Luca Tomassini
email: tomassin@axp.mat.uniroma2.it.

Stefano Viaggiu
email: viaggiu@axp.mat.uniroma2.it.

Physically motivated uncertainty relations at the Planck length for an emergent non commutative space-time

Keywords: numbers: 04.20.-q, 02.40.Gh, 04.60.-m, 03.65.-w

We derive new space-time uncertainty relations (STUR) at the fundamental Planck length L P from Quantum Mechanics and General Relativity (GR), both in flat and curved backgrounds. Contrary to claims present in the literature, our approach suggests that no minimal uncertainty appears for lengths, but instead for minimal space and four-volumes. Moreover, we derive a maximal absolute value for the energy density. Finally, some considerations on possible commutators among quantum operators implying our STUR are done.

Introduction

The idea that limitations on the localizability of space-time events should appear at very small distances is very old and can perhaps be traced back to Wigner and Saleker [1,2]. In a thorough investigation on the impact of Quantum Mechanics on the precision of measurements of distances between points and hence of the metric on curved space-times, the first author remarked that "the establishment of a close network of points in space-time requires a reasonable energy density (...). However, it is not necessary to discuss this in detail because the measurement of the distances between points of the network gives more stringent requirements than the establishment of the network".

Later, the attention shifted precisely on the conditions imposed on "the establishment of the network" itself, that is on the very possibility of giving an operational definition of the concept of space-time event. A variety of strategies were devised to attack the problem (see for example [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18] and the next Section for a brief discussion), but as far as we know it was Mead [4] who first observed that any attempt to determine the position of an object in a space-time region O of size comparable with the error (uncertainty) one is willing to accept entails, by the Heisenberg uncertainty principle, the localization of a non zero amount of energy in O itself. It is then then reasonable, Mead argued, to expect that a too high precision could trigger the formation of an event horizon representing a black hole surrounding O, thus preventing any signal to go back to the observer and making the measurement itself meaningless. In this context, we point out the remark of Einstein himself [19] that one cannot put an arbitrary amount of energy in a given finite spatial volume becomes relevant.

The relevant scale for the appearance of such effects was determined as the one of the Planck length

L P = G c 3
1, 6.10 -33 cm.

(

) 1
This point of view has been revived and made more precise in [5] (see [20] for a nice review and introduction to more recent developments), where the following program (for flat Minkowski space) was proposed:

1) Derive operationally meaningful uncertainty relations between coordinates of space-time events from the requirement of gravitational stability under localization experiments.

2) Promote the above mentioned coordinates to the status of quantum operators and commutation relations among them from which the uncertainty relations follow.

Making use of a newtonian approximation of the gravitational potential associated to the localizing energy (see Appendix 2), the following uncertainty relations (DFR relations from now on) were obtained:

c∆t ∆x 1 + ∆x 2 + ∆x 3 ≥ L 2 P , (2)
∆x 1 ∆x 2 + ∆x 1 ∆x 3 + ∆x 2 ∆x 3 ≥ L 2 P . (3)
The authors also succeeded in making Step 2) above, thus obtaining a physically motivated model of a non commutative (see [START_REF] Connes | Noncommutative Geometry[END_REF]) quantum space-time. Moreover, the foundations were laid down for the construction of quantum fields on it. Last but not least, it was first observed that in all previous works the necessity of a minimal uncertainty length could be traced back to the fact that only spherically symmetric geometries are usually considered.

More on this point in the next Section. Notice that equation ( 3) implies that we can determine with arbitrary precision up to two spatial coordinates, say x 1 , x 2 , meaning ∆x 1 , ∆x 2 < with > 0 arbitrarily small. Obviously, we have to pay the price of an arbitrarily large ∆x 3 . This is where we see the extra freedom of non spherical geometries come into action. However, if we take only ∆x 1 < , then ∆x 2 and ∆x 3 might well remain finite (say of order L P ). This would imply, again by the Heisenberg uncertainty relations and in agreement with the results of [START_REF] Bahns | [END_REF], that an infinite amount of energy could be localized inside an arbitrarily small space volume. Since event horizons have been ruled out by assumption, an unphysical naked singularity would arise.

This last observation shows the necessity of going beyond the newtonian approximation and was the starting point of our investigations. Indeed, in this paper we take exactly the same point of view of [5], but in our case GR enters the scene through (a refinement of) the so-called hoop conjecture (see below), first formulated by Thorne in [START_REF] Thorne | Magic Without Magic[END_REF]. We will see in the following that in our proposed space-time uncertainty relations the above pathologies do not occur and accordingly we will find an upper limit for the space-volume density of energy.

This paper is organized as follows. In section 2 we fix some notations and concepts. In section 3 we review some gedanken experiments present in the literature. In section 4 we derive the STUR with a minkowskian background metric together with some physical consequences. Section 5 is devoted to an attempt to generalize the STUR to a general curved background. Section 6 collects some final remarks and conclusions. Finally, in Appendix 1 we show that the approximations needed to obtain our STUR entail no change in their physical consequences, while in Appendix 2 we analyze those leading to relations (2), (3).

We conclude this introduction with two more remarks. First, we make no attempt to tackle Step 2). Second, we do not claim we provide proves, our aim is only to convince the reader by sound and hopefully robust physical arguments that our uncertainty relations are a realistic proposal: being there no chance of a direct experimental verification, they can only be tested by their consequences.

Notation and preliminaries

It is well known (see for example [START_REF] Landau | [END_REF]) that position, momentum and energy are described in Quantum Mechanics by self-adjoint operators acting on some Hilbert space H, while the time t is only a parameter. In what follows, the symbol ∆(A) indicates the uncertainty of the operator A in some quantum state ω (for example a vector state ω(A) = (φ, Aφ) where φ ∈ H and (•, •) indicates the scalar product) as defined by the formula (variance)

∆(A) = ∆ ω (A) = ω((A -ω(A)I) 2 = ω(A 2 ) -ω(A) 2 (4)
while (t being a parameter) ∆t = t 2 -t 1 where for the sake of definiteness t 1 , t 2 should be thought of as the times at which the measurements determining the region O are performed). The symbol I of course indicates the identity on H. 1 Once the average is taken, the resulting quantities behave as classical ones. Note that since the functional ∆ is not linear, Λ ν µ ∆(x ν ) = ∆(Λ ν µ x ν ), where Λ ν µ is any Lorentz transformation and x µ are the coordinates (operators) with x 0 = t. Thus, it is not the uncertainty relations that have to be covariant but the quantum operators they come from.

Suppose now an observer wants to perform a measurement of position and/or time in, say, a flat Minkowsky space-time and he is willing to accept an error ∆x i (i = 1, 2, 3), ∆t. Then, he will have to concentrate a certain amount of matter-energy in a space-time region O of corresponding size. Should this region be sufficiently small, it is reasonable to expect that quantum effects should no more be negligible. One is thus led to consider the usual Heisenberg uncertainty relations

∆x i ∆p j ≥ δ ij 2 i, j = 1, 2, 3 ( 5 )
which, in our relativistic setting, should be complemented with the timeenergy relations

∆t∆E ≥ 2 , (6)
and draw the conclusion that the measurement entails an energy transfer to the (dynamic) space-time. In particular, we see that very high energies will correspond to smaller errors in time. If in addition one requires small space uncertainties, a black hole could be produced thus preventing any signal to come out from the localization region and making the measurement itself meaningless.

As remarked in the introduction, this line of reasoning seems to have a long history and was first put on quantitative grounds by Mead in 1964 [4]. There, under the assumption of spherical symmetry (i.e. ∆R ∆x 1 ∆x 2 ∆x 3 ), the author derived the relation ∆R ≥ L P (7) and concluded from it that an absolute minimal length had to exist. It is interesting to note that the same result was obtained in a string theoretical context (see for example [25]) and by model independent considerations in [26]. In both cases the starting point were the Generalized Uncertainty Relations (GUP)

∆R ≥ ∆p + α ∆p ( 8
)
(α is a constant depending on Regge's slope and the gravitational constant G), from which once more under the assumption of spherical symmetry (7) follows by minimization of the second member. Finally, we stress that the position-momentum uncertainty relations are of a completely different nature than the time-energy one. More precisely, the first ones are relations between specific averages (variances) of quantum operators while the second is obtained by evaluating the transition probability between states with energies differing by an amount ∆E. The difficulty stems from the impossibility to associate a bona fide self-adjoint operator to time in the case of a non trivial (i.e. lower bounded) energy spectrum, but it has been circumvented in [27] by making use of the concept of operator valued measures (see [START_REF] Ludwig | Foundations of Quantum Mechanics[END_REF] for a thorough discussion). The time-energy uncertainty relation (6) was then recovered by a slight modification of the usual position-momentum argument. However, the following arguments should be considered independent of such details.

Minimal uncertainty length in the literature: a review

In this section we give a brief and certainly incomplete account of how the concept of minimal uncertainty is addressed in the literature. We already pointed out the observation first made in [5]: very often the conclusion that a minimal uncertainty exists can be traced back to the (implicit or explicit) assumption that the geometry of the problem be spherical.

We start with the papers [17,18]. Since they build on the ideas of [1,2], the main concern of the authors are limitations on the measurement of distances between different space-times points. However, they give a quantitative estimate of the uncertainty in the localization of each single point which is obtained by invoking the Schwarzschild metric.

