Surprising Structures Hiding at Penrose's Future Null Infinity

TL;DR

The paper introduces new coordinate systems near Penrose's null infinity that reveal hidden structures, allowing definitions of physical quantities like dipole moments and deriving classical mechanics results from asymptotic spacetime properties.

Contribution

It presents novel shear-free null congruence coordinates at null infinity and connects asymptotic gravitational data to classical mechanics and electrodynamics results.

Findings

Defined complex dipole moments from asymptotic Weyl tensor components.

Derived classical mechanics relations from asymptotic spacetime structures.

Identified new coordinate systems revealing hidden structures at null infinity.

Abstract

Since the late1950s, almost all discussions of Asymptotically Flat (Einstein-Maxwell) Space-Times have taken place in the context of Penrose's Null Infinity, $\mathcal{I}^{+}.$\ $\ $In addition,\ almost all calculations have used the Bondi coordinate and tetrad systems. \ We show - first, that there are other natural coordinate systems, near $\mathcal{I}^{+},$ (analogous to light-cones in flat-space) that are based on (asymptotically) shear-free null geodesic congruences (analogous to the flat-space case). \ Using these new coordinates and their associated tetrad, we \textit{define the complex dipole moment, i.e., as the mass dipole plus i times angular momentum,} from the $l=1,\ $harmonic coefficient of a component of the asymptotic$\ $Weyl tensor. Second, from this definition, from the Bianchi Identities and from the Bondi mass and linear momentum, we show that \ there exists a large number of results - identifications and dynamics - identical to those of classical mechanics and electrodynamics. They include, among many others, \textbf{P}=M\textbf{v}+..., \textbf{L}=\textbf{r}x\textbf{P}, \ spin, Newtons 2nd Law with the Rocket force term (\.{M}\textbf{v}) and radiation reaction, angular momentum conservation and others. \ All these relations take place in the rather mysterious H-Space.

Equations

$$x^a=u{t}^a+r{\hat{l}}^a(\zeta ,\overline{\zeta })$$

$${\hat{l}}^a(\zeta ,\overline{\zeta })$$

$$t^a$$

$${\hat{n}}^a$$

$${\hat{m}}^a$$

$${\overline{\hat{m}}}^a$$

$$P$$

$$d{s}^2=d{u}^2+2dudr-4r^2{P}^{-2}d\zeta d\overline{\zeta }$$

$$d{\tilde{s}}^2={\tilde{g}}_{ab}d{x}^{a}d{x}^b={\Omega }^{2}d{u}^2+{\Omega }^{2}2dudr-4P^{-2}d\zeta d\overline{\zeta }.$$

$$x^a={\xi }^a(t)+r^{\ast }{\hat{l}}^{\ast a}({\zeta }^{\ast },{\overline{\zeta }}^{\ast }).$$

$$u=G_F(t,\zeta ,\overline{\zeta })\equiv {\xi }^a(t)l_a(\zeta ,\overline{\zeta })=\frac{\sqrt{2}}{2}{\xi }^0(t)+\frac{1}{2}{\xi }^i(t)Y_{1i}^0(\zeta ,\overline{\zeta }),$$

$$L=\eth G_{F},\ \ \overline{L}=\overline{\eth}G_{F}\$$

$${\hat{l}}^{\ast a}$$

$${\hat{m}}^{\ast a}$$

$${\hat{n}}^{\ast a}$$

$$b$$

$${ð}^2{G}_F=0,$$

$${ð}^{2}G={\sigma }^0(G,\zeta ,\overline{\zeta })$$

$${\sigma }^0(u,\zeta ,\overline{\zeta })={\xi }^{ij}(u)Y_{2ij}^2+....$$

$$u$$

$$z^a$$

$$u=\frac{\tau }{\sqrt{2}}-\frac{1}{2}{\xi }^i(\tau )Y_{1i}^0(\zeta ,\overline{\zeta })+{\xi }^{ij}({\xi }^a(\tau ))Y_{ij}^0(\zeta ,\overline{\zeta })+..$$

$${\xi }^{ij}=\frac{\sqrt{2}G}{24c^4}(Q_{Mass}^{ij\prime \prime }+i{Q}_{spin}^{ij\prime \prime }).$$

$$L$$

$$\overline{L}$$

$$l^{\ast a}$$

$$b$$

$$\tau$$

$${\xi }^a(\tau )$$

$$G({\xi }^a(\tau ),\zeta ,\overline{\zeta })=G_R({\xi }_R^a(t,\lambda ),{\xi }_I^a(t,\lambda ),\zeta ,\overline{\zeta })+i{G}_I({\xi }_R^a(t,\lambda ),{\xi }_I^a(t,\lambda ),\zeta ,\overline{\zeta }).$$

$$\lambda =\Lambda (t,\zeta ,\overline{\zeta }),$$

$$u=G_R({\xi }_R^a(t,\Lambda ),{\xi }_I^a(t,\Lambda ),\zeta ,\overline{\zeta }).$$

