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Mathematical Structures and Modeling 2022. N. 1 (61). PP. 34-40
UDC 514.142 DOI 10.24147/2222-8772.2022.1 .34-40
MACROCAUSALITY IMPLIES LORENZ GROUP: A PHYSICS-RELATED COMMENT ON GUTS'S RESULTS
Abstract. It is known that, in the space-time of Special Relativity, causality implies Lorenz group, i.e., if we know which events can causally influence each other, then, based on this information, we can uniquely reconstruct the affine structure of space-time. When the two events are very close, quantum effects, with their probabilistic nature, make it difficult to detect causality. So, the following question naturally arises: can we uniquely reconstruct the affine structure if we only know causality for events which are sufficiently far away from each other? Several positive answers to this question were provided in a recent paper by Alexander Guts. In this paper, we describe a very simple answer to this same question.
Keywords: causality, special relativity, Alexandrov-Zeeman theorem.
1. Introduction
Causality in Special Relativity reminder. According to Special Relativity Theory, an event a = (t,xl,x2,x3) can causally influence an event b = (s,yl,y2,y3) (we will denote it by a ^ b) if and only if a signal originated in a can reach b while traveling with a speed not exceeding the speed of light c, i.e., if
Causality implies Lorenz group: main result. The famous result by A.D. Alexandrov (see, e.g., [1-3]) shows that any bijection f : IR4 ^ IR4 of the 4-D space-time that preserves causality - i.e., for which a ^ b if and only if f (a) ^ f (b) - is a composition of rotations, shifts, scalings a • X • a, Lorenz transformations, and, if needed, spatial inversion (t,xl,x2,x3) ^ (t, —xl, -x2, -x3).
Causality implies Lorenz group: physical consequences. Thus, if we know, for
every two events a and b, whether a can causally influence b, then, based on this information, we can uniquely reconstruct the linear (= affine) structure on the 4-D space-time.
O. Kosheleva
Ph.D. (Phys.-Math.), Associate Professor, e-mail: olgak@utep.edu
V. Kreinovich Ph.D. (Phys.-Math.), Professor, e-mail: vladik@utep.edu
University of Texas at El Paso, El Paso, USA
s - t ^ c • -\J(xi - yx)2 + (x2 - y2)2 + (x3 - y3)2.
(1)
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Quantum effects lead to an additional complexity. A usual way to test a causal relation between the events is to make some change in a and observe the resulting effect in b. This can be done in a deterministic situation. However, when the events a and b are very close to each other, with microworld-size differences between their coordinates, then quantum effects prevail, the relation becomes probabilistic, and detecting causality becomes difficult. As a result, we can only observe causal relation when the event b is sufficiently far away from the event a.
Related natural question and Guts's answers. Can we still reconstruct the linear structure of space-time based on this observable (macro) causality?
This question was studies in a recent paper [3] by Alexander Guts that provides many positive answers to this question.
What we do in this paper. The main objective of this paper is to provide yet another positive answer - an answer that probably follows from Guts's results, but that is so simple that we believe it is worth describing.
2. Analysis of the Problem and the Main Result
Let us be as general as possible - without abandoning simplicity. While our main interest is in the causal relation (1), we follow a natural mathematical tendency of formulating this result in the most general form - as long as this desire for generality does not make things more complicated.
So, instead of relation (1), let us consider any relation a ^ b that is described by a closed convex cone F: a ^ b if and only if b — a e F.
Natural requirement on macrocausality relation. In addition to the "theoretical" causality relation a ^ b, we assume that there is also an additional observable (= macro) causality relation a c b. Of course, when we can observe that a causally influence b, this means that a ^ b in the theoretical sense as well, i.e., that a c b implies a ^ b.
If a observably influences b (a c b), this means that b is sufficiently far away from a. Thus, if b ^ c, this means that c is even further from a than b - so we should also be able to detect that a influences c as well. In other words, if a c b and b ^ c, then a c c.
Finally, since we are considering a homogeneous space-time, it is reasonable to require that the macrocausality relation c if shift-invariant, i.e., that a c b implies a + c c b + c.
Now, we are ready to formulate our result.
Comment. To make this paper understandable to as many readers as possible, we add as many definitions as needed - even though they are most probably familiar to many readers.
Definition 1. Let R be a binary relation on a set S. We say that a bijection f : S m- S preserves the relation R if for every a,b e S, we have aRb if and only if f (a)Rf (b).
Definition 2. A set S с IRra is called a convex cone if for every two elements si,s2 € S and for every two non-negative real numbers c1 and c2, the element c1 • s1 + c2 • s2 also belongs to S.
Proposition. Let F с IRra be a convex cone which is a closed set, let a ^ b mean b — a € F, and let с be a binary relation on IRra that is satisfied by at least one pair (ao,bo) and that satisfies the following properties for all a, b, and c:
1. if а с b, then a ^ b;
2. if а с b and b ^ c, then а с с; and
3. if а с b, then a + с с b + с.
