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Are there 3+1D spacetimes that lead to a $\delta$-function in curvature? Are there any examples that one can provide?

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    $\begingroup$ I am not sure about that, but I think the curvature of a Black Hole is divergent at the centre of the black hole. I think one can show that $\endgroup$
    – schris38
    Jun 28 at 7:24
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    $\begingroup$ The curvature of a non-rotating black hole has a measure of curvature called the Kretschmann scalar, which has the form $K=R_{abcd}R^{abcd}=\frac{48 M^{2}}{r^{6}}$. No delta-distribution present there. Generally, I think to see a delta-like curvature you'd need to solve the EFEs in a setting that is not at least $\mathcal{C}^{3}$, where the three-times differentiability requirement of the metric follows from the Riemann tensor involving second derivatives of the metric and the differential Bianchi identity adds another derivative. $\endgroup$
    – K.T.
    Jun 28 at 8:16
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    $\begingroup$ Might post a proper answer later, but the Israel junction conditions allow for "thin shells" in spacetime, which are hypersurfaces along which the curvature tensor exhibits a Dirac delta singularity concentrated on the hypersurface. More importantly, this is perfectly rigorous (or at least can be made so). There was a paper by Geroch and Traschen where it was proven that only these singular solutions of the EFE exist in general, hence any other form of singular sources like point masses or strings where the energy -momentum tensor has a delta term should be treated with suspicion. $\endgroup$ Jun 28 at 12:50
  • $\begingroup$ I don't remember however if they had concluded that definitely no other singular solution exists or only that they don't exist "generally". $\endgroup$ Jun 28 at 12:51
  • $\begingroup$ This paper might be of interest for this question: aip.scitation.org/doi/abs/10.1063/1.524568?journalCode=jmp $\endgroup$
    – Gold
    Jun 28 at 13:40

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A good analysis of this subject is contained in the paper R. Geroch, J. Traschen, Strings and other distributional sources in general relativity.

They conclude that thin shells (delta singularity on a codimension 1 hypersurface) are well-defined but higher codimension surfaces supporting delta singularities in the curvature are not.

I will not be completely rigorous with my use of distributions here, since it takes quite a lot of effort to state how the theory works on manifolds correctly. I will also not be completely general in the formulation in the sense that I will consider timelike hypersurfaces only (the spacelike case is identical with some sign changes here and there), even though with small modifications, the considerations here are also valid for general hypersurfaces with no assumptions on its causal type.

Let $M$ be a smooth four dimensional manifold (four dimensionality is not strictly required though) with a Lorentzian metric tensor $g$ on it and a hypersurface $\Sigma\subseteq M$ such that $g$ is smooth away from $M$ but is merely assumed to be continuous on $\Sigma$. It is further assumed that $\Sigma$ partitions $M$ into two disconnected subdomains $M_+$ and $M_-$ with $\Sigma$ being the common boundary surface separating the two.

Note that the induced metric $h=i^\ast g$ ($i:\Sigma\rightarrow M$ is the inclusion) on $\Sigma$ is not only unambigous under these assumptions, but is smooth.

One may then derive (this is the so-called Israel formalism for thin shells) that the curvature tensor exists as a tensor distribution on $M$ in the following form: $$ \mathrm{Riem}=\mathrm{Riem}_+\Theta_\Sigma+\mathrm{Riem}_-\left(1-\Theta_\Sigma\right)+Q\delta_\Sigma, $$ where 1) $$ \Theta_\Sigma(p)=\left\{\begin{matrix} 1 & p\in M_+ \\ 0 & p\in M_- \end{matrix}\right. $$ is the step function associated with the hypersurface and the Dirac delta $\delta_\Sigma$ associated with the hypersurface is defined by $$\langle \delta_\Sigma,\varphi\rangle=\int_\Sigma\varphi\mu_h $$ for any test function $\varphi$ (smooth with compact support) on $M$ with $\mu_h$ being the induced volume form of the metric $h$ on $\Sigma$, 2) $\mathrm{Riem_\pm}$ are the curvature tensors calculated from the smooth metrics $g_{\pm}$ in $M_\pm$ and 3) $Q$ is a smooth tensor field defined only along $\Sigma$.

Letting $(U,x^\mu),\ \mu=0,1,2,3$ be an "admissible" chart (i.e. one which passes through $\Sigma$ smoothly) on $M$ intersecting $\Sigma$, the coefficients $Q_{\kappa\lambda\mu\nu}$ of the tensor field $Q$ are $$ Q_{\kappa\lambda\mu\nu}=n_{\kappa}\left[K_{\lambda\mu}\right]n_{\nu}-n_{\kappa}\left[K_{\lambda\nu}\right]n_{\mu}+n_{\lambda}\left[K_{\kappa\nu}\right]n_{\mu}-n_{\lambda}\left[K_{\kappa\mu}\right]n_{\nu}, $$ where $$ [K_{\mu\nu}](p)=\lim_{q\in M_+\rightarrow p}K^+_{\mu\nu}(q)-\lim_{q\in M_-\rightarrow p}K^{-}_{\mu\nu}(q) $$ is the difference of the extrinsic curvature tensors of $\Sigma$ as calculated from the two sides.

This formalism gives a large class of metrics whose curvature tensors are singular.


