Non-linear Plane Gravitational Waves as Space-time Defects
F. L. Carneiro, S. C. Ulhoa, J. W. Maluf, J. F. da Rocha-Neto
NNon-linear Plane Gravitational Waves as Space-time Defects
F. L. Carneiro, ∗ S. C. Ulhoa,
2, 3, † J. W. Maluf, ‡ and J. F. da Rocha-Neto § Instituto de F´ısica, Universidade de Bras´ılia70.919-970 Bras´ılia DF, Brazil International Center of Physics, Instituto de F´ısica,Universidade de Bras´ılia, 70910-900, Bras´ılia, DF, Brazil Canadian Quantum Research Center,204-3002 32 Ave Vernon, BC V1T 2L7 Canada
Abstract
We consider non-linear plane gravitational waves as propagating space-time defects, and construct theBurgers vector of the waves. In the context of classical continuum systems, the Burgers vector is a measureof the deformation of the medium, and at a microscopic (atomic) scale, it is a naturally quantized object.One purpose of the present article is ultimately to probe an alternative way on how to quantize planegravitational waves. ∗ [email protected] † [email protected] ‡ Electronic address: [email protected] § rocha@fis.unb.br a r X i v : . [ g r- q c ] J a n . INTRODUCTION Non-linear gravitational waves constitute a class of exact solutions of Einstein’s field equationsof general relativity. Several of these exact solutions form a subset of solutions known as planegravitational waves, or pp-waves (plane-fronted gravitational waves). These waves, in turn, arecharacterized by a local (in space and in time) deformation of an otherwise flat space-time. For thisreason, it is possible to think of non-linear plane gravitational waves as propagating space-timedefects, since the latter are also locally flat.Non-linear plane gravitational waves and space-time defects share many features. Both fieldconfigurations (i) are established over a flat space-time background, (ii) induce a local deformationin the background geometry, (iii) may have an axial symmetry (along the z axis, for instance), (iv)may have a singularity along an axis (the z axis, for instance). Therefore, it is possible to defineand evaluate the Burgers vector for non-linear plane gravitational waves. The Burgers vector in acrystalline lattice or inside a metal determines the nature of the defect.Metals may be deformed both elastically and plastically. Elastic deformations take placeat low external stresses, and are reversible, whereas plastic deformations are irreversible. Thelatter deformations are described by moving, dynamical dislocations, which imprint a permanentdeformation in the metal. The two ordinary types of dislocations in a metal are the screw andedge dislocations. A screw dislocation occurs when a half plane moves, or slips, relatively toone adjacent half plane (consider two infinite, adjacent and parallel planes, the upper and lowerplanes; one upper half plane remains attached to the adjacent lower half plane, whereas the otherupper half plane slips with respect to the adjacent lower half plane), and an edge dislocationis characterized by a missing half plane in an otherwise perfect lattice. We refer the reader toChapter 5 of Ref. [1], where a clear explanation of these defects is presented. The Burgers vector isconstructed by first establishing a Burgers circuit, which is a circuit around the defect. The idea isto first consider a closed, regular circuit in a perfect crystalline lattice. If the lattice is deformed byone type of dislocation, the circuit established in the perfect lattice fails to close in the deformedmedium, if the circuit encompasses the defect. The vector that must be added in order to closethe circuit, in the presence of a defect, is precisely the Burgers vector. At the atomic scale in aphysical lattice, the Burgers vector is quantized, i.e., it is a multiple of a minimum atomic distance.In the context of the 4-dimensional space-time geometry, space-time (or topological) defects havealready been investigated in some depth, see for instance Refs. [2–6].Non-linear gravitational waves are relatively simple solutions of Einstein’s equations, as we2earn from the well known review by Ehlers and Kundt [7]. These waves have been recentlyreconsidered in connection with the memory e ff ect [8–11]. The memory e ff ect is the permanentdisplacement of massive particles and ordinary objects of a physical system