (80451) Alwoods 2.29 0.09 5.70 114.6 351.1 1.6 2.09 1.16 11.18 15.96 0.17 4 72 18.54 0.04 0.52 ± ± 81985 (81985) 2000 QV122 3.01 0.23 7.82 333.5 292.2 5.4w3.13 2.14 0.38 13.89 0.20 4 86 18.01 0.02 0.59 ± ± 83524 (83524) 2001 ST148 2.98 0.10 8.28 332.6 4.2 4.5w3.26 2.27 0.79 14.23 0.15 4 83 18.62 0.04 0.58 ± ± 84852 (84852) 2003 AN38 3.06 0.08 9.79 144.5 71.1 7.6w2.95 1.96 1.40 14.97 0.21 2 34 18.96 0.05 0.52 ± ± 88367 (88367) 2001 PY8 2.39 0.12 5.80 296.4 32.5 2.0 2.63 1.66 5.01 15.50 0.28 2 20 18.97 0.06 0.55 ± ± 90232 (90232) 2003 BD27 3.12 0.19 17.62 155.4 334.3 7.8w2.55 1.57 2.69 14.59 0.29 4 78 17.87 0.02 0.83 ± ± 90997 (90997) 1998 BC 2.49 0.04 8.05 329.1 263.4 3.8w2.47 1.48 0.64 15.04 0.27 4 88 17.89 0.02 0.93 ± ± 93114 (93114) 2000 SE58 2.52 0.07 1.95 189.0 246.0 2.9 2.45 1.47 2.74 15.70 0.26 4 66 18.75 0.04 0.84 ##LTLine## ##LTLine## ##LTLine##

text: Table

text: Obj

text: ID

text: Designation

text: a

text: e

text: i

text: Ω

text: ω

text: D

text: r

text: α

text: HR

text: n

text: m

text: PTFR

text: m

text: P-L

text: Alwoods

text: BC

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: – 95 – Table 4—Continued (cid:52) (cid:52) Obj ID Designation a e i Ω ω D r α HR n m PTFR m Note. — ∆m is the lightcurve variation over our four-night observing- time span. Also, see note and footnotes associated with Table 2 for nomenclature and explanation. ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Table

text: Obj

text: ID

text: Designation

text: a

text: e

text: i

text: Ω

text: ω

text: D

text: r

text: α

text: HR

text: n

text: m

text: PTFR

text: m

text: Note.

text: ∆m

text: is

text: the

text: lightcurve

text: variation

text: over

text: our

text: four-night

text: observing-

text: time

text: span.

text: Also,

text: see

text: note

text: and

text: footnotes

text: associated

text: with

text: Table

text: for

text: nomenclature

text: and

text: explanation.

text: ##LTLine##

font: default

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: I. INTRODUCTION A foamy structure of space-time on short time and distance scales has been proposed in quantum gravity [1–3]. The non-trivial space-time can lead to the violation of Lorentz invariance at the Planck scale. Probing the Lorentz invariance violation (LIV) eﬀects provides a useful way to test the validity of quantum gravity theories, such as loop quantum gravity [4, 5], string theory [6, 7] and double special relativity [8]. According to the LIV eﬀect, massless particles have energy-dependent velocities. Hence, the velocity of a photon propagating in vacuum may have a tiny deviation from the trivial value, c. The delay of arrival time induced by the LIV eﬀect is a monotonically increasing function of the photon energy and the distance of source. We need plenty of distant energetic photons in order to observe the LIV eﬀect. As the most energetic explosions in the Universe, gamma-ray bursts (GRBs) can be detectable up to distances as far as tens of Gpc away from us. It has been proposed that GRBs provide an eﬀective way to probe the LIV eﬀect because of their cosmological distance and rapid emissions of energetic photons [9, 10]. In fact, GRBs have already been widely used to constrain the LIV eﬀect [9–29]. However, since we have little knowledge of the emission mechanism of GRBs, we cannot distinguish the LIV-induced time delay from the intrinsic time delay. Ellis et al. [16] performed a linear regression analysis of GRBs with measured redshift. The data were ﬁtted by a straight line with a slope corresponding to the quantum gravity scale and the intercept representing the possible intrinsic time delay inherited from the sources. They found a strong correlation between the parameters characterizing an intrinsic time delay and a distance-dependent propagation eﬀect. Their work based on the assumption that all GRBs had the same intrinsic emission mechanism and the intrinsic time delay. Since the durations of GRBs span about 6 orders of magnitude, there is no persuasive reason to believe that two high energy photons from two diﬀerent GRBs (or two photons with diﬀerent energy from the same GRB) have the same intrinsic time delay relative to the trigger time of low energy photons. As an improvement, Zhang and Ma [21] ﬁtted the data with straight lines of the same slope but diﬀerent intercepts. However, photons on the same line must have the same intrinsic time delay. This is not always true, because photons on the same line may come from diﬀerent GRBs and have very diﬀerent energy. Chang et al. [11] used the magnetic jet model to estimate the intrinsic time delay between emissions of low and high energy photons. Unfortunately, the magnetic jet model depends on some unobservable parameters, and thus introduces large uncertainties. Vasileiou et al. [17] adopted maximum-likelihood analysis to test quantum gravity model in which photons had normally distributed velocities. They used the 2

text: I.

label: section

text: INTRODUCTION

text: A

text: foamy

text: structure

text: of

text: space-time

text: on

text: short

text: time

text: and

text: distance

text: scales

text: has

text: been

text: proposed

text: in

text: quantum

text: gravity

text: The

text: non-trivial

text: space-time

text: can

text: lead

text: to

text: the

text: violation

text: of

text: Lorentz

text: invariance

text: at

text: the

text: Planck

text: scale.

text: Probing

text: the

text: Lorentz

text: invariance

text: violation

text: (LIV)

text: eﬀects

text: provides

text: a

text: useful

text: way

text: to

text: test

text: the

text: validity

text: of

text: quantum

text: gravity

text: theories,

text: such

text: as

text: loop

text: quantum

text: gravity

text: string

text: theory

text: and

text: double

text: special

text: relativity

text: According

text: to

text: the

text: LIV

text: eﬀect,

text: massless

text: particles

text: have

text: energy-dependent

text: velocities.

text: Hence,

text: the

text: velocity

text: of

text: a

text: photon

text: propagating

text: in

text: vacuum

text: may

text: have

text: a

text: tiny

text: deviation

text: from

text: the

text: trivial

text: value,

text: c.

text: The

text: delay

text: of

text: arrival

text: time

text: induced

text: by

text: the

text: LIV

text: eﬀect

text: is

text: a

text: monotonically

text: increasing

text: function

text: of

text: the

text: photon

text: energy

text: and

text: the

text: distance

text: of

text: source.

text: We

text: need

text: plenty

text: of

text: distant

text: energetic

text: photons

text: in

text: order

text: to

text: observe

text: the

text: LIV

text: eﬀect.

text: As

text: the

text: most

text: energetic

text: explosions

text: in

text: the

text: Universe,

text: gamma-ray

text: bursts

text: (GRBs)

text: can

text: be

text: detectable

text: up

text: to

text: distances

text: as

text: far

text: as

text: tens

text: of

text: Gpc

text: away

text: from

text: us.

text: It

text: has

text: been

text: proposed

text: that

text: GRBs

text: provide

text: an

text: eﬀective

text: way

text: to

text: probe

text: the

text: LIV

text: eﬀect

text: because

text: of

text: their

text: cosmological

text: distance

text: and

text: rapid

text: emissions

text: of

text: energetic

text: photons

text: In

text: fact,

text: GRBs

text: have

text: already

text: been

text: widely

text: used

text: to

text: constrain

text: the

text: LIV

text: eﬀect

text: However,

text: since

text: we

text: have

text: little

text: knowledge

text: of

text: the

text: emission

text: mechanism

text: of

text: GRBs,

text: we

text: cannot

text: distinguish

text: the

text: LIV-induced

text: time

text: delay

text: from

text: the

text: intrinsic

text: time

text: delay.

text: Ellis

text: et

text: al.

text: performed

text: a

text: linear

text: regression

text: analysis

text: of

text: GRBs

text: with

text: measured

text: redshift.

text: The

text: data

text: were

text: ﬁtted

text: by

text: a

text: straight

text: line

text: with

text: a

text: slope

text: corresponding

text: to

text: the

text: quantum

text: gravity

text: scale

text: and

text: the

text: intercept

text: representing

text: the

text: possible

text: intrinsic

text: time

text: delay

text: inherited

text: from

text: the

text: sources.

text: They

text: found

text: a

text: strong

text: correlation

text: between

text: the

text: parameters

text: characterizing

text: an

text: intrinsic

text: time

text: delay

text: and

text: a

text: distance-dependent

text: propagation

text: eﬀect.

text: Their

text: work

text: based

text: on

text: the

text: assumption

text: that

text: all

text: GRBs

text: had

text: the

text: same

text: intrinsic

text: emission

text: mechanism

text: and

text: the

text: intrinsic

text: time

text: delay.

text: Since

text: the

text: durations

text: of

text: GRBs

text: span

text: about

text: orders

text: of

text: magnitude,

text: there

text: is

text: no

text: persuasive

text: reason

text: to

text: believe

text: that

text: two

text: high

text: energy

text: photons

text: from

text: two

text: diﬀerent

text: GRBs

text: (or

text: two

text: photons

text: with

text: diﬀerent

text: energy

text: from

text: the

text: same

text: GRB)

text: have

text: the

text: same

text: intrinsic

text: time

text: delay

text: relative

text: to

text: the

text: trigger

text: time

text: of

text: low

text: energy

text: photons.

