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Gravity, Strings and Particles pp 49–60Cite as

Space, Time, and Space–Time

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Abstract

Let me start this chapter with a somewhat “philosophic” introduction, concerning the so-called reality of physical models. As a physicist, I am a bit reluctant to deal with similar topics which—strictly speaking—are beyond my professional expertise.

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Notes

  1. 1.

    The times t 1, t 2 and the positions x 1, x 2 are obviously referred to the clock and to the coordinate system of a given observer. For a different observer the corresponding values of times and of the spatial coordinates could be different. What matters, in this example, is only the variation of the position with time.

  2. 2.

    This scenario is also called “Emerging Block Universe,” see for instance the contribution of G.F. R. Ellis to the book Springer Handbook of Spacetime [32].

  3. 3.

    Obviously, what is meant for the past and future may vary according to the considered observer, and may also depend on the local geometric properties characterizing the considered portion of space–time.

  4. 4.

    This scenario is also called “Block Universe”, or frozen Universe. See for instance P. C. W. Davies [33].

  5. 5.

    See also the discussion in the book by J. B. Barbour [34].

  6. 6.

    Such observers, for kinematic or geometric reasons, would be “causally disconnected” from us, namely they could not send us information through signals that do not exceed the speed of light.

  7. 7.

    Discussed in a recent paper by J. D. Barrow and D. J. Shaw [35].

  8. 8.

    We are referring to the so-called four-velocity vector (or to the associated four momentum vector) representing, geometrically, the tangent to the world-line of the given object.

  9. 9.

    According to this principle, an antiparticle of positive energy corresponds to a particle of negative energy which propagates backward in time.

  10. 10.

    Unless, for some reasons, “superluminal” matter with modulus of the space–time velocity equal to 2c, 3c, etc. does not interact at all with the “subluminal” matter present in our world, and is thus completely disconnected by our physical evidence.

  11. 11.

    I wish to thank my colleague Paolo Cea for an interesting discussion on this point.

  12. 12.

    Here “different” means, in particular, “not diffeomorphic,” i.e., geometries which cannot be linked by coordinate transformations of differentiable and invertible type.

  13. 13.

    See, e.g., the paper by N. Kaloper and K. A. Olive [36].

  14. 14.

    See, e.g., the review paper by E. Caianiello [37]. Phase space is a higher-dimensional space in which the number of dimensions is doubled, since to any given coordinate is associated a new one corresponding to the so-called canonically conjugate momentum.

  15. 15.

    The so-called geodesic trajectories are the world-lines determined by the space–time geometry. The set of all geodesics fully identifies a given space–time and uniquely determines its physical and geometric properties.

  16. 16.

    A space–time manifold is called “geodesically complete” when all the physical geodesic trajectories can be arbitrarily extended forward and backward in time, without limits and without running into any singularity.

  17. 17.

    They are transformations in which the metric is multiplied by an appropriate scalar function of the coordinates. Using this type of transformations it is possible to pass from a geometry where the trajectory of a given particle is not a geodesic to a new geometry where the same trajectory is instead a geodesic.

  18. 18.

    This example was discussed in a paper I wrote in 2004 [38].

  19. 19.

    Not us, but our descendants! Because the possible future singularity would be distant in time billions of years.

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  1. Dipartimento di Fisica, Università di Bari, Bari, Italy

    Maurizio Gasperini

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Gasperini, M. (2014). Space, Time, and Space–Time. In: Gravity, Strings and Particles. Springer, Cham. https://doi.org/10.1007/978-3-319-00599-7_4

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