BMS symmetry in spacetime

Phy 155 Report

BMS symmetry in spacetime

BMS symmetry in spacetime Zheng Liang Lim 915633481 Physics Department, UC Davis, CA 95616 USA.

Contents 1 Abstract

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2 Introduction

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3 Definition of asymptotic flatness 3.1 Conformal compactification of Minkowski spacetime . . . . . . . . . . 3.2 BMS gauge in 4 dimension . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Physics of asymptotically flat spacetime . . . . . . . . . . . . . . . .

3 3 5 6

4 Supertranslation and Superrotation 4.1 Cauchy data at null infinity . . . . 4.2 Vector fields preliminaries . . . . . 4.3 Supertranslation and superrotation

7 7 8 9

symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Gravitational Memory effects

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6 Conclusion

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7 Acknowledgement

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8 Appendix 8.1 Covariant derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Lie derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Killing vectors of Minkowski spacetime in spherical coordinates . . . .

15 15 17 19

References

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Abstract

This work studies the properties of BMS symmetry with the special focus on asymptotic spacetime. We will start off with the work of Penrose on conformal compactification of asymptotically flat spacetime to understand the notion of the asymptotic structure. After which, the metric and the Einstein field equation of BMS will be explored to obtained the desired properties of BMS radiative field. Moreover, the idea of both supertranslation and superrotations will be discussed as well together with the notion of the gravitational memory effects and supertranslated black hole as examples to further understand the physical impact the symmetries have on spacetime. Note that mathematical derivations were avoided unless necessary and the physical aspects of BMS will be our main focus.

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Introduction

One of the main research areas in general relativity in the eighties is the study of asymptotic symmetries in spacetime. It is the study of non-trivial exact symmetries or conserved charges of any system with an asymptotic region or boundary. If a system containing gravitational sources is surrounded by an infinite vacuum region, the geometry of the spacetime will tend to approach the flat Minkowski geometry of special relativity at infinity (large distance). Such spacetimes are known as asymptotically flat spacetimes. One of the earliest examples appeared in the pioneering work of Bomdi, Vander Burg, Metzner and Sach (BMS), who attempted to recover the well known Lie group; the Poincare group as the symmetry group of asymptotically flat spacetime. They postulated that had this been the case, general relativity could essentially be reduced to special relativity at large distance and in regions where the gravitational potential is small; the weak field limit. However, to their surprised, that did not happened and their discovered an even larger group; the infinite dimensional BMS group, in which the finite dimensional Poincare group is a subgroup of it. The BMS group is a group of vector fields that leave the Bondi metric asymptotically invariant. The Bondi metric is the metric that describe the asymptotically flat spacetime which the angular metric component can be expanded into subleading terms. The BMS group consists of supertranslations and superrotations symmetries ; which are the infinite class of transformations that is generated similarly to translation and can be seen as a extension of translations itself.

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Definition of asymptotic flatness

3.1

Conformal compactification of Minkowski spacetime

An easy way to introduce the notions of asymptotic infinities and flatness in Minkowski spacetime is to introduce the concept of Penrose compactification, which globally and conformally (preserving angles) maps the spacetime into another Lorentzian manifold with finite extent and boundary and is differentiable everywhere except at infinity [1]. In spherical coordinates (t, r, θ, φ), the metric of the Minkowski diagram can be written as, ds2 = −dt2 + dr2 + r2 (dθ2 + sin2 θdφ2 ).

(1)

By contracting the angular components one can rewritten this as, ds2 = −dt2 + dr2 + r2 γAB dxA dxB

(2)

where γAB is the unit metric of S 2 . To represent the whole of spacetime in some finite sheet, one can perform stereographic project; shining a light on a S 2 sphere such that every point on this sheet actually represents a S 2 sphere except at the origin. The origin can be viewed as the North pole of the sphere which is not well defined. To study the motion of spacetime along the radial and timelike direction, null coordinates such as the advanced and retarded coordinates were introduced and they are given by v = t + r and u = t − r respectively. So the metric line element becomes, 1 (3) ds2 = −dudv + (u − v)2 γAB dxA dxB . 4 A further transformation can be made to the new coordinates U and V where U = tan−1 u and V = tan−1 v where U, V ∈ ( −π , π ). Suppose that we let the 2 2 conformal factor be defined as Ω(U, V ) = cos U cos V and redefine out space and timlike coordinates as T = U + V and R = V − U , our new metric can be written as

