7 Gauge (BMS) Invariance

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The issue of gauge invariance, the understanding of which is not obvious or easy, must now be addressed. The claim is that the work described here is in fact gauge (or BMS) invariant.

First of all we have, ℑ+ℂ, or its real part, ℑ+. On ℑ+ℂ, for each choice of spacetime interior and solution of the Einstein–Maxwell equations, we have its UCF, either in its complex version, &tidle; uB = X (τ,ζ,ζ), or its real version, Equation (291View Equation). The geometric picture of the UCF is a one-parameter family of slicings (complex or real) of + ℑ ℂ or + ℑ. This is a geometric construct that has a different appearance or description in different Bondi coordinate systems. It is this difference that we must investigate. We concentrate on the complex version.

Under the action of the supertranslation, Equation (67View Equation), we have:

&tidle; ^ &tidle; &tidle; &tidle; X (τ,ζ,ζ) → X (τ, ζ,ζ) = X (τ,ζ,ζ) + α(ζ,ζ), (332 )
with α (ζ, &tidle;ζ) an arbitrary complex smooth function on (complexified) S2. Its effect is to add on a constant to each spherical harmonic coefficient. The special case of translations, with
a α (ζ, &tidle;ζ) = d ˆla(ζ, &tidle;ζ), (333 )
simply adds to the l = (0,1) harmonic components the complex constants da, so, via Equation (261View Equation), we have the (complex) Poincaré translations,
ξa → ^ξa = ξa + da. (334 )

The action of the homogeneous Lorentz transformations, Equation (68View Equation),

^uB = KuB, (335 ) -- K = --------------1-+-ζζ--------------, (336 ) (aζ + b)(aζ + b) + (cζ + d)(cζ + d) aζ + b ^ζ = ------; ad − bc = 1. (337 ) cζ + d iλ cζ-+-d- e = cζ-+ d- (338 )
is considerably more complicated. It leads to
^ X^(τ, ^ζ,&tidle;ζ) = KX (τ,ζ,ζ&tidle;). (339 )

Before discussing the relevant effects of the Lorentz transformations on our considerations we first digress and describe an important technical issue concerning representation of the homogeneous Lorentz group.

The representation theory of the Lorentz group, developed and described by Gelfand, Graev and Vilenkin [16Jump To The Next Citation Point] used homogeneous functions of two complex variables (homogeneous of degrees, n − 1 1 and n2 − 1) as the representation space. Here we summarize these ideas via an equivalent method [21Jump To The Next Citation Point15] using spin-weighted functions on the sphere as the representation spaces. In the notation of Gelfand, Graev and Vilenkin, representations are labeled by the two numbers (n1,n2) or by (s,w ), with (n1,n2 ) = (w − s + 1,w + s + 1). The ‘s’ is the same ‘s’ as in the spin weighted functions and ‘w’ is the conformal weight [44] (sometimes called ‘boost weight’). The different representations are written as D (n1,n2). The special case of irreducible unitary representations, which occur when (n1, n2) are not integers, play no role for us and will not be discussed. We consider only the case when (n1,n2) are integers so that the (s,w ) take integer or half integer values. If n1 and n2 are both positive integers or both negative integers, we have, respectively, the positive or negative integer representations. The representation space, for each (s,w ), are the functions on the sphere, -- η(s,w)(ζ,ζ), that can be expanded in spin-weighted spherical harmonics, -- sYlm (ζ, ζ), so that

∞ -- ∑ -- η(s,w)(ζ,ζ) = η(lm) sYlm(ζ,ζ). (340 ) l=s

Under the action of the Lorentz group, Equation (335View Equation), they transform as

