CC BY
30
UDC 531.55+514.85
Tachyon Gas as a Candidate for Dark Matter Y. A. Rylov
Institute for Problems in Mechanics, Russian Academy of Sciences 101-1, Vernadskii Ave., Moscow, 119526, Russia
In the physical geometry (i.e. in geometry, described completely by its world function) identical geometric objects have identical description in terms of the world function. As a result spacelike straight segment is a three-dimensional surface even in the space-time geometry of Minkowski. Tachyons have two unexpected properties: (1) a single tachyon cannot be detected and (2) the tachyon gas can be detected by its gravitational influence. Although molecules (tachyons) of the tachyon gas moves with superluninal velocities, the mean motion of these molecules appears to be underluminal. The tachyon gas properties differs from those of usual gas. The pressure of the tachyon gas depends on the gravitational potential and does not depend on temperature. As a result the tachyon gas may form huge halos around galaxies. These halos have almost constant density, and this circumstance can explain the law of star velocities at the periphery of a galaxy. Properties of the tachyon gas admit one to consider it as a dark matter.
Key words and phrases: discrete geometry, tachyon, dark matter, dark energy, rotation curves.
At the metric approach to geometry the space-time geometry is described in terms of the world function and only in terms of the world function. All geometrical objects and all geometrical quantities are expressed in terms of the world function a. Such a representation of geometrical quantities will be referred as a-immanent representation. At the metric approach two different regions R1 and R2 of the space-time may have different geometries described relatively be world functions a1 and a2. Let a physical body, having the shape G1 = g1(v1) in the region R1 evolves as a free moving body and appears in the region R2 with other geometry. The shape of the body is described now as G2 = g2(a2). How are functions g1 and g2 connected? As far as the physical geometry is a monistic construction, which is described completely by the only quantity (world function), the only possibility may take place
The conventional (Riemannian) space-time geometry is pluralistic. It is described by several basic geometrical quantities, whose properties are described by axioms. In the pluralistic conception of geometry it is very difficult to consider the problem of geometrical objects identification in different geometries. This problem is not considered in the general relativity, which uses different geometries for different regions of the space-time. The only geometric object which is considered in the general relativity is the world line of a free pointlike body. It is supposed that the world line of a free body is a geodesic.
In the framework of Riemannian space-time geometry the shape of a geodesic is determined by the metric tensor. This conventional definition of the world line of a free body agrees with the definition (1) for timelike world lines. However it disagrees with (1) for spacelike world lines, because in the physical geometry a spacelike straight segment is not a one-dimensional line. It is a three-dimensional surface. It is easy to verify, using definition of the straight segment 7[p0 p^ between points P0 and P1
1. Introduction
g i(^) = 92(^) = g(v)-
(1)
Received 11th June, 2012.
Indeed, in the 4-dimensional space-time one equation (2) describes 3-dimensional surface, in general. For timelike distances this surface degenerates into one-dimensional line, because in this case the distance satisfies the anti-triangle axiom
V2a(Po,P2) + < ^2cj(Po,Pi), a(Pt,Pk) > 0, i,k = 0,1, 2. (3)
For spacelike distances the triangle axiom (3) is not fulfilled, and the set of points R satisfying equation (2) is 3-dimensional.
Of course, points of any segment of the "straight" line x = vt + x0, |v|2 > c2 satisfy the relation (2), but it is only a small part of points R satisfying (2).
Our conceptual logical consideration disagrees with the general opinion that the segment of straight is a one-dimensional set in any geometry. For instance, Blumental constructed the distance geometry [1], where he used metric approach to geometry with distance which does not satisfy the triangle axiom. Blumental failed to construct a curve in the framework of the metric approach. He was forced to define a curve as continuous mapping of a segment of the numerical axis onto the space, where the geometry is given. According to this definition the straight line is a one-dimensional set, that cannot be formulated in terms of a distance. It is a remnant of the pluralistic geometric conception.
Ellipsoid ££p0ptp3 is defined in terms of distance
£CPoPlP3 = {R^2a(Po,R) + ^j2a(Pl,R) = + , (4)
where points P0, Pi are focuses of the ellipsoid, and P3 is some point on the surface of the ellipsoid.
Degenerated ellipsoid, where the point P3 on its surface coincides with one of focuses is by definition segment T[p0p1] = ££p0p1p1 of straight between focuses P0, Pi. In the geometry, where distance satisfies the triangle axiom the degenerated ellipsoid is a one-dimensional set. However, when triangle axiom is not satisfied the degenerated ellipsoid is a (n — 1)-dimensional surface in n-dimensional space.
The straight segment is defined in the Euclidean geometry by the relation (2). In the same form it is defined in the space-time geometry of Minkowski. In the proper Euclidean geometry any smooth curve line is defined as a limit of a broken line, when lengths of its links (straight segments) tend to zero. In the physical geometry a curve is defined in the same form. If the curve describes a world line of a free particle, the vectors describing adjacent links of the broken line are equivalent. Equivalence of vectors means that vectors are in parallel and their lengths are equal. For timelike world line these conditions lead to one-dimensional straight line. For the spacelike world line (tachyon) these conditions lead to a world chain with wobbling links. Amplitude of this wobbling is infinite and any link is an infinite three-dimensional surface. A single tachyon described by such a world chain cannot be detected. However, the tachyon gas may be detected by its gravitational field.
