# Static solution to the Friedmann equation [duplicate]

I want to find the static solution for the Friedmann equation:

$$\bigg(\frac{\dot{a}}{a}\bigg)^2=H_0^2\bigg(\Omega_m\bigg(\frac{a_0}{a}\bigg)^3+\Omega_v+\Omega_k\bigg(\frac{a_0}{a}\bigg)^2\bigg)$$

When there is no radiation. $$\Omega_m$$ is the matter parameter, $$\Omega_v$$ is the vacuum energy and $$\Omega_k$$ is the curvature parameter. Since I wanted a static solution I tried $$\dot{a}=0$$ and tried to solve for $$a$$, but this gives me a horrible equation with no clear solution.

Any suggestions?

When the universe is static (i.e., $$\ddot a = \dot a = 0$$), the Hubble parameter is always zero since

$$H = \frac{{\dot a}}{a} = 0.$$

In this case, the dimensional densities, $${\Omega _i} = \frac{{8\pi G}}{{3{H^2}}}{\rho _i}$$, are infinite, so, the right hand side of your equation is indefinite (actually, this is a common question about deriving Einstein's static universe from this equation). Therefore, it is not appropriate to use the Friedmann equation in terms of the density parameters in this case. In fact, you need to work with the original Einstein field equations (the standard Friedmann equations) as

$$\frac{{{{\dot a}^2} + k{c^2}}}{{{a^2}}} = \frac{{8\pi G\rho + \Lambda {c^2}}}{3},$$ and $$\frac{{\ddot a}}{a} = - \frac{{4\pi G}}{3}\left( {\rho + \frac{{3p}}{{{c^2}}}} \right) + \frac{{\Lambda {c^2}}}{3}.$$

The right hand side of the second equation has to be vanishes in order to have a static universe. So, using the second equation and assuming $$\ddot a = \dot a = 0$$ together with $$p=0$$ for matter, you immediately arrive at

$$4\pi G{\rho _{m}} = \Lambda {c^2},$$

where $${\rho _{m}}$$ is the energy density of the matter in the universe. According to the first (Friedmann) equation, $$k$$ has to be positive in the Einstein's static universe (in order to the universe be static), so using this equation the radius of Einstein's universe is obtained as

$${\cal R} = \frac{c}{{\sqrt {4\pi G{\rho _m}} }} = \frac{1}{{\sqrt \Lambda }}.$$

These conditions define the Einstein's static universe. If you have trouble understanding the last relationship or if you want more information with additional discussions about Einstein's static universe (such as instability), you can see this introductory article.

In addition, by a simple search ("Einstein static universe"), you could find a number of good relevant discussions in Physics Stack Exchange, for example, see these:

especially this one: