If I start with the standard FRW cosmology equations,(adsbygoogle = window.adsbygoogle || []).push({});

$${\eqalign{

3\dot a^2/a^2&=8\pi\rho-3k/a^2\cr

3\ddot a/a&=-4\pi\left(\rho+3P\right),}}$$

and set [/tex]\rho=P=0[/itex] (or $T^{\mu\nu}=0$), I have

$${\eqalign{

3\dot a^2/a^2&=-3k/a^2\cr

3\ddot a/a&=0.}}$$

The second equation gives $$\ddot a=0,$$ but $$\dot a$$ seems to depend on the value of k.

Namely, if I set k=0, then $$\dot a=0$$ and this leads to an ordinary Minkowski space metric. If I choose k=+1, then a is complex and that doesn't seem physical, but if I set k=-1, then I can get $$3\dot a^2/a^2=3/a^2~~~~\Longrightarrow~~~~\dot a=1~~~~\Longrightarrow~~~~a=t,$$ which I suppose describes a spatially hyperbolic universe (k=-1) with no energy/matter content, where spatial distances increase linearly in time.

Do we just ignore this solution based on the assumption that nonzero k implies the presence of mass/energy by definition, or have I gone wrong in my reasoning somewhere?

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# Can empty space curve?

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