$$l^{\ast a}=l^a-\frac{L}{r}{\overline{m}}^a-\frac{\overline{L}}{r}{m}^a+0(r^{-2}).$$

$$u=\frac{\hat{\tau }}{\sqrt{2}}+{\xi }^{ij}({\xi }^a(\hat{\tau }))Y_{ij}^2(\zeta ,\overline{\zeta })+..$$

$${\sigma }^0={\xi }^{ij}(u)Y_{2ij}^2(\zeta ,\overline{\zeta })+...,$$

$$\text{case }1$$

Full text

This leads to the enigma: ”why do these well known relations of classical mechanic take place in $H-$space?” and ”What is the physical meaning of $H-$space?”

1 Introduction

The modern era of the study of Gravitational Radiation began in the 1950s with the pioneering work of Hermann Bondi[1]. This was quickly expanded by the major contributions of Rainer Sachs[2] and Roger Penrose[3][4][5][9] among many others.  After years of further developments, theoretical, numerical and observational, we had its culmination with the observation and analytic understanding of the collision and merger of the pair of black holes that produced the gravitational wave signal, GW105,[6], that was seen by LIGO in 2016. Gravitational wave theory now could play a major role in astrophysics and physics.

Bondi’s work began with integrating the Einstein Equations in the asymptotic region - in the vicinity of future null infinity.  This involved the important step of using special null surfaces as part of the coordinate system referred to as Bondi coordinates, ($r,u$,$\zeta,\overline{\zeta}$). (The Bondi coordinates are defined uniquely up to a group of transformations known as the BMS group.[3][7]) The idea of working near or even at infinity, (though at the beginning was slightly nebulous), was formalized (by Penrose[3]) by bringing infinity into a finite region of the space-time by its conformal compactification, (rescaling and contraction). Future null Infinity was then represented by a null three-surface in space-time (referred to as\ \mathcal{I}^{+},\vocalized by SCRI+).  $\mathcal{I}^{+}$,  a null 3-surface with the topology of $\mathcal{S}^{2}\times\mathcal{R}$,  (visualized as a light-cone at future null infinity, apex at time-like infinity) is  coordinatized by the complex stereographic coordinates, ($\zeta=e^{i\phi}\cot\frac{\theta}{2},\overline{\zeta}=e^{-i\phi}\cot\frac{\theta}{2}$),\on the \mathcal{S}^{2}\ \part,\(labeling the null generators) and with $u$ on the$\ \mathcal{R\ }$part (labeling the cross-sections of $\mathcal{I}^{+}\mathcal{)}$.  The BMS group can be described as coordinate transformations among the coordinates ($u$,$\zeta,\overline{\zeta}).$ In addition to the introduction of null surfaces, Bondi’s other insight was the realization that several components of the asymptotic Weyl tensor could be identified with the * mass/energy* and linear momentum of the *source *and their loss - in analogy, in Maxwell theory, to charge and charge conservation as integrals of the fields at infinity.  These Weyl tensor components come from the harmonic components of the leading coefficients in the $r^{-1}\ ~{}$expansion in the spin-coefficient version of the Weyl tensor and are thus functions just on \mathcal{I}^{+},\i.e., are functions of ($u$,$\zeta,\overline{\zeta}$).

An important idea to recognize is that the leading far-field components of the Weyl tensor (in the spin-coefficient formulation[9]), depend very much on the choice of the tetrad and coordinates systems to be used on \mathcal{I}^{+}.\ \In the past almost all Weyl tensor components were chosen in a Bondi system. However, in the present work, the major new ingredient that leads to our results is the introduction of totally new coordinates and tetrad systems on \mathcal{I}^{+}\and its neighborhood, (very different from Bondi’s) that very closely mimic certain natural coordinate systems on the Minkowski space \mathcal{I}^{+}.\ \This allows us to describe other functions on \mathcal{I}^{+}\(arising from other Weyl  tensor coefficients) that yield - with the Bianchi identities - a variety of additional physical quantities, such as angular momentum, center of mass, its position and velocity, their evolution, as well as force laws and electric and magnetic dipole moments and center of charge. In other words these results and relationships from classical mechanics simply appear as components of the Weyl tensor at infinity.

The prime idea involved in the choice of these systems is use of the special null surfaces that are associated with asymptotically *shear-free null geodesic congruences  - *as directly opposed to Bondi’s null surfaces which do have non-vanishing asymptotic shear - the time-*integral *of the Bondi news function.