Tachyon gas is considered here, because the tachyon gas has characteristic properties of so-called dark matter. On one hand, one failed to detect single particles of the dark matter. On the other hand, the dark matter form a huge halos around galaxies with almost constant mass distribution inside the halo. Existence of such halos is discovered by its gravitational influence on the star velocities in the galaxy periphery. Tachyon gas has similar properties. A single tachyon cannot be detected according to geometric properties. Besides, tachyon gas has almost constant mass density in the gravitational field of a galaxy.
Tachyon is a hypothetical faster-than-light particle. Its rest mass is imaginary. Such particles have not been detected. First such particles were considered by A. Sommerfeld [2]. Particles with negative and imaginary masses were investigated by Ya. P. Terletsky [3]. Tachyons were investigated also by other investigators [4-7]. One considered not only tachyons, but also tachyonic fields which are results of the tachyon quantization.
Unfortunately, effective description of tachyons is possible only in a discrete spacetime geometry. Conventional consideration of tachyons in the continuous Riemannian space-time geometry leads to conclusion that tachyons do not exist, whereas investigations of tachyons in the framework of a discrete space-time geometry leads only to the conclusion that a single tachyon cannot be detected. Impossibility of the tachyon detection does not mean that tachyons do not exist. Tachyons may exist, but one cannot detect a single tachyon, even it will appear that a tachyon may interact with some elementary particle. For instance, neutron decays spontaneously into proton, electron and neutrino. However, one cannot be sure that this decay is not a result of collision with tachyon, because the tachyon gas may fill the whole universe with almost constant density. In this relation the tachyon gas properties remind the vacuum properties.
Such unusual properties of tachyons are conditioned by the fact that in the discrete space-time geometry there are world chains instead of smooth world lines. Links of the tachyon world chain are spacelike segments. Two adjacent points of the tachyon world chain are divided by very large spatial distance. Discovering one point of this world chain, one cannot detect the another point of the world chain.
Crucial point of our investigation is a use of the discrete space-time geometry, whose properties differ strongly from properties of the Riemannian geometry and other continuous geometries. Conventional mathematical technique of differential geometry is inadequate in the discrete geometry. Linear vector space, which is a foundation of the differential geometry, cannot be introduced in the discrete geometry. Introducing the linear vector space formalism in the discrete geometry, one obtains multivalence of such operations as summation of vectors and decomposition of a vector into components. The only quantity which is common for continuous geometry and the discrete one is
the distance d or the world function a = ^d2.
If one considers a discrete space-time geometry, one may not use quantum principles, because for usual particles of positive rest mass (tardions) the quantum principles are corollaries of the space-time geometry discreteness. Consideration of quantum principles in the discrete space-time geometry reminds description of Brownian motion in terms of thermogen (in terms of axiomatic thermodynamics). If the elementary length Ao of discrete space-time geometry is connected with the quantum constant h by means of the relation A2 = h/bc (constants h, b, c are universal constants), the quantum effects for tardions can be explained as geometrical effects of the discrete space-time geometry [8]. In such a situation it is useless to quantize tachyons and to consider tachyonic fields. One should consider tachyons as classical particles in the discrete space-time geometry.
Mathematical technique of differential (continuous) geometry cannot be applied to a discrete geometry. In the discrete geometry there are no continuous world lines, there are no differential equations and differential relations. One may not use the phase space of coordinates and momenta for description of the particle state, because the momentum is a result of differentiation along the continuous world line. But one cannot use differentiation in the discrete geometry. In the discrete space-time geometry the particle state is described by two points Ps, Ps+1. Vector PSPS+1 describes the geometric momentum of a particle, and its geometric mass ^ = |PSPS+1| determines the usual particle mass m by the relation
m = b^, (5)
where b is an universal constant. The particle dynamics in the discrete space-time geometry is described by the skeleton conception [9], where instead of the continuous world line one uses the world chain C (broken line), whose links are vectors PsPs+1 of the same length ^
C =U PsPs+1, |PsPs+11 = » = const, s = ... 0,1, 2,
5
For free particle the adjacent vectors PsPs+i and Ps+iPs+2 are equivalent (PsPs+i = Ps+iPs+2). It means that
((PsPs+i.Ps+i Ps+2) = |PsPs+i | ■ |Ps+iPs+2|) A (|PsPs+i| = |Ps+i Ps+2|). (7)
If the vector PsPs+i is fixed the equivalence relation (7) determines the adjacent vector Ps+iPs+2 ambiguously, provided the space-time geometry is discrete. As a result the world chain wobbles. Amplitude of this wobbling is of the order of the elementary length A0 for tardions (y2 > 0). This wobbling is a reason of quantum effects. For tachyons (^2 < 0) amplitude of this wobbling is infinite.
For tachyons the spatial distance between adjacent points Ps and Ps+i is random, and it may be infinitely large. As a result one cannot detect a single tachyon. In other words, single tachyons were not discovered in experiments, because they are unobservable, but not because they do not exist.
However, if one cannot detect a single tachyon, it does not mean that one cannot observe the gravitational influence of the tachyon gas, consisting of many unobserv-able tachyons. Unobservable tachyons may form so-called dark matter, which form large spherical halo around some galaxies. Existence of such a halo is necessary for explanation of the rotational velocities of stars (rotation curves) in some galaxies [10]. In these galaxies the rotational velocities of stars do not depend practically on the distance r from the galaxy core. Sometimes the star velocities increase arise with increasing of the distance r. If the gravitating mass is concentrated in the galaxy core, then the Newtonian force of gravitation is proportional to r-2, and rotational velocity is to be proportional r-i/2. Inside the gravitating sphere with uniform distribution of the mass the Newtonian gravitation force is proportional to r, and the rotational velocity is proportional to r.
