How to include fermions into general relativity by exotic smoothness

Abstract

The purpose of this paper is two-fold. At first we will discuss the generation of source terms in the Einstein–Hilbert action by using (topologically complicated) compact 3-manifolds. There is a large class of compact 3-manifolds with boundary such as a torus given as the complement of a (thickened) knot admitting a hyperbolic geometry, denoted as hyperbolic knot complements in the following. We will discuss the fermionic properties of this class of 3-manifolds, i.e. we are able to identify a fermion with a hyperbolic knot complement. Secondly we will construct a large class of space-times, the exotic ${\mathbb {R}}^{4}$, containing this class of 3-manifolds naturally. We begin with a topological trivial space, the ${\mathbb {R}}^{4}$, and change only the differential structure to obtain many nontrivial 3-manifolds. It is known for a long time that exotic ${\mathbb {R}}^{4}$’s generate extra sources of gravity (Brans conjecture) but here we will analyze the structure of these source terms more carefully. Finally we will state that adding a hyperbolic knot complement will result in the appearance of a fermion as source term in the Einstein–Hilbert action.

Appendices

Appendix A: Connected and boundary-connected sum of manifolds

Now we will define the connected sum $\#$ and the boundary connected sum $\natural$ of manifolds. Let $M,N$ be two $n$-manifolds with boundaries $\partial M,\partial N$. The connected sum $M\#N$ is the procedure of cutting out a disk $D^{n}$ from the interior $int(M)\setminus D^{n}$ and $int(N)\setminus D^{n}$ with the boundaries $S^{n-1}\sqcup \partial M$ and $S^{n-1}\sqcup \partial N$, respectively, and gluing them together along the common boundary component $S^{n-1}$. The boundary $\partial (M\#N)=\partial M\sqcup \partial N$ is the disjoint sum of the boundaries $\partial M,\partial N$. The boundary connected sum $M\natural N$ is the procedure of cutting out a disk $D^{n-1}$ from the boundary $\partial M\setminus D^{n-1}$ and $\partial N\setminus D^{n-1}$ and gluing them together along $S^{n-2}$ of the boundary. Then the boundary of this sum $M\natural N$ is the connected sum $\partial (M\natural N)=\partial M\#\partial N$ of the boundaries $\partial M,\partial N$.

Appendix B: Spin $\frac{1}{2}$ from space a la Friedman and Sorkin

As shown by Friedman and Sorkin [35], the calculation of the angular momentum in the ADM formalism is connected to special diffeomorphisms $R(\theta )$ (rotation parallel to the boundary w.r.t. the angle $\theta$). So, one can speak of spin $\frac{1}{2}$, in case of $R(2\pi )=-1$. Interestingly, these diffeomorphisms are well-defined on all hyperbolic 3-manifolds.

In the following we made use of the work [35] in the definition of the angular momentum in ADM formalism. In this formalism, one has the 3-manifold $\varSigma$ together with a time-like foliation of the 4-manifold $\varSigma \times {\mathbb {R}}$. For simplicity, we consider the interior of the 3-manifold or we assume a 3-manifold without boundary. The configuration space ${\mathcal {M}}$ in the ADM formalism is the space of all Riemannian metrics of $\varSigma$ modulo diffeomorphisms. On this space we define the linear functional $\psi :\mathcal {M}\rightarrow \mathbb {C}$ calling it a state. In case of a many-component object like a spinor one has the state $\psi :\mathcal {M}\rightarrow \mathbb {C}^{n}$. Let $g_{ab}$ be a metric on $\varSigma$ and we define the generalized position operator

$$\begin{aligned} \hat{g}_{ab}\psi (g)=g_{ab}\psi (g) \end{aligned}$$

together with the conjugated momentum

$$\begin{aligned} \hat{\pi }^{ab}\psi (g)=-i\frac{\delta }{\delta g_{ab}}\psi (g). \end{aligned}$$

