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About symmetries once again

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I have been analyzing the basic visions of TGD trying to identify weak points. Symmetries are central in TGD. The basic motivation for TGD was the loss of Poincare invariance in GRT and in the following I will analyze the claim of Phil Gibbs that one obtains energy conservation in GRT. Quite recently, I learned that the generalized holomorphies of space-time surfaces define non-trivial conserved charges. Also a generalization of Super-Kac-Moody charges associated with certain embedding space isometries emerges. This suggests a very close connection with string models and provides a possibility to provide answers to the longstanding questions relating to the identification of the isometry group of the “world of classical worlds” (WCW).

About the proposal of Phil Gibbs

The following considerations are inspired by discussions with Marko Manninen and related to whether in general relativity it could be possible to define conserved quantities associated with at least some general coordinate transformations as proposed by Phil Gibbs (see this). This is certainly in conflict with the general idea that the choice of coordinates cannot have any physical effect and personally I am skeptic. I however decided to analyze the proposal in detail and found that it relates to a possible generalization of the notion of Newtonian gravitational flux, which gives the gravitational mass of the system.

  1. Gibbs’ proposal for Noether charges associated with general coordinate transformations says nothing detailed about charges and the straightforward application of the basic formula gives vanishing charges since these currents turn out to be proportional to T-G which vanishes by Einstein’s equations. However, the action includes a term containing second derivatives of the metric. Could this give an anomalous contribution to the Noether charge?
  2. In electrodynamics and gauge theories, charges are obtained in connection with gauge transformations that become constant at a distance. The gauge charge density is a total divergence and gauge charge can be expressed as an electric flux across a very large sphere. On the other hand, in Newton’s theory, the gravitational flux far enough from the system gives its mass. Could mass correspond to a time translation as a symmetry? Could the transformation of the charge into total divergence generalize to other general coordinate transformations?
  3. Einstein action (curvature scalar) contains terms proportional to the second order partial derivatives of the metric: these terms come from the part of the curvature scalar linear in Riemann connection, which serves as analogs of non-abelian gauge potentials. However, this does not give third derivatives to the equations of motion. The reason is that the second derivatives occur linearly. If the square of the curvature tensor would define the action as an analog of Yang-Mills action, the situation would be different. This term is analogous to a dissipative and might relate to the general features of GRT dynamics (blackholes as asymptotic states).

Is the divergence term taken into account automatically in the straightforward Noetherian guess for the conserved currents? Or could the charges associated with the general coordinate transformations emerge as analogs of electric charge as a flux integral over a very large sphere. This would certainly contradict the fact that general coordinate transformations do nothing to the system, so that they cannot relate physically non-equivalent configurations.

  • The deduction of field equations involves transformation of the terms containing derivatives for the variation of the metric so that only terms involving only the variation of the metric remains besides total divergences, which must vanish for symmetries leaving the action invariant. This gives an explicit formula for the conserved Noether currents. In the case of the curvature scalar, the first term in the conserved current comes from the variation of the first derivatives of the metric. The second term comes from the variation of the second derivatives of the metric tensor and an explicit expression can be deduced for it. This gives a total divergence. Is this term automatically included in the term proportional to T-G? This seems very likely. Just for curiosity, let us consider the possibility that the total divergence term is not included in T-G and must be included as an additional term.
    1. As a total divergence this term can be transformed into a surface integral and is proportional to the vector field generating the transformation. This term could give a non-vanishing contribution as an integral over the boundary at infinity which can be regarded as an infinitely large sphere. If the space-time is asymptotically Minkowskian, the counterparts of 4-momentum, angular momentum and also charges associated with Poincare transformation are obtained. Also the charges associated with arbitrary general coordinate transformations are obtained but these are not in general conserved.
    2. The explicit form for the conserved current associated with infinitesimal general coordinate transformation generated by the vector field jμ is
  • Jμ(j)= Lαβμ Dgαβ + LαβμννDgαβ

    =LαβμDgαβ +∂νLαβμ Dgαβ – ∂ν[LαβμDgαβ],

    where one has

    Lαβμ= ∂ L/∂(∂μ gαβ),
    Lαβμν= ∂ L/∂(∂μνgαβ),
    Dgαβ=jρρ gαβ .

    The third term at the second line is a total divergence and this contribution, call it Q3(j) to the expression for the charge as a 3-D integral of the μ=t component of the current can be transformed to a surface integral.

    Q3(j)= -∫S2[∂ L/∂(∂trρ gαβ)] ∂ρ gαβjρ]dS .

