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Energy-momentum balance in particle - domain wall perforating collision

We investigate the energy-momentum balance in the perforating collision of a point particle with an infinitely thin planar domain wall within the linearized gravity in arbitrary dimensions. Since the metric of the wall increases with distance, the wall and the particle are never free, and their energy-momentum balance involves not only the instantaneous kinetic momenta, but also the non-local contribution of gravitational stresses. However, careful analysis shows that the stresses can be unambiguously divided between the colliding objects leading to definition of the gravitationally dressed momenta. These take into account for gravity in the same way as the potential energy does in the non-relativistic theory, but our treatment is fully relativistic. Another unusual feature of our problem is the non-vanishing flux of the total energy-momentum tensor through the lateral surface of the world tube. In this case the zero divergence of the energy-momentum tensor does not imply conservation of the total momentum defined as the integral over the space-like section of the tube. But one can still define the conservation low infinitesimally, passing to time derivatives of the momenta. Using this definition we establish the momentum balance in terms of the dressed particle and wall momenta.


INTRODUCTION
In the standard theory of particle collisions, both classical and quantum, one assumes the existence of asymptotic states in which the particles can be regarded as non-interacting. This gives rise to the energy-momentum conservation playing crucial role in understanding of such processes. For this picture to be valid, the interaction force between the colliding objects must fall down sufficiently fast with the distance. Meanwhile, in various physically interesting situations this is not so, the notable example being interaction between quarks.
To explore possibility of the energy-momentum definition in absence of asymptotical free states we consider here collision of the gravitationally interacting infinitely thin domain wall and point particle. Such a problem is of interest for applications in the standard [1][2][3][4][5][6] and the Rundall-Sundrum type [7][8][9][10] cosmology, string theory [11], in studying brane -black hole composites [12,13], black hole escape from branes [14][15][16] and in other situations. Gravitational force exerted upon the particle by the plane domain wall does not fall with distance [17,18], so the particle can not be considered free at any moment. If the domain wall is viewed as a fixed source of gravity, the particle moves along the geodesic in the space-time generated by the domain wall, and the notion of the gravitational potential energy can be introduced. But if one wants to treat both objects on equal footing, the interaction potential can not be introduced. The domain wall -particle scattering problem, however, is well posed within the linearized gravity, where it can be formulated in close parallel to the case of two gravitating particles. Moreover, while the head-on collision of particles is a singular problem even in the linearized gravity, our process is still tractable, since the gravitational force acting upon the particle remains finite when it comes unto contact with the wall. Recently we have shown [19,20] that the perforation of the domain wall by the particle can be well described in linearized gravity in terms of distributions. A novel feature of this situation is due to existence of the internal dynamics of the domain wall which gets excited after the perforation in the form of the spherical branon wave.
Here we would like to show that the problem of the energy-momentum conservation in the domain wallparticle interaction is also tractable going beyond the linear theory up to the second order in the gravitational constant. This is needed in order to introduce the effective gravitational stress-tensor which has to be taken into account in establishing the energy-momentum balance. Such stress-tensor obtained by expanding the Einstein tensor up to the second order in metric deviations is non-local. But as we will show, careful analysis allows to unambiguously split it between the domain wall and the particle leading to definition of gravitationally dressed colliding objects. This dressing resembles introduction of the potential energy in the non-relativistic theory, but an essential difference is that now the treatment is fully relativistic and both objects are considered on equal footing. Gravitational dressing does not mean taking into account proper gravitational field of each object, but rather accounting for gravitational field of the partner. So our dressing must not be confused with the self energy problem.

