Gravitationally Bound Bose Condensates with Rotation

In the weak field approximation tp10 ; sc16 , $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ where $|h_{\mu\nu}|\ll|\eta_{\mu\nu}|$, if we set $\sqrt{-g}\approx 1+\frac{1}{2}h$, where $h=\eta^{\mu\nu}h_{\mu\nu}$ is the trace of $h_{\mu\nu}$, then the scalar field action can be written as the sum of three terms,

$$S_{\phi}=S_{0}[\phi]-\int d^{4}x\left[\frac{1}{2}h\left(\eta^{\alpha\beta}(\nabla_{\alpha}\phi)^{*}(\nabla_{\beta}\phi)+V(|\phi|)\right)-h^{\alpha\beta}(\nabla_{\alpha}\phi)^{*}(\nabla_{\beta}\phi)\right]$$

(4)

to linear order in $h_{\mu\nu}$. The zeroeth order action, $S_{0}[\phi]$, is the action in (1), but on a flat background. We will call the trace term $S_{1}[h,\phi]$ and the second correction term $S_{2}[h_{\alpha\beta},\phi]$. In the non-relativistic limit, the field can be described by a complex wavefunction $\psi$ according to

$$\phi=\frac{\hbar}{\sqrt{2m}}e^{-imc^{2}t/\hbar}\psi,$$

(5)

which, at low temperatures, describes a condensed, $N$-particle state and will be normalized as $\int d^{3}{\vec{r}}|\psi|^{2}=N$ in what follows. We will also take the scalar field potential to be of the form

$$V(|\phi|)=\frac{mc^{2}}{2}|\psi|^{2}+{\widetilde{V}}(|\psi|).$$

(6)

Expanding the zeroeth order action in the non-relaivistic limit, one finds

$$S_{0}[\phi]\approx\int dt\int d^{3}{\vec{r}}\left[\frac{i\hbar}{2}\psi^{*}\overleftrightarrow{\nabla_{t}}\psi-\frac{\hbar^{2}}{2m}|\nabla\psi|^{2}-{\widetilde{V}}(|\psi|)\right]$$

(7)

where we have dropped terms quadratic in $\dot{\psi}/c$, using the fact that, in the non-relativistic approximation, the difference between the total energy and the rest mass energy is supposed to be small. In the same approximation,

$$S_{1}[\phi]\approx\int dt\int d^{3}{\vec{r}}~h\left[\frac{i\hbar}{2}\psi^{*}\overleftrightarrow{\nabla_{t}}\psi-\frac{\hbar^{2}}{2m}|\nabla\psi|^{2}-{\widetilde{V}}(|\psi|)\right].$$

(8)

Assuming the gravitational field is weak ($|h|\ll 1$), this term may therefore be dropped, as also the last term in the expansion of $S_{2}[h_{\alpha\beta},\phi]$,

	$\displaystyle S_{2}[h_{\alpha\beta},\phi]$	$\displaystyle=$	$\displaystyle\int d^{4}x~h^{\alpha\beta}(\nabla_{\alpha}\phi)^{*}(\nabla_{\beta}\phi)$		(9)
		$\displaystyle=$	$\displaystyle\int d^{4}x\left[h^{tt}|\dot{\phi}|^{2}+h^{ti}\left(\dot{\phi}^{*}\nabla_{i}\phi+\nabla_{i}\phi^{*}\dot{\phi}\right)+h^{ij}\nabla_{(i}\phi^{*}\nabla_{j)}\phi\right]$		(11)
		$\displaystyle\approx$	$\displaystyle\int dt\int d^{3}{\vec{r}}\left[\frac{1}{2}mc^{4}h^{tt}|\psi|^{2}+\frac{i\hbar c^{2}}{2}h^{ti}\left(\psi^{*}\nabla_{i}\psi-\nabla_{i}\psi^{*}\psi\right)\right].$		(13)

