- #1
Vincentius
- 78
- 1
Hi, could anyone help me out?
The FLRW metric in spherical coordinates is:
[itex]\;\;[/itex] ds2 = dt2 - a(t)2(dr2 + r2dΩ2) [itex]\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;[/itex] (1)
I am considering a similar metric of the format:
[itex]\;\;[/itex] ds2 = [itex]\frac{1}{a(t')^{2}}[/itex]dt'2 - a(t')2(dr2 + r2dΩ2) [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!
The FLRW metric in spherical coordinates is:
[itex]\;\;[/itex] ds2 = dt2 - a(t)2(dr2 + r2dΩ2) [itex]\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;[/itex] (1)
I am considering a similar metric of the format:
[itex]\;\;[/itex] ds2 = [itex]\frac{1}{a(t')^{2}}[/itex]dt'2 - a(t')2(dr2 + r2dΩ2) [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!