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4 Associations Between Glycemia and Cognitive Performance in Adults with Type 1 Diabetes (T1D) using Continuous Glucose Monitoring (CGM) and Ecological Momentary Assessment (EMA)
- Olivia H Wang, Miranda Zuniga-Kennedy, Luciana Mascarenhas Fonseca, Michael Cleveland, Zoe W. Hawks, Lanee Jung, Jane D. Bulger, Elizabeth Grinspoon, Shifali Singh, Martin Sliwinski, Alandra Verdejo, Ruth S. Weinstock, Laura Germine, Naomi Chaytor
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- Journal:
- Journal of the International Neuropsychological Society / Volume 29 / Issue s1 / November 2023
- Published online by Cambridge University Press:
- 21 December 2023, pp. 792-793
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Objective:
Despite associations between hypoglycemia and cognitive performance using cross-sectional and experimental methods (e.g., Insulin clamp studies), few studies have evaluated this relationship in a naturalistic setting. This pilot study utilizes an EMA study design in adults with T1D to examine the impact of hypoglycemia and hyperglycemia, measured using CGM, on cognitive performance, measured via ambulatory assessment.
Participants and Methods:Twenty adults with T1D (mean age 38.9 years, range 26-67; 55% female; 55% bachelor’s degree or higher; mean HbA1c = 8.3%, range 5.4% - 12.5%), were recruited from the Joslin Diabetes Center at SUNY Upstate Medical University. A blinded Dexcom G6 CGM was worn during everyday activities while completing 3-6 daily EMAs using personal smartphones. EMAs were delivered between 9 am and 9 pm, for 15 days. EMAs included 3 brief cognitive tests developed by testmybrain.org and validated for brief mobile administration (Gradual Onset CPT d-prime, Digit Symbol Matching median reaction time, Multiple Object Tracking percent accuracy) and self-reported momentary negative affect. Day-level average scores were calculated for the cognitive and negative affect measures. Hypoglycemia and hyperglycemia were defined as the percentage of time spent with a sensor glucose value <70 mg/dL or > 180 mg/dL, respectively. Daytime (8 am to 9 pm) and nighttime (9 pm to 8 am) glycemic excursions were calculated separately. Multilevel models estimated the between- and within-person association between the night prior to, or the same day, time spent in hypoglycemia or hyperglycemia and cognitive performance (each cognitive test was modeled separately). To evaluate the effect of between-person differences, person-level variables were calculated as the mean across the study and grand-mean centered. To evaluate the effect of within-person fluctuations, day-level variables were calculated as deviations from these person-level means.
Results:Within-person fluctuations in nighttime hypoglycemia were associated with daytime processing speed. Specifically, participants who spent a higher percentage of time in hypoglycemia than their average percentage the night prior to assessment performed slower than their average performance on the processing speed test (Digit Symbol Matching median reaction time, b = 94.16, p = 0.042), while same day variation in hypoglycemia was not associated with variation in Digit Symbol Matching performance. This association remained significant (b = 97.46, p = 0.037) after controlling for within-person and between-person effects of negative affect. There were no significant within-person associations between time spent in hyperglycemia and Digit Symbol Matching, nor day/night hypoglycemia or hyperglycemia and Gradual Onset CPT or Multiple Object Tracking.
Conclusions:Our findings from this EMA study suggest that when individuals with T1D experience more time in hypoglycemia at night (compared to their average), they have slower processing speed the following day, while same day hypoglycemia and hyperglycemia does not similarly impact processing speed performance. These results showcase the power of intensive longitudinal designs using ambulatory cognitive assessment to uncover novel determinants of cognitive variation in real world settings that have direct clinical applications for optimizing cognitive performance. Future research with larger samples is needed to replicate these findings.
