# Action for extended objects

Take a spacetime $$M$$, with some $$k$$-manifold embedding

$$X : \Sigma \to M$$

The image of $$X$$ represents some extended object (a $$k$$-brane as the string theory people say). If we only care about the dynamic of $$X$$, we can simply write its action using, say, the Nambu-Goto action :

$$S[X ; U] = \int_{X^{-1}(U)} d\mu[X_*g]$$

which is the volume of the induced metric (ie the pushforward of $$g$$ by $$X$$), or in the usual coordinates,

$$S[X ; U] = \int_U \sqrt{\det(g_{\mu\nu}(X(\sigma)) \partial_a X^\mu(\sigma) \partial_b X^\nu(\sigma) )} d^k\sigma$$

But if we want to consider the metric tensor as being itself dynamic (for instance if we're interested in the Nambu-Goto action to treat idealized topological defects, point particles, thin-shell spacetimes, etc), or if we want to couple it to another more traditional field (ie a charged point particle for instance), we generally want everything to be written in terms of an integral over the target space. But how to do this?

The various ways I tried to achieve this are :

The naive way

Simply add the two actions together :

$$$$S[X,g] = \int_M R_g d\mu[g] + \int_\Sigma d\mu[X_* g]$$$$

Pros : Somewhat accurate, up to issues that I'll get into later regarding the class of $$g$$

Cons : Not useful to actually get the Euler-Lagrange equation

The physicist way

When done in physics papers, it's usually represented as

$$$$S[X, g ; U] = \int_U R_g + \left[\int_{X^{-1}(U)} \delta(x - X(\sigma)) \sqrt{\det(g_{\mu\nu}(x) \partial_a X^\mu(\sigma) \partial_b X^\nu(\sigma) )} d^k\sigma\right] d^n x$$$$

This lets people use the variation of the metric tensor directly, and gives out the appropriate stress-energy tensor $$T \approx m\delta(x(\tau))$$ for point particles.

Pros : Good enough for calculations

Cons : Not terribly rigorous regarding the use of distributions

De Rham currents

The common way to deal with distributions on manifolds is the use of currents, which are linear functionals on $$k$$-forms with compact support. If we take the measure form $$\omega = \sqrt{-g} \bigwedge dx_i$$ and the integration current $$[U]$$. The Nambu-Goto action should be, I believe

$$$$S[X,g; U] = \langle X_* [U], \omega[g] \rangle$$$$

with $$X_* [U]$$ the pushforward of the integration over $$U$$ by $$X$$. This is indeed equal to $$\langle [U], X^* \omega[g] \rangle$$ via the properties of currents, which I think is the original Nambu-Goto action. As everything is linear, I think the action should be something like

$$$$S[X,g; U] = \langle [U], R_g \omega[g] + X^* \omega[g] \rangle$$$$

Pros : Somewhat rigorous, but see cons

Cons : The Euler-Lagrange equation of this quantity doesn't seem trivial to compute (I'm not quite sure how the fundamental lemma of variational calculus would crop up here). Also by the Geroch-Traschen theorem, $$R$$ can't be a smooth $$0$$-form (or even a distribution, depending on the brane), and so shouldn't be on this side

Generalized functions

To work with algebras of distributions, one can use a class of generalized functions (Colombeau generalized functions or asymptotic generalized functions), so that the metric, Riemann tensor and stress-energy tensor can all be expressed as generalized functions that are allowed to be singular.

Pros : All of that

Cons : It's not evident how the action should be written down for a distribution-valued action, or how the variation would work. Also, for physical reasonableness, the standard part of any measurable quantity should exist, and I don't know if that is guaranteed in any reasonable case.

Is there any method to deal with this?