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PHYSICS ESSAYS 25, 3 (2012)
Relativistic mechanics in multiple time dimensions
Milen V. Velev
a)
Ivailo Street, No. 68A, 8000 Burgas, Bulgaria
(Received 19 March 2012; accepted 28 June 2012; published online 21 September 2012)
Abstract: This article discusses the motion of particles in multiple time dimensions and in
multiple space dimensions. Transformations are presented for the transfer from one inertial
frame of reference to another inertial frame of reference for the case of multidimensional time.
The implications are indicated of the existence of a large number of time dimensions on
physical laws like the Lorentz covariance, CPT symmetry, the principle of invariance of the
speed of light, the law of addition of velocities, the energy-momentum conservation law, etc.
The Doppler effect is obtained for the case of multidimensional time. Relations are derived
between energy, mass, and momentum of a particle and the number of time dimensions in
which the particle is moving. The energy-momentum conservation law is formulated for the
case of multidimensional time. It is proven that if certain conditions are met, then particles
moving in multidimensional time are as stable as particles moving in one-dimensional time.
This result differs from the view generally accepted until now [J. Dorling, Am. J. Phys. 38, 539
(1970)]. It is proven that luxons may have nonzero rest mass, but only provided that they move
in multidimensional time. The causal structure of space-time is examined. It is shown that in
multidimensional time, under certain circumstances, a particle can move in the causal region
faster than the speed of light in vacuum. In the case of multidimensional time, the application
of the proper orthochronous transformations at certain conditions leads to movement
backwards in the time dimensions. It is concluded that the number of different antiparticles in
the k-dimensional time is equal to 3
k
2
k
. Differences between tachyons and particles moving
in multidimensional time are indicated. It is shown that particles moving faster than the speed
of light in vacuum can have a real rest mass (unlike tachyons), provided that they move in
multidimensional time. Ó2012 Physics Essays Publication. [DOI: 10.4006/0836-1398-25.3.403]
R´
esum ´
e: L’article traite du mouvement des particules dans un temps et espace multi-
dimensionnels. Les transformations de r ´
ef ´
erentiels inertiels d’un syst `
eme `
a l’autre sont d ´
eduites
dans un temps multidimensionnel. Les cons ´
equences de l’existence d’un plus grand nombre de
dimensions temporelles sur les lois physiques sont d ´
emontr ´
ees: sur l’invariance de Lorentz, sur
la sym ´
etrie CPT, sur le principe de l’invariance de la vitesse de la lumi `
ere, sur la loi
d’accumulation des vitesses, sur la loi de conservation de l’ ´
energie-impulsion etc. L’effet
Doppler est obtenu dans un temps multidimensionnel. Les corr ´
elations entre l’ ´
energie, la
masse, l’impulsion d’une particule donn ´
ee sont d ´
eduites, ainsi que le nombre des dimensions
temporelles dans lesquelles cette particule se meut. Formul ´
ee `
a´
et ´
e la loi de conservation de
l’ ´
energie-impulsion en cas de temps multidimensionnel. Il est d ´
emontr ´
e que si ont ´
et ´
e satisfaites
certaines conditions, les particules qui se d ´
eplacent dans un temps multidimensionnel sont tout
aussi stables que les particules se d ´
epla ¸cant dans un temps unidimensionnel. Se r ´
esultat tranche
avec le point de vue adopt ´
e jusqu’ `
apr´
esent [J. Dorling, Am. J. Phys. 38, 539–540 (1970)]. Il est
d´
emontr ´
e que les luxons peuvent avoir en repos une masse non ´
egale `
az
´
ero, mais `
ala
condition qu’ils se meuvent dans un temps multidimensionnel. La structure causale de
l’espace-temps est ´
etudi ´
ee. Il est d ´
emontr ´
e que dans un temps multidimensionnel, dans
certaines conditions, une particule peut se mouvoir dans le champ causal plus rapidement que
la vitesse de la lumi `
ere dans du vide. En cas de temps multidimensionnel, l’application des
propres transformations orthochrones m `
ene, dans certaines conditions, `
a une marche en
arri `
ere dans la mesure du temps. Nous atteignons la conclusion que le nombre des diff ´
erentes
antiparticules dans un temps k-dimensionnel est ´
egal `
a(3
k
2
k
). Les diff ´
erences entre les
tachyons et les particules se mouvant dans un temps multidimensionnel sont montr ´
ees. Il est
d´
emontr ´
e que les particules qui se meuvent plus rapidement que la vitesse de la lumi `
ere dans
du vide peuvent avoir une masse r ´
eelle en repos ( `
a la diff ´
erence des tachyons), mais `
ala
condition qu’elles se meuvent dans un temps multidimensionnel.