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Fig 1: A representation of the celestial sphere of 2 dimension on a Penrose diagram of Minkowski space. The red line is the line of constant radius, while the green line represents the outgoing radial light ray. sin2 R γAB dxA dxB 4 The Penrose diagram includes following structures at the boundaries; Ω2 ds2 = −dT 2 + dR2 +

(4)

• Past timelike infinity (i− ): It represents the asymptotic sphere at t → −∞, while keeping r fixed. It is also the starting point of any maximally extended timelike geodesic. • Future timelike infinity (i+ ): It represents the region of the asymptotic sphere where t → ∞, at constant radius r. It is also the end point of any maximally extended timelike geodesic. • Spatial infinity (i0 ): It represents the region of the asymptotic sphere where r → ∞ at constant t.

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• Past null infinity (F − ): It represents the surface formed by the starting points of the ingoing null geodesic at r → ∞ where u is fixed. • Future null infinity (F + ): It is the future counterpart of the past null infinity and is the surface of the outgoing null geodesics where r → ∞ at fixed v. The BMS group is completely determined by the action that is taken globally throughout the whole of spacetime on F + .

3.2

BMS gauge in 4 dimension

Now, we will slowly introduce gravity into our spacetime structure. In general, we can define the notion of asymptotic flatness by adapting a similar coordinate system as the one we have before and specify the fall off conditions. Mathematically, this particular spacetime will describe a family of out going null hypersurfaces where u is constant [16]. While physically, it represents the footprint left behind by radiation such as the gravitational waves as it approaches infinity. The normal co-vector kµ = −∂µ u of these null hypersurfaces satisfies the relation g µν (∂µ u)(∂ν u) = 0.

(5)

This implies that g uu = 0. Moreover, for a given angular coordinate xA , we have k µ ∂µ xA = −g µν (∂µ u)(∂ν xA ) = 0.

(6)

This implies that g uA = 0. When BMS first reviewed the problem, the line element of the metric of the asymptotically flat spacetime was written in the form V (7) ds2 = − e2α du2 − 2e2α dudr + gAB (dxA − U A du)(dxB − U B du) r where α is a function of order of inverse r2 as well as U A . For U A , upon solving the Einstein field equation, gives us something called the angular momentum aspect. It is defined as the angular density of the angular momentum with respect to the origin + O(r−2 ) at r = 0. On the other hand, the Vr term turns out to be equal to −1 + 2M r where M is the Bondi mass aspect. Similar to the angular momentum aspect, it gives the angular density of the energy of spacetime as measured from a point at future infinity F + in the direction pointing out along the angular coordinate xA . One can imagine how this could been done by drawing a circular loop along constant U in figure 1. In addition, gAB is be expanded as follow, 1 EAB gAB = r2 γAB + rCAB + DAB + γAB C EF CEF + + O(r−2 ). 4 r 5

(8)

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where CAB , DAB and EAB are functions that depends on u, θ and φ which describes terms of higher order leading corrections which are also known as the cauchy data which contains all the information about the gravitational radiation around F + . Since, asymptotic symmetries are by definition, the gauge transformations that preserves the fall off condition of the metric, we can choose a gauge condition such that ∂r det( grAB 2 ) = 0. This not only ensure that the tensors CAB , DAB and EAB are all traceless but also simplify our calculations of the angular terms when solving the Einstein field equation. It is noteworthy that this gauge we have chosen is arbitrary and it is not ”real”. Therefore, the general allowed line element for the Bondi metric is, ds2 = − du2 − 2dudr + r2 γAB dxA dxB +

2M 2 du + rCAB dxA dxB r

1 4 +DB CAB dudxA + [ (NA + u∂A M ) r 3 1 1 BC − ∂A (CBC C )]dudxA + γAB CCD C CD dxA dxB .... 8 4

(9)

where “....” represents the subleading terms. Note that our metric require two additional constraint for it to truly satisfy the Einstein field equation. They are obtained by setting the Riemann curvature tensor RuA and Ruu to zero. They are essentially the expressions for the partial derivative of the Bondi mass aspect and the angular which contains terms that includes the matter stress tensor. One can also see that as r → ∞, guu = −1, gur = −2 as α → 0 and gAB → r2 γAB . As a result, we have obtained the Minkowski spacetime as written in (1). This means that near future infinity, asymptotic spacetime appears to be the same as flat spacetime. There are some important physics behind asymptotically flat spacetime which I will dive into it deeply in the next section.

