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In my course on Lagrangian/Hamiltonian mechanics I noticed that we dealt with finding the stationary point of the change in action $ \delta S $ and we were never really doing anything with $ S $ itself. What are the uses of $ S $ itself? Is it something we can solve for and what makes it useful?

Personal investigations:

So far my investigations have I've only heard of two uses. I asked my TA if there were any uses for the action $S$ and he, noting my interest in statistical mechanics, said that the action can be used in the construction of a partition function, and he said he it's of a form like $Z = e^{-\beta S} $. He also mentioned it was a similarly used in QFT for constructing partition functions. I'm still pretty confused about why that is the case though...

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I can think of two immediate uses of the action $S$ itself in general relativity. The first to derive the entropy of a Schwarzschild black hole, and the second to compute the nucleation rate of instantons which emerge due to tunneling processes from a false to true vacuum.


Bekenstein-Hawking Entropy

To compute the entropy, we use a semi-classical approach. Normally, one defines a partition function,

$$Z\sim \mathrm{tr} \, e^{-\beta H}$$

with $\beta^{-1}=k_BT$, where $H$ is the Hamiltonian. The approach is rather ad hoc, but notice roughly,

$$H = \int d^3x \, \mathcal{H} \sim \frac{1}{\beta}\int d^4x \, \mathcal{H} \sim \frac{1}{\beta}I_E$$

where $I_E$ is the Euclidean action which for our case reads,

$$I_E = \frac{1}{16\pi G}\int_M d^4x \, \sqrt{|g|} R + \frac{1}{8\pi G} \int_{\partial M} d^3x \, \sqrt{|h|} K$$

which is the Einstein-Hilbert action supplemented by the Gibbons-Hawking term to account for the contribution from the boundary of the manifold. For Einstein gravity, we expect $Z$ to be dominated by classical solutions, that is,

$$Z \sim \sum_{\mathrm{class \, solns}} e^{-\beta I_E}$$

For both the Kerr and Schwarzschild black holes, the Ricci scalar vanishes. However, they have a non-zero boundary term which will contribute to the partition function. So, our metric is,

$$ds^2 = \left( 1-\frac{2GM}{r}\right)d\tau^2 + \left( 1-\frac{2GM}{r}\right)^{-1}dr^2 + r^2 d\Omega^2$$

where $\tau$ is a periodic coordinate, with periodicity $8\pi GM$. Temporarily, we impose a cutoff $R >> GM$, and the metric on the boundary is simply the Schwarzschild metric without the $dr^2$ term, evaluated at the cutoff, $R$. The extrinsic curvature is given by the divergence of the normal,

$$K=\nabla_a n^a = -\frac{1}{r^2} \frac{\partial}{\partial r} r^2 \sqrt{1-\frac{2GM}{r}} = -\frac{2}{r}\sqrt{1-\frac{2GM}{r}} - \frac{GM}{r^2}\frac{1}{\sqrt{1-\frac{2GM}{r}}}$$

evaluated at $R$. The integrand of the action is independent of $\tau,\theta,\phi$, and the integration is trivial. One obtains a dependence on the cutoff, $R$, and to obtain a finite limit as $R\to\infty$ one subtracts the Euclidean action of Minkowski space with the same boundary. Eventually, one finds,

$$I_E = \frac{1}{2}\beta M.$$

Plugging it into the partition function and computing,

$$S = -\beta^2 \frac{\partial}{\partial \beta} \beta^{-1} \ln Z$$

one finds the result $S = 4\pi G M^2$, or in terms of the area, $A/4G$ in natural units.


Bubble Nucleation

The second application is bubble nucleation, first studied by Coleman and de Luccia. The general idea is one has a field theory which has a true vacuum and a false vacuum, but the difference in potential is minute. In each region, the stress-energy tensor will be different, and hence the spacetime geometry. Coleman and de Luccia studied tunneling processes from the false vacuum to the true vacuum, which involved the nucleation of an instanton or bubble. In their original paper, the Lagrangian was given by,

$$\mathcal{L} = \frac{1}{2}(\partial \phi)^2 - \frac{\lambda}{2}(\phi^2-\eta^2)^2 - \frac{\epsilon}{2\eta}(\phi-\eta)$$

The classical vacuum $\phi = \eta$ is a true vacuum, but $\phi=-\eta$ is not, though the difference is $\epsilon$. The action enters the picture because it can be used to evaluate the rate,

$$\Gamma \propto e^{-I_B}$$

where $I_B$ is the bounce action, which is the action computed for the wall minus the action for the space inside the wall. Coleman and de Luccia found that if one includes gravitational effects, they may stop the nucleation of the bubble because the warping of the geometry of spacetime may stop the bubble from achieving the right volume to surface area ratio to ensure the net energy is zero.

Later work by R. Gregory et al. on the nucleation of such instantons in a de Sitter-Schwarzschild space reveals the nucleation rate is slightly higher compared to the Coleman instanton. For a detailed pedagogic calculation, see Classical Solutions in Quantum Field Theory by Weinberg, or the papers:

  • Black holes as bubble nucleation sites, R. Gregory et al, [hep-th/1401.0017v1].
  • Gravitational effects on and of vacuum decay, S. Coleman, F. de Luccia, Phys. Rev. D 21, 3305.
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