Borde–Guth–Vilenkin theorem

Here is an example of derivation of the BGV theorem for an expanding homogeneous isotropic flat universe (in units of speed of light c=1).[6] Which is consistent with ΛCDM model, the current model of cosmology. However, this derivation can be generalized to an arbitrary space-time with no appeal to homogeneity or isotropy.[6]

The Friedmann–Lemaître–Robertson–Walker metric is given by

${\displaystyle ds=dt^{2}-a^{2}(t)dx_{i}dx^{i}}$,

where t is time, xⁱ (i=1,2,3) are the spatial coordinates and a(t) is the scale factor. Along a timeline geodesic x_i = constant, we can consider the universe to be filled with comoving particles. For an observer with proper time τ following the world line x^μ(τ), has a 4-momentum ${\displaystyle P^{\mu }=mdx^{\mu }/d\tau =(E,\mathbf {p} )}$, where ${\displaystyle E={\sqrt {p^{2}+m^{2}}}}$ is the energy, m is the mass and p=|p| the magnitude of the 3-momentum.

From the geodesic equation of motion, it follows that ${\displaystyle p(t)=p_{\rm {f}}\;a(t_{\rm {f}})/a(t)}$ where p_f is the final momentum at time t_f. Thus

${\displaystyle \int _{t_{\rm {i}}}^{t_{\rm {f}}}H(\tau )d\tau =\int _{a(t_{\rm {i}})}^{a(t_{\rm {f}})}{\frac {mda}{\sqrt {m^{2}a^{2}+p^{2}a(t_{\rm {f}})}}}=F(\gamma _{\rm {f}})-F(\gamma _{\rm {i}})\leq F(\gamma _{\rm {f}})}$,

where t_i is an initial time, ${\displaystyle H={\dot {a}}/a}$ is the Hubble parameter, and

${\displaystyle F(\gamma )={\frac {1}{2}}\ln \left({\frac {\gamma +1}{\gamma -1}}\right)}$,

γ being the Lorentz factor. For any non-comoving observer γ>1 and F(γ)>0.

The expansion rate averaged over the observer world line can be defined as

${\displaystyle H_{\rm {av}}={\frac {1}{\tau _{\rm {f}}-\tau _{\rm {i}}}}\int _{t_{\rm {i}}}^{t_{\rm {f}}}H(\tau )d\tau }$.

Assuming ${\displaystyle H_{\rm {av}}>0}$ it is follows that

${\displaystyle \tau _{\rm {f}}-\tau _{\rm {i}}\leq {\frac {F(\gamma _{\rm {f}})}{H_{\rm {av}}}}}$.

Thereby any non-comoving past-directed timelike geodesic satisfying the condition ${\displaystyle H_{\rm {av}}>0}$, must have a finite proper length, and so must be past-incomplete.