Spacetimes as Topological Spaces, and the Need to Take Methods of General Topology More Seriously

Why is the manifold topology in a spacetime taken for granted? Why do we prefer to use Riemann open balls as basic-open sets, while there also exists a Lorentz metric? Which topology is a best candidate for a spacetime: a topology sufficient for the description of spacetime singularities or a topology which incorporates the causal structure? Or both? Is it more preferable to consider a topology with as many physical properties as possible, whose description might be complicated and counterintuitive, or a topology which can be described via a countable basis but misses some important information? These are just a few from the questions that we ask in this chapter, which serves as a critical review of the terrain and contains a survey with remarks, corrections and open questions.

[1]  Zeeman-Göbel Topologies , 1988, The British Journal for the Philosophy of Science.

[2]  Konstantin Mischaikow,et al.  Reconstructing Functions from Random Samples , 2013, 1310.3390.

[3]  E. C. Zeeman,et al.  The topology of Minkowski space , 1967 .

[4]  Zhangxin Chen.  Reservoir Simulation: Mathematical Techniques in Oil Recovery (CBMS-NSF Regional Conference Series in Applied Mathematics) , 2007 .

[5]  B. Papadopoulos,et al.  The order on the light cone and its induced topology , 2017, 1710.05177.

[6]  Andrew King,et al.  A new topology for curved space–time which incorporates the causal, differential, and conformal structures , 1976 .

[7]  L. Susskind,et al.  Cool horizons for entangled black holes , 2013, 1306.0533.

[8]  Kyriakos Papadopoulos.  On Properties of Nests: Some Answers and Questions , 2017, 1710.06855.

[9]  G. R. Lee,et al.  Production of Sigma (1385)(+/-) and Xi (1530)(0) in proton-proton collisions at root s=7 , 2015 .

[10]  Leonard Susskind,et al.  Dear Qubitzers, GR=QM , 2017, 1708.03040.

[11]  B. Papadopoulos,et al.  On two topologies that were suggested by Zeeman , 2017, Mathematical Methods in the Applied Sciences.

[12]  Roger Penrose,et al.  On the structure of causal spaces , 1967, Mathematical Proceedings of the Cambridge Philosophical Society.

[13]  The large-scale structure of the ambient boundary , 2015, 1512.09216.

[14]  Kyriakos Papadopoulos.  On the Orderability Problem and the Interval Topology , 2018, 1803.09643.

[15]  Black hole entropy: a spacetime foam approach , 1997, gr-qc/9706030.

[16]  Roger Penrose,et al.  A complete guide to the laws of the universe , 2005 .

[17]  S. Cotsakis.  TALKING ABOUT SINGULARITIES , 2007, gr-qc/0703084.

[18]  R. Penrose.  Techniques of Differential Topology in Relativity , 1972 .

[19]  Kyriakos Papadopoulos.  Nests, and Their Role in the Orderability Problem , 2017, 1801.00700.

[20]  K. Hofmann,et al.  A Compendium of Continuous Lattices , 1980 .

[21]  K. Papadopoulos,et al.  A topological characterization of ordinals: van Dalen and Wattel revisited , 2012 .

[22]  R. Penrose.  Gravitational collapse and spacetime singularities , 1965 .

[23]  D. Garfinkle,et al.  The 1965 Penrose singularity theorem , 2014, 1410.5226.

[24]  Zeeman topologies on space-times of general relativity theory , 1976 .

[25]  I. Antoniadis,et al.  Topology of the ambient boundary and the convergence of causal curves , 2015, 1505.04737.

[26]  N. Bohr.  II - Can Quantum-Mechanical Description of Physical Reality be Considered Complete? , 1935 .

[27]  On the possibility of singularities on the ambient boundary , 2017, 1702.02459.

[28]  Ravi P. Agarwal,et al.  Asymptotic periodicity for some evolution equations in Banach spaces , 2011 .

[29]  Cosmological Singularities , 2002, gr-qc/0201067.

[30]  H. Kleinert,et al.  Uncertainty relation on a world crystal and its applications to micro black holes , 2009, 0912.2253.

[31]  R. Low.  Spaces of paths and the path topology , 2016 .

[32]  E. C. Zeeman,et al.  Causality Implies the Lorentz Group , 1964 .

[33]  Robert T. Jantzen,et al.  The eleventh Marcel Grossmann Meeting : on recent developments in theoretical and experimental general relativity, gravitation, and relativistic field theories : proceedings of the MG11 meeting on general relativity, Berlin, Germany 23-29 July 2006 , 2008 .

[34]  I. Antoniadis,et al.  Ambient cosmology and spacetime singularities , 2014, 1409.2220.

[35]  The causal order on the ambient boundary , 2016, 1603.08117.

[36]  Mark Van Raamsdonk.  Building up spacetime with quantum entanglement , 2010 .

[37]  E. Minguzzi.  Limit curve theorems in Lorentzian geometry , 2007, 0712.3942.

[38]  Albert Einstein,et al.  The Particle Problem in the General Theory of Relativity , 1935 .

[39]  Dynamical topology change in M theory , 2000, hep-th/0010207.