caused by the passageof a non-linear gravitational wave (although the memory e ff ect is also considered in the contextof linearised gravitational waves). In particular, the dynamical state of the massive particles isdi ff erent before and after the passage of the wave [12–15], in view of the velocity memory e ff ect.The “permanent displacement” mentioned above may be understood as a plastic deformation ofthe physical medium, that is constituted by massive particles, and in this sense propagating defectsin metals (and crystalline lattices) and non-linear gravitational waves share relevant features. Onthe other hand, linearised gravitational waves may be understood as elastic deformations of thespace-time.In this article we will consider several relevant circuits in the space-time of pp-waves andconstruct the corresponding Burgers vector. The graphical distribution of Burgers vectors in thethree dimensional space allows an alternative understanding and characterization of these waves.We will also evaluate the gravitational pressure that the pp-waves exert on certain surfaces, thatallows to obtain the gravitational force applied on idealized particles. The evaluation of thegravitational pressure is carried out with the definitions established in the teleparallel equivalentof general relativity. Under certain approximations, the resulting expressions of the gravitationalpressure are quite simple and interesting.This article is divided as follows. In section II the Teleparallelism Equivalent to GeneralRelativity is described. In section III the generalized pp-waves are introduced. The strain tensorand the Burgers vector are evaluated also in section III. In section IV, the gravitational pressureand the force exerted by the wave are calculated. Finally in the last section the conclusions arepresented. We use geometrical unities system where G = c = . II. TELEPARALLELISM EQUIVALENT TO GENERAL RELATIVITY (TEGR)
In this section we briefly introduce the ideas of Teleparallelism Equivalent to General Relativity(TEGR) along the lines of reference [16]. In this approach, the gravitational field is representedin terms of the dynamic tetrad field e a µ , but at the same time it establishes the reference systemby choosing the six additional components when compared to the metric tensor. The geometricframework of TEGR is such that absolute parallelism is a fundamental attribute of space-time.3his condition is determined by the Weitzenb ¨ock connection Γ µλν = e a µ ∂ λ e a ν which has a vanishing curvature and a torsion tensor defined by T a λν = ∂ λ e a ν − ∂ ν e a λ . (1)The Weitzenb ¨ock connection is related to the Christo ff el’s symbols, Γ µλν , identically by Γ µλν = Γ µλν + K µλν , (2)where K µλν is the contortion tensor, and is given by K µλν =
12 ( T λµν + T νλµ + T µλν ) , (3)with T µλν = e a µ T a λν . The expression (2) induces a direct relationship between Ricci scalar and aquadratic combination of torsions. It reads eR ( e ) ≡ − e ( 14 T abc T abc + T abc T bac − T a T a ) + ∂ µ ( eT µ ) . (4)It should be noted that the left-hand side of the above expression is the Hilbert-Einstein Lagrangian.Thus the TEGR Lagrangian density is given by L ( e a µ ) = − κ e ( 14 T abc T abc + T abc T bac − T a T a ) − L M ≡ − κ e Σ abc T abc − L M , (5)where κ = / (16 π ), L M is the Lagrangian density of matter fields and Σ abc is defined as Σ abc =
14 ( T abc + T bac − T cab ) +
12 ( η ac T b − η ab T c ) , (6)with T a = e a µ T µ = e a µ T ν ν µ . Hence the field equations that are equivalent to Einstein’s equationsread ∂ ν (cid:16) e Σ a λν (cid:17) = κ e e a µ ( t λµ + T λµ ) , (7)where t λµ = κ (cid:104) Σ bc λ T bc µ − g λµ Σ abc T abc (cid:105) , (8)is the gravitational energy-momentum tensor. Such an expression goes one step further towardsthe solution of the longstanding problem of gravitational energy.4ue to the anti-symmetric feature of Σ a λν , it is possible to obtain ∂ λ ∂ ν (cid:16) e Σ a λν (cid:17) ≡ . (9)Then the energy-momentum vector is P a = (cid:90) V d x e e a µ ( t µ + T µ ) , (10)this can be equivalently expressed as P a = k (cid:90) V d x ∂ ν (cid:16) e Σ a ν (cid:17) . (11)It should be noted that the energy-momentum vector is invariant under coordinate transformationsof the 3-dimensional space, and under