Table Of ContentWhat Dimensions Do the Time and Space Have: Integer or Fractional?
L.Ya.Kobelev
Department of Physics, Urals State University
Lenina Ave., 51, Ekaterinburg 620083, Russia
Electronic address: [email protected]
A theory of time and space with fractional dimensions (FD) of time and space (dα, α = t,r)
0 defined on multifractal sets is proposed. The FD is determined (using principle of minimum the
0 functionals of FD) by the energy densities of Lagrangians of known physical fields. To describe
0 behaviour of functions defined on multifractal sets the generalizations of the fractional Riemann-
2 Liouville derivatives Dd(t) are introduced with the order of differentiation (depending on time and
t
n coordinate) beingequalthevalueof fractional dimension. Fordt =const thegeneralized fractional
a derivatives (GFD) reduce to ordinary Riemann-Liouville integral functionals, and when dt is close
J to integer, GFD can be represented by means of derivatives of integer order. For time and space
7 with fractional dimensions a method to investigate the generalized equations of theoretical physics
1 by means of GFD is proposed. The Euler equations defined on multifractal sets of time and space
are obtained using the principle of the minimum of FD functionals. As an example, a generalized
]
h Newton equation is considered and it is shown that this equation coincide with the equation of
p classical limit of general theory of relativity for dt → 1. Several remarks concerning existence of
- repulsive gravitation are discussed. The possibility of geometrization all the known physical fields
e
and forces in the frames of thefractal theory of time and space is demonstrated.
c
a
01.30.Tt, 05.45, 64.60.A; 00.89.98.02.90.+p.
p
s
.
s
c
i
s I. INTRODUCTION use the approximation in which the description of each
y multifractalintervalofthesesetswillbecharacterizedby
h
The problem concerning the nature of space and time middletimemomenttandrefertoeachoftheseintervals
p
[ is one of the most interesting problems of the modern as ”points”) is characterizedby global fractal dimension
physics. Arethespaceandtimecontinuous? Whyistime (FD) dt(r(t),t), and for different intervals FD are differ-
1 irreversible? What dimensions do space and time have? ent( because of the time dependence and spatial coordi-
v
5 How is the nature of time in the equations of modern natesdependenceofdt ). FormultifractalsetsSt (orSr)
3 physicsisreflected? Differentapproaches(quantumgrav- each set is characterizedby global FD of this set and by
0 ity,irreversiblethermodynamics,synergeticsandothers) local FD of this set( the characteristics of local FD of
1 provide us with different answers to these questions. In time and space sets in this paper we do not research).
0 this paper the hypothesis about a nature of time and In this case the classical mathematical calculus or frac-
0 space based on an ideas of the fractal geometry [1] is tional (say, Riemann - Liouville) calculus [3] can not be
0 offered. The corresponding mathematical methods this appliedto describea smallchangesofacontinuousfunc-
/
s hypothesismakesuseofarebasedonusingtheideaabout tion of physical values f(t), defined on time subsets St,
c fractionaldimensions(FD)asthemaincharacteristicsof because the fractional exponent depends on the coordi-
i
s time and space and in connection with this the gener- nates andtime. Therefore,we haveto introduceintegral
y alization of the Riemann-Liouville fractional derivatives functionals (both left-sided and right-sided) which are
h