text: As

text: an

text: improvement,

text: Zhang

text: and

text: Ma

text: ﬁtted

text: the

text: data

text: with

text: straight

text: lines

text: of

text: the

text: same

text: slope

text: but

text: diﬀerent

text: intercepts.

text: However,

text: photons

text: on

text: the

text: same

text: line

text: must

text: have

text: the

text: same

text: intrinsic

text: time

text: delay.

text: This

text: is

text: not

text: always

text: true,

text: because

text: photons

text: on

text: the

text: same

text: line

text: may

text: come

text: from

text: diﬀerent

text: GRBs

text: and

text: have

text: very

text: diﬀerent

text: energy.

text: Chang

text: et

text: al.

text: used

text: the

text: magnetic

text: jet

text: model

text: to

text: estimate

text: the

text: intrinsic

text: time

text: delay

text: between

text: emissions

text: of

text: low

text: and

text: high

text: energy

text: photons.

text: Unfortunately,

text: the

text: magnetic

text: jet

text: model

text: depends

text: on

text: some

text: unobservable

text: parameters,

text: and

text: thus

text: introduces

text: large

text: uncertainties.

text: Vasileiou

text: et

text: al.

text: adopted

text: maximum-likelihood

text: analysis

text: to

text: test

text: quantum

text: gravity

text: model

text: in

text: which

text: photons

text: had

text: normally

text: distributed

text: velocities.

text: They

text: used

text: the


text: of the dispersion relation, i.e., the n = 2 and n = 3 cases in Eq.(2). We get a conservative limit × × M > 3.90 105 GeV for the quadratic correction, and M > 3.48 102 GeV for the cubic QG QG correction. The constraint on the LIV eﬀect in the quadratic and cubic cases is much looser than in the linear case. IV. SUMMARY In summary, we have used 20 short GRBs with redshift measurements from the Swift data archive to constrain the possible LIV eﬀect that was predicted by some quantum gravity theories. Based on the fact that the LIV-induced time delay should not be longer than the duration of a GRB, ∼ we derived the lower limit of quantum gravity energy scale at the order of M 1014 GeV. The QG × strictest limit was given by GRB 140622A, i.e., M > 5.05 1014 GeV. Our constraint on M , QG QG although not as tight as previous results, is safer and more reliable than previous studies. GRBs with higher redshift and shorter duration give a tighter constraint on M . Future observations QG at higher and broader energy bands will also tighten the constraint. We are grateful to D. Zhao for useful discussions. This work has been supported by the Na- tional Natural Science Foundation of China under grant Nos. 11375203, 11305181, 11322545 and 11335012, and by Knowledge Innovation Program of The Chinese Academy of Sciences. ∗ Corresponding author at: sangyu@ihep.ac.cn [1] G. Amelino-Camelia, J. Ellis, N. E. Mavromatos, and D. V. Nanopoulos, Int. J. Mod. Phys. A 12, 607 (1997). [2] J. Ellis, N. E. Mavromatos, and D. V. Nanopoulos, Phys. Lett. B 665, 412 (2008). [3] J. Ellis, N. E. Mavromatos, and D. V. Nanopoulos, Int. J. Mod. Phys. A 26, 2243 (2011). [4] R. Gambini and J. Pullin, Phys. Rev. D 59, 124021 (1999). [5] J. Alfaro, H. A. Morales-T´ecotl, and L. F. Urrutia, Phys. Rev. D 65, 103509 (2002). [6] J. Ellis, N. Mavromatos, and D. Nanopoulos, Phys. Lett. B 293, 37 (1992). [7] J. Ellis, N. Mavromatos, and D. Nanopoulos, Chaos, Solitons & Fractals 10, 345 (1999). [8] G. Amelino-Camelia, Int. J. Mod. Phys. D 11, 35 (2002). [9] G. Amelino-Camelia, J. Ellis, N. Mavromatos, D. V. Nanopoulos, and S. Sarkar, Nature 393, 763 (1998). [10] J. Ellis and N. E. Mavromatos, Astropart. Phys. 43, 50 (2013). [11] Z. Chang, Y. Jiang, and H.-N. Lin, Astropart. Phys. 36, 47 (2012). [12] L. Shao, Z. Xiao, and B.-Q. Ma, Astropart. Phys. 33, 312 (2010). [13] S. Zhang and B.-Q. Ma, Astropart. Phys. 61, 108 (2014). 9 ##LTLine## ##LTLine## ##LTLine##

text: of

text: the

text: dispersion

text: relation,

text: i.e.,

text: the

text: n

text: and

text: n

text: cases

text: in

text: We

text: get

text: a

text: conservative

text: limit

text: M

text: GeV

text: for

text: the

text: quadratic

text: correction,

text: and

text: M

text: GeV

text: for

text: the

text: cubic

text: QG

text: QG

text: correction.

text: The

text: constraint

text: on

text: the

text: LIV

text: eﬀect

text: in

text: the

text: quadratic

text: and

text: cubic

text: cases

text: is

text: much

text: looser

text: than

text: in

text: the

text: linear

text: case.

text: IV.

text: SUMMARY

text: In

text: summary,

text: we

text: have

text: used

text: short

text: GRBs

text: with

text: redshift

text: measurements

text: from

text: the

text: Swift

text: data

text: archive

text: to

text: constrain

text: the

text: possible

text: LIV

text: eﬀect

text: that

text: was

text: predicted

text: by

text: some

text: quantum

text: gravity

text: theories.

text: Based

text: on

text: the

text: fact

text: that

text: the

text: LIV-induced

text: time

text: delay

text: should

text: not

text: be

text: longer

text: than

text: the

text: duration

text: of

text: a

text: GRB,

text: we

text: derived

text: the

text: lower

text: limit

text: of

text: quantum

text: gravity

text: energy

text: scale

text: at

text: the

text: order

text: of

text: M

text: GeV.

text: The

text: QG

text: strictest

text: limit

text: was

text: given

text: by

text: GRB

text: i.e.,

text: M

text: GeV.

text: Our

text: constraint

text: on

text: M

text: QG

text: QG

text: although

text: not

text: as

text: tight

text: as

text: previous

text: results,

text: is

text: safer

text: and

text: more

text: reliable

text: than

text: previous

text: studies.

text: GRBs

text: with

text: higher

text: redshift

text: and

text: shorter

text: duration

text: give

text: a

text: tighter

text: constraint

text: on

text: M

text: Future

text: observations

text: QG

text: at

text: higher

text: and

text: broader

text: energy

text: bands

text: will

text: also

text: tighten

text: the

text: constraint.

text: We

text: are

text: grateful

text: to

text: D.

text: Zhao

text: for

text: useful

text: discussions.

text: This

text: work

text: has

text: been

text: supported

text: by

text: the

text: Na-

text: tional

text: Natural

text: Science

text: Foundation

text: of

text: China

text: under

text: grant

text: Nos.

text: and

text: and

text: by

text: Knowledge

text: Innovation

text: Program

text: of

text: The

text: Chinese

text: Academy

text: of

text: Sciences.

text: Corresponding

text: author

text: at:

text: sangyu@ihep.ac.cn

text: G.

text: Amelino-Camelia,

text: J.

text: Ellis,

text: N.

text: E.

text: Mavromatos,

text: and

text: D.

text: V.

text: Nanopoulos,

text: Int.

text: J.

text: Mod.

text: Phys.

text: A

text: J.

text: Ellis,

text: N.

text: E.

text: Mavromatos,

text: and

text: D.

text: V.

text: Nanopoulos,

text: Phys.

text: Lett.

text: B

text: J.

text: Ellis,

text: N.

text: E.

text: Mavromatos,

text: and

text: D.

text: V.

text: Nanopoulos,

text: Int.

text: J.

text: Mod.

text: Phys.

text: A

text: R.

text: Gambini

text: and

text: J.

text: Pullin,

text: Phys.

text: Rev.

text: D

text: J.

text: Alfaro,

text: H.

text: A.

text: Morales-T´ecotl,

text: and

text: L.

text: F.

text: Urrutia,

text: Phys.

text: Rev.

text: D

text: J.

text: Ellis,

text: N.

text: Mavromatos,

text: and

text: D.

text: Nanopoulos,

text: Phys.

text: Lett.

text: B

text: J.

text: Ellis,

text: N.

text: Mavromatos,

text: and

text: D.

text: Nanopoulos,

text: Chaos,

text: Solitons

text: Fractals

text: G.

text: Amelino-Camelia,

text: Int.

text: J.

text: Mod.

text: Phys.

text: D

text: G.

text: Amelino-Camelia,

text: J.

text: Ellis,

text: N.

text: Mavromatos,

text: D.

text: V.

text: Nanopoulos,

text: and

text: S.

text: Sarkar,

text: Nature

text: J.

text: Ellis

text: and

text: N.

text: E.

text: Mavromatos,

text: Astropart.

text: Phys.

text: Z.

text: Chang,

text: Y.

text: Jiang,

text: and

text: H.-N.

text: Lin,

text: Astropart.

text: Phys.

text: L.

text: Shao,

text: Z.

text: Xiao,

text: and

text: B.-Q.

text: Ma,

text: Astropart.

text: Phys.

text: S.

text: Zhang

text: and

text: B.-Q.

text: Ma,

text: Astropart.