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- By Rose Teteki Abbey, K. C. Abraham, David Tuesday Adamo, LeRoy H. Aden, Efrain Agosto, Victor Aguilan, Gillian T. W. Ahlgren, Charanjit Kaur AjitSingh, Dorothy B E A Akoto, Giuseppe Alberigo, Daniel E. Albrecht, Ruth Albrecht, Daniel O. Aleshire, Urs Altermatt, Anand Amaladass, Michael Amaladoss, James N. Amanze, Lesley G. Anderson, Thomas C. Anderson, Victor Anderson, Hope S. Antone, María Pilar Aquino, Paula Arai, Victorio Araya Guillén, S. Wesley Ariarajah, Ellen T. Armour, Brett Gregory Armstrong, Atsuhiro Asano, Naim Stifan Ateek, Mahmoud Ayoub, John Alembillah Azumah, Mercedes L. García Bachmann, Irena Backus, J. Wayne Baker, Mieke Bal, Lewis V. Baldwin, William Barbieri, António Barbosa da Silva, David Basinger, Bolaji Olukemi Bateye, Oswald Bayer, Daniel H. Bays, Rosalie Beck, Nancy Elizabeth Bedford, Guy-Thomas Bedouelle, Chorbishop Seely Beggiani, Wolfgang Behringer, Christopher M. Bellitto, Byard Bennett, Harold V. Bennett, Teresa Berger, Miguel A. Bernad, Henley Bernard, Alan E. Bernstein, Jon L. Berquist, Johannes Beutler, Ana María Bidegain, Matthew P. Binkewicz, Jennifer Bird, Joseph Blenkinsopp, Dmytro Bondarenko, Paulo Bonfatti, Riet en Pim Bons-Storm, Jessica A. Boon, Marcus J. Borg, Mark Bosco, Peter C. Bouteneff, François Bovon, William D. Bowman, Paul S. Boyer, David Brakke, Richard E. Brantley, Marcus Braybrooke, Ian Breward, Ênio José da Costa Brito, Jewel Spears Brooker, Johannes Brosseder, Nicholas Canfield Read Brown, Robert F. Brown, Pamela K. Brubaker, Walter Brueggemann, Bishop Colin O. Buchanan, Stanley M. Burgess, Amy Nelson Burnett, J. Patout Burns, David B. Burrell, David Buttrick, James P. Byrd, Lavinia Byrne, Gerado Caetano, Marcos Caldas, Alkiviadis Calivas, William J. Callahan, Salvatore Calomino, Euan K. Cameron, William S. Campbell, Marcelo Ayres Camurça, Daniel F. Caner, Paul E. Capetz, Carlos F. Cardoza-Orlandi, Patrick W. Carey, Barbara Carvill, Hal Cauthron, Subhadra Mitra Channa, Mark D. Chapman, James H. Charlesworth, Kenneth R. Chase, Chen Zemin, Luciano Chianeque, Philip Chia Phin Yin, Francisca H. Chimhanda, Daniel Chiquete, John T. Chirban, Soobin Choi, Robert Choquette, Mita Choudhury, Gerald Christianson, John Chryssavgis, Sejong Chun, Esther Chung-Kim, Charles M. A. Clark, Elizabeth A. Clark, Sathianathan Clarke, Fred Cloud, John B. Cobb, W. Owen Cole, John A Coleman, John J. Collins, Sylvia Collins-Mayo, Paul K. Conkin, Beth A. Conklin, Sean Connolly, Demetrios J. Constantelos, Michael A. Conway, Paula M. Cooey, Austin Cooper, Michael L. Cooper-White, Pamela Cooper-White, L. William Countryman, Sérgio Coutinho, Pamela Couture, Shannon Craigo-Snell, James L. Crenshaw, David Crowner, Humberto Horacio Cucchetti, Lawrence S. Cunningham, Elizabeth Mason Currier, Emmanuel Cutrone, Mary L. Daniel, David D. Daniels, Robert Darden, Rolf Darge, Isaiah Dau, Jeffry C. Davis, Jane Dawson, Valentin Dedji, John W. de Gruchy, Paul DeHart, Wendy J. Deichmann Edwards, Miguel A. De La Torre, George E. Demacopoulos, Thomas de Mayo, Leah DeVun, Beatriz de Vasconcellos Dias, Dennis C. Dickerson, John M. Dillon, Luis Miguel Donatello, Igor Dorfmann-Lazarev, Susanna Drake, Jonathan A. Draper, N. Dreher Martin, Otto Dreydoppel, Angelyn Dries, A. J. Droge, Francis X. D'Sa, Marilyn Dunn, Nicole Wilkinson Duran, Rifaat Ebied, Mark J. Edwards, William H. Edwards, Leonard H. Ehrlich, Nancy L. Eiesland, Martin Elbel, J. Harold Ellens, Stephen Ellingson, Marvin M. Ellison, Robert