3.3

Physics of asymptotically flat spacetime

Previously, we have introduced he concept of Bondi mass aspect and the angular momentum aspect. TheR Bondi mass is obtained after performing an integration over the S 2 sphere; M 0 (u) = S d2 ΩM (u, xA ) where d2 Ω = sinθdθdφ. The angular momentum can also be obtained in a similar manner with the integral performed together with the generators of rotation [2] (See [2] for more details). Since the Bondi mass is a function of our retarded time u, one can then consider the derivative of M’(u). This gives us the famous Bondi Sach mass loss formula which is given by, Z Z 1 0 2 A d2 Ω(∂u C AB )(∂u CAB ). (10) ∂u M (u) = ∂u d ΩM (u, x ) = − 8 S 6

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Since, ∂u M 0 ≤ 0, energy or mass of spacetime is decreasing with time for either pure gravity or gravity that is coupled to matter. Physically, this means that when the bulk emits gravitational radiation or electromagnetic radiation, it escapes as F + and decreases the spacetime energy as u evolves. As u → −∞, the Bondi mass is equal to the total energy of a Cauchy slices which is a plane at the instant of u = −∞ in which contains the initial conditions that determines the future (and the past) uniquely. The initial conditions will be further explored in the next section. Similarly, the energy of spacetime can be described in the classical mechanics and quantum mechanics sense, in which the energy can be represented by its Hamiltonian that follows the ADM prescription. Furthermore, the time derivative of the tensor CAB is closely related to supertranslation, it the so called Bondi news tensor which is usually denoted by ∂u (CAB ) = NAB . It is the analog of electromagnetic or gravitational field strength and the square of it is proportional to the energy flux across F + . Since we are only considering spacetime with an asymptotic behaviour and do not require them to be near flat in the deep interior, the Bondi news trivially determines CAB up to an integration function by integrating it with respect to u. On the contrary, when there is no news in the system, both energy and mass are conserved.

4

Supertranslation and Superrotation symmetries

These section contains some unprecedented terminology to some people. Readers that are not familiar with the concept of covariant derivative and Lie derivative may read the appendix for a physical intuition of what they are about.

4.1

Cauchy data at null infinity

Let us now study the details of the data are the early time of I + ; I++ and the late time denoted as I−+ . The equation obtained by setting both RuA and Ruu to zero are given by, 1 1 M ) ∂u M = DA DB NAB − NAB N AB + 4πlimr→∞ (r2 Tuu 4 8 1 M ∂u NA = − DB (DB DC CAC − DA DC CBC ) + u∂A ∂u M − 8πlimr→∞ [r2 TuA ] 4 1 1 − ∂A (CBC N BC ) − CAB DC N BC . 4 2

(11)

These are the set of equations that describes the rate of change of the mass and M the angular momentum aspect with respect to the retarded time u. Here, Tab is the matter stress tensor while DA is the covariant derivative associated to the angular 7

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component of the metric. Since, we are expanding with respect to the region at I + , we are thus, only interested in the matter near infinity. Problems arises when NAB does not fall off fast enough as u approaches positive and negative infinity as we will have infinite energy. However, both Christodoulou and Klainerman [3] have discov1 near the past and future boundaries of I + , ered that if the news falls off faster than |u| I−+ and I++ (see figure 2), one can prevent the energy from blowing up and also obtain a geodesically complete solution. In addition, Cauchy data such as C(xA ), M (xA ) and NA (xA ) at null infnity are important quantities to keep track of conservation of charges (supertranslation), energy and the analogue of angular momentum; superrotation charges in the absence the Bondi news. For example, whenever the fields are sufficiently weak near spatial infinity, Lorentz and CPT invariant conditions requires that C(xA )|I−+ = C(xA )|I+− , M (xA )|I−+ = M (xA )|I+−

(12)

while conservation of the superrotation charges requires NA (xA )|I−+ = NA (xA )|I+− .

(13)

These matching conditions physically implies that there are infinite amount of supertranslation charge, energy and superrotation charges taking place in processes like gravitational scattering. This was postulated and anwsered by Strominger in 2014 [4]. The notion of infinite conservation laws will be addressed in the following subsection where we start to discuss the concept of supertranslation and superrotation.