If we call $h^{ti}=-A^{i}/c^{2}$, $h^{tt}=-2\Phi_{G}/c^{4}$, then this correction term looks like

$$S_{2}[h_{\alpha\beta},\phi]\approx\int dt\int d^{3}{\vec{r}}\left[-m\Phi_{G}|\psi|^{2}-J\cdot A\right],$$

(14)

where $\Phi_{G}$ is the gravitational potential energy (to be obtained from the Einstein equations) and

$$J_{i}=\frac{i\hbar}{2}\psi^{*}\overleftrightarrow{\nabla_{i}}\psi$$

(15)

is the scalar current. Putting everything together, the non-realtivistic, weak field GPP action for the scalar field is

$$S_{\psi}\approx\int d^{4}x\left[\frac{i\hbar}{2}\psi^{*}\overleftrightarrow{\nabla_{t}}\psi-\frac{\hbar^{2}}{2m}|\nabla\psi|^{2}-{\widetilde{V}}(|\psi|)-m\Phi_{G}|\psi|^{2}-J\cdot A\right].$$

(16)

The third term in the action above represents the self-interaction of the non-relativistic field, the fourth term represents its interaction with the gravitational field and the last term is the gravitomagnetic term, which incorporates frame dragging.

The non-relativistic action may be put in a more suggestive form if we define $A_{\mu}=(-\Phi_{G},A_{i})$ and the “covariant” derivative

$$D_{\mu}\psi=\left(\nabla_{\mu}-\frac{im}{\hbar}A_{\mu}\right)\psi.$$

(17)

Then,

$$\frac{i\hbar}{2}\psi^{*}\overleftrightarrow{D_{t}}\psi=\frac{i\hbar}{2}\psi^{*}\overleftrightarrow{\nabla_{t}}\psi+mA_{t}|\psi|^{2}=\frac{i\hbar}{2}\psi^{*}\overleftrightarrow{\nabla_{t}}\psi-m\Phi_{G}|\psi|^{2}$$

(18)

and

$$-\frac{\hbar^{2}}{2m}(D_{i}\psi)^{*}(D^{i}\psi)=-\frac{\hbar^{2}}{2m}|\nabla\psi|^{2}-J\cdot A$$

(19)

upon dropping terms that are quadratic in the gravitational field. Thus our action reads,

$$S_{\psi}\approx\int d^{4}x\left[\frac{i\hbar}{2}\psi^{*}\overleftrightarrow{D_{t}}\psi-\frac{\hbar^{2}}{2m}|D_{i}\psi|^{2}-{\widetilde{V}}(|\psi|)\right],$$

(20)

so, in the weak field limit, the interaction of the scalar field with the gravitational field has the (well-known) form of an “electromagnetic” coupling.

Turning to the gravitational part of the action, we first examine the Einstein equations of motion to determine the general form of the metric. It is convenient to work in the harmonic gauge, defining ${\overline{h}}_{\mu\nu}=h_{\mu\nu}-\frac{1}{2}\eta_{\mu\nu}h$ and imposing the condition ${{\overline{h}}_{\mu\nu}}^{,\nu}=0$. The Einstein tensor is $G_{\mu\nu}=\frac{1}{2}\Box{\overline{h}}_{\mu\nu}$ and Einstein’s equations are

$$\Box{\overline{h}}_{\mu\nu}=\frac{16\pi G}{c^{4}}T_{\mu\nu}$$

(21)

The gauge condition, ${{\overline{h}}_{\mu\nu}}^{,\nu}=0$ tells us that the stress tensor is conserved on a flat background, so $T_{\mu\nu}$ is to be evaluated for the field on a flat background,

$$T_{\mu\nu}=\nabla_{\mu}\phi^{*}\nabla_{\nu}\phi+\nabla_{\nu}\phi^{*}\nabla_{\mu}\phi+\eta_{\mu\nu}\mathcal{L}.$$

(22)

