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Mathematical Structures and Modeling 2020. N. 1(53). PP. 49-55
UDC 514.763.6 DOI 10.24147/2222-8772.2020.1.49-55
ON GEOMETRY OF FINSLER CAUSALITY: FOR CONVEX CONES, THERE IS NO AFFINE-INVARIANT LINEAR ORDER (SIMILAR TO COMPARING VOLUMES)
Olga Kosheleva
Ph.D. (Phys.-Math.), Associate Professor, e-mail: olgak@utep.edu
Vladik Kreinovich
Ph.D. (Phys.-Math.), Professor, e-mail: vladik@utep.edu
University of Texas at El Paso, El Paso, Texas 79968, USA
Abstract. Some physicists suggest that to more adequately describe the causal structure of space-time, it is necessary to go beyond the usual pseudo-Riemannian causality, to a more general Finsler causality. In this general case, the set of all the events which can be influenced by a given event is, locally, a generic convex cone, and not necessarily a pseudo-Reimannian-style quadratic cone. Since all current observations support pseudo-Riemannian causality, Finsler causality cones should be close to quadratic ones. It is therefore desirable to approximate a general convex cone by a quadratic one. This can be done if we select a hyperplane, and approximate intersections of cones and this hyperplane. In the hyperplane, we need to approximate a convex body by an ellipsoid. This can be done in an affine-invariant way, e.g., by selecting, among all ellipsoids containing the body, the one with the smallest volume; since volume is affine-covariant, this selection is affine-invariant. However, this selection may depend on the choice of the hyperplane. It is therefore desirable to directly approximate the convex cone describing Finsler causality with the quadratic cone, ideally in an affine-invariant way. We prove, however, that on the set of convex cones, there is no affine-covariant characteristic like volume. So, any approximation is necessarily not affine-invariant.
Keywords: space-time geometry, Finsler spaces, causality.
1. Formulation of the Corresponding Physical Problem: Analysis of Finsler Causality Relations
Geometric description of physical causality: a brief reminder (for details, see, e.g., [5]). In Newton's mechanics, a space-time event (t,x) can causally influence an event (t',x') if and only if t ^ t'. In Special Relativity Theory, a space-time event (t,x), with x = (x\,...,xm), can causally influence an event (t',x') with x' = (x[,... ,x'm) if the difference vector
t, /\X i, ... , ^^) — (t t, X i Xi, . . . , Xr^^ X^^)
belongs to the quadratic cone
C = {(At, AXi,..., Axm) : At ^ 0 & (At)2 - (Axi)2 - ... - (Axm)2 ^ 0}.
According to the General Relativity Theory, similar cones describe local causality: a space-time event (t,x), with x = (xj_,... ,xm), can causally influence an event (t + dt,x + dx) with dx = (dxi,... ,dxm) if the difference vector (dt,dxi,... ,dxm) belongs to the cone
C = {(dt,dxi,..., dxm) : dt ^ 0&(dt)2 - (dxi)2 - ... - (dxm)2 ^ 0}.
Finsler causality. Some physical theories use more general - not necessarily quadratic - convex cones to describe causality; see, e.g., [1,4,6]. This generalization of the usual causality is known as Finsler causality.
It is desirable to approximate Finsler causality with the usual one. As of
now, all observations are consistent with the usual causality relation, namely with the locally quadratic causality cones of General Relativity theory. Thus, even if the actual causality relation is a Finsler one, it is close to the usual quadratic one. So, to be able to efficiently deal with Finsler causality relations, it is therefore desirable to be able to approximate Finsler causality cones with quadratic ones. This way, we will be able to use the formulas corresponding to the usual quadratic causality as the first approximation, and thus to concentrate our analysis on the -empirically small - differences between these causality relations. How can we find such an approximation?
A possible approach to the desired approximation. A convex cone is uniquely determined by its intersection with a hyperplane - which is a convex body. For the quadratic cone that corresponds to the usual causality relation, this intersection is a bounded quadratic body, i.e., an ellipsoid - and vice versa, for each ellipsoid, the corresponding cone is a quadratic cone. For general convex cones, the intersection is a generic convex body.