  • In the stationary case, the gtt component of the metric includes the gravitational potential and its radial derivative gives a 1/r2 term whose flux over the spherical surface is non-vanishing and gives the same result as gravitational flux in Newton’s theory. Therefore there is a 1/r2 term in the curvature tensor, which is analogous to the electric field. This interpretation requires that the space is asymptotically Minkowski space, so it is possible to talk about Poincare symmetry as an asymptotic symmetry. Constant time shift corresponds to mass.
  • The flux contribution to the charge must be linear in Christoffel symbols and involve the indices t and r. A good guess is that the charge is proportional to
  • Q3(j)= ∫S2 C(t,rρ) jρdS .

    where C(t,rρ) denotes Christoffel symbol. For Schwarzschild metric this gives Newtonian gravitational flux for time translation jρρ,t.

  • One must pose additional conditions guaranteeing that these charges do not flow radially out of the infinite sphere. This becomes a condition that the second derivatives of the metric with respect to the radial coordinate r approach zero faster than 1/r2. This holds true very generally. Note however that the flux associated with arbitrary j need not be conserved. Consider as an example generalized coordinate transformations which approach trivial transformations in the future and non-trivial transformations in the past.
  • Year or two ago there was a lot of talk about an infinite number of charges that can be connected in this way as asymptotic charges to conformal transformations of an infinitely large sphere. These charges could be a special case of pseudo charges described above. To conclude, there are two options. Personally I am convinced that the T-G option is the right one. For the T-G=0 option, the total charges related to general coordinate transformations are therefore zero. One could however say that the total Noether charges are always zero but that they can be divided into interior and flux parts according to holography and cancel out each other. Flux part would correspond to what is called gravitational charge. In this sense the charges related to general coordinate transformations or at least Poincare transformations can be assigned with the system via holography as flux integrals. This could perhaps be interpreted within the framework of holography. These fluxes would characterize the asymptotic behavior giving in turn information about the dynamics in the interior.
  • The conserved charges associated with holomorphies

    Generalized holomorphy not only solves explicitly the equations of motion but, as found quite recently, also gives corresponding conserved Noether currents and charges.

    1. Generalized holomorphy algebra generalizes the Super-Virasoro algebra and the Super-Kac-Moody algebra related to the conformal invariance of the string model. The corresponding Noether charges  are conserved. Modified Dirac action allows to construct the supercharges having interpretation as WCW gamma matrices. This suggests an answer to a longstanding question related to the isometries of the “world of the classical worlds” (WCW).
    2. Either the generalized holomorphies or the symplectic symmetries of H=M4× CP2 or both together define WCW isometries and corresponding super algebra. It would seem that symplectic symmetries induced from H are not necessarily needed and might actually correspond to symplectic symmetries of WCW. This would give a close similarity with the string model, except that one has half-algebra for which conformal weights are proportional to non-negative integers and gauge conditions only apply to an isomorphic subalgebra. These are labeled by positive integers and one obtains a hierarchy.
    3. By their light-likeness, the light cone boundary and orbits of partonic 2-surfaces allow an infinite-dimensional isometry group. This is possible only in dimension four. Its transformations are generalized conformal transformations of 2-sphere (partonic 2-surface) depending on light-like radial coordinate such that the radial scaling compensates for the usual conformal scaling of the metric. The WCW isometries would thus correspond to the isometries of the parton orbit and of the boundary of the light cone! These two representations could provide alternative representations for the charges if the strong form of holography holds true and would realize a strong form of holography. Perhaps these realizations deserve to be called inertial and gravitational charges.

    Can these transformations leave the action invariant? For the light-cone boundary, this looks obvious if the light-cone is sliced by a surface parallel to the light-cone boundary. Note however that the tip of this surface might produce problems. A slicing defined by the Hamilton-Jacobi structure would be naturally associated with partonic orbits.

  • What about Poincare symmetries? They would act on the center of mass coordinates of causal diamonds (CDs) as found already earlier (see this). CDs form the “spine” of WCW, which can be regarded as fiber space with fiber for a given CD containing as a fiber the space-time surfaces inside it. The super-symmetric counterparts of holomorphic charges for the modified Dirac action and bilinear in fermionic oscillator operators associated with the second quantization of free spinor fields in H, define gamma matrices of WCW. Their anticommutators define the Kähler metric of WCW. There is no need to calculate either the action defining the classical Kähler action defining the Kähler function or its derivatives with respect to WCW complex coordinates and their conjugates. What is important is that this makes it possible to speak about WCW metric also for number theoretical discretization of WCW with space-time surfaces replaced with their number theoretic discretizations.
  • For a summary of earlier postings see Latest progress in TGD.

    For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.


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