Then every bijection that preserves с also preserve
Conclusion. For the case when ^ is the Special Relativity causality relation (1) and с is macrocausality, this result means that every bijection that preserves macrocausality is a composition of rotations, shifts, scalings, Lorenz transformations, and, if needed, spatial inversion.
Thus, even if we only know observable causality, we can still uniquely reconstruct linear structure on IRra.
Proof. To prove the Proposition, let us show that the relation ^ can be described in terms of с, namely, that for all b and c:
b ^ с ^Уа (а с b ^ а с с). (2)
This will imply that any bijection that preserved с preserves ^ as well.
Indeed, the left-to-right part of (2) follows from the second property listed in the Proposition. So, to complete the proof, we need to prove the right-to-left implication. Indeed, let us assume that
Уа (а с b ^ а с с). (3)
Due to the third property (shift-invariance), we have а с b if and only if 0 с b — a, i.e., if and only if b — a € M = {c : 0 с с}. We know that a0 с b0.
• Thus, due to shift-invariance, we have a0 + (b — b0) с b, and therefore, due to (3), we have
a0 + (b — 60) с с.
• Then, due to shift-invariance, we have a0 + (b — b0) + (b — с) ^ b and therefore, due to (3), we have
a.0 + (b — 60) + (b — с) с с.
• Then, due to shift-invariance, we have a0 + (b — b0) + 2(b—с) ^ b and therefore, due to (3), we have
a.0 + (b — 60) + 2(b — с) с с,
etc.
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By induction, we can prove that for every natural number m, we have
a0 + (b — b0) + m • (b — c) C c. Due to the first property, this implies that
^o + (b — bo) + m • (b — c) ^ c. By definition of the relation this means that
c — (a0 + (b — bo)+ m • (b — c)) = (m +1) • (c — b) + (b0 — a0) G F. Since F is a convex cone, we also have
1 ((m +1) • (c — b) + (b0 — a0)) = c — b + ^^^ G F.
m+1 m+1
In the limit m ^ œ, these elements tend to c — b. Since F is a closed set, we thus have c — b G F, which means that b ^ c. The proposition is proven.
Acknowledgments
This work was supported in part by the National Science Foundation grants 1623190 (A Model of Change for Preparing a New Generation for Professional Practice in Computer Science), and HRD-1834620 and HRD-2034030 (CAHSI Includes), and by the AT&T Fellowship in Information Technology.
It was also supported by the program of the development of the Scientific-Educational Mathematical Center of Volga Federal District No. 075-02-2020-1478, and by a grant from the Hungarian National Research, Development and Innovation Office (NRDI).
References
1. Alexandrov A.D. On Lorentz transformations. Uspekhi Math. Nauk, 1950, vol. 5, no. 1, pp. 187 (in Russian).
2. Alexandrov A.D. and Ovchinnikova V.V. Remarks on the foundations of Special Relativity. Leningrad University Vestnik, 1953, no. 11, pp. 94-110 (in Russian).
3. Guts A.K. Non-connected order in affine space and its automorphisms. Mathematical Structures and Modeling, 2021, vol. 4(60), pp. 17-38 (in Russian).
4. Zeeman E.C. Causality implies the Lorentz group. Journal of Mathematical Physics, 1964, vol. 5, no. 4, pp. 490-493.
МАКРОПРИЧИННОСТЬ ВЛЕЧЕТ ГРУППУ ЛОРЕНЦА: КОММЕНТАРИЙ, СВЯЗАННЫЙ С ФИЗИКОЙ РЕЗУЛЬТАТА ГУЦА
О. Кошелева
к.ф.-м.н., доцент, e-mail: olgak@utep.edu В. Крейнович
к.ф.-м.н., профессор, e-mail: vladik@utep.edu
Техасский университет в Эль-Пасо, Эль-Пасо, США
Аннотация. Известно, что в пространстве-времени специальной теории относительности причинность влечёт группу Лоренца, т. е. если мы знаем, какие события могут причинно влиять друг на друга, то на основе этой информации мы можем однозначно реконструировать аффинную структуру пространства-времени. Когда два события очень близки, квантовые эффекты с их вероятностной природой затрудняют обнаружение причинно-следственной связи. Итак, естественно возникает следующий вопрос: можем ли мы однозначно реконструировать аффинную структуру, если мы знаем причинность только для событий, достаточно удалённых друг от друга? Несколько положительных ответов на этот вопрос было дано в недавней статье Александра Гуца. В этой статье мы опишем очень простой ответ на этот же вопрос
Ключевые слова: причинность, специальная теория относительности, теорема Александрова-Зимана.
Дата поступления в редакцию: 14.02.2022