are introduced. The method and theory are developed suitabletodescribethe dynamicsoffunctions definedon
p
: todescribedynamicsoffunctionsdefinedonmultifractal multifractal sets (see [1]). Actually, this functionals are
v sets of time and space with FD. simple and natural generalizationthe Riemann-Liouville
Xi Following [2], we will consider both time and space as fractional derivatives and integrals:
r anonlymaterialfieldsexistingintheUniverseandgener- d n t f(t′)dt′
a ating allother physicalfields. Assume thateachofthem Dd f(t)= (1)
a+,t dt Γ(n−d(t′))(t−t′)d(t′)−n+1
consists of a continuous, but not differentiable bounded (cid:18) (cid:19) Za
set of small elements. Let us suppose a continuity, but
notadifferentiability,ofsetsofsmalltimeintervals(from Dd f(t)=(−1)n d n b f(t′)dt′
whichtimeconsist)andsmallspaceintervals(fromwhich b−,t dt Γ(n−d(t′))(t′−t)d(t′)−n+1
(cid:18) (cid:19) Zt
space consist). First,let us consider setof smalltime in-
(2)
tervals S (for the set of small space intervals the way of
t
reasoning is similar). Let time be defined on multifrac- where Γ(x) is Euler’s gamma function, and a and b are
tal set of such intervals (determined on the carrier of a some constants from [0,∞). In these definitions, as usu-
measure Rn ). Each of intervals of this set (further we ally, n = {d}+1 , where {d} is the integer part of d
t
1
if d ≥ 0 (i.e. n −1 ≤ d < n) and n = 0 for d < 0. A=aε(t)f(t) (7)
Functions under the integral sign we will consider to be
Thus we obtain
generalized functions defined on the space of finite func-
tions [4]. Similar expressions can be written down for ∂ ∂
GFDoffunctionsf(r,t)withrespecttospatialvariables D01++εf(t)= ∂tf(t)+ ∂t[aε(r(t),t)f(t)] (8)
r, with f(r,t) being defined on the elements of set Sr
where a is a regularization parameter. For the sake of
whose dimension is dr.
independence of GFD from this constant it is useful in
For an arbitrary f(t) it is useful to expand the gen-
eralized function 1/(t−t′)ε(t′) under the integral sign in tphreopfoolrltoiwoninaglttooac−ho1.osIetβcain(obnewshhoicwhnεtdheaptefnodrslalringeeatritlhye)
′
(1)-(2)intoapowerseriesinε(t)whend=n+ε, ε→+0
exactexpressionsforthetermsin(1)-(2)proportionalto
and write
ε are very close to the approximate expression given by
d n t f(t′) (8) provided a special choice for the parameter α is is
Dad+,tf(t)= dt Γ(n−d(t′))(t−t′) (3) made (t=t0+(t−t0), t−t0 ≪t0, α∼lnt∼lnt0)
(cid:18) (cid:19) Za
′ ′ ′
×(1+ε(t)ln(t−t)+...)dt
II. EQUATIONS OF PHYSICAL THEORIES IN
d n b f(t′) MULTIFRACTAL TIME AND SPACE
Dbd−,tf(t)=(−1)n dt Γ(n−d(t′))(t′−t) (4)
(cid:18) (cid:19) Zt Equations describing dynamics of physical fields, par-
′ ′ ′
×(1+ε(t)ln(t −t)+...)dt
ticles and so on can be obtained from the principle of
minimum of fractional dimensions functionals. To do
Takinginto accountthat allfunctions herearerealfunc-
this, introduce functionals of fractional dimensions of
tions and 1/t = P(1/t)±πiδ(t), singular integrals here
space and time F (...|d (r)), α = t,r. These function-
can be defined through the rule α α
als are quite similar to the free energy functionals,but
t f(t′) now it is fractional dimension (FD) that plays the role
′
dt =af(t) (5)
t−t′ of an order parameter (see also [2]). Assume further
Z0 that FD d is determined by the Lagrangian densities
α
whereaisarealregularizationfactor. Agoodagreement L ,(i = 1,2,..., α = t,r) of all the fields ψ , describ-
α,i α,i
of (3)-(4)with the exactvalues givenby expressions(1)- ing the particles and Φ describing the interactions in
α,i
(2) can be obtained at large time by fitting the value of the point (r)
a.