In the non-relativistic limit, keeping only leading order terms, we find

	$\displaystyle T_{tt}$	$\displaystyle\approx$	$\displaystyle mc^{4}|\psi|^{2}$		(23)
	$\displaystyle T_{ti}$	$\displaystyle\approx$	$\displaystyle c^{2}J_{i}$		(24)
	$\displaystyle T_{ij}$	$\displaystyle\approx$	$\displaystyle 0$		(25)

This shows that we can take ${\overline{h}}_{ij}=0$. In this case, ${\overline{h}}=-h=-{\overline{h}}_{tt}/c^{2}$. If we take ${\overline{h}}_{tt}=-4\Phi_{G}$ then ${\overline{h}}=4\Phi_{G}/c^{2}$ and $h_{tt}=-2\Phi_{G}$. The remaining metric coefficients are

$$h_{ij}={\overline{h}}_{ij}-\frac{1}{2}\eta_{ij}{\overline{h}}=-\frac{2\Phi_{G}}{c^{2}}\delta_{ij},~~h_{ti}={\overline{h}}_{ti}=-A_{i}$$

(26)

and the line element,

$$ds^{2}=c^{2}(1+2\Phi_{G}/c^{2})dt^{2}+2A_{i}dtdx^{i}-(1-2\Phi_{G}/c^{2})\delta_{ij}dx^{i}dx^{j},$$

(27)

is subject to the gauge conditions,

$${{\overline{h}}_{t\mu}}^{,\mu}=\frac{4\dot{\Phi}_{G}}{c^{2}}-\nabla\cdot A=0,~~{{\overline{h}}_{i\mu}}^{,\mu}=\frac{1}{c^{2}}\dot{A}_{i}=0.$$

(28)

In the non-relativistic limit, we may ignore the term $\dot{\Phi}_{G}/c^{2}$, so our gauge conditions become $\dot{A}_{i}=0=\nabla\cdot A$.

To set up the total action it is convenient to compare the line element in (27) to the standard line element of Arnowitt, Deser and Misner (ADM) adm62

$$ds^{2}=N^{2}dt^{2}-N_{i}dtdx^{i}-\gamma_{ij}dx^{i}dx^{j}$$

(29)

and write the bulk Lagrangian for the gravitational field in terms of the lapse function, $N$, the shift, $N_{i}$, the first fundamental form, $\gamma_{ij}$, the intrinsic curvature scalar of spatial hypersurfaces, ${}^{(3)}R$, and the extrinsic curvature (of spatial hypersurfaces), $K_{ij}$,

$$\mathfrak{L}_{G}=\frac{c^{3}N}{16\pi G}\sqrt{\gamma}\left[{}^{(3)}R+K_{ij}K^{ij}-K^{2}\right]$$

(30)

where $K$ is the trace of $K_{ij}$. This gives us the lapse, shift and first fundamental form respectively in linear approximation,

	$\displaystyle N\approx c\left(1+\frac{\Phi_{G}}{c^{2}}\right)$		(31)
	$\displaystyle N_{i}=-A_{i}$		(32)
	$\displaystyle\gamma_{ij}=\delta_{ij}\left(1-\frac{2\Phi_{G}}{c^{2}}\right),$		(33)

from which we find the intrinsic curvature of the spatial hypersurfaces up to second order

$${}^{(3)}R=\frac{4}{c^{2}}\nabla^{2}\Phi_{G}+\frac{2}{c^{4}}\left[3(\nabla\Phi_{G})^{2}+8\Phi_{G}\nabla^{2}\Phi_{G}\right]$$

(34)

and, up to first order, the extrinsic curvature

$$K_{ij}=\frac{1}{2N}\left[\dot{\gamma}_{ij}-\nabla_{(i}N_{j)}\right]\approx\frac{1}{2c}{\nabla_{(i}A_{j)}}=\frac{f_{ij}}{2c},$$

(35)

where we have defined $f_{ij}=\nabla_{(i}A_{j)}$. In this approximation, the trace of the extrinsic curvature, $K=\eta^{ij}K_{ij}$ vanishes by virtue of the gauge condition. After some algebra, we find the gravitational action

$$S_{G}=\beta\int dt\int d^{3}{\vec{r}}\left(-(\nabla\Phi_{G})^{2}+\frac{c^{2}}{8}f_{ij}f^{ij}\right),$$

