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DOI 10.1007/s10958-015-2214-y Journal of Mathematical Sciences, Vol. 204, No. 6, February, 2015

POLYNUMBERS, NORMS, METRICS, AND POLYINGLES R. R. Aidagulov and M. V. Shamolin UDC 517.925

CONTENTS 1. Vector Spaces and Algebras of 2. Polylinear Functions and Metr 3. Polyingles . . . . . . . . . . . References . . . . . . . . . . . En ics .. .. d . . . omorph .... .... .... is . . . m . . . s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 742 745 751 758

1.

Vector Spaces and Algebras of Endomorphisms

Let Rn b e a vector space of dimension n over the field of real numb ers. This means that Rn is equipp ed with a commutative and associative op eration called addition with resp ect to which Rn is a group and an op eration of multiplication of elements of Rn by real numb ers (this op eration is always assumed to b e a distributive linear transform). Moreover, there exist n elements e1 ,e2 ,... ,en such that any element x Rn can b e uniquely represented in the form
n

x=
i=1

xi ei = xi ei ,

xi R ,

Here we accept the Einstein summation convention: when an index variable app ears twice in a single term, it implies summation of that term over all the values of the index (which is usually clear from the context). An additive mapping f : Rn Rk is a homomorphism (of commutative groups with resp ect to addition) b etween the spaces Rn and Rk . A linear mapping f : Rn Rk is a homomorphism (of modules over R) b etween the spaces Rn and Rk . In the finite-dimensional case, the notions of a linear mapping and a continuous additive mapping coincide. Definitions of multiplication in modules, rings, and algebras include distributivity, which means the additivity of the multiplication transform from the left and from the right. Instead, we require the linearity of multiplication (from the left and from the right), which is a stronger condition. Linear mappings f : Rn Rn are called endomorphisms of the space Rn . They form not only a vector space of dimension n2 , but also an associative algebra Mn with unity. The multiplication op eration in this algebra is not commutative. If f is a linear mapping from Rn to Mn , then in the space Rn we can define multiplication by the rule x y = f (x)(y ). However, for an arbitrary linear mapping, the obtained structure of an algebra on R commutative nor associative. The associativity condition is equivalent to the condition f (x y ) = f (x)f (y ),
Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 88, Geometry and Mechanics, 2013.
n

is neither

742

1072­3374/15/2046­0742 c 2015 Springer Science+Business Media New York


i.e., the condition that f is a mapping of rings. Then f maps Rn into an n-dimensional subalgebra of the algebra Mn . If in an algebra there is no preimage of the identity matrix, we can introduce such an element increasing the dimension of the algebra by 1. We prop ose a physical interpretation of these constructions in the framework of the sp ecial theory of relativity (STR). By Rn we mean coordinates of events in some inertial reference system. A transition to another inertial reference system corresp onds to a linear (invertible) transform of coordinates of events. The commutativity of the algebra is equivalent to the commutativity of the Lorentz addition of velocities, when the reference systems 3 and 4 coincide (the reference system 3 is a system that moves with velocity u with resp ect to the reference system 1, which, in turn, moves with velocity v, and the reference system 4 is a system that moves with velocity v with resp ect to the reference system 2, which, in turn, moves with velocity u). In principle, noncommutativity of multiplication in an algebra is allowed. Velocities corresp ond to the indicatrix, so that multiplication (or Lorentz addition) of velocities determines a Lie group of dimension n - 1. Actually, in the STR, all transforms of a transition to another inertial reference system are not closed with resp ect to multiplication. If we take all p ossible sums and products of such elements, then we obtain a subalgebra M4 that commutes with the op erator of multiplication by the matrix I , which in an appropriate basis has the form iO I= , Oi where O= 00 , 00 i= 01 . -1 0

It is isomorphic to the algebra M2 (C) of dimension 8 with resp ect to the set of real numb ers R. In the STR, multiplication (Lorentz addition) is defined indep endently of multiplication in M2 (C) (the algebra obtained is neither a subalgebra nor a quotient algebra). The algebra obtained contains the rotation group (with resp ect to multiplication) G. Moreover, for any transform f (x) and any element g G, there exists another element g1 G such that gf (x) = f (x)g1 , and the condition f (x1 )g1 = f (x2 )g2 , implies x1 = x2 , g1 = g2 . For any two elements x1 and x2 , there exist x3 ,g G such that f (x1 )f (x2 ) = f (x3 )g. This allows one to define multiplication of velocities similar to multiplication in the quotient group x1 x2 = x3 , which defines multiplication up to Thomson rotations. We obtain ln(f (x3 )) = ln(f (x1 )) + ln(f (x2 )) mod J, where J is the ideal corresp onding to G in the Lie algebra of the group of invertible transforms M2 (C). However, this multiplication is not related to the addition op eration by the distributivity law. The genuine algebra of the SRT is the Clifford algebra Cl(3) M2 (C), which also has the biquaternion = representation (see [13]). Corresp ondingly, there is nothing wrong in the fact that the dimension of the algebra is greater than the dimension of the representation space. Generally sp eaking, multiplications 743 x1 = 0,


that corresp ond to velocities do not define the algebra completely. We must also introduce multiplications by v , R+ , and several multiplications that corresp ond to the reversal of orientation of time and/or space and p ermutation of directions in space. In p olynumb ers (i.e., finite-dimensional associative algebras over R with unities), the op eration of exp onential mapping can b e introduced: exp(0) = 1, exp(a1) = exp(a)1, a R.