Instead of usual integrals and usual partial deriva- dα =dα[Lα,i(r,t)] (9)
tives, in the frames of multifractional time hypothe-
Equationsthatgovernd behaviorcanbe foundbymin-
sis it is necessary to use GFD operators to describe α
imizing this functional and lead to the Euler’s equations
small alteration of physical variables. These function-
written down in terms of GFD defined in (1)-(2)
als reduce to ordinary integrals and derivatives if space
acinddetwimitehdtihmeeRnsiieomnasnanr-eLitoaukvenilletofrbaectiinotneaglero,paenradtocrosini-f D+dα,Lα,i(x)dα−D−dα,xD+dα,L′α,i(x)dα =0 (10)
di =const. Iffractionaldimensioncanberepresentedas Substitution in this equation GFD for usual derivatives
di = n+εi(r(t),t), |ε| ≪ 1, it is also possible to reduce and specifying the choice for F dependence on dα and
GFD to ordinary derivatives of integer order. Here we relationsbetweend andL (thelattercancorrespondto
α α
show this only for the case when d=1−ε<1 thewellknownquantumfieldtheoryLagrangians)makes
possibletowritedownthefunctionaldependenceF[L]in
∂ t ε(τ)f(τ)dτ
D1+εf(t)= (6) theform(a,b,careunknownfunctionsofLorconstants,
0+ ∂tZ0 Γ(1+ε(τ))(t−τ)1−ε(τ) L0 is infinitely large density of the measure carrier Rn
∂ t ε(τ)f(τ)dτ energy)
≈
∂t (t−τ)1−ε(τ)
Z0 F(...|d )= dL 1[a(L )∂dα]2
α α α
Though for ε 6= 0 the last integral is well defined and Z (cid:26)2 ∂Lα
is real-valued, expanding it in power series in ε leads to b(L )
+ α (L −L )d2 +c(L )d (11)
singular integrals like (5) 2 α α,0 α α α
(cid:27)
t ε(t′)f(t′) or
′
A= dt
t−t′ 1 ∂d
Z0 F(...|d )= d4L [a(L ) α]2
α α α
2 ∂L
Toregularizethisintegralwewillconsiderittobedefined Z (cid:26) α
on the space of finite main functions ϕ(t′)/2πi and take + b(Lα)(L −L )d2 + 1c(L )d4 (12)
the real part of the common regularization procedure 2 α α,0 α 4 α α
(cid:27)
2
The equations that determine the value of fractional di- any principal difficulties. As Lagrangians L one can
α,i
mensionfollowfromtakingthevariationof(11)-(12)and choose any of the known in the theoretical physics La-
read grangians of fields and their sums, taking into account
interactions between different fields.
∂ ∂d
a(L) t,α +b(L)(L−L )d +c(L)d2 =0 (13) From Eq.(17) it is possible to obtain generalizations
∂L ∂L 0 α α
(cid:18) (cid:19) of all known equations of physics (Newton, Shroedinger,
or Dirac,Einsteinequationsandetc.),andthesimilarequa-
tions for fractional space dimensions (α = r). Such
∂ ∂d
a(L ) t,α +b(L )(L −L )d2 +c(L )d4 =0 generalized equations extend the application of the cor-
∂Lα (cid:18) α ∂Lα (cid:19) α α 0,α α α α responding theories for the cases when time and space
are defined on multifractal sets, i.e. these equations
(14)
would describe dynamics of physical values in the time
For nonstationary processes one have to substitute the andthespacewithfractionaldimensions.TheMinkowski-
timederivativeofd intotheright-handsideofEqs.(13)- like space-time with fractional (fractal) dimensions for
α
(14). Neglectingthediffusionofdα processesinthespace the case dt ∼ 1 can be defined on the flat continuous
withenergydensities givenby the LagrangiansLwe can Minkowskispace-time(thatis,the measurecarrieristhe
define L − L = L˜ ≪ L with L˜ having sense Minkowski space-time R4). These equations can be re-
α α,0 α α,0 α
of over vacuum energy density and for the simplest case duced to the well known equations of the physical theo-
(13) gives (α=t, L ≡L ) ries for small energy densities, or, which is the same, for
t,i i
smallforces(d →1)ifweneglectthecorrectionsarising
t
d =L˜ =1+ β L (t,r,Φ ,ψ ) (15) duetofractalityofspaceandtimedimensions(anumber
t t i i i i
of examples from classical and quantum mechanics and
i
X
general theory of relativity were considered in [2]).For
Morecomplicateddependencies ofdα onLα,i areconsid- statistical systems of many classical particles the GFD
ered in [2]. Note that relation (15) (and similar expres- help to describe an influence of fractal structures arising
sion for dr does not contain any limitations on the value in systems on behavior of distribution functions.