In this paper we try to calculate parameters of the tachyon gas in order to determine, whether the tachyon gas can fill the halo of galaxies with necessary density.
2. Discrete Space-Time Geometry
Discrete geometry is obtained as a generalization of the proper Euclidean geometry Qe, which is constructed usually as a logical construction. Conventionally one uses the Euclidean method, when all statements of QE are deduced from a system of axioms, describing properties of simplest geometrical objects of QE. The Euclidean method is inadequate for construction of the discrete geometry Qd. Inadequacy of the Euclidean method is connected with the fact, that one does not know how the simplest geometrical objects of Qe look in other geometries. For instance, the straight segment T[p0p1] between the points P0 and Pi is one-dimensional line in QE, whereas 7[p0p1] is a surface in Qd. There is only one quantity, which is common for QE and Qd. It is the distance d(P0,Pi) between two arbitrary points P0 and Pi of the point set Q, where
the geometry is given. It is more effective to use the world function a = ^ d2 instead
of the distance d, because the world function a is always real (even in the geometry of Minkowski, where d may be imaginary).
The world function a is a real single-valued function. It is defined by the relation
a : Q x Q ^ R, a(P,Q) = a(Q,P), a(P,P) = 0, VP,Q e fi. (8)
To generalize QE onto Qd, one needs to describe QE in terms of the Euclidean world function aE. Thereafter replacing aE by the world function ad of Qd in all statements of Qe, one obtains all statements of Qd. The world function ad of Qd may be taken in the form
\2
ad(P,Q) = au(P,Q) + Ysgn(°u(P,Q)), VP,Q e Q, (9)
where ctm is the world function of the Minkowski geometry QM, and A0 is the elementary length. Due to relation (9) in Qd all distances satisfy the relation
IPd(P,Q)l = | V2<Td(P,Q)| / (0,Ao), vP,Q e Q. (10)
Being presented in terms of the world function cte, the proper Euclidean geometry Qe contains two kinds of relations: (1) general geometric relations, which contains only world function cte, and (2) special relations of the geometry which are constraints, imposed on the world function cte. The approach, when a geometry is described in terms and only in terms of the world function, will be referred to as metric approach. Any geometry described completely by the world function will be referred to as a physical geometry.
Let us adduce some general geometric definitions which are important in the particle dynamics:
Vector PQ is an ordered set {P, Q} of two points P, Q (but not an element of the linear vector space as usually). Scalar product (P0P1.Q0Q1) of two vectors P0P1 and QoQ1 is defined by the relation
(P0P1.Q0Q1) = ct(P0, Q1) + CT(P1,Q0) - CT(P0,Q0) - CT(P1,Q1). (11)
The length |PQ| of the vector PQ is defined by the relation
|PQ|2 = (PQ.PQ) = 2a(P, Q), (12)
n vectors P0P1, P0P2, .... P0P„ are linear dependent, if and only if the Gram determinant
Fn(Pn) = det ||(P0P,.P0Pfc)|| , i,k = 1,2,.. .n, Vn = {P0,P2,...Pn} (13) vanishes
Fn(Vn) = 0. (14)
Two vectors P0P1 and Q0Q1 are equivalent (equal) (P0P1 eqv Q0Q1), if the vectors are in parallel
(P0P1 n Q0Q1): (P0P1.Q0Q1) = IP0P1I ■ IQ0Q1I , (15)
and their lengths are equal
a(P0,P1 ) = a(Q0,Q1). (16)
According to (15), (16) the equivalence definition has the form (7)
P0P1eqvQ0Q1 : (P0P1.Q0Q1) = IP0P112 A IP0P1I2 = IQ0Q1I2 . (17)
All general geometric relations (11)-(17) are obtained as properties of the linear vector space. However, they do not contain any reference to the linear vector space. They are written in terms of the world function cte of the proper Euclidean geometry, and they may be used in any physical geometry even in the case, when one cannot introduce linear vector space in this geometry. To use the relations (11)-(17) in a discrete geometry, it is sufficient to use the world function ad of the discrete geometry Qd in them.
Formally general geometric relations (11)-(17) realize some processing of information, contained in the world function. Such a processing is to be universal, i.e. it is uniform for all generalized geometries. This method of processing is known for the proper Euclidean geometry QE. It may be applied for construction of general geometric relations for other generalized geometries. In the case, when one can introduce linear vector space, such a processing admits one to construct the particle dynamics
in the space-time geometry, equipped by the linear vector space. As far as the general geometric relations (11)-(17) are universal in the sense that they do not refer to the linear vector space, they may be used for construction of the particle dynamics in those space-time geometries, where introduction of the linear vector space is impossible.
Such a construction of geometry is very effective, because it does not need proofs of numerous theorems and a test of the axioms compatibility. Besides, the geometry can be constructed in the coordinateless form. Monistic character of the geometry (description in terms of one basic quantity — world function) provides automatically a correct connection between all secondary quantities in all physical geometries. Ascertainment of a connection between different geometric quantities is the main problem of a pluralistic construction of a geometry, which is based on a use of several independent basic quantities.
