approaching space-time through velocity in doubly special relativity

PHYSICAL REVIEW D 70, 125012 (2004)

Approaching space-time through velocity in doubly special relativity

R. Aloisio*INFN, Laboratori Nazionali del Gran Sasso, 67010 Assergi, L’Aquila, Italy

A. Galante†

Dipartimento di Fisica dell’Universita di L’Aquila, 67100 L’Aquila, Italyand INFN, Laboratori Nazionali del Gran Sasso, 67010 Assergi, L’Aquila, Italy

A. F. Grillo‡

INFN, Laboratori Nazionali del Gran Sasso, 67010 Assergi, L’Aquila, Italy

E. LuzioDipartimento di Fisica dell’Universita di L’Aquila, 67100 L’Aquila, Italy

F. Mendezx

INFN, Laboratori Nazionali del Gran Sasso, 67010 Assergi, L’Aquila, Italy(Received 27 October 2004; published 17 December 2004)

*Electronic†Electronic‡ElectronicxElectronic

1550-7998=20

We discuss the definition of velocity as dE=djpj, where E, p are the energy and momentum of aparticle, in doubly special relativity (DSR). If this definition matches dx=dt appropriate for the space-timesector, then space-time can in principle be built consistently with the existence of an invariant length scale.We show that, within different possible velocity definitions, a space-time compatible with momentum-space DSR principles cannot be derived.

DOI: 10.1103/PhysRevD.70.125012 PACS numbers: 11.30.Cp, 03.30.+p, 04.60.–m, 12.90.+b

I. INTRODUCTION

Doubly special relativity (DSR) is a generalization of thespecial relativity which, besides the speed of light c, in-troduces a second invariant scale [1–5]. The reason to lookfor such generalization can be traced back to an heuristicargument of quantum gravity (QG): if quantum effects ofgravity become important at certain distances (typicallythe Planck length) or energies, then these scales should beobserver independent.

With this in mind, the DSR proposal tries to give a (atleast phenomenological) answer to the question raised bythe previous argument, that is, if it is possible to find adifferent symmetry that guarantees another invariant scale(which will be eventually related with the standard QGscale). A concrete realization of such ideas was given in thespace of energy and momentum where the deformedboosts, dispersion relation, and composition law werewritten [4–6].

DSR can also be understood as a nonlinear realization ofthe Lorentz group in the momentum space [4,7,8], i.e.,apart from the physical variables, we can consider auxil-iary variables that define a space where the Lorentz group

address: [email protected]: [email protected]: [email protected]: [email protected]

04=70(12)=125012(9)$22.50 125012

acts linearly.1 However the program has not been com-pleted yet. One of the most pressing problems on thissubject is to find an explicit realization of these principlesin the space-time. In fact, the connection between theexistence of an energy or momentum (or both) invariantscale and the consequences in the physical space-time isnot clear. This is a necessary step to undergo since it is inthe actual space-time sector where the experiments areperformed, the instruments collect data and, finally, ourphysical description has to apply.

One possible way to approach this problem is by notic-ing that in the usual case (relativistic and nonrelativistic)there are quantities, like the velocity of a particle, that canalso be written in terms of variables of the energy-momentum space. In the standard description, the velocityis the derivative of the spatial coordinate with respect totime and it is also the derivative of the energy with respectto momentum.

In this work, we analyze two possible definitions for thevelocity of a particle and we test their implications for thespace-time (problem treated for first time in [9]). The firstcase studied corresponds to the standard definition men-

1Since the auxiliary variables transform according to thestandard Lorentz group, we will refer to them and to the spacewhere they are defined as classical variables and classicalmomentum space, respectively.

-1 2004 The American Physical Society

ALOISIO, GALANTE, GRILLO, LUZIO, AND MENDEZ PHYSICAL REVIEW D 70, 125012 (2004)

tioned above which, when physical processes are consid-ered, gives rise to inconsistencies (also reported in[10,11]).

As a different possibility, we have used the map thatconnects the momentum space with the classical momen-tum space and, since both approaches give rise to the sameexpression for velocities (see also the discussion in [12]),we conclude that attempts to define the space-time of DSRin terms of a classical space-time shall give inconsistentresults.

Another possible definition of velocities analyzed in thiswork is related to a deformation of the definition of de-rivatives. In this approach we adopt again the notion ofvelocity as the rate of change of the energy with respect tothe change of momentum, but the notion of change adoptedis now DSR compatible. That is, the difference (for energyand also for momenta) is covariant under a DSR boost.This definition of velocity does not show the inconsisten-cies described before while it is still connected to a limitedthree-momentum; however also in this case it seemsinconsistent with a continuous differential space-timemanifold.