A wealth of localization "gedanken experimenten" is presented in [7], where fluctuations of the space-time metric are analyzed making use of photons instead of target massive particles (together with some considerations on the Einstein's equations). We will do the same in the following. In the first experiment a microscope-type construction is presented, namely a photon scattered by a target particle. Here, the Newtonian gravitational interaction between the photon and the target particle is calculated together with the suggested modification of the Heisenberg relation leading to the relation ( 8) without invoking black hole physics. In any case, the Newtonian gravitational field is given in the form ∼ 1 r 2 and so spherical symmetry is implicitly assumed. The same assumption affects the description of the time resolution of synchronizing clocks in terms of the gravitational field generated by photons: a time-time component of a spherically symmetric metric is introduced. In the second example, a minimal uncertainty length is analyzed within a path integral formulation of quantum gravity. At a certain point the author denotes with "the radius of the region" and so, once again, spherical symmetry comes on stage. In [13] another interesting localizing experiment based on clocks and on the Heisenberg uncertainty principle is presented and an uncertainty relation involving the volume ∆V of the region to be localized and the time coordinate is found. The author denotes ∆V = (∆x) 3 and imposes that the "Schwarzschild radius should not exceed the uncertainty ∆x": thus the condition ∆x = ∆y = ∆z is implicitly assumed. Even the minimal uncertainty found in [14] using the GUP ( 8) is obtained using a spherical black hole argument. More references presenting arguments plagued by the assumption of spherical geometry can be found in [20].

The necessity of considering more general geometries is however clearly stressed in [12], where an interesting localizing experiment using an interferometer is proposed. The authors provide sound arguments to show that the uncertainty in one single coordinate can be reduced to arbitrarily small values, but are only able to provide quantitative results in spherical case where a minimal uncertainty is once again found.

We conclude this section by mentioning that not only uncertainties in coordinates and distances have been considered in the literature. For example, much in the spirit of the investigations of Wigner himself, in [16], [15] the main interest are the quantum fluctuations of the metric. Spherical symmetry is assumed in [15], while in [16] only asymptotic flatness of static space-time is needed. In both papers a Heisenberg type uncertainty relation is found between the metric and a "conjugate" quantity, so that there is no minimal uncertainty.

Minkowskian background

Spherical case

As it should now be clear, localization in some space-time region O will mean producing a (quantum) state confined there. We start by considering an experiment performed by an observer in a flat minkowskian space-time. It should be no surprise that in what we call spherical case (that is ∆x 1 ∆x 2 ∆x 3 ∆R/ √ 3, ∆R being the radius of the localization sphere) a completely satisfactory answer can be given. We start by presenting a separate treatment of this situation.

The metric outside the space volume V (3) where an energy ∆E is concentrated is:

ds 2 exp = dr 2 1 -2G∆E c 4 r + r 2 dΩ 2 -1 - 2G∆E c 4 r dt 2 . (9)
Of course, we set ∆E = c 2 ∆M , ∆M being then the ADM mass energy of the experiment. Since the configuration is spherical, gravitational radiation is absent so that the exterior metric is static. Notice that, due to spherical symmetry, this will be true irrespective of the duration ∆t of the experiment, where of course t is the time measured by an observer at spatial infinity. Making use of the Komar integral, it is clear that for any r > ∆R the massenergy of the experiment is simply ∆M . From (9), the condition that no black hole is formed reads:

G∆E c 4 < ∆R 2 . ( 10
)
As remarked above, our observer measures the time t and the energy in terms of it. He can therefore use the Heisenberg relation (6) to estimate the average energy ∆E (recall the discussion in section 2) of the configuration, so that condition (10) becomes:

c∆R∆t ≥ L 2 P . (11)
This is the time-space uncertainty relation. Since ∆E ≥ c∆p, from (5) and condition (10) we get:

∆R∆x 1 ∆x 2 ∆x 3 ≥ L 2 p (∆x 1 ) 2 (∆x 2 ) 2 + (∆x 2 ) 2 (∆x 3 ) 2 + (∆x 1 ) 2 (∆x 3 ) 2 . ( 12
)
In the spherical case, this reads ∆x i ≥ L P , i = 1, 2, 3, i.e. a minimal uncertainty length appears in agreement with (7) and (8). Furthermore, in this particular situation, our results agree with the ones in (3).

General case

The main problem when considering more general geometries than the spherical one is the absence of general theorems guaranteeing that no black holes are formed, independently to the dynamics governing the energy-matter motion. In fact solving the global Cauchy problem remains a formidable task at the present day, apart from the remarkable work of Christodulou [START_REF] Christodulou | [END_REF]30] about spherical collapse in the presence of classical scalar fields. For this reason, we resort to the so called hoop conjecture (see [START_REF] Thorne | Magic Without Magic[END_REF]31,32,33,34,35,36,37]). In its original version it states that • Horizons form when and only when a mass M is concentrated into a region whose circumference in every direction satisfies C ≤ 4πM G c 2 .

For example, this would imply that a mass confined in a cylinder with a small base radius but a sufficiently large height would not produce an horizon. The statement is purposely imprecise. As the mass M we could take the ADM mass or a quasi-local one such as the Tolmann mass [START_REF] Tolman | Relativity, Thermodynamics and Cosmology[END_REF]. The study of event horizon formation is very complicated since it is a global concept available only to "omniscient" observers (see [32]). For these reasons, in the literature, geometrical and more tractable objects as apparent horizons and trapped surfaces (see [START_REF] Hawking | The Large Scale Structure of Space-Time 1973[END_REF][START_REF] Kriele | [END_REF]41]) are often used. In fact, the formation of trapped surfaces and apparent horizons represents an indication of the formation of black holes, via the cosmic censorship conjecture [START_REF] Penrose | Seminar on Differential Geometry[END_REF].

In any case the hoop conjecture can also be stated in terms of the surface of the horizon A H and the mass M , according to the so called Penrose's isoperimetric type conjecture (see [33,[START_REF] Gibbons | [END_REF]). For a stationary black hole we have [31]:

A E < A H ≤ 16π G 2 c 4 M 2 , (13)
A E being the apparent horizon. As a result, the existence of an apparent horizon implies the existence of an event horizon enclosing it. More generally, denoting with A i the minimal area of a surface enclosing an apparent horizon, Penrose's isoperimetric conjecture states that:

A i ≤ 16π G 2 c 4 M 2 . ( 14
)
To ensure that no black hole is formed, we need to invert inequality (14). Furthermore, we study situations in which the mass-energy M is localized within a region of area A with A ≥ A i . As a result, we can with no loss of generality consider the inequality:

A ≥ 16π G 2 c 4 M 2 , (15)
so that in no case the formation of an event horizon enclosing the localizing area A is possible. This is our starting point. By considering a surface ∆A enclosing a mass ∆M and again putting ∆E = c 2 ∆M , we have:

∆A ≥ 16π G 2 c 8 (∆E) 2 . ( 16
)
Suppose now the localization region is a box with edges ∆x 1 , ∆x 2 , ∆x 3 . Then:

∆A ∼ ∆x 1 ∆x 2 + ∆x 1 ∆x 3 + ∆x 2 ∆x 3 α 2 , ( 17
)
where the normalization constant α will be determined later.

Concerning the energy ∆E, from equation ( 6) we again have:

∆E ≥ 2∆t . (18)
Furthermore, since the localization experiment is performed by concentration of mass-energy, we have:

(∆E) 2 ≥ c 2 (∆p x 1 ) 2 + (∆p x 2 ) 2 + (∆p x 3 ) 2 , (19)
and from (5) we deduce that:

(∆E) 2 ≥ c 2 2 4 1 ∆x 1 2 + 1 ∆x 2 2 + 1 ∆x 3 2 . ( 20
)
Note that inequality (20) holds for any type of localizing mass-energy. From ( 17), ( 18) and ( 16) we obtain:

(c∆t) 2 ∆x 1 ∆x 2 + ∆x 1 ∆x 3 + ∆x 2 ∆x 3 ≥ 4πα 2 L 4 P . (21)
Comparison with the spherical case gives 4πα 2 = 1. We can use inequality (20) to obtain the space-space uncertainty relation. To this purpose first notice that the following inequality holds:

1 ∆x 1 + 1 ∆x 2 + 1 ∆x 3 2 ≥ 1 ∆x 1 2 + 1 ∆x 2 2 + 1 ∆x 3 2 . ( 22
)
Estimating the energy ∆E by the left hand side of ( 22) we get a less precise result, i.e. we ask that the area ∆A be greater than a quantity which in turn is greater than the energy. Nevertheless, we argue in Appendix 1 that this causes no change in the physical content of the uncertainty relation. Summing up, our final expression for the STUR is:

(c∆t) 2 ∆x 1 ∆x 2 + ∆x 1 ∆x 3 + ∆x 2 ∆x 3 ≥ L 4 P , (23)
(∆x 1 ) 2 (∆x 2 ) 2 (∆x 3 ) 2 ≥ L 4 P ∆x 1 ∆x 2 + ∆x 1 ∆x 3 + ∆x 2 ∆x 3 . (24)
Furthermore, it easy to see that making use of the following crude estimates for ( 23) and ( 24) respectively:

(a + b + c) 2 ≥ ab + bc + ac, (25)
(ab + bc + ac) 3 ≥ a 2 b 2 c 2 , ( 26
)
with a = ∆x 1 , b = ∆x 2 , c = ∆x 3 , we get the DFR relations ( 2) and (3).

Some physical consequences

To start with, rewrite [START_REF] Landau | [END_REF] in the form

∆x 1 ∆x 2 ∆x 3 ≥ L 4 P 1 ∆x 1 + 1 ∆x 3 + 1 ∆x 3 . ( 27
)
If ∆x 1 < ( > 0), the other uncertainties being arbitrary, we obtain:

∆x 1 ∆x 2 ∆x 3 ≥ L 4 P . ( 28
)
We see then that if an uncertainty on a single spatial coordinate is arbitrarily small, the spatial volume must increase accordingly. Not only no minimal uncertainty length arises, but localization of an infinite energy in a finite volume is forbidden.