(36)

where $\beta=(8\pi G)^{-1}$. The resulting total effective action,

$$S=\int dt\int d^{3}{\vec{r}}\left[\frac{i\hbar}{2}\psi^{*}\overleftrightarrow{\nabla_{t}}\psi-\frac{\hbar^{2}}{2m}|\nabla\psi|^{2}-{\widetilde{V}}(|\psi|)-m\Phi_{G}|\psi|^{2}-J\cdot A-\beta(\nabla\Phi_{G})^{2}+\frac{\beta c^{2}}{8}f_{ij}f^{ij}\right],\\$$

(37)

now allows us to use the non-linear Schroedinger equation to describe large scale structure, so long as $\psi(t,{\vec{r}})$ is interpreted as a Schroedinger field, normalized to the particle number, as mentioned earlier.

Extremizing the action in (37) with respect to variations in $\psi$ and employing the gauge conditions gives non-linear Schroedinger equation for $\psi$

$$i\hbar\mathcal{D}_{t}\psi=-\frac{\hbar^{2}}{2m}\nabla^{2}\psi+m\Phi_{G}\psi+{\widetilde{V}}^{\prime}(|\psi|)\psi$$

(38)

where $\mathcal{D}_{t}=\left(\nabla_{t}-A^{i}\nabla_{i}\right)$ is the transport derivative, which takes into account frame dragging due to the rotation. Likewise, varying $A_{i}$ and $\Phi_{G}$ give

$$\nabla_{j}f^{ji}=\nabla^{2}A^{i}=-\frac{16\pi G}{c^{2}}{J^{i}}$$

(39)

(employing the gauge condition) and

$$\nabla^{2}\Phi_{G}=4\pi Gm|\psi|^{2},$$

(40)

up to first order. These are readily solved by

	$\displaystyle\Phi_{G}({\vec{r}})=-4\pi Gm\int d^{3}{\vec{r}}^{\prime}~\frac{\psi^{*}(t,{\vec{r}}^{\prime})\psi(t,{\vec{r}}^{\prime})}{|{\vec{r}}-{\vec{r}}^{\prime}|}$		(41)
			(42)
	$\displaystyle A_{i}({\vec{r}})=\frac{8\pi iG\hbar}{c^{2}}\int d^{3}{\vec{r}}^{\prime}~\frac{\psi^{*}(t,{\vec{r}}^{\prime})\overleftrightarrow{\nabla^{\prime}}_{i}\psi(t,{\vec{r}}^{\prime})}{|{\vec{r}}-{\vec{r}}^{\prime}|}$		(43)

respectively.

It is often useful to treat the BEC as a superfluid. We now consider the hydrodynamic analogy by employing the Madelung transformation m27 to rewrite equation (38). Setting phc11 ; bh07 ; phc16

$$\psi(t,{\vec{r}})=\sqrt{\nu(t,{\vec{r}})}~e^{iS(t,{\vec{r}})}$$

(44)

where $\nu$ is the number density of particles and $S$ is the real action. By comparing the real and imaginary parts of the Schroedinger equation for $\psi$ we find

	$\displaystyle{\mathcal{D}}_{t}\nu+\frac{\hbar}{m}\left(\nabla\nu\cdot\nabla S+\nu\nabla^{2}S\right)=0$		(45)
			(46)
	$\displaystyle{\mathcal{D}}_{t}S+\frac{\hbar}{2m}(\nabla S)^{2}+\frac{m}{\hbar}\Phi_{G}+\frac{1}{\hbar}{\widetilde{V}}^{\prime}(\nu)+Q(\nu)=0$		(47)

where $Q$ is the “quantum potential”

$$Q=-\frac{\hbar}{4m}\left[\frac{\nabla^{2}\nu}{\nu}-\frac{1}{2}\left(\frac{\nabla\nu}{\nu}\right)^{2}\right]$$