Ellsberg, Jean Bethke Elshtain, Eldon Jay Epp, Peter C. Erb, Tassilo Erhardt, Maria Erling, Noel Leo Erskine, Gillian R. Evans, Virginia Fabella, Michael A. Fahey, Edward Farley, Margaret A. Farley, Wendy Farley, Robert Fastiggi, Seena Fazel, Duncan S. Ferguson, Helwar Figueroa, Paul Corby Finney, Kyriaki Karidoyanes FitzGerald, Thomas E. FitzGerald, John R. Fitzmier, Marie Therese Flanagan, Sabina Flanagan, Claude Flipo, Ronald B. Flowers, Carole Fontaine, David Ford, Mary Ford, Stephanie A. Ford, Jim Forest, William Franke, Robert M. Franklin, Ruth Franzén, Edward H. Friedman, Samuel Frouisou, Lorelei F. Fuchs, Jojo M. Fung, Inger Furseth, Richard R. Gaillardetz, Brandon Gallaher, China Galland, Mark Galli, Ismael García, Tharscisse Gatwa, Jean-Marie Gaudeul, Luis María Gavilanes del Castillo, Pavel L. Gavrilyuk, Volney P. Gay, Metropolitan Athanasios Geevargis, Kondothra M. George, Mary Gerhart, Simon Gikandi, Maurice Gilbert, Michael J. Gillgannon, Verónica Giménez Beliveau, Terryl Givens, Beth Glazier-McDonald, Philip Gleason, Menghun Goh, Brian Golding, Bishop Hilario M. Gomez, Michelle A. Gonzalez, Donald K. Gorrell, Roy Gottfried, Tamara Grdzelidze, Joel B. Green, Niels Henrik Gregersen, Cristina Grenholm, Herbert Griffiths, Eric W. Gritsch, Erich S. Gruen, Christoffer H. Grundmann, Paul H. Gundani, Jon P. Gunnemann, Petre Guran, Vidar L. Haanes, Jeremiah M. Hackett, Getatchew Haile, Douglas John Hall, Nicholas Hammond, Daphne Hampson, Jehu J. Hanciles, Barry Hankins, Jennifer Haraguchi, Stanley S. Harakas, Anthony John Harding, Conrad L. Harkins, J. William Harmless, Marjory Harper, Amir Harrak, Joel F. Harrington, Mark W. Harris, Susan Ashbrook Harvey, Van A. Harvey, R. Chris Hassel, Jione Havea, Daniel Hawk, Diana L. Hayes, Leslie Hayes, Priscilla Hayner, S. Mark Heim, Simo Heininen, Richard P. Heitzenrater, Eila Helander, David Hempton, Scott H. Hendrix, Jan-Olav Henriksen, Gina Hens-Piazza, Carter Heyward, Nicholas J. Higham, David Hilliard, Norman A. Hjelm, Peter C. Hodgson, Arthur Holder, M. Jan Holton, Dwight N. Hopkins, Ronnie Po-chia Hsia, Po-Ho Huang, James Hudnut-Beumler, Jennifer S. Hughes, Leonard M. Hummel, Mary E. Hunt, Laennec Hurbon, Mark Hutchinson, Susan E. Hylen, Mary Beth Ingham, H. Larry Ingle, Dale T. Irvin, Jon Isaak, Paul John Isaak, Ada María Isasi-Díaz, Hans Raun Iversen, Margaret C. Jacob, Arthur James, Maria Jansdotter-Samuelsson, David Jasper, Werner G. Jeanrond, Renée Jeffery, David Lyle Jeffrey, Theodore W. Jennings, David H. Jensen, Robin Margaret Jensen, David Jobling, Dale A. Johnson, Elizabeth A. Johnson, Maxwell E. Johnson, Sarah Johnson, Mark D. Johnston, F. Stanley Jones, James William Jones, John R. Jones, Alissa Jones Nelson, Inge Jonsson, Jan Joosten, Elizabeth Judd, Mulambya Peggy Kabonde, Robert Kaggwa, Sylvester Kahakwa, Isaac Kalimi, Ogbu U. Kalu, Eunice Kamaara, Wayne C. Kannaday, Musimbi Kanyoro, Veli-Matti Kärkkäinen, Frank Kaufmann, Léon Nguapitshi Kayongo, Richard Kearney, Alice A. Keefe, Ralph Keen, Catherine Keller, Anthony J. Kelly, Karen Kennelly, Kathi Lynn Kern, Fergus Kerr, Edward Kessler, George Kilcourse, Heup Young Kim, Kim Sung-Hae, Kim Yong-Bock, Kim Yung Suk, Richard King, Thomas M. King, Robert M. Kingdon, Ross Kinsler, Hans G. Kippenberg, Cheryl A. 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Yee, Viktor Yelensky, Yeo Khiok-Khng, Gustav K. K. Yeung, Angela Yiu, Amos Yong, Yong Ting Jin, You Bin, Youhanna Nessim Youssef, Eliana Yunes, Robert Michael Zaller, Valarie H. Ziegler, Barbara Brown Zikmund, Joyce Ann Zimmerman, Aurora Zlotnik, Zhuo Xinping