In this case, if elements x and y commute, then exp(x + y ) = exp(x)exp(y ). Images of the exp onential mapping are vectors (subsequently, we identify a vector with its matrix representation) corresp onding to p ositive eigenvalues. First, we consider the case of a commutative algebra. In this case, all transforms f (x) have the same basis of eigenvectors and can b e diagonalized in this basis, which reduces the algebra to the direct sum of the algebras of real and complex numb ers, Rn-2k + Ck , where k is the numb er of pairs of complex conjugate eigenvectors for the basis. Therefore, any function of vectors invariant with resp ect to a transition to another basis is a symmetric function on eigenvalues (i.e., a function of diagonal elements in a sp ecial basis). This is also valid in the noncommutative case. Groups of coordinate transformations act on functions of vectors (elements of the algebra). There is the canonical Galois corresp ondence b etween transformation groups and the sets of functions that remain invariant under the actions of these groups (see [5]). The minimal transformation group of interest is the group of automorphisms of the set of p olynumb ers, i.e., the group of invertible linear mappings for which a(xy ) = a(x)a(y ). From the p oint of view of physics, functions that are not preserved even under the action of automorphisms of the algebra are not of interest. A wider group is obtained by complementing automorphisms with antiautomorphisms (which can b e called odd automorphisms), i.e., invertible linear transformations for which a(xy ) = a(y )a(x). Clearly, the product of two antiautomorphisms is an automorphism. Therefore, functions that are preserved under automorphisms but are not preserved under the extension of the group by the antiautomorphisms describ ed ab ove, can b e called odd functions, whereas functions that are also preserved under antiautomorphism can b e called even functions. A wider group is obtained if we consider all invertible linear transformations (module automorphisms instead of algebra automorphisms). Function that are preserved under the action of this group are said to b e invariant. There are no invariant vectors. Unit vectors are preserved under b oth automorphisms and antiautomorphisms, i.e., they are even vectors. At the same time, there exists an invariant covector (i.e., linear function on vectors), for example, the trace Tr and any prop ortional covector. The (n - 1)-dimensional subspace of vectors that are annihilated by this covector is also invariant. An invariant covector allows one to define the "real" and "imaginary" parts of a vector as follows: 1 Re(x) = Tr(x), Im(x) = x - Re(x) 1. n The conjugate (symmetric) vector for a vector x is defined as 2 1Re(x) - x, where 1 denotes the unit vector. In algebras that are obtained by several iterations of the doubling procedure starting from some commutative algebra, the transition to the conjugate element is an involution (i.e., an antiautomorphism whose square is the identity automorphism). The doubling is p erformed so that the extended 744


conjugation remains an involution in the doubled algebra. In physics, the "real" part usually corresp onds to the time coordinate and the "imaginary" part to the spatial coordinate. The p olynomial functions Tr(xm ), m = 1,... ,n, are invariant functions. In the algebras Hn = Rn (the direct sums of n copies of the set of real numb ers), there exist automorphisms that swap n "unities" of each comp onents. This set of n unities is also invariant with resp ect to automorphisms and their sum--the unity of the algebra--is invariant with resp ect to automorphisms and antiautomorphisms. Therefore, any invariant function in metrics of Berwals­Moor or Chernov (with symmetric functions) typ e can b e expressed as a function of the ab ove p olynomials. Such functions of several vectors can also b e expressed in the same form as a function of p olynomials of several variables. In the case of noncommutative multiplication, the difference is only in the use of noncommutative p olynomials. However, the trace is indep endent of the order of factors and hence we obtain the same invariant functions despite the fact that matrices of multiplication are distinct. One can construct homogeneous metrics using homogeneous p olynomials of such functions. For example, using three constants, we can construct a metric of rank 3: ds3 = a Tr(xy z )+ b Tr(x)Tr(yz )+Tr(y )Tr(xz )+Tr(z )Tr(xy )+ c Tr(x)Tr(y )Tr(z ) . 2. Polylinear Functions and Metrics f : V k R, V =R (1)

Polylinear functions of rank k can b e defined as functions
n

linear with resp ect to each of the variables. These functions can also b e defined in the form f (x1 ,... ,xk ) = gi
1

...i

k

xi1 ... xik . 1 k

If a functions does not change its value under an arbitrary p ermutation of arguments, then it is said to b e symmetric. As metric tensors, only symmetric p olylinear functions can b e used. The length of a vector is defined as the p ower of order 1/k of a symmetric p olylinear function f (·,... , ·) (with the same arguments). Introducing the indices jm = im + m - 1, 1 j1 < ··· < jk n + k - 1, i1 ··· ik .
k Therefore, the numb er of indep endent comp onents of a symmetric tensor of rank k is equal to Cn+k-1 . Actually, tensors that are usually used for constructing metrics p ossess an additional symmetry of numeration of coordinates. In the case of total symmetry with resp ect to numeration, the numb er of indep endent comp onents is expressed by the well-known partition function (see [7]) p(k, n) 2k-1 ; it is equal to the numb er of representations of a numb er k as the sum of no more than n natural numb ers. For n k (this condition holds in all cases) all values are p(k, n) = p(k ) and

and taking symmetry into account, we can reduce the indices of a tensor to the form

p(1) = 1, p(2) = 2, p(3) = 3, p(4) = 5, p(5) = 7, p(6) = 11,

2 = 1 + 1, 3 = 2 + 1 = 1 + 1 + 1, 4 = 3 + 1 = 2+2 = 2+1 + 1 = 1+1 + 1+1, 5 = 4 + 1 = 3+2 = 3+1 + 1 = 2+2 + 1 = 2 + 1+1+ 1 = 1+1 + 1+1+ 1, etc. 745


