johne1618
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Starting with the Robertson-Walker metric
\large ds^2 = -dt^2 + a^2(t) [ \frac{dr^2}{1-kr^2} + r^2(d\theta^2+\sin^2\theta d\phi^2]
Consider a light ray emitted at the Big Bang traveling radially outwards from our position.
Therefore we have:
ds = 0
d\theta = d\phi = 0
Substituting into the above metric we have
dt = a(t) \frac{dr}{\sqrt{1-kr^2}}
Integrating both sides and assuming a(0)=0 we have
t = a(t) \int{\frac{dr}{\sqrt{1-kr^2}}}
Thus we must have a linear cosmology
a(t) \propto t
\large ds^2 = -dt^2 + a^2(t) [ \frac{dr^2}{1-kr^2} + r^2(d\theta^2+\sin^2\theta d\phi^2]
Consider a light ray emitted at the Big Bang traveling radially outwards from our position.
Therefore we have:
ds = 0
d\theta = d\phi = 0
Substituting into the above metric we have
dt = a(t) \frac{dr}{\sqrt{1-kr^2}}
Integrating both sides and assuming a(0)=0 we have
t = a(t) \int{\frac{dr}{\sqrt{1-kr^2}}}
Thus we must have a linear cosmology
a(t) \propto t
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