THE SETUP
Our system consists of a point particle moving along the world-line x M = z M (τ ) and an infinitely thin domain wall filling the world-volume V D−1 given by the embedding equations x M = X M (σ µ ) in D− dimensional spacetime with the metric g MN , M = 0, ..., D − 1, µ = 0, ...D − 2 of the signature (+, −, ..., −). The action can be written as where S p (z M , e) is the particle action in the Polyakov form (e(τ ) is the ein-bein on the particle world-line), S dw (X M , γ µν ) is the domain wall geometrical action where X M µ = ∂X M /∂ σ µ are the tangent vectors and γ µν is the inverse metric on the domain wall world-volume V D−1 , γ = detγ µν , and S grav (g M ) is the Einstein-Hilbert action Variation of (2.3) with respect to X M and γ µν gives the brane equation of motion in the covariant form 5) and the constraint equation whose solution defines γ µν as the induced metric on V D−1 : Similarly, variation of (2.2) with respect to z M (τ ) and e(τ ) and gives the geodesic equation in arbitrary parametrization with the source terms due to the domain wall 10) and the particle (the corresponding quantities here and below will be labeled by bar): Einstein equations with the source term (2.10) have some exact non-singular solutions [17,18,[21][22][23], while no such solution are possible for the point particle source. Actually, reasonable exact solutions exist for branes time-dependent solutions describing the collision, which can only be constructed perturbatively. Actually we work in the linearized gravity assuming smallness of deviation of the space-time metric from Minkowskian: but we keep the full Einstein action to be able to extract the gravitational stress tensor as the second order expansion term of the Einstein tensor: where H = H M M , ✷ = η MN ∂ M ∂ N and S MN stands for the quadratic terms in H MN : In this definition there is the following subtlety. The metric deviation H MN is defined initially as generally covariant quantity with lower indices and then identified with the Minkowskian tensor whose indices are raised with the inverse Minkowski metric. The quadratic tensor S MN , which is also further regarded as the Minkowskian tensor, is obtained expanding the Einstein tensor with upper indices, all internal contractions of metric deviations being performed with Minkowski metric. The full set of variables in our problem consists of z M (τ ) , e(τ ) , X µ (σ) , γ µν and H MN (x). To treat the problem perturbatively we expand all of them in powers of κ and derive the system of iterative equations. The D−dimensional cartesian coordinates of the embedding space-time are split as x M = (x µ , z), x µ = (t, r), and the particle is assumed to move along z, i.e. normally to the domain wall. In the zeroth order the particle is assumed to move with the constant velocity so the world-line and the ein-bein are corresponding to the parametrization in terms of the proper time. The wall in the zeroth order is assumed to be plane, unexcited and being at rest at z = 0 in the chosen Lorentz frame: where Σ M µ are (D − 1) constant Minkowski vectors normalized as Obviously, this is a solution to the Eq. (2.5) for κ = 0, and the corresponding induced metric is the fourdimensional Minkowski metric γ µν = η µν . Thus it is convenient to fix Σ M µ = δ M µ without loss of generality. The moment of perforation of the wall by the particle that occurs at z = 0 is t = τ = 0.
The metric deviation must be further expanded in κ : where the first order term is split into the sum of contributions of the wall h MN and of the particleh MN . These obey the linear equations where the sources must be constructed in terms of the above zeroth order quantities and the Fock-de Donder gauge ∂ N h MN − 1/2 ∂ M h = 0 is assumed. The next order metric deviation δH MN does not split anymore on separate contributions and obeys (in the same gauge) the d'Alembert equation with the source Generically the gravitational stress-tensor is non-local quantity, but, as we will show below, within the perturbation theory it can still be split into two contributions which may be attributed to the wall and the particle separately. This is how the idea of gravitational dressing is implemented. The domain of validity of our perturbation theory is somewhat subtle and worth to be discussed in detail. Gravity force exerted by the wall upon the particle is repulsive and we consider the case when the initial velocity of the particle is large enough to reach the wall and to perforate it. After the perforation the particle gets accelerated by the wall's gravitational repulsion and goes away. Since the metric deviations caused by the wall in the linearized gravity are growing with z, one can treat the collision perturbatively only in some vicinity of the perforation moment. From the particle energy E = mγ, the wall's tension µ (of dimensionality length −(D−1) ) and the gravitational coupling constant κ 2 , having in D dimensions the dimensionality length D−2 , one can form two length parameters (in the units c = 1): the first of which corresponds to the curvature radius of the bulk generated by the wall, while the second is the gravitational radius of the energy E. The wall's gravity is small at the distances from the wall z < l , while gravity of the particle is small for z 2 + r 2 > r 2 S . If r S ≪ l, these conditions intersect within some matching zone. But it turns out [19] that, assuming the linearized gravity to be true gravity theory for point particles elsewhere one is still able to treat the collision up to z = 0 (the perforation point) consistently in terms of distributions (i.e. in the formal limit r S = 0. Here we will show that this treatment is consistent with the energy-momentum conservation in the perforation process in the linear order in κ , thus giving further evidence for validity of our approach. Note that this is quite different from the more singular case of the head-on collision of two gravitating point particles which can not treated within the linearized gravity.