Let $\phi _{\alpha }$ with $\alpha =1,2,3$ be vector fields fulfilling the commutator rules $[\phi _{\alpha },\phi _{\beta }]=-\epsilon _{\alpha \beta \gamma }\phi _{\gamma }$ generating an isometric realization of the $SO(3)$ group on the 3-manifold $\varSigma$. The angular momentum corresponding to the initial point $(g_{ab},\pi ^{ab})$ with the conjugated momentum $\pi ^{ab}=(16\pi )^{-1}(-K^{ab}+g^{ab}K)\sqrt{g}$ (in the ADM formalism) and the extrinsic curvature $K_{ab}$ is given by

$$\begin{aligned} J_{\alpha }=-\int \limits _{\varSigma }\mathcal {L}_{\phi _{\alpha }}(g_{ab})\pi ^{ab}\, d^{3}x \end{aligned}$$

with the Lie derivative ${\mathcal {L}}_{\phi _{\alpha }}$ along $\phi _{\alpha }$. The action of the corresponding operator $\hat{J}_{\alpha }$ on the state $\psi (g)$ can be calculated to be

$$\begin{aligned} \hat{J}_{\alpha }\psi (g)=-i\frac{d}{d\theta }\psi \circ R_{\alpha }(\theta )^{*}(g)|_{\theta =0} \end{aligned}$$

where $R_{\alpha }(\theta )$ is a 1-parameter subgroup of diffeomorphisms generated by $\phi _{\alpha }$. Then a rotation will be generated by

$$\begin{aligned} \exp (2\pi i\hat{J})\psi =\psi \circ R(2\pi )^{*}. \end{aligned}$$

Now a state $\psi$ carries spin $\frac{1}{2}$ iff $\psi \circ R(2\pi )^{*}=-\psi$ or $R(2\pi )=-1$. In this case the diffeomorphism $R(2\pi )$ is not located in the component of the diffeomorphism group which is connected to the identity (or equally it is not generated by coordinate transformations).

Appendix C: Scalar curvature and energy density

Let us consider a Friedmann–Robertson–Walker-metric

$$\begin{aligned} ds^{2}=dt^{2}-a(t)^{2}h_{ik}dx^{i}dx^{k} \end{aligned}$$

on $N\times [0,1]$ with metric $h_{ik}$ on $N$ and the Friedmann equation

$$\begin{aligned} \left( \frac{\dot{a}(t)}{c\cdot a(t)}\right) ^{2}+\frac{k}{a(t)^{2}}=\kappa \frac{\rho }{3} \end{aligned}$$

with the scaling factor $a(t)$, curvature $k=0,\pm 1$ and $\kappa =\frac{8\pi G}{c^{2}}$. As an example we consider a 3-dimensional submanifold $N$ with energy density $\rho _{N}$ and curvature $R_{N}$ (related to $h$) fixed embedded in the space-time. Next we assume that the 3-manifold $N$ possesses a homogenous metric of constant curvature. For a fixed time $t$, the scalar curvature of $N$ is proportional to

$$\begin{aligned} R_{N}\sim \frac{3k}{a(t)^{2}} \end{aligned}$$

and by using the Friedmann equation above, one obtains

$$\begin{aligned} \rho _{N}=\frac{1}{\kappa }R_{N}+\rho _{c} \end{aligned}$$

with the critical density

$$\begin{aligned} \rho _{c}=\frac{3}{\kappa }\left( \frac{\dot{a}(t)}{c\cdot a(t)}\right) ^{2}=\frac{3H^{2}}{\kappa } \end{aligned}$$

and the Hubble constant $H$

$$\begin{aligned} H=\frac{\dot{a}}{c\cdot a}. \end{aligned}$$

The total energy of $N$ is given by

$$\begin{aligned} E_{N}=\int \limits _{N}\rho _{N}\,\sqrt{h}d^{3}x=\frac{1}{\kappa }\int \limits _{N}R_{N}\sqrt{h}d^{3}x+\rho _{c}\cdot vol(N)\,. \end{aligned}$$

(21)

For a space with constant curvature $R_{N}$ we obtain

$$\begin{aligned} E_{N}=\left( \frac{1}{\kappa }R_{N}+\rho _{c}\right) \cdot vol(N) \end{aligned}$$

(22)