So, to approximate a generic convex cone by a quadratic one, it is sufficient to approximate a general convex body by an ellipsoid. This is indeed possible. For example, to every bounded convex body B c IRra with a non-empty interior, we can associate a unique ellipsoid E enclosing this body: namely, out of all ellipsoids E that contain the set B, we can select the ellipsoid E0 with the smallest possible volume V(E) (see, e.g., [2,3]):
V(E0) = min{V(E) : E is an ellipsoid and B c E}.
This definition makes perfect sense, since for ellipsoids, E c E' and E = E' imply that V(E) < V(E'). Thus, the fact that the ellipsoid E0 has the minimal possible volume implies that there is no sub-ellipsoid E' c E0 that contains the original body B.
Strictly speaking, to describe the volume, we need to fix (affine) coordinates in the corresponding hyperplane. However, while the actual values of the volume
depend on the coordinates, the resulting ellipsoid is the same no matter what coordinates we use. The reason for this is that the volume is affine-covariant: for any affine transformation T that describes the transition to new affine coordinates, and for every ellipsoid E, we have V(TE) = c(T) ■ V(E) for some constant c(T). As a result, for every two ellipsoids E and E', we have V(E) < V(E') V(TE) < V(TE'), i.e., which ellipsoid has larger volume and which has smaller volume does not depend on the choice of the coordinates.
Limitations of the usual approximation, and the resulting problem. The
problem with the above approximation idea is that to follow this idea, we must first choose a hyperplane, and there is no guarantee that for a different hyperplane, we will not get a different approximation.
To make the approximation more physically meaningful, it is therefore desirable to avoid such an arbitrary choice, and instead of approximating a body by an ellipsoid, to directly approximate a general convex cone C by a quadratic cone Q. Out of all possible quadratic cones Q that contain C, it is desirable to select the "smallest" one in the sense of an appropriate total order ^ on the class of all quadratic ellipsoids. This selection should not depend on the choice of coordinates. To guarantee that, it is desirable to make sure that the corresponding order is affine-covariant, i.e., that Q <Q' if and only if TQ < TQ'.
What we do in this paper. In this paper, we show that for quadratic cones, there are no affine-invariant volume-like characteristics. Moreover, we show that there is no affine-invariant ordering relation between quadratic cones.
The physical meaning of this result is that it is not possible to approximate Finsler causality by quadratic causality in an affine-invariant way.
2. Definitions and the Main Result Definition 1. Let n ^ 2.
• By an affine transformation T : IRra ^ IR", we mean a reversible mapping
n
of the type Xi ^ Uj ■ ■
3 = 1
• By a quadratic cone, we mean an image TQ0 of the standard quadratic cone
Qo = ... ,xn) : xi ^ 0&x\ ^ x\ + ... + x2n} under an affine transformation.
Notation. For each n ^ 2, let Qn denote the set of all n-dimensional quadratic cones.
Definition 2. By a linear (total) pre-order we mean a transitive relation ^ for which, for every two objects a and b, we have a ^ b or b ^ a.
Comment. For some objects a and b, it is possible to have both a ^ b and b ^ a. For example, for the corresponding relation on the ellipsoids E ^ E' ^ V(E) ^ V(E'),
when we have two ellipsoids E and E' of the same volume, then we have both E ^ E' and E' ^ E.
Notation. We will denote:
• the situation when a ^ b and b ^ a by a < b, and
• the situation when a ^ b and b ^ a by a ~ b.
Definition 3. Let n ^ 2.
• A linear pre-order on the set Qn is called c-consistent if Q c Q' and Q = Q' imply that Q < Q'.
• A linear pre-order on the set Qn is called affine-invariant if for each affine transformation T, Q ^ Q' implies TQ < TQ'.
Proposition. For each n ^ 2, no linear order on the set Qn of all n-dimensional quadratic cones is both c-consistent and affine-invariant.