The special relations of the proper Euclidean geometry have the form [11]:
I. Definition of the metric dimension:
3Vn = {Po,Pi,...Pn} C Q, Fn(Vn) = 0, Fk(Qfc+i) = 0, k > n, (18)
where Fn(Vn) is the n-th order Gram's determinant (13). Vectors P0P^, i = 1, 2,.. .n are basic vectors of the rectilinear coordinate system Kn with the origin at the point P0. The covariant coordinates of the point P in the coordinate system Kn are defined by the relation
xl(P) = (PoPl.PoP), i = 1, 2,... ,n. (19)
The metric tensors gik(Vn) and glk(Vn), i,k = 1, 2,... ,n in Kn are defined by the relations
k=n
^ gik (Vn)gik(Vn) = sf, gu (Vn) = (PoPl.PoPi), i,l = 1, 2,...,n. (20)
fc=i
II. Linear structure of the Euclidean space:
i,k=n
MP,Q) = ^ ^ 9tk(Vn){xi(P) — Xi(Q))(xk(P) — xk(Q)), VP,Q e Q, (21)
i,k=i
where coordinates Xi(P), Xi(Q), i = 1, 2,... ,n of the points P and Q are covariant coordinates of the vectors P0P, P0Q respectively in the coordinate system K. III: The metric tensor matrix g\k (Vn) has only positive eigenvalues gk
gk > 0, k = 1,2,...,n. (22)
IV. The continuity condition: the system of equations
(PoPi.PoP) = Vi e R, i = 1,2,... ,n (23)
considered to be equations for determination of the point P as a function of coordinates y = {Vi}, i = 1, 2,...,n has always one and only one solution. Conditions I-IV contain a reference to the dimension n of the Euclidean space, which is defined by the relations (18).
Special relations of the proper Euclidean geometry QE may be not valid for other physical geometries. In some cases these relations may used partly. For instance, the metric dimension may be defined locally. Instead of constraint (18) one uses the condition
VPo e Q, 3Vn = {Po,Pi,...Pn}c Q, Fn(Vn) = 0, Fk(Vk) = 0, k > n, (24)
where all skeletons Vn contain only infinitely close points. The conditions (24) determine the metric dimension for locally flat (Riemannian) geometry.
All relations I-IV are written in terms of the world function. They are constraints on the form of the world function of the proper Euclidean geometry.
The proper Euclidean geometry looks in the ^-representation quite different, than in conventional representation on the basis of the linear vector space. For instance, such a quantity as dimension has two different meanings in the ^-representation. On one hand, the metrical dimension nm is the maximal number of linear independent vectors, which is determined by the relations (18). On the other hand, the coordinate dimension nc, is the number of coordinates, which is used at the description of the point set Q. In the proper Euclidean geometry the coordinate dimension coincides with the metric dimension (nc = nm), and this fact is a corollary of special (not general geometric) relations (18), (19).
In general, the coordinate labelling of points of Q has no relation to the geometry. In the proper Euclidean geometry the two dimensions coincide, because the coordinate dimension nc is determined by the special conditions (18), (19), which are characteristic for the proper Euclidean geometry. In the geometry Q& the number nm of linear independent vectors is more, than the number of coordinates nc. For instance, for six points V5 = {P0, P1,..., P5} and five vectors
P0P1 = {1,0,0, 0} , P0P2 = {0,1,0,0,0} , P0P3 = {0, 0,1, 0} , P0P4 = {0,0,0,/} , P0P5 = {a, 0, 0, 0} ,
the Gram determinant F5(V5) does not vanish in the geometry Qd with the world function (9). One obtains for the case d = A0/2 ^ a2, I2
^5(^5) = d(-a2l6 + 3al7 - 18) + 0(d2). (25)
For five points V4 = {P0, P1...P4}one obtains
F4(^4) = -I8 - 4l6d + 0(d2). (26)
It means that, in general, the metric dimension nm ^ 5 in Qd. In Qd the metric dimension nm cannot coincide with the coordinate dimension nc. It means essentially that one cannot introduce a finite number of linear independent basic vectors and expand space-time vectors over these basic vectors. It is very unexpected, because the conventional construction of a differential geometry (for instance, the Riemannian one) starts, giving n-dimensional manifold with a coordinate system on it. Of course, one assumes, that the maximal number of linear independent basic vectors at any point is equal to n = nm = nc. Only in this case one can expand vectors over basic vectors and use operations, defined in the linear vector space. In the case of a discrete space-time geometry, where nm = nc, the linear vector space cannot be introduced, although the coordinate system can be introduced, and the coordinate dimension nc = 4 as in the space-time geometry of Minkowski. Four coordinates x = {^x°,x1,x2,x3}, xk e R are defined as usually.
Note, that the conditions (18), defining metric dimension nm contain a lot of constraints, and all they are special conditions of Qe. It means that there is a lot of physical geometries, where nm = nc, and one cannot introduce a linear vector space there. In the limit d ^ 0, F5(V5) = 0 in (25), and transforms to QM. In this case the metric dimension nm = 4 coincides with the coordinate dimension nc = 4. It means that one may use approximately the space-time geometry QM in the case, when typical lengths I of vectors is much greater, than the elementary length A0. In this case one may set approximately A0 = 0, and suppose that nm = nc.