These special null surfaces, which are standard and easily understood surfaces in Minkowski space, are of two types. The first are the null cones (with generators automatically shear-free and twist-free) with apex on arbitrary time-like curves - a special case being time-like geodesics. The second type arise (formally) from complex light-cones with apex on complex world-lines[7][10]. In this case, the associated real congruence, though shear-free, is now twisting, They will be first reviewed and described in Sec. II. Sec.III will be devoted to their generalization (in asymptotically flat spaces) to* asymptotically shear-free congruences*, i.e., their definition and construction.  Though at first it appears that there is a serious impediment to their construction, it turns out that by a slight zigzag or maneuver, the impediment can be overcome and the construction can be completed in exact analogy to the flat-space case.  We will have coordinate systems on (asymptotically flat) $\mathcal{I}^{+}$, very closely matching those in Minkowski space. The asymptotic generators (the null geodesics) of the complex surfaces (by construction) will be *real *and asymptotically shear-free but, in general, they will be twisting. In Secs.IV and V we will, by using earlier work, show how the asymptotic Weyl tensor components, expressed in the new coordinates with their associated tetrads, naturally yield a large number of functions determining the interior space time properties as mentioned earlier.  In the discussion, the strange appearance of $H$-space-coordinates is addressed.  We emphasize that though complex ideas are used, we are dealing with real space-times.

For close to 50 years the coordinatization of \mathcal{I}^{+}\by Bondi coordinates has been almost sacrosanct  - nevertheless we present (what we believe is a strong argument) that other choices of coordinates on \mathcal{I}^{+}\have considerable value and their use should be seriously considered.

2 \mathcal{I}^{+}\of Minkowski space

Using standard Minkowski coordinates, x^{a},\with metric \eta_{ab}\and signature (+,-,-,-), the family of null cones with apex at the origin is described parametrically, ($u,r$,$\zeta,\overline{\zeta}$), by

[TABLE]

with \widehat{l}^{a}(\zeta,\overline{\zeta})\a null vector that sweeps out the null cone as \zeta=\cot\frac{\theta}{2}e^{i\phi}\varies over the sphere of null directions, i.e.,

[TABLE]

The r\ \is the affine parameter along the null generators of the cone and u\the time at the spatial origin (or the retarded time on the cone itself).

An alternate interpretation of Eq.(1) is that it is the coordinate transformation between the x^{a}\and the null coordinates ($u,r$,$\zeta,\overline{\zeta}$).  We will refer to the coordinate transformation (and coordinates) given by Eq.(1), as well as its later generalization in Sec.III, as ”static null coordinates”.

Aside:  The full null tetrad, ($\widehat{l},\widehat{n},\widehat{m},\overline{\widehat{m}}$), associated with Eq.(2), is given by

[TABLE]

The metric tensor in these new coordinates, given by

[TABLE]

can be conformally transformed (rescaled) by \Omega^{2}=r^{-2},\leading to

[TABLE]

The surface defined by $\Omega=0,$ a null surface,\is identified as the \mathcal{I}^{+}\of Minkowski space.\It can be thought of as the intersection of the endpoints of the future null cones that have apex on the world-line x^{a}=ut^{a},\with  future null infinity. It is coordinatized by\ (u,\zeta,\overline{\zeta})\and is a special case of Bondi coordinates.

These null coordinates can be generalized to a new set, ($t,r^{\ast},\zeta^{\ast},\overline{\zeta}^{\ast}$), by basing them on null cones with apex on an arbitrary time-like world-line, x^{a}=\xi^{a}(t),\by the   parametric form, ($t,\zeta^{\ast},\overline{\zeta}^{\ast}$), t\a real parameter,

[TABLE]

\widehat{l}^{\ast a}\ \is again a null vector sweeping out the null directions on the cone. By equating the right sides of Eqs.(1) and  (8),\multiplying by the four null tetrad vectors associated with \widehat{l}^{a}(\zeta,\overline{\zeta}),\Eq.(4),  and passing to the limit $r=>\infty,\ ($or \Omega=0)\we find the relationship, (the light-cone cuts)[11][12],

[TABLE]

that describes, in Bondi coordinates, the intersection of the null cones, apex on \xi^{a}(t),\with $\mathcal{I}^{+}$.  Eq.(9), thus defines a one real parameter, t,\family of ’slicings’ of $I^{+}.$

At a point on \mathcal{I}^{+},\ (u,\zeta,\overline{\zeta}),\where the two null vectors \widehat{l}^{\ast a}\and \widehat{l}^{a}\meet, the null angle (on their past light-cone) between them (stated in stereographic coordinates, L\and $\overline{L}\ ),$ is given by

[TABLE]

Asymptotically, the two null vectors (and associated tetrads), \widehat{l}^{\ast a}\and $\widehat{l}^{a},$ are related by

[TABLE]

It is useful to distinguish between the null coordinates based Eq.(1), ($u,r$,$\zeta,\overline{\zeta}$), referred to as static null coordinates and those based on Eq.(8), ($t,r^{\ast},\zeta^{\ast},\overline{\zeta}^{\ast}$),  referred to as dynamic or comoving null coordinates.