The latter approach is close to the velocity definition in �Poincare (KP) and � Minkowski (KM) [13–15] and to theDSR approach where a KM space-time is assumed [16,17].

It is important however to note that the definitions ofvelocity we will describe in detail in the rest of the paperare formal and it is not clear if and how they are related tothe rate of change of space with time and, consequently,their phenomenological implications.

This work is connected to some early efforts in theapproach to a space-time formulation of DSR. Followingthe close relation between DSR and the algebraic sector ofthe so-called �-Poincare (KP) deformation of the Poincaregroup [18], some authors have introduced the idea that theDSR compatible space-time should be a noncommutativespace-time, as it happens with the space-time associated toKP [19]. Following the approaches similar to DSR, otherauthors have investigated possible nonlinear realizations ofthe Lorentz group, directly on the space-time [20].

The paper is organized as follows. In the next section wewill review the formulation of DSR as a nonlinear realiza-tion of the Lorentz group in the momentum space. InSec. III, the standard definition of velocities will be re-viewed and analyzed. Sec. IV is devoted to the analysis inthe so-called classical space. The definition of velocities

2We omit the indices in order to simplify the notation.3In the following we will always refer to this specific realization.

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with a DSR inspired derivative is given in Sec. V. The lastsection is devoted to the discussion and conclusions.

II. DSR FORMULATION

Considering the extensive literature on this topic [1–9,21,22], we only briefly summarize what we will defineas a DSR. Understanding DSR as a nonlinear realization ofLorentz symmetry we consider two momentum spaces:one, that we call the classical momentum space �, withcoordinates � � f;�g, where the Lorentz group actslinearly, and another, the physical space P, with coordi-nates p � fE;pg. There exists a function F:P! �, suchthat � � F�p�.2 This function is invertible and depends ona parameter �. The image of the point p � E;pmax�(where pmax is a vector with modulus jpmaxj � 1=�) underF is infinity; this requirement ensures that 1=� is aninvariant momentum scale.

Boost transformations and the Casimir elements in P arethe inverse images of the boosts and Casimir in the classi-cal space. That is, the boost in the P space is given by

B � F�1 � F; (1)

where � is an element of the Lorentz group.The Casimir in the classical space is 2 � �2 � 2,

which can be written in the P space asF0�p�2 � F�p�2 � 2.

For the DSR1 model,3 the explicit form of F and itsinverse is [7]

F�1�x; y� �1� ln�x�

��1� �2x2 � y2�

p�

y��x��1� �2x2 � y2�

p��1

" #; (2)

F�x; y� �1� �sinh�x� �

�22 y

2e�x�ye�x

" #: (3)

It is then possible to write the explicit formulas for theboosts and the Casimir elements. Because of the rotationinvariance we will always reduce the problem to 1+1dimensions so, without loss of generality, we can writethe boost for energy and momentum

E�� E�1

�ln�1� �px sinh�� 1� cosh��

�

�sinh�E�e��E �

�2p2

2

��; (4)

px�� px cosh�� sinh��1 sinh�E�e��E � �p2

2 ��

1� �px sinh�� 1� cosh��sinh�E�e��E � �2p2

2 �: (5)

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APPROACHING SPACE-TIME THROUGH VELOCITY IN. . . PHYSICAL REVIEW D 70, 125012 (2004)

Here � parametrizes the elements of the Lorentz group andtherefore �1< �<1.

The Casimir turns out to be

cosh�E� ��2p2

2e�E � cosh�m�; (6)

withm � m� the physical mass, that is, it coincides withthe mass of the particle in the limit �! 0.

Finally, in order to have a full description of the way inwhich the measurements made by one observer are relatedto the measurements made by another observer boostedwith respect to the first, it is necessary to know the relationbetween � and the relative velocity V. This will allow tocheck the consistency of the definition of velocities, aquantity that in principle can be measured by experiments.

III. ON THE DEFINITION OF RELATIVEVELOCITY

Our final aim would be to investigate the relation be-tween measurements made by two observers in relativemotion and, through that, the structure of the space-time.To complete this program it is first necessary to discuss thedefinition of velocities and its relation with the boostparameter.

This section is devoted to investigate such topics. Wewill adopt the standard definition of velocity [1] and studyits consequences from the point of view of measurementsmade by two inertial observers.