text: Phys.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 14 3. Intermediate (ISCO) limit be agnostic to both, and simply name the Killing ﬁelds H , H± and W , demanding only the SL(2, R) commu- 0 0 ∓ In order to avoid these diﬃculties one could instead tation relations [H0, H±] = H±, [H+, H−] = 2H0 and deﬁne an alternate limit by keeping (A2) but replacing the U(1) generator W0 commuting with everything. (A3) with For any complex number {h and in}teger m,≥an inﬁnite- dimensional representation ψ with k 0 may be h,m,k − constructed as follows. The member ψ should satisfy r r − t − t h,m,0 x˜ = + δ 2/3, t˜= δ2/3, φ˜ = φ , (A7) the highest-weight condition, r 2M 2M + L which will keep the ISCO radius and frequency ﬁnite. LH+ψh,m,0 = 0 (A9a) In this limit t(cid:40)he metric becomes (cid:41) ψ = hψ, (A9b) H0 h,m,0 L ds2 = 2M2Γ − x˜2dt˜2 + dx˜2 + dθ2 + Λ2[dφ˜ + x˜dt˜]2 , together with W0ψh,m,0 = imψ. The remaining mem- x˜2 bers of the representation are formed by repeated appli- cation of H−, (A8) L where Γ and Λ are given below (A4).(cid:29)This metric ψh,m,k = ( H−)kψh,m,0. (A10) agrees with the near metric (A4) when x¯ (cid:15)¯ and with (cid:28) the far metric (extremal Kerr) when x 1. It too can L be derived the pedestrian way, that is to say, Eq. (A8) Here is the Lie derivative. Since SL(2, R) is not may also be obtaine(cid:28)d by (cid:28)using the coordinates (A3) with compact, this tower does not terminate and the repre- δ = 1 and using (cid:15) x 1 in the metric components, sentation is inﬁnite-dimensional. keeping to leading order in each component. The ISCO Solutions to the wave equation may be organized in limit is thus intermediate between the near and far re- representations of the isometry group. For simplicity, we gions, corresponding to the middle region in the throats work in the case of a scalar ﬁeld Φ and drop circumﬂexes diagram, Fig. 2. in order to be agnostic between the coordinates (A4) and Eq. (A8) is a non-singular spacetime; in fact it is diﬀeo- (A8). Adopting the decomposition morphic to (A4). Eq. (A8) is generally called “NHEK” or“Poincare NHEK” and it is the form originally discov- ered in [5] as a limit of precisely extremal Kerr. This met- Φ = Q(x, t)S(θ)eimφ, (A11) ric also approximates the near-horizon region of precisely extremal black hole in the pedestrian sense. Poincare with S satisfying the angular equation (22), the wave NHEK and near-NHEK cover diﬀerent patches of the maximally extended spacetime. equ(cid:2)ation implies: (cid:3) Despite its adaptation to the ISCO, the limit (A7) does not appear to be useful in calculating the radiation from L L − − L L − a particle orbiting there. Since the metric does not agree H0( H0 1) H− H+ Φ = (K 2m2)Φ. (A12) with near-extreme Kerr at the horizon or at inﬁnity, it cannot be used to impose boundary conditions at either The operator on the LHS is the Casimir of SL(2, R); − place. We have found it more useful to include the ISCO thus Φ is an eigenstate of the Casimir with eigenvalue K region in our “near” region in the m(cid:28)ain body for the pur- 2m2. If Φ is also highest-weight (A9) then this becomes poses of calculation, deﬁned as x 1 without deciding ∼ ∼ between x (cid:15) (the near-horizon region) and x (cid:15)2/3 − − (the intermediate region). Note that the ISCO(cid:29)is not in h(h 1) + 2m2 K = 0, (A13) our region of overlap, since that region has x (cid:15)2/3 (as (cid:28) well as x 1). Correspondingly, the wave equation as- sociated with the ISCO limit, that is the NHEK wave which is solved by (cid:114) equation, does not appear explicitly in our calculation. Note however the connection to the extremal calculation of [2], discussed in the Introduction and in Section , which 1 ± − 1 h = K 2m2 + . (A14) involved solving explicitly the NHEK wave equation. 2 4 4. Symmetry Group and Conformal Weights (In the text we deﬁne h to be the plus branch of (A14). × − The NHEK spacetime has an enhanced SL(2, R) U(1) The minus branch appears as 1 h in (e.g.) (33).) The isometry group. The explicit form of the Killing ﬁelds de- highest-weight condition thus replaces the second-order pends on the coordinates and the choice of basis. We will diﬀerential equation (A12) with two ﬁrst-order equations ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Intermediate

text: (ISCO)

text: limit

text: be

text: agnostic

text: to

text: both,

text: and

text: simply

text: name

text: the

text: Killing

text: ﬁelds

text: H

text: H±

text: and

text: W

text: demanding

text: only

text: the

text: R)

text: commu-

text: In

text: order

text: to

text: avoid

text: these

text: diﬃculties

text: one

text: could

text: instead

text: tation

text: relations

text: H±]

text: H±,

text: [H+,

text: H−]

text: and

text: deﬁne

text: an

text: alternate

text: limit

text: by

text: keeping

text: but

text: replacing

text: the

text: generator

text: commuting

text: with

text: everything.

text: with

text: For

text: any

text: complex

text: number

text: {h

text: and

text: in}teger

text: m,≥an

text: inﬁnite-

text: dimensional

text: representation

text: ψ

text: with

text: k

text: may

text: be

text: h,m,k

label: equation

text: constructed

text: as

text: follows.

text: The

text: member

text: ψ

text: should

text: satisfy

text: r

text: r

text: t

text: t

text: x˜

text: δ

text: t˜=

text: φ˜

text: φ

text: the

text: highest-weight

text: condition,

text: r

text: L

text: which

text: will

text: keep

text: the

text: ISCO

text: radius

text: and

text: frequency

text: ﬁnite.

text: In

text: this

text: limit

text: metric

text: becomes

text: ψ

text: hψ,

text: L

text: Λ2[dφ˜

text: together

text: with

text: imψ.

text: The

text: remaining

text: mem-

text: bers

text: of

text: the

text: representation

text: are

text: formed

text: by

text: repeated

text: appli-

text: cation

text: of

text: H−,

text: L

text: where

text: Γ

text: and

text: Λ

text: are

text: given

text: below

text: metric

text: ψh,m,k

text: agrees

text: with

text: the

text: near

text: metric

text: when

text: x¯

text: and

text: with

text: the

text: far

text: metric

text: (extremal

text: Kerr)

text: when

text: x

text: It

text: too

text: can

text: L

text: be

text: derived

text: the

text: pedestrian

text: way,

text: that

text: is

text: to

text: say,

text: Eq.

text: Here

text: is

text: the

text: Lie

text: derivative.

text: Since

text: R)

text: is

text: not

text: may

text: also

text: be

text: by

text: the

text: coordinates

text: with

text: compact,

text: this

text: tower

text: does

text: not

text: terminate

text: and

text: the

text: repre-

text: δ

text: and

text: using

text: x

text: in

text: the

text: metric

text: components,

text: sentation

text: is

text: inﬁnite-dimensional.

text: keeping

text: to

text: leading

text: order

text: in

text: each

text: component.

text: The

text: ISCO

text: Solutions

text: to

text: the

text: wave

text: equation

text: may

text: be

text: organized

text: in

text: limit

text: is

text: thus

text: intermediate

text: between

text: the

text: near

text: and

text: far

text: re-

text: representations

text: of

text: the

text: isometry

text: group.

text: For

text: simplicity,

text: we

text: gions,

text: corresponding

text: to

text: the

text: middle

text: region

text: in

text: the

text: throats

text: work

text: in

text: the

text: case

text: of

text: a

text: scalar

text: ﬁeld

text: Φ

text: and

text: drop

text: circumﬂexes

text: diagram,

text: Fig.

text: in

text: order

text: to

text: be

text: agnostic

text: between

text: the

text: coordinates

text: and

text: Eq.

text: is

text: a

text: non-singular

text: spacetime;

text: in

text: fact

text: it

text: is

text: diﬀeo-

text: Adopting

text: the

text: decomposition

text: morphic

text: to

text: Eq.

text: is

text: generally

text: called

text: “NHEK”

text: or“Poincare

text: NHEK”

text: and

text: it

text: is

text: the

text: form

text: originally

text: discov-

text: ered

text: in

text: as

text: a

text: limit

text: of

text: precisely

text: extremal

text: Kerr.

text: This

text: met-

text: Φ

text: Q(x,

text: t)S(θ)eimφ,

text: ric

text: also

text: approximates

text: the

text: near-horizon

text: region

text: of

text: precisely

text: extremal

text: black

text: hole

text: in

text: the

text: pedestrian

text: sense.

text: Poincare

text: with

text: S

text: satisfying

text: the

text: angular

text: equation

text: the

text: wave

text: NHEK

text: and

text: near-NHEK

text: cover

text: diﬀerent

text: patches

text: of

text: the

text: maximally

text: extended

text: spacetime.

text: implies:

text: Despite

text: its

text: adaptation

text: to

text: the

text: ISCO,

text: the

text: limit

text: does

text: not

text: appear

text: to

text: be

text: useful

text: in

text: calculating

text: the

text: radiation

text: from

text: L

text: L

text: L

text: L

text: a

text: particle

text: orbiting

text: there.