time reparametrizations. On the other hand, it transformsas a vector under SO(3,1) symmetry.From equations (7) and (9) one obtains ddt (cid:90) V d x e e a µ ( t µ + T µ ) = − (cid:73) S dS j (cid:104) e e a µ ( t j µ + T j µ ) (cid:105) , which indicates an energy-momentum flux, where the surface S delimits the volume V . Thegravitational energy-momentum flux is defined as Φ ag = (cid:73) S dS j ( e e a µ t j µ ) , (12)and the energy-momentum flux of matter fields as Φ am = (cid:73) S dS j ( e e a µ T j µ ) . (13)Then, it is possible to write dP a dt = − (cid:16) Φ ag + Φ am (cid:17) = − k (cid:73) S dS j ∂ ν ( e Σ aj ν ) . (14)The momentum flux is given by the spatial part of the above equation, thus dP ( i ) dt = − (cid:73) S dS j φ ( i ) j , (15)where φ ( i ) j = k ∂ ν ( e Σ ( i ) j ν ) . (16)Equation (15) has the nature of force, therefore equation (16) represents the pressure on a direction( i ) over a surface oriented towards j . 5 II. THE GENERALIZED PP-WAVES
The line element of the pp waves can be described in double null coordinates u , v by thegeneralized line element ds = H ( u , x , y ) du + dx + dy + dudv − a ( u , x , y ) dudx − a ( u , x , y ) dudy . (17)The surfaces u = constant are flat and the wave propagates along the null direction v . The functions a , ( u , x , y ) may be eliminated locally by an appropriate choice of coordinates, therefore they maybe chosen as zero. However, some topological properties of the space-time may be lost when sucha choice is made [17]. Topological defects manifests as a global e ff ect, thus it is worth consideringthe generalized form of the pp-waves metric and particularizing it latter as special cases.The line element (17) may be written in Cartesian coordinates by means of the relations u = z − t √ , v = z + t √ . (18)The line element becomes ds = (cid:18) H − (cid:19) dt + dx + dy + (cid:18) H + (cid:19) dz − Hdtdz + √ a dtdx − √ a dxdz + √ a dtdy − √ a dydz . (19)The regular pp-waves may be obtained by choosing a , ( u , x , y ) =
0, as mentioned above.A tetrad field adapted to a stationary observer and related to the line element above can be canbe written as e a µ = − A a / √ A a / √ A − H / A − a / √ A − a / √ A / A , (20)where a and µ denote lines and rows, respectively, and A = (cid:112) − H / x = ρ cos φ , y = ρ sin φ , a = − J ρ sin φ , (21) a = J ρ cos φ . (22)Then, the line element (19) becomes ds = (cid:18) H − (cid:19) dt + d ρ + ρ d φ + √ Jdtd φ − √ Jdzd φ + (cid:18) + H (cid:19) dz − Hdtdz . (23)The function J is related to the spinning nature of the gyratons. Similarly, the tetrad field adaptedto a stationary observer for the above metric may be rewritten as e (cid:48) a µ = − A J √ A − H / A φ ) − ρ sin( φ ) 00 sin( φ ) ρ cos( φ ) 00 0 − J √ A / A . (24)It should be noted that for J = H and J is given by ∇ H = ρ (cid:16) ∂ u ∂ φ J (cid:17) , (25)where ∇ ≡ ∂ ρ ∂ ρ + ρ ∂ ρ + ρ ∂ φ ∂ φ in cylindrical coordinates. For J = J ( u ) ⇒ ∂ φ J = ∇ H = , (26)then it is possible to solve the equation (25) to obtain the following classes of solutions H = − C ln (cid:32) x + y a (cid:33) f ( u ) , (27) H + = − C + (cid:16) x − y (cid:17) f ( u ) , (28) H × = − C × (cid:0) xy (cid:1) f ( u ) , (29) H + = − C + x − y (cid:0) x + y (cid:1) f ( u ) , (30) H × = − C × xy (cid:0) x + y (cid:1) f ( u ) . (31)Here, the multiplicative factors have a proper dimension to leave H dimensionless, for instance C + and C × have dimension of inverse squared distance, C + and C × have dimension of squared7istance, while C is dimensionless. The constant a in (27) delimits the validity region of thesolution in vacuum, i.e., the radius of the source. These functions are not the only possiblesolutions, but they have a well establish physical meaning. The function f ( u ) is arbitrary andestablishes the form of the pulse, usually chosen as a Gaussian. In the next subsections, e ff ectssuch as deformations and distortions associated with generalized pp-waves will be analyzed. A. The strain tensor