For example, the symmetric bilinear function discussed at the end of the previous section are functions of such typ e. In particular, for k = 2 we have two symmetric bilinear functions, one of which j j corresp onds to gij = i and the other to gij = 1 - i . All other such functions are linear combinations of these two functions. In the case where an index symmetry occurs for all indices except for one of them (which corresp onds to time), the numb er of indep endent comp onents is p1 (k, n) = p(k, n - 1) + p(k - 1,n - 1) + ··· + p(1,n - 1) + 1. However, in the algebras Hn = Rn (the direct sum of n copies of the set of real numb ers), each comp onent is contained symmetrically (although there is a selected vector, namely, the sum of the comp onents of the unity). Therefore, metrics that are related to the algebra must b e completely symmetric. Polylinear metrics of rank k n related to the p olynumb er structure (i.e., invariant with resp ect to the action of the automorphism group of p olynumb ers) are defined by p(k) parameters. Such metrics are said to b e p olynumb er metrics. The numb er of parameters of a metric in a general-relativity analog is still greater by n2 ; it is related to the choice of n generators of the algebra in the n-dimensional space. However, not all n2 parameters are indep endent. For example, an analog of the Berwald­Moor metric ds4 = de1 de2 de3 de4 , ei = ai t + bi x + ci y + di z, i = 1, 2, 3, 4, contains 16 parameters related to multiplication by numb ers ri , i = 1, 2, 3, 4, r1 r2 r3 r4 = 1.

This leads to 13 parameters of the orientation of the metric for the Berwald­Moor metric; moreover, all these parameters can change from one p oint to another. In a sp ecial-relativity analog, all parameters are constant. The general form of the metric in the SRT is as follows: ds = dt 1 - P2 (v )+ P3 (v )+ ... , where dxi dt and Pk (v ) is a symmetric p olynomial of degree k of the coordinates of the velocity v . This can b e justified as follows: any symmetric function of coordinates (only if they are invariant with resp ect to all automorphisms) can b e represented as a function of symmetric functions. If we k , factor out dt, we obtain a homogeneous function of degree 0, i.e., a function of the ratios (1 )k k = 2,... ,n, where k is the kth symmetric p olynomial of generators. We show b elow that they are symmetric functions of the velocities. More precisely, there exist metrics invariant with resp ect to automorphisms whose expansions in (2) starts from terms of order higher than 2. However, they are degenerate critical metrics that can b e reduced to a noncritical form by a small deformation and can b e written in the form (2) in an appropriate basis. The coordinates of velocity can b e found up to rotations from the expression for P2 (v ) (see (2)). Except for degenerate cases, due to the contributions of higher degrees of velocity, the symmetry of coordinates is reduced to p ermutations of coordinates. Consider homogeneous metrics starting from the case k = 2 (the case where k = 1, ds = dt, is not of interest). We have vi = ds2 = a dx1 + ··· + dx 746
n 2

P2 (v ) =

12 2 v + ··· + vn 21

-1

,

(2)

+ b dx2 + ··· + dx2 . 1 n

(3)


We pass to the following coordinates: x1 + ··· + xn x1 + ··· + xn , yi = xi + qxn - r , y0 = n n The inverse transition is p erformed by the formulas 1 (rn - r - n)y0 + xn = q (n - 1) - 1 xi = yi + ry0 - Taking first differentials dxi = dyi - s1 + dy0 , where s
m n -1

i = 1,... ,n - 1.

y
i=1

i

,
n -1

q (rn - r - n)y0 + q (n - 1) - 1 i = 1,... ,n - 1,
n -1

y
i=1

i

.

dxn = fs1 + gdy0 ,

=
i=1

(dyi )m

is a symmetric form of degree m of the variables dyi , i = 1,... ,n - 1, we obtain f = (n - 1) - 1, and
n i=1

g = n - (n - 1)

(dxi )2 = s2 - 2s2 +2s1 dy0 +(n - 1)(dy0 - s1 )2 +(fs1 + gdy0 )2 . 1

Equating the coefficients of s2 and s1 dy0 in the last expression to zero, we have 1 2 n(n - 1) - 2n +1 = 0, Thus, we obtain the following relations: = 1 = g, = 1 , n n 1 f = ± , n q= -1 = -r. n n ( - 1)n(1 + - n) = 0.

Substituting these expressions into (3), we have
n i=1 2 (dxi )2 = s2 + ndy0 , n -1 i=1

ds = (a + bn)(dy0 ) + b The metric defined by the last relation is nondegenerate when b = 0, Finally, we have 1 xi = y i + y 0 - n n 1 y0 = n
n n -1

2

2

(dyi )2 .

a = -bn.
n -1

yj ,
j =1

i < n,

1 xn = y 0 ± n
n

yj ,
j =1

(4)

xj ,
j =1

xn 1 + y i = xi - 1 n n(1 n)

xj .
j =1

Now we show that precisely these coordinates, up to scaling of time and length (spatial measure), corresp ond to the separation of the time and spatial coordinates for any metric invariant under arbitrary automorphisms of the algebra Hn . More precisely, there is arbitrariness in the elimination of 747


one variable (in our case, xn ; this is similar to the choice of one affine parameter, which is a coordinate in a pro jective manifold), arbitrariness in the choice of the sign ± n, and arbitrariness of ordering of the variables yi (due to symmetry, the metric itself is indep endent of the last two arbitrary factors), and, finally, arbitrariness of scaling of time and space. We assume that summing of yi is p erformed over i = 1,... ,n - 1. We denote their symmetric forms (with resp ect to the variables y ) as follows: sj =
j dyi ,



m

=
i1 <··· m

dyi1 ... dyim .

We denote the symmetric form with resp ect to all variables x by Sj = dxj , i m =
i1 <··· m

dxi1 ... dxim .

