Abstract:
The Light-cone—Once, Nonce, and Thence
Hermann Minkowski’s (1908) lecture on “Space and Time” introduced the concept of the light-cone, which has belonged, since then, to the basic kit of every physics student. In this talk, I will discuss the evolution of the concept with reference to three authors: Leonhard Euler, Immanuel Kant, and Albert Einstein. In the first part, I present some recent historical work on three kinematic proofs (Hyder, 2019), each of which involves the same components: i. the Principle of Relativity, ii. a Definition of Simultaneity, and iii. the kinematic parallelogram-law. The conclusion of Euler’s proof is i. the Principle of Relativity, whereas for both Kant and Einstein, i. is a premise, and the conclusion is iii. the kinematic parallelogram law. By contrast, while Kant’s Definition of Simultaneity employs instantaneous causal connections, Einstein’s uses causal signals with only finite speed.
In the second part, I show that this early relativistic kinematics was always associated with two causal principles: the first, “Law of Causality” asserts that all future events are causally determined by past events, and not conversely; the second asserts that all simultaneous events are bicausally linked by instantaneous causes, such as gravitation. The logical analysis of both of these principles forms the core of Kant’s work. Drawing on work of Eberhard (1972), I show that the conjunction of either Kant’s or Einstein’s kinematic proofs with the Law of Causality generates one of two structures: “our” light-cone if the kinematics is hyperbolic, the “flat” light-cone if it is Euclidean. But that difference in the space-time geometry depended only on the Definition of Simultaneity we used as the input to our kinematic proof. Thus their logical structure is also altered—only one of the causal laws can now be true.
In the concluding section, I discuss modern methods in modal logic for analyzing the causal structure of the light-cone, drawing on work by David Lewis on “counterfactual” causation.
References:
Bombelli, L., J.-H. Lee, D. Meyer, and R. Sorkin. (1987) Space-time as a Causal Set, Physical Review Letters 59: 521–524
Euler, Leonhard. Mechanica, 2 vols. St. Petersburg: Academy of Sciences, 1736.
Hyder, David. (2019) The Kinematics of the Metaphysical Foundations of Natural Science, Kant-Studien 110(3): 477-497. https://doi.org/10.1515/kant-2019-3007
Knuth, K.H. and N. Bahrenyi. (2010) A Derivation of Special Relativity from Causal Sets. arXiv:1005.4172v2 [math-ph]
-------- (2014) A Potential Foundation for Emergent Space-time, J. of Math. Phys. 55: 112501.1-35.
Lewis, David. (1973) Causation, Journal of Philosophy 70(17): 556-567.
Minkowski, Hermann (1909) Raum und Zeit, Jahresbericht der Deutschen Mathematiker-Vereinigung 18: 75–88. English translation.
Eberhard, P.H. “Bell's Theorem and the Different Concepts of Locality” Nuovo Cimento 46(11): 392-418.