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.
- 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?
- 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?
- 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.
- 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.
- The explicit form for the conserved current associated with infinitesimal general coordinate transformation generated by the vector field jμ is
Jμ(j)= Lαβμ Dgαβ + Lαβμν ∂νDgαβ
=LαβμDgαβ +∂νLαβμ Dgαβ - ∂ν[LαβμDgαβ],
where one has
Lαβμ= ∂ L/∂(∂μ gαβ),
Lαβμν= ∂ L/∂(∂μνgαβ),
Dgαβ=jρ∂ρ 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/∂(∂tr ∂ρ 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 jρ=δρ,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.
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.