of β L (t,r,Φ ,ψ ) unless such limitations are imposed
i i i i
on the corresponding Lagrangians, and therefore d can
t
reach any whatever high or small value. III. GENERALIZED NEWTON EQUATIONS
The principle of fractal dimension minimum, consist-
ing in the requirement for F variations to vanish under
α Below we write down the modified Newton equations
variationwithrespecttoanyfield,inthistheoryproduce
generated by the multifractal time field in the presence
the principle of energy minimum (for any type of frac-
of gravitationalforces only
tional dimension dependency on the Lagrangian densi-
teireasl)i.zeIdtfarlalocwtiosntaolrdeecreiivvaetEivuelsefro’sr-fluiknecetiqounastfio(nys(xw)i,tyh′(gxe)n)-, D−dt,(tr,t)D+dt,(tr,t)r(t)=D+dr,rΦg(r(t)) (20)
that describe behaviour of physical value f depending
on physical variables y and their generalized fractional b2
derivatives y′ =D+dα,xf D−dr,rD+dr,rΦg(r(t))+ 2gΦg(r(t))=κ (21)
δFt,yi ∼δdt,yi =0 (16) In(21)theconstantb−1 isoforderofthesizeoftheUni-
g
verse and is introduced to extend the class of functions
δ d (f)=δ L (f)=0, α=r,t (17) onwhichgeneralizedfractionalderivativesconceptisap-
yi α yi α,i
plicable. These equations do not hold in closed systems
becauseofthefractalityofspatialdimensions,andthere-
D+dα,yi(x)f −D−dα,xD+dα,yi′(x)f =0 (18) fore we approximate fractional derivatives as D0d+r ≈ ∇.
The equations complementary to (20)-(21) will be given
The boundary conditions will have the form in the next paragraph. Now we can determine d for the
t
x1 distances much larger than gravitational radius r0 (for
D+dα,y′(x)f =0 (19) the problem of a body’s motion in the field of spherical-
i (cid:12)x0 symmetric gravitating center) as (see (11) and [5] for
(cid:12)
In these equations the vari(cid:12)ables x stand for either t or more details)
r (the latter takes into account fractality of spatial di-
mensions), yi = {Φi,ψi}, (i = 1,2,...), Lα,i are the La- dt ≈1+βgΦg (22)
grangiandensities of the fields and particles. Here f can
be of any mathematical nature (scalar, vector, tensor, Neglecting the fractality of spatial dimensions and the
spinor,etc.),andmodificationoftheseequationsforfunc- contribution from the term with b−g1, and taking βg =
tionsf ofmorecomplicatedstructuredoesnotencounter 2c−2), from the energy conservation law (approximate
3
stoinocepeonursythsteeomrys)awndemobattahienmaticalapparatusapplyonly D−dt,tD+dt,tΦe(t(r))+ b22etΦe(t(r))=er (26)
2 These equations should be solved together with the gen-
2γM ∂r(t)
1− eralized Newton equations (20)-(21) for r(t(r),r).
c2r ∂t
(cid:20) (cid:21)(cid:18) (cid:19) With the general algorithm proposed above, it is easy
2γM ∂ϕ(t) 2 2mc2 to obtain generalized equations for any physical theory
+ 1− r2 − =2E (23)
c2r ∂t r in terms of GFD. From these considerations it also fol-
(cid:20) (cid:21) (cid:18) (cid:19) lows that for every physical field originating from the
Here we used the approximate relation between gener- time with fractional dimensions there is the correspond-
alizedfractionalderivativeanusualinteger-orderderiva- ing fieldarisingdue to the fractionaldimensionofspace.