A. Definition of velocity and its relation with �

The physical scenario consists of two observers in rela-tive uniform motion. S0 is the reference frame where weconsider particles at rest (it corresponds to � � 0), while Sis another reference frame whose motion with respect to S0

is described by a nonzero boost parameter �.Since at present stage we only know the DSR trans-

formation laws in the energy-momentum sector, we willconsider only measurement of these quantities. Clearlywhat we do not know are the transformation laws in thespace-time sector and, in fact, nothing guarantees us eventhat the space-time is a continuous and differentiablemanifold.

In undeformed relativity (and also in the Galilean case)it is possible to express the velocity in terms of the mo-mentum and the energy. This is provided by the relation

V �dEdp

; (7)

where E is the energy of the particle, V � jvj and p � jpj.This expression, that gives the right results for the

standard cases mentioned above, might not be correct inour case. One must note that (7) is based on the fact that(a) there exists an expression for the energy in terms ofcoordinates and momenta (in general, one assumes that it

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is possible to define a Hamiltonian that generates timetranslations) and (b) there exists a canonical simplecticstructure [23].

Both ingredients are independent and, regardingpoint (a), here we assume that the energy in terms ofmomenta is given by (6). Note also that if we follow thestandard approach, what we have called the velocity shouldbe identified with the variation of coordinates with respectto time: this is the outcome of the Hamilton equations.

Starting from the above working hypothesis, from thedispersion relation (6) we can calculate the velocity pre-viously defined

Vp� ��p

r2�p�

1�

C2�m�p�

C2�m�p� � r2�p�

; (8)

with r2�p� � 1� p2�2 and C2�m�p� �

cosh2�m� � cosh�m��cosh2�m� � r2�p�

p.

Now let us assume that there is a particle of mass m atrest in S0 (where rest means that p � 0) and we observe itfrom S. Since we can relate momenta measured in S withthe same quantities measured in S0 by using (4), we are ableto express the velocity of the particle in terms of its massand the parameter �. We obtain

V��

��1� r2�m��

qr2�m��

1�

C2�m��

C2�m�� r2�m��

; (9)

with

r2�m�� 1�sinh2�m�sinh2��

�cosh�m� � sinh�m� cosh��2;

C 2�m�� cosh2�m� � cosh�m�

��cosh2�m� � r2�m��

q:

At first sight, (9) seems to have a pole for C2�m��

r2�m��, which occurs form � 0. This is not true and indeedthe m! 0 limit is well defined and gives V�� tanh��.Notice that this is not the velocity of a photon since itcorresponds to the limit p0

� 0, and therefore describesthe motion of a geometrical point.

A fundamental property of (9) is its mass dependence(see also discussions in [10,11]). In order to discuss theconsequences of the definition, as well as the meaning ofthis mass dependence, in the next subsection we willanalyze some limits and peculiarities of the previousexpression.

B. Special limits and the mass dependence of therelative velocity

In (9) we see that the mass always appears in thecombination �m. Since � is the parameter controlling thedeparture from standard Lorentz invariance it is interestingto consider the large mass limit (�m� 1), which we callthe macroscopic bodies limit, and the limit of small masses

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�m� <1 that will be referred to as the microscopicbodies limit.

1. Microscopic bodies limit (�m� 1)

It is expected that for particles with masses far below theinvariant scale, the relativistic limit must be recovered. It isnot difficult to show that, for any value of �

V�m�1 � tanh��1� tanh�� sinh��m

�1

2tanh2��m�2 � � � �

�: (10)

We see that the zero order term corresponds to the relativ-istic case.

It is also interesting to note that, in this microscopiclimit, the momentum of the particle seen by S, is given by

px �m sinh�� m2 cosh�� sinh�� O�2�; (11)

and we see again that the first term is the standard relativ-istic one.

Finally, let us note that since all the relativistic limits arerecovered for microscopic bodies, if we consider now�! 0 we reproduce the Galilean limit for momentum aswell as for the relative velocity.

2. Macroscopic bodies limit (�m� 1)

Following the previous analysis, it is natural to considerthe limit �m� 1: this limit is not forbidden because, inDSR, particles have a maximum momentum attainable butthe energy (and the mass) can be as large as we want.

The velocity in this limit becomes

V�m�1 � sinh�� 2e�2�m tanh�� : (12)

From here it is clear that the relativistic limit is not recov-ered, instead, for �! 0 we recover the Galilean one.