- Edited by Daniel Patte, Vanderbilt University, Tennessee
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- The Cambridge Dictionary of Christianity
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- 05 August 2012
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- 20 September 2010, pp xi-xliv
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Vulcan petawatt: Design, operation and interactions at 5 × 1020 Wcm−2
- C.N. DANSON, P.A. BRUMMITT, R.J. CLARKE, J.L. COLLIER, B. FELL, A.J. FRACKIEWICZ, S. HAWKES, C. HERNANDEZ-GOMEZ, P. HOLLIGAN, M.H.R. HUTCHINSON, A. KIDD, W.J. LESTER, I.O. MUSGRAVE, D. NEELY, D.R. NEVILLE, P.A. NORREYS, D.A. PEPLER, C.J. REASON, W. SHAIKH, T.B. WINSTONE, R.W.W. WYATT, B.E. WYBORN
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- Laser and Particle Beams / Volume 23 / Issue 1 / March 2005
- Published online by Cambridge University Press:
- 02 June 2005, pp. 87-93
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The Vulcan Nd:glass laser at the Central Laser Facility (CLF) is a petawatt (1015 Watts) interaction facility, designed to deliver irradiance on target of 1021W.cm−2 for the UK and international user community. The facility came online to users in 2002 and considerable experience has been gained operating Vulcan in this mode. The facility delivers a wide-ranging experimental program in fundamental physics and advanced applications. This includes the interaction of ultrahigh intensity light with matter, fast ignition fusion research, photon induced nuclear reactions, electron and ion acceleration by light waves, and the exploration of the exotic world of plasma physics dominated by relativity. We report on the first year's operation of the facility and the highlights of the experimental campaigns.
Influence of a monetary incentive upon attentional modification of short-lead prepulse inhibition and long-lead prepulse facilitation of acoustic startle
- LARRY W. HAWK, JOSHUA S. REDFORD, JOSEPH S. BASCHNAGEL
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- Psychophysiology / Volume 39 / Issue 5 / September 2002
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- 20 August 2002, pp. 674-677
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- September 2002
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Short-lead prepulse inhibition and long-lead prepulse facilitation of startle are greater during attended than ignored prestimuli. The present work examined whether this attentional modification is influenced by monetary incentive. Participants (43 college students) were randomly assigned to receive a small performance-based monetary incentive or were instructed to try their best. The task was to judge the duration of tones of one of two pitches during a series of 48 tones. Prepulse inhibition of startle eyeblink EMG was assessed at 60, 120, and 240 ms, and prepulse facilitation was assessed at 4,500 ms following tone onset. Short-lead percent prepulse inhibition was greater during attended than ignored prestimuli only at 120 ms among paid participants. Long-lead prepulse facilitation was greater for attended than ignored tones, but this effect did not vary with incentive condition. This study demonstrates that attentional modification of short-lead prepulse inhibition is sensitive to a monetary incentive and provides a basis for further examination of motivational effects on early attentional processing.