The same conclusion can be drown in the case ∆x

1 < 1 , ∆x 2 < 2 ( 1 , 2 > 0)
, but now we also see that there is no minimal surface in the sense that the product of two linear uncertainties can be made arbitrarily small. In fact, we have:

∆x 1 ∆x 2 ∆x 3 ≥ L 4 P 1 1 + 1 2 . ( 29
)
Moreover, by taking the third power of ( 24) and estimating the resulting right hand side by (26) we easily get:

∆x 1 ∆x 2 ∆x 3 ≥ L 3 P , (30)
i.e. a lower bound for the space volume appears. Finally, ( 23) and ( 24) together imply:

c∆t∆x 1 ∆x 2 ∆x 3 ≥ L 4 P . (31)
meaning that an analogous result holds for the four-volume. On the other hand, if we put ρ∆x 1 ∆x 2 ∆x 3 = ∆M (ρ being thus the density) in ( 15) and take advantage of ( 16) and ( 24) we get:

ρ < 1 4 √ π c 2 GL 2 P ∼ ρ P , (32)
where ρ P indicates Planck's density. Thus, independently on the geometry of the localization zone, an upper limit for the density appears. In view of (30) this should be no surprise, but we stress that GR alone cannot account for this result: to rule out singular distributions of matter-energy one needs Quantum Mechanics. Concerning relation [START_REF] Thorne | Magic Without Magic[END_REF], when the "spatial" part ∆x 1 ∆x 2 + ∆x 1 ∆x 3 + ∆x 2 ∆x 3 goes to infinity, ∆t can be made arbitrarily small. Conversely, when ∆x 1 ∼ ∆x 2 ∼ ∆x 3 ∼ L p , we have c∆t ≥ L p , showing that, limited to spherical case, we have a minimum uncertainty also in the time coordinate. Our next considerations concern the possible underlying quantum structure. The determination of commutators among quantum operators that could give rise to our STUR is certainly not a simple task. Nevertheless, some partial conclusions can be drown thanks to the fact that in our approach the DFR relations appear as an approximation to the STUR themselves and moreover provide exactly the same result in the spherical case. We look at this fact as a hint that the commutation relations obtained in [5] could be viewed as a limiting case of the ones giving rise to the STUR. In view of the generalization to curved backgrounds, we conclude this section by the following remarks. Introducing the surfaces

∆A 0 = (c∆t) 2 , ∆A 1 = ∆x 2 ∆x 3 , ∆A 2 = ∆x 1 ∆x 3 , ∆A 3 = ∆x 1 ∆x 2 . ( 33
)
the STUR ( 23) and ( 24) become:

∆A 0 ∆A 3 + ∆A 1 + ∆A 2 ≥ L 4 P , (34)
∆A 3 ∆A 1 ∆A 2 ≥ L 4 P ∆A 3 + ∆A 1 + ∆A 2 . (35)
Moreover, by making use once more of ( 26), we get

∆A 3 ∆A 2 + ∆A 3 ∆A 1 + ∆A 2 ∆A 1 ≥ L 4 P , (36)
Notice that, together with (34), this is exactly the same structure of the DFR relations, albeit in terms of the surfaces ∆A 0 , ∆A 1 , ∆A 2 , ∆A 3 . We believe that this might be more than a simple formal manipulation, since we have seen that GR (more precisely condition ( 16)) and Quantum Mechanics seem to lead naturally to terms proportional to L 4 P . More importantly, the commutators [A α , A β ] (where α, β = 0, . . . , 3) between the quantum operators A α are known, being formally identical to the commutators between the space-time coordinates present in [5]. The next step would be to find explicit expressions for the coordinate operators t, x, y, z in terms of the A α 's and calculating the corresponding commutators.

Curved background: an attempt

The extension of our results to the case of a general curved background presents one main difficulty. Of course, we still have condition (16) at our disposal, but unfortunately we cannot use Heisenberg relations ( 6) and ( 5) any more. The main problem is that, thanks to the equivalence principle, there is not a simple manner to isolate the energy contribution inside the localizing "box" (the experiment) from the rest of the mass-energy distribution, i.e for a non-isolated system. However, in the literature (see [44,45,46,47] and for a comprehensive review [48] and references therein) the concept of quasi-local mass has been introduced to measure the energy of a system within a closed compact orientable space-like two-surface by means of a energy-momentum four-vector depending on the first fundamental form on the two-surface itself, the second fundamental form and null normals which define a preferred basis for the fibers of the normal bundle. It is a common opinion that the most promising was proposed in [47] by means of the Hamiltonian formulation of GR. Obviously, the most important requirements for a quasi-local mass are:

1. The ADM and Bondi mass should be recovered respectively at spatial and null infinity.

2. When the surface degenerates to a point, correct limits must be obtained.

3. The quasi-local mass must be non negative and zero for any space-like two surface in flat minkowskian background.

The conditions above are not trivial to fulfill. We do not enter in more details and thereafter denote with the symbol E ql the quasi-local energy. Furthermore, for our purposes, it seems to be reasonable to use the formalism introduced in [49] and [50]. Then, g µν being the background metric, we define:

h µν = g µν + γ µν , γ 0 = 1 |g 00 | , γ i = 0, γ i = g 0i |g 00 | , (37)
ds 2 = dσ 2 -c 2 dT 2 , (38)
where

dσ 2 = h ik dx i dx k , (39)
with T given by

T = |g 00 |dt + g 0i c |g 00 | dx i , i = 1, 2, 3. (40)
In this way the line element is written only in terms of observable quantities, i.e. T and σ. As a consequence, we propose to substitute the inequality (6) for a flat background with:

∆E ql ∆T ≥ 2 . (41)
Expression ( 41) is generally well defined and reduces to inequality (6) in a flat background. As a consequence, using (41) and (16) we can write the time-space uncertainty relation as:

∆A 00 ∆A ≥ L 4 P , ∆A 00 = (c∆T ) 2 , ( 42
)
where ∆A is now a surface depending on the non-flat background metric [START_REF] Tolman | Relativity, Thermodynamics and Cosmology[END_REF] and which cannot be expressed in the simple form (17). The space-space relation is much more involved and we have not been able to find a reasonable generalization for a curved background. However, some interesting inequalities can be suggested. First of all, if we consider a stationary background, h ik does not depend on the time T and as a result, we can write ( 23) and ( 24) in a form more appropriate for a curved background without any explicit reference to the spatial coordinates. We get:

c∆T ∆V (3) ≥ L 4 P , (43)
(∆V (3) ) 2 ≥ L 4 p ∆A. ( 44
)
Obviously, expressions [START_REF] Gibbons | [END_REF] and (44) reduce to [START_REF] Thorne | Magic Without Magic[END_REF] and [START_REF] Landau | [END_REF] in a minkowskian background by writing the spatial volume and the total surface of the localizing "box" in terms of Cartesian coordinates. We regard expressions (42), ( 43) and ( 44) as an indication that proper times, surfaces and volumes may be the natural object to be quantized. Taking into account that inequality (30) must still hold in a stationary general background (∆V (3) ≥ L 3 P ), [START_REF] Gibbons | [END_REF] together with (42) entail an upper limit to the proper density ρ. In fact, in analogy with the flat case, we can write the proper mass (see [36]) ∆M p (the mass inside the proper three volume ∆V (3) ) as ρ∆V (3) = ∆M p . As a consequence, recalling (16) we see that inequality (32) is still valid independently on the geometry of the configuration. Oncer more, we stress that this result is a consequence of both GR and Quantum Mechanics. In any case, for a general non-stationary background it seems reasonable to take advantage of the results in (31) to obtain:

∆V (4) ≥ L 4 P , (45)
∆V (4) being the invariant four-volume. Our method allows no further generalizations, since in a non-stationary background the four-volume ( 45) cannot be put in the form [START_REF] Gibbons | [END_REF], because of the time dependence of the metric h ik .

Conversely, if we adopt inequality (43) also for a non-stationary background, expression (45) cannot be directly inferred. Summarizing, for a stationary background space-time we use ( 42), ( 43) and (44). Obviously they are not independent. In fact, starting with (44) and using [START_REF] Penrose | Seminar on Differential Geometry[END_REF], we obtain inequality [START_REF] Gibbons | [END_REF]. For a general time dependent spacetime, we use ( 42) and (45). Let us conclude this section with a remark. Recently, in [51] it has been advocated that energy can be localized in GR by means of a generalization of the Tolman mass ( [START_REF] Tolman | Relativity, Thermodynamics and Cosmology[END_REF]) to a generic non-static background. The expressions for the energy and momentum proposed there are:

Ẽ = c 4 4πG R µ ν u ν u µ √ -gd 4 x, µ, ν = (0, 1, 2, 3), (46) pα
= c 4 4πG R α ν u ν √ -gd 4 x, α = (1, 2, 3), (47)
where u ν stands for the four-velocity of the observer. The main idea is that the energy is encoded by the Ricci tensor R µν . We stress that if ( 46) and ( 47) are correct, gravitational waves do not carry energy. For a free-falling observer we have u ν = u ν = (1, 0, 0, 0). Since ( 46) is an invariant four scalar, it is clear that he computes the Tolman mass (multiplied by its proper time), i.e. the mass "without gravity". As a consequence, in [51] the following uncertainty relations are proposed:

∆E∆t ≥ 2 → ∆ Ẽ ≥ 2 , (48)
∆p α ∆x α ≥ 2 → ∆p α ≥ 2 . (49)
Thanks to the observations above concerning a free-falling observer, to impose condition ( 16) to (48) we can divide the quantity ∆ Ẽ by the chronoinvariant time given by [START_REF] Kriele | [END_REF]. As a consequence, using (48) the time-space uncertainty relation becomes the one given by [START_REF] Penrose | Seminar on Differential Geometry[END_REF]. For the space-space relation, we could use expression (49), but this cannot be written in the form (20) since the norm of a vector is given in terms of the background metric. Hence, we are forced to use again the arguments leading to ( 43) and (45).