-- isλ w -- ^η(s,w)(^ζ,^ζ) = e K η(s,w)(ζ,ζ ). (341 )
These representations, in general, are neither irreducible nor totally reducible. For us the important point is that many of these representations possess an invariant finite-dimensional subspace which (often) corresponds to the usual finite dimensional tensor representation space. Under the transformation, Equation (341View Equation), the finite number of coefficients in these subspaces transform among themselves. It is this fact which we heavily utilize. More specifically we have two related situations: (1) when the (n1,n2 ) are both positive integers, (or w ≥ |s|), there will be finite dimensional invariant subspaces, + D (n1,n2), which are spanned by the basis vectors -- sYlm(ζ,ζ), with l given in the range, |s| ≤ l ≤ w. All the finite dimensional representations can be obtained in this manner. And (2) when the (− n1,− n2) are both negative integers (i.e., we have a negative integer representation) there will be an infinite dimensional invariant subspace, D − (−n1,−n2), described elsewhere [21Jump To The Next Citation Point]. One, however, can obtain a finite dimensional representation for each negative integer case by the following construction: One forms the factor space, D (−n1,− n2)∕D −(−n1,−n2). This space is isomorphic to one of the finite dimensional spaces associated with the positive integers. The explicit form of the isomorphism, which is not needed here, is given in Held et al. [21Jump To The Next Citation Point16].

Of major interest for us is not so-much the invariant subspaces but instead their interactions with their compliments (the full vector space modulo the invariant subspace). Under the action of the Lorentz transformations applied to a general vector in the representation space, the components of the invariant subspaces remain in the invariant subspace but in addition components of the complement move into the invariant subspace. On the other hand, the components of the invariant subspaces do not move into the complement subspace: the transformed components of the compliment involve only the original compliment components. The transformation thus has a non-trival Jordan form.

Rather than give the full description of these invariant subspaces we confine ourselves to the few cases of relevance to us.

I. Though our interest is primarily in the negative integer representations, we first address the positive integer case of the s = 0 and w = 1, [(n1,n2 ) = (2,2 )], representation. The harmonics, l = (0,1 ) form the invariant subspace. The cut function, -- X (τ,ζ,ζ), for each fix values of τ, lies in this space.

We write the GCF as

a -- u = X (ξ (τ),ζ,ζ), (342 ) aˆ -- ∑ lm a -- = ξ la(ζ, ζ) + H (ξ )Ylm (ζ,ζ). l,|m|≤l

After the Lorentz transformation, the geometric slicings have not changed but their description in terms of -- (u,ζ,ζ) has changed to that of ′ ′-′ (u ,ζ,ζ ). This leads to

u′ = KX, (343 ) = X ′, -′ ∑ --′ = ξ ′aˆl′a(ζ ′,ζ ) + H ′lm(ξ′a)Ylm (ζ′,ζ ). l,|m|≤l

Using the transformation properties of the invariant subspace and its compliment we see that the coordinate transformation must have the form;

′a a b lm b ξ = F (ξ,H (ξ ),...), (344 )
in other words it moves the higher harmonic coefficients down to the l = (0,1) coefficients. The higher harmonic coefficients transform among themselves;
H ′lm(ξ′a) = F lm(...,Hl ′m ′(ξb),...). (345 )

Treating the ξaand ξ′a as functions of τ, we have

a aˆ a ∑ lm a -- V = ∂τX = v X,a = v la + v H ,a (ξ )Ylm (ζ,ζ), (346 ) l,|m|≤l dξa dξ′a dξb va = ---, v′a = ----= Fa, b----= F a,b vb, ′ d τ dτ dτ V = KV,( ) ∑ -- = v′a ( ˆl′ + H ′lm, Y (ζ′,ζ′)) , a a lm ( l,|m |≤l ) ∑ -- = vbFa, b( ˆl′+ H ′lm, aYlm (ζ′,ζ′)) . a l,|m |≤l
Our ℌ-space coordinates, za = ξa, and their τ-derivatives, va, are the coefficients of the l = (0,1) harmonics in respectively the X and the V expansions. A Lorentz transformation induces a specific coordinate transformation and associated vector transformation on these coefficients.

