Gauge Fixity Towards Hyperbolicity: General Theory of Relativity and Superpotentials. Part 1.

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Gravitational field is described by a pseudo-Riemannian metric g (with Lorentzian signature (1, m-1)) over the spacetime M of dimension dim(M) = m; in standard General Relativity, m = 4. The configuration bundle is thence the bundle of Lorentzian metrics over M, denoted by Lor(M) . The Lagrangian is second order and it is usually chosen to be the so-called Hilbert Lagrangian:

LH: J2Lor(m) → ∧om(M)

LH: LH(gαβ, Rαβ)ds = 1/2κ (R – 2∧)√g ds —– (1)

where

R = gαβ Rαβ denotes the scalar curvature, √g the square root of the absolute value of the metric determinant and ∧ is a real constant (called the cosmological constant). The coupling constant 1/2κ which is completely irrelevant until the gravitational field is not coupled to some other field, depends on conventions; in natural units, i.e. c = 1, h = 1, G = 1, dimension 4 and signature ( + , – , – , – ) one has κ = – 8π.

Field equations are the well known Einstein equations with cosmological constant

Rαβ – 1/2 Rgαβ = -∧gαβ —— (2)

Lagrangian momenta is defined by:

pαβ = ∂LH/∂gαβ = 1/2κ (Rαβ – 1/2(R – 2∧)gαβ)√g

Pαβ = ∂LH/∂Rαβ = 1/2κ gαβ√g —– (3)

Thus the covariance identity is the following:

dα(LHξα) = pαβ£ξgαβ + Pαβ£ξRαβ —– (4)

or equivalently,

α(LHξα) = pαβ£ξgαβ + PαβεξΓεαβ – δεβ£ξΓλαλ) —– (5)

where ∇ε denotes the covariant derivative with respect to the Levi-Civita connection of g. Thence we have a weak conservation law for the Hilbert Lagrangian

Div ε(LH, ξ) = W(LH, ξ) —– (6)

Conserved currents and work forms have respectively the following expressions:

ε(LH, ξ) = [Pαβ£ξΓεαβ – Pαε£ξΓλαλ – LHξε]dsε = √g/2κ(gαβgεσ – gσβgεα) ∇α£ξgβσdsε – √g/2κξεRdsε = √g/2κ[(3/2Rαλ – (R – 2∧)δαλλ + (gβγδαλ – gα(γδβ)λβγξλ]dsα —– (7)

W(LH, ξ) = √g/κ(Rαβ – 1/2(R – 2∧)gαβ)∇(αξβ)ds —– (8)

As any other natural theory, General Relativity allows superpotentials. In fact, the current can be recast into the form:

ε(LH, ξ) = ε'(LH, ξ) + Div U(LH, ξ) —– (9)

where we set

ε'(LH, ξ) = √g/κ(Rαβ – 1/2(R – 2∧)δαββ)dsα

U(LH, ξ) = 1/2κ ∇[βξα] √gdsαβ —– (10)

The superpotential (10) generalizes to an arbitrary vector field ξ, the well known Komar superpotential which is originally derived for timelike Killing vectors. Whenever spacetime is assumed to be asymptotically fiat, then the superpotential of Komar is known to produce upon integration at spatial infinity ∞ the correct value for angular momentum (e.g. for Kerr-Newman solutions) but just one half of the expected value of the mass. The classical prescriptions are in fact:

m = 2∫ U(LH, ∂t, g)

J = ∫ U(LH, ∂φ, g) —– (11)

For an asymptotically flat solution (e.g. the Kerr-Newman black hole solution) m coincides with the so-called ADM mass and J is the so-called (ADM) angular momentum. For the Kerr-Newman solution in polar coordinates (t, r, θ, φ) the vector fields ∂t and ∂φ are the Killing vectors which generate stationarity and axial symmetry, respectively. Thence, according to this prescription, U(LH, ∂φ) is the superpotential for J while 2U(LH, ∂t) is the superpotential for m. This is known as the anomalous factor problem for the Komar potential. To obtain the expected values for all conserved quantities from the same superpotential, one has to correct the superpotential (10) by some ad hoc additional boundary term. Equivalently and alternatively, one can deduce a corrected superpotential as the canonical superpotential for a corrected Lagrangian, which is in fact the first order Lagrangian for standard General Relativity. This can be done covariantly, provided that one introduces an extra connection Γ’αβμ. The need of a reference connection Γ’ should be also motivated by physical considerations, according to which the conserved quantities have no absolute meaning but they are intrinsically relative to an arbitrarily fixed vacuum level. The simplest choice consists, in fact, in fixing a background metric g (not necessarily of the correct Lorentzian signature) and assuming Γ’ to be the Levi-Civita connection of g. This is rather similar to the gauge fixing à la Hawking which allows to show that Einstein equations form in fact an essentially hyperbolic PDE system. Nothing prevents, however, from taking Γ’ to be any (in principle torsionless) connection on spacetime; also this corresponds to a gauge fixing towards hyperbolicity.

Now, using the term background for a field which enters a field theory in the same way as the metric enters Yang-Mills theory, we see that the background has to be fixed once for all and thence preserved, e.g. by symmetries and deformations. A background has no field equations since deformations fix it; it eventually destroys the naturality of a theory, since fixing the background results in allowing a smaller group of symmetries G ⊂ Diff(M). Accordingly, in truly natural field theories one should not consider background fields either if they are endowed with a physical meaning (as the metric in Yang-Mills theory does) or if they are not.

On the contrary we shall use the expression reference or reference background to denote an extra dynamical field which is not endowed with a direct physical meaning. As long as variational calculus is concerned, reference backgrounds behave in exactly the same way as other dynamical fields do. They obey field equations and they can be dragged along deformations and symmetries. It is important to stress that such a behavior has nothing to do with a direct physical meaning: even if a reference background obeys field equations this does not mean that it is observable, i.e. it can be measured in a laboratory. Of course, not any dynamical field can be treated as a reference background in the above sense. The Lagrangian has in fact to depend on reference backgrounds in a quite peculiar way, so that a reference background cannot interact with any other physical field, otherwise its effect would be observable in a laboratory….

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