FIRST ORDER PERTURBATIONS
For reader's convenience we briefly reproduce here the results obtained in [19]. The full metric deviation in the first order is the sum ofh MN generated by the unperturbed particle motion and h MN representing gravity of the unperturbed wall at rest. The first reads explicitly for D > 3: where r = δ ij σ i σ j is the radial distance on the wall from the perforation point. This is just the Lorentzcontracted D-dimensional Newton field of the uniformly moving particle. In what follows we will also need the corresponding Fourier-transform The metric deviations due to the wall grow linearly with the distance the corresponding Fourier-transform reads The first order correction to the particle motion δz M in the field of the wall (3.3) depends on the choice of the parameter on the world-line. Specifying it so that the deviation of the ein-bein is zero, we obtain from the geodesic equation (2.7) observing that the force is repulsive as expected. Integrating (3.7) twice with initial conditions δz M (0) = 0, δż M (0) = 0, one has Substituting (3.8) back into (3.6) one can check that the gauge condition δe = 0 holds indeed. In order to find perturbations of the domain wall embedding functions δX M due to gravitational interaction with the particle one has to derive the linearized perturbation of the Nambu-Goto equation specifying the world-volume metric as an induced metric where brackets denote symmetrization over indices with the factor 1/2. Then linearizing the rest of the Eq.
(2.5), after some rearrangements one obtains the following equation for deformation of the wall: where ✷ D−1 ≡ ∂ µ ∂ µ and Π MN is the projector onto the (one-dimensional) subspace orthogonal to V D−1 . The source term in (3.10) reads: Using the aligned coordinates on the brane σ µ = (t, r), we will have δ M µ = Σ M µ , so the projector Π MN reduces the system (3.10) to a single equation for M = z component. Generically, the transverse coordinates of the branes can be viewed as Nambu-Goldstone bosons (branons) which appear as a result of spontaneous breaking of the translational symmetry [24]. These are coupled to gravity and matter on the brane in the brane-world models via the induced metric [25]. In our case of co-dimension one there is only one such branon. The remaining components of the perturbation δX M can be removed by suitable transformation of the coordinates on the world-volume, so the equality δX µ = 0 is nothing but the choice of gauge. Note that in this gauge the perturbation of the induced metric δγ µν does not vanish, as it was for the perturbation of the particle ein-bein e.
Denoting the physical component as δX z ≡ Φ(σ µ ) we obtain the branon (D − 1)-dimensional wave equation: with the source term J ≡ J z . Substituting (3.1) into the eq. (3.11) we obtain: (3.14) The retarded solution to the Eq. (3.12) consists of two parts Φ = Φ a + Φ b , where the first is antisymmetric in time and represents an eventual deformation of the wall correlated with the particle motion. The second part is the spherical branon wave starting at the moment of perforation and propagating to infinity with the velocity of light. This wave is not the solution of the homogeneous branon equation, but is has a jump at t = 0 ensuring continuity of the full solutions. The explicit expressions of both parts were presented in [19], they depend on the dimension of the space-time. Here we will not need their explicit form, so we give only their integral representations suitable for later use:

CONSERVATION OF THE ENERGY-MOMENTUM
In the first order in κ the total energy-momentum tensor consists of three contributions (2.18 and satisfies the conservation equation (2.19). To convert the latter into the the energy-momentum balance equation one has to integrate over the world-tube Ω: bounded by the closed hypersurface consisting of two space-like hypersurfaces associated with the moments of time t 0 , t f (usually chosen orthogonal to the time-axis), and the closing lateral hypersurface Σ ∞ at spatial infinity. To get the usual energy-momentum conservation equation two conditions should hold: i) finiteness of the the integral of τ MN over Σ t which is interpreted as the D−momentum vector, and ii) vanishing of the lateral flux τ MN through the time-like hypersurface Σ ∞ . This is usually guaranteed by the sufficient fall-off of the integrand at infinity. In the case of domain wall both conditions are not satisfied. First, the wall is considered an infinite and having finite mass-density, so the total energy in the zero order in κ diverges. We will see shortly that the corresponding contribution diverges also in the linear in κ order. Secondly, the lateral flux for the wall is non-zero since the integrand does not fall fast enough at spatial infinity. So in our case the momentum equation reads According to the split of the total energy-momentum tensor (2.19) we can write where are the first-order kinetic momenta carried by the particle 1 and the wall, while is the momentum carried by their gravitational field. The lateral flux at the right hand side of the Eq. (4.4) can also be split into similar three contributions. The boundary hypersurface Σ ∞ in the (D − 1)-dimensional space consists of three components Thus the difference between the momenta defined in a standard way as the integrals over space-like hypersurfaces is related to some integral over the corresponding time interval. Another unpleasant feature is that the integrals (4.7) are divergent for an infinite wall. To cure both these drawbacks one could introduce the cut-off volume for the wall, but this make the analysis more complex. Instead we pass to time derivatives of the momenta which are all finite. In other words we check the momentum conservation between the infinitely close moments of time. Then the Eq. (4.10) gives: 11) where the integral at the right hand side will be called the lateral momentum flux. This term looks like an external force acting upon the system, but in fact it is due to an additional loss of the wall momentum. In principle it could be absorbed by the redefinition of the wall momentum at the left hand side, but we prefer to keep the usual definition (4.7).