Conclusions

In this paper we obtained the STUR from Quantum Mechanics and classical GR. More precisely, we made use of the Heisenberg uncertainty relations and of the observation that the formation of black holes due to concentration of energy makes the measurement of position operationally meaningless. As advocated in [5], the STUR then follow from a sound physical input.

To start with, we obtained the STUR in a flat background. Then, we have generalized them to general curved backgrounds with particular attention to the stationary case. Our STUR appear completely physically reasonable, in the sense that an infinite energy can never be localized in a finite volume. Furthermore, in agreement with [5], we found no minimal uncertainty while a natural lower bound for the Lorentz invariant (or general invariant in the case of GR) four-volume appears together with an upper limit for the space-volume density. We stress that our results coincide, in the case of a spherical geometry, with the ones obtained by making use of the GUP, that is by allowing the Heisenberg relations to be modified by gravity. Some indications were given for the possible underlying quantum structure, but we were not able to find the commutators implying our STUR. This will be the subject of a future work. in agreement with [START_REF] Ludwig | Foundations of Quantum Mechanics[END_REF]. For ∆x 1 < 1 , ∆x 2 < 2 ( 1 > 0, 1 > 0) and ∆x 3 > ∆x 2 , ∆x 2 we have, this time in agreement with [START_REF] Christodulou | [END_REF]

The case ∆x i < i and i arbitrarily small (i = 1, 2, 3) is forbidden by (51).

To estimate the three-volume first notice that we can suppose with no loss of generality that ∆x 1 ∆x 2 = max{∆x 1 ∆x 2 , ∆x 2 ∆x 3 , ∆x 1 ∆x 3 }. Then from (51) we get:

(∆x 1 ) 2 (∆x 2 ) 2 (∆x 3 ) 2 ≥ L 4 P 3 ∆x 1 ∆x 2 ≥ ≥ L 4 P 9 ∆x 1 ∆x 2 + ∆x 1 ∆x 3 + ∆x 2 ∆x 3 . (55)
By taking the third power of both sides and using (26), we arrive at: and recall that in the weak approximation φ(r) becomes the Newtonian potential, we obtain:

(
φ = ln 1 - 2GM c 2 - GM c 2 r . ( 64
)
Since (remember that in the weak field approximation |φ| << 1) the condition for no trapped surfaces becomes GM c 2 r < 1, which is wrong only by a factor 2.

  , ∆x 1 ∆x 2 ∆x 3 ≥ L 4 P 2(∆x 2 ∆x 3 + ∆x 3 ∆x 1 )

							×	∆x 2 ∆x 3 ∆x 1 +	∆x 3 ∆x 1 ∆x 2	=
	=	L 4 P 2	∆x 2 ∆x 1 (∆x 1 + ∆x 2 )	+	∆x 1 ∆x 2 (∆x 1 + ∆x 2 )	≥
		>	L 4 P 2	∆x 2 (∆x 1 + ∆x 2 ) 2 +	∆x 1 (∆x 1 + ∆x 2 ) 2 =
		=	L 4 P 2	1 (∆x 1 + ∆x 2 )		≥	L 4 P 2( 1 + 2 )	.

  ∆x 1 ) 2 (∆x 2 ) 2 (∆x 3 ) (∆x 1 ∆x 2 + ∆x 1 ∆x 3 + ∆x 2 ∆x 3 ) 2 .and by the same assumption then for the three-volume:(∆x 1 ) 2 (∆x 2 ) 2 + (∆x 2 ) 2 (∆x 3 ) 2 + (∆x 1 ) 2 (∆x 3 ) 2 (∆x 1 ∆x 2 + ∆x 1 ∆x 3 + ∆x 2 ∆x 3 ) 2 ≥ ≥ ∆x 1 ∆x 2 ∆x 1 ∆x 2 + ∆x 1 ∆x 3 + ∆x 2 ∆x 3

	so that	c∆t∆x 1 ∆x 2 ∆x 3 ≥	L 4 P 3	.	(59)
		2 3	≥	L 12 P 9 3 (∆x 1 ) 2 (∆x 2 )	2 (∆x 3 ) 2 ,	(56)
	which gives
		∆x 1 ∆x 2 ∆x 3 ≥	L 3 P 3 √ 3	.	(57)
	Finally, for the four-volume we get:
		(c∆t) 2 (∆x 1 ) 2 (∆x 2 ) 2 (∆x 3 )	2 ≥	(58)
	P L 8	(∆x 1 )
						2	≥	1 9	,

2 (∆x 2 ) 2 + (∆x 2 ) 2 (∆x 3 ) 2 + (∆x 1 ) 2 (∆x 3 )

2

For any spectral value of an operator A, there is a state ω with arbitrary small uncertainty: we see that an absolute lower bound in the uncertainty of one coordinate makes its description by means of an operator impossible. At the same time, in[START_REF] Bahns | [END_REF] operators describing distances, space and four-volumes defined by independent space-time events were defined in terms of coordinates obeying (2) and (3). Lower bounds were found on euclidean distances and four-volumes.

Acknowledgements

We would like to thank Gerardo Morsella and Nicola Pinamonti for several interesting discussions and suggestions. L T would also like to thank Sergio Doplicher for his constant support and encouragement and Sebastiano Carpi for many useful hints. Finally, we would like to thank the referees for their observations and suggestions.

Research supported by GNAMPA-INDAM and the German Research Foundation (Deutsche Forschungsgemeinschaft (DFG)) through the Institutional Strategy of the University of Göttingen.

Appendix 1

The exact form of our STUR (i.e. without making use of the approximation [START_REF] Bahns | [END_REF]) is:

First of all observe that we can write (51) as:

Consider now the case ∆x 1 < ( > 0), ∆x 2 , ∆x 3 > ∆x 1 . Then

Appendix 2

In [5] the condition that no trapped surface is formed is -φ < 1, φ being of course the gravitational potential in the newtonian approximation. By considering a constant energy density ρ(-t) at negative times x 0 = -t over a volume centered at the origin with sides ∆x 1 + t, ∆x 2 + t, ∆x 3 + t, the authors obtain φ ∼ ∆E × I, with

Notice that

Putting ∆A DF R = I -2 , condition (60) becomes:

Expression ( 62) is now comparable (apart from an inessential constant) with our starting condition (16). Much in the spirit of [5], will take ∆E as given by ( 18) and (20).

To begin with, it is easy to see that in the spherical case ∆A DF R ∼ (∆x 1 )

2 so that (61) and ( 17) differ only by a positive constant thus giving rise to the same physical consequences. For ∆x 1 = ∆x 2 >> ∆x 3 (a large disk), the result is ∆A DF R ∼ (∆x 1 ) 2 and ∆A ∼ (∆x 1 ) 2 (where of course ∆A is given in ( 17)), so that in this case too the two methods agree. However, for a cylindrical geometry with ∆x 1 >> ∆x 2 ∼ ∆x 3 we find ∆A DF R ∼ ∆A ∼ (∆x 1 ) 2 and we see that ∆A DF R is an overestimation compared to our calculations. In a sense, the approximation considered in [5] amounts to identifying a cylinder with the sphere containing it. Of course, a greater ∆A DF R means a smaller |φ| so that the preceding overestimation appears to be a consequence of the weak field approximation. We conclude with the following remark. The fact that the use of the weak field approximation can give sound results should be no surprise. For example, if we take the spherical static metric: ds 2 = -e 2φ(r) c 2 dt 2 + e -2ψ(r) dr 2 + r 2 dΩ 2 , (63)


Maarten Van De Meent
email: m.vandemeent@uu.nl

Piecewise Flat Gravitational Waves

We examine the continuum limit of the piecewise flat locally finite gravity model introduced by 't Hooft. In the linear weak field limit we find the energy-momentum tensor and metric perturbation of an arbitrary configuration of defects. The energymomentum turns out to be restricted to satisfy certain conditions. The metric perturbation is mostly fixed by the energy-momentum except for its lightlike modes which reproduce linear gravitational waves, despite no such waves being present at the microscopic level.

I. INTRODUCTION

Einsteinian gravity in 2 + 1 dimensions is in many ways much simpler than in 3 + 1 dimensions. In 2 + 1 dimensions the Einstein equation completely fixes the Riemann tensor in terms of the energy-momentum. In particular, there are no local gravitational degrees of freedom. [START_REF] Deser | Three-dimensional Einstein gravity: Dynamics of flat space[END_REF] This makes quantization of gravity in 2 + 1 dimensions possible as was done non-perturbatively by Witten in 1988 [START_REF] Witten | 2 + 1 dimensional gravity as an exactly soluble system[END_REF] (perturbative quantization was shown the following year by Deser, McCarthy, and Yang [START_REF] Deser | Polynomial formulations and renormalizability in quantum gravity[END_REF]). Local degrees of freedom can only be added by including matter. A system of point particles interacting gravitationally has a locally finite number of degrees of freedom and may be quantized. [START_REF] Hooft | Canonical quantization of gravitating point particles in (2+1)-dimensions[END_REF][START_REF] Waelbroeck | Canonical quantization of (2+1)-dimensional gravity[END_REF][START_REF] Hooft | Quantization of Point Particles in 2+1 Dimensional Gravity and Space-Time Discreteness[END_REF] The geometry of a system of point particles in 2 + 1 dimensions is that of a piecewise flat manifold consisting of blocks of flat spacetime glued together along their faces. [START_REF] Hooft | Causality in (2+1)-dimensional gravity[END_REF] All curvature is concentrated along the edges of the blocks, which coincide with the paths of the particles. In 2008, 't Hooft suggested [START_REF] Hooft | A locally finite model for gravity[END_REF] that to obtain a locally finite theory of gravity in 3 + 1 dimensions, one should impose the rule that empty space has no local structure and is therefore locally (Riemann) flat. The combination of this rule with Einstein's equation implies that the only local degrees of freedom that may be added are curvature defects of co-dimension 2, i.e. lines propagating at constant velocity. The result is a model that can be regarded as a generalization of gravity in 2 + 1 dimensions, which we studied in more detail in [START_REF] Van De Meent | Collisions in piecewise flat gravity in 3+1 dimensions[END_REF], where it was shown that a configuration of defect lines in 3 + 1 dimensions can be described as a piecewise flat manifold.