The reduction to the form (2) is equivalent to the expression of Sj , j > 1, of j , j > 1, through the variables 1 z0 = dy0 = S1 , s1 ,s2 ,... , n as follows:
j

Sj = (z0 + fs1 ) +(n - 1)(z0 - z1 ) +
j j m=1

m sm Cj (z0 - z1 )j j j = nz0 + z0

-m 2 Cj n(n - 1)2 - 2n +1 + ... ,

-2 2 s1

i.e., for f2 1 1 , = = 1-f ±n n n the second term of the expansion considered vanishes for any j . Therefore, the following relation holds: f=
j

Sj =

j nz0

+
m=3

C

m j -m m s1 j z0

f

m

+(n - 1)(-) + m(-)
m

m-1

j

+
m=2

m Cj sm (z0 - s1 )j

-m

... . (5)

As in to the Newton­Girard formulas, we can obtain metrics expressed through p ermanents and calculate the corresp onding anisotropy coefficients. The basic metrics used in Finsler geometry are p ermanent, i.e., they can b e expressed in the form dsk = k , k =
i1 <··· k

dxi1 ... dxik .

We present expressions for the time coordinate for this case. These formulas can b e obtained from the ab ove by using the Newton­Girard formulas. This yields the value a = 3/2 for the anisotropy (it dep ends only on n, which is equal to 4 in our case) of the Chernov metric and two values a and b for the Berwald­Moor metric. Similarly, we obtain
m-1 m

m = (z0 + fs1 )
k =0

C

k m-n+k m-1-k

(z0 - s1 )k +
k =0

C

k m-1-n+k m-k

(z0 - s1 )k .

(6)

This implies that any noncritical metric invariant under automorphisms of p olynumb ers Hn can b e reduced to the form (2) by scaling the time variable y0 and the space variables yi (t = c1 y0 , zi = c2 yi ) under certain inequalities for parameters. 748


Arbitrariness of the choice of variables leading to the form (2) in the case where P2 (v ) is an arbitrary symmetric p olynomial of degree 2 is substantially wider and it does not allow one to define spatial relations (length). We present the final expressions for Sj , j , j = 1, 2, 3, 4: S1 = nz0 ,
2 S2 = nz0 + s2 , 3 S3 = nz0 +3s2 z0 + A31 s3 + A32 s1 s2 + s3 , 1 3 4 2 S4 = nz0 +6z0 s2 +4z0 A31 s3 +3s2 s1 + s 1

+ P4 ,

P4 = A41 s4 +62 s2 s2 +4s3 s1 + s4 , 1 1 A31 = 1 2± n 1 A32 = -3 = -3 n± n -1 1 A41 = 3± (n ± n)3 n
k

1 (n ± n)2

1 3 n8 -3 , 2 1 + 2. n

,

(7)

In the case of a metric of degree k , we similarly obtain that dsk = (dt)k k 1 - V2 + 2 Pi (V ) ,
i=3

Vm =
i

dyi dt

m

,

(8)

where Pi (V ) are homogeneous p olynomials of degree i of the spatial coordinates of velocities: P3 (V ) = a31 V13
p(k )

+ a32 V1 V2 + a33 V3 ,

... ,

Pk (V ) =
i=1

aki Ti (V ).

Precisely these terms lead to anisotropy. In the Berwald­Moor and Chernov metrics defined by p ermanents, the coefficients of anisotropic terms cannot b e adjusted and are uniquely calculated from (8). Therefore, this does not allow one to introduce a small anisotropy. To remove this shortcoming, one introduces metrics with arbitrary coefficients that preserve their form under arbitrary automorphisms of p olynumb ers; they are called p olynumb er metrics. Using (7), we can reduce a cubic metric to the form
3 3 ds3 = a1 S1 + a2 S1 S2 + a3 S3 = b1 z0 + b2 z0 s2 + a3 A31 s3 + A32 s1 s2 + s 1 3

,

b1 = a1 n + a2 n + a3 n,

3

2

b2 = na2 +3a3 .

After normalization with resp ect to the time and space coordinates y , the metric contains only one parameter adjusting anisotropy: 3 (9) ds3 = dt3 - dt dx2 + dy 2 + dz 2 + aP3 (dx, dy , dz ), 2 where P3 (dx, dy , dz ) is a given symmetric p olynomial of degree 3. The last term causes anisotropy. In particular, under reversal of space orientation, this term changes its sign. It is hard to imagine that the reversal of the orientation of a ruler changes its length. That is why the authors hold the view that the spatial metric (without time) is Euclidean. The sp eed of light and the so-called Lorentz transforms (transforms of one inertial reference system into another) dep end on direction (near singularities). In the latter case, anisotropy app ears only in terms of the third order with resp ect to velocities (the sp eed of light has order 1 but dep ends on direction). In the case k = 4, the metric has two anisotropy parameters: ds4 = dt4 - 2 dt
2

dx2 + dy 2 + dz

2

+ adtP3 (dx, dy , dz ) + bP4 (dx, dy , dz ).

(10) 749


For a = 0, the anisotropy is even and prop erties do not change under the reversal of spatial orientation. By a rank-k metric, one can obtain a metric of lower rank by a convolution with fixed vectors. Let m > k and vectors Ai , l = 1,... ,m - k, b e given. Then a metric of rank m is obtained as l follows: im-k gj1 ...jm = gi1 ...im-k j1 ...jm Ai1 ... Am-k 1 (here summation over rep eated indices is assumed). Metrics obtained in this way with p ositive (exp onentially representable) vectors are called accompanying metrics. Accompanying metrics are invariant with resp ect to automorphisms of the algebra if all vectors Ai are prop ortional to the unit vector 1i = 1. Such accompanying metrics are called subl ordinated metrics. For example, the Chernov metric is a sub ordinated metric for the Berwald­Moor metric. A symmetric metric whose accompanying metrics of rank 2 have the signature (+, -, -,... , -) and which is negative definite on the hyp erplane Tr(x) = 0 and p ositive on the unit vector 1 is called a Lorentz-typ e metric. This notion is similar to the notion of the hyp erb olicity of a p olynomial in the sense of G°rding, which defines a metric with resp ect to a unit vector [3]. a Metrics that are sub ordinated to the Berwald­Moor metric are called metrics of p ermanent typ e. They are conformally equivalent to k . In the following section, we present remarkable prop erties of such metrics. Here we mention only the following obvious prop erty (which follows directly from the definition): an accompanying metric for a Lorentz-typ e metric is also a Lorentz-typ e metric. In view of the imp ortance of the case n = 4 for physics (in this case, in addition, irrationalities in transition expressions disapp ear), we rewrite for this case all typ es of anisotropy of metrics of third and fourth orders, which dep end on n. The case n = 4 is remarkable not only by the disapp earance of irrationalities, but also by the fact that, in the case of p ositive sign, transitions are realized sup ersymmetrically (this holds only for n = 4) and anisotropy p olynomials are of the simplest form. Therefore, the case n = 4 is remarkable. We rescale the spatial variables as follows: y1 y2 z2 t = y0 , x = , y = , z = . 2 2 2 Then we have x1 = t + x - y - z, Therefore, 1 = 4dt, 2 = 2 3dt2 - dx2 - dy 2 - dz
2 2 2 2 2 2 2

