Let γ be a timelike worldline in a strongly causal time-oriented GR spacetime.
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Let p,q∈γ with p earlier than q (p<q). Then I+(p)∩I(q) is a causal diamond.
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Causal diamonds are relativistic counterparts of the pre-relativistic “thick” present.
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Several objections to this claim are considered and answered.
Abstract
Richard Arthur (2006) and I (Savitt, 2009) proposed that the present in (time-oriented) Minkowski spacetime should be thought of as a small causal diamond. That is, given two timelike separated events p and q, with p earlier than q, we suggested that the present (relative to those two events) is the set I+(p)∩I-(q). Mauro Dorato (2011) presents three criticisms of this proposal. I rebut all three and then examine two more plausible criticisms of the Arthur/Savitt proposal. I argue that these criticisms also fail.
Section snippets
Causal diamonds
At the end of the twentieth century, it looked as if one question at the intersection of physics and metaphysics had been settled. What is the present in Minkowski spacetime, M? The upshot of a series of well-known papers beginning in the 1960s seemed to prove that one had a very limited choice. The present, at or for a spacetime point e∈M could be either the whole spacetime M or just the point e itself. The choice is no wider if one allows the present to be defined relative to a spacetime
Dorato contra diamonds
Arthur׳s and my proposal was criticized in Dorato (2011). The aim of this paper is to evaluate these criticisms and then to add a few further thoughts of my own. In the course of this discussion a more detailed understanding of the proposal under fire will emerge.
Dorato crisply sums up his arguments on page 391 of his paper:
(i)
[Causal diamonds have] no important applications in physical theories;
(ii)
it does not seem a plausible, strong and non-arbitrary explanation of the extendedness of our
Region-relative becoming
I spoke at the beginning of this paper of theorems that seem to show that the present for a given event in Minkowski spacetime could only be either the event itself or the whole of the spacetime. If that claim is correct, isn’t the Arthur/Savitt proposal straightforwardly ruled out?9 My answer will be: no, I don’t think so. How could that be? Well, theorems have conditions, and it may be possible to introduce causal diamond presents by
Maxwell’s equations represented by differential operators describe the local dependence between both electric and magnetic fields in every location of space and time. This description responds to the field concept proposed by Michael Faraday and formalized by James Clerk Maxwell. The discretized version in the form of the finite-difference time-domain (FDTD) technique relies on a proposal of local computation of the respective differential operators making use of the central difference approximation of the second derivative of functions through the Taylor series expansion. This article introduces a novel time-symmetric “non-local” technique from the mathematical formalism of electromagnetic potentials in the wave equation and its physical interpretation in the Minkowski spacetime. In the proposed case study, the “non-local” proposal is 1643 faster than FDTD, i.e., more than three orders of magnitude, and it uses a time-step that is 4096 times greater than the Courant–Friedrichs–Lewy limit without encountering stability problems. New electromagnetic potentials are calculated from the previous and distant ones located at vertices of spatiotemporal regions called causal diamonds that tessellates spacetime without the need for computations inside them. We show that the performance gain is proportional to the size of the spacetime tessellations because the ratio “domain to boundary” increases as the domain extension does. Consequently, our novel “non-local” approach provides a reduction in computational complexity and a more comprehensible explanation of their fundamental physical aspects, without contradicting the principles of the successful classical field theory.
Nature of problem: The Courant–Friedrichs–Lewy (CFL) condition imposes a numerical bound to the size of the Yee cell in the FDTD approach. This restriction causes excessive oversampling, degrading the computation time of this technique.
Solution method: We applied the parallelogram rule to solve the wave equation for the case of
time–space dimensions. With this rule, we proposed a novel algorithm that allows encoding potentials along the characteristic lines that tessellate the spacetime. The new technique produces higher performance gains than those achieved through traditional FDTD.