Looking Backward, Looking Forward: MLA Members Speak
- April Alliston, Elizabeth Ammons, Jean Arnold, Nina Baym, Sandra L. Beckett, Peter G. Beidler, Roger A. Berger, Sandra Bermann, J.J. Wilson, Troy Boone, Alison Booth, Wayne C. Booth, James Phelan, Marie Borroff, Ihab Hassan, Ulrich Weisstein, Zack Bowen, Jill Campbell, Dan Campion, Jay Caplan, Maurice Charney, Beverly Lyon Clark, Robert A. Colby, Thomas C. Coleman III, Nicole Cooley, Richard Dellamora, Morris Dickstein, Terrell Dixon, Emory Elliott, Caryl Emerson, Ann W. Engar, Lars Engle, Kai Hammermeister, N. N. Feltes, Mary Anne Ferguson, Annie Finch, Shelley Fisher Fishkin, Jerry Aline Flieger, Norman Friedman, Rosemarie Garland-Thomson, Sandra M. Gilbert, Laurie Grobman, George Guida, Liselotte Gumpel, R. K. Gupta, Florence Howe, Cathy L. Jrade, Richard A. Kaye, Calhoun Winton, Murray Krieger, Robert Langbaum, Richard A. Lanham, Marilee Lindemann, Paul Michael Lützeler, Thomas J. Lynn, Juliet Flower MacCannell, Michelle A. Massé, Irving Massey, Georges May, Christian W. Hallstein, Gita May, Lucy McDiarmid, Ellen Messer-Davidow, Koritha Mitchell, Robin Smiles, Kenyatta Albeny, George Monteiro, Joel Myerson, Alan Nadel, Ashton Nichols, Jeffrey Nishimura, Neal Oxenhandler, David Palumbo-Liu, Vincent P. Pecora, David Porter, Nancy Potter, Ronald C. Rosbottom, Elias L. Rivers, Gerhard F. Strasser, J. L. Styan, Marianna De Marco Torgovnick, Gary Totten, David van Leer, Asha Varadharajan, Orrin N. C. Wang, Sharon Willis, Louise E. Wright, Donald A. Yates, Takayuki Yokota-Murakami, Richard E. Zeikowitz, Angelika Bammer, Dale Bauer, Karl Beckson, Betsy A. Bowen, Stacey Donohue, Sheila Emerson, Gwendolyn Audrey Foster, Jay L. Halio, Karl Kroeber, Terence Hawkes, William B. Hunter, Mary Jambus, Willard F. King, Nancy K. Miller, Jody Norton, Ann Pellegrini, S. P. Rosenbaum, Lorie Roth, Robert Scholes, Joanne Shattock, Rosemary T. VanArsdel, Alfred Bendixen, Alarma Kathleen Brown, Michael J. Kiskis, Debra A. Castillo, Rey Chow, John F. Crossen, Robert F. Fleissner, Regenia Gagnier, Nicholas Howe, M. Thomas Inge, Frank Mehring, Hyungji Park, Jahan Ramazani, Kenneth M. Roemer, Deborah D. Rogers, A. LaVonne Brown Ruoff, Regina M. Schwartz, John T. Shawcross, Brenda R. Silver, Andrew von Hendy, Virginia Wright Wexman, Britta Zangen, A. Owen Aldridge, Paula R. Backscheider, Roland Bartel, E. M. Forster, Milton Birnbaum, Jonathan Bishop, Crystal Downing, Frank H. Ellis, Roberto Forns-Broggi, James R. Giles, Mary E. Giles, Susan Blair Green, Madelyn Gutwirth, Constance B. Hieatt, Titi Adepitan, Edgar C. Knowlton, Jr., Emanuel Mussman, Sally Todd Nelson, Robert O. Preyer, David Diego Rodriguez, Guy Stern, James Thorpe, Robert J. Wilson, Rebecca S. Beal, Joyce Simutis, Betsy Bowden, Sara Cooper, Wheeler Winston Dixon, Tarek el Ariss, Richard Jewell, John W. Kronik, Wendy Martin, Stuart Y. McDougal, Hugo Méndez-Ramírez, Ivy Schweitzer, Armand E. Singer, G. Thomas Tanselle, Tom Bishop, Mary Ann Caws, Marcel Gutwirth, Christophe Ippolito, Lawrence D. Kritzman, James Longenbach, Tim McCracken, Wolfe S. Molitor, Diane Quantic, Gregory Rabassa, Ellen M. Tsagaris, Anthony C. Yu, Betty Jean Craige, Wendell V. Harris, J. Hillis Miller, Jesse G. Swan, Helene Zimmer-Loew, Peter Berek, James Chandler, Hanna K. Charney, Philip Cohen, Judith Fetterley, Herbert Lindenberger, Julia Reinhard Lupton, Maximillian E. Novak, Richard Ohmann, Marjorie Perloff, Mark Reynolds, James Sledd, Harriet Turner, Marie Umeh, Flavia Aloya, Regina Barreca, Konrad Bieber, Ellis Hanson, William J. Hyde, Holly A. Laird, David Leverenz, Allen Michie, J. Wesley Miller, Marvin Rosenberg, Daniel R. Schwarz, Elizabeth Welt Trahan, Jean Fagan Yellin
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- PMLA / Publications of the Modern Language Association of America / Volume 115 / Issue 7 / December 2000
- Published online by Cambridge University Press:
- 23 October 2020, pp. 1986-2078
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- December 2000
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The Anisotropy of the Universe at Large Times
- S. W. Hawking
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- Symposium - International Astronomical Union / Volume 63 / 1974
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- 07 February 2017, pp. 283-286
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- 1974
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The most important cosmological observation in the last forty years has undoubtedly been the discovery of the microwave background. As well as confirming the existence of a hot early phase of the Universe, by its spectrum, its remarkable isotropy indicates that the Universe must be very nearly spherically symmetric about us. Because of the revolution of thought brought about by Copernicus, we are no longer vain enough to believe that we occupy any special position in the Universe. We must assume, therefore, that the radiation would appear similarly isotropic in any other place. One can show that the microwave radiation can be exactly isotropic at every point only if the Universe is exactly spatially homogeneous and isotropic, that is to say, it is described by one of the Friedmann models. (Ehlers et al., 1968). Of course, the Universe is neither homogeneous nor isotropic locally. This must mean that the background radiation is not exactly isotropic, but only isotropic to within the very good limits set by the observations (about 0.1%). One would like to know, however, what limits the observations place on the large-scale anisotropies and inhomogeneities of the Universe. One would also like to know why it is that the Universe is so nearly, but not exactly, isotropic.