tive(10)witha=0.5andnotationscorrespondingtothe Thesenewfieldswerereferredtoas”temporalfields”and
conventionaldescription of motion of mass m near grav- obeyEqs.(14)-(16)withx=r, α=r. Thenthequestion
itating center M. The value a = 0.5 follows from the arises,do these equations have any physicalsense or can
regularization method used and alters if we change the these new fields be discovered in certain experiments? I
latter. Eq.(23) differs from the corresponding equation wont to pay attention on the next fact: if Lt,i ≈ Lr,i no
in general theory of relativity by presence of additional new fields are generated. This is the case when fractal
terminthefirstsquarebrackets. Thistermdescribesve- dimensionoftime andspaced cannotbe dividedon
t,/br
locity alteration during gyration and is negligible while dt+dr,thetimeandspacefractalsetscannotbedivided
¯
perihelium gyration calculations. If we are to neglect it, too.The FD time and space are common and defined by
Eq.(23) reduces to the corresponding classical limit of value given by L (the latter can be chosen in the form
i
equationsofgeneralrelativityequation. Forlargeenergy of usual Lagrangiansin the known theories).
densities (e.g., gravitationalfield at r <r ) Eqs.(7) con-
0
tainnodivergences[2],sinceintegrodifferentialoperators
ofgeneralizedfractionaldiferentiationreduceto general- V. CAN REPULSIVE GRAVITATIONAL FORCES
ized fractional integrals (see (1)). EXIST?
Note,thatchoosingforfractionaldimensiondrinGFD
D0d+r LagrangiandependenceintheformLr,i ≈Lt,i gives In general theory of relativity no repulsive gravita-
for(17)additionalfactorof0.5insquarebracketsin(23) tional forces are possible without a change of the Rie-
and it can be compensating by fitting factor βg. mann space curvature (metric tensor changes). But in
the frames of multifractal time and space model, even
whenwecanneglectthe fractalityofspatialcoordinates,
IV. FIELDS ARISING DUE TO THE from (24) it follows (for spherically-symmetric mass and
FRACTALITY OF SPATIAL DIMENSIONS electric charge distributions)
(”TEMPORAL” FIELDS)
∂2t(r) ∂ m k e2
r r
m = Φ (t)≈− ± (27)
If we are to take into account the fractality of spatial r ∂r2 ∂t i c2t2 ct2
i
dimensions (d 6= 1, d 6= 1, d 6= 1), Eqs.(17)-(19), we X
x y z
arrivetoanewclassofequationsdescribingcertainphys- with accuracy of the order of b2. Here m is the ana-
r
icalfields(weshallcallthem”temporal”fields)generated
logue of mass in the time space and corresponds to spa-
bythespacewithfractionaldimensions. Theseequations
tial inertia of object alteration with time changing (it
arequitesimilarto the correspondingequationsthatap-
is possible that m coincides with ordinary mass up to
r
pear due to fractality of time dimension and were given
a dimensional factor). Eq.(27) describes the change of
earlier. In Eqs.(10)-(12) we must take x = r,/;α = r
the time flow velocity from space point to space point
and fractal dimensions dr(t(r),r) will obey (14) with t depending on the ”temporal” forces and indicates that
beingreplacedbyr. Forexample,fortime t(r(t),t)) and
in the presence of physical fields time does not flow uni-
potentials Φg(t(r),r) and Φe(t(r),r) (analogues of the formly in different regions of space, i.e the time flow is
gravitationalandelectric fields) the equations analogous
irregular and heterogeneous (see also Chapter 5 in [2]).