An important point for us is to note that the cuts u=G_{F}(t,\zeta,\overline{\zeta})\satisfy the (so-called) flat-space good cut equation,

[TABLE]

namely the condition for the null normals to the ’cuts’ to define null vectors that are *shear-free[7][3][9]. *In the following section dealing with asymptotically flat spaces, this equation will be generalized to the good cut equation,

[TABLE]

where \sigma^{0}(u,\zeta,\overline{\zeta})\is the asymptotic shear (the time integral of the Bondi news function).

Remark 1

For relevance and analogy with the following section, we point out that in this flat space discussion we could have taken the \xi^{a}(\tau)\to be a complex world-line[7][10]. The L=\eth G_{F}\and its complex conjugate, via Eq.(11), would still lead to $\widehat{l}^{\ast a}$ being shear-free but now it would be twisting.  The cuts however would be intrinsically complex and their real parts would have to be chosen - but only after the differentiation.

3 Sec.III, \mathcal{I}^{+}\of Asymptotically Flat Space

Turning from Minkowski space to asymptotically flat spaces, we begin with \mathcal{I}^{+}\constructed from fixed but arbitrary Bondi coordinates, ($u,r,\zeta,\overline{\zeta}$), and Bondi tetrad, (l^{a},m^{a},\overline{m}^{a},n^{a}),\with$\ l^{a}$ tangent to the Bondi null surfaces, m^{a},\overline{m}^{a},\ tangent to the Bondi cuts at \mathcal{I}^{+}\and n^{a}\ \tangent to the \mathcal{I}^{+}\null generators, with n,m,\overline{m}\ parallel propagated down the null geodesics on $u$.  The radiation free-data is

[TABLE]

We now mimic the construction of the shear-free congruences of the previous section.

It is known[7][8] that the general regular solution of Eq.(13) depends on four complex parameters, z^{a},\(defining $H$-space) that can be taken as functions of the complex parameter \tau\ \and written as z^{a}=\xi^{a}(\tau),.\i.e., as an arbitrary (to be determined) complex world-line in $H$-space.\ \The solution (via coordinate conditions on the first four harmonics,$\ l=0,1$) can be written in the form

[TABLE]

with the quadrupole term \xi^{ij},\arising from the data, $\sigma^{0}=\xi^{ij}Y_{ij}^{2}(\zeta,\overline{\zeta})+..$. From the freedom to rescale \tau,\i.e.,\ \tau^{\ast}=F(\tau),\we set \xi^{0}(\tau)=\tau.\ This is the velocity normalization and the slow motion approximation, with\ \xi^{a\ \prime}\equiv v^{a},\ \xi^{0\ \prime}=\sqrt{1+v^{i2}}\approx 1\ .\ \Eq.(14) then becomes

[TABLE]

(The \xi^{ij}\turn out to be the time-derivatives of the gravitational quadrupoles:

[TABLE]

The idea is now to generalize the flat-space cuts, Eq.(9), to a one-parameter family of real cuts in the asymptotically flat situation, via Eq.(16). Unfortunately this does not work immediately since,* in general*, for arbitrary \sigma^{0}(u,\zeta,\overline{\zeta}),\the G^{\ast}(\tau,\zeta,\overline{\zeta})\ \will be complex and there will essentially be no real cuts.  Before we see a way around this problem we mention that IF the \sigma^{0}\ \was of pure electric type[13] then real cuts could be found and the situation would resemble Eq.(9).  For general type of \sigma^{0}\the **Remark **of the previous section becomes relevant, i.e., it is the analogue of the present case.

The way around the problem of the complexity of the cuts, Eq. (14, is the following: treating \tau\as complex, we first construct the null angles,

[TABLE]

to produce an asymptotically shear-free null vector field, l^{\ast a},\via Eq.(11),

[TABLE]

Note: We use\ \overline{L}\ \and do not use $\widetilde{L}=\overline{\eth}G(\xi^{a},\zeta,\overline{\zeta}).$

Next, using

[TABLE]

we decompose G^{\ast}(\tau,\zeta,\overline{\zeta})\into its real and imaginary parts,

[TABLE]

By setting G_{I}=G_{I}(\xi_{R}^{a}(t,\lambda),\xi_{I}^{a}(t,\lambda),\zeta,\overline{\zeta})=0\and  solving it for $\lambda,$ i.e.,

[TABLE]

and substituting \lambda\into G_{R}\ ,\we have the* one real parameter family of real cuts,*

[TABLE]

This construction can be done under fairly general conditions assuming $\partial G_{I}/\partial\lambda\neq 0.$

With the L\and \overline{L},\of Eqs.(17) and (18) evaluated on the real cuts, the associated *real but twisting shear-free null vector field *is,

[TABLE]

We now have the situation in asymptotically flat spaces that is totally analogous to the situation we had in Minkowski space. We refer to the $H$-space coordinates,\ z^{a},\as complex pseudo-Minkowski coordinates, since in the flat-limit they are the complexified Minkowski coordinates.  The \mathcal{I}^{+}\coordinates associated with these pseudo-Minkowski coordinates i.e., the ($\tau,\zeta,\overline{\zeta}$), will be referred to as pseudo-Minkowski cuts.

