$L^{1}$ energy-momentum tensors in general relativity; semi-classical gravity

Question

I was unsure whether to pose this question in a physics or mathematics forum, but it is an interesting idea I have been thinking about for some time.

In any (semi-)classical field theory it is often assumed that the Lagrangians of the theory $\mathcal{L}$ are themselves smooth functions of some kind over a Lorentzian (Riemannian) manifold $\Omega$, but could we permit the matter piece of the theory to be, say, distributional? Typically we will always begin with an action:

$\mathcal{S} = \int_{d \Omega} d^{4} x \sqrt{-g} (\mathcal{L}_{Field} + \mathcal{L}_{Matter})$

Euler-Lagrange and least action principles leads to field equations with $\delta \mathcal{S} = 0$.

In particular, I am interested in general relativity wherein we have $\mathcal{L}_{Field}$ is just the Einstein-Hilbert Lagrangian $\mathcal{L}_{Field} = R$ leading to the field equations (avoiding constants):

$R_{\mu \nu} - \frac {1} {2} g_{\mu \nu} R = T_{\mu \nu}$

Where, using the Hilbert construction, $T^{\mu \nu} = 2 \frac {\delta \mathcal{L}_{Matter}} {\delta g_{\mu \nu}} + g^{\mu \nu} \mathcal{L}_{Matter}$.

What if we now allow $T_{\mu \nu} \in L^{1}(\Omega)$ or $T_{\mu \nu} \in H^{1}(\Omega)$? That is to say we reduce the demands of regularity on the the stress energy tensor and allow it to reside in a less regular space.

Interesting examples occur to me of the form:

i) $T_{\mu \nu} \propto \rho(t,r) \delta{(t - t_{0})}$; could this permit an insertion of a cosmological singularity relating to the big bang or something of this nature? Perhaps generalised Friedmann equations could be developed relating to this and could provide a semi-classical analysis of the big bangs development.

ii) Typically a (perfect) fluid is modelled by the Euler stress tensor $T_{\mu \nu} = \rho v_{\mu} v_{\nu} + P \delta_{\mu \nu}$ where one then lets $\rho$ and $P$ describe the matter distribution of the star. If one instead employs a distributional object instead, say even a Gaussian and having the densest matter closer to the core or something like that could this yield a fruitful alternative approach? One could investigate stellar evolution (or at least place bounds) on stars with (somewhat) unspecified matter ensembles.

iii) Brownian motion in the vicinity of black holes or other dense objects; one could take $T_{\mu \nu}$ to resemble a Wiener process allowing for certain atomic anomalies to divert the particles path as it falls towards these objects. This may have consequences for things such as gravitational wave radiation.

There is some beautiful mathematics that permits some analysis relating to these objects using Sobolev spaces and embeddings; (Benilan et al, "An $L^1$-Theory of Existence and Uniqueness of Solutions of Nonlinear Elliptic Equations"; A. Prignet "Remarks on existence and uniqueness of solutions of elliptic problems with right-hand side measure"; work done by Boccardo & Gallouet, etc.) Naturally these papers refer to elliptic problems, but presumably one can employ their methods using 3+1 ADM splits or something of that nature. I have seen bits and pieces of this machinery used in the context of electromagnetism but they seem to be largely avoided by the physics community.

Has any work been done in relation to these kinds of ideas?

Thanks for any input!

Answer 1

I'll try to rephrase your question first. The Einstein Field Equations

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=8\pi T_{\mu\nu}$

are (as is well known) a system of hyperbolic partial differential equations (derivable from an action principle, though I fail to see the relevance of this point here). As such, they give rise to an initial value problem, namely given a set of initial conditions, usually a riemannian manifold together with extrinsic curvature data and extra data from matter fields, to find a lorentzian manifold satisfying Einstein Equations such that the inital data corresponds to an given hypersurface (and fields defined within it). In other words we need results for the existence and uniqueness of solutions.

The classical result from Y. Choquet-Bruhat and R. Geroch provides global existence and uniqueness for smooth initial data. Here, I guess, lies your question, viz. can we relax the smoothness assumption and work on lower regularity initial data spaces?