to Newton’s will read (here spatial coordinates play the Note, that introducing equations like (27) in the time
role of time) space is connected with the following from our model
consequences (see (17)-(19)) about equivalence of time
D−dr,(rr,t)D+dr,(rr,t)t(r)=D+dt,t Φg(t(r))+erm−r1Φe(t(r)) and space and the possibility to describe properties of
time (a real field generating all the other fields except
(cid:0) (2(cid:1)4)
”temporal”) by the methods used to describe the char-
acteristics of space. Below we will show that taking into
b2 consideration usual gravitational field in the presence of
D−dt,tD+dt,tΦg(t(r))+ 2gtΦg(t(r))=κr (25) its ”temporal” analogues gives way to the existence of
gravitational repulsion proportional to the third power
4
of velocity. Indeed, the first term in the right-hand side time and space (12-14). Therefore there exists a com-
of (27) is the analogue of gravity in the space of time plete geometrization of all physical fields, based on the
(”temporal” field). Neglecting fractional corrections to ideaoftimeandspacewith(multi)fractionaldimensions,
the dimensions and taking into account both usual and the hypothesis about minimum of functional of fractal
”temporal”gravitation,Newtonequationshavetheform dimensions and GFD calculus used in this model. The
origin of all physical fields is the result and consequence
mdd2t2r =Fr+Ft =−γmr2M + mcrtk2r ddrt 3 (28) ospfatchee.aOpnpeeacrainngsaoyftthheatfraacctoiomnpalletdeimgeeonmsioentrsizoafttioimneofanaldl
(cid:18) (cid:19)
thefieldsthattakesplaceinourmodeloffractaltimeand
The criteria for the velocity, dividing the regions of at- space is the consequence of the inducing (and describing
traction and repulsion reads by GFD) composed structure of multifractal time and
space as the multifractal sets of multifractal subsets S
dr 3 γmM −1 m k t
= r r (29) andSrwithglobalandlocalFD.Thefractionalityofspa-
(cid:18)dt(cid:19) (cid:18) ¯r2 (cid:19) ct2 tial di¯mensions dr also leads to a new class of fields and
forces (see (17)-(19) with α = r). For the special case
Here r(t) must also satisfy Eq.(20). Introducing gravita-
of integer-valued dimensions (dt = 1, dr = 3) the mul-
tionalradii r and t (the latter is the ”temporal” gravi-
0 0 tifractal sets of time and space St and Sr coincide with
tationalradius,similartotheconventionalradiusr0),we the measure carrier R4. From (14) it follows then that
can rewrite (23) as follows
neither particles nor fields exist in such a world. Thus
the four-dimensional Minkowski space becomes an ideal
dr =c3 ct2 t0 (30) physicalvacuum (for FD dα >1 the exponent of Rn has
(cid:12)dt(cid:12) s r2r0 value n > 4). On this vacuum, the multifractal sets of
In the last two exp(cid:12)(cid:12)(cid:12)res(cid:12)(cid:12)(cid:12)sionsr is the distance froma body ttiimonealanddimsepnascieon(sS,tanandditS¯rg)enaerreadteesfinoeudr wwiotrhldthweiitrhfrtahce-
withmassmtothegravitatingcenter,tisthetimediffer- physical forces and particles.
encebetweenthepointswherethebodyandthegravitat- Now the following question can be asked: what is the
ing center are situated, mr =m/c, κr =κt/c. If we ad- reason for the dependence in the considered model of
mit that r0 andt0 arerelatedto eachother asr0 =t0/c, fractal theory of time and space of fractionality of di-
the necessarycondition for the dominance of gravitation mensions on Lagrangian densities? One of the simplest
repulsion will be c < rt−1. It is not clear whether this hypothesis seems to assume that the appearing of frac-
criteria is only a formal consequence of the theory or it tional parts in the time and space dimensions with de-
has something to do with reality and gravitational re- pendenceonLagrangiandensitiesoriginatesfromcertain
pulsion does exist in nature. What is doubtless, that in deformations or strains in the spatial and time sets of
the frames of multifractal theory of time and space it the measure carrier caused by the influence of the real
is possible to introduce (though, may be, only formally) time field on the realspace field and vise versa(generat-
dynamic gravitational forces of repulsion (as well as re- ing of physical fields caused by deformations of complex
pulsive forces of any other nature, including nuclear). manifoldsdefinedintwistorspaceiswellknown[6]). As-
sumingthenthatmultifractalsetsSt andSr arecomplex
manifolds (complex-valued dimensions of time and spa-
VI. THE GEOMETRIZATION OF ALL tialpointscanbecompacted),deformation,forexample,
PHYSICAL FIELDS AND FORCES ofcomplex-valuedsetS undertheinfluenceofthespatial
t
points set Sr would result in appearing of spatial energy
Themultifractalmodeloftimeandspaceallowstocon- densitiesintimedimension,thatisgeneratingofphysical
siderthefractionaldimensionsoftimedtandspacedr(or fields (see [6]). Fractionaldimensions of space appearing
undivided FD dtr as the source of all physical fields (see (under the influence of set St deformations) yields new
(11)) (including, in particular, the case when flat (not class of fields and forces (or can also not yield). It can
fractal) Minkowski space-time R4 is chosen as the mea- be shown also that for small forces (e.g., for gravity - at
sure carrier). From this point of view, all physical fields distances much larger then gravitational radius) gener-
are consequences of fractionality (fractality) of time and alized fractionalderivatives (1)-(2) canbe approximated
space dimensions. So all the physical fields and forces through covariant derivatives in the effective Riemann
areexistin consideredmodelof multifractalgeometryof space [2] and covariant derivatives of the space of the
time and space as far as the multifractal fields of time standard model in elementary particles theory [7] (with
andspaceare exists. Within this point ofview,allphys- the corrections taking into account fields generating and
icalfields arerealasfaras ourmodelofrealmultifractal characterizing the openness of the world in whole [8,9]).