For the reality considerations, we have two separate cases: (1)  the comoving choice of an arbitrary \xi^{a}(\tau),\in Eq.(16),  to be determined by choosing it to be the complex center of mass or (2) by the choice of the static pseudo-Lorentzian \mathcal{I}^{\mathcal{+}}\coordinates, i.e., by taking (a ‘straight’ $H$-space world-line) \xi^{a}=\widehat{\tau}\delta_{0}^{a},\for the cuts:

[TABLE]

If we assume that both \lambda\is small and the slow motion approximation, these constructions can be explicitly carried out.  They lead, via the assumed form of the asymptotic shear,

[TABLE]

to the linearized expressions (that we need),

[TABLE]

[TABLE]

4 Review and Further Developments

As mentioned earlier, much of the material described here will involve functions or structures that ‘live’ on $\mathcal{I}^{+}$  and originate with the leading Weyl and Maxwell tensor components. Our major interest will center on the asymptotic behavior, the physical meaning, the evolution and transformation properties of these tensors.  Using Bondi coordinates and tetrad, the five complex self-dual NP components of the Weyl tensor and three complex Maxwell components are[4]:

[TABLE]

[TABLE]

From the radial asymptotic Bianchi identities and Maxwell equations, we have their asymptotic behavior (the ’peeling’ theorem)[4]:

[TABLE]

[TABLE]

with

[TABLE]

The non-radial Bianchi Identities and Maxwell equations yield the evolution equations for these leading terms (our basic variables):

[TABLE]

[TABLE]

The u-derivative is denoted by the overdot.

After the final coordinate transformation to the static pseudo-Minkowski coordinates and static pseudo-Minkowski cuts, the Eqs. (39-44) are seen to contain our classical (mechanical) equations of motion.

The quantity \sigma^{0}(u,\zeta,\overline{\zeta})\(referred earlier to as the asymptotic shear), is the leading term in the shear of the geodesic congruence, $l^{a};$ i.e.$.,$

[TABLE]

while its first $u$-derivative  is the Bondi news function.  We consider \sigma^{0}(u,\zeta,\overline{\zeta})\as a free function but take it only up to the quadrupole terms, Eq.(25). It, as such, plays a significant role in what later follows. From the spin-coefficient equations one finds that

Body Math

[TABLE]

4.1 Physical Identifications

From the definition of the mass aspect, ${\large\Psi,}$ (real from field equations) by

[TABLE]

Bondi defined the asymptotic mass, $M_{B},$ and 3-momentum, P_{B}^{i}\ \as the\ l=0\&\ l=1\harmonic coefficients of \Psi.\Specifically,

Definition 1

[TABLE]

By rewriting Eq.(39), replacing the $\Psi_{2}^{0\,}$ by $\Psi$ via Eq.(46), we have

[TABLE]

Immediately we have the Bondi mass/energy loss theorem:

[TABLE]

the integral taken over the unit 2-sphere at constant $\ u$.  This relationship is at the basis of almost all the contemporary work on the detection of gravitational radiation.

**Definition  2   **Though it has been a controversial subject and there is no general agreement, we adopt the definition (which comes from linear theory) of the complex mass dipole moment, $(D_{(complex)}^{i}=D_{(mass)}^{i}+ic^{-1}J^{i})$, as the l=1\harmonic component of \Psi_{1}^{0},\

[TABLE]

D^{i}\the mass dipole and J^{i},\the total angular momentum, as ”seen” at null infinity.  The main defense of this definition is that it works extremely well.

**Definition  3  **Our identification - which is standard - for the complex E&M dipole, (electric and magnetic dipoles, $(D_{Elec}^{i}+iD_{Mag})$) as the $l=1$ harmonic component of \phi_{0}^{0}\is:

[TABLE]

Later we will connect these three physical identifications with the complex center of mass.*  *

Actually, for the general situation there is the independent complex center of charge. Here, however for simplicity, we assume that they coincide.  This is not necessary but is a simplifying restriction.[7]

**Comment. **For later use we note that from the asymptotic Maxwell equations, Eqs.(43) and (44), we have that[7]

[TABLE]

with the Q\s representing known quadrupole terms.