x2 = t + y - x - z,

x3 = t + z - x - y,

x4 = t + x + y + z.

(11)

, +12dx dy dz , +8dt dx dy dz + dx + dy + dz .
4 4 4

3 = 4dt dt - dx - dy - dz 4 = dt - 2dt
4 2 2 2

(12)

dx + dy + dz

- 2 dx2 dy 2 + dy 2 dz 2 + dz 2 dx

Here 2 corresp onds to the Minkowski metric (we still have freedom of choice of the spatial and time scale), 3 corresp onds to the Chernov metric (with the simplest anisotropy for n = 4), and 4 corresp onds to the Berwald­Moor metric. All these metrics are p ermanent. If there is a chosen vector 1, from a metric of rank k we can obtain metrics of lower rank by replacing some vectors (missing in scalar products of lower rank) by the vector 1 in the scalar product of rank k. For example, if we have a trilinear scalar product (A, B , C ), then it is p ossible to obtain the associated bilinear scalar product (A, B ) = (A, B , 1) by using the chosen vector. 750


In p olynumb er metrics, there always exists a chosen By means of the unit vector one can obtain metrics of such a sub ordinated metric. Moreover, the case k = n 1 x1 x1 2 x2 x2 1 2 det 1 ... ... n! xn xn 1 2

vector 1 corresp onding to the identity matrix. lower rank and define angles of lower rank in corresp onds to the Berwald­Moor metric: ... x1 n ... x2 n ... ... ... xn n

(the p ermanent of the coordinates of vectors in a sp ecific basis). It also is a norm of a vector as in numb er theory. Thus, the norm of the product of p olynumb ers is equal to the product of their norms. Therefore, p olynumb er metrics of lower rank can b e obtained by filling n - k columns with 1's. In particular, the Chernov metric is obtained in this way from the Berwald­Moor metric by replacing one column by a column consisting of 1's. This procedure, up to a conformal factor, is equivalent to the rejection of the highest anisotropic term in the metric, reduction by z0 , and decreasing the degree of ds by 1. A general metric of rank 3 in the case n = 4 can b e reduced by scaling (when the signs of a, 2a + b, and c in the expression a1 2 + b3 + c3 coincide) to the form 1 ds3 = dt dt2 - dx2 - dy 2 - dz ds4 = dt4 -2dt
2 2

+ adxdy dz .

(13)

Therefore, in the general case a metric of rank 4 (under appropriate conditions) has the form dx2 + dy 2 + dz 2 +adtdxdy dz +b dx4 + dy 4 + dz 4 +c dx2 dy 2 + dy 2 dz 2 + dz 2 dx2 . (14) The numb er of parameters is equal to p(4) - 2 = 3, where p(4) is the total numb er of parameters minus two parameters owing to spatial and time scaling. A spatial metric (a definition of lengths) can b e introduced as follows: dr 2 = lim
dt

dt2 - ds

2

.

The degree 2 in this formula corresp onds to all nondegenerate (noncritical) metrics and can b e replaced by a higher degree only in the degenerate case. Moreover, in all nondegenerate cases, this metric is the usual Euclidian metric. Using this metric, it is p ossible to calculate the sp eed of light in various directions by (12). For the Chernov and Berwald­Moor metrics, the sp eed of light in the directions of the spatial axes x, y , and z is equal to 1. The minimal sp eed corresp onds to the direction (-1, -1, -1); for the Chernov metric it is equal to 0.823, and for the Berwald­Moor metric to 1/ 3. In the last case, the sp eed of light reaches the maximal value ( 3) in the direction (1, 1, 1); however, for the Chernov metric, the inequality ds = 0 holds for this direction (it seems that light does not propagate). This shows that one need examine metrics whose anisotropy is less than the anisotropy of the metrics considered ab ove. 3. Polyingles

We discuss several approaches to the notion of p olyingles b etween k vectors in a space with a metric of rank k, which generalizes the notion of an angle (bingle) b etween two vectors for quadratic metrics. A p olyingle must b e defined as a function of k vectors {A1 ,... ,Ak } in a space with a metric of rank k satisfying the following conditions: (1) the function is symmetric, i.e., invariant under any p ermutation of arguments: {A1 ,A2 ,A3 ,... ,Ak } = {A2 ,A1 ,A3 ,... ,Ak }, {A1 ,A2 ,... ,Ak
-1