Key words: Multidimensional Time; Special Relativity; Mass-Energy Equivalence; Energy-Momentum Conservation
Law; Antiparticles; Tachyons; Lorentz Transformations; Invariance of the Speed of Light.
a)
milen.velev@gmail.com, milenvp@abv.bg
0836-1398/2012/25(3)/403/36/$25.00 Q2012 Physics Essays Publication403
I. INTRODUCTION
The concept of multidimensional time has been
introduced and has become more important in contem-
porary physical theories.
1–5
According to the inflation
theory of the big bang, the visible universe is only a small
part of the multiverse, and it is possible that many other
universes have emerged in which conditions are entirely
different from the conditions of our universe.
6
Up to now,
no physical principle or law has been found that
determines the possible number of spatial dimensions
and of temporal dimensions (or limits the number of
spatial and temporal dimensions to a value which differs
from the observed number in our universe). Due to this
fact, the number of spatial and temporal dimensions in
our universe is more probably a result of chance than a
result of unknown processes acting during the initial
development phases of the universe. Through the
anthropic principle it is explained that we live in a
universe with more than three dimensions of space (or 10
dimensions, as predicted by M-theory) but only one
dimension of time. Therefore, in the other universes that
are part of the multiverse it is quite possible that space
and time have entirely different numbers of dimensions
than the dimensions in our universe. We can assume the
existence of universes having two, three, four, or more
temporal dimensions. The relation between the anthropic
principle and the number of spatial and temporal
dimensions is considered by Tegmark.
5
Here we do not
discuss this matter.
Asshowninsomestudies,
1,2
it is possible to
formulated physically meaningful theories with two time
dimensions. Bars noted that ‘‘two-time physics could be
viewed as a device for gaining a better understanding of
one-time physics, but beyond this, two- time physics
offers new vistas in the search of the unified theory while
raising deep questions about the meaning of spacetime.’’
2
For systems that are not yet understood or even
constructed, such as M-theory, two-time physics points
to a possible approach for a more symmetric and more
revealing formulation in 11 þ2 dimensions
a
that can lead
to deeper insights, including a better understanding of
space and time. The two-time physics approach could be
one of the possible avenues to construct the most
symmetric version of the fundamental theory.
1,2
As noted by Tegmark, ‘‘Even when m.1, there is no
obvious reason why an observer could not, none the less,
perceive time as being one-dimensional, thereby main-
taining the pattern of having ‘thoughts’ in a one-
dimensional succession that characterizes our own reality
perception. If the observer is a localized object, it will
travel along an essentially one-dimensional (timelike)
world line through the (nþm)-dimensional space-time
manifold.’’
5,b
Thus it is fully reasonable to ask the
question ‘‘What relations, effects, and features would
exist if we examined an object moving in multidimen-
sional time?’’
In order to find experimental evidence for the
existence of particles moving in multidimensional time,
it is necessary to know their physical properties. As noted
by Recami in another context—the experimental search
for the hypothetical particles named tachyons—‘‘it is not
possible to make a meaningful experiment without a good
theory.’’
7
The main objective of this article is to generalize the
special theory of relativity (STR) for the cases of
multidimensional time and multidimensional space. There
is a need to clarify not only the mathematical but also the
physical meaning of multidimensional time.
In this respect, the study raises several basic tasks:
deriving transformations for the transition between
inertial frames of reference for the case where the
number of time dimensions is greater than one;
establishing the implications arising from the
existence of a large number of dimensions of time
on physical laws—the Lorentz covariance, CPT
symmetry, the constancy of the speed of light, the
law of addition of velocities, the energy-momentum
conservation law, etc.;
deriving the Doppler effect for the case of
multidimensional time;
examining the causal structure of space-time;
deriving formulas for momentum and energy for
the case of more than one time dimension;
establishing the exact relationship between the
energy of a particle and the number of time
dimensions in which the particle is moving;
formulating the energy-momentum conservation
law;
considering antiparticles in multidimensional time;
and
distinguishing between tachyons and particles
moving in multidimensional time.
The problem with the generalization of STR for the
case of multidimensional time is still not sufficiently
studied and is only briefly mentioned in different studies
concerning the topic. The consequences on physical laws
of the existence of multidimensional time have also not
been well studied. Up to now there has been no
distinction between tachyons and particles moving in
multidimensional time.
II. GENERAL CONSIDERATIONS
Important for this study is following question: Are
there physical arguments and grounds allowing general
conclusions concerning the dimension of time? Related to
this question is another: Is Minkowski space-time real
and should we accept time as the fourth dimension, given
the fact that STR can be equally formulated in a three-
dimensional or a four-dimensional language? As noted by
Petkov, of course, we have to solve this issue before
seriously discussing a theory involving a large number of
a
Two-time physics introduces one additional space dimension and
one additional time dimension.
b
Here mis the number of time dimensions and nis the number of
space dimensions.
404 Phys. Essays 25, 3 (2012)