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I have two questions regarding the FLRW metric, it is more about its interpretation.

The metric reads:

##dl²=dt²-a²(\frac{dR²}{1-kR²}+R²d\Omega²)## where ##a## is the radius of the 3-sphere (universe), and ##R=r/a## a normalized radial coord.

What I don't understand is this statement: "##k=+1## then the universe is closed".

I understand that when ##k=+1## then if ##R>1## the metric changes its signature which does not make sense in GR. Therefore ##R<1## if ##k=+1##.

My question is then:

If ##R=r/a##, then ##R<1## means that ##r<a##. Which just means you just don't observe things outside the universe of radius ##a##. I don't see why the radius of the universe ##a## could not be infinite.

The same problem the other way around: When ##k=-1##, ##r## can take an arbitrary value. Meaning it can be bigger than ##a##. How can the radial coord be bigger than the radius of the universe ?

How does the radial coord. tells us anything about the finiteness of the universe?

Also that FLRW metric seems like a huge regression from the Schwarchild's metric which has been built to keep the signature. Why not just use the Schwarchild's metric or something more general like a simple spherical metric (which is used to find the Schwarchild's metric) ?

I am obviously missing something. Maybe the

Schwarchild's metric does not fit the problems addressed by Friedmann and Lemaître but I don't see how...

Thank you and have a great weekend!

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# Questions about the FLRW metric

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