,Ak } = {Ak ,A1 ,A2 ,... ,Ak

-1

}; 751


(2) the k -ingle is indep endent of the lengths of vectors and dep ends only on their directions, i.e., the function is homogeneous of degree 0 with resp ect to all arguments: {A1 ,... ,Ak } = {A1 ,... ,Ak }, (3) if c is an isometry of the space, then {c(A1 ),... ,c(Ak )} = {A1 ,... ,Ak }; (4) for p ositive (i.e., exp onentially representable) vectors, p olyingles are always defined and continuously dep end on variations of vectors in this domain. Actually, these conditions imply that prop erty (3) is also valid for conformal transforms. Therefore, if conformal transforms transitively act on the set of vectors (this is valid for p olynumb er metrics), then {A,... ,A} = {B ,... ,B } = const. By analogy to bingles, this constant can b e made equal to 1 by normalization. (5) A p olyingle is not a constant function for p ositive vectors. However, these requirements do not suffice to define k-ingles. Assume that there exist a transform u preserving a k-ingle and changing only the first two vectors so that it is p ossible to decrease the angle b etween them to zero. Using such transforms rep eatedly, we sup erp ose all the vectors into a single vector. This can b e easily proved for the transform A1 + A2 . (A1 ,A2 ) (B, B ), B = 2 In this case, the sum of the vectors is preserved and, therefore, in the limit, all the vectors b ecome equal to their arithmetic mean value and, b earing in mind the remark after condition (3), this leads to a constant value. In the general case, one can similarly reduce a p olyingle to a constant value. The following lemma can b e proved similarly. Lemma 1. If the symmetry group of a k-ingle is the general linear group GL(n), then this k -ingle is constant, i.e., contradicts condition (5). Proof. Let k vectors {A1 ,... ,Ak } b e linearly indep endent. Then we take any vector from this set, say A1 , and apply the transforms A1 + Ai , i = 1. Ai 2 Further, applying such transforms rep eatedly (the vectors still remain indep endent and hence such transforms b elong to the group Gl(n)), we obtain a contradiction with condition (5). The case where the initial vectors {A1 ,... ,Ak } are linearly dep endent can b e reduced to the ab ove case by taking indep endent vectors arbitrarily close to given vectors. The mapping, which associates with any set of k vectors V k Rn their k-ingle, stratifies the given set into levels, and the transformation group G preserving k-ingles acts so that each orbit b elongs to a certain level set. Thus, we find that the group G cannot contain elements that change only two vectors and preserves the other vectors. Some orbits of this group may not contain a "standard" typ e that annihilates bingles. For example, if all k vectors are equal, then it is unlikely that the group G enables their "disconnection" so that they b ecome pairwise orthogonal. Most likely, such an orbit is degenerate and contains only all sets of identical k vectors. Moreover, for some metrics, the fact that a set of k vectors contains a vector orthogonal to all other vectors probably implies that the corresp onding k-ingle vanishes. This means that counting of the numb er 752 R+ ;


of conditions and the reduction of k vectors to the standard form, provided that the numb er of these conditions is less than the dimension of the group, does not imply the p ossibility of such a reduction. One way of defining k -ingles is as follows. First, we define this notion for coinciding vectors and then introduce formulas that describ e the change of k-ingles under the change of one of the vectors in a direction p erp endicular to the other vectors. The analysis of various definitions of k-ingles allows one to reject most of them, whereby only definitions based on homogeneous functions are adopted. Taking into account the fact that not all spaces contain a selected vector, we conclude that the definition of k-ingles should b e rather general and indep endent of this prop erty. Therefore, the general form of a k -ingle is a combination of the scalar products of k vectors taken in various combination. It is p ossible to construct 2k-1 typ es of combinations of k vectors (see the previous section). However, many typ es of combinations related to p ermutations give the same results; they corresp ond to various partitions of the numb er k into the sums p(k ). For example, we obtain two typ es of partitions for k = 2: 1+1, 2; therefore, (A, B ) is the first typ e and (A, A), is the second typ e. For k = 3, we have 1+1 + 1, therefore, (A, B , C ) is the first typ e, (A, A, B ), is the second typ e, and (A, A, A), (B, B, B ), (C, C, C ) is the third typ e. As the first typ e, we take the representation of k as k unities. In this typ e, we have a unique scalar product. For all typ es we introduce the following quotient: i = ((Aj (1) ,Aj (2) ,... ,Aj (k) ))1/mi ; |A1 |· |A2 | ·· · ··|Ak | |A| = (A,A,... ,A)1/k (the case definition If for a a k-ingle. i = p(k) yields the unity (1)). All these values satisfy the necessary requirements for the of k -ingles. k -ingle there exist several functions, then any function of such k-ingles can also b e taken as Therefore, any function of p(k) - 1 arguments i , defines a k-ingle. 753 i > 1, p
(k )

(B, B )

2+ 1,

3;

(A, A, C ),

(B, B, A),

(B, B, C ),

(C, C, A),

(C, C, B )

the denominator contains the norms of the vectors:

1,


Note that if all vectors are the same (Ai = A), then i 1 and, therefore, we can assume that the k -ingle is equal to 1. In the case where one vector is B and all the other vectors are the same and equal to A, then (A,... ,A,B ) , 1 = |A|k-1 |B | (A,... ,A,B ) k (A,... ,A,B ,B ) k 2 = , |A|k-1 |B |
k -2 1

...

Since the function p(k) rapidly increases as k grows, the numb er of functionally indep endent quantities can b e substantially less than their total numb er. Therefore, it is advisable to op erate with the smallest quantity of these numb ers. For m1 = 1, we can find 1 from the definition of bingles. For an even numb er m2 = k (k - 1) of changes of the comp osition of vectors, we can calculate the value 2 corresp onding to the partition 2+1 + 1+ ··· +1 (k - 2 unities and one deuce). If a k-ingle can b e expressed through only one of these numb ers, there is no need to take a function of it. For bingles, we use a trigonometric function (or, in the case of a pseudo-Euclidean metric, a hyp erb olic function), since the rotation group that changes the value of a bingle is known. This can b e done by various methods and hence one cannot uniquely recover the method used in the construction of a k-ingle of k equal vectors. Moreover, in the case of tringles, there are two numb ers 1 and 2 , but even two numb ers cannot determine a unique way of construction of an initial triple of vectors from a triple of equal vectors. The form of 1 resembles the form of a bingle: 1 = (A1 ,... ,Ak ) , |A1 | ... |Ak | |Ai | = (Ai ,... ,Ai )1/k . (15)