fields of time and space correctly predicts and describes All this allows to speak about natural insertion of the
the physicalreality. Butsince in this modelall the fields offeredmathematicaltoolsofGFD,atleastforε≪1,in
are determined by the value of fractal dimension of time the structure of all modern physical theories (note here,
and space, they appear as geometrical characteristics of that the theory of gravitationas the theory of real fields
5
withaspin2isinventedin[10]). Notealso,thatnumber Earth’ssurfaceitisaboutd −1∼10−12)makespossible
t
ofproblemswithintheframeworkofthe theoryofmulti- to neglect it and use conventional models of the physics
fractaltime and space (classicalmechanics,nonrelativis- of closed systems.
tic and relativistic quantum mechanics) were considered c) The model allowsto consider all fields andforces of
in [2,9,11]. the real world as a result of the geometrization of time
andspace(maybemoreconvenienttheterm”fractaliza-
tion”oftimeandspace)inthetermsoffractalgeometry.
VII. CONCLUIONS Itisfractionaldimensionsoftimeandspacethatgenerate
allfieldsandforcesthatexistintheworld. Themodelin-
Inourmodelwepostulatetheexistenceofmultifractal troducesanewclassofphysicalfields(”temporal”fields),
space and time and treat vacuum as Rn space which is which originates from the fractionality of dimensions of
the measure carrier for the sets of multifractal time and space. These fields are analogous to the known physi-
space. Fractionality of time dimension leads then to ap- calfields and forces andcan ariseor not arise depending
pearing of space-time energy densities L(r(t),t), that is on certain conditions. Thus the presented model of time
generating of the known fields and forces, and fraction- and space is the first theory that includes all forces in
ality of space dimensions gives new time-space energy single theory in the frames of fractal geometry. Repeat
densities L(t(r),r) and a new class of ”temporal” fields. once more: the model allows to consider all the fields
Note, that the roles of dt and dr in distorting accord- andforcesoftheworldastheresultofgeometrizationin-
ingly space and time dimensions is relative and can be cluding them in FD of time and space. It is non-integer
interchanged. Apparently,onecanconsiderthe ”united” dimensions of time and space that produce the all ob-
dimension dt,r - the dimension of undivided onto time servable fields. The new class of fields naturally comes
and space multifractal continuum in which time and co- intoconsideration,originatingsolelyfromthe fractional-
ordinatesarerelatedtoeachotherbyrelationslikethose ityofspacedimensionsandwiththeequationssimilarto
for Minkowskispace, not using the approximaterelation those of the usual fields. The presented model of space
utilized in this paper dtt,r=dt+dr. Moreover in some and time is the first theory that allows to consider all
cases it seems to be even impossible to separate space physical fields and forces in terms of a unique approach.
and time variables, and then dt and dr can be chosen to d)Basingonthemultifractalmodeloftimeandspace,
beequaltoeachother,i.e.,therewouldbeonlyonefrac- one can develop a theory of ”almost inertial” systems
tional dimensiond =d =1+ β L (r(t),t;t(r),r) [11,13,14]whichreducestothespecialtheoryofrelativity
t rx,y,z i i
describing the whole space-time. In this case one would when we neglect the fractional corrections to the time
have to calculate generalized fractiPonal derivatives from dimension. In such ”almost inertial” frames of reference
thesameLagrangians,andnew”temporal”fieldswillnot motion of particles with any velocity becomes possible.