In this respect the model is very similar to the piecewise flat approach to gravity introduced by Regge in 1961. [START_REF] Regge | General Relativity without Coordinates[END_REF] Regge calculus has since then been used as a tool both in classical numerical general relativity as well as in quantum gravity. (See [START_REF] Williams | Regge calculus: A Bibliography and brief review[END_REF] and [START_REF] Regge | Discrete structures in gravity[END_REF] for a review and references.) A key difference is that Regge calculus allows defects of any signature, whereas 't Hooft's model insists on interpreting the defects as propagating physical degrees of freedom. Imposing causality then implies that only non-spacelike defects may appear.

In 't Hooft's model all physical degrees of freedom, including the gravitational ones, are defects and therefore treated as matter. As a result, by design, the model has no local gravitational structure. This implies that -a priori -there are no long range gravitational fields and gravitational waves, unlike the world that we observe, and therefore raises the question how physical this model can be. A resolution may come from the fact that the model allows both positive and negative energy defects, i.e. defects with a positive or negative deficit angle respectively. 1 One could imagine a configuration of defects that on large scales has zero average energy-momentum, but has a non-trivial average Weyl curvature. In other words, we might hope that the vacuum structure found in general relativity is recovered in the continuum limit.

The continuum limit of piecewise flat gravity models has been studied extensively in the context of Regge calculus. [START_REF] Cheeger | Lattice gravity or Riemannian structure on piecewise linear spaces[END_REF][START_REF] Cheeger | On the curvature of piecewise flat spaces[END_REF][START_REF] Feinberg | Lattice gravity near the continuum limit[END_REF][START_REF] Friedberg | Derivation of Regge's action from Einstein's theory of general relativity[END_REF] It is known that the space of Regge manifolds is dense in the space of solutions of general relativity. In particular, Regge calculus contains approximations to any vacuum solution of general relativity. However, this does not answer our question since we restrict to piecewise flat configurations that only contain physical (i.e. non-spacelike) defects. It is not clear whether this subset of configurations is still dense. This question will be addressed in this article.

Our primary interest in this model has been to handle it as a precursor to a possible theory of quantum gravity. Just as 2 + 1 dimensional gravity, it may teach us interesting lessons about the quantization of gravity, regardless whether it serves well as a model for real world physics. The emphasis on causality and the fact that the studied configurations are exact solutions of general relativity, may however also make this model interesting as an approximation scheme for classical inquiries in general relativity. For example, it would be interesting to see if it could be used as a test case for questions about the averaging problem in cosmology. For such applications it is crucial to know what kind of matter sources may be approximated. This will be a second line of inquiry.

The description of a configuration of defects simplifies dramatically in the limit where the energies of all defects are considered to be infinitesimal. This allows us to study continuous distributions of infinitesimal defects. Section II explains how to describe such a configuration in this limit using a density function on the configuration space of an individual defect and introduces the notation used in the rest of the article.

Section III then constructs the energy-momentum tensor produced by an arbitrary configuration of physical defects. It finds the conditions that the energy-momentum will obey (and consequently must be obeyed by any theory that is to be approximated by this model in the limit of weak fields).

The metric perturbation produced by a general configuration of physical defects is obtained in section IV. After which, in section V, the results of the preceding sections are combined to find the metric perturbations that can be produced by a configuration with vanishing energy-momentum. We find that it is possible to reproduce the complete spectrum of gravitational waves found in linearized Einstein gravity.

II. DEFECT CONFIGURATIONS IN WEAK FIELD

In the previous articles [START_REF] Hooft | A locally finite model for gravity[END_REF][START_REF] Van De Meent | Collisions in piecewise flat gravity in 3+1 dimensions[END_REF] two approaches to describing a general configuration of propagating defects were described. Both of which share that they rapidly increase in complexity as the number of defects increases, because the description of the state of a defect involves the states of other defects as well. This convolution of the description is this model's manifestation of the non-linear nature of general relativity.

However, in this paper we want to look at geometries generated by configurations with large numbers of defects, which makes the previously employed methods prohibitively complex. Fortunately, the description of a configuration of defects drastically simplifies in the limit that all defect angles are very small. In this limit each defect can be treated as a linear perturbation to a Minkowski background. Moreover, any new intermediate defects that would be created by the collision of two defects are higher order in the defect angles of the colliding defects and can be neglected. Consequently, in the linear limit the state of a defect line can be described while ignoring the presence of other defects. A configuration of defects can therefore be completely described by giving the number of defects in any given state. Since the effect of two defects with the same state is simply that of a single defect with the combined energy of the two defects, we can completely describe the configuration by giving the energy in each possible state. That is, if M is the state space of a single defect with unit energy density, a configuration of defects can be described by a distribution ρ : M → R giving the energy density in each state.

To parameterize the state space M, recall that in [START_REF] Hooft | A locally finite model for gravity[END_REF] it was shown that the state of a single defect line with a density ρ could be described by the following data: a vector p that gives the position of the defect, a vector d that gives its direction, and a vector v that gives its velocity.

The triple p, d, v is enough to uniquely identify the state of the defect. However, this characterization is not unique, since the triple

p + α d, β d, v + γ d (1)
describes exactly the same state. In fact, the triples describing the same state as p, d, v can be completely parameterized by the numbers α, β, and γ. The state space of a line defect with unit density M is thus obtained from R 3 × R 3 × R 3 by modding out by the equivalence relation [START_REF] Deser | Three-dimensional Einstein gravity: Dynamics of flat space[END_REF]. A set of unique representatives for each equivalence class in M can be formed by taking a triple p, d, v that satisfies the following conditions, v ⊥ d, p ⊥ d, and d = 1.

(2) Such a representative triple is unique up to a sign of d, i.e. to get a unique representative d should be viewed as an element of the real projective plane, RP 2 .

In this article it will be convenient to decompose density functions ρ on M in a set of canonical functions that we call laminar plane waves. A laminar plane wave is a configuration where all defects have the same direction d0 and velocity v 0 and the density ρ is a plane wave function with wave vector k 0 with respect to the position p of the defects. That is, the density function ρ lam [ k 0 , d0 , v 0 ] corresponding to a laminar plane wave with wave vector k 0 , direction d0 , and velocity v 0 can be written as a function of p, d, v representing an equivalence class in M as follows,

ρ lam [ k 0 , d0 , v 0 ]( p, d, v) = e 2πi k 0 • p δ( d -d0 )δ( v -v 0 ). 2 (3)
Since p en v are always perpendicular to d it is sufficient to consider only laminar plane waves with k 0 and v 0 perpendicular to d0 . The set M of laminar plane waves with parameters k 0 , d0 , v 0 is complete in the sense that any density function ρ( p, d, v) on M can be written as

ρ( p, d, v) = M d k 0 d d0 d v 0 ρ lam [ k 0 , d0 , v 0 ]( p, d, v)ρ( k 0 , d0 , v 0 ) (4) = k⊥ d d k e 2πi k• p ρ( k, d, v), (5)
where ρ( p, d, v) is a density function on M. The function ρ( k, d, v) can therefore be viewed as a partial Fourier transform of ρ( p, d, v). 3It will also be convenient to split the parameter v in a component collinear with k and a component perpendicular to both k and d,

v = ω k k 2 + v k × d k 2 , (6)
where the ambiguous direction of d is chosen such that v is non-negative. The usefulness of this split becomes apparent when we do a Lorentz transform. The Lorentz transform of a laminar plane wave with parameters k, d, ω, v is again a laminar plane wave, but with different parameters. It can be shown that the combination k µ = (ω, k) transforms as a 4-vector under the Lorentz transformation. In section III we will see that this is the wave vector of the corresponding energy-momentum tensor. In the remainder of this article will denote the parameters of a laminar plane wave as k µ , d, v .

The physicality condition that we impose on the defects implies that the velocity of each defect must be smaller than or equal to c = 1. 4 By squaring equation [START_REF] Hooft | Quantization of Point Particles in 2+1 Dimensional Gravity and Space-Time Discreteness[END_REF] we see that this implies that

ω 2 + v 2 k 2 ≤ 1. (7)
Consequently, we see that for a laminar plane wave of physical defects, the wave vector k µ must be spacelike or lightlike and v must be smaller than or equal to k µ k µ .

III. ENERGY-MOMENTUM

In this section we will derive the energy-momentum tensor generated by an arbitrary configuration of physical defects in the limit that the distribution is continuous and all defect angles are small. This will tells us what conditions are imposed on the energy-momentum tensor by the requirement that the defects are physical (i.e. non-tachyonic). This puts limits on the kinds of models which can be found as the continuum limit of the piecewise linear model considered here.