Quantum Aspects of Accretion onto Black Holes in the Early Universe
- S. W. Hawking, G. W. Gibbons
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- Symposium - International Astronomical Union / Volume 64 / 1974
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- 07 February 2017, p. 185
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- 1974
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It is argued that for black holes of mass less than roughly 1017 g formed in the early Universe, the accretion process must be calculated using quantum mechanics. It is also argued that primordial black holes must be more massive than 1017 g if they are to possess charge.
Contents
- S. W. Hawking, G. F. R. Ellis
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- The Large Scale Structure of Space-Time
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- 26 January 2010
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- 10 May 1973, pp vii-x
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Preface
- S. W. Hawking, G. F. R. Ellis
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- The Large Scale Structure of Space-Time
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- 10 May 1973, pp xi-xii
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Summary
The subject of this book is the structure of space–time on length-scales from 10–13 cm, the radius of an elementary particle, up to 1028 cm, the radius of the universe. For reasons explained in chapters 1 and 3, we base our treatment on Einstein's General Theory of Relativity. This theory leads to two remarkable predictions about the universe: first, that the final fate of massive stars is to collapse behind an event horizon to form a ‘black hole’ which will contain a singularity; and secondly, that there is a singularity in our past which constitutes, in some sense, a beginning to the universe. Our discussion is principally aimed at developing these two results. They depend primarily on two areas of study: first, the theory of the behaviour of families of timelike and null curves in space–time, and secondly, the study of the nature of the various causal relations in any space–time. We consider these subjects in detail. In addition we develop the theory of the time-development of solutions of Einstein's equations from given initial data. The discussion is supplemented by an examination of global properties of a variety of exact solutions of Einstein's field equations, many of which show some rather unexpected behaviour.
This book is based in part on an Adams Prize Essay by one of us (S. W. H.). Many of the ideas presented here are due to R. Penrose and R. P. Geroch, and we thank them for their help. We would refer our readers to their review articles in the Battelle Rencontres (Penrose (1968)), Midwest Relativity Conference Report (Geroch (1970c)), Varenna Summer School Proceedings (Geroch (1971)), and Pittsburgh Conference Report (Penrose (1972b)).
3 - General Relativity
- S. W. Hawking, G. F. R. Ellis
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- The Large Scale Structure of Space-Time
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- 26 January 2010
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- 10 May 1973, pp 56-77
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Summary
In order to discuss the occurrence of singularities and the possible breakdown of General Relativity, it is important to have a precise statement of the theory and to indicate to what extent it is unique. We shall therefore present the theory as a number of postulates about a mathematical model for space–time.
In §3.1 we introduce the mathematical model and in §3.2 the first two postulates, local causality and local energy conservation. These postulates are common to both Special and General Relativity, and thus may be regarded as tested by the many experiments that have been performed to check the former. In §3.3 we derive the equations of the matter fields and obtain the energy–momentum tensor from a Lagrangian.
The space–time manifold
The third postulate, the field equations, is given in §3.4. This is not so well established experimentally as the first two postulates, but we shall see that any alternative equations would seem to have one or more undesirable properties, or else require the existence of extra fields which have not yet been detected experimentally.