4.2

Vector fields preliminaries

Previously, we have briefly reviewed the definition of asymptotic symmetries as gauge transformation the preserves the fall of condition of the BMS metric. Such transformations are in reality infinitesimal diffeomorphism which can be described mathematically by the Lie derivative of the components of the BMS metric line element. Preserving the Bondi gauge requires, gAB ) = 0. (14) r2 Solving these sets of equations will give us the relation of the gauge parameter ξ r , ξ u and ξ A which are important in finding the “net” vector field, which is formed by the linear superposition of these gauge parameters, that is associated to the diffeomorphism. Applying our Lie derivative equation yields, Lξ grr = 0, Lξ grA = 0, Lξ ∂r (

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u

A

ξ = f (u, x ),

B

B

Z

A

ξ = Y (u, x ) − ∂A f

r ξ = − (DA ξ A − U C ∂C f ) 2

∞

dr0 g AB e2α ,

r

(15)

r

respectively, where f (u, xA ) and Y B (u, xA ) are the arbitrary integration functions that depends on (u, θ, φ). On the other hand, preserving the boundary condition at infinity requires, Lξ guu = O(r−1 ), Lξ gur = O(r−2 ), Lξ guA = O(1), Lξ gAB = O(r)

(16)

and solving these new sets of equation will give us the constraints that can be applied to (15). Only two constraints are applicable to (15) which are given by, 1 (17) ∂u f = DA Y A , ∂u Y B = 0. 2 the last equation simply obeys the conformal Killing equation on the 2-sphere which will not be elaborated in this paper (see [5] for the details). The second constraint suggest that Y A is independent of u. Hence, the two relevant constraints implies that, B

B

Z

A

ξ =Y (x ) − ∂A f

∞

dr0 g AB e2α

r (18) u C ξ =F (x ) + DC Y 2 Thus, summing up the asymptotic gauge parameters will give us the total vector field which is, u

A

ξ =ξ u ∂u + ξ A ∂A + ξ r ∂r (19) 1 1 u '(F + DC Y C )∂u + (Y B − DB F )∂B − (rDC Y C − DC DC f )∂r 2 r 2 where the higher orders of inverse r are neglected since our calculations are focused on regions were r → ∞.

4.3

Supertranslation and superrotation

The total vector field contains both the terms that represent the supertranslation symmetry and the superrotation symmetry. The part that describes the supertranslation consists of, 9

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1 1 (20) ξT = F (xC )∂u − DA F (xC )∂A + DA DA F (xC )∂r ... r 2 where ... are the subleading terms. Note that the associated ξT are constructed from the first 4 spherical harmonics, decomposed from F, where F (xC ) = a0 Y00 (xC ) + am Y1m (xC ), where am ∈ < and m = {−1, 0, 1}. The first coefficients represents the translation of the retarded time u while the second coefficient implies spatial translation. Time translation means that energy of spacetime is conserved while the three spatial translation implies that the ADM momentum is conserved instead. It is responsible for the presence of supertranslation charges. As non-trivial diffeomorphisms, supertranslation act on the asymptotically flat spacetime phase space, transforming one geometry into a new physically distinct geometry. To see this, consider Bob, who is located at the future null infinity, is observing pulses of gravitational or electromagnetic radiation. Initially, he sees that there is an outgoing pulse of radiation crossing the south pole of I + , and another pulse crossing the north pole of I + both at the retarded time of u = 200. However, suppose that Bob, gained superpowers and decided to supertranslate the south pole by u = 50 and the north pole by zero. Now, Bob will see that the outgoing radiation at the north pole will be crossing at u = 250, while the one at the south pole will only cross at u = 200 [6]. Hence, it is apparent that supertranslation can bring about effects that can be observed even in the classical level. This is demonstrated by figure 2.

Fig 2: A representation of supertranslation along the surface of a celestial 2 sphere in Minkowski spacetime. The black lines that cuts across the diagram are the so called cauchy slices where our data are stored. The red lines denotes the supertranslation 10

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which the slices underwent. I−+ , I++ denotes the early and late retarded times of I + while I+− and I−− denotes the late and early time of I − respectively. The cauchy slices undergo supertranslation by some retarded time u at every angle independently along the future null infinity I + . The transformation by the supertranslation or sometimes called the action can be computed mathematically by taking the Lie derivative of the BMS metric (11). It turns out that the Bondi news tensor NAB , the field tensor CAB and the Bondi mass aspect transform as, LF CAB =F ∂u CAB − 2DA DB F + γAB DC DC F 1 LF M =F ∂u M + [N AB DA DB F + 2DA F DB N AB ] 4 LF NAB =F ∂u NAB .