In order to understand this result, we note that in DSR1there exists a maximum momentum (pmax � 1=�) which isan invariant dimensional scale. Therefore the condition�m� 1 means m� pmax, something that resembleswhat occurs in undeformed relativity when the transitionfrom the relativistic to the nonrelativistic regime is consid-ered. In this sense, we can expect the relation (12) betweenV and � to be Galilean-like in the limit �! 0 as indeedhappens.

We could expect a similar behavior to hold also for themomentum variable. The macroscopic limit for the mo-mentum is given by

p �sinh��

��1� cosh��Oe��m�: (13)

For large values of � the momentum correctly goes to pmax.As a function of the velocity, the previous expression

turns out to be

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p �V

�1��1� V2

p��Oe��m�; (14)

for any value of �. In particular, when �! 0—and there-fore the velocity (12) is the Galilean one—we see that

p � mV: (15)

Then, the relations we derived for �m� 1 are perfectlyacceptable from a DSR1 point of view, but they are incon-sistent with everyday experience for macroscopic bodies.

This is related to the so-called soccer ball problem,according to which it would be impossible to have macro-scopic bodies with momentum grater than the Planck’smomentum (if we identify 1=� � ppl), since all the parti-cles in this body have a limited momentum and the DSR1compatible composition law allows only bodies with mo-menta no greater than pmax.

The problem resides in the fact that this result is notconsistent with our everyday experience, and the reason isthat we are trying to use DSR principles (which we expectto be relevant as a quantum gravity effect) in the oppositeextreme limit (the macroscopic one) where quantum ef-fects and especially quantum gravity effects are expectedto be irrelevant. A true space-time description should curethese apparent inconsistencies.

3. The mass dependence of V

In the standard Lorentz theory the velocity, defined asthe derivative of the energy with respect to the momentum,is a function of the boost parameter without any depen-dence on the particle mass or energy. Since the boostparameter has a geometrical significance, irrespective ofthe particle content of the system, this guarantees that allparticles with the same velocity in a reference frame willhave the same velocity in any other reference frame.

Now we try to give a physical interpretation to expres-sion (9) which, comparing to the standard Lorentz case, hasthe evident peculiarity to define a mass dependent velocity.This can be stated in a different way by noting that,inverting the relation (9), the parameter of the boost de-pends on the mass of the particle.

Can this result be accepted? To answer this, let usconsider two observers, one in S0 and the other in S andtwo particles at rest in S0 with different masses m1 and m2;a given value for � describes the relative motion of theobservers. In this framework the observer in Swill measuretwo different velocities for the two particles and, conse-quently, there could be events observed by S but that do nothappen according to S0. As an example, if we imagine thetwo particles at rest at a given distance in S0, since m1 andm2 have different velocities in S, it might be possible toobserve a collision according to S, something which willnever occur according to S0. In the previous argument, it isimplicit the fact that the relation between velocity and thespace-time coordinates is the usual one (in the sense that

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two particles at a finite distance and finite relative velocityalong a given axis will collide in a finite time interval).

In other words, if we try to establish the relative velocitybetween two reference frames measuring the velocities ofparticles of different mass at rest in one of the referenceframes we will clearly get different values. To better under-stand the physical origin of this apparently inconsistentresult we can consider again our two-particle system. Thetotal energy and momentum of the system is knownthrough the composition laws which are compatible withthe DSR1 principles. DSR1 composition laws, at first orderin � are

Etot � E1 � E2 � �p1p2; (16)

ptot � p1 � p2 � �E1p2 � E2p1�; (17)

which can be expressed in terms of the rapidity � by using(4) at first order.

Etot�� M cosh�� M2

2sinh2��; (18)

ptot�� M sinh�� M2 sinh�� cosh��; (19)

with M � m1 �m2. From here, using (7), the velocity ofthe two-particle system (1� 2) is

V1�2� � tanh��1� �M tanh�� sinh�� : (20)

This expression coincides with (10) after the substitutionm! M: this is a necessary consistency check between thedefinition of velocity and the momentum compositionlaws. The point is that the presence of more than oneparticle turns out to introduce problems in interpretingexperimental results. If the particles have different massesor if we deal with multiparticle systems, we get differentresults for relative velocities of reference system corre-sponding to a given boost parameter.

This resembles the spectator problem that arises in�-Poincare models [18] when we consider the deformedcomposition law for four-momentum. In that case thecomposition law is asymmetric under the interchange ofparticles and a particle that does not participate in a reac-tion process (the spectator) can modify (simply by itspresence, without any direct interaction) the thresholdenergy for the process [22].