The set of the Gram determinants values Fn(Vn), n = 2, 3,... may be such, that one cannot introduce the metric dimension nm. Apparently, the discrete space-time geometries are geometries without a definite metric dimension. Such "dimensionless"
geometries look especially exotic. Contemporary researchers deal only with the Euclidean method, which uses only space-time geometries of definite dimension. They can hardly conceive properties of "dimensionless" space-time geometries. On the other hand, the classical particle dynamics does not work in microcosm, described by the geometry of Minkowski. As far as the discrete ("dimensionless") space-time geometries are not known for most researchers, they use quantum dynamics, which imitates the discrete geometry properties. This imitation is arbitrary and desultory. Besides, this imitation is not complete. There are such properties of real particle dynamics, which cannot be imitated by quantum dynamics in the space-time of Minkowski.
We see that coincidence of metric dimension nm with the coordinate dimension nc and a construction of a smooth manifold with the dimension n = nm = nc is a special property of the proper Euclidean geometry QE, which is not a general geometric property. The conventional Euclidean method of the differential geometry construction starts from the definition of a smooth manifold with fixed dimension. Such a method is not a general method of the generalized geometries construction, because it uses special properties of QE, which, generally speaking, are not characteristic for all generalized geometries. In general, a use of the coordinate description for the generalized geometries construction is a use of special properties of the proper Euclidean geometry Qe for such a construction. Such an approach cannot be a general method of the generalized geometries construction. Using special properties of QE, one obtains only a part of possible generalized geometries. In particular, a use of the coordinate description does not admit one to construct geometries with indefinite metric dimension and with intransitive equality relation. However, the coordinate labelling of points of Q has nothing to do with a construction of a manifold. The coordinate labelling of points may be used always, and it has no relation to a construction of generalized geometries. The coordinate labelling becomes to deal with the generalized geometry construction, when one imposes the condition nc = nm.
The relation nc = nm is a special property of the proper Euclidean geometry QE, and it may be wrong for many physical geometries, because physical geometries may have no definite metric dimension. Using the relation nc = nm at the construction of a generalized geometry, one may meet such a situation, when the real space-time geometries appear beyond the scope of consideration.
3. Dynamics of Particle with Two-Point Skeleton
In the discrete space-time geometry the state of a particle (physical body) is described by its skeleton Vn = {P0,Pi,... ,Pn}, consisting of n + 1 space-time points, connected rigidly. The skeleton may be considered as a discrete analog of a frame connected rigidly with a physical body (particle). Tracing the motion of the skeleton one may trace the motion of the particle. The state of a pointlike particle is described by two-point skeleton Vi = {P0,Pi}. The vector P0Pi describes energy-momentum of the particle, and ^ = |P0Pi| is a geometric mass of the particle, connected with usual mass by the relation (5). Information on position of two skeleton points is sufficient for description of the state of a pointlike particle. Dynamics of the pointlike particle skeleton Vi is described by the world chain (6), (7). According to these relation and definition of the scalar product (11) the dynamic equations for the pointlike particle are written in the form
a(Ps-i,Ps) = a(Ps,Ps+i), s= ... 0,1, 2 ..., (27)
a(Ps-i,Ps+i) = 4a(Ps-i,Ps), s = ... 0,1, 2 .... (28)
In the inertial coordinate system of the Minkowski geometry, where = 1, the points P0,Pi, P2 have coordinates
Po = {xo, x} , Pi = {xo + Po, x + p} , P2 = {xo + 2po + ao, x+2p + a} . (29) The 4-vector a = {ao, a} is a discrete analog of the acceleration vector.
Let us choose world function in the form, which it has in the extended general relativity [12,13] with slight gravitational field described by the gravitational potential V (x)
au(x, x>) = 2 ((c2 - 2V(y)) (X0 - 4)2 - (x - x')2), y = ^^, (30)
where V = V(y) is a gravitational potential at the point y, and the world function ad has the form (9). One obtains in ^d
(c2 - 2V)(P0 + «0)2 - (p + a)2 + eA0 = (c2 - 2V)p20 - p2 + £A0 = »2,
£ = sgn(^2) (31)
((c2 - 2^)(2P0 + «0)2 - (2p + a)2) + eA2 = 4((C2 - 2^)$ - p2 + eA2),
e = sgn(^2). (32)
Here quantities x = {^0, x} , p = {p0, p} are supposed to be given and 4-vector a = {a0,a} is to be determined from dynamic equations (31), (32). It is supposed that
2
\/p2 + £ H2 - eA0
P0 = Vc2 - 2^ . (33)
The dynamic equations have the same form for timelike (^2 > 0, s > 0) and spacelike (^2 < 0, e < 0) world chains. We have two equations for four components of 4-vector a. As a result the solution is not unique, in general.
After transformation of equations (31), (32) one obtains two relations
2ap + 3£Ap
= 2p0(c2 - 2V) , (34)
/ c2 - 2^ - W 3£A0 \2 2 2
£ -o-77- «11---^-^7v ) + £OL\ = r , (35)
V C2 - 2V J \ 11 2P0(c2 - 2V - v2) ) i ' v ;
where r is the radius of the sphere, where ends of the 4-vectors a are placed
„2 = o„\2 , f 3A0 ^
\2P0^(c2 - 2V))
2 ^ 2v2 - (c2 - 2^)
2p^ (c2 - 2^ W (c2 - 2^ - ^2) '
r2 = 3^2 + --„ ,,2 ot, , (36)
pV(c2 - 2^) (37)
v = , v = |v| , (37)
VP2 + e M + eA0
(av) 2 (^)2 «V 2 2 /QtA «11 = , = a - «lb «11 = , all = ~, v = v. (38)
Here ay is the component of 3-vector a which is in parallel with the vector v, whereas ai is the component of 3-vector a, which are perpendicular to the vector v.