(21)

From these relations, it is easy to see that supertranslation cannot create inertial mass or gravitational radiation. In particular, suppose that we were to supertranslate the flat Minkowski spacetime where M = NAB = CAB = 0, then the above equations imply that the inhomogenous term vanishes. Thus, there is no change in the Bondi mass and Bondi news tensor. The only field which can be transformed is CAB which is related to the supertranslation memory field often denoted by C(xA ). In fact C transforms under the following equation, LF C = F

(22)

which represents the spontaneous breaking of supertranslation invariance between the gravitational vacua, where C is the Goldstone boson that facilitates the breaking. In addition, since supertranslation commute with time translation as they are related via the terms in the special harmonics, their associated charges will commute with the Hamiltonian and this means that all the degenerate states have the same energy [4]. On the other hand, if we set F = 0, we can obtain the vector field that corresponds to superrotation. It is given by, u r (23) ξR = Y A (xC )∂A − DC Y A (xC )∂r + DA Y A (xC )∂u ... 2 2 Where Y A is the vector field that satisfies the conformal killing equation of a 2sphere (for advanced students see appendix). Most of the details of superrotations involves group theory and hence, we will not delve into it. One simply has to know that superrotation is a symmetry that gives rise to superrotation charges. Previously, we have also described briefly the concept of infinite supertranslation charge and energy at the anti-podal points of the the asymptotically flat spacetime. This is 11

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because, recall that from (19) F is any arbitrary function that depends on xA or φ, θ. Hence, we can possibly have an infinite amount of such a function at every angle in the celestial sphere at null infinity which we could use in to construct our conservation laws such as the supertranslation charge conservation. Similar argument also applies to energy and superrotation charges. Moreover, supertranslation symmetry have been applied to black holes to study the perturbation theory and determine if at both the classical and quantum level, whether the charge and energy are conserved [7]. For an unmodified Schwarzschild metric line element in the advanced Bondi coordinate, 2M )dv 2 + 2dvdr + r2 γAB dxA dxB . (24) r To exclude superrotations, we require the components to be bounded in a local orthornomal coordinate system as r → ∞. Unlike the vector fields in retarded time coordinate, some of the terms gained a negative sign and becomes, ds2 = −(1 −

1 1 (25) ξT = F ∂v + DF ∂A − DA DA F ∂r . r 2 where DA DA can be rewritten as D2 . By computing the action of supertranslation on the Schwarzschild metric, one obtains, M D2 F )dv 2 + 2dvdr − DA (2V F + D2 F )dvdxA + r2 (r2 γAB + 2rDA DB F − rγAB D2 F )dxA dxB .

ds2 + LF ds2 = − (V −

(26)

where V = (1 − 2M ). Similarly, one can apply the same treatment to the r Schwarzschild metric in retarded time coordinate and obtain, M D2 F )du2 − 2dudr − DA (2V F + D2 F )dudxA + r2 (r2 γAB − 2rDA DB F + rγAB D2 F )dxA dxB .

ds2 = − (V +

(27)

Initially, at the past null infinity, the advanced coordinate is being utilised and hence, the radius of the event horizon rS is given by MI − = 21 (rS − 12 D2 f ). However, at the future null infinity MI + = 12 (rS + 12 D2 f ). This means that for some positive function f, the effective mass of the black hole is smaller in the retarded coordinates than the advanced coordinate. This is the statement of mass conservation at null infinity. Furthermore, since supertranslation charge commute with the Hamiltonian (Abelian), supertranslating a black hole does not add supertranslation charges, just 12

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as an ordinary translation does not add momentum. In fact, the modification of the original Schwarzschild line element is deemed as dressing the Schwarzschild black hole with linearised supertranslation hair. This is an allusion to the famous no hair theorem which states that for a Schwarzschild black hole, the only conserved quantities are the mass, charge and angular momentum [8]. This theorem seems to present the notion that information about anything that is absorbed by the black hole will be lost. However, the modification of the Schwarschild solution tells us that the supertranslation of a black hole which is equivalent to the excitation of the black hole by the absorption of an incoming gravitational radiation or gravitons which carry superrotation charges, will cause the black hole to gain supertranslation hairs that are encoded at the boundary. Thus, information is not lost and is stored within the black hole instead. The effects of supertranslation on a black hole can be measured by observing the gravitational memory effect.