**Comment. **The indices 0,i,j,k…. have direct geometric meaning coming from the position or tangent vectors in $H$-space. They also can be interpreted as vectors in a representation space of the Lorentz group that has its origin via a subgroup of the BMS group acting on $\mathcal{I}^{+}$.[7]

4.2 Modus Operandi

Our operation now consists of taking the Weyl tensor components (mainly $\Psi_{1}^{0\,}$and $\Psi_{2}^{0\,}$) and transferring them from the Bondi tetrad ($l^{a},\overline{m}^{a},m^{a},n^{a}$) to the pseudo-Minkowski tetrad ($l^{\ast a},m^{\ast a},\overline{m}^{\ast a},n^{\ast a}$) (actually doing this two times) via

[TABLE]

and

[TABLE]

The L\and its complex conjugate, $\overline{L},$ are determined by Eq.(17) with G\given by, Eq.(16)

[TABLE]

We perform this tetrad rotation twice: first with the dynamic pseudo-Minkowski coordinates, the \xi^{a}(\tau)\to be determined by a center of mass condition and then again for the static pseudo-Minkowski coordinates, \xi^{a}=\tau\delta_{0}^{a}\done to put the final results in a pseudo-Lorentzian frame.

In addition, in each case, we must change the I^{+}\coordinates from Bondi to the ($\tau,\zeta,\overline{\zeta}$) slicing via Eq.(14).

Aside: Eventually the ($\tau,\zeta,\overline{\zeta}$)* will be changed to the real I^{+}\coordinates ($t,\zeta,\overline{\zeta}$).*

These transformations will be implemented in several stages.

Stage I. We go from the Bondi coordinates to the pseudo-Minkowski situation via Eq.(16). After both the tetrad and coordinate change, for constant \tau,\we concentrate on the l=1\ \component, i.e., \Psi_{1}^{\ast 0i}\of \Psi_{1}^{\ast 0}.\Using  the definition of the complex mass dipole moment, the complex center of mass ‘position’ is determined (defined) by setting the $\Psi_{1}^{\ast 0i}$ equal to zero. This allows us to determine the three components of \xi^{i},\(the center of mass),\with$\ \xi^{0}=\tau.$

**Stage II. With the \xi^{a}(\tau)\**now known we can go back and find the l=1\spherical harmonic  component of the Bondi $\Psi_{1}^{0i}(u)$ in terms of the complex center of mass. Using, in the Bondi frame, the evolution equations, Eqs.(39)-(44), with the just found center of mass position, \xi^{a}(u),\many of our mechanical equations are obtained but now expressed as functions of our $u$.

**Stage III. With all these relations, including the complex center of mass and the dynamics - expressed in the Bondi frame - we do the transformation back to the pseudo-Minkowski system but now to the static frame to obtain our final results. **The ”static frame” mimics the ordinary flat-space Lorentzian frame.

The calculations involved in these three stages were rather involved. They included the coordinate and tetrad transformations between the Bondi frame and the pseudo-Minkowski frames several times, they often involved Taylor expansions up to the quadrupole terms and the frequent use of Clebsch-Gordon expansion of the spherical harmonics products.  Since much of this has been completed and appeared in published and refereed literature,[7], we will not redo them but simply refer to these results for our present use.

5  Results

5.1 **Stage I **

We begin with an asymptotically flat space-time in a Bondi coordinate and tetrad system (previous section) and perform the null rotation, Eq.(55), to the shear-free null vector field

[TABLE]

with $L=\eth G(\xi^{a}(\tau),\zeta,\overline{\zeta}),$ and its complex conjugate,\ \overline{L}.\The G\ \is given by Eq.(61),

[TABLE]

with \xi^{a}(\tau)\ \an unknown world-line in $H$-space - to be determined. The relevant (for us) Weyl tensor component, \Psi_{1}^{0},\transforms, Eq.(57), as

[TABLE]

Considering $L$ and \sigma^{0}\as first order and M_{B}\in Eq.(46), as zero order, with the Mass Aspect $\Psi,$

[TABLE]

we have

[TABLE]

Our procedure for finding the complex center of mass now centers on Eq.(66). The right-side, which is a function of both u\and $\tau,$ is transformed to a function of only \tau\via Eq.(63).    All the variables on the right side are then expanded in spherical harmonics and simplified by Clebsch-Gordon expansions. We separate out and set to zero the l=1\harmonics on the right side, i.e., we force $\Psi_{1}^{\ast 0i}$ =0\for constant $\tau$ slices. This is a lengthy and difficult task and approximations are needed.

From this result there are two things that could be done: (1) from the three $l=1$ terms we could solve for the three \xi^{i}(\tau)\or, as we do, (2) go backwards and solve for the original \Psi_{1}^{0i}\but now in terms of the \xi^{i}(\tau).\

After considerable work we have \Psi_{1}^{0}\ ,\found  first as a function of \tau\and then finally, via the (approximate) inverse function to Eq.(63), i.e.,

[TABLE]

then expressed as a function of u.\After its spherical harmonic expansion, we finally have the $l=1$ harmonic coefficient,$\ \Psi_{1}^{0i}$  - (needed for our definition 2, Eq.(51), of the complex mass dipole), namely

[TABLE]

Considering that many of the quadrupole terms involve high powers of\ c^{-1}\and we can consider quadrupole-quadrupole interactions as weak, we approximate \Psi_{1}^{0i}\simply  as

[TABLE]

5.2 Stage 2

Equating our definition 2

[TABLE]

of the complex mass dipole with Eq.(68), using

[TABLE]

we obtain our

Result:1

[TABLE]

or

[TABLE]

The first term in D_{(mass)}^{i}\is the standard dipole definition while the second term is identical to a dipole term in the relativistic angular-momentum tensor[14][16].