The mathematical model we shall use for space–time, i.e. the collection of all events, is a pair (ℳ, g) where ℳ is a connected four-dimensional Hausdorff C∞ manifold and g is a Lorentz metric (i.e. a metric of signature + 2) on ℳ.
8 - Space–time singularities
- S. W. Hawking, G. F. R. Ellis
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- The Large Scale Structure of Space-Time
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- 10 May 1973, pp 256-298
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Summary
In this chapter, we use the results of chapters 4 and 6 to establish some basic results about space–time singularities. The astrophysical and cosmological implications of these results are considered in the next chapters.
In §8.1, we discuss the problem of defining singularities in space–time. We adopt b-incompleteness, a generalization of the idea of geodesic incompleteness, as an indication that singular points have been cut out of space–time, and characterize two possible ways in which b-incompleteness can be associated with some form of curvature singularity. In §8.2, four theorems are given which prove the existence of incompleteness under a wide variety of situations. In §8.3 we give Schmidt's construction of the b-boundary which represents the singular points of space–time. In §8.4 we prove that the singularities predicted by at least one of the the theorems cannot be just a discontinuity in the curvature tensor. We also show that there is not only one incomplete geodesic, but a three-parameter family of them. In §8.5 we discuss the situation in which the incomplete curves are totally or partially imprisoned in a compact region of space–time. This is shown to be related to non-Hausdorff behaviour of the b-boundary. We show that in a generic space–time, an observer travelling on one of these incomplete curves would experience infinite curvature forces. We also show that the kind of behaviour which occurs in Taub–NUT space cannot happen if there is some matter present.
2 - Differential geometry
- S. W. Hawking, G. F. R. Ellis
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- The Large Scale Structure of Space-Time
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- 10 May 1973, pp 10-55
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Summary
The space–time structure discussed in the next chapter, and assumed through the rest of this book, is that of a manifold with a Lorentz metric and associated affine connection.
In this chapter, we introduce in §2.1 the concept of a manifold and in §2.2 vectors and tensors, which are the natural geometric objects defined on the manifold. A discussion of maps of manifolds in §2.3 leads to the definitions of the induced maps of tensors, and of sub-manifolds. The derivative of the induced maps defined by a vector field gives the Lie derivative defined in §2.4; another differential operation which depends only on the manifold structure is exterior differentiation, also defined in that section. This operation occurs in the generalized form of Stokes' theorem.
An extra structure, the connection, is introduced in §2.5; this defines the covariant derivative and the curvature tensor. The connection is related to the metric on the manifold in §2.6; the curvature tensor is decomposed into the Weyl tensor and Ricci tensor, which are related to each other by the Bianchi identities.
In the rest of the chapter, a number of other topics in differential geometry are discussed. The induced metric and connection on a hypersurface are discussed in §2.7, and the Gauss–Codacci relations are derived. The volume element defined by the metric is introduced in §2.8, and used to prove Gauss' theorem.
Appendix A - Translation of an essay by P. S. Laplace
- S. W. Hawking, G. F. R. Ellis
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- The Large Scale Structure of Space-Time
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- 26 January 2010
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- 10 May 1973, pp 365-368
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4 - The physical significance of curvature
- S. W. Hawking, G. F. R. Ellis
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- The Large Scale Structure of Space-Time
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- 26 January 2010
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- 10 May 1973, pp 78-116
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Summary
In this chapter we consider the effect of space–time curvature on families of timelike and null curves. These could represent flow lines of fluids or the histories of photons. In §4.1 and §4.2 we derive the formulae for the rate of change of vorticity, shear and expansion of such families of curves; the equation for the rate of change of expansion (Raychaudhuri's equation) plays a central role in the proofs of the singularity theorems of chapter 8. In §4.3 we discuss the general inequalities on the energy–momentum tensor which imply that the gravitational effect of matter is always to tend to cause convergence of timelike and of null curves. A consequence of these energy conditions is, as is seen in §4.4, that conjugate or focal points will occur in families of non-rotating timelike or null geodesics in general space–times. In §4.5 it is shown that the existence of conjugate points implies the existence of variations of curves between two points which take a null geodesic into a timelike curve, or a timelike geodesic into a longer timelike curve.
Timelike curves
In chapter 3 we saw that if the metric was static there was a relation between the magnitude of the timelike Killing vector and the Newtonian potential. One was able to tell whether a body was in a gravitational field by whether, if released from rest, it would accelerate with respect to the static frame defined by the Killing vector.