In the linear weak field limit the energy-momentum tensor of a configuration of defects can be found by adding together the energy momentum tensors generated by the individual defects. Therefore, if Tµν p, d, v (x κ ) is the energy-momentum generated by a single defect with position p, direction d and velocity v with unit energy density, then the total energymomentum of a configuration of defects given by a density function ρ( p, d, v) on M is

T [ρ] µν (x κ ) = M d p d d d v ρ( p, d, v) Tµν p, d, v (x κ ). ( 8
)
Consequently, if we have an explicit expression for Tµν p, d, v (x κ ), we can compute the energy-momentum tensor for any configuration of defects. Alternatively, since the laminar plane waves form a complete basis for all configurations, the total energy-momentum of a configuration can also be obtained from the energy-momentum, T lam µν k µ , d, v (x κ ), of a laminar plane wave with wave vector k µ , direction d and perpendicular velocity v through,

T [ρ] µν (x κ ) = M dk d d dv ρ(k µ , d, v) T lam µν k µ , d, v (x κ ). (9)
We will now derive the energy-momentum tensor T lam µν k µ , d, v (x κ ) generated by a single laminar plane wave. In [START_REF] Hooft | A locally finite model for gravity[END_REF] 't Hooft derived the energy-momentum tensor of a single stationary defect through the origin and directed along the z-axis, a result that was already well-known from the context of cosmic strings (see for example [START_REF] Brandenberger | Lensing and CMB Anisotropies by Cosmic Strings at a Junction[END_REF]). With unit energy density the result is,

Tµν 0, ẑ, 0 (x κ ) =     1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1     δ(x)δ(y). (10)
The dependence on the position p = (p x , p y , 0) 5 can be obtained by performing appropriate shifts, which yields

Tµν (p x , p y , 0), ẑ, 0 (x κ ) =     1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1     δ(x -p x )δ(y -p y ). ( 11
)
5 Remember that p should be perpendicular to d.

By combining this result with equations ( 3) and ( 8) we obtain the energy-momentum generated by a stationary laminar plane wave with wave vector k µ = (0, k), direction ẑ and zero velocity,

T lam µν k λ , ẑ, 0 (x κ ) = M d p d d d v ρ lam [k λ , ẑ, 0]( p, d, v) Tµν p, d, v (x κ ) = p⊥ẑ d p Tµν p, ẑ, 0 (x κ )e 2πi k• p =     1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -1     e 2πi k• x . ( 12
)
Any other laminar plane wave can be obtained by applying appropriate Lorentz transformation Λ µ ν . To write the result we first notice that the right hand side of equation to ( 12) can be written as

u µ u ν + u 2 d µ d ν e 2πik•x , (13)
if we introduce the 4-vectors u µ = (1, 0, 0, 0) and d µ = (0, 0, 0, 1). After applying Λ µ ν and a linear redefinition 6

d ν = αΛ µ ν d µ + βΛ µ ν u ν , u µ = γΛ µ ν d µ + δΛ µ ν u µ (14)
such that

d µ d µ = 1, d 0 = 0, u µ d µ = 0, u 0 = 1, (15)
the direction d and velocity v of the new laminar plane wave can be found as

u µ = (1, v), and
d µ = (0, d). ( 16
)
Applying the same Lorentz transformation and linear redefinition to equation ( 13) yields the energy-momentum of the new laminar plane wave

T lam µν [k λ , d, v](x κ ) = - 1 u 2 u µ u ν + u 2 d µ d ν e 2πik•x . ( 17
)
This gives us the energy-momentum for all laminar plane waves with k µ k µ > 0. Notice that a laminar plane wave of defects with wavevector k µ only contributes to the Fourier mode of the energy-momentum tensor with wavevector k µ . As a result the Fourier transform of the 6 The 4-vectors u µ and d µ span the plane of a single defect in the laminar plane wave. Applying a Lorentz transformation yields vectors spanning the plane of a defect in the transformed plane wave. However, these vectors will not correspond to the (non-covariant) parameters we introduced to describe the velocity and direction of the defect. To find a pair of 4-vectors u µ and d µ that span the same plane but correspond to the velocity v and direction d of the defect we need to do a linear transformation.

total energy-momentum of a configuration specified by the density function ρ on M has the especially simple form,

T µν [ρ](k λ ) = d⊥ k d d √ k λ k λ 0 dv ρ(k λ , d, v) T pl µν [k λ , d, v]. (18)
The energy-momentum tensor for laminar plane wave with k µ k µ = 0, can be found as a limit of equation [START_REF] Brandenberger | Lensing and CMB Anisotropies by Cosmic Strings at a Junction[END_REF]. There are two possibilities, 1. k µ → 0 corresponding to a constant "wave" of defects with the same direction and velocity.

2. u µ u µ → 0 corresponding to a laminar plane wave of lightlike defects.

The first case is easy enough to compute since equation ( 17) is regular in this limit. However, in the second limit the 1/u 2 factor blows up. This can fixed by noting that the normalization of T lam µν is arbitrary, and we are therefore free to rescale it by a factor -u 2 changing equation (17

) to T lam µν [k λ , d, v](x κ ) = u µ u ν + u 2 d µ d ν e 2πik•x , (19)
which is regular in the limit that u 2 goes to zero. This fixes T lam 00 to be 1, which has the additional advantage of giving ρ the physical interpretation of the energy density present in a particular mode of laminar plane wave.

We now have all the ingredients we need to calculate the energy-momentum tensor of a general configuration of physical defects and can explore what conditions this will satisfy. In particular, we are interested in what conditions are imposed on the energy-momentum tensor by the restriction that all defects must be physical.

One immediate condition that we observe from equation ( 17) is that k µ T lam µν = 0 because k µ d µ = k µ u µ = 0. Consequently, equation [START_REF] Carroll | Can the dark energy equation-of-state parameter w be less than -1?[END_REF] implies that the total energy-momentum must satisfy

k µ T µν [ρ] = 0, (20)
for any distribution ρ on M. That is, energy is conserved, as one expects from any reasonable physical theory.

Other conditions can be obtained by examining T µν [ρ](k λ ) mode for mode. We have already observed that laminar plane waves with wave vector k µ only contribute to modes of the energy-momentum tensor with the same wave vector. Therefore, since laminar plane waves of physical defects must have k µ k µ ≥ 0, it is impossible for a configuration of defects to generate an energy-momentum tensor with Fourier modes with k µ k µ < 0.

For modes with k µ k µ > 0 we can assume by Lorentz invariance that, k µ = (0, k, 0, 0). The direction d, which must be perpendicular to k, can then be parameterized by a single angle φ with φ = 0 corresponding to the direction of the ẑ-axis. The Fourier transform of equation ( 19) then becomes,

T lam µν [k κ , d, v] =     1 0 v k cos φ v k sin φ 0 0 0 0 v k cos φ 0 v 2 k 2 -sin 2 φ cos φ sin φ v k sin φ 0 cos φ sin φ v 2 k 2 -cos 2 φ     . ( 21
) If we expand ρ(k µ , d, v) as ρ(k µ , d, v) = ∞ n=0 (2n + 1) π P n ( 2v √ k µ kµ -1) r 0n + 2 ∞ m=1
(r mn cos mφ + rmn sin mφ) ,

where the P n are Legendre polynomials and the coefficients r mn are understood to be functions of k µ , then the total contribution of the k µ -mode of ρ to the energy-momentum (18) becomes,

T µν [ρ](k µ ) =    
2r 00 0 r 11 + r 10 r11 + r10 0 0 0 0 r 11 +r 10 0 -1 3 r 00 +r 01 + 1 3 r 02 +r 20 r20 r11 +r 10 0 r20

-1 3 r 00 +r 01 + 1 3 r 02 -r 20     . (23)
If we compare this to the most general form of a mode energy-momentum tensor with wave vector k µ = (0, k, 0, 0) that satisfies k µ T µν = 0,

T µν (k κ ) =     T 00 0 T 01 T 02 0 0 0 0 T 01 0 T 22 T 23 T 02 0 T 23 T 33     , (24)
then we find that any such energy-momentum tensor can be generated by appropriate choices of the coefficients r mn . We therefore find the physicality condition on the defects puts no further restrictions on the modes of the energy-momentum tensor with k µ k µ > 0.

In the special case that k µ is lightlike, we can assume, without loss of generality, that k µ = (ω, ω, 0, 0). In this limit (19) becomes,

T lam µν [k µ , d, v] =     1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0     , (25)
and the total contribution of a distribution of defects ρ to the Fourier mode of the energymomentum with k µ = (ω, ω, 0, 0) becomes

T µν [ρ](k µ ) =    
r 00 r 00 0 0 r 00 r 00 0 0 0 0 0 0 0 0 0 0

    . ( 26
)
Apparently additional restrictions apply to the lightlike modes of the energy-momentum tensor generated by a configuration of physical defects. Not only do these modes have to be transverse, but they also cannot have any pressure or momentum perpendicular to their direction of propagation. This restriction can be formalized in the following way: For any lightlike mode of the energy-momentum tensor T µν (k κ ) and any lightlike vector l µ , the contraction l µ T µν (k κ ) is a non-spacelike vector.

The other special case, k µ = 0 is somewhat different since any direction d will be perpendicular to k = (0, 0, 0). The total contribution to the zero mode of the energy-momentum tensor will thus be found by integrating equation ( 19) over all mutually perpendicular v and d. That is,

T µν [ρ](0) = - v⊥ d d vd d ρ(0, d, v)(u µ u ν + u 2 d µ d ν ). (27)
Since tensors of the form u µ u ν + u 2 d µ d ν span the space of symmetric 2-tensors, we can conclude that any zero mode of the energy-momentum tensor may be produced by a configuration of defects. The condition that it contains only physical defects poses no further restrictions.

We therefore obtain the following restrictions that the energy-momentum tensor of a configuration of physical defects will satisfy in the continuum weak field limit

1. k µ T µν (k λ ) = 0 for all k µ . 2. T µν (k λ ) = 0 for all k µ with k µ k µ < 0.
3. l µ T µν (k λ ) is a non-spacelike vector for all lightlike k µ and l µ .

The first and third condition hold for most physically reasonable theories. The first expresses conservation of energy-momentum, while the last is normally imposed as part of the null dominant energy condition which is employed in cosmology to ensure vacuum stability while allowing negative vacuum energy. [START_REF] Carroll | Can the dark energy equation-of-state parameter w be less than -1?[END_REF] The second condition is satisfied by various simple matter models used in general relativity, such as dusts. However, it is typically violated in classical wave like systems. For example, consider a standing wave solution of the Klein-Gordon equation, φ = cos(ωt) cos( k • x). Even if k 2 > ω 2 , the energy-momentum tensor -which behaves like the square of φwill have terms which are proportional to cos(2ωt) and consequently will violate the second condition.