In the case of timelike vector P0P1 £ = 1 and v2 < c2. In the nonrelativistic case, when v2 ^ c2, V ^ c2, equation (35) has the form
(«11 - ^ 2 + Oi = r2, r2 ^ 3A0 + 0(A2). (39)
Solution of this equation has the form
3A0
ay = + \/3Ao cos a_i_i = \/3Ao sin$cos p, a_i_2 = \/3Ao sin$ sin p, (40) 2 ß
3AA0v2 + V3Ao^ cos$ + 3 A 2ß 2 ß
ao = + ^/3Aov cos $ + - , (41)
Here are arbitrary real numbers. It means that the difference between adjacent vectors P0Pi and PiP2, described by the 4-vector a, is determined nonuniquely. The particle world chain wobbles with amplitude of the order of A0. Statistical description of this wobbling leads to the Schrodinger equation [8], provided AO = h/(bc).
In the case of tachyon, when vector P0P1 is spacelike, e = —1 and v2 > c2. Equation (35) takes the form
' + 2 V — c2 \ ( 3A0 \2 2 2 (42)
all — o^/ ..2 , OTZ — a2 = r > (42)
(v2 + 2V — c2 \ f v C2 — 2V ) V
c2 — 2V J \ 11 2po(v2 + 2V — c2)
2
2 = 3A2 3Ao ^ 2 «2 + 2V — ^ (43)
r =3Ao — Uo^—WyJ ( «2 +2V — C2) . (43) Solution of equation (42) is also nonunique
3Ao r^c2 — 2V , Q
a =-^-to-TYV +--/ == cosh (44)
11 2po(v2 — ( c2 — 2V )) v2 — c2 + 2V ) V ;
P py/( c2 — 2V )
a± i = rsinh$ cos a± 2 = rsinh$ sin v= — = — , (45)
Po vV — Ißl2 + A0
2ap - 3A2 ^ ())' + ^^^gjS^) - 3A2 (4fi)
a° 2p0{c2 - 2V) 2p0(c2 - 2V) ' (6)
Here •&, p are arbitrary real numbers. But now the wobbling amplitude is infinite because of functions cosh and sinh. The wobbling amplitude is infinite even in the case of space-time geometry of Minkowski, when Ao = 0. In this case equation (42) takes the form
V + 2V - 2 2 n .
«2 - a± = 0, (47)
C2 — 2V )
and its solution has the form
fv2 + 2V — c2 \ v C2 — 2V )
I c2 — 2V I c2 — 2V
a = V ,2 + 2V — C2 r0' a0 = V y2 + 2V — C2 r0V, (48)
a_|_ 1 = r0 cos p, a_i_2 = r0 sin p.
Here p is an arbitrary real number, and r0 is an arbitrary positive number. In this case the wobbling amplitude is infinite because of infinite values of r0.
In relations (44)-(47), (48) one chooses the positive sign for radical
,/ c2 — 2V = 1 V V2 + 2V — c2 y i
2 2V
2 + 2V — 2
Such a choice corresponds to the fact that tachyon does not turn into antitachyon in the process of its motion. This is an additional constraint. It is connected with a definite direction of the time arrow (direction of the time flow). Apparently, the time arrow does exist, but it is taken into account in the solutions, but not in dynamic equations. Equation (42) is invariant with respect to transformation t ^ —t.
Arguments in virtue of the time arrow are as follows. The Maxwell equations are invariant with respect to transformation t ^ —t. However, real electromagnetic interaction between the charged particles is described by the retarded Lienard-Wiechert potential, whereas the advanced potential is not used. This fact may be interpreted as an existence of the time arrow (causality principle). Besides, the universe is asymmetric with respect to existence of particles and antiparticles. Particle differs from antipar-
ticle by direction of the time flow along the world line. Fixed sign of
prevents from permanent transition from tachyon to antitachyon along the world chain of tachyon.
c2 — 2V v2 + 2^ - c2
4. Averaging of the Dynamic Equations Solutions
We try to obtain parameters of the tachyon gas, averaging solution of dynamic equations for single tachyons over distributions (44)-(46). We shall average over parameters
D = ^ (cu)!, (o^) = r2 sinh* (49)
o(r, V, y)
The norm N has the form
e 2^
N = J sinh § dtf J dp = 2^(cosh 6 — 1) = 4^ sinh2 6 . (50)
0 0
It is supposed, that 6 ^ to at the end of averaging. According to (44)
e
(ay) = N-! J sinh •& dtfx
0 2tT
V_3A0_w + cosh
0 \2Po{v2 — (C2 — 2V)) ^2 — (C2 — 2V)) )
3\20v r^(c2 — 2V)
o A + / (cosh 6 + 1). (51)
2P0 {v2 — (c2 — 2y^ 2yJ{V2 — (C2 — 2V))
As a result one obtains
(ay) = rV(c2 — 2ZL= cosh 6 + O(60), ((a±)i) = ((a±h) = 0. (52) 2^ (v2 — (c2 — 2V))
According to (46) one obtains
(«0) = — (ay) + ^(60) =-cosh 6 + 0(60). (53)
P0 2p0^ {v2 — (c2 — 2^))
One obtains for the module u of the mean 3-velocity vector u = / P + (a\ \,
\po + (ao) /
P + (« Kpo + (ao)
defined by the mean vector (PiP2)
/ ^ \ \ ~dt /
(ao) p
l— (c2 - 2V) + 0(6-1), (54)
According to (33) and (54) the mean velocity is less, than speed of the light
u = ?!(C2 - 2V) = J\ - ^ - A° <c. (55)
p y p2
The mean tachyon gas velocity u = 0, if p2 = |p|2 - AO.