The expression for angular momentum has (1), the intrinsic spin $\overrightarrow{S}$ (same as for the Kerr metric)[7] and (2) the conventional orbital angular momentum term.

We find the details, arising from the definitions 1 and 2 and the Einstein Equations, to be rather remarkable - and strange.  Note that the z^{a}\are $H$-space coordinates.

Continuing, we substitute Eq.(68) into the evolutionary Bianchi Identity, Eq.(40),

[TABLE]

and using definition 1 and Eq.(54), we find, directly from the real part (with no manipulation), that

Result: 2

[TABLE]

The prime denotes the u-derivative.

We obtain, for the Bondi momentum, the familiar kinematic $M_{B}\overrightarrow{v}$ term and a term familiar from electrodynamics, the radiation reaction contribution to the linear momentum.

From the imaginary part of the Bianchi Identity we have the momentum loss equation;

Result: 3

[TABLE]

There are several things to note: (1) the first term on the right side can be moved to the left, simply changing the definition of J^{i}\ \to

[TABLE]

i.e., adding an electromagnetic part to the spin term S^{i},\and  (2) Landau & Lifschitz [15] have a special case of Eq.(79). They omit the \xi_{I}^{i}\terms.

Finally substituting the relevant terms into the evolutionary Bianchi Identity, Eq.(39),

[TABLE]

Result: 4

[TABLE]

The first term is the standard Bondi quadrupole mass loss (now including the spin quadrupole contribution to the loss - maybe new), the second and third terms are the classical E&M dipole and quadrupole energy loss - including the correct numerical factors.

The l=1\terms lead to the momentum loss expression,

Result: 5

[TABLE]

where F_{recoil}^{i}\is composed of many non-linear radiation terms involving the time derivatives of the gravitational quadrupole and the E&M dipole and quadrupole moments. These terms are known and given[7] but not relevant to us now.  Instead we substitute Eq.(77) into Eq.(83) leading to Newton’s second law;

[TABLE]

Result: 6

We find this rather astonishing - there is exactly the classical mechanics standard rocket mass loss expression and the classical radiation reaction term.  The last term is just the momentum recoil force - also known explicitly.

In the context of these results we mention two further automatic results:

1. From earlier results,

[TABLE]

and the classical classical gyromagnetic ratio \gamma=\frac{q}{2Mc}g,\we see

[TABLE]

and discover the Dirac value of the $g$-factor to be $g$=$2$.

2. In classical relativistic mechanics [16] one has the definition of the relativistic angular momentum tensor, $M^{ab}$,

[TABLE]

so that

[TABLE]

Using our results, S^{i}=cS^{\ast i}=Mc\xi_{I}^{i},\ S_{0}=0,\ V_{0}\sim 1,\ V_{k}\sim 0\and multiplying by \epsilon_{ijk},\we have agreement with our Eq.(71).

Then from

[TABLE]

we have agreement with our Eq.(70), i.e., with our \overrightarrow{P}\mathrm{x}\overrightarrow{S}\term.

The relativistic angular momentum tensor (unrelated to physical space-time) is sitting quietly and unobserved in our Weyl tensor.

5.3 Stage 3

Going from the Bondi slicings of stage 2 to the ”static” real pseudo-Minkowski slicing of stage 3 is easy. From Eq.(30), we see that the l=1\ harmonics are now missing

[TABLE]

From our approximations, i.e., disregarding the quadrupoles, we can simply replace our u\ with \widehat{t},\so that all our **Results, **1-6, remain unchanged. We emphasize that these Results are for real time and real coordinates even though most of the calculations involved complex variables.

6 VI. Discussion

Several Remarks are in order.

1. The present work is the third in a developing series of papers dealing with unusual structures hiding on \mathcal{I}^{+}.\In the earlier work[7][14] the coordinates as well as the time (for the classical equations of mechanics), were, in general complex - here they are real.  Furthermore the \mathcal{I}^{+}\cuts were complex and were open to the perplexing question ”What was the significance of the complex Bondi slicing? Why was their use important?” Here we have the answer. They were not important. The important observation is that here we are using slicings (pseudo-Minkowskian) that mimic (or are as close to) real flat-Lorentzian slicing as possible - but give the same results as in the Bondi slicing. This does clear away one of the enigmas associated with the earlier work[14] - why was Bondi slicing important?