(48)

If we introduce the velocity field, $u=\hbar\nabla S/m$, then the first of the above equations,

$${\mathcal{D}}_{t}\nu+\nabla\cdot(\nu u)=0,$$

(49)

has the form of a continuity equation. The second can also be put in an interesting form: write it as

$$\nabla_{t}u-\nabla(A\cdot u)+\frac{1}{2}\nabla(u)^{2}+\nabla\Phi_{G}+\nabla\left(\frac{1}{m}{\widetilde{V}}^{\prime}(\nu)\right)+\frac{\hbar\nabla Q}{m}=0$$

(50)

and notice that $u$ is irrotational, so $\nabla_{i}u_{j}=\nabla_{j}u_{i}$ implying that $\nabla(u^{2})=2(u\cdot\nabla)u$. Likewise $\nabla_{i}(A\cdot u)=(A\cdot\nabla)u_{i}+u_{j}\nabla_{i}A_{j}$, therefore

$${\mathcal{D}}_{t}u_{i}+(u\cdot\nabla)u_{i}=-\frac{1}{m\nu}\nabla_{i}P-\nabla_{i}\Phi_{G}+u_{j}\nabla_{i}A_{j}-\frac{\hbar}{m}\nabla_{i}Q$$

(51)

where $m\nu$ represents the mass density of the BEC and $P$ is the pressure given by

$$\nabla_{i}P=\nu\nabla_{i}{\widetilde{V}}^{\prime}(\nu).$$

(52)

This is the general form of the equation of state. For example, for quartic interactions, ${\widetilde{V}}(\nu)=\frac{\lambda}{4}\nu^{2}$, we find bh07

$$P=\frac{\lambda}{4}\nu^{2}.$$

(53)

Thus, not surprisingly, a repulsive interaction ($\lambda>0$) leads to a positive pressure and an attractive interaction ($\lambda<0$) to a negative pressure. The effect of frame dragging is contained in the transport derivative, ${\mathcal{D}}_{t}$. In terms of the particle density and real action, the gravitational potential and shift are given by

	$\displaystyle\Phi_{G}({\vec{r}})=-4\pi Gm\int d^{3}{\vec{r}}^{\prime}~\frac{\nu(t,{\vec{r}}^{\prime})}{|{\vec{r}}-{\vec{r}}^{\prime}|}$		(54)
			(55)
	$\displaystyle A_{i}({\vec{r}})=-\frac{16\pi Gm}{c^{2}}\int d^{3}{\vec{r}}^{\prime}~\frac{\nu(t,{\vec{r}}^{\prime})u_{i}(t,{\vec{r}}^{\prime})}{|{\vec{r}}-{\vec{r}}^{\prime}|}$		(56)

respectively.

For simplicity, we will analyze the situation when all particles are in the $k=1=l$ eigenstate of angular momentum, i.e., we take

$$\psi(t,{\vec{r}})=wF(r/R)\sin\theta e^{i\varphi}e^{i\mu t}.$$

(59)

The procedure below can be extended to arbitrary angular momentum states. The normalization condition gives,

$$w=\sqrt{\frac{3N}{8\pi R^{3}C_{22}}}$$

(60)

where $N$ is the number of bosons in the BEC and $C_{22}=\int_{0}^{\infty}d\xi~\xi^{2}F^{2}(\xi)$. To find an expression for $\Phi_{G}$, we use the expansion of the Green function in spherical harmonics with vanishing boundary conditions at the origin and at infinity,

$$\frac{1}{|{\vec{r}}-{\vec{r}}^{\prime}|}={4\pi}\sum_{l,m}\frac{1}{2l+1}\frac{r_{<}^{l}}{r_{>}^{l+1}}Y^{*}_{lm}(\theta^{\prime},\varphi^{\prime})Y_{lm}(\theta,\phi)$$