II. A second important example concerns the mass aspect, (where we have introduced the 0 Y 0 for simplicity of treatment of numerical factors)

0 0 i 0 ij 0 Ψ ≡ Ψ(0,−3) = Ψ Y0 + Ψ Y 1i + Ψ Y2ij + ... (347 )
Ψ is an s = 0 and w = − 3, [(n1, n2) = (− 2, − 2)] quantity. The invariant subspace has a basis set of the harmonics with l ≥ 2. The factor space is isomorphic to the finite dimensional positive integer space, [(n1,n2 ) = (2,2 )] and hence the harmonic coefficients of l = (0,1) lie in this finite dimensional representation space. From this isomorphism we know that functions of the form, Ψ0Y 00+ ΨiY 01i, have four coefficients proportional to the Bondi four-momentum
P a = (M c,P i), (348 )
a Lorentzian four-vector. Note that we have rescaled the 0 Ψ in Equation (347View Equation) by the 0 Y0, differing from that of Equation (63View Equation) in order to give the spherical harmonic coefficients immediate physical meaning without the use of the factors in equations Equation (64View Equation) and Equation (65View Equation).

This is the justification for calling the l = (0,1) harmonics of the mass aspect a Lorentzian four-vector. Technically, the Bondi four-momentum is a co-vector but we have allowed ourselves a slight notational irregularity.

III. The Weyl tensor component, ψ01, has s = 1 and w = − 3, [(n1, n2) = (− 3,− 1)]. The associated finite dimensional factor space is isomorphic to the finite part of the s = − 1, w = 1,[(n1,n2) = (3,1)] representation. We have that

0 0 0 1 0 1 ψ1 ≡ ψ 1(1,− 3) = ψ 1iY1i + ψ 1ijY2ij + ... (349 )
leads to the finite-dimensional representation space
finite space = ψ01iY1−i1. (350 )

The question of what finite tensor transformation this corresponds to is slightly more complicated than that of the previous examples of Lorentzian vectors. In fact, it corresponds to the Lorentz transformations applied to (complex) self-dual antisymmetric two-index tensors [29Jump To The Next Citation Point]. We clarify this with an example from Maxwell theory: from a given E and B, the Maxwell tensor, Fab, and then its self-dual version can be constructed:

ab+ ab ∗ab W = F + iF .

A Lorentz transformation applied to the tensor, ab+ W, is equivalent [30] to the same transformation applied to

0 i ψ 1 ↔ (E + iB )i. (351 )

These observations allow us to assign Lorentzian invariant physical meaning to our identifications of the Bondi momentum, a P and the complex mass dipole moment and angular momentum vector, DiMass Dipole + iJi.

IV. Our last example is a general discussion of how to construct Lorentzian invariants from the representation spaces. Though we will confine our remarks to just the cases of s = 0, it is easy to extend them to non-vanishing s by having the two functions have respectively spin-weight s and − s.

Consider pairs of conformally weighted functions (s = 0), W (w),G (−w−2), with weights respectively, (w, − w − 2). They are considered to be in dual spaces. Our claim is that the integrals of the form

∫ I = G (− w−2)W (w)dΩ (352 )
are Lorentz invariants.

We first point out that under the fractional linear transformation, ′ ζ ↔ ζ, Equation (337View Equation), the area element on the sphere

-- 4idζ∧d ζ d Ω = -------2- (353 ) (1 + ζζ )
transforms as [21Jump To The Next Citation Point]
dΩ′ = K2d Ω. (354 )

This leads immediately to

∫ ′ ′ ′ I = G (−w− 2)W (w)dΩ (355 ) ∫ = K −w−2G (−w−2)KwW (w)K2d Ω, ∫ = G (−w− 2)W (w)dΩ,
the claimed result.

There are several immediate simple applications of Equation (352View Equation). By choosing an arbitrary w = − 2,s = 0 function, say -- G (−2)(ζ,ζ ), we immediately have a Lorentzian scalar,

∫ ∫ N ≡ G(−2)dΩ = G ′(−2)d Ω′. (356 )

If this is made more specific by chosing G (−2) = V −2, we have the remarkable result (proved in Appendix D) that this scalar yields the ℋ-space metric via

∫ 8π(gabvavb)−1 ≡ V −2dΩ. (357 )

A simple variant of this arises by taking the derivative of (357View Equation) with respect to va,and multiplying by an arbitrary vector, a w leading to

∫ 8πwava (gcdvcvd)−2 ≡ V −3waZ,a dΩ. (358 )

Many other versions can easily be found.

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