The mean vector (P1P2) is timelike although vector P1P2 is spacelike. It means that averaged particles of the tachyon gas look as tardions. In other words, the mean tachyon is a tardion. It is unexpected, but the tachyon gas may be considered as a usual gas.
One obtains for components of the mean squared velocity
2 \ 2 (a"' '"oN 2 OTM2 , „.2 ,
((£)) = 8 = (?)' ^ -)2 + °<e-1> = °2 + °<e-1>' (56) ( (it )2} = W + W = (7)2 ^ - ^ + 2^^ - 2^) + °<e-1>. <57)
Taking into account (37), one obtains
2\ («^1) + (a2±2) + (a2)
«n
11 =(c2 - 2V) + 0(0-1). (58)
The pressure P of the tachyon gas is defined by the relation
1 " " ^
№) -< I dx I >*)■ (59)
* = 3 Mil ^
Here p = p(x) is the tachyon gas mass density. It follows from (58) and (54) that
P(x) = 3p(x) (c2 - 2V(x) - u2(x)). (60)
3
All tachyon gas parameters p, u, P are considered as function of the space-time points x = {:e0, x}.
In the gravitational field of a galaxy the tachyon gas is at rest (u = 0), if the balance condition is fulfilled
VP = pVV. (61)
According (60) this condition is written in the form
1 5
3(c2 - 2^(x))Vp =3pVV(x). (62)
Equation (53) is integrated in the form
u
Poc5
\/lc2 - 2V(x)|5
(63)
Here p0 = const.
In the case of spherically symmetric gravitational field of a galaxy one obtains instead of (63)
p(r)
poc5
^lc2 - 2V(r)|£
(64)
If the gravitational field is not strong and V(r) ^ c2, the potential V(r) may be approximated by the expression
V( )
GM 4nG -+
3
-por
(65)
Here G is the gravitational constant and M is the mass of the galaxy. The expression (65) takes the form
p(r) =
PoCu
^\c2 - ^ - 8-iGpor2
Po[ 1 +
5GM 20kG
+
2
3 2
~Por
(66)
If po is large enough and 20^por2 > 15Mr-1, the density p(r) may even increase with increase of r. At any rate the second term in (66) slacks the decrease of density p(r) with increase of .
It follows from (63) that the tachyon gas density is larger in regions with larger gravitational potential. It means that the tachyon gas is attracted to massive bodies as usual tardion gas. Besides, the tachyon gas density changes rather slowly with the change of the gravitational potential, whereas in isothermal atmosphere this dependence is exponential. Slowly dependence the tachyon gas density on the gravitational potential facilitates formation of halo with the almost constant tachyon gas density.
Remark. Averaging solutions of the dynamic equations, one supposed, that the gravitational potential V was constant. In general, one should take into account the fact that potential V depends on coordinates and, hence, on the 4-vector a. We hope that our approximation does not change the tachyon gas properties essentially. Two main properties of the tachyon gas (its strong mobility and very high pressure) depend slightly on the form of gravitational potential.
P
5
5. Dark Energy
There is an impression that many cosmological problems are connected with a use of Riemannian space-time geometry, which is inadequate in application to general relativity, because the methods of differential geometry describe only a small part of possible space-time geometries. Observation of accelerated expansion of universe is explained usually by so-called dark energy. There are different version of the dark energy nature [14,15], but all these versions try to explain cosmic antigravitation which is a reason of of accelerated expansion of universe. Conventional general relativity, based on the Riemannian space-time geometry can explain antigravitation only by means of negative mass, by negative pressure or by so-called A-term, taken with a proper sign.
The expanded general relativity (EGR) uses more general class of possible spacetime geometries. In the physical geometries of EGR [12,13] a spherical dust cloud of radius R and of the mass M cannot collapse and form a black hole. Decreasing radius R, the parameter s = 2GM/(c2R) becomes large enough, a region of antigravitation
arises in the center of the cloud. The antigravitation prevents from appearance of the dark hole. Impossibility of collapse prescribes another scenario for gravitational contraction of the dust cloud, than the conventional scenario. When the radius R of the cloud decreases as a result of gravitational contraction, the parameter e increases, and inside the cloud the region of antigravitation appears which prevents from the further contraction. However, the cloud contraction continue by inertia. When parameter s becomes large enough, the contraction stops, and the opposite process begins. The central region of the cloud begin to expand. There are different stages of expansion. At some stages this expansion may be accelerated. At other stages the speed of expansion may be decreased. It is possible that different parts of the central region of the cloud may be at different stages of expansion. It is important that there is no necessity to invent mythical essences like negative pressure and quintessence. One needs only to construct a true model of the universe expansion, based on a correct conception of the space-time geometry.