text: Since

text: the

text: metric

text: does

text: not

text: agree

text: H−

text: H+

text: Φ

text: (K

text: with

text: near-extreme

text: Kerr

text: at

text: the

text: horizon

text: or

text: at

text: inﬁnity,

text: it

text: cannot

text: be

text: used

text: to

text: impose

text: boundary

text: conditions

text: at

text: either

text: The

text: operator

text: on

text: the

text: LHS

text: is

text: the

text: Casimir

text: of

text: R);

text: place.

text: We

text: have

text: found

text: it

text: more

text: useful

text: to

text: include

text: the

text: ISCO

text: thus

text: Φ

text: is

text: an

text: eigenstate

text: of

text: the

text: Casimir

text: with

text: eigenvalue

text: K

text: region

text: in

text: our

text: “near”

text: region

text: in

text: the

text: body

text: for

text: the

text: pur-

text: If

text: Φ

text: is

text: also

text: highest-weight

text: then

text: this

text: becomes

text: poses

text: of

text: calculation,

text: deﬁned

text: as

text: x

text: without

text: deciding

text: between

text: x

text: (the

text: near-horizon

text: region)

text: and

text: x

text: (the

text: intermediate

text: region).

text: Note

text: that

text: the

text: not

text: in

text: h(h

text: K

text: our

text: region

text: of

text: overlap,

text: since

text: that

text: region

text: has

text: x

text: (as

text: well

text: as

text: x

text: Correspondingly,

text: the

text: wave

text: equation

text: as-

text: sociated

text: with

text: the

text: ISCO

text: limit,

text: that

text: is

text: the

text: NHEK

text: wave

text: which

text: is

text: solved

text: by

text: equation,

text: does

text: not

text: appear

text: explicitly

text: in

text: our

text: calculation.

text: Note

text: however

text: the

text: connection

text: to

text: the

text: extremal

text: calculation

text: of

text: discussed

text: in

text: the

text: Introduction

text: and

text: in

text: Section

text: which

text: h

text: K

text: involved

text: solving

text: explicitly

text: the

text: NHEK

text: wave

text: equation.

text: Symmetry

text: Group

text: and

text: Conformal

text: Weights

text: (In

text: the

text: text

text: we

text: deﬁne

text: h

text: to

text: be

text: the

text: plus

text: branch

text: of

text: The

text: NHEK

text: spacetime

text: has

text: an

text: enhanced

text: R)

text: The

text: minus

text: branch

text: appears

text: as

text: h

text: in

text: (e.g.)

text: The

text: isometry

text: group.

text: The

text: explicit

text: form

text: of

text: the

text: Killing

text: ﬁelds

text: de-

text: highest-weight

text: condition

text: thus

text: replaces

text: the

text: second-order

text: pends

text: on

text: the

text: coordinates

text: and

text: the

text: choice

text: of

text: basis.

text: We

text: will

text: diﬀerential

text: equation

text: with

text: two

text: ﬁrst-order

text: equations

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 5 Eq. (31) is the “near” equation and its solutions will (cid:28) which is independent of the frequency Ω and hence carry the label “near”. The equations agree when x (cid:28) 0 independent of (cid:15). To leading order Eq. (16) becomes x 1, becoming (cid:48)(cid:48) (cid:48) − σ = 2(cid:15) (23) x2R + 2xR + [2m2 K]R = 0. (32) −3 − n = mx (cid:15) 1, (24) 4 0 This is the “region of overlap” and the solutions are where we have used Eq. (6) to get the second relation. ∼ − − − For the ISCO we see that n diverges as n (cid:15) 1/3. In Roverlap = Pxh 1 + Qx h (33) App. n is related to the frequency conjugate to the time of the near-horizon metric. The radial equation (14) becomes for constants P and Q, where h is given in (3). This → region corresponds to the x 0 behavior of solutions of → ∞ (cid:48)(cid:48) (cid:48) − the far equation (30) and the x behavior of solu- x(x + 2(cid:15))R + 2(x + (cid:15))R + Vˆ R = Nx δ(x x ), (25) 0 0 tions to the near equation (31). Thus each solution of (30) or (31) is characterized by values of P and Q ob- tained by looking at the appropriate asymptotic region. where we introduce A pair of solutions approximates a single smooth solution √ to Eq. (25) (and hence (14)) when the solutions have the − 3 q same P and Q. N = 2 M S(cid:96)m(π/2) (26) C. Far Solutions The far equation (30) is a conﬂuent hypergeometric and equation and its solutions can be written in a number of equivalent ways. We parameterize the general solution − (1mx(x + 2) n(cid:15))2 − by P and Q, Vˆ = 2 + m2 K. (27) x(x + 2(cid:15)) − − Rfar = Pxh 1e imx/2 F (h + im; 2h; imx) (34) 1 1 We can also simplify (20), (21) using (6), − − − − + Qx he imx/2 F (1 h + im; 2(1 h); imx) . 1 1 E 1 | ∞| ˙∞ = m2 C 2 (28) 8 That is, at small x we have E − 3 | | ˙ = x m2 CH 2. (29) H 16 0 → − − → Rfar Pxh 1 + Qx h, x 0. (35) E Notice that ˙ < 0, indicating that these modes are H superradiant. → − Notice that the two solutions are related by h 1 h. B. Matched Asymptotic Expansions Overview For large x the asymptotic behavior is (cid:29) For x x , Eq. (25) becomes 0 → ∞ − Rfar C eimx/2x 1+im (cid:48)(cid:48) (cid:48) − ∞ − − − → ∞ x2R + 2xR + [m2(2 + x + x2/4) K]R = 0. (30) + D e imx/2x 1 im, x . (36) ∼ (Note that x n(cid:15) by (24).) This is the “far” equation with 0 (cid:28) and its solutions will carry the label “far”. For x 1 Eq. (25) instead becomes − − − ∞ (im) h+imΓ(2h) (im)h 1+imΓ(2(1 h)) C = P + Q − (cid:20) (cid:21) Γ(h + im) Γ(1 h + im) x(x + 2(cid:15))R(cid:48)(cid:48) + 2(x + (cid:15))R(cid:48) (31) ∞ (−im)−h−imΓ(2h) (−im)h−1−imΓ(2(1 − h)) D = P − + Q − − (mx + n(cid:15))2 − − Γ(h im) Γ(1 h im) + + m2 K R = Nx δ(x x ). x(x + 2(cid:15)) 0 0 ∞ To be outgoing at inﬁnity we must have D = 0 or ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Eq.

text: is

text: the

text: “near”

text: equation

text: and

text: its

text: solutions

text: will

text: which

text: is

text: independent

text: of

text: the

text: frequency

text: Ω

text: and

text: hence

text: carry

text: the

text: label

text: “near”.

text: The

text: equations

text: agree

text: when

text: x

text: independent

text: of

text: To

text: leading

text: order

text: Eq.

text: becomes

text: x

text: becoming

text: σ

text: K]R

text: −3

text: n

text: mx

text: This

text: is

text: the

text: “region

text: of

text: overlap”

text: and

text: the

text: solutions

text: are

text: where

text: we

text: have

text: used

text: Eq.

text: to

text: get

text: the

text: second

text: relation.

text: For

text: the

text: ISCO

text: we

text: see

text: that

text: n

text: diverges

text: as

text: n

text: In

text: Roverlap

text: Pxh

text: Qx

text: h

text: App.

text: n

text: is

text: related

text: to

text: the

text: frequency

text: conjugate

text: to

text: the

text: time

text: of

text: the

text: near-horizon

text: metric.

text: The

text: radial

text: equation

text: becomes

text: for

text: constants

text: P

text: and

text: Q,

text: where

text: h

text: is

text: given

text: in

text: This

text: region

text: corresponds

text: to

text: the

text: x

text: behavior

text: of

text: solutions

text: of

text: the

text: far

text: equation

text: and

text: the

text: x

text: behavior

text: of

text: solu-

text: x(x

text: Vˆ

text: R

text: Nx

text: δ(x

text: x

text: tions

text: to

text: the

text: near

text: equation

text: Thus

text: each

text: solution

text: of

text: or

text: is

text: characterized

text: by

text: values

text: of

text: P

text: and

text: Q

text: ob-

text: tained

text: by

text: looking

text: at

text: the

text: appropriate

text: asymptotic

text: region.

text: where

text: we

text: introduce

text: A

text: pair

text: of

text: solutions

text: approximates

text: a

text: single

text: smooth

text: solution

text: to

text: Eq.

text: (and

text: hence

text: when

text: the

text: solutions

text: have

text: the

text: q

text: same

text: P

text: and

text: Q.

text: N

text: M

text: C.

text: Far

text: Solutions

text: The

text: far

text: equation

text: is

text: a

text: conﬂuent

text: hypergeometric

text: and

text: equation

text: and

text: its

text: solutions

text: can

text: be

text: written

text: in

text: a

text: number

text: of

text: equivalent

text: ways.

text: We

text: parameterize

text: the

text: general

text: solution

text: by

text: P

text: and

text: Q,

text: Vˆ

text: K.

text: x(x

text: Rfar

text: Pxh

text: F

text: (h

text: im;

text: imx)

text: We

text: can

text: also

text: simplify

text: using

text: Qx

text: he

text: F

text: h

text: im;

text: h);

text: imx)

text: E

text: C

text: That

text: is,

text: at

text: small

text: x

text: we

text: have

text: E

text: x

text: CH

text: H

text: Rfar

text: Pxh

text: Qx

text: h,

text: x

text: E

text: Notice

text: that

text: indicating

text: that

text: these

text: modes

text: are

text: H

text: superradiant.

text: Notice

text: that

text: the

text: two

text: solutions

text: are

text: related

text: by

text: h

text: h.