What is remarkable is that in � Poincare this problemhas its origin in the asymmetry of the momentum compo-sition law and, at the end, this property gives rise to thenoncommutative structure of the space-time [24]. Here wedeal with a symmetric composition law of momenta but,assuming (7), we find some inconsistency similar to the�-Poincare spectator problem. With a speculative attitudeit could be interpreted as a signal that the space-time mighthave a nonstandard structure.

The above discussion points out the impossibility tophysically accept these results. The inconsistencies are

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based on the assumption that a unique boost parameter isassociated with a transformation between different refer-ence frames, independent of their particle content. Thisstill leaves an open door: let us impose the physical re-quirement that the velocities measured by S are equal whenthe corresponding particles are at rest in S0. The abovecondition reads V1 � V2 and if m1 � m2 this necessarilyimplies that the transformation parameter has to be differ-ent for particles of different mass, i.e.,

V�1; �m1� � V�2; �m2�: (21)

The above formula relates the � parameter for particlesof different mass. For example, at first order in �m, in thecase of two particles, we have

�2 � �1 � �sinh3�1�m1 �m2� �O��2m1 �m2�2�:

(22)

As a particular case we can express the �i of the i particleas a function of its mass and a boost parameter (�) thatcorresponds to them! 0, p! 0 limit discussed at the endof Sec. III A. This will allow us to write

�i � �� misinh3�� O�2m2

i �; (23)

i.e., to explicitly express all the mass dependence of theboost parameter for any particle. In this picture the boostparameter turns out to be particle (mass) dependent under aprecise physical requirement: different particles will havethe same velocity in another reference frame if they are atrest in a given frame.

We can expect this to turn into contradiction whenconsidering multiparticle states. In fact the DSR four mo-mentum composition laws are formulated to be covariantunder boosts when the same boost parameter is consideredfor each particle. The latter condition is not any moresatisfied if we assume (21) as can be verified in the nextexample. Consider two particles labeled 1 and 2 at rest inthe reference frame S0. At first order in � their momentumin the S reference frame can be written in terms of the zeromass boost parameter � and the particle masses

pi�� mi sinh��1� �micosh3�� O�2�; (24)

Ei��mi cosh��1��mi cosh��sinh2��1� tanh2��

�O�2�: (25)

Now we consider these two particles as a unique system.In the reference frame S0 we use the composition law (16)to get that the total momentum is zero and the energy isM � m1 �m2. Then we can go to the reference system Sand write that the total momentum of the system is simplygiven by (24) with mi replaced by the total mass M.

We can redo the calculations first considering the mo-mentum of each particle in the S reference system and thencomposing it [again using the composition law (16)] to getthe total momentum. For consistency the latter should

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coincide with the total momentum previously obtained.This is not the case. Composing the momentum in S we get

Ptotal�� M sinh��1� �Mcosh3��

� 2�m1m2sinh3�� cosh�� O�2�;

Etotal�� M cosh��1� �M cosh��sinh2��1

� tanh2�� 2�m1m2sinh2��cosh2��

� �3� tanh2�� O�2�:

which differs from the previous result due to the last termin the right-hand side of both equations.

We conclude that both possibilities, i.e., to consider aunique boost parameter for all particles (like in the stan-dard Lorentz formulation) as well as to change it to keepvelocities equal in all reference systems for particles at restin a given reference frame, are not compatible with thevelocity definition we adopted at the beginning of thissection.

4See, for instance, G. Amelino-Camelia in gr-qc/0309054.5Notice that as an argument of E through the dispersion

relation p has to be considered as a numerical parameter andit is correct to write p2 � p1 � ".

IV. VELOCITY IN THE CLASSICAL SPACE

Another way to obtain the relation between V and theboost parameter �, and also to get information about thestructure of the space-time, is to study the relation betweenquantities defined in the real space and in the classicalspace.

The velocity in the classical space � is naturally definedthrough (7), but in terms of the classical coordinates

� �dd�

: (26)

Then, it is possible to write the velocity (7) in the realspace, in terms of variables in the classical space by usingp � F��, with F defined in (2) and tanh�� . Then

V ��@E� @�E�@p� @�p

: (27)

This velocity trivially coincides with (9) because, at thislevel, we have only made a change of variables.

