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

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
Staff Emeritus
Gold Member
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