text: B.

text: Matched

text: Asymptotic

text: Expansions

text: Overview

text: For

text: large

text: x

text: the

text: asymptotic

text: behavior

text: is

text: For

text: x

text: x

text: Eq.

text: becomes

text: Rfar

text: C

text: x

text: K]R

text: D

text: e

text: im,

text: x

text: (Note

text: that

text: x

text: by

text: This

text: is

text: the

text: “far”

text: equation

text: with

text: and

text: its

text: solutions

text: will

text: carry

text: the

text: label

text: “far”.

text: For

text: x

text: Eq.

text: instead

text: becomes

text: (im)

text: (im)h

text: h))

text: C

text: P

text: Q

text: Γ(h

text: im)

text: h

text: im)

text: x(x

text: h))

text: D

text: P

text: Q

text: (mx

text: Γ(h

text: im)

text: h

text: im)

text: K

text: R

text: Nx

text: δ(x

text: x

text: x(x

text: To

text: be

text: outgoing

text: at

text: inﬁnity

text: we

text: must

text: have

text: D

text: or

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 7 F. Retarded Solution − − − − W (z) = lim F (µ ν + 1, µ ν + 1; c; 1 c) ν,µ c→∞ 2 1 2 2 z × − To construct the retarded solution we demand pure e z/2zν (61) ingoing at the horizon, pure outgoing at inﬁnity, and the − z proper match at the delta-function source at x = x0 in Mν,µ(z) = bl→im∞ 2F1(µ ν + 12, b; 1 + 2µ; b ) Eq. (31). This is given by × − e z/2zµ+12 . (62) Rnear(x )Rnear(x ) Rnear(x) = Nx in < up > Speciﬁcally, we h(cid:16)ave (cid:17) ret 0 x(x + 2(cid:15))W[Rnear(x), Rnear(x)] in up = αA(N1x−02h)Rinnear(x<)Runpear(x>), (55) Rnear(x ) = x−i2n x0 + 1 i(n2 −m) in 0 0 2(cid:15) × (cid:18) − (cid:19) e3im/4(3im/2) imW − (3im/2) (63) im,h 1 where x and x are the lesser and greater of x and 2 x , respect<ively. W>e have used (50) and (49) to evaluate − 2(cid:15) i(n2 −m) 0 Rnear(x ) = (3imx /2) h + 1 the Wronskian. N 0 0 x 0 G. Large-n Asymptotics × e3im/4M − (3im/2), (64) im,h 1 2 Thus far we have considered exact solutions of the near − equation (31), where (cid:15) (and hence x and n) is treated where we have again used that n(cid:15) = 3mx /4. 0 0 as ﬁnite.We now simplify further using the smallness of H. Horizon ﬂux (cid:15), which by (24) corresponds to large n. We ﬁrst simplify the expressions for A, B, α, and b. To compute the power radiated down the event horizon → For this we need the following asymptotic approximation, we need to examine the x 0 behavior of our solution and extract the coeﬃcient CH deﬁned in (18). For x < x 0 the solution is given by Γ(p + z) − → ∞ = zp q, z , (56) Γ(q + z) Nx Rnear(x) = −0 Rnear(x)Rnear(x ). (65) ret αA(1 2h) in up 0 which holds for any complex p, q, z. From (45) and (46) we have Thus the horizon coeﬃcient is (cid:0) (cid:1) − Nx A = ΓΓ((h2h− im1))(−in)−im(2(cid:15))−in/2 32im(cid:0)x0 1−h(cid:1) (57) CrHet = αA(1 −02h)CiHnRunpear(x0), (66) − Γ(1 2h) − − − B = − − ( in) im(2(cid:15)) in/2 3imx h , (58) where CH is the horizon coeﬃcient for the in solu- Γ(1 h im) 2 0 tion. Howeivner, from (41) we have simply CH = 1. Using in Eqs. (50) and (59), a (cid:18)more convenient expression is (cid:19) and from (51) and (52) we have Nx b CH = −0 BRnear(x ) + − Rnear(x ) . − ret AB(1 2h) N 0 1 b in 0 α = B(1/b 1) (59) (67) − − Γ(1 2h)2 Γ(h im)2 − b = − − − (3m2x /2)2h 1. (60) Γ(2h 1)2 Γ(1 h im)2 0 Using Eqs. (26), (57), (58), (60), (63), and (64) and simplifying, we ﬁnd (cid:18) (cid:19) (We repeat Eq. (51) for convenience.) In obtaining − − these equations we have used that n(cid:15) = 3mx0/4 from 2i (cid:18) − 2(cid:15) i(n2(cid:19)m) Eq. (24). CrHet = N 3me3im/4( in)im(2(cid:15))in/2 1 + x 0 When the radial functions are evaluated at x = x , − − − 0 × Γ(h im) b Γ(1 h im) the 2F1 hypergeometrics in Eqs. (39) and (40) reduce to M + − − W , Γ(2h) 1 b Γ(1 2h) Whittaker W and M functions via the conﬂuence iden- (68) tities: ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: F.

text: Retarded

text: Solution

text: W

text: (z)

text: lim

text: F

text: (µ

text: ν

text: µ

text: ν

text: c;

text: c)

text: ν,µ

text: c→∞

text: z

text: To

text: construct

text: the

text: retarded

text: solution

text: we

text: demand

text: pure

text: e

text: ingoing

text: at

text: the

text: horizon,

text: pure

text: outgoing

text: at

text: inﬁnity,

text: and

text: the

text: z

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text: match

text: at

text: the

text: delta-function

text: source

text: at

text: x

text: in

text: Mν,µ(z)

text: bl→im∞

text: ν

text: b;

text: b

text: Eq.

text: This

text: is

text: given

text: by

text: e

text: Rnear(x

text: )Rnear(x

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text: Nx

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text: up

text: Speciﬁcally,

text: we

text: ret

text: x(x

text: Rnear(x)]

text: in

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text: Rnear(x

text: −m)

text: in

text: imW

text: im,h

text: where

text: x

text: and

text: x

text: are

text: the

text: lesser

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text: of

text: x

text: and

text: x

text: respect<ively.

text: W>e

text: have

text: used

text: and

text: to

text: evaluate

text: −m)

text: Rnear(x

text: h

text: the

text: Wronskian.

text: N

text: x

text: G.

text: Large-n

text: Asymptotics

text: im,h

text: Thus

text: far

text: we

text: have

text: considered

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text: solutions

text: of

text: the

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text: hence

text: x

text: and

text: n)

text: is

text: treated

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text: we

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text: again

text: used

text: that

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text: ﬁnite.We

text: now

text: simplify

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text: using

text: the

text: smallness

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text: H.

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text: by

text: corresponds

text: to

text: large

text: n.

text: We

text: ﬁrst

text: simplify

text: the

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text: for

text: A,

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text: α,

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text: b.

text: To

text: compute

text: the

text: power

text: radiated

text: down

text: the

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text: this

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text: need

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text: following

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text: we

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text: the

text: x

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text: coeﬃcient

text: CH

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text: Γ(p

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text: from

text: we

text: have

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text: Eqs.

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text: have

text: Nx

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text: CH

text: −0

text: BRnear(x

text: Rnear(x

text: ret

text: N

text: b

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text: α

text: Γ(h

text: b

text: h

text: Using

text: Eqs.

text: and

text: and

text: simplifying,

text: we

text: ﬁnd

text: (We

text: repeat

text: Eq.

text: for

text: convenience.)

text: In

text: obtaining

text: these

text: equations

text: we

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text: used

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text: Eq.

text: CrHet

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text: the

text: radial

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text: Γ(h

text: im)