COMPUTATION OF THE MOMENTA
We proceed in analyzing various contributions to the differential conservation equation (4.11). Note that in the zero order in κ the particle and the wall kinetic momenta are simplȳ where V br is the world-volume introduced for normalization. These quantities are constant which can be omitted from the further analysis.
A. Kinetic momenta The first order particle stress tensor is obtained expanding the general expression (2.11) in κ : The first order stress-tensor of the wall is obtained substituting the first-order metric deviation (3.1) due to the particle and the first-order perturbations of the wall world-volume into the Eq. (2.10): Again, the delta-functions in the integrand indicate on its localization on the unperturbed wall world-volume. Due to the kinematics of the collision, the first order kinetic momenta also have non-zero only the 0 and z components. The particle momentum is calculated substituting the wall metric deviation h MN (τ ) given by (3.3) and the particle world-line deviation δz M (τ ) given by (3.8) into (5.1) and integrating with the help of the delta-function: Now calculate the time component of the wall momentum. Substituting the deviation δX M = δ M z Φ with Φ given by (3.15) into the integrand of (4.7) we get Since Φ is the function of the world-volume coordinates (t, r) only, the term Φ δ ′ (z) vanishes upon integration over z, so δP 0 does not depend on Φ. Substituting into the second quantity the particle metric deviation (3.1) one gets so the first order zero component will read where the integral of χ over r including the volume factor with a 2 = γ 2 (z − vt) 2 , linearly diverges at the upper limit. This is not surprising taking into account an infinite extension of the wall. To avoid a cumbersome normalization procedure, we pass from the momentum to its time derivative. This will be sufficient to define gravitational dressing of the kinetic momenta which is our main goal here. The derivativeQ is finite and the corresponding integral is easily evaluated by the substitution 1 + (r/a) 2 = 1/y leading to Euler's beta-function: Since the unperturbed momentum of the wall is constant (also infinite), we can interpret the resulting quantity as describing the derivative of the full momentum up to the first order simply omitting δ: The computation of the spatial component of the wall momentum is more involved. The flux T z0 can be simplified as follows: where the wall perturbation is the sum of two terms Φ = Φ a + Φ b , the first describing the regular deformation induced by the particle gravitational field, and the second corresponding to the shock branon wave emerging at the moment of piercing and then freely propagates outwards along the wall. Substituting (3.15) into (5.11) and taking into account that I a | t=0 = I b | t=0 2 , one can verify the absence of terms proportional to δ(t) in the time derivative of the total perturbation Φ ,0 . Thus one can write δP z (t) = δP z a (t) + δP z b (t) with Let us start with the 'regular' part P z a . Substituting I a (3.16) and performing an integration over the sphere we obtain: one obtains again the divergent quantity now with a 2 = γ 2 v 2 t 2 . Passing to the time derivative we use the fact that δ(t)İ a = 0 in the distributional sense. Taking into account that there is no zero order contribution to P z , we can writė Now we present the corresponding quantities for the particle, differentiating the sum of the zero and the first order (5.3) momenta:

