The Physical Basis of The Direction of Time (The Frontiers ...

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168 5 The Time Arrow of Spacetime Geometry infinite-dimensional superspace that it defines by its quadratic form is super- Lorentzian (with signature + −−−...). 4 This fact has important consequences. In the familiar case of mechanics, vanishing kinetic energy, E − V = 0, describes turning points of the motion. However, since there are no forbidden regions for indefinite kinetic energy, the boundary V = V − E = 0 does not force the trajectories to come to a halt and reverse direction here. Rather, this condition now describes a smooth transition between ‘subluminal’ and ‘superluminal’ directions in superspace (not in space!), as can be seen in Fig. 5.7. A trajectory would be reflected from an infinite potential ‘barrier’ only if this were either negative at a time-like boundary, or positive at a space-like one. Reversal of the cosmic expansion at a max requires the vanishing of an appropriate V eff (α) that includes the actual kinetic energy of the other degrees of freedom (similar to the effective radial potential in the Kepler problem). It is evident that this behavior must be important for a reversal of time and its arrow. In the Friedmann model, a point on the trajectory in configuration space determines Friedmann time t (that could be read from comoving test clocks) – except where the curve intersects itself. In a mini-superspace with more than two degrees of freedom (adding a material clock, for example), physical time on a trajectory is generically unique, since intersections could occur only accidentally. This demonstrates that the essential requirement for the state to represent a carrier of information about time is reparametrization invariance of the dynamical laws – not its spacetime-geometric interpretation. Although a time parameter is in general physically meaningless in these theories, it is often misused for an inappropriate interpretation. An example is Veneziano’s (1991) string model, based on a dilaton field Φ. Its equations of motion lead to a time dependence of the form f(t − t 0 ), with an integration constant t 0 that determines the value of the time parameter at the big bang (where α = −∞). A translation t 0 → t 0 + T would thus be meaningless (as already pointed out by Leibniz). The solution for t
5.4 Geometrodynamics and Intrinsic Time 169 The shift functions N i of (5.27) can be chosen to vanish even when spatial symmetries are absent. The secondary momentum constraints H i := ∂ ˜H/∂N i = 0, which warrant conservation of vanishing canonical momenta p Ni , and which are fulfilled automatically for the Friedmann solution because of its symmetry, then have to be solved explicitly. The lapse function N(x, y, z, t) now determines genuine many-fingered time (as a spatial field on the dynamically evolving hypersurface) with respect to the coordinate t. If N is nonetheless chosen as a function of t alone, the foliation proceeds everywhere according to physical time (normal to the hypersurface, with fixed ‘comoving’ coordinates). This may not always be a convenient choice. For example, observers coming very close to a black hole horizon would observe the stars moving very fast through a little hole that remains in the sky above the horizon because of their extreme time dilation. In a universe that is bound to recontract they could reach the contraction era within very short proper times. This renders the immediate vicinity of horizons very sensitive to a conceivable cosmic final condition, which may even exclude black hole horizons and singularities (see Zeh 1983, 2005a, and Sect. 6.2.3). In this case, a foliation according to York time, mentioned in Sect. 5.1, may be preferable, since it arrives ‘simultaneously’ at all final singularities. Note, however, that the external curvature scalar K, which defines York time, is not a function of state, f( (3) G). Among the simplest inhomogeneous models are the spherically symmetric ones, with a metric ds 2 = −N(χ, t) 2 dt 2 + L(χ, t) 2 dχ 2 + R(χ, t) 2( dθ 2 +sin 2 θ dφ 2) . (5.36) They contain one remaining spatial gauge function, that has to be eliminated by means of the momentum constraint H χ = 0. This is analogous to Gauß’s law in electrodynamics, as it similarly refers to the radial coordinate. Qadir (1988) proposed an illustrative toy model for such an inhomogeneous universe (Fig. 5.8). It forms a generalization of the Oppenheimer–Snyder model for the gravitational collapse of a homogeneous spherical dust cloud (see Misner, Thorne and Wheeler 1973, Chap. 32). The latter model pastes (or ‘sutures’) a comoving spherical surface surrounding part of a contracting closed Friedmann solution (representing the dust cloud) consistently to the external region a Schwarzschild–Kruskal solution. Qadir then pastes this Schwarzschild solution in turn to another (much larger) partial Friedmann solution with much smaller energy density (his universe proper). This pasting at two spatial boundaries, with Friedmann radial coordinate values χ 1 and χ 2 , say, is consistent only if the total masses of the two partial Friedmann universes are identical, and can thus be identified with the Schwarzschild mass M characterizing the partial vacuum solution. The latter forms a strip from Fig. 5.2 between two non-intersecting geodesics that lead from the past to the future Kruskal singularity (big bang and big crunch). In order to comply with the Weyl tensor hypothesis as much as possible, Qadir assumed the ‘Schwarzschild corridor’ to be absent at the big bang.
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the frontiers collection
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H. Dieter Zeh THE PHYSICAL BASIS OF
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Preface Four previous editions of t
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Contents Introduction .............
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Introduction The asymmetry of Natur
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Introduction 3 tion of time (thus a
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Introduction 5 1. Radiation. In mos
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Introduction 7 language is loaded w
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Introduction 9 between temporal and
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12 1 The Physical Concept of Time i
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14 1 The Physical Concept of Time r
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16 1 The Physical Concept of Time a
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18 2 The Time Arrow of Radiation Th
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20 2 The Time Arrow of Radiation by
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22 2 The Time Arrow of Radiation t
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24 2 The Time Arrow of Radiation or
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26 2 The Time Arrow of Radiation wa
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28 2 The Time Arrow of Radiation th
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30 2 The Time Arrow of Radiation m
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32 2 The Time Arrow of Radiation v
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34 2 The Time Arrow of Radiation wh
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36 2 The Time Arrow of Radiation th
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38 2 The Time Arrow of Radiation 'i
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40 3 The Thermodynamical Arrow of T
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100 4 The Quantum Mechanical Arrow
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Page 208 and 209: Epilog 201 The role of time is very
Page 210 and 211: Appendix: A Simple Numerical Toy Mo
Page 212 and 213: Notebook: A Toy Model Appendix: A S
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Page 216 and 217: References References marked with (
Page 218 and 219: References 211 Bläsi, B., and Hard
Page 220 and 221: References 213 Davies, P.C.W., and
Page 222 and 223: References 215 Gerlach, U.H. (1988)
Page 224 and 225: References 217 Horwich, P. (1987):
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References 219 Levine, H., Moniz, E
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References 221 Pearle, Ph. (1976):
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References 223 Smoluchowski, M. Rit
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References 225 Zeh, H.D. (1971): On
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Index absorber, 18, 24-28, 36-38, 5
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Index 229 forward determinism, 66,
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Index 231 stochastic process, see d
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