This indicates that the model cannot represent all types of matter at linear order. At this level all interactions are neglected, and we end up with a system that is very similar to a dust of non-interacting point particles. Beyond linear order defect lines will collide in a non-trivial manner, as was discussed in [START_REF] Van De Meent | Collisions in piecewise flat gravity in 3+1 dimensions[END_REF]. The energy-momentum tensor corresponding to the resolution of a collision will typically violate the second condition.

IV. METRIC PERTURBATIONS

We now turn to the effect that a configuration of defects ρ(k µ , d, v) has on the metric. In the limit of weak fields the metric g µν can be separated in a static Minkowski background η µν and a small perturbation h µν ,

g µν (x λ ) = η µν + h µν (x λ ). ( 28
)
If we consider only the linear perturbations caused by the presence of a defect, then the total perturbation caused by a continuous distribution of defects can be found as the integral of the perturbations of individual components. Consequently, if ĥlam µν [k λ , d, v](x κ ) is the perturbation of the metric caused by a laminar plane wave with wave vector k µ , direction d, and perpendicular velocity v, then the total perturbation caused by a configuration ρ is given by,

h µν [ρ](x κ ) = M dk d d dv ρ(k λ , d, v) ĥlam µν k λ , d, v (x κ ). (29)
Therefore, if we know ĥlam µν k λ , d, v for any combination of the parameters (k λ , d, v), than h µν [ρ] can be computed for any configuration ρ.

To calculate ĥlam µν k λ , d, v (x κ ) we fix its gauge freedom with the condition ∂ µ h µν = 0. 7 With this choice the linearized Einstein equation becomes (in its Fourier transformed form),

T µν (k κ ) = 2π 2 k 2 h µν -h(η µν - k µ k ν k 2 ) , (30)
where h is the trace of h µν . When k µ k µ = 0, equation (30) can be inverted to obtain the linear metric perturbation as a function of the energy-momentum tensor. In particular, if T lam µν k λ , d, v is the Fourier mode of the energy-momentum tensor generated by a laminar plane wave, then the metric perturbation generated by that laminar plane wave is given by a single Fourier mode,

ĥlam µν [k λ , d, v] = 1 2π 2 k 2 δ α µ δ β ν -1 2 (η µν -kµkν k 2 )η αβ T lam αβ [k λ , d, v]. (31)
We can therefore study the effects of a distribution of defects ρ(k µ , φ, v) on a per mode basis. By plugging in the T lam µν [k λ , d, v] from equation ( 19), we find that,

ĥlam µν [k λ , d, v] = 1 2π 2 k 2 u µ u ν -u 2 (η µν -d µ d ν - k µ k ν k 2 ) , (32)
where u µ = (1, v) and d µ = (0, d).

When k µ k µ > 0, we can assume by Lorentz invariance that k µ = (0, k, 0, 0). Parameterizing d as (0, -sin φ, cos φ) and v as (0, v cos φ, v sin φ), we obtain the explicit expression,

ĥlam µν [k λ , d, v] = 1 2π 2 k 2     v 2 k 2 0 v k cos φ v k sin φ 0 0 0 0 v k cos φ 0 cos 2 φ cos φ sin φ v k sin φ 0 cos φ sin φ sin 2 φ     . (33)
The total metric perturbation caused by a configuration of defects given by a distribution ρ on M can be obtained from ĥlam µν [k λ , d, v] through the integral,

h µν [ρ](k λ ) = d d dv ρ(k λ , d, v) ĥlam µν k λ , d, v . (34)
Performing this integral in the case that k µ = (0, k, 0, 0) and applying the expansion of ρ as given in (22) yields,

h µν [ρ](k κ ) = 1 2π 2 k 2     2 3 r 00 + r 01 + 1 3 r 02 0 r 10 + r 11 r10 + r11 0 0 0 0 r 10 + r 11 0 r 00 + r 20 r20 r10 + r11 0 r20 r 00 -r 20     . ( 35
)
7 There is some residual gauge freedom for the lightlike modes of the metric perturbation as we will discuss later on.

When k µ k µ = 0 the linearized Einstein equation (30) cannot be inverted and equation (32) cannot be applied directly. However, we may obtain the metric perturbation for these modes as a limiting case of the modes with k µ k µ > 0.

In the case that k µ becomes lightlike we can assume due to Lorentz invariance that it goes to k µ = (ω, ω, 0, 0). Such a laminar plane wave is the limit of waves with momentum k µ = (ω, κ, 0, 0) as κ → ω. Since physicality requires that 0 ≤ v ≤ √ κ 2 -ω 2 , v must simultaneously go to zero. Simply setting v = 0 and using k µ = (ω, κ, 0, 0), u µ = (1, ω/κ, 0, 0), and d µ = (0, 0, -sin φ, cos φ) in equation (32) yields,

ĥlam µν [k λ , φ, 0] = - 1 2π 2 κ 2     0 0 0 0 0 0 0 0 0 0 cos 2 φ cos φ sin φ 0 0 cos φ sin φ sin 2 φ     . (36)
Since only the pre-factor depends on κ the limit as κ goes to ω is straight forward.

Letting v go to zero by another route will lead to a result that (in the κ → ω limit) differs from the above by a term of the following form

    ξ 1 ξ 1 ξ 2 ξ 3 ξ 1 ξ 1 ξ 2 ξ 3 ξ 2 ξ 2 0 0 ξ 3 ξ 3 0 0     , (37)
where the ξ i are arbitrary (possibly infinite) parameters.

Such a contribution can be gauged away. Under an infinitesimal coordinate transformation given by a vector field ξ µ , the h µν transforms as,

h µν → h µν + ∂ µ ξ ν + ∂ ν ξ µ . (38)
The gauge condition ∂ µ h µν = 0 implies that

k µ k µ ξ ν (k λ ) + k ν k µ ξ µ (k λ ) = 0. ( 39
)
This completely fixes ξ µ (k λ ) for k µ k µ = 0. However, when k µ k µ = 0 it only implies that k µ ξ µ (k λ ) = 0. The residual gauge transformation subject to that condition for k µ = (ω, ω, 0, 0) takes the form of equation (37). Contributions of that form can therefore be gauged away, and we are free to adopt (36) as the gauge fixed form of ĥlam µν [(ω, ω, 0, 0), φ, 0]. Inserting (36) in the integral (34) and using the expansion (22) for ρ yields the lightlike modes of the metric perturbation caused by a configuration of defects given by a distribution ρ

h µν [ρ](ω, ω, 0, 0) = 1 2π 2 ω 2     0 0 0 0 0 0 0 0 0 0 r 00 + r 20 r20 0 0 r20 r 00 -r 20     . (40)
In the case that k µ → 0, the equation (32) diverges. This signals a breakdown of the linear perturbation approach in the limit of constant fields. This is not unexpected, since constant non-zero energy densities will typically lead to non-trivial effects on the global level with respect to the topology or causal structure.

V. GRAVITATIONAL WAVES

In the previous two sections we have obtained the energy-momentum and metric perturbation caused by a continuous distribution of defects at the linear level. At the microscopic level of individual defects the energy-momentum tensor completely fixed the metric (up to gauge transformations) due to the ad hoc rule we imposed that the vacuum should be completely flat. We are now able to answer the question whether this property persists in the continuum limit.

Since we are considering the linear limit, it is enough to consider what additional metric structure maybe present for a configuration with zero energy-momentum. In section III we obtained a complete expression for the energy-momentum of a configuration of defects described by a distribution ρ on M. Requiring that this expression (equations ( 23) and ( 26)) vanishes implies the following conditions on the coefficients r nm (k µ ) of ρ in the expansion (22);

r 00 (k µ ) = 0, r 01 (k µ ) + 1 3 r 02 (k µ ) = 0, r20 (k µ ) = 0, r 11 (k µ ) + r 10 (k µ ) = 0, r 20 (k µ ) = 0, r11 (k µ ) + r10 (k µ ) = 0,      for k µ k µ > 0, and (41)
r 00 (k µ ) = 0, for k µ k µ = 0. ( 42
)
The metric perturbation caused by a vacuum configuration of defects can now be found by applying these conditions to the complete expressions (35) and (40), found in section IV for the metric perturbation caused by a configuration of defects. This yields that for k µ k µ > 0, h µν (k λ ) vanishes when T µν vanishes, as one would expect since the linearized Einstein equation is invertible for k µ k µ = 0.

However, for k µ k µ = 0, the requirement that T µν (k λ ) vanishes only fixes r 00 (k µ ) to be zero. The coefficients r 20 (k µ ) and r20 (k µ ) are unconstrained. Consequently, for each lightlike k µ there exists a two parameter family of vacuum metric structures. When k µ = (ω, ω, 0, 0) these are described by

h µν [ρ](ω, ω, 0, 0) = 1 2π 2 ω 2     0 0 0 0 0 0 0 0 0 0 r 20 r20 0 0 r20 -r 20     , (43)
where we immediately recognize r 20 and r20 as the coefficients of the familiar + and × polarizations of gravitational waves.

This answers the question we posed in the introduction of whether any of the vacuum structure of general relativity would be recovered in the continuum limit of our model. The answer turns out to be affirmative. In fact, we find that all vacuum (i.e. Ricci flat) metrics that are available in linearized general relativity may be found as the continuum limit of a sequence of configurations of physical defects.

VI. CONCLUSIONS AND OUTLOOK

We have studied continuous distributions of physical defects. In the limit of weak fields, considering only linear contributions, the energy-momentum tensor of such a distribution turns out to satisfy certain conditions. These conditions can be expressed as follows for the Fourier transform of the energy-momentum tensor, 1. k µ T µν (k λ ) = 0 for all k µ . 2. T µν (k λ ) = 0 for all k µ with k µ k µ < 0.