B. Branon contribution
On can check that the shock wave (branon) part of the wall's perturbation Φ b does not give contribution to the zero-component of the momentum (the energy). However there is still the branon contribution to P z arising after the perforation. Substituting the integral representation (3.17) for I b into the Eq. (5.12), one obtains Integration over k is performed using the integral [26] ∞ 0 The latter expression contains Γ − D−5 2 which has a simple pole at odd D 5. Thus (5.19) drastically depends upon the parity of D.
First let us consider odd D 5. Applying the distributional limit [27, §3.5, eq. (1)] where the order of derivative is integer and we pass to the differentiation over t 2 . Integrating trivially over r and next differentiating with respect to |t| 2 according to easily verified for integer λ, one obtains finally: Now consider even D 4: the Gamma-function is regular now and we represent Substituting (5.25) into (5.19) one obtains The variable change y = r 2 /t 2 leads us again to the beta-function B D−2 2 , 1 2 so Applying (5.22) for integer D−4 2 , the momentum carrying by branon reads Substituting Λ (3.15), one arrives at the same expression (5.23), as for odd space-time dimensionality 3 . Notice that final result keeps this form also for D = 2, 3. This gives rise to another problem: while all other contributions to momenta transfer are proportional to sign functions of time, (5.24) is proportional to the Heaviside function. Since sgn(t) and θ(t) are linearly independent, this extra contribution can not fix the above non-conservation problem. We will see shortly, that this time-asymmetric part is related to non-zero lateral flux of the momentum. Using the first order Einstein equation forh MN one can see that these are localized on the particle's world-line.
So apparently non-local stresses localize on the wall and the particle world-volumes. From the above calculations is clear that it happens because one deals with the products of two Coulomb-like fields which are tight to the sources without the retardation. Thus we present the integral of S z0 as the sumS z and S z according to their localization: where explicitlyS with χ defined by (5.6).

(5.33)
The second integral is evaluated by integration by parts over z and then using the arising delta-function. The radial integral diverges as before: The corresponding derivative is finite: Now consider the S 00 component. The following contributions are non-zero: • Terms with first derivatives of both h MN andh MN : These are localized to the wall integrating by parts over z; • Terms with the second z-derivatives ofh MN : These are localized to the wall integrating over z by parts twice ; • Second derivatives of h MN are directly localized on the wall: • Boxes ofh MN :S 00 ≡ −3 h 00 ✷h 00 + 1 2 h 00 ✷h + h P Q ✷h P Q .
These are localized on the particle world-line after application of linearized Einstein equations.
Thus the last contribution gives the finite energȳ the corresponding derivative beingṠ The first three contributions attributed to the wall are integrated exactly as before leading to divergent total energy, but finite time derivativė (5.39)

GRAVITATIONAL DRESSING
Let us briefly summarize basic features of the particle-wall piercing collision in the perturbative approach. In zero order in gravitational coupling κ the wall is plane, unexcited and extending to spatial infinity. Its total momentum is constant and infinite. The particle is moving with the constant velocity orthogonally to the wall, its momentum is constant and finite. Gravitational interaction between them is repulsive and cause deceleration of the particle before the moment of perforation at t = 0 and acceleration after the perforation. The perturbation of the particle world-line is strictly time antisymmetric. The action of the particle gravity upon the wall is more complicated: the wall's deformation consists of the time antisymmetric component due to continuously varying gravitational force and a shock-wave component which arises after the perforation.
In the first order in κ the total conserved (in Minkowskian sense) energy-momentum tensor consists of three parts: two kinetic terms and the stress-tensor of the gravitational field. Since the latter is constructed from the metric deviations generated by time-independent sources, the presumably non-local gravitational stresses in fact localize at the unperturbed particle's world-line and the wall's word-volume. Therefore we can associate the corresponding gravity contributions with kinetic terms obtaining "dressed" momenta of the particle and the wall. The associated integrated total momenta are infinite due to slow fall-off of deformations at spatial infinity. To get rid of infinities we passed to time derivatives of momenta, which actually represent the total forces acting on the particle and the wall. These latter are finite and we can explore the energy-momentum balance in the form of the third Newton's law. Note that the contribution of the shock wave makes the balance non-symmetric in time. This contribution, however applies only to spatial component of momentum, and does not influence the energy balance. Now we show that on can construct the gravitationally dressed momenta of the particle and the wall such that the total momentum (4.5) satisfying the balance equation (4.11) be the sum of two but not three quantities For this it is enough to split the contribution of gravitational stresses between the particle and the wall according to their localization revealed in the previous section.