Most certainly yes! A somewhat old reference is Klainerman and Rodnianski from 2001, that prove results for inital data $H^{2+\epsilon}$. Note that the energy-mommentum tensor is not part of the initial data as such, but it is indirectly , as far as it is given by field configurations. As a side note, a rough guide says that you'll need $T_{\mu\nu}$ to be at least differentiable, because you need to enforce local conservation of energy $\nabla^\mu T_{\mu\nu}=0$. This statement is the source of the common phrase that the metric should be at least $C^3$, given that the curvature contains second derivatives.

I'm not very informed on this particular area of research, but last year I attended a conference where Piotr Chrúsciel gave a talk on improving this results for even less regular inital data, so I guess that it remains a good topic in mathematical relativity.

EDIT: The talk Chrúsciel gave was based on this paper, that discuss lorentzian causality for metrics that are only continuous, but not differentiable (causality results are essential in the Choquet-Bruhat-Geroch paper). Interestingly it shows that $C^0$ metrics have strange causal structure, namely there are open sets with points accessible to null curves but not timelike. Pictorially it means that the light cone can get "fat". Therein you will find lots of references to newer results for low regular initial data, supplementing the Klainerman Rodnianski result. A good companion to this paper is this one, also from Chrúsciel, that gives the state of the art results in causality theory, indicating how regular the metric must be for each theorem.

Concerning the potential interest, at least from my perspective, the idead is that with less regularity we have better control of the behavior of Einstein Equations, meaning that solutions are less dependent on details of the initial data. The ones who would feel most reassured then would be the Numerical Relativity folks.

From your list I would comment thusly

i)The object you mention does not correspond to singularities in general relativity. The major point of singularities are the geodesical incompletness of the manifold, and a lot of the techniques completely ignore the PDE question in it, see e. g. Wald's "General Relativity". From the physical point of view the problem when working with singularities is not a question of low regularity of the solution, but rather that we expect quantum gravity to be relevant. Therefore I think is fair to say that no one expects the singularities per se to have a resolution in classical relativity. Even if such approach existed it is not clear it would be physically relevant.

ii)Currently (one of) the greatest problem in stellar evolution is the extreme sensitiveness of mass limits and such things with respect to the state equation $\rho(p)$. Therefore most of the research is devoted to nuclear and sub-nuclear physics (I've found this slides, they give a general idea of the race to constrain the enormous number of different equations of state). But it may be an interesting approach, although I'm not sure where you could go, I would certainly be interested in hearing about it.

iii)Relativist brownian motion is a thorny subject, given the loss of markovian property. I'm not sure where this would go either.

As a final comment, you are using a (apparently) different definition of classical and semi-classical field theories, at least from what I usually see in the gravitation community. Classical Field Thoeries are those that do not rely on quantum mechanics in any form, be they relativistic, like eletromagnetism, or non-relativistic, like the classical string problem. Semi-classical field theory means treating the matter quantum mechanically but gravity classicaly, a famous result is Hawking radiation. In this context, quantum fields over classical spacetimes, distributional energy-mommentum tensors are prevalent, due to the quantum nature of the matter involved. Dealing with the problems that appear from having this distributions is doing renormalization in curved spacetimes. If you're fond of functional analysis in this context you could try to take a look in Fulling's book.

Answer 2

There are many cases where we do consider a stress-energy tensor which is not a smooth function but rather a distribution. In particular, in gravitational brane theories, often $T_{\mu\nu} \propto \delta(w)$ where $w$ is a coordinate transverse to the brane. This allows one to define a surface stress-energy tensor,

$$S_{\mu\nu} \sim \int_{-\infty}^\infty dw \, T_{\mu\nu}$$

of the brane, and constraints on the extrinsic curvature are related to this stress-energy, through the Israel junction conditions, who also considered shells of dust which admit similar, non-smooth stress-energy tensors.

The most famous setup is probably the Randall-Sundrum model, but there are plenty, such as for example the embedding of a de Sitter brane in $\mathrm{AdS_5}$, which gives rise to a 5D stress-energy,

$$T^M_N = -\Lambda_5 \delta^M_N + \delta(w)\times \mathrm{const}.$$

As for the mathematical validity I am unsure, but these models are certainly studied and the existing previous answer certainly demonstrates we can consider other classes of stress-energy.