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Gravitational Memory effects

When a pulse of gravitational radiation or energy propagate through spacetime, they produce a gravitational field which moves a couple of nearby inertial observers called detectors that are travelling near future null infinity I + . As a result, the final positions of a pair of detectors are displaced irreversibly relative to their initial positions. This effect is known as the gravitational memory effect or the Christodoulou effect [11,17]. Suppose that the radiation starts of at u = ui and ends off at u = uf , then the Bondi news tensor and the matter stress tensor are identically zero since on I + , it is asymptotically approximated by the Schwarszchild metric. Hence, the detector which moves along a timelike trajectory experiences the radiation only during the time interval 4u = uf − ui . The detectors are displaced when the field tensor CAB changes. Since NAB = ∂u CAB , by integrating the first term of (11) by parts with respect to u, one obtains Z uf 1 A B duTuu . (28) 4M = D D 4CAB − 4 ui where 4M and 4CAB are the change in the Bondi mass aspect and the field tensor from ui to uf . Recall that CAB is also related to the supertranslation memory field C. One can then establish the change in the supertranslation memory field in terms of the change in the mass aspect. It is given by, Z uf 1 2 1 2 M − (D + 2)D 4C = 4M + [ NAB N AB + 4πlimr→∞ (r2 Tuu )]. (29) 4 8 ui Firstly, note that the equation shows that supertranslation that are caused by 13

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gravitational waves are non local. If gravitational wave is passing through the north pole, the effects appears to be larger in between the north and the south pole of I + as compared to the ends.

Fig 3: A representation of the memory effect. The radiation which is in red, is turned on and passes a pair of detectors near located I + . As a result, their relative position starts to oscillate and they got shifted permanently away from their initial positions. This effect becomes part of the memory of spacetime just like how students remember physics concept before a test. Secondly, the change in the Bondi mass aspect from ui to uf is sometimes called the ordinary memory. This occurs when a single mass is separated into 2 bi-polar components. Sometimes, electromagnetic radiation can cause the displacement memory effect. It is known as the null memory effect. Lastly, the gravitational memory can be seen as how one can effectively measure the information that is embedded within the black hole caused by supertranslation. For example, suppose we are interested in studying a dying star which is collapsing into a black hole. We can place a myriad of evenly spaced inertial observers near I + , Due to the memory effect, the positions of these detectors will eventually shifts. In fact, we expect to measure infinitely many information or conserved charges instead of the originally postulated ten Poincare’ charges for the system. This is particularly useful in studying black holes using computer simulations.

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Conclusion

In conclusion, we have introduced the general diffeomorphism and its action for both superrotation and supertranslation using the general expression of gAB . Moreover, we have imposed Lorentz and CPT invariance on supertranslation and conservation of ADM angular momentum on superrotation. Doing so, allowed us to establish certain matching conditions that relate future and past null infinity during events 14

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like gravitational scattering. The matching conditions, simply imply conservation of supertranslation and superrotation charges. Being theoretical physicists, we are interested in how diffeomorphism in supertranslation can affect the general Schwarzschild metric. It turns out that supertranslating a black hole does modify the metric to a certain extent that it can be known as implanting supertranslation hair. Ironically, gravitational waves that cause supertranslations does not add supertranslation charges but rather, carries infinite number of superrotation charges. The effects of supertranslations are measured by observing the so-called memory effects. Furthermore, such findings may be useful in solving the black hole information paradox [12] and reveal flawed assumptions from the past. We now know that black holes in reality, carry an infinite number of conserved charges which generalises the ten Poincare’ charges. These concerns notwithstanding, we also recognise that there exists a great deal of potential for future work on this topic. In particular, the question remains on how does one relates supertransformations to the entanglement entropy of the black hole. While some entropy bound conjectures has been motivated with regard to BM S gauge theory [13,14], much queries have remained yet to be answered.

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Acknowledgement

I would like to thank Professor Veronica Hubeny for the nice discussion on the physical intuition of certain mathematical tools such as the covariant derivative and Lie derivative. This work is not supported by the NSF nor the Simon Foundation.

