Original derivation of Papapetrou equations

1. Aug 23, 2014

WannabeNewton

Hi all. As you may be aware, there are a handful of modern derivations of the Papapetrou equations i.e. the equations of motion for a pole-dipole particle in curved space-times. These are usually along the lines of a Hamiltonian or Lagrangian derivation (c.f. Racine et al 2009 and Anandan et al 2003) and of course one has the famous yet extremely dense series of papers on EOM and covariant multipole moments for extended bodies in curved space-times by Dixon.

However I wanted to see the original derivation by Papapetrou, as it would be more suited to my needs. There are however a few steps in the paper that I cannot follow. The paper is here: http://rspa.royalsocietypublishing.org/content/209/1097/248 but I don't know how many of you will be able to access it so let me just attach an image of the single page relevant to my preliminary questions.

Equation (2.2) is itself perplexing to me. I've stared at it for hours and I can't see how he gets it so I must be missing something very simple. When he refers to the "dynamical equation" he is talking about $\nabla_{\beta}T^{\alpha\beta} = 0$ (he uses $\mathfrak{T}^{\alpha\beta}$ for $T^{\alpha\beta}$ and note the time-like component is the 4th as opposed to zeroth component in his calculations). But this is equal to $\partial_{\beta}T^{\alpha\beta} + \Gamma^{\alpha}_{\mu\nu}T^{\mu\nu} + \Gamma^{\beta}_{\beta \gamma}T^{\alpha\gamma} = 0$. He doesn't have the $\Gamma^{\beta}_{\beta \gamma}T^{\alpha\gamma}$ term and certainly it doesn't vanish in general so why does it vanish for the derivation of the single-pole and pole-dipole equations of motion?

My next concern is regarding the power series (2.6) for the connection coefficients inside the particle worldtube. The integrals (2.4) and (2.5) that (2.6) is inserted into is, according to Papapetrou, over "the three dimensional space $t = \text{const.}$". Does this refer to an entire space-like hypersurface of space-time or just the intersection of the space-like hypersurface with the worldtube of the particle? If it is the former then I don't see how it is valid to insert (2.6) into the integrals given that the power series is only valid inside the worldtube whereas the integrals extend outside of it. If it is the latter then inserting (2.6) into the integrals would be valid but the equalities (2.4) and (2.5) themselves would not be valid for, considering for example (2.4), the equality only holds through the divergence theorem if $T^{\alpha\beta}$ has compact support on the integration domain which won't be true if the domain is the intersection of the space-like hypersurface with the particle worldtube: $$\frac{d}{dt}\int_{\Sigma} T^{\alpha 0}d^3 x = \int_{\Sigma} \partial_0 T^{\alpha 0}d^3 x = -\int \partial_i T^{\alpha i}d^3 x - \int_{\Sigma} \Gamma^{\alpha}_{\mu\nu}T^{\mu\nu}d^3 x = -\int_{\partial \Sigma} T^{\alpha i}n_i d^2 x - \int_{\Sigma} \Gamma^{\alpha}_{\mu\nu}T^{\mu\nu}d^3 x$$ and $\int_{\Sigma} T^{\alpha i}n_i d^2 x = 0$ only if $T^{\alpha\beta}$ has compact support on $\Sigma$.

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2. Aug 23, 2014

samalkhaiat

3. Aug 23, 2014

WannabeNewton

Ah that's perfect Sam, thank you! It looks like my issue was in interpreting the notation in Papapetrou's paper.

So just to clarify, $\mathfrak{T}^{\alpha\beta}$ is the tensor density $\mathfrak{T}^{\alpha\beta} = \sqrt{-g}T^{\alpha\beta}$ and not the energy-momentum tensor itself, yes?

And is the "small" 3-voulme $V$ of integration assumed to be small enough so that the series expansion for $\Gamma^{\alpha}_{\beta\gamma}$ about the central worldline $X^{\alpha}$ is valid inside the integral but large enough so that $T^{\alpha\beta} = 0$ on $\partial V$ so as to have compact support on $V$ for the divergence theorem?

I'll let you know if I have any more questions about the pole-dipole calculation in the Papapetrou paper, thanks.

4. Aug 23, 2014

samalkhaiat

Yes, this is why you don't see a term with $\Gamma^{a}_{ab}$ in it.

Yes, see the paragraph after equation (3).

Good Luck

Sam