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I reviewed this question but sometimes I'm unsure about delta ($\Delta$) versus differential ($d$) notation. Does the expression "$ds^2=-c^2dt^2+a^2(t)[dr^2 + S_k^2(r)d\Omega^2 ]$" mean the same thing as "$\Delta s^2=-c^2\Delta t^2+a^2(t)[\Delta r^2 + S_k^2(r)\Delta \Omega^2 ]$" ? It seems like it may, but my understanding is that the former may be more appropriate for infinitesimal distance.

ProfRob
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bblohowiak
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1 Answers1

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The first one is exact, but the second would be an approximation for a non-infinitesimal (i.e. finite) interval. In other words $$ ds^2=-c^2dt^2+a^2(t)[dr^2 + S_k^2(r)d\Omega^2] $$ but $$\Delta s^2 \simeq-c^2\Delta t^2+a^2(t)[\Delta r^2 + S_k^2(r)\Delta \Omega^2 ] $$ where the expression is evaluated for some intermediate/average value of $t$, and $r$.

ProfRob
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