What is interesting to note is that this result suggests thatthere could be also a map between a classical space-timeand the real space-time. In fact, let us assume that this mapexists. That is, there exists a classical space-time � withcoordinates �, � and two functions A��; ��, B��; �� whichalso depend on the parameter � (they are analogous to thecomponents of the function F�1 in the momentum space),such that

x � A��; ��; t � B��; ��: (28)

In this case, posing � � d�=d�, the relation between thevelocity in the classical space and the velocity in the realspace will be given by

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V �dxdt

��@�A� @�A

�@�B� @�B; (29)

which, by hypothesis, does not depend on m.This is inconsistent with (9) where a mass dependence is

present and therefore A and B defined above do not exist.Let us point out that this is in agreement with the result

in [25], where it is shown that it is not possible to definesuch map and keep the notion of invariant length scale.

V. DEFORMED DERIVATIVES

In this last section we will consider another definition ofvelocity as was first discussed by Lukierski and Nowicki inthe context of �-Poincare formulation [13].

Let us consider the function Ep� that is the solution forthe energy that comes from the Casimir defined in (6). Thevelocity, as we have been discussing up to now, is thederivative of such function with respect to p, that is,

V � limp1!p2

Ep2� � Ep1�

p2 � p1:

This definition involves the difference of the energy (eval-uated in two different points) and a difference of momenta,but this operation is only well defined through the use ofcomposition law of four momenta,4 that is,

"p � pa�pb; � F�1�Fpa� � Fpb��; (30)

with F defined in (2) for DSR1.Then, a DSR1 inspired definition of velocities is

V � limp1!p2

Ep2��Ep1�

p2�p1; (31)

� limp1!p2

"E

"p: (32)

That is, the comparison of the quantities are expressed interms of covariant differences.

In order to take the limit we use (30).5 The result is

"p � p2�p1�;

� F�1��Fp1 � "� � Fp1��;

� F�1��@F@E

��p1�p"0 �

@F@p

��p1�p"1

�; (33)

where we have restored the indices explicitly (for simplic-ity we are in 1� 1 dimensions, therefore � 0; 1).

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6We are calling ‘‘time’’ the canonical conjugate of theHamiltonian.


Using the Casimir relation

CE;p� � cosh�E� ��2

2p2e�E � cosh�m�;

namely, the fact that free particles move on the orbits ofCE;p� � 0 and the variations we are considering mustsatisfy "CE; p� � 0,

"0 � �

@C@p

@C@E

�1"1;

�dEdp

"1;

� V": (34)

Now we can write the velocity of the particle as

V � lim"!0

F�1�0�@pF� V@EF�"�

F�1�1�@pF� V@EF�"�: (35)

For DSR1, using the function F defined in (2) we get

V ��p cosh�m�

C4�m�p�

�2C2�m�p� � r2�m�p��

2

f2C2�m�p� � r2�m�p��C

2�m�p� � 1�g

;

(36)

with r�m�p� and C2�m�p� defined in (8).

The relation with the boost parameter � can be obtainedby replacing the momentum p�� given in (4), in theprevious expression. The result is

V�� tanh��; (37)

which is the standard relation between the boost parameterand the velocity in the undeformed relativistic case.However one can see that the relation between the boostparameter and the momenta is quite different comparedwith the undeformed case:

E�� 1

�lncosh�m� � sinh�m�%��; (38)

p�� 1

�%��V��

�%�� coth�m��; (39)

with %�� 1� V2��1=2 � cosh��.The fact that the relation between the velocity and the

boost parameter coincides with the standard of undeformedrelativity—with the definition of velocity adopted here—might suggest that there are no differences between DSRproposal and the standard relativity.

However, the interpretation of this velocity as the rate ofchange of the space coordinate with respect to time is notguaranteed, because we do not know which deformation (ifany) of the Poisson brackets (simplectic structure) betweenthe coordinates of the phase space x; p� could permit towrite this deformed velocity as the result of a Hamiltonequation.

Therefore, it would be premature to conclude that, withthis definition of velocity, there are no testable phenome-

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nological consequences, in particular, for the propagationof light. In fact these consequences rely on finite time anddistance measurements which are still undefined, even inthis approach.

VI. DISCUSSION AND CONCLUSIONS

In (undeformed) special relativity (SR) the assumptionof constancy of speed of light and the homogeneity of thespace (linearity of transformations), together with the defi-nition of velocity for massive bodies and its identificationwith frame velocity, allow one to constructively define(through gedanken experiments) space-time.