6. Concluding Remarks
Our conclusions depend on existence and properties of tachyons, and these properties seem to be rather unexpected. This surprise is conditioned by a fundamental change of approach to geometry. Here one uses the metric approach to geometry, when geometry is considered as a science on the shape and dispositions of geometric objects. At such an approach any geometry is described completely by its world function and only by its world function. Although nobody deny the metric approach, the mathematical formalism of differential geometry is based on the idea that any geometry is a logical construction, and all statements of a geometry can be deduced from several geometric axioms. The logical structure of a geometry is considered as a principal property of geometry. So-called symplectic geometry is considered as a geometry, because its logical structure reminds the logical structure of the Euclidean geometry, although the symplectic geometry has no relation to a description of geometric objects.
Mathematical technique adequate to metric approach was unknown. Attempts of constructing such a technique failed [1,16]. Formalism of world function was suggested by J.L. Synge, who used it for description of the Riemannian space-time geometry [17]. But he failed to obtain coordinateless description of space-time geometry.
Tachyons and their properties can be effectively described only in the framework of a discrete space-time geometry. However, the discrete geometry is nonaxiomati-zable geometry, and it cannot be constructed by the Euclidean method as a logical construction. As a result tachyons appeared outside the scope of the space-time geometry, and one considered them as hypothetical objects, and their properties were unknown.
Now tachyon gas is a real gas, whose gravitational influence can be identified with the gravitational influence of the mysterious dark matter. One succeeded to construct the tachyon gas statics only due to developed coordinateless technique of metric approach to space-time geometry. The tachyon gas dynamics is not yet constructed. Possibility of the tachyon existence follows from the mathematical formalism based on a use of the world function. No new hypotheses on properties of tachyons were used.
References
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2. Sommerfeld A. Simplified Deduction of the Field and the Forces of an Electron Moving in Any Given Way // Knkl. Acad. Wetensch. — 1904. — Vol. 7. — Pp. 345367.
3. Terletsky Y. P. Positive, Negative and Imaginary Rest Masses // J. de Physique at le Radium. — 1963. — Vol. 23, No 11. — Pp. 910-920.
4. Bilaniuk O.-M. P., Deshpande V. K, Sudarshan E. C. G. "Meta" Relativity // American Journal of Physics. — 1962. — Vol. 30, No 10. — P. 718.
5. Feinberg G. Possibility of Faster-Than-Light Particles // Physical Review. — 1967. — Vol. 159, No 5. — Pp. 1089-1105.
6. Feinberg G. Lorentz Invariance of Tachyon Theories // Phys. Rev. D. — 1978. — Vol. 17. — P. 1651.
7. Komar A., Susskind L. Superluminal Behavior, Causality, and Instability // Phys. Rev. — 1969. — Vol. 182, No 5. — Pp. 1400-1403.
8. Rylov Y. A. Non-Riemannian Model of the Space-Time Responsible for Quantum Effects // Journ. Math. Phys. — 1991. — Vol. 32, No 8. — Pp. 2092-2098.
9. Rylov Y. A. Discrete Space-Time Geometry and Skeleton Conception of Particle Dynamics // Int. J. Theor. Phys. — 2012. — Vol. 51, No 6. — Pp. 1847-1865. — See also e-print 1110.3399v1.
10. Merritt D. et al. Empirical Models for Dark Matter Halos. I. Nonparametric Construction of Density Profiles and Comparison with Parametric Models // The Astronomical Journal. — 2006. — Vol. 132, No 6. — Pp. 2685-2700. — DOI: 10.1086/508988.
11. Rylov Y. A. Geometry without Topology as a New Conception of Geometry // Int. Jour. Mat. & Mat. Sci. — 2002. — Vol. 30, No 12. — Pp. 1847-1865. — See also e-print math.MG/0103002.
12. Rylov Y. A. General Relativity Extended to Non-Riemannian Space-Time Geometry. — e-print 0910.3582v7.
13. Rylov Y. A. Induced Antigravitation in the Extended General Relativity // Gravitation and Cosmology. — 2012. — Vol. 18, No 2. — Pp. 107-112.
14. Чернин A. Д. Физический вакуум и космическая антигравитация // УФН. — 2008. — Т. 178, № 3. — С. 267. [Chernin A.D. Physical Vacuum and Cosmic Antigravitation // Uspechi Fizicheskich Nauk. — 2008. — V. 178, No 3. — P. 267 ]
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УДК 531.55+514.85
Тахионный газ как кандидат на тёмную материю
Ю. А. Рылов
Институт проблем механики, РАН Россия, 119526, Москва, Пр. Вернадского, 101-1
В физической геометрии (т. е. геометрии полностью описываемой её мировой функцией) тождественные объекты имеют одинаковое описание в терминах мировой функции. В результате пространственно-подобный отрезок прямой представляет собой трёхмерную поверхность даже в пространственно-временной геометрии Минковского. В дискретной геометрии пространства—времени тахионы имеют два неожиданных свойства: 1 — отдельный тахион не может быть обнаружен; 2 — тахионный газ может быть обнаружен по его гравитационному воздействию. Хотя молекулы (тахионы) тахионного газа движутся со сверхсветовыми скоростями, средняя скорость движения этих молекул оказывается досветовой. Свойства тахионного газа отличаются от свойств обычного газа. Давление тахионного газа зависит от гравитационного потенциала и не зависит от температуры. В результате тахионный газ может образовывать огромные гало вокруг галактик. Эти гало имеют почти постоянную плотность, и это обстоятельство может объяснить кривые вращения звёзд на периферии Галактики. Свойства тахионного газа позволяют рассматривать его как тёмную материю.
Ключевые слова: дискретная геометрия, тахион, тёмная материя, тёмная энергия, кривые вращения.