Comment. The formulation of this proposition of ours first appeared in Geombina-torics journal.
Proof. We will prove this result by contradiction. Let us assume that a linear order ^ is a c-consistent and affine-invariant linear order on the set Qn, and let us deduce a contradiction from this assumption.
1°. Let us first consider the case n = 2.
1.1°. Let us first consider the following quadratic cone
Qi =f {(x\,x2) : x\ ^ 0& x2 ^ 0}. An affine transformation Ti : (x\,x2) ^ (-x2,x\) maps this cone into TiQi = {(x\,x2) : x\ ^ 0&x2 ^ 0}. If we apply the same transformation Ti again, we get
T?Qi = {(xux2) : xi ^ 0&x2 ^ 0}, T?Qi = {(xhX2) : xi ^ 0&X2 ^ 0},
and TfQr = Qr.
1.2°. Since ^ is a linear order, we have either Qx ^ TXQX or TXQX ^ Qx. Let us consider these two possibilities one by one.
1.2.1°. If Qi ^ TiQi, then, since the order ^ is affine-invariant, we get TiQi ^ TfQx, TfQx ^ TfQx, and TfQx ^ Qx. Thus, by transitivity, we get T\Q\ ^ Qi and hence,
Qi - m,.
1.2.2°. If TlQl ^ Ql, then we similarly get Ql ^ TlQl and thus, Ql — T1Q1. So, in both cases, we have Ql — TlQl.
1.3°. Let us now consider a different affine transformation
T2(Xl,X2) = (Xl + X2,X2).
For this transformation, T2Ql = {(xl,x2) : 0 ^ x2 ^ x]} and
T2TlQl = {(xl,x2) : X2 ^ 0&Xl ^ X2}.
Since Ql — TlQl, we have T2Q- — T2TxQi.
On the other hand, since T2Ql Ç Ql and T2Ql = Ql, we have T2Ql < Ql. Similarly, since T-Q- ç T2T1Q1 and T-Q- = T2TiQi, we have TlQ- < T2T1Q1. From T2Ql < Ql, Q- — T-Q-, and T-Q- < T2T1Q1, we conclude that
T2Ql < T2T1Q1,
which contradicts to the previous conclusion that T2Ql — T2TlQl. This contradiction shows that for n = 2, indeed, no Ç-consistent and affine-invariant is possible on the class Qn.
2°. Let us now consider the case n> 2. In this case, we consider a cone
Ql = {(xl,..., xn) : xl ^ 0 & xl ■ x2 ^ x"2 + ... + x2n}, and transformations
Ti(X],X2,X3, ... ,xn) = (
%2, Xl, X3, . . . , Xn)
and
T2(xi,x2,x3, . . . ,Xn) = (xi + X2,X2,X3, . . . ,Xn).
Thus, the inverse transformations take the form
' / ' i (<"V»' rvJ ryJ ry* \ - (<"V»' _rvJ ry* ryJ
± i (■L1, ■L2, . . . , xn) — (■h2, . . . , xn)
and
Tl( rfj rfj rfj rfj - ( rfj _ rfj /yJ ry'
2 (xl, x2, x3, . . . , xn) — (xi
Here too, TfQi = Qi, thus Qi - TxQx. The set T2Q1 the form
= {x' :
T-lx' E Qx} has
T2Q1 = {(x\,... ,xn) : Xl — X2 ^ 0& (xi — X2) ■ X2 ^ x\ + ... + x2}.
The corresponding inequality
(xl — x2) ■ x2 ^ x\ + ... + X,
is equivalent to
and thus, implies that
+ ... + x„
s. 2 \ 2 | | 2
^1 * ^1 ^ * ^2 ^ % ^^ * * * ^^ Xn,
so T2Q1 C Qi and T2Qi = Qi and hence, T2Qi < Qv
Similarly, the set T2T1Q1 consists of all the points (x1, * * * ,xn) for which
i.e., for which
-x2 • (Xi - x2) " x\ + ... + xl
+ 2 2 I I 2
_ X^ "" x3 I . . . I .
