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 :

\begin{equation} S[X,g] = \int_M R_g d\mu[g] + \int_\Sigma d\mu[X_* g] \end{equation}

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

\begin{equation} 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 \end{equation}

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

\begin{equation} S[X,g; U] = \langle X_* [U], \omega[g] \rangle \end{equation}

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

\begin{equation} S[X,g; U] = \langle [U], R_g \omega[g] + X^* \omega[g] \rangle \end{equation}

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?


1 Answer 1


The short answer is that the naive, physicist, and de Rham way are all equivalent. Specifically the de Rham way is defined precisely so that it is the "physicist" way when expressed in terms of a specific coordinate system (atlas). (I believe something is up with your volume forms in the "physicist way" in the version of your post I see.) The physicist way should reduce to the naive way once you integrate out the delta function. Coordinate-free expressions may make certain algebraic identities clearer and connect them to more general mathematical structures, but that simply does not make them more rigorous. "Rigorous" does not mean "the way a pure mathematician would prefer to express it".

The problem is, as you have noted, that by using these actions you do not get a meaningful classical field theory. Thus, you can understand it as an action for a quantum field theory, or as an action for a classical effective field theory, where in both cases the divergences are handled by some sort of regularization/renormalization procedure. In fact, the latter has been done in the case of classical post-Newtonian dynamics of compact objects, where it is technically simpler to work with effective point-particle actions along with covariant regularization procedures such as zeta-function, Hadamard, or dimensional regularization. (See e.g. the Living Review Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries by Luc Blanchet.)


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