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Appendix

8.1

Covariant derivative

A covariant derivative which is often denoted as ∇ or D/dt is an operator that acts on vectors along another vector field. It can be seen as a generalisation of the directional derivative of a vector field. To get a better understanding, suppose that we have a path, need not be a geodesic that lies in some spacetime manifold M. From our basic calculus, a derivative operator can be written as the limit of some difference in the function at two given points divided by their parameter t being taken to zero. Similarly, the covariant derivative operating on some vector field Y can be defined as, ∇X Y = limt→

Y (t) − Y|| (t) t

One can visualise it as shown below; 15

(30)

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Fig A: A representation of a covariant derivative of a vector field Y at point p, along another vector field X which is in green. The red and blue vectors are the deviated vectors that originates from Y as Y is taken along the path denoted by the curve along X. The covariant derivative, in essence can be seen as the measurement of the deviation of the vector field Y’ and Y” along the path from the original vector field Y which is perpendicular to the vector field X at the point p. If there is no deviation, we say that Y is always perpendicular to X along the path and hence, Y are considered to be parallel to one another throughout the path. In this case the covariant derivative is zero and this phenomenon is called parallel transportation or parallel translation. A geodesic is the path in which satisfy the condition of parallel transport. For a given vector V in the tangent plane of spacetime, one can follow our common notation in electromagnetism and relativity and write it as, V = V µ eµ

(31)

By applying the covariant derivative along some vector c0 (t) = aν (t)eν , one can apply Leibniz rule and the equation becomes, 16

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DV D µ D d = (V eµ ) = (V µ )eµ + V µ (eµ ) dt dt dt dt dV µ eµ + V µ ∇c0 (eµ ) = dt dV µ = eµ + aν ∇eν eµ V µ dt dV l = el + aν (t)Γlνµ V µ el dt Where one just simply treat V µ as a function and so, the covariant derivative on V µ becomes the ordinary derivative. Γlνµ is called the Christoffel symbol. It is related to the our metric tensor via, 1 Γlµν = g lk (−∂k gµν + ∂ν gkµ + ∂µ gkν ) (32) 2 where g lk is the components of the inverse of the matrix of glk . Moreover, one has to note that covariant derivative may not be locally defined. Thus, we need the concept of Lie derivative.

8.2

Lie derivative

In Lie derivative you relate the spaces over nearby points by something called the flow of a vector field v. This flow, defined by q(s), generates 1-1 map from manifold to itself locally (diffeomorphism). In particular, suppose we have a small curve w(t) at the starting point x and move to another point x’ along the flow, or another curve q(s). Let Y be the tangent vector at x and Y’ be the tangent vector at the point x’. Since, the tangent vector to this curve at x’ can be moved to tangent vector to the curve at x via parallel transport (transport back) as well, we can measure the difference between the tangent vector drawn along the curve w(t) at x and that by parallel transport from Y’ at x’. After that, we can apply our definition of limits from calculus to define a new set of derivative. This derivative is no other than the Lie derivative which is commonly written as Lv Y . The Lie derivative of a vector field Y is defined as, 1 (33) Lv Y = lims→0 [(q−s ) ∗ Y |qs (x) − Y |x ] s for some x that lies in the spacetime Manifold. While the Lie derivative of a function can be defined as,

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1 Lv f (x) = lims→0 [f (qs (x)) − f (x)]. s One can visualise it as shown below;

(34)

Fig B: A representation of a Lie derivative of a vector field Y along the flow q(s) of another vector field v. The blue lines represents the tangent vector along the curve w(t) at x’ that is parallelly transported to x. The differences between the mapped vector field (also in blue) and the ”original” vector field Y (in black) are then measured. By Taylor expanding the vector field of the new corresponding vector field that is associated with the flow and using the definition of the Lie derivative, one can show that, Lv Y = v α ∂α Y µ − Y ν ∂ν v µ = [v, Y ].

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where [.] is called the Lie bracket [15]. A similar expression can also be written for the Lie derivative of a one form by perform Taylor expansion and using the definition of the Lie derivative by noting that a scalar function f can also be written as f = v α ωα . It is given by, Lv ωα = v µ ∂µ ωα + ων ∂α v ν .

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One can then generalise the result of the Lie derivative of the one-form to higher rank tensors by writing the rank two tensor gab as the exterior product of two one forms ca and db . By applying Leibniz’s rule for two forms, one can show that the Lie 18

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derivative of a metric tensor gαβ along the flow of the vector field ξ can be expressed as, Lξ gαβ = ξ µ ∂µ gαβ + gµβ ∂α ξ µ + gαµ ∂β ξ µ

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In some cases, Lξ gαβ can be evaluated to zero. In that case, we say that the metric satisfies the Killing’s equation which is defined by, Lξ gαβ = 0.