2. It is the collection of results, Eqs. (70), (71), (77),(79),(82) and (84), mimicking or imitating classical mechanics, that constitute our main contribution - in conjunction with our novel pseudo-Minkowskian slicing. Nevertheless there still remains the major enigma, ”Why and how is it that the (now) real $H$-space coordinates play so accurately the role of space-time coordinates. Though they appear to be space-time equations of motion there is no physical space-time that is associated with the equations. They (as we said earlier) take place on the strange $H$-space.  Is this just a giant coincidence? We find that difficult to believe.  What possible meaning can one give to them - and to the $H$-space?  What is the  physical meaning of the decomposition, \xi^{i}(\tau)=\xi_{R}^{i}(\tau)+i\xi_{I}^{i}(\tau).\Why should the imaginary part of the $H$-space coordinate be related to spin? Is there something here of real significance? We do not know - but it is suggestive.

We also observe the curious result - in our construction - that when we have spin-angular momentum, we also have the case of the twisting sear-free congruence. Twist seems happily to be related to spin.

3. There is the suggestion that $H$-space might play the role of some sort of observation or optical image space. Due to curvature affects one can not look straight back and expect to see distant objects along the line of sight. Is this a manifestation of this image forming? This is pure speculation and not at all clear.

4. Among the 6 Results of the previous section, all but Result #3 are identical to standard classical (mechanical and E&M) equations. The special case of Result #3 where spin is omitted is given in reference ([15]). We have not been able to find any reference that contains our complete result.  A related question is - if our conservation of angular momentum result is new, can we consider it to be a prediction.

5. We point out the utter simplicity of the origin of the radiation reaction term in the expression for $P_{B}^{i}$.  In Eq.(40)

[TABLE]

the$\ P_{B}^{i}$ , is sitting in the \eth\Psi_{2}^{0\,},\the q\sits in the $\phi_{1}^{0}$ , while \xi^{i\ \prime\prime}\ \is in the $\overline{\phi}_{2}^{0}.$  The numerical factors are there. The radiation reaction term is just sitting there - no assumptions, no derivation, it is just waiting to be observed in the equation.

6.  We emphasize several things;

a. We are using only the standard Einstein equations coupled in the standard way to the standard Maxwell equations. Furthermore we are using the standard asymptotically flat solutions of Bondi, Sachs, Penrose and Newman-Unti - with nothing further added.  We note that in this standard framework we search for shear-free null geodesic congruences via the so-called ”Good Cut Equation” - our main research tool.  From the null surfaces associated with these congruences and their intersections with the asymptotically flat $\mathcal{I}^{+}$, we construct -(mimicking the virtually identical flat-space construction) the one-parameter family of real cuts.

b. On these cuts, using the tetrad adapted to the cuts (different from the Bondi tetrad), we defined the complex center of mass in terms of a Weyl tensor component.  From that definition and that of the Bondi mass and momentum, all of our Results followed.  No formal lengthy derivations were needed; the Results were just sitting there in the Weyl tensor and Bianchi Identities.

c. The subtlety of some of the results were rather surprising: e.g., finding the rocket force, the Abraham-Lorentz radiation reaction force as well as the electromagnetic energy loss from both the dipole and quadrupole radiation (both of electric and magnetic type, and all with correct coefficients) and the angular momentum loss.  Some of the results clearly arise from the inclusion of the Maxwell stress-energy tensor in the Einstein-Maxwell equations. There however was no straight matter stress tensor used.

7. Though it is not obvious where to go next, we notice that there is a chance to expand on Penrose’s Asymptotic Twistor theory via twistor curves on the new family of shear-free cuts of  \mathcal{I}^{+}.\In the past they were defined on the associated $H$-space.  Now it could possibly be done on the physical space itself. This must be investigated.

8. The natural question also arises - can our method of obtaining equations of motion be extended and applied to a two-body problem? At the moment we see possibilities but not good ones.

9.  We have not yet explored how the BMS group interacts with the present results - via representation theory.

10. And finally we fully acknowledge that we have no good idea for the physical meaning of the $H$-space and its coordinates – why their real parts should be appropriate for the description of the center-of-mass motion or why their imaginary parts are associated with spin? Does the $H$-space metric play any role? Is all this an empty accident? or is there something profound? It is sufficiently crazy and far-off the mainstream that it might well be profound.  In this context, we note that instead of looking at the leading Weyl tensor terms, we could - getting the same results - just as well used weighted integrals (powers of r\and spherical harmonics) over the ”shear-free” spheres (cuts) at \mathcal{I}^{+}\that so resemble Minkowski space light-cone cuts.  This seems to help pick out our Lorentzian-like mechnical results.

7 VII. Acknowledgements

We gratefully thank Timothy Adamo for both a careful critical reading of the manuscript, for hours of enlightening discussions and collaboration on an earlier manuscript[7] where many of the present ideas were developed. We are deeply indebted to Roger Penrose for years of friendship and patient explanations and encouragement on exploring the present ideas.

8 BIBLIOGRAPHY