Can FLRW metric be taken to describe stretching in time rather than space?

[themos]

I hope the title makes sense. I see a factor a(t)^2 in from of the space part of the metric. Is it equivalent to scaling the time part? If so, is there an advantage in abandoning notion of "expansion" of space in favour of time "speeding up" or "slowing down"?

 

[George Jones]

I hope the title makes sense. I see a factor a(t)^2 in from of the space part of the metric. Is it equivalent to scaling the time part? If so, is there an advantage in abandoning notion of "expansion" of space in favour of time "speeding up" or "slowing down"?

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Conformal time is related to this, but I don't think this is what you mean. When conformal time is used, "time" and space are scaled by the same factor. Take a flat universe with

$$ds^2 = dt^2 - a \left(t\right)^2 \left( dx^2 + dy^2 + dz^2 \right)$$

and define conformal time $\eta$ by

$$dt = a \left(t\right) d \eta .$$

Then,

$$ds^2 = a \left(t\right)^2 \left( d\eta^2 - dx^2 + dy^2 + dz^2 \right).$$

On spacetime diagrams that use conformal time, light rays are straight lines just like in special relativity, which is often useful. For light

$$ds^2 = 0$$

gives

$$d \eta = \pm \sqrt{dx^2 + dy^2 + dz^2}$$

or, for example,

$$\eta = x + const$$

for light in the x-direction.

 

[Chalnoth]

I hope the title makes sense. I see a factor a(t)^2 in from of the space part of the metric. Is it equivalent to scaling the time part? If so, is there an advantage in abandoning notion of "expansion" of space in favour of time "speeding up" or "slowing down"?

The reason why we usually take space as expanding and not time is that we consider atoms to be stable.

If we were to instead take space as static, then yes, we would have to scale the time coordinate. But then we would find that atoms change in size with time. That's not to say that this is wrong, but it is contrary to our usual conception of atoms, and would require some serious rewriting of the laws of physics to get everything to work out properly. So I'm not entirely sure that this would be useful.