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## Main Question or Discussion Point

Hi, could anyone help me out?

The FLRW metric in spherical coordinates is:

[itex]\;\;[/itex] ds

I am considering a similar metric of the format:

[itex]\;\;[/itex] ds

Are (1) and (2) equivalent? Is it just a matter of substituting dt'/a for dt in (1)?

Would the new time coordinate simply be t'=[itex]\int[/itex]a(t)dt ?

Is (2) a known parametrization? Has it a name? What kind of time would t' represent?

The background of my question is that the FLRW metric (1) does not reflect time dilation, as e.g. in the Schwarzschild metric, while I would expect time dilation to go along with expansion of space. In the Schwarzschild metric, t is coordinate time. In the FLRW metric t is proper time already. The parallel with the Schwarzschild metric (when using isotropic coordinates) suggests a metric of format (2), or something alike.

Thanks for any help!

The FLRW metric in spherical coordinates is:

[itex]\;\;[/itex] ds

^{2}= dt^{2}- a(t)^{2}(dr^{2}+ r^{2}dΩ^{2}) [itex]\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;[/itex] (1)I am considering a similar metric of the format:

[itex]\;\;[/itex] ds

^{2}= [itex]\frac{1}{a(t')^{2}}[/itex]dt'^{2}- a(t')^{2}(dr^{2}+ r^{2}dΩ^{2}) [itex]\;\;\;\;\;\;\;[/itex] (2)Are (1) and (2) equivalent? Is it just a matter of substituting dt'/a for dt in (1)?

Would the new time coordinate simply be t'=[itex]\int[/itex]a(t)dt ?

Is (2) a known parametrization? Has it a name? What kind of time would t' represent?

The background of my question is that the FLRW metric (1) does not reflect time dilation, as e.g. in the Schwarzschild metric, while I would expect time dilation to go along with expansion of space. In the Schwarzschild metric, t is coordinate time. In the FLRW metric t is proper time already. The parallel with the Schwarzschild metric (when using isotropic coordinates) suggests a metric of format (2), or something alike.

Thanks for any help!