text: b

text: h

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text: the

text: hypergeometrics

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text: ACCEPTED BY IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING 1 Compressed Sensing of Multi-Channel EEG Signals: The Simultaneous Cosparsity and Low Rank Optimization Yipeng Liu*, Member, IEEE, Maarten De Vos, Member, IEEE, and Sabine Van Huﬀel, Fellow, IEEE Abstract Goal: This paper deals with the problems that some EEG signals have no good sparse representation and single channel processing is not computationally eﬃcient in compressed sensing of multi-channel EEG signals. Methods: An optimization model with L0 norm and Schatten-0 norm is proposed to enforce cosparsity and low rank structures in the reconstructed multi-channel EEG signals. Both convex relaxation and global consensus optimization with alternating direction method of multipliers are used to compute the optimization model. Results: The performance of multi-channel EEG signal reconstruction is improved in term of both accuracy and computational complexity. Conclusion: The proposed method is a better candidate than previous sparse signal recovery methods for compressed sensing of EEG signals. Signiﬁcance: The proposed method enables successful compressed sensing of EEG signals even when the signals have no good sparse representation. Using compressed sensing would much reduce the power consumption of wireless EEG system. Index Terms alternating direction method of multipliers (ADMM), compressed sensing, cosparse signal recovery, low rank matrix recovery, multi-channel electroencephalogram (EEG). I. Introduction Wireless body sensor networks take spatially distributed sensors to acquire physiological signals, and transmit them over wireless links to a central unit for signal processing [1]. The electroencephalogram (EEG) signal is one of the most frequently used biomedical signals. It has important applications in medical healthcare, brain computer interfacing (BCI), and so on [2]. Continuous EEG monitoring usually requires large amount of data to be sampled and transmitted, which leads to large size of batteries. The recording unit of the wireless portable EEG systems is powered with batteries, and the physical size of the batteries sets the overall device size and operational lifetime. A physically too large device would not be portable; and excessive battery power consumption would make the long time wireless recording very hard [3] [4] [5]. Compressed sensing (CS) was proposed to deal with this challenge. Rather than ﬁrst sample the analog signal at Nyquist rate and discard most in the compression, it directly acquires the digital compressed measurements at a lower sampling rate, and recovers the digital signals by nonlinear algorithms from the compressed measurements [6]. CS relies on the assumption that the signal vector x is compressed by a ∈ × random matrix Φ RM N (measurement or sampling matrix) in discrete form as [6] [7]: y = Φx, (1) This work was supported by FWO of Flemish Government: G.0108.11 (Compressed Sensing); Belgian Federal Science Policy Oﬃce: IUAP P7/19/ (DYSCO, ‘Dynamical systems, control and optimization’, 2012-2017); ERC Advanced Grant: BIOTENSORS (339804). Asterisk indicates corresponding author. *Yipeng Liu is with School of Electronic Engineering / Center for Robotics / Center for Information in BioMedicine, University of Electronic Science and Technology of China (UESTC), Xiyuan Avenue 2006, Western High-Tech Zone, Chengdu, 611731, China. Most of this work has been done when he was with ESAT-STADIUS Division / iMinds Medical IT Department, Dept. of Electrical Engineering, University of Leuven, Kasteelpark Arenberg 10, box 2446, 3001 Leuven, Belgium. (e-mail: yipengliu@uestc.edu.cn) Maarten De Vos is with Institute of Biomedical Engineering, Department of Engineering, University of Oxford, Oxford, United Kingdom. Sabine Van Huﬀel is with ESAT-STADIUS Division / iMinds Medical IT Department, Dept. of Electrical Engineering, University of Leuven, Kasteelpark Arenberg 10, box 2446, 3001 Leuven, Belgium. Copyright (c) 2014 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending an email to pubs-permissions@ieee.org.

text: ACCEPTED

text: BY

text: IEEE

text: TRANSACTIONS

text: ON

text: BIOMEDICAL

text: ENGINEERING

text: Compressed

label: title

text: Sensing

text: of

text: Multi-Channel

text: EEG

text: Signals:

text: The

text: Simultaneous

text: Cosparsity

text: and

text: Low

text: Rank

text: Optimization

text: Yipeng

label: author

text: Liu*,

text: Member,

text: IEEE,

text: Maarten

text: De

text: Vos,

text: Member,

text: IEEE,

text: and

text: Sabine

text: Van

text: Huﬀel,

text: Fellow,

text: IEEE

text: Abstract

text: Goal:

label: abstract

text: This

text: paper

text: deals

text: with

text: the

text: problems

text: that

text: some

text: EEG

text: signals

text: have

text: no

text: good

text: sparse

text: representation

text: and

text: single

text: channel

text: processing

text: is

text: not

text: computationally

text: eﬃcient

text: in

text: compressed

text: sensing

text: of

text: multi-channel

text: EEG

text: signals.

text: Methods:

text: An

text: optimization

text: model

text: with

text: norm

text: and

text: norm

text: is

text: proposed

text: to

text: enforce

text: cosparsity

text: and

text: low

text: rank

text: structures

text: in

text: the

text: reconstructed

text: multi-channel

text: EEG

text: signals.

text: Both

text: convex

text: relaxation

text: and

text: global

text: consensus

text: optimization

text: with

text: alternating

text: direction

text: method

text: of

text: multipliers

text: are

text: used

text: to

text: compute

text: the

text: optimization

text: model.

text: Results:

text: The

text: performance

text: of

text: multi-channel

text: EEG

text: signal

text: reconstruction

text: is

text: improved

text: in

text: term

text: of

text: both

text: accuracy

text: and

text: computational

text: complexity.

text: Conclusion:

text: The

text: proposed

text: method

text: is

text: a

text: better

text: candidate

text: than

text: previous

text: sparse

text: signal

text: recovery

text: methods

text: for

text: compressed

text: sensing

text: of

text: EEG

text: signals.

text: Signiﬁcance:

text: The

text: proposed

text: method

text: enables

text: successful

text: compressed

text: sensing

text: of

text: EEG

text: signals

text: even

text: when

text: the

text: signals

text: have

text: no

text: good

text: sparse

text: representation.

text: Using

text: compressed

text: sensing

text: would

text: much

text: reduce

text: the

text: power

text: consumption

text: of

text: wireless

text: EEG

text: system.

text: Index

text: Terms

text: alternating

text: direction

text: method

text: of

text: multipliers

text: (ADMM),

text: compressed

text: sensing,

text: cosparse

text: signal

text: recovery,

text: low

text: rank

text: matrix

text: recovery,

text: multi-channel

text: electroencephalogram

text: (EEG).

text: I.

text: Introduction

text: Wireless

text: body

text: sensor

text: networks

text: take

text: spatially

text: distributed

text: sensors

text: to

text: acquire

text: physiological

text: signals,

text: and

text: transmit

text: them

text: over

text: wireless

text: links

text: to

text: a

text: central

text: unit

text: for

text: signal

text: processing

text: The

text: electroencephalogram

text: (EEG)

text: signal

text: is

text: one

text: of

text: the

text: most

text: frequently

text: used

text: biomedical

text: signals.

text: It

text: has

text: important

text: applications

text: in

text: medical

text: healthcare,

text: brain

text: computer

text: interfacing

text: (BCI),

text: and

text: so

text: on

text: Continuous

text: EEG

text: monitoring

text: usually

text: requires

text: large

text: amount

text: of

text: data

text: to

text: be

text: sampled

text: and

text: transmitted,

text: which

text: leads

text: to

text: large

text: size

text: of

text: batteries.

text: The

text: recording

text: unit

text: of

text: the

text: wireless

text: portable

text: EEG

text: systems

text: is

text: powered

text: with

text: batteries,

text: and

text: the

text: physical

text: size

text: of

text: the

text: batteries

text: sets

text: the

text: overall

text: device

text: size

text: and

text: operational

text: lifetime.

text: A

text: physically

text: too

text: large

text: device

text: would

text: not

text: be

text: portable;

text: and

text: excessive

text: battery

text: power

text: consumption

text: would

text: make

text: the

text: long

text: time

text: wireless

text: recording

text: very

text: hard

text: Compressed

text: sensing

text: (CS)

text: was

text: proposed

text: to

text: deal

text: with

text: this

text: challenge.

text: Rather

text: than

text: ﬁrst

text: sample

text: the

text: analog

text: signal

text: at

text: Nyquist

text: rate

text: and

text: discard

text: most

text: in

text: the

text: compression,

text: it

text: directly

text: acquires

text: the

text: digital

text: compressed

text: measurements

text: at

text: a

text: lower

text: sampling

text: rate,

text: and

text: recovers

text: the

text: digital

text: signals

text: by

text: nonlinear

text: algorithms

text: from

text: the

text: compressed

text: measurements

text: CS

text: relies

text: on

text: the

text: assumption

text: that

text: the

text: signal

text: vector

text: x

text: is

text: compressed

text: by

text: a

text: random

text: matrix

text: Φ

text: RM

text: N

text: (measurement

text: or

text: sampling

text: matrix)

text: in

text: discrete

text: form

text: as

text: y

text: Φx,

text: This

text: work

text: was

text: supported

text: by

text: FWO

text: of

text: Flemish

text: Government:

text: (Compressed

text: Sensing);

text: Belgian

text: Federal

text: Science

text: Policy

text: Oﬃce:

text: IUAP

text: (DYSCO,

text: ‘Dynamical

text: systems,

text: control

text: and

text: optimization’,

text: ERC

text: Advanced

text: Grant:

text: BIOTENSORS

text: Asterisk

text: indicates

text: corresponding

text: author.

text: *Yipeng

text: Liu

text: is

text: with

text: School

text: of

text: Electronic

text: Engineering

text: Center

text: for

text: Robotics

text: Center

text: for

text: Information

text: in

text: BioMedicine,

text: University

text: of

text: Electronic

text: Science

text: and

text: Technology

text: of

text: China

text: (UESTC),

text: Xiyuan

text: Avenue

text: Western

text: High-Tech

text: Zone,

text: Chengdu,

text: China.

text: Most

text: of

text: this

text: work

text: has

text: been

text: done

text: when

text: he

text: was

text: with

text: ESAT-STADIUS

text: Division

text: iMinds

text: Medical

text: IT

text: Department,

text: Dept.

text: of

text: Electrical

text: Engineering,

text: University

text: of

text: Leuven,

text: Kasteelpark

text: Arenberg

text: box

text: Leuven,

text: Belgium.

text: (e-mail:

text: yipengliu@uestc.edu.cn)

text: Maarten

text: De

text: Vos

text: is

text: with

text: Institute

text: of

text: Biomedical

text: Engineering,

text: Department

text: of

text: Engineering,

text: University

text: of

text: Oxford,

text: Oxford,

text: United

text: Kingdom.

text: Sabine

text: Van

text: Huﬀel

text: is

text: with

text: ESAT-STADIUS

text: Division

text: iMinds

text: Medical

text: IT

text: Department,

text: Dept.

text: of

text: Electrical

text: Engineering,

text: University

text: of

text: Leuven,

text: Kasteelpark

text: Arenberg

text: box

text: Leuven,

text: Belgium.

text: Copyright

text: (c)

text: IEEE.

text: Personal

text: use

text: of

text: this

text: material

text: is

text: permitted.