I might update this answer with a concrete example in the future, but the book E. Poisson, A relativist's toolkit contains a number of worked out examples (for example Section 3.9).

The example in Section 3.9 considers a timelike hypersurface $\Sigma$ which is spatially spherical with the "interior" region $M_-$ being Minkowski spacetime and the "exterior" region $M_+$ being Schwarzschild spacetime. A singular dust energy-momentum tensor is placed on $\Sigma$. The resulting spacetime is a valid, albeit weak, solution of the Einstein field equations in which the curvature tensor has a Dirac delta singularity.

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The simplest example I can think of is to have a "bumpy" metric (also here):

\begin{equation} g_{\mu \nu} = \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 + \theta (x - x_{0}) & 0 & 0 \\ 0 & 0 & 1 + \theta (x - x_{0}) & 0 \\ 0 & 0 & 0 & 1+ \theta (x - x_{0}) \\ \end{pmatrix} \end{equation}

where $x_{0}$ the proper distance after which the spatial directions of the metric experience a sudden "jump". Any Christoffel symbol involving the derivative $\partial _{\mu} g_{ii} \, , \, i = 1,2,3$ yields a delta function $\delta (x - x_{0})$ times other stuff depending on the derivative at hand. Since some Christoffel symbols now have a delta function, so will the Riemannn curvature tensor:

\begin{equation} R^{\mu} {}_{ \rho \nu \sigma} = \nabla _{\nu} \Gamma ^{\mu}{}_{\rho \sigma} - \nabla_{\sigma} \Gamma ^{\mu}{}_{\nu \rho} \end{equation}

where $\nabla _{\rho}$ the covariant derivative (which itself will include a Christoffel symbol). Subsequently the Ricci tensor:

\begin{equation} R_{\rho \sigma} = R^{\mu}{}_{\rho \mu \sigma} \end{equation}

will also have components that contain a delta function.

The scalar curvature $R = g^{\rho \sigma} R_{\rho \sigma}$ will carry those delta functions trivially, since the inverse metric $g^{\rho \sigma}$ will have the Heaviside theta function in the denominator. The resulting curvature will be 0 everywhere except the pole $x = x_{0}$ where it diverges.

There is no physical example that I know of which exhibits any similar behaviour, but you can always find an infinite amount of metrics such as the one above, which lead to curvature tensors including delta functions in some of their components.

An important stipulation of course is that this kind of abrupt "jumps" and "bumps" on the metric are technically not well-defined mathematically when we perform differential geometry, since then the manifold you examine (your spacetime) is not smooth, thus a lot of the assumptions we take for granted in GR (General relativity) don't hold.

The above then remain relevant as a crude approximation of a metric that is smooth, but with abrupt changes at specific points.

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    $\begingroup$ -1: As you yourself said, it's not physically realistic and the OP is not merely asking for a manifold with a metric with a delta curvature. So you too aren't answering the question - just one you've decided to answer as an appropriate reformulation. He asked for a 3+1d spacetime. Secondly, the manifold can still be smooth, it's merely the metric that isn't. $\endgroup$ Jun 28 at 11:09
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    $\begingroup$ The OP asked for spacetimes that lead to curvature which includes the delta function, they did not specify whether they want a physically realistic model or a spacetime that may describe a real system. For the second point, it is correct, but I assume we have the case of a metric whose topology is meant to describe the topology of the manifold itself. $\endgroup$
    – rhomaios
    Jun 28 at 11:30
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    $\begingroup$ This also seems to be the best I can think of as well. I was thinking more along the lines of global monopoles hoping they might give me a delta singularity but they don't. $\endgroup$
    – user44690
    Jun 28 at 11:46
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    $\begingroup$ It seems either you can have a conical singularity which is effectively 2+1 D or a discontinuous metric. $\endgroup$
    – user44690
    Jun 28 at 11:46
  • $\begingroup$ The post pertains to GR, so it qualifies as physics. It is perhaps a more mathematical question or more abstract/less realistically applicable, but that doesn't mean it's not physics. $\endgroup$
    – rhomaios
    Jun 28 at 14:14
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Delta functions are discontinuous at the origin. A curvature that is exactly describable by a delta function isn't possible in spacetime as spacetime is seen to be continuous and smooth and hence curvature must be too.

As delta functions have a point that is infinite, the appropriate forms of spacetime, given the criticism above, are spacetimes with an infinite point of curvature.

Such spacetime are given by the Schwarzschild spacetimes that describe black holes and which have essential singularities where geodesics end. These are points of infinite curvature. This could well be one good reason for including torsionful gravity as they do not have singularities such as the Einstein-Cartan theory. It's worth noting that supergravities, which superstring theories reduce to in the weak energy limit, have torsion and so they too do not have singularities.

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  • $\begingroup$ This doesn't have a $\delta$-function $\endgroup$
    – Eletie
    Jun 28 at 8:59
  • $\begingroup$ Also less related: do you have a good source for the lack of singularities in Einstein-Cartan? From what I've read, most people only claim it can be singularity free, not that this a generic property (and in fact people struggle to construct these with realistic parameters). Would be interested in any sources that have some physically realistic scenarios! Thanks $\endgroup$
    – Eletie
    Jun 28 at 9:06

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