be generated. e) On the grounds of the considered fractal theory of
The considered model of multifractal time and space time and space very natural but very strong conclusion
offers a new look (both in mathematical and philosophi- can be drawn: all the theory of modern physics is valid
calsenses)ontothepropertiesofspaceandtimeandtheir only for weak fields and forces, i.e. in the domain where
descriptionandontothenatureofallthefieldstheygen- fractionaldimensionisalmostintegerwithfractionalcor-
erate. This gives way to many interesting results and rections being negligibly small.
conclusions, and detailed discussion of several problems f) The problem of choosing the proper forms of defor-
can be found in [2]. Here we restrict ourselves with only mation that would define appearing of fractional dimen-
brief enumerating of the most important ones. sions also remains to be solved. So far there is no clear
a)Themodeldoesnotcontradicttotheexistingphys- understanding now which type of fractal dimensions we
ical theories. Moreover, it reduces to them when the mustuse, dt anddr or dt,r. Obviously,solvingnumerous
potentials and fields are small enough, and gives new differentproblemswilldependonthischoiceastheresult
predictions (free of divergencies in most cases) for not of different points of view on the nature of multifractal
small fields. Though, the question about applicability structures of time and space.
of the proposed relation between fractal dimension and The author hopes that new ideas and mathematical
Lagrangiandensities still remains open. tools presented in this paper will be a good first step on
b) We consider time and space to be material fields thewayofinvestigationsoffractalcharacteristicsoftime
which are the basis of our material Universe. In such and space in our Universe.
a Universe there exist absolute frames of reference, and
all the conservation laws are only good approximations
valid for fields and forces of low energy density since
the Universe is an open system, defined on certain mea-
sure carrier(the latter probably being the 4-dimensional
Minkowski space). Smallness of fractional corrections to
[1] Mandelbrot B. The fractal geometry of nature (W.H.
the value of time dimension in many cases (e.g., on the
Freeman, NewYork,1982)
6
[2] KobelevL.Ya.Fractaltheoryoftimeandspace,(Konross,
Ekaterinburg,1999) (in Russian)
[3] S.G.Samko,A.A.Kilbas,O.I.Marichev, Fractional Inte-
gralsandDerivatives-TheoryandApplications(Gordon
and Breach, NewYork,1993)
[4] I.M.Gelfand, G.E.Shilov, Generalized Functions (Aca-
demic Press, NewYork,1964)
[5] Kobelev L.Ya. Generalized Riemann-Liouville fractional
derivatives for multifractal sets, Urals State University,
Dep.v VINITI,25.11.99, No.3492-B99
[6] Penrose R.J., J.Math. Phys., 8, 345 (1967)
[7] Kobelev L.Ya., Multifractal time and space, covariant
derivativesandgauge invariance,UralsStateUniversity,
Dep.v VINITY,09.08.99, No.2584-B99 (in Russian);
[8] Yu.L.Klimontovich Statistical Theory of Open Systems
(Kluwer, Dordrecht,1995)
[9] KobelevL.Ya,Multifractaltimeandirreversibilityindy-
namic systems, Urals State University, Dep. v VINITY,
09.08.99, No.2584-99 (in Russian)
[10] A.A.Logunov, M.A.Mestvirishvili, Theoretical and
Mathematical Physics, 110, 1 (1997) (in Russian)
[11] KobelevL.Ya.,Multifractality ofTimeandSpecialThe-
oryofRelativity,UralsStateUniversity,Dep.vVINITY
19.08.99, No.2677-99 (in Russian)
[12] Kobelev L.Ya., Fractal Theory of Time and Space, Ural
StateUniv.,Dep.vVINITY22.01.99, No.189-99(inRus-
sian)
[13] KobelevL.Ya.,CanaParticleVelocityExceedtheSpeed
of Light in Empty Space? gr-qc/0001042, 15 Jan 2000
[14] Kobelev L.Ya., Physical Consequences of Moving Faster
than Light in Empty Space. gr-qc/0001043, 15 Jan 2000
7