(61)

and find

	$\displaystyle\Phi_{G}(r,\theta)$	$\displaystyle=$	$\displaystyle-\frac{3GmN}{2C_{22}R}\left\{\frac{2}{3}\left[\frac{R}{r}\int_{0}^{r/R}d\eta\eta^{2}F^{2}(\eta)+\int_{r/R}^{\infty}d\eta\eta F^{2}(\eta)\right]\right.$		(64)
			$\displaystyle\left.\hskip 42.67912pt+\frac{1}{15}(1-3\cos^{2}\theta)\left[\frac{R^{3}}{r^{3}}\int_{0}^{r/R}d\eta~\eta^{4}F^{2}(\eta)+\frac{r^{2}}{R^{2}}\int_{r/R}^{\infty}\frac{d\eta}{\eta}F^{2}(\eta)\right]\right\}.$

Now from (43) it follows that only $A_{\phi}$ survives (in this stationary state), and

$$A_{\phi}(r,\theta)=\frac{4\hbar}{mc^{2}}~\Phi_{G}(r,\theta).$$

(65)

Putting these results into (57), and taking, for simplicity, quartic interactions of arbitrary sign, $V(|\psi|)=\frac{\lambda}{4}|\psi|^{4}$, we find the four terms in the total energy to be

	$\displaystyle H_{K}(R)$	$\displaystyle=$	$\displaystyle\frac{N\hbar^{2}}{2mR^{2}C_{22}}\int_{0}^{\infty}d\xi\left[\xi^{2}F^{\prime 2}(\xi)+2F^{2}(\xi)\right]$		(66)
	$\displaystyle H_{V}(R)$	$\displaystyle=$	$\displaystyle\frac{3\lambda N^{2}}{40\pi R^{3}C_{22}^{2}}\int d\xi~\xi^{2}F^{4}(\xi)$		(68)
	$\displaystyle H_{\Phi}(R)$	$\displaystyle=$	$\displaystyle-\frac{Gm^{2}N^{2}}{2RC_{22}^{2}}\left[\int_{0}^{\infty}d\xi~\xi F^{2}(\xi)\int_{0}^{\xi}d\eta~\eta^{2}F^{2}(\eta)+\int_{0}^{\infty}d\xi~\xi^{2}F^{2}(\xi)\int_{\xi}^{\infty}d\eta~\eta F^{2}(\eta)\right.$		(72)
			$\displaystyle\hskip 28.45274pt\left.+\frac{1}{25}\int_{0}^{\infty}\frac{d\xi}{\xi}F^{2}(\xi)\int_{0}^{\xi}d\eta~\eta^{4}F^{2}(\eta)+\frac{1}{25}\int_{0}^{\infty}d\xi~\xi^{4}F^{2}(\xi)\int_{\xi}^{\infty}\frac{d\eta}{\eta}F^{2}(\eta)\right]$
	$\displaystyle H_{A}(R)$	$\displaystyle=$	$\displaystyle\frac{3\hbar^{2}GN^{2}}{c^{2}C_{22}^{2}R^{3}}\left[\int_{0}^{\infty}\frac{d\xi}{\xi}F^{2}(\xi)\int_{0}^{\xi}d\eta~\eta^{2}F^{2}(\eta)+\int_{0}^{\infty}d\xi F^{2}(\xi)\int_{\xi}^{\infty}d\eta~\eta F^{2}(\eta)\right]$		(74)

where $\xi=r/R$ is a dimensionless variable. The second term in the integral in the expression for the kinetic energy, $H_{K}$, represents the contribution of the azimuthal motion of the condensate. $H_{\Phi}$ and $H_{A}$ are the contributions of the gravitational field. These expressions may be put into simpler form in terms of the coefficients