text: However,

text: permission

text: to

text: use

text: this

text: material

text: for

text: any

text: other

text: purposes

text: must

text: be

text: obtained

text: from

text: the

text: IEEE

text: by

text: sending

text: an

text: email

text: to

text: pubs-permissions@ieee.org.


text: 11        R−2 (rc−Ra1 )2 0 0  R2 R1 rc2 − µ¯¯ = ε¯¯ = 0 −R2(R2 R−1)a2 0 (40a) (R2 R1)2a2 R12R22a˙2 − 0 0 −R2(R2 R−1)a2 (R2 R1)2a2 R12R22a˙2       0 0 0  − eγ¯¯ = −hγ¯¯ = 0 0 (R2−RR1)12Ra222−aa˙R12R22a˙2 (40b) 0 − R1R22−aa˙ 0 (R2 R1)2a2 R12R22a˙2 in a local frame of comoving coordinates, and     −  R2R−2R1 (rp rp2R1)2 − 0 − 0  µ¯¯ = ε¯¯ = 0 (−R2 R1)R2(a2 −rp2a˙2) 0 (41a) (R2 R1)2a2+R22(rp R1)2a˙2 − − 0 0 (−R2 R1)R2(a2 −rp2a˙2) (R2 R1)2a2+R22(rp R1)2a˙2       0 0 0  eγ¯¯ = −hγ¯¯ = 0 − 0 − −Ra1a˙ (R22+(RR12r−p−R21)R22ar2p+)Ra222+(rRp22−rRp(1r)p2−a˙2R1)a˙2 (41b) 0 R1a˙ (R22+R1r−p 2R2rp)a2+R22−rp(rp R1)a˙2 0 a (R2 R1)2a2+R22(rp R1)2a˙2 in the local frame of physical coordinates. It may be readily seen that setting a(t) = 1 returns us to Minkowski spacetime where the results for both comoving and physical coordinates reduce to the ordinary ﬂat spacetime Pendry cloak. D. Coordinate Transformations Finally, we can now show that the TO procedure and the resulting transformation medium is generally covariant, meaning that knowing the coordinate transformation to switch between comoving and physical coordinates, we can relate the coordinate expressions of χ by (see Appendix ) (cid:12) (cid:12) (cid:12) (cid:48) (cid:48) (cid:48) (cid:48) χα(cid:48)β(cid:48)µ ν (xp) = Lαα(cid:48)Lββ(cid:48)χαβµν(xc)Jµ µJν ν →C (42) xc (xc) (cid:48) C where Jµ µ is the Jacobian matrix of the Cco−ordinate transformation xp = (xc) and Lαα(cid:48) is the Jacobian matrix of the inverse coordinate transformation x = 1(x ). As applied to a tensor, the coordinate transformation is enacted c p by contracting the Jacobian matrices of the coordinate transformation and inverse coordinate transformation with the tensor. However, the Jacobian matrices of the coordinate transformation, by construction, refer to the coordinate basis of the tangent bundle, hence by themselves can only transform from the coordinate basis representation in one coordinate system to the coordinate basis representation in the other coordinate system. Instead, what we would really like is to transform from the local frame of comoving coordinates to the local frame of physical coordinates. As shown in Appendix , the coordinate and frame transformations may be combined such that the components of χ˜ in one local frame may be directly transformed to the other local frame (cid:12) (cid:12) (cid:48) (cid:48) (cid:48) (cid:48) χA(cid:48)B(cid:48)C D (xp) = KA(cid:48)AKB(cid:48)BMC C MD D χABCD(xc) xc→T (xc) (43) with ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: µ¯¯

text: ε¯¯

text: 0

text: eγ¯¯

text: −hγ¯¯

text: in

text: a

text: local

text: frame

text: of

text: comoving

text: coordinates,

text: and

text: (rp

text: µ¯¯

text: ε¯¯

text: 0

text: eγ¯¯

text: −hγ¯¯

text: a

text: in

text: the

text: local

text: frame

text: of

text: physical

text: coordinates.

text: It

text: may

text: be

text: readily

text: seen

text: that

text: setting

text: a(t)

text: returns

text: us

text: to

text: Minkowski

text: spacetime

text: where

text: the

text: results

text: for

text: both

text: comoving

text: and

text: physical

text: coordinates

text: reduce

text: to

text: the

text: ordinary

text: ﬂat

text: spacetime

text: Pendry

text: cloak.

text: D.

text: Coordinate

text: Transformations

text: Finally,

text: we

text: can

text: now

text: show

text: that

text: the

text: TO

text: procedure

text: and

text: the

text: resulting

text: transformation

text: medium

text: is

text: generally

text: covariant,

text: meaning

text: that

text: knowing

text: the

text: coordinate

text: transformation

text: to

text: switch

text: between

text: comoving

text: and

text: physical

text: coordinates,

text: we

text: can

text: relate

text: the

text: coordinate

text: expressions

text: of

text: χ

text: by

text: (see

text: Appendix

text: ν

text: (xp)

text: µJν

text: ν

text: →C

text: xc

text: (xc)

text: C

text: where

text: Jµ

text: µ

text: is

text: the

text: Jacobian

text: matrix

text: of

text: the

text: Cco−ordinate

text: transformation

text: xp

text: (xc)

text: and

text: is

text: the

text: Jacobian

text: matrix

text: of

text: the

text: inverse

text: coordinate

text: transformation

text: x

text: As

text: applied

text: to

text: a

text: tensor,

text: the

text: coordinate

text: transformation

text: is

text: enacted

text: c

text: p

text: by

text: contracting

text: the

text: Jacobian

text: matrices

text: of

text: the

text: coordinate

text: transformation

text: and

text: inverse

text: coordinate

text: transformation

text: with

text: the

text: tensor.

text: However,

text: the

text: Jacobian

text: matrices

text: of

text: the

text: coordinate

text: transformation,

text: by

text: construction,

text: refer

text: to

text: the

text: coordinate

text: basis

text: of

text: the

text: tangent

text: bundle,

text: hence

text: by

text: themselves

text: can

text: only

text: transform

text: from

text: the

text: coordinate

text: basis

text: representation

text: in

text: one

text: coordinate

text: system

text: to

text: the

text: coordinate

text: basis

text: representation

text: in

text: the

text: other

text: coordinate

text: system.

text: Instead,

text: what

text: we

text: would

text: really

text: like

text: is

text: to

text: transform

text: from

text: the

text: local

text: frame

text: of

text: comoving

text: coordinates

text: to

text: the

text: local

text: frame

text: of

text: physical

text: coordinates.

text: As

text: shown

text: in

text: Appendix

text: the

text: coordinate

text: and

text: frame

text: transformations

text: may

text: be

text: combined

text: such

text: that

text: the

text: components

text: of

text: χ˜

text: in

text: one

text: local

text: frame

text: may

text: be

text: directly

text: transformed

text: to

text: the

text: other

text: local

text: frame

text: D

text: (xp)

text: C

text: MD

text: D

text: χABCD(xc)

text: xc→T

text: (xc)

text: with

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 16 (cid:34)(cid:18) (cid:19) (cid:18) (cid:19) (cid:35) r2ω 2 u ± 2 1/2 r(τ) = 0 0 + r r u cτ . (A15) u c 0 u c t t t ± It is easy to verify that for purely radial geodesics, in which case ω = 0 and u = c, r(τ) = r cτ as expected. 0 r 0 Knowing r(τ), Eq. (A11) can now be integrated(cid:20) to ﬁnd the pa(cid:21)rametriza(cid:20)tion of(cid:21)ϕ ± ± − r u u2c2τ ∓ − u ϕ(τ) = tan 1 0 r t tan 1 r + ϕ . (A16) r2ω r ω 0 0 0 0 0 Choosing an initial point (t , r , π/2, ϕ ) and initial velocity with respect to τ (u , u , 0, ω ) we then have an 0 0 0 t r 0 explicit expression for the curve. Furthermore, one can readily convert this to Cartesian coordinates with x(τ) = r(τ) cos(ϕ(τ)), and y(τ) = r(τ) sin(ϕ(τ)) if, for example, one wanted to make a parametric plot with something like Mathematica. These become − x = (u cos ϕ r ω sin ϕ )τ + r cos ϕ r 0 0 0 0 0 0 (A17) y = (u sin ϕ + r ω cos ϕ )τ + r sin ϕ . r 0 0 0 0 0 0 Note that if the chosen initial point on the trajectory coincides with the point of closest approach, then r = b 0 is just the impact parameter, u = 0 because at the point of closest approach the velocity is purely tangential, r and ϕ = ϕ + π/2 where ϕ is the angle at which the trajectory crosses the x-axis (special care is needed for 0 in in trajectories with initial point at the origin). If one is unconcerned with following the wavefront and only cares about → the geometrical path, then rescaling τ r ω τ allows us to write 0 0 − x = τ cos ϕ b sin ϕ in in (A18) y = τ sin ϕ + b cos ϕ . in in 2. Null geodesics in FLRW: comoving Cartesian coordinates a. Null geodesics In comoving Cartesian coordinates the FLRW metric is (cid:0) (cid:1) − ds2 = c2dt2 + a2(t) dx2 + dy2 + dz2 . (A19) We parameterize a curve with aﬃne parameter τ, along with it’s velocity and acceleration, in the same manner as Eqs. (A1a)–(A1c). It should be borne in mind that the scale factor a(t) appearing in the metric, which is a scalar function, is diﬀerent from the acceleration vector aµ. Note that with this parameterization, the scale factor is also parametrized by τ through its dependence on t, in other words a = a(t(τ)). We will suppress the explicit dependence on t or τ except where necessary for clarity. Dot notation will denote derivatives with respect to τ and primes will denote derivatives with respect to t (cid:48) (cid:48) da da (cid:48) a = a (t) = , a˙ = = a t˙. (A20) dt dτ For a null geodesic we require − g vαvβ = c2t˙2 + a2(x˙2 + y˙2 + z˙2) = 0. (A21) αβ ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: u

text: r(τ)

text: r

text: r

text: u

text: cτ

text: u

text: c

text: u

text: c

text: t

text: t

text: t

text: It

text: is

text: easy

text: to

text: verify

text: that

text: for

text: purely

text: radial

text: geodesics,

text: in

text: which

text: case

text: ω

text: and

text: u

text: c,

text: r(τ)

text: r

text: cτ

text: as

text: expected.