In solid mechanics, the deformation of materials is an important feature in understanding theirproperties. When an elastic deformation is present, the strain tensor quantifies the relative amountof change during deformation. In the case of plastic deformations such as dislocations, whereHooke’s law does not apply everywhere, a dislocation core is constructed and outside this corethe Hooke’s law is applied. In a plastic deformation, the components of the strain tensor can bewritten as a function of the dislocation intensity, i.e., the Burgers vector.We can import this concept into space-time. Thus we introduce the strain tensor, understoodas the di ff erence between the geometries before and after a given event, for instance, the passageof a gravitational wave. Therefore the strain tensor is defined as [21] ε µν ≡ (cid:16) g µν − ¯ g µν (cid:17) (32)where ¯ g µν is the flat space-time metric in an arbitrary coordinate system and g µν is the metrictensor of the gravitational wave. Usually, in the case of metals, this tensor is built in threedimensions. Here it is always possible to take the three-dimensional part of the deformationtensor for comparison.The strain tensor calculated from the metric (19) is given by ε µν = H − √ a − √ a H − √ a − √ a − √ a − √ a H − √ a − √ a H . (33)A deformation is a measurable e ff ect, that is, it is not dependent on the chosen coordinate system.8ence, it is necessary to project such a quantity on the Lorentz symmetry indices. We get then ε ab = e a µ e b ν ε µν = H / A − √ a / A − √ a / A H / A − √ a / A − √ a / A − √ a / A − √ a / AH / A − √ a / A − √ a / A H / A . (34)As a consequence the 3D strain tensor of a gyratonic wave is ε ( i )( j ) = √ J ρ A sin φ − √ J ρ A cos φ √ J ρ A sin φ − √ J ρ A cos φ H / A , (35)therefore it is possible to see that regular pp-waves are responsible for longitudinal deformationswhile gyratonics also cause transversal shearing. B. The Burgers vector
In a spacetime with torsion, the Burgers vector is defined as b a = (cid:90) S T a µν dx µ ∧ dx ν , (36)where ∧ is the exterior product. This means that torsion is the dislocation superficial density. Dueto the torsion symmetry and the properties of tetrad (20), the spatial components of the Burgersvector can be written as b ( i ) = (cid:73) C e ( i ) j dx j , where C is a path delimited by the surface S . It is worth noting that the result of this integraldepends on the path taken.In order to construct the Burgers circuit, a plane can be chosen. First, let us consider a squarewith side 2 L in the YZ plane, centered at an arbitrary point ( t , x , y , z ). Thus b ( i ) = (cid:90) y + Ly − L e ( i ) 2 ( t , x , y , z + L ) dx + (cid:90) z − Lz + L e ( i ) 3 ( t , x , y − L , z ) dx + (cid:90) y − Ly + L e ( i ) 2 ( t , x , y , z − L ) dx + (cid:90) z + Lz − L e ( i ) 3 ( t , x , y + L , z ) dx . (37)9hen, the only non-vanishing component of the Burgers vector is b (3) = − √ (cid:90) y + Ly − L a A (cid:12)(cid:12)(cid:12)(cid:12) z = z + Lz = z − L dy + (cid:90) z + Lz − L A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y = y + Ly = y − L dz . The result for a regular pp-wave can be obtained by choosing a , =
0, yielding b (3) = (cid:90) z + Lz − L A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y = y + Ly = y − L dz . (38)We must note that the Burgers vector is calculated locally, therefore depends on the circuit shapeand the location of its center. This means that we have a distribution of vectors in space, i.e.,a vector field. Each set of coordinates ( t , x , y , z ) will have its own Burgers vector within itsneighborhood. In the figures below we show the form of this distribution for some choices of H ,all for a regular pp-wave. All figures were obtained for 2 L = .
002 and f ( u ) = e − u . In all of thembelow, the side bar indicates a color scale for the vector modulus. - - y - - - z × - × - × - × - × - × - × - × - × - × - FIG. 1: Burgers vector for H with t = x = C =
1. The points were distributed with y ranging from −
10 to − + +
10, in steps of 0 .
05; and z ranging from − +
3, in steps of 0 .
1. The regionaround the propagation axis was excluded from integration.
If we choose a similar circuit in the plane XZ, we obtain b (3) = − √ (cid:90) L − L a A (cid:12)(cid:12)(cid:12)(cid:12) z = Lz = − L dx + (cid:90) L − L A (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x = Lx = − L dz . - y - - - z × - × - × - × - × - × - × - × - × - × - FIG. 2: Burgers vector for H + with t = x = C + =
1. The points were distributed with y rangingfrom − − . + . +
5, in steps of 0 .