	$\displaystyle C_{mn}=\int_{0}^{\infty}d\xi~\xi^{m}F^{n}(\xi)$		(75)
			(76)
	$\displaystyle B_{mn}=\int_{0}^{\infty}d\xi~\xi^{m}F^{\prime n}(\xi)$		(77)
			(78)
	$\displaystyle D_{mn}=\int_{0}^{\infty}d\xi~\xi^{m}F^{2}(\xi)\int_{0}^{\xi}d\eta~\eta^{n}F^{2}(\eta)$		(79)
			(80)
	$\displaystyle A_{mn}=\int_{0}^{\infty}d\xi~\xi^{m}F^{2}(\xi)\int_{\xi}^{\infty}d\eta~\eta^{n}F^{2}(\eta).$		(81)

We find

	$\displaystyle H(R)$	$\displaystyle=$	$\displaystyle H_{\Phi}(R)+H_{K}(R)+H_{V}(R)+H_{A}(R)$		(82)
		$\displaystyle=$	$\displaystyle-\frac{Gm^{2}N^{2}}{2RC_{22}^{2}}\left(D_{12}+A_{21}+\frac{1}{25}(D_{-14}+A_{4,-1})\right)+\frac{N\hbar^{2}(B_{22}+2C_{02})}{2mR^{2}C_{22}}$		(86)
			$\displaystyle\hskip 113.81102pt+\frac{3\lambda N^{2}C_{24}}{40\pi R^{3}C_{22}^{2}}+\frac{3G\hbar^{2}N^{2}}{c^{2}R^{3}C_{22}^{2}}\left(D_{-12}+A_{01}\right).$

$H(R)$ does not include the rest mass energy, $Nmc^{2}$, of the condensate and the total energy can be written as $E(R)\approx Nmc^{2}+H(R)$. If $H_{B}(R)$ is the binding energy per particle, then $E(R)=N|mc^{2}-H_{B}(R)|$. We can therefore identify $|H_{B}(R)|=|H(R)|/N$. In fact, using previous results esvw14 , let us define the dimensionless parameters $\rho$ and $n$ by

$$R=\frac{M_{p}}{m}\sqrt{\frac{|\lambda|}{\hbar c}}~\rho,~~N=\frac{nM_{p}}{m^{2}}\left(\frac{\hbar}{c}\right)^{3/2}\frac{c}{\sqrt{|\lambda|}},$$

(87)

where $M_{p}=\sqrt{\hbar c/G}$ is the Planck mass. Then the binding energy per unit mass may be given as,

$$\mathcal{H}(\rho)=\frac{H}{mN}=\frac{\hbar^{3}c}{|\lambda|M_{p}^{2}}\left[\frac{A}{\rho}+\frac{B}{\rho^{2}}+\frac{C}{\rho^{3}}\right],$$

(88)

where

	$\displaystyle A=-\frac{n}{2C_{22}^{2}}\left(D_{12}+A_{21}+\frac{1}{25}(D_{-14}+A_{4,-1})\right)$		(89)
	$\displaystyle B=\frac{1}{2}\left(\frac{B_{22}+2C_{02}}{C_{22}}\right)$		(90)
	$\displaystyle C=n\left({\text{sgn}}(\lambda)\frac{3C_{24}}{40\pi C_{22}^{2}}+\frac{3\hbar^{3}}{|\lambda|cM_{p}^{2}C_{22}^{2}}(D_{-12}+A_{01})\right).$		(91)

The coefficient $A$ characterizes the contribution of the gravitational potential energy to the total energy and is negative and $B$ represents the contribution of the kinetic energy of the bosons. The effects of the rotation are contained in a contribution to the kinetic energy through the coefficient $C_{02}$ in $B$ and in the last term in the expression for $C$, which combines the contributions of the scalar potential and frame dragging. This term is proportional to $3n/b^{2}$, where $b$ is a dimensionless parameter, $b^{2}=|\lambda|cM_{p}^{2}/\hbar^{3}$, characterizing the strength of the self interactions.