text: r

text: Knowing

text: r(τ),

text: Eq.

text: can

text: now

text: be

text: to

text: ﬁnd

text: the

text: r

text: u

text: u

text: ϕ(τ)

text: tan

text: r

text: t

text: tan

text: r

text: ϕ

text: r

text: ω

text: Choosing

text: an

text: initial

text: point

text: (t

text: r

text: ϕ

text: and

text: initial

text: velocity

text: with

text: respect

text: to

text: τ

text: (u

text: u

text: ω

text: we

text: then

text: have

text: an

text: t

text: r

text: explicit

text: expression

text: for

text: the

text: curve.

text: Furthermore,

text: one

text: can

text: readily

text: convert

text: this

text: to

text: Cartesian

text: coordinates

text: with

text: x(τ)

text: r(τ)

text: cos(ϕ(τ)),

text: and

text: y(τ)

text: r(τ)

text: sin(ϕ(τ))

text: if,

text: for

text: example,

text: one

text: wanted

text: to

text: make

text: a

text: parametric

text: plot

text: with

text: something

text: like

text: Mathematica.

text: These

text: become

text: x

text: (u

text: cos

text: ϕ

text: r

text: ω

text: sin

text: ϕ

text: )τ

text: r

text: cos

text: ϕ

text: r

text: y

text: (u

text: sin

text: ϕ

text: r

text: ω

text: cos

text: ϕ

text: )τ

text: r

text: sin

text: ϕ

text: r

text: Note

text: that

text: if

text: the

text: chosen

text: initial

text: point

text: on

text: the

text: trajectory

text: coincides

text: with

text: the

text: point

text: of

text: closest

text: approach,

text: then

text: r

text: b

text: is

text: just

text: the

text: impact

text: parameter,

text: u

text: because

text: at

text: the

text: point

text: of

text: closest

text: approach

text: the

text: velocity

text: is

text: purely

text: tangential,

text: r

text: and

text: ϕ

text: ϕ

text: where

text: ϕ

text: is

text: the

text: angle

text: at

text: which

text: the

text: trajectory

text: crosses

text: the

text: x-axis

text: (special

text: care

text: is

text: needed

text: for

text: in

text: in

text: trajectories

text: with

text: initial

text: point

text: at

text: the

text: origin).

text: If

text: one

text: is

text: unconcerned

text: with

text: following

text: the

text: wavefront

text: and

text: only

text: cares

text: about

text: the

text: geometrical

text: path,

text: then

text: rescaling

text: τ

text: r

text: ω

text: τ

text: allows

text: us

text: to

text: write

text: x

text: τ

text: cos

text: ϕ

text: b

text: sin

text: ϕ

text: in

text: in

text: y

text: τ

text: sin

text: ϕ

text: b

text: cos

text: ϕ

text: in

text: in

text: geodesics

text: in

text: FLRW:

text: comoving

text: Cartesian

text: coordinates

text: a.

text: geodesics

text: In

text: comoving

text: Cartesian

text: coordinates

text: the

text: FLRW

text: metric

text: is

text: We

text: parameterize

text: a

text: curve

text: with

text: aﬃne

text: parameter

text: τ,

text: along

text: with

text: it’s

text: velocity

text: and

text: acceleration,

text: in

text: the

text: same

text: manner

text: as

text: Eqs.

text: It

text: should

text: be

text: borne

text: in

text: mind

text: that

text: the

text: scale

text: factor

text: a(t)

text: appearing

text: in

text: the

text: metric,

text: which

text: is

text: a

text: scalar

text: function,

text: is

text: diﬀerent

text: from

text: the

text: acceleration

text: vector

text: aµ.

text: Note

text: that

text: with

text: this

text: parameterization,

text: the

text: scale

text: factor

text: is

text: also

text: parametrized

text: by

text: τ

text: through

text: its

text: dependence

text: on

text: t,

text: in

text: other

text: words

text: a

text: a(t(τ)).

text: We

text: will

text: suppress

text: the

text: explicit

text: dependence

text: on

text: t

text: or

text: τ

text: except

text: where

text: necessary

text: for

text: clarity.

text: Dot

text: notation

text: will

text: denote

text: derivatives

text: with

text: respect

text: to

text: τ

text: and

text: primes

text: will

text: denote

text: derivatives

text: with

text: respect

text: to

text: t

text: da

text: da

text: a

text: a

text: (t)

text: a˙

text: a

text: t˙.

text: dt

text: dτ

text: For

text: a

text: geodesic

text: we

text: require

text: g

text: vαvβ

text: αβ

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: 17 The geodesic equation yields (cid:48) aa t¨+ (x˙2 + y˙2 + z˙2) = 0, (A22a) c2 (cid:48) 2a x¨ + t˙x˙ = 0. (A22b) a The equations for y and z are identical to that of x, thus it is suﬃcient to consider only the equations for t and x. (cid:48) Using the null condition Eq. (A21) and a t˙ = a˙, the t and x equations become a˙ t¨+ t˙ = 0, (A23a) a a˙ x¨ + 2 x˙ = 0 (A23b) a with solutions (cid:90) 1 t(τ) = u a dτ, (A24a) t 0 a (cid:90) 1 x(τ) = u a2 dτ, (A24b) x 0 a2 and similarly for y and z. The coeﬃcients (u , u , u , u ) give the direction of propagation of the geodesic at τ = 0, | t x y z i.e. vµ = (u , u , u , u ), and satisfy the null condition τ=0 t x y z − c2u2 + a2(u2 + u2 + u2) = 0. (A25) t 0 x y z b. Conformal ﬂatness It is well known that FLRW is conformally ﬂat, which means that FLRW is equivalent to a conformal rescaling of Minkowski space. In other words, transforming to “conformal time” 1 dη = dt, (A26) a(t) the metric becomes (cid:0) (cid:1) − ds2 = a(t) c2dη2 + dx2 + dy2 + dz2 . (A27) Since the null character of a tangent vector (and hence the null character of a geodesic) is invariant to conformal transformations, null geodesics of Minkowski spacetime are conformally mapped to null geodesics of FLRW spacetime. In Minkowski spacetime, a deﬁning characteristic of null geodesics is that ∆x = c ∆t for light propagating along the ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: The

text: geodesic

text: equation

text: yields

text: aa

text: t¨+

text: x¨

text: t˙x˙

text: a

text: The

text: equations

text: for

text: y

text: and

text: z

text: are

text: identical

text: to

text: that

text: of

text: x,

text: thus

text: it

text: is

text: suﬃcient

text: to

text: consider

text: only

text: the

text: equations

text: for

text: t

text: and

text: x.

text: Using

text: the

text: condition

text: Eq.

text: and

text: a

text: t˙

text: a˙,

text: the

text: t

text: and

text: x

text: equations

text: become

text: a˙

text: t¨+

text: t˙

text: a

text: a˙

text: x¨

text: x˙

text: a

text: with

text: solutions

text: t(τ)

text: u

text: a

text: dτ,

text: t

text: a

text: x(τ)

text: u

text: dτ,

text: x

text: and

text: similarly

text: for

text: y

text: and

text: z.

text: The

text: coeﬃcients

text: (u

text: u

text: u

text: u

text: give

text: the

text: direction

text: of

text: propagation

text: of

text: the

text: geodesic

text: at

text: τ

text: t

text: x

text: y

text: z

text: i.e.

text: vµ

text: (u

text: u

text: u

text: u

text: and

text: satisfy

text: the

text: condition

text: t

text: x

text: y

text: z

text: t

text: x

text: y

text: z

text: b.

text: Conformal

text: ﬂatness

text: It

text: is

text: well

text: known

text: that

text: FLRW

text: is

text: conformally

text: ﬂat,

text: which

text: means

text: that

text: FLRW

text: is

text: equivalent

text: to

text: a

text: conformal

text: rescaling

text: of

text: Minkowski

text: space.

text: In

text: other

text: words,

text: transforming

text: to

text: “conformal

text: time”

text: dη

text: dt,

text: a(t)

text: the

text: metric

text: becomes

text: a(t)

text: Since

text: the

text: character

text: of

text: a

text: tangent

text: vector

text: (and

text: hence

text: the

text: character

text: of

text: a

text: geodesic)

text: is

text: invariant

text: to

text: conformal

text: transformations,

text: geodesics

text: of

text: Minkowski

text: spacetime

text: are

text: conformally

text: mapped

text: to

text: geodesics

text: of

text: FLRW

text: spacetime.

text: In

text: Minkowski

text: spacetime,

text: a

text: deﬁning

text: characteristic

text: of

text: geodesics

text: is

text: that

text: ∆x

text: c

text: ∆t

text: for

text: light

text: propagating

text: along

text: the

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##