So, if (xi,... ,xn) g TiQi, i.e., if
then
X2 • Xi " x3 ^^ . . . ^^
+ 2 2 I I 2
_ "" ^3 I . . . I
and thus, x e T2T1Q1. So here too, T1Q1 c T2TxQ1 and T1Q1 = T2T1Q1, hence T1Q1 < T2T1QL
Similarly to Part 1 of this proof, from T2Q1 < Q1, Q1 ~ T1Q1, and
T-Q- <T2T1Q1,
we conclude that T2Ql < T2TlQl, which contradicts to the previous conclusion that T2Ql ~ T2TlQl. This contradiction shows that for any n, no Ç-consistent and affine-invariant is possible on the class Qn.
Acknowledgments
This work was supported in part by the National Science Foundation grants HRD-0734825 and HRD-1242122 (Cyber-ShARE Center of Excellence) and DUE-0926721. The authors are thankful to D. G. Pavlov and to all the participants of the 5th International Conference on Finsler Extensions of Relativity Theory FERT, (Moscow-Fryazino, Russia) for valuable discussions.
References
1. Asanov S.S. Finsler Geometry, Relativity and Gauge Theories. Springer Verlag, Berin, Heidelberg, New York, 1985.
2. Busemann H. Metric Methods of Finsler Spaces and in the Foundations of Geometry. Princeton University Press, Princeton, New Jersey, 1943.
3. Busemann H. Geometry of Geodesics. Dover Publ., New York, 2005.
4. Chang Z. and Li V. Modified Friedmann model in Randers-Finsler space of approximate Berwald type as a possible alternative to dark energy hypothesis. Physics Letters, 2009, vol. B676, pp. 173-176.
5. Misner C.W., Thorne K.S., and Wheeler J.A. Gravitation. Freeman Publ., San Francisco, California, 1973.
6. Pavlov D.G., Atanasiu Gh., and Balan V. (eds.) Space-Time Structure. Algebra and Geometry, Lilia-Print, Moscow, 2007.
О ГЕОМЕТРИИ ПРИЧИННОСТИ ФИНСЛЕРА: ДЛЯ ВЫПУКЛЫХ КОНУСОВ НЕ СУЩЕСТВУЕТ АФФИННО-ИНВАРИАНТНОГО ЛИНЕЙНОГО ПОРЯДКА (АНАЛОГИЧНОГО СРАВНЕНИЮ ОБЪЁМОВ)
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, США
Аннотация. Некоторые физики предполагают, что для более адекватного описания причинной структуры пространства-времени необходимо выйти за пределы обычной псевдоримановой причинности и перейти к более общей причинности Финслера. В этом общем случае множество всех событий, на которые может влиять данное событие, является локально выпуклым конусом общего типа, а не обязательно квадратичным конусом псевдориманова типа. Поскольку все текущие наблюдения подтверждают псевдориманову причинность, конусы причинности Финслера должны быть близки к квадратичным. Поэтому желательно аппроксимировать общий выпуклый конус квадратичным. Это можно сделать, если мы выберем гиперплоскость и аппроксимируем пересечения конусов и этой гиперплоскости. В гиперплоскости нам нужно аппроксимировать выпуклое тело эллипсоидом. Это можно сделать аффинно-инвариантным способом, например, выбрав среди всех эллипсоидов, содержащих тело, тот, который имеет наименьший объем; так как объем является аффинно-ковариантным, этот отбор является аффинно-инвариантным. Однако, этот отбор может зависеть от выбора гиперплоскости. Поэтому желательно непосредственно аппроксимировать выпуклый конус, описывающий причинность Финслера, квадратичным конусом, в идеале аффинно-инвариантным образом. Однако, мы доказываем, что на множестве выпуклых конусов не существует аффинно-ковариантной характеристики, такой как объем. Поэтому любое приближение не обязательно является аффинно-инвариантным.
Ключевые слова: геометрия пространства-времени, финслеровы пространства, причинность.
Дата поступления в редакцию: 06.12.2019