The extrema of ${\mathcal{H}}$ in (88) are readily found to be given by

$${\rho_{\text{eq}}}=\frac{B}{|A|}\mp\sqrt{\frac{B^{2}}{|A|^{2}}+\frac{3C}{|A|}},$$

(92)

assuming it is real, i.e., provided that $B^{2}>3|A||C|$ when $C<0$. In what follows, we set

	$\displaystyle A$	$\displaystyle=$	$\displaystyle an$		(93)
	$\displaystyle C$	$\displaystyle=$	$\displaystyle qn\left(\frac{d}{qb^{2}}+{\text{sgn}}(\lambda)\right)$		(94)

where the parameters $a$, $q$ and $d$ are obtained from the constants defined in (91), viz.,

	$\displaystyle a=-\frac{1}{2C_{22}^{2}}\left(D_{12}+A_{21}+\frac{1}{25}(D_{-14}+A_{4,-1})\right)$		(95)
			(96)
	$\displaystyle q=\frac{3C_{24}}{40\pi C_{22}^{2}}$		(97)
			(98)
	$\displaystyle d=\frac{3}{C_{22}^{2}}\left(D_{-12}+A_{01}\right).$		(99)

The first , $a$, characterizes the strength of the gravitational interaction, the second, $q$, characterizes the strength of the self interaction and the last, $d$, the effect of the frame dragging. In terms of these, the dimensionless equilibrium radius is

$${\rho_{\text{eq}}}=\frac{B}{|a|n}\left[1+\sqrt{1+\frac{3|a|q}{B^{2}}\left(\frac{d}{qb^{2}}+{\text{sgn}}(\lambda)\right)n^{2}}\right].$$

(100)

For it to exist, we must require the quantity under the radical to be non-negative. If the kinetic energy of the bosons is ignored, as in the Thomas-Fermi approximation, $B$ approaches zero and the equilibrium radius approaches a constant independent of the number of particles. In the absence of rotation ($d=0$) this equilibrium is possible only for repulsive self interactions. If one does not ignore the kinetic energy of the bosons then the equilibrium radius decreases with increasing $n$.

When $C>0$ an absolute minimum of the energy exists, as the interactions together with the rotations are able to stabilize the condensate. When $C<0$ only attractive interactions are permitted and the energy of the system is unbounded from below, but we have found a local minimum in (100), not an absolute minimum, of the energy. This local minimum exists only when the number of particles is below a certain critical value, ${n_{\text{c}}}$, which will be determined below. It is important to discuss the validity of this solution.

There are three criteria that it must satisfy in order to be self consistent, the first two arising from the GEM approximation. First, the de Broglie wavelength of the bosons should be much larger than their Compton wavelength to ignore special relativistic corrections. Second, the size of the condensate at the minimum has to be much greater than the Schwarzschild radius of the corresponding mass, to justify not using general relativistic corrections. The impact of these two conditions on the range of the parameters is determined below and the conditions are satisfied by the examples we give. When the energy is unbounded from below (in the case $C<0$, $\lambda<0$) the local minimum we have found could be unstable. This can be avoided if additional stabilizing terms exist in the effective potential (as is the case for QCD axions, where the effective potential is given by the Bessel function $J_{0}$ esvw14 ). Irrespective of whether there is a stabilization term, the configuration at the local minimum could be a viable state, provided it has a lifetime greater than the age of the universe. This would occur if the probability of tunneling out of it is sufficiently small. The lifetime of metastable BECs formed by bosonic atoms with attractive interactions, confined by a harmonic trap, was estimated using instanton methods in st97 ; frar99 . One can use similar techniques to compute the lifetimes of gravitationally bound BEC systems against tunneling out of the metastable local minimum state. The result is that the tunneling rate behaves approximately as $\sim\exp(-2NJ/\hbar)$, where $N$ is the number of particles and $J$ is the WKB expression for the exponent of the tunneling rate. In the case of an astrophysical BEC, the number of particles is very large ($N\sim 10^{58}$ – $10^{100}$) and the WKB expression, $J$, is not vanishingly small unless the fraction $f=n/{n_{\text{c}}}$ is commensurately close to unity. Thus so long as $f<1$ the lifetime of the metastable state grows exponentially with the particle number.