05; and z ranging from − +
3, in steps of 0 .
1. Theregion around the propagation axis was excluded from integration. The bars on the right side indicate themodulus of the Burgers vector on both sides.
In Figures 1, 2, 3 and 4, a non-vanishing distribution of Burgers vectors for the regular pp-wave isdisplayed. They are consistent with the polarization of the chosen H solution. It is worth notingthat the Burgers vector has a direct relationship with the strain tensor, despite the latter comingfrom the metrical tensor while the first comes from the space-time torsion. With J = H is the onlydeterminant for the evolution of the system.The most interesting solution is obtained by choosing a circuit perpendicular to the axis ofpropagation Z. In order to present this solution, we consider a circular path, centered at thepropagation axis, and focus only on the gyratonic wave. The Burgers vector is b ( i ) = (cid:90) π e (cid:48) ( i ) 2 d φ = (cid:90) π e (cid:48) (3) 2 d φ , (39)11 - y - - - z × - × - × - × - × - × - × - × - FIG. 3: Burgers vector for H × with t = x = . C + =
1. The points were distributed with y rangingfrom − − . + . +
5, in steps of 0 .
05; and z ranging from − +
3, in steps of 0 .
1. Theregion around the propagation axis was excluded from integration. The bars on the right side indicate themodulus of the Burgers vector on both sides. where e (cid:48) is given by (24). Similarly to the above cases, the only non-vanishing component is b (cid:48) (3) = b = − √ (cid:90) π JA d φ . (40)We can see that the Burgers vector is zero if evaluated for a regular pp-wave in this circuit, i.e., J =
0. An interesting result occurs when we have an axially symmetric wave, i.e., J = J ( u ) and H = H ( u , ρ ). In this particular case it is possible to evaluate the integral analytically, obtaining b = − π J √ A . (41)We can define a dislocation core and write the strain tensor as a function of the Burgers vector.The strain tensor can be transformed into polar coordinates as ε ( φ )( z ) = − sin φ ε (1)(3) + cos φ ε (2)(3) . (42)Thus, using (35) and (41), we obtain ε ( φ )( z ) = b πρ , (43)12 - - y - - - z × - × - × - × - × - × - × - × - FIG. 4: Burgers vector for H + with t = x = C + = a =
1. The points were distributed with y rangingfrom − . .
5, in steps of 0 .
05; and z ranging from − +
3, in steps of 0 .
1. The bars on the right sideindicate the modulus of the Burgers vector on both sides.
The result above is exactly the same result observed in a crystal with a screw dislocation, as canbe seen in equation (5.3) of [1]. The obtained stress field, outside the dislocation core, falls o ff as ρ − , therefore consists in a long range field. This fact can cause a particle to feel the e ff ects ofdislocation even outside the core. 13 V. GRAVITATIONAL PRESSURE
In this section we aim to calculate the gravitational force imparted by the gyratonic gravitationalwaves. We calculate the torsion components. The non-vanishing components are [22] T (0)(0)(1) = − T (3)(1)(3) = − ρ A (cid:16) √ φ∂ t J − sin φ∂ φ H + ρ cos φ∂ ρ H (cid:17) , T (0)(0)(2) = − T (3)(2)(3) = − ρ A (cid:16) − √ φ∂ t J + cos φ∂ φ H + ρ sin φ∂ ρ H (cid:17) , T (0)(0)(3) = T (3)(0)(3) = − A ∂ t H , T (0)(1)(3) = ρ A (cid:16) √ φ∂ t J − sin φ∂ φ H + ρ cos φ∂ ρ H (cid:17) , T (0)(2)(3) = ρ A (cid:16) − √ φ∂ t J + cos φ∂ φ H + ρ sin φ∂ ρ H (cid:17) , T (3)(0)(1) = − √ ρ A sin φ∂ t J , T (3)(0)(2) = √ ρ A cos φ∂ t J . Then, using the above quantities, we have Σ (0)01 = Σ (3)01 = − √ ∂ ρ H √ − H Σ (0)02 = Σ (3)02 = − √ ρ ∂ φ H √ − H + ∂ t J ρ √ − H Σ (1)01 = ∂ t H − H cos φ Σ (1)02 = − ρ ∂ t H − H sin φ Σ (1)03 = − ρ ∂ φ H sin φ − ρ∂ ρ H cos φ − H Σ (2)01 = ∂ t H − H sin φ Σ (2)02 = ρ ∂ t H − H cos φ Σ (2)03 = ρ ∂ φ H cos φ + ρ∂ ρ H sin φ − H . The non-null components of the energy-momentum tensor are t = t = t = − k ρ A (cid:104) ρ ( ∂ ρ H ) + ( ∂ φ H ) + ∂ u J ∂ φ H (cid:105) . (44)Finally, the gravitational pressure defined by (16) is given by φ (3)3 = − k ρ A (cid:104) ρ ( ∂ ρ H ) + ( ∂ φ H ) + ∂ u J ∂ φ H (cid:105) . (45)14ne