3. l µ T µν (k λ ) is a non-spacelike vector for all lightlike k µ and l µ .

The first and third conditions are satisfied by most reasonable classical theories. The second condition, shows an inability to model wave-like phenomenon at the linear order. This property is shared by other non-interacting matter models, like dusts. Beyond linear order line defects collide in a non-trivial manner causing violations of this second condition. It would be very interesting to see if any restrictions remain once interactions are included.

The metric perturbation caused by a configuration of defects turns out to be mostly fixed by its energy-momentum. That is if two continuous distributions of defects have the same energy-momentum tensor, they also produce the same linear metric perturbation. At least for most modes.

The lightlike modes of the metric perturbation form an exception, as they are only partially fixed by the energy-momentum tensor. The metric perturbations of two configurations with the same energy-momentum may differ by a transverse traceless lightlike mode, i.e. by a gravitational plane wave. We thus see that the piecewise flat model of propagating defects -even though it does not contain any a priori vacuum structure -recovers the vacuum structure of general relativity in the continuum limit, at least at the linearized level.

The analysis in this article is possible because the description of the model of propagating line defects simplifies dramatically in the linear limit. Of course, it would be interesting to see what happens to these results if we go beyond the linear level. Besides seeing what restrictions may persist on the energy-momentum tensor obtained in the continuum limit, it would be particularly interesting to see whether the non-linear vacuum structures of general relativity -like black hole horizons -are recovered.

Going beyond the linear level is difficult however. Not only do the descriptions of the defect lines become intertwined with the presence of other defect lines, but we also need to account for collisions of the line defects. The resolution of such collisions was discussed in our previous paper [START_REF] Van De Meent | Collisions in piecewise flat gravity in 3+1 dimensions[END_REF], where we saw that these resolutions are not unique (and therefore require extra physical input) and sometimes require the appearance of superluminal nodes, which go against the spirit of the model which insists that all features propagate at subluminal speeds.

A general analysis of configurations, as done in this paper, therefore seems unlikely at the non-linear level. It might however be possible to produce specific examples of nonlinear vacuum structure by taking intuition from the linear limit. For example, the linear analysis showed that only lightlike defects moving in a single direction are needed to form a gravitational plane wave. This suggests that the constituent defects of a gravitational wave do not collide with each other. This simplifies the description of such a solution. One might therefore hope to form a family of exact piecewise flat solutions, which have zero energy-momentum when average over some region and which approximate a family of exact gravitational plane wave solutions like the one found by Bondi et al. [START_REF] Bondi | Gravitational Waves in General Relativity. III. Exact Plane Waves[END_REF] Some progress has been made in this direction.

Similarly, one might hope to use spherical symmetry to produce a piecewise flat vacuum approximation to a Schwarzschild black hole.

The appearance of negative energy degrees of freedom may appear unnatural. However, since in our treatment gravitational excitations are included in the energy-momentum side of Einstein's equation, they may appear with an opposite sign. Classical positivity conditions will, at best, only be valid on larger scales in this model.

The density function of course needs to be real. The expansion in complex exponentials is for the sake of convenience only. To obtain physical results we should consider only the real part.

Conversely, ρ( k, d, v) may be obtained from ρ( p, d, v) by an inverse partial Fourier transform.

Throughout this article we use natural units such that c = h = 8πG = 1 and metric signature (-+ ++).

Acknowledgments

The author would like to thank his advisor, Gerard 't Hooft, for many valuable discussions.


Julien Fatome

Bertrand Kibler

Esben R Andresen

Hervé Rigneault

Christophe Finot
email: christophe.finot@u-bourgogne.fr

All-fiber transform-limited spectral compression by selfphase modulation of amplitude-shaped pulses

We demonstrate efficient spectral compression of picosecond pulses in an all-fiber configuration at telecommunication wavelengths. Thanks to parabolic pulse shaping, a spectral compression by a factor 12 is achieved with an enhanced Strehl ratio.

Introduction

Due to the strong light confinement over a large distance, optical fibers are an ideal medium to observe and take advantage of the self phase modulation (SPM) induced by Kerr nonlinearity [START_REF] Finot | Nonlinear Fiber Optics[END_REF]. Highly coherent supercontinuum and ultrashort pulses have therefore benefited from the large spectral expansion experienced in highly nonlinear fibers. A less-known potential application of SPM relies on the configuration where the initial pulse has a negative linear chirp corresponding to an anomalous dispersive medium. In such a configuration, the SPM-induced chirp cancels the initial frequency modulation so that long and short wavelengths redistribute toward the center wavelength, leading to a spectral narrowing of the pulse [START_REF] Planas | Spectral narrowing in the propagation of chirped pulses in singlemode fibers[END_REF][START_REF] Oberthaler | Spectral narrowing of ultrashort laser pulses by self-phase modulation in optical fibers[END_REF]. The performances of the spectral compression process crucially depends on the initial pulse shape and routinely used hyperbolic secant or Gaussian temporal intensity profiles lead to the presence of large spectral pedestals. To overcome this issue, the optimum pulse shape has been theoretically found to be a parabola, which has recently been experimentally validated by using femtosecond pulses evolving in a passive photonic crystal fiber around 1 µm [START_REF] Andresen | Transform-limited spectral compression by self-phase modulation of amplitude shaped pulses with negative chirp[END_REF]. In this contribution, we provide a new illustration of the benefits brought by an initial shaping of the temporal intensity profile. Our experiments explore a completely different range of input parameters (initial wavelength, duration and peak power as well as typical fiber lengths) and a high-quality spectral compression by a factor above 10 is demonstrated in a passive fiber.

Experimental set-up

The experimental setup is sketched in Fig. 1a. A 2.4 ps pulse train is delivered by a mode-locked fiber laser (MLL) running at a repetition rate of 10 GHz at a central wavelength of 1552.5 nm. The pulses have a transform-limited hyperbolic secant profile. The repetition rate is then decreased down to 1.25 GHz by means of an intensity modulator (IM) and a phase modulator (PM) is used to avoid Brillouin backscattering. The shaping of the amplitude spectral profile is achieved by a liquid crystal on silicon device ((Finisar WaveShaper) and the anomalous linear chirp is imprinted by a 3900-m segment of standard single mode fiber (SMF). The initial pulses are thus stretched to a temporal duration of 70 ps and are characterized in intensity by a fast oscilloscope (see Fig. 1b) before being amplified up to 30 dBm of average power by a high-power erbium doped fiber amplifier (EDFA). The output power is varied by an optical variable attenuator (OVA) controlled by a computer (PC) and synchronized to the optical spectrum analyzer (OSA). Finally, the spectral narrowing takes place in a 1-km long highly nonlinear fiber (HNLF) with an attenuation of 1 dB/km, a nonlinear coefficient of 10/W/km and a normal dispersion.

Experimental results

Figures 2ab describe the evolution of the output spectral profiles as a function of the input pulse peak-power for the two pulse shapes under investigation. These figures outline the significant influence of the initial shape on the compression behavior. If the spectral width and the power required for optimum compression are comparable, the presence of sidelobes and oscillations is clearly accentuated in the case of a sech 2 profile. One can also make out that the spectral brightness is enhanced using a parabolic input pulse. Moreover, the spectral expansion following the point of optimum compression differs significantly.

Details of the output spectrum at the optimum compression point (Fig. 2c) show that compared to input pulses, the power density has then been increased by a factor 9 for parabolic pulses, which represents a twofold improvement of the spectral brightness compared to the sech 2 pulses. In both cases, a compression factor ratio of 12.2 is measured and we have checked that the temporal intensity profiles of the output pulses were not affected by the compression process. Therefore, the resulting time-bandwidth product (TBP) is found to be 0.745. We are thus only 12% higher than the bandwidth limited case (theoretical TBP of a parabolic pulse is 0.66). On the other side, for hyperbolic secant pulses, the resulting TBP is more than twofold above the transform limit condition (0.315).

Figure 2d describes more quantitatively the differences between the two initial pulse shapes: we have plotted as a function of the input peak-power the evolution of Strehl ratio defined as the ratio between the actual spectral peak power and the spectral peak power obtained assuming a flat temporal phase of the pulse. The Strehl ratio demonstrates once again the superiority of the parabolic-shaped pulses with a Strehl ratio three times higher than in the sech pulse case. Finally, we show that the root-mean-square (rms) spectral width measured for the parabolic pulse shape as a function of input peak power (Fig 2e ) follows a simple but non-trivial parabolic law, confirming an analogy that can be drawn between the spectral compression of a parabolic pulse and the impact of second order dispersion in the temporal domain.

Fig. 2 : (a-b) Experimental evolution of the spectral profile as a function of the input peak power for initial sech and parabolic profiles. Spectra are normalized so that the energy of the spectra equals 1. (c) Optical spectra obtained at the optimum compression point (solid line) for a sech pulse (grey line) and parabolic shaped pulse (black line) and compared to the input spectra (dashed line). (d) Evolution of the Strehl ratio according to the initial peak-power. (e) Evolution of the rms spectral width according to the input peak power for a parabolic pulse.

Conclusion

We have demonstrated the spectral compression of picosecond pulses in an all-fiber configuration at telecommunication wavelengths. Using widely available standard single mode fiber as a stretcher to impose the initial linear chirp, a spectral narrowing by a factor above 12 has been achieved for output pulses close to the Fourier transform limit. This compression has been based on a kilometer long highly nonlinear fiber with W-level peak powers and has involved pulse energies well below the nJ level, which is compatible with GHz operation. An initial parabolic pulse shaping has also been shown to significantly improve the results. The quality of the compressed spectra has indeed been enhanced with much lower substructures and an improved Strehl ratio.

Fig. 1 :

 1 Fig. 1 : (a) Experimental setup. (b) Intensity profiles after the SMF with and without shaping compared with sech 2 and parabolic fits.