Should Einstein's Field Equations be modified when using conformal time?
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Let us consider the FRW metric for flat space expressed in terms of conformal time ηη and cartesian spatial co-ordinates x,y,zx,y,z:
ds2=a2(η){dη2−dx2−dy2−dz2}.ds2=a2(η){dη2−dx2−dy2−dz2}.
As in the standard FRW co-ordinate system one can see that if two observers are separated by a constant co-moving interval dxdx then the interval of proper distance between them, dsds, is given by:
ds=a(η) dx.ds=a(η) dx.
Thus we have an expanding universe as expected.
But, contrary to the standard FRW co-ordinates, an interval of proper time dτdτmeasured by a co-moving observer using conformal time ηη is given by:
dτ=a(η) dη.dτ=a(η) dη.
Thus the co-moving observer's clock is going slower as the universe expands. This can be understood if one imagines that the co-moving observer uses a lightclock that measures a unit of time by bouncing a pulse of light off a mirror placed some distance away. When one uses the standard time co-ordinate one assumes that such a mirror is at a constant proper distance from the observer. But when one uses conformal time then one implicitly assumes that the mirror is at a constant co-moving distance from the observer. Thus he is using a clock whose unit of time is getting longer as the Universe expands
hope it helps....
ds2=a2(η){dη2−dx2−dy2−dz2}.ds2=a2(η){dη2−dx2−dy2−dz2}.
As in the standard FRW co-ordinate system one can see that if two observers are separated by a constant co-moving interval dxdx then the interval of proper distance between them, dsds, is given by:
ds=a(η) dx.ds=a(η) dx.
Thus we have an expanding universe as expected.
But, contrary to the standard FRW co-ordinates, an interval of proper time dτdτmeasured by a co-moving observer using conformal time ηη is given by:
dτ=a(η) dη.dτ=a(η) dη.
Thus the co-moving observer's clock is going slower as the universe expands. This can be understood if one imagines that the co-moving observer uses a lightclock that measures a unit of time by bouncing a pulse of light off a mirror placed some distance away. When one uses the standard time co-ordinate one assumes that such a mirror is at a constant proper distance from the observer. But when one uses conformal time then one implicitly assumes that the mirror is at a constant co-moving distance from the observer. Thus he is using a clock whose unit of time is getting longer as the Universe expands
hope it helps....
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The use of conformal Gaussian systems in conjunction with the conformal Einstein field equations renders a particularly attractive system of evolution.
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