can calculate the total force by integrating over a surface S , F ( i ) = − (cid:90) φ ( i ) j dS j . Hence, if a surface whose normal vector is oriented towards the z axis is chosen, then F z ≡ F (3) = (cid:90) φ (3)3 dS = k (cid:90) d ρ d φ ρ ( ∂ ρ H ) + ( ∂ φ H ) + ∂ u J ∂ φ H ρ A . (46)We see that the direction of the force is longitudinal, and therefore the gravitational wave impartsa force on hypothetical particles along the direction of the propagation of the wave.In the case of an axially symmetric wave, i.e., H = H ( u , ρ ) and J = J ( u ), the integral (46) can beevaluated analytically, where the surface of integration S is chosen as a disc centered in the z axis,with ρ and ρ as the inner and outer radii, respectively. In this case, where the solution of theEinstein equation (26) is given by (27), we obtain F z = − C (cid:32) A ( u , ρ ) − A ( u , ρ ) (cid:33) , (47)where we considered a = ff erent solutions. In figure 5 wesee the respective graphs as a function of the z coordinate, specifically at t =
0. We can see that thisis a negative force for all cases considered. There is a deformation parallel to the Burgers vectoritself, as indicated by the strain tensor component ε (3)(3) . - - - z - - - - - F z H H + H x FIG. 5: Forces for solutions (27,30,31) with C = / C + = C X = a = t = f ( u ) = e − u = J . . CONCLUSION In this article we analyzed how generalized pp-waves can be interpreted as topological defects.For this purpose, we calculated the strain tensor associated with such waves, as well as thedislocation determined by the Burgers vector. In particular, we chose a square path in the YZ andXZ planes and obtained a distribution of Burgers vectors for several H solutions. With that wecould see that there is a well defined Burgers vector for regular pp-waves. In the same sense, wechose a circular path perpendicular to the z axis that allowed us to obtain the strain tensor as afunction of the modulus of the Burgers vector. Thus we compare the result with a dislocation in acrystal. Surprisingly, we saw that the gyratonic wave shares similarities with a crystal endowedwith a topological defect with cylindrical symmetry. We conclude that in order to describe a givenmetric as a topological defect, it is necessary to take into account both the strain tensor and theBurgers vector.The results obtained with the gyratonic waves are very similar to those of crystals, mainlybetween the gyratonic wave and a crystal with a screw dislocation. The qualitative di ff erencesarise due to the presence of a normal component σ ( z )( z ) in the strain tensor of the pp-waves, while inthe case of a crystal screw dislocation, only the shear component σ ( φ )( z ) is present. For gravitationalwaves, the existence of the normal σ ( z )( z ) component is inherent to its type, while the existenceof the shear component σ ( φ )( z ) depends on the existence of the gyratonic term J . Therefore, inthe space-time of a gravitational wave we may have compression and shear when a longitudinalforce is applied. Most interesting, when we dismiss the gyratonic term J in the line element, thepp-waves space-time looses its capacity to shear.The characterization of waves as topological defects can be applied in an attempt to quantizepp-waves. The quantization of a space-time may be performed by the geometric assumption ofthe Burgers vector being an integer of the Planck’s length [23, 24] and its quantization parameterscan be measured by analyzing the interaction of the gravitational field with particles [25]. Thus,interpreting pp-waves as space-time defects may provide a way to quantize these waves. Forinstance, in the case of an axially symmetric gravitational wave, we have b = − √ π J / A , thusimposing b = nb [25], where b is the fundamental scale of the defect and n an integer, we have − √ π J / A = nb . This feature will be further investigated elsewhere. [1] R. E. Smallman, “Modern Physical Metallurgy” Third Edition, (Butterworths, London, 1976).
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