- #1
copernicus1
- 99
- 0
If I start with the standard FRW cosmology equations,
$${\eqalign{
3\dot a^2/a^2&=8\pi\rho-3k/a^2\cr
3\ddot a/a&=-4\pi\left(\rho+3P\right),}}$$
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?
$${\eqalign{
3\dot a^2/a^2&=8\pi\rho-3k/a^2\cr
3\ddot a/a&=-4\pi\left(\rho+3P\right),}}$$
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?