10 - The initial singularity in the universe
- S. W. Hawking, G. F. R. Ellis
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- The Large Scale Structure of Space-Time
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- 26 January 2010
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- 10 May 1973, pp 348-364
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Summary
The expansion of the universe is in many ways similar to the collapse of a star, except that the sense of time is reversed. We shall show in this chapter that the conditions of theorems 2 and 3 seem to be satisfied, indicating that there was a singularity at the beginning of the present expansion phase of the universe, and we discuss the implications of space–time singularities.
In §10.1 we show that past-directed closed trapped surfaces exist if the microwave background radiation in the universe has been partially thermalized by scattering, or alternatively if the Copernican assumption holds, i.e. we do not occupy a special position in the universe. In §10.2 we discuss the possible nature of the singularity and the breakdown of physical theory which occurs there.
The expansion of the universe
In §9.1 we showed that many stars would eventually collapse and produce closed trapped surfaces. If one goes to a larger scale, one can view the expansion of the universe as the time reverse of a collapse. Thus one might expect that the conditions of theorem 2 would be satisfied in the reverse direction of time on a cosmological scale, providing that the universe is in some sense sufficiently symmetrical, and contains a sufficient amount of matter to give rise to closed trapped surfaces. We shall give two arguments to show that this indeed seems to be the case. Both arguments are based on the observations of the microwave background, but the assumptions made are rather different.
9 - Gravitational collapse and black holes
- S. W. Hawking, G. F. R. Ellis
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- The Large Scale Structure of Space-Time
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- 26 January 2010
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- 10 May 1973, pp 299-347
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Summary
In this chapter, we shall show that stars of more than about 1½ times the solar mass should collapse when they have exhausted their nuclear fuel. If the initial conditions are not too asymmetric, the conditions of theorem 2 should be satisfied and so there should be a singularity. This singularity is however probably hidden from the view of an external observer who sees only a ‘black hole’ where the star once was. We derive a number of properties of such black holes, and show that they probably settle down finally to a Kerr solution.
In §9.1 we discuss stellar collapse, showing how one would expect a closed trapped surface to form around any sufficiently large spherical star at a late stage in its evolution. In §9.2 we discuss the event horizon which seems likely to form around such a collapsing body. In §9.3 we consider the final stationary state to which the solution outside the horizon settles down. This seems to be likely to be one of the Kerr family of solutions. Assuming that this is the case, one can place certain limits on the amount of energy which can be extracted from such solutions.
For further reading on black holes, see the 1972 Les Houches summer school proceedings, edited by B. S. de Witt, to be published by Gordon and Breach.
Notation
- S. W. Hawking, G. F. R. Ellis
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- The Large Scale Structure of Space-Time
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- 26 January 2010
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- 10 May 1973, pp 381-384
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Index
- S. W. Hawking, G. F. R. Ellis
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- The Large Scale Structure of Space-Time
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- 26 January 2010
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- 10 May 1973, pp 385-391
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5 - Exact solutions
- S. W. Hawking, G. F. R. Ellis
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- The Large Scale Structure of Space-Time
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- 26 January 2010
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- 10 May 1973, pp 117-179
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1 - The role of gravity
- S. W. Hawking, G. F. R. Ellis
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- The Large Scale Structure of Space-Time
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- 26 January 2010
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- 10 May 1973, pp 1-9
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Summary
The view of physics that is most generally accepted at the moment is that one can divide the discussion of the universe into two parts. First, there is the question of the local laws satisfied by the various physical fields. These are usually expressed in the form of differential equations. Secondly, there is the problem of the boundary conditions for these equations, and the global nature of their solutions. This involves thinking about the edge of space–time in some sense. These two parts may not be independent. Indeed it has been held that the local laws are determined by the large scale structure of the universe. This view is generally connected with the name of Mach, and has more recently been developed by Dirac (1938), Sciama (1953), Dicke (1964), Hoyle and Narlikar (1964), and others. We shall adopt a less ambitious approach: we shall take the local physical laws that have been experimentally determined, and shall see what these laws imply about the large scale structure of the universe.
There is of course a large extrapolation in the assumption that the physical laws one determines in the laboratory should apply at other points of space–time where conditions may be very different. If they failed to hold we should take the view that there was some other physical field which entered into the local physical laws but whose existence had not yet been detected in our experiments, because it varies very little over a region such as the solar system.