This is b ecause for k = 2 we have (k) = 2 and the two formulas coincide. Actually, if we take all functions satisfying the required conditions (1)­(5), then we can supplement this by all k-ingles with other symmetric metrics and l-ingles (l = k ) with symmetric metrics of rank l. We will henceforth consider arbitrary functions of such p olyingles. The Berwald­Moor metric p ossesses a continuous isometry group I , whereas the isometry groups of the other symmetric metrics do not contain this isometry group isometry I . Therefore, in this case, other p olyingles b ecome unusable. For other metrics (e.g., for the Chernov metric without a continuous isometry group) we could significantly expand the class of p ossible p olyingles. In this case, only the restriction not to use other symmetric metrics except for the initial metric enables us to restrict ourselves to only these p olyingles. Now we prove some prop erties of these p olyingles. Lemma 2. Let gij be a symmetric metric of rank 2 al l of whose off-diagonal elements are zero. If al l diagonal elements are nonnegative, then the fol lowing inequality holds: (A, B ) (A, A)(B, B ), where (A, B ) = If g11 > 0, then for time-like vectors A1 > 0, and also (A, B ) (A, A)(B, B ). 754 B 1 > 0, (A, A) > 0, (B, B ) > 0, gii 0, i > 1, gij Ai B j .


Proof. Consider the quadratic expression f (x) = (xA + B ,xA + B ) = (A ,A )x2 +2(A ,B )x +(B ,B ), where A i = |Ai |, B i = |B i |. In the first case, the quadratic form is p ositive-definite and hence the discriminant D = (A ,B )2 - (A ,A )(B ,B ) of the form is nonp ositive. Therefore, we have the estimates (A, B )2 (A ,B )2 (A ,A )(B ,B ) = (A, A)(B, B ). In the second case, the quadratic form is not p ositive-definite and hence the discriminant is nonnegative. Therefore, (A, B )2 (A ,B )2 (A ,A )(B ,B ) = (A, A)(B, B ). The lemma is proved. In fact, in this proof we have not used the diagonality of gij . The main thing here is not the p ositive definiteness of the corresp onding quadratic form, but the existence of x such that (Ax - B, Ax - B ) 0. Lemma 3. If a metric of rank k is of Lorentz type, then for any positive vectors Ai , l = 1,... ,k, the l fol lowing inequality holds : (A1 ,... ,Ak )2 (A1 ,... ,Ak )(A1 ,... ,Ak ). (16)

Proof. This lemma immediately follows from the second case of Lemma 2 with using accompanying metrics. Lemma 4. Any linear combination of metrics of Lorentz type with positive coefficients is also a metric of Lorentz type. This lemma follows directly from the definition. Lemma 5. Permanent metrics (i.e., metrics subordinated to the Berwald­Moor metric and conformal ly equivalent to the metric dsk = k ) are metrics of Lorentz type. Proof. This lemma is implied by the fact that all accompanying metrics of rank 2 can b e represented as linear combinations with p ositive coefficients of metrics of Lorentz typ e. Lemma 3 implies the following assertion. Prop osition. Al l k -ingles take values not less than 1 on positive vectors. Proof. Note that on p ositive vectors, metrics of Lorentz typ e satisfy prop erty (16). Therefore, successively amalgamating the partition of k into summands, we decrease the product in the geometric mean. For simplicity, we illustrate this by examples of tringles and "quadringles." For tringles, we write the following prop erty: (A, B , C )6 (A, A, C )(B, B, C )(A, B , A)(C, B , C )(A, B , B )(A, C, C ). Raising to the p ower 1/6 and dividing by |A||B ||C |, we obtain the following inequality: 1 2 . Similarly, we write inequality (16) for six terms: (A, A, C )2 (A, C, C )(A, A, A), (B, B, C )2 (B, C, C )(B, B, B ), ... . 755


Multiplying them and dividing the left-hand side by the right-hand side (all terms are p ositive for p ositive vectors), we obtain the inequality 1 2 1. Further, for k = 4 we enumerate the partitions of 4: (1) 1 +1+1 +1; (2) 2 +1+1; (3) 2 + 2; (4) 3 + 1; (5) 4. For any of the six variants we write an inequality of the form (16) (A, B , C, D )2 (A, A, C, D )(B, B, C, D ),... and multiply them; we obtain 1 2 . Multiplying the twelve inequalities of the form (A, A, C, D )2 (A, A, C, C )(A, A, D , D ), we have 2 3 . Further, multiplying the six inequalities of the form (A, A, B , B )2 (A, A, A, B )(A, B, B, B ), we have 3 4 . Finally, multiplying the six inequalities of the form (A, A, A, B )4 (A, A, A, A)2 (A, A, B , B )2 (A, A, A, A)(A, A, A, B )(A, B, B, B ), we obtain the final inequalities for quadringles: 1 2 3 4 1. Inequalities for higher k-ingles can b e obtained similarly. Now we consider calculations of k -ingles up to second order of smallness in the case where the vectors do not greatly differ from the unit vector: Ai = 1 + xi , l l |xi | l 1. (18) (17)

This corresp onds to the case of vectors whose spatial coordinates satisfy the inequality max(|dx|, |dy |, |dz |) v 1. dt,

i.e., vectors corresp onding to small velocities (compared to the sp eed of light):

We denote the sub ordinated second-order metric by gij , the first-order metric by gi =
j

gij ,

and the zero-order metric by g=
i

gi .