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where ξ is the so called Killing vector field. We can further evaluate this by writing, 0 =ξ µ ∂µ gab + gµb ∂a ξ µ + gaµ ∂b ξ µ =ξ µ ∂µ gab + [∂a (gµb ξ µ − ξ µ ∂a gµb ] + [∂b (gaµ ξ µ − ξ µ ∂b gaµ ] =ξ µ ∂µ gab + ∂a ξb − ξ µ ∂a gµb − ξ µ ∂b gaµ 1 1 =∂a ξb − ξ µ (∂a gµb + ∂b gaµ − ∂µ gab ) + ∂b ξa − ξ µ (∂a gµb + ∂b gaµ − ∂µ gab ) 2 2 =∂a ξb − ξµ Γµba + ∂b ξa − ξµ Γµab =Da ξb + Db ξa

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where D is the covariant derivative. Solving this gives us the ten isometries which form the well known Poincare’group of symmetries of special relativity for Minkowski spacetime [10].

8.3

Killing vectors of Minkowski spacetime in spherical coordinates

The isometries of the Minkowski metric in cartesian coordinates ds2 = −dt2 + dx2 + dy 2 + dz 2 with the metric signature η αβ = (−1, 1, 1, 1) are given by the solutions of the Killing equation, Dν ξµ + Dµ ξν = 0,

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where D is the co-variant derivative. The most general solution to the Killing’s equation can be written as an independent vector fields given by, ξα = aα + bβµ xµ

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where bβµ = −bµβ for arbitrary constants aα and bβµ . If we let aα = 0 and bβµ = 1, Then we can construct the generators of boosts and rotations ξβµ with the linear combination of the independent vector fields [9]. 19

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ξβµ = (η αβ bβµ xµ )∂α

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where α = β. Thus, this gives rise to ten different Killing vectors, ξ0 = ∂t , ξ1 = ∂x , ξ2 = ∂y , ∂3 = ∂z ξ01 = iKx = x∂t + t∂x , ξ02 = iKy = y∂t + t∂y , ξ03 = iKz = z∂t + t∂z ξ21 = iJz = x∂y − y∂x , ξ13 = iJx = −x∂z + z∂x , ξ32 = iJy = y∂z − z∂y . In spherical coordinates with retarded null time coordinate u, we require t = u + r, x = rsinθcosφ, y = rsinθsinφ, z = rcosθ.

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Then, the ten Killing vectors becomes, ξ0 =∂u , 1 sinφ ξ1 = − sinθcosφ∂u + sinθcosφ∂r + (cosθcosφ∂θ − ∂φ ) r sinθ 1 cosφ ξ2 = − sinθsinφ∂u + sinθsinφ∂r + (cosθsinφ∂θ + ∂φ ) r sinθ 1 ξ3 =cosθ∂u − cosθ∂r + sinθ∂θ r u u sinφ ξ01 = − usinθcosφ∂u + (r + u)sinθcosφ∂r + (1 + )cosθcosφ∂θ − (1 + ) ∂φ r r sinθ u u cosφ ξ02 = − usinθsinφ∂u + (r + u)sinθsinφ∂r + (1 + )cosθsinφ∂θ + (1 + ) ∂φ r r sinθ u ξ03 =ucosθ∂u − (r + u)cosθ∂r + (1 + )sinθ∂θ r ∂ ξ21 = ∂φ cosθ ξ32 = − sinφ∂θ − cosφ ∂φ sinθ cosθ ξ13 =cosφ∂θ − sinφ ∂φ sinθ The first four vector fields describes temporal and and spatial translation symmetries, the next three represents rotation along the 3 spatial directions and the last three represents Lorentz boost. 20

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[14] Raphael Bousso (2016), “Asymptotic Entropy Bounds”, arxiv: 1606.02297 [hep-th] [15] Nakahara, M. (2005). Geometry, topology, and physics. Bristol: Inst, of Physics Publishing. [16] Madler, T., Winicour, J. (2016), “Bondi-Sachs Formalism”, arxiv:1609.01731 [gr-qc] [17] A. Strominger., A. Zhiboedov, A., “Gravitational memory, BMS supertranslations and soft theorems”, JHEP 01 (2016) 086, arXiv:1411.5745 [hep-th]

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