Is the spacetime interval between the Big Bang and a given event dependent only on the time component?

Question

My thinking is this - at the time closest to the Big Bang when physics doesn't break down, the universe would have had a volume that was still very close to zero. If that were so, wouldn't the space component contribution be negligible, meaning that the formula becomes nearly $s^2=-c^2t^2$?

I'm asking this because if it were so, and since spacetime intervals are invariant to all observers, it would make the time passed since the Big Bang also invariant to all observers, kind of like an absolute clock.

But that really depends on there not being any space contribution, because the volume of the universe is nearly zero at the time of the Big Bang, and I'm not sure if that's true.

Answer 1

Even in special relativity it isn't correct to say:

$$ s^2=-c^2t^2 $$

because the line element isn't simply proportional to time. If there is no movement in space, i.e. $dx$, $dy$ and $dz$ are all zero, then in special relativity it would be true to say:

$$ ds^2=-c^2dt^2 $$

where $dt$ is the infinitesimal change in $t$, but the Big Bang isn't described by special relativity so even this is wrong. Instead you'd have:

$$ ds^2=g_{\alpha\beta}dx^\alpha dx^\beta $$

where the elements of the metric (actually the FLRW metric) $g_{\alpha\beta}$ are not constants i.e. $g_{00}$ is not $-c^2$.

If you imagine some particle shortly after the Big Bang it's certain true that it can't have moved far in space, but then it won't have moved far in time either so there's no reason to suppose the spatial movement $dx$, $dy$ and $dz$ would be smaller than the time movement $dt$. In fact for a photon emitted soon after the big bang the spatial and time motion would be equal because $ds^2 = 0$.

Having said this, the Big Bang is normally described by the FLRW metric, as I mentioned above, and this uses the notion of a co-moving observer. This observer considers themself stationary so their $dx$, $dy$ and $dz$ are indeed zero and $ds^2$ does equal $-c^2dt^2$. Read the article on the FLRW metric for more on this.