Doubly special relativity theories are constructed inmomentum space by requiring the existence of an invariantmomentum (and/or energy) scale which assumes the mean-ing of maximum momentum. The reason for this is theexpectation that quantum gravity introduces a minimum(invariant) length scale. However the construction of thespace-time sector is still in its infancy, although there areinteresting connections with the quantum deformed ap-proaches. We have in mind to try to repeat the undeformedSR construction, which has to pass through a realisticdefinition of velocity for both massless and massiveparticles.

In this work we have discussed three different definitionsof velocity in a DSR1 scenario as an approach to the space-time compatible with these principles: (a) the velocity asthe derivative of the energy with respect to the momentum,(b) the classical space-time approach, and (c) a deforma-tion of the concept of derivatives, induced by the deforma-tion of the addition law in the momentum space.

Our first expression for the velocity is obtained by thenatural assumption

v �dEdp

;

which is a consequence of two requirements, as was dis-cussed in the first part of this work: (i) a Hamiltonian(energy) function, which is the generator of time trans-lations and (ii) a canonical simplectic structure. Therefore,when we adopt the above expression, necessarily we mustunderstand v as the change of the spatial coordinate withrespect to time,6 and then, we can extract information fromgedanken experiments by using the standard kinematics.

An unavoidable consequence of this approach, however,is the dependence of the velocity on the mass of boostedparticles [10] (for photons, the dependence of the speed oflight on the energy) which implies, for example, that twoparticles of different mass at rest in some reference framewill have different velocities as seen by another observerboosted with respect to the first one. As a consequence,these two masses could interact (a collision, for instance)

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for one observer while continue to be at rest for the other.The principle of relativity is then violated. Since we requirethat DSR respects the relativity principle, our assumptionfor the definition of velocity must be discarded. Anotherway of describing this result is by saying that referenceframes cannot be unambiguously attached to massive par-ticles or objects.

A possible way out is to impose, as a physical require-ment, that all bodies which are at rest according to oneobserver, move with the same velocity for any other ob-server boosted with respect to the first. This solves theproblem with the relativity principle, but it is not compat-ible with the composition laws of momenta in DSR, andmakes it impossible to associate in a unique way the boostparameter with a given reference frame.

Another possibility, which also permits investigating apossible approach to DSR in the space-time as a nonlinearrealization of the Lorentz group, is to define the velocity inthe classical space as defined in Sec. IV and then to map itinto the real space.

We have shown that it is not possible to construct afunction that maps a classical space-time, in which theLorentz group acts linearly, into the real space-time. Wehave shown that such function would not be universal forall particles because it will depend on the mass of eachparticle. This result is in agreement with a previous oneobtained in [25].

Finally, let us comment on our last result. We havechosen a deformed definition for the derivative because,as we argued, the difference (composition law) of energyand momentum has to be modified in order to be invariantunder DSR. We have only one definition for the differenceof energy and momentum that is compatible with DSRprinciples, which is [26]

"p � pb�pa � pb�Spa� � F�1�Fpb� � Fpa��;

where we have used the antipodal map S�p� �F�1��Fp��.

This law, inherited from the composition law, should bethe right one that must appear in the definition of deriva-tives. An example of this kind of deformed derivative canbe found in the definition of velocities in �-Poincarescenario as given in [13]. However there are two differ-ences: (i) in DSR the composition law for the energy ismodified, while in � Poincare it is the primitive one, and(ii) the composition law of momenta in DSR is symmetricand therefore there is only one possible definition for

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derivative; instead in KP we have two possible choiceswhich, in fact, give rise to two possible velocities.

The result for DSR1 is a definition of (three-)velocitywhich is that of undeformed special relativity, while stillmaintaining an invariant momentum scale; therefore weconclude that DSR proposal has, as it must be, two invari-ant scales, namely c, the speed of light and 1=�, themaximum momentum attainable for a particle.

The above defined velocity also gives information aboutthe space-time. If we follow the same argument for the KPscenario, we will see that the velocity found in this case isin agreement with the definition of velocity in terms of aHamiltonian and a deformed simplectic structure. Then, invery speculative sense, we would say that this velocity forDSR could correspond to a deformed simplectic structureand, at the end, to a space-time with a nontrivial structure.

A posteriori our results are not unexpected since inmomentum space the limit of infinitesimal incrementalratios makes sense even in the presence of an (invariant)maximum momentum, while this is clearly not so forspace-time increments in the presence of a minimumlength. This is at the base of the failure of attempts toconstruct directly a (continuous) space-time with an invari-ant length scale as described in [25].