756


The following relations hold:
k n

(A1 ,... ,Ak ) = g +
l=1 i=1

gi xi + l
1l
gij xi xj + O 3 , lm

(19)

and |Al | = g + k
n i=1

gi xi + l

k(k - 1) 2
1/k

1/k gij xi xj ll
i,j n i=1

+ O 3 k-1 2g k-1 2g2
2

+ O 3 .

=g

1+ 1 g

gi xi + l

gij xi xj - ll
i,j

gi x
i

i

Dividing the k-scalar product by the product of norms, we see that k -ingles are identically equal to 1 up to the first order of smallness. This implies the following: 1 = 1 + 1 g gij xi xj - lm
1l
k-1 2g

gij xi x l
l,i,j

j l 2

k-1 + 2g2

gi xi l
l i

-

1 g2

gi x
1l
i l i

gi xi + O 3 . (20) m

Other k-ingles have a similar form. However, only k-ingles calculated up to the second order of smallness are invariant under Lorentz transforms that preserve sub ordinated metrics up to the second order of smallness. Clearly, the symmetry group of a function calculated with less accuracy is not less than the exact symmetry group (the exact symmetry preserves approximations of all orders). Therefore, the question arises, what is the exact symmetry group. To this end, we can calculate k-ingles up to third order and recognize which symmetries of second order are also symmetries of third order. However, this approach is too cumb ersome. The following approach is simpler. Assume that the symmetry group G of a certain function is broader than the group K of conformal transformations. If G contains a certain rotation g (circular or hyp erb olic), then it also contains all rotations ag a-1 for any automorphism a of the algebra of p olynumb ers. It is easy to show that the minimum group containing the group of Bervald­Moor conformal transformations and elements g and ag a-1 is the complete group Gl(n). As was noted ab ove (see Lemma 1), such a function must b e a constant, which contradicts the necessary condition (5) and the definition of a k-ingle. Therefore, this remark completely solves the problem of p olyingles. It remains to verify that the tringles 1 ,2 ,... introduced ab ove are functionally indep endent. This question is intuitively clear, and to prove indep endence of the tringles mentioned, we restrict ourselves to sub ordinated metrics for the Bervald­Moor metrics. In such p olynumerical metrics, the scalar products are calculated as p ermanents and represent some average values. The k-ingles themselves represent a certain typ e of correlation (more exactly, the value reciprocal to the correlation) b etween the values of the coordinates. In the case where all coordinates are p ositive, they are b ounded from b elow by 1 (the strong correlation) and they are always greater than 1 if the set contains miscellaneous vectors. Moreover, for p ositive vectors we can order these k -ingles as follows: 1
p(k )-1

··· 2 1 . 757


First, we consider the case of H2 . As was mentioned ab ove, we obtain in this case the Minkowski metric x1 + x2 x1 - x2 ds2 = dt2 - dx2 , dt = , dx = . 2 2 Therefore, bingles characterize hyp erb olic rotations; they are not defined for two arbitrary vectors. The same situations occurs also for the case k > 2. Calculating tringles, we can combine one vector C with the unit vector by a conformal transformation; then the vectors A and B b ecome A and B , resp ectively, and the tringle can b e calculated by the sub ordinated scalar product of rank 2: 1 = (A ,B , 1) . |A ||B ||1|

However, the norms in the denominator are calculated by a metric of rank 3, which need not coincide with the norms in the sub ordinated metric of rank 2. Consider the case H3 , k = n = 3. We have 1 = 2 = per(x, y , z ) , [p er(x, x, x)p er(y, y, y )p er(z, z, z )]1/3 [p er(x, x, y )p er(x, x, z )p er(x, y , y )p er(y, y, z )p er(x, z , z )p er(y, z, z )]1/6 . [p er(x, x, x)p er(y, y, y )p er(z, z, z )]1/3

To prove the functional indep endence of these quantities, it suffices to present examples of sets of vectors for which one quantity is the same whereas the other is not. Consider an arbitrary set of three vectors x, y , and z one of whose coordinates is equal to 0 and the other two coordinates are p ositive numb ers. Then 1/1 = 0 for any such sets whereas the quantities of 2 can differ, which proves their functional indep endence. The case of the Chernov metric (k = 3, n = 4) is reduced to the previous case if we consider sets of vectors in which one fixed coordinate is equal to 0. The case H4 , k = n = 4, is reduced to the case H3 if one of the four vectors is E = (1, 1, 1, 1). In this case, we obtain four typ es of 4-ingles. One can prove their functional indep endence and that they are ordered for sets with p ositive coordinates. The question also arises, aren't elliptic functions used for calculation of tringles for the Berwald­ Moor metric of rank 3? An analysis shows that in this case elliptic functions do not play a key role. Elliptic functions and the theory of Ab elian varieties (in the case where a variety is a commutative group whose op eration is represented by a rational function of coordinates of the summands) are applicable for the so-called Penrose isotropic cones (see [1, 2, 4, 6, 8­12, 14­27]). Acknowledgment. This work was partially supp orted by the Russian Foundation for Basic Research (Pro ject No. 12-01-00020-a). REFERENCES 1. M. Anastasiei and R. Miron, "Preface for generalized Finsler metrics," in: Finsler Geom.: J. Summer Res. Conf., Seattle, Wash., July 16­20, 1995, Providence, Rhode Island (1996), pp. 157­159. 2. A. K. Aringazin and G. S. Asanov, "Problems of finslerian theory of gauge fields and gravitation," Repts. Math. Phys. Warsz., 25, 183­241 (1988). 3. E. F. Beckenbach and R. Bellman, Inequalities, Springer-Verlag, Berlin­G¨ ottingen­Heidelb erg (1961). 4. T. Q. Binh, "Cartan-typ e connections and connection sequences," Publ. Math., 35, Nos. 3-4, 221­229 (1985). 5. P. M. Cohn, Universal Algebra, Reidel, Dordrecht, Netherlands (1981). 758


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