A. Dressed particle momentum
Both kinetic and gravitational contributions to the particle momentum are finite, so we introduce the total dressed momentum as the sumP Substituting here (5.3) and (5.32) we obtain the following non-zero components: Note that gravitational stresses contribution to the energy (5.32) is negative, so the total first order contribution to the dressed energy may have negative sign depending on the particle velocity.

B. Dressed wall momentum
For the wall we write similarly where the kinetic contribution consists of the sum of the regular and the branon parts δP M = δP M a + δP M b . Actually the branon part δP M b is non-zero only for the spatial component M = z, while for the time component we have with Q(a) given by (5.8) with a 2 = γ 2 v 2 t 2 . This is a divergent quantity, but its time derivative is finite. Using the Eq. (5.9) it is easy to establish the identity showing that the change of the dressed wall's energy per unit time is opposite to the change of the dressed particle's energy. For the spatial component P M we have two complications. First, the shock wave contribution δP M b is nonzero. Second, for this component the lateral flux of momentum is also non-zero. It turns out, that the regular and the branon parts as the functions of time are linearly independent, so the balance equation (4.11) must hold for them separately. So consider first the regular kinetic part δP z a . Summing up the expressions (5.15) and (5.34) one gets Computing the time derivative of the difference between the wall and the particle momenta we find where the limit r → ∞ is to be taken. This results in This compensates the extra terms in (6.9).
Branon part. As was explained above, the contributions of the branon wave to the time derivative of the wall momenta and the lateral flux must balance each other independently, we are going to check it now. The only non-zero are z-components, δP z b is defined in (5.12) with the corresponding derivative  On the other hand, the flux over the lateral surface dS r is determined by the corresponding T zr −component of the brane's stress-energy tensor and reads while the rate of its change is given by (after the trivial z−integration) Differentiating it with the help of (6.11), one arrives at that exactly compensates (6.16): The computation of F z b in the closed form is presented in the Subsection 5 B, hence: It is worth noting that combining two components of the lateral force, one obtains In other words, the total lateral z−force is continuous and constant 4 : The same concerns the total z−component of momentum.

CONCLUSIONS
In this paper we have analyzed the collision problem between the point particle and the domain wall in which no free momenta of colliding objects can be defined, and the energy-momentum conservation involves at any moment the contribution of the field stresses. Generically, the stresses are non-local, but it turns out that within the perturbation theory their contribution can be unambiguously split into two parts which are effectively localized and can be prescribed to the particle and the wall separately, leading to the the notion of gravitational dressing. This is somewhat similar to introduction of the potential energy in the non-relativistic theory, but our treatment is fully relativistic. The dressed particle momentum involves its kinetic momentum plus its "potential" momentum in the field of the wall; similarly, the wall dressed momentum involves its "potential" momentum in the field of the particle. Thus our dressing is very different from the usual dressing in the sense of adding the contribution of the proper field. We think that such a picture may be useful also in other situations in which the free states of the colliding objects can not be defined.
The second novel feature of the particle-wall collision which we have revealed here is the non-zero momentum flux through the "lateral" surface of the world tube. Because of this flux, the divergence-free stress tensor does not define the conserved energy-momentum charges as the integral over time-like sections of the world tube, since the lateral momentum flux is integrated over the time. One can still consider the change of such integrals between the infinitesimally closed surfaces, thus passing to the time derivatives of these charges. Then taking into account the lateral flux we establish the instantaneous energy-momentum balance in terms of the dressed particle and wall momenta. Actually the non-vanishing flux arises for the space component of the momentum orthogonal to the wall, while the energy is still conserved in the usual sense.
The third feature, which is also fully tractable within our model, is the excitation of the wall under the collision. Contrary to the case of colliding particles, the wall has the internal degrees of freedom which get excited in form of the branon wave. This excitation consists of two parts: one is the direct deformation of the wall in the gravitational field of the particle, which depends on their separation; another is the shock branon wave which starts after the perforation and propagates freely outward along the wall with the velocity of light. The latter gives a separate contribution to the energy-momentum which satisfies our balance equation with account for the lateral momentum flux.
Our procedure of gravitational dressing as relativistic counterpart to the potential energy seems to be applicable to collisions of particles and branes interacting via other fields. In fact the linearized gravity is similar to electrodynamics or any other linear field theory. The reason for "localization" of field stresses is that within the perturbational treatment of collision the first order field perturbations entering the field stress tensor satisfy D'Alembert equations with localized sources. This is the general features of classical relativistic collision problems.