All our considerations have been discussed in the frame-work of DSR1, a theory with limited momentum but un-limited energy. Similar discussions can be carried out indifferent DSR flavors.

As a last, but very important remark, let us call attentionto the fact that, since we do not know the relation betweenwhat we have called velocity and dx=dt, it is hard to saywhether the DSRs can be verified or disproved experimen-tally, for instance, by studying the time of flight of photonsof different energies from distant sources; in particular, weare not allowed to conclude that, since the deformed (three-)velocity definition is identical with undeformed specialrelativity, there are no effects, since what one is reallymeasuring are time and distances, for which we have atthe moment no definition.

ACKNOWLEDGMENTS

We would like to thank G. Amelino-Camelia, J. L.Cortes, J. Gamboa and J. Kowalski-Glikman for usefuldiscussions on this topic. Part of this work was developedduring a Mini-Workshop at LNGS in September 2004.F. M. thanks INFN for support.

[1] G. Amelino-Camelia, Int. J. Mod. Phys. D 11, 35 (2002).[2] G. Amelino-Camelia, Int. J. Mod. Phys. D 11, 1643

(2002).

[3] G. Amelino-Camelia, Nature (London) 418, 34 (2002).[4] J. Magueijo and L. Smolin, Phys. Rev. Lett. 88, 190403

(2002).

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[5] D. V. Ahluwalia-Khalilova, Int. J. Mod. Phys. D 13, 335(2004).

[6] N. R. Bruno, G. Amelino-Camelia, and J. Kowalski-Glikman, Phys. Lett. B 522, 133 (2001).

[7] J. Lukierski and A. Nowicki, Int. J. Mod. Phys. A 18, 7(2003).

[8] J. Magueijo and L. Smolin, Phys. Rev. D 67, 044017(2003).

[9] G. Amelino-Camelia, in New Developments in Funda-mental Interaction Theories, edited by J. Lukierski et al.(American Institute of Physics, New York, 2001), pp. 137–150.

[10] S. Mignemi, Phys. Lett. A 316, 173 (2003).[11] A. A. Deriglazov and B. F. Rizzuti, hep-th/0410087.[12] A. Granik, hep-th/0207113.[13] J. Lukierski and A. Nowicki, Acta Phys. Pol. B 33, 2537

(2002).[14] P. Kosinski and P. Maslanka, Phys. Rev. D 68, 067702

(2003).[15] T. Tamaki, T. Harada, U. Miyamoto, and T. Torii, Phys.

Rev. D 66, 105003 (2002).[16] M. Daszkiewicz, K. Imilkowska, and J. Kowalski-

Glikman, Phys. Lett. A 323, 345 (2004).[17] J. Kowalski-Glikman, Mod. Phys. Lett. A 17, 1 (2002).

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[18] J. Lukierski, H. Ruegg, A. Nowicki, and V. N. Tolstoi,Phys. Lett. B 264, 331 (1991); J. Lukierski, H. Ruegg, andA. Nowicki, Phys. Lett. B 271, 321 (1991); J. Lukierski,H. Ruegg, and W. Ruhl, Phys. Lett. B 313, 357 (1991).

[19] J. Kowalski-Glikman and S. Nowak, Int. J. Mod. Phys. D12, 299 (2003); A. Blaut, M. Daszkiewicz, J. Kowalski-Glikman, and S. Nowak, Phys. Lett. B 582, 82 (2004).

[20] A. A. Deriglazov, Phys. Lett. B 603, 124 (2004); S.Mignemi, gr-qc/0403038; D. Kimberly, J. Magueijo, andJ. Medeiros, Phys. Rev. D 70, 084007 (2004); C. Heuson,gr-qc/0312034; S. Gao and Xiao-ning Wu, gr-qc/0311009.

[21] For a recent review see J. Kowalski-Glikman, hep-th/0405273 and references therein.

[22] See also the approach by F. Girelli and E. R. Livine, gr-qc/0407098.

[23] S. Mignemi, Phys. Rev. D 68, 065029 (2003).[24] S. Majid and H. Ruegg, Phys. Lett. B 334, 348 (1994);

S. Zakrzewski, J. Phys. A 27, 2075 (1994).[25] R. Aloisio, A. Galante, A. Grillo, E. Luzio, and F. Mendez

(to be published).[26] R. Aloisio, J. M. Carmona, J. L. Cortes, A. Galante, A. F.

Grillo, and F. Mendez, J. High Energy Phys. 05 (2004)028.

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approaching space-time through velocity in doubly special relativity

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