- #1
johne1618
- 371
- 0
For simplicity I assume a flat radial FRW metric (with [itex]c=1[/itex]):
[tex]
ds^2 = -dt^2 + a^2(t)\ dr^2
[/tex]
Now let us consider the path of a light ray, a null geodesic, with [itex]ds=0[/itex] so that we have:
[tex]
dt = a(t)\ dr
[/tex]
Now at the present time [itex]t_0[/itex] we define [itex]a(t_0)=1[/itex] so that we have:
[tex]
dt_0 = dr
[/tex]
The interval of radial co-ordinate [itex]dr[/itex] does not depend on time so that it is the same in both equations.
We combine the two equations to eliminate [itex]dr[/itex] giving:
[tex]
dt = a(t)\ dt_0
[/tex]
If [itex]dt[/itex] is always a fixed interval of cosmological time [itex]t[/itex] then from our perspective at the present time [itex]t_0[/itex] it is represented by the time interval [itex]dt_0[/itex] given by:
[tex]
dt_0 = \frac{dt}{a(t)}
[/tex]
Thus one second measured in the future at time [itex]t[/itex] is equivalent to [itex]1/a(t)[/itex] seconds of our present time [itex]t_0[/itex].
Therefore in an expanding Universe clocks speed up from the perspective of our present time.
Is this effect real or apparent?
Are there any observations that could decide between the two?
The conventional view is that this effect is only apparent and is the cause of the cosmological redshift.
[tex]
ds^2 = -dt^2 + a^2(t)\ dr^2
[/tex]
Now let us consider the path of a light ray, a null geodesic, with [itex]ds=0[/itex] so that we have:
[tex]
dt = a(t)\ dr
[/tex]
Now at the present time [itex]t_0[/itex] we define [itex]a(t_0)=1[/itex] so that we have:
[tex]
dt_0 = dr
[/tex]
The interval of radial co-ordinate [itex]dr[/itex] does not depend on time so that it is the same in both equations.
We combine the two equations to eliminate [itex]dr[/itex] giving:
[tex]
dt = a(t)\ dt_0
[/tex]
If [itex]dt[/itex] is always a fixed interval of cosmological time [itex]t[/itex] then from our perspective at the present time [itex]t_0[/itex] it is represented by the time interval [itex]dt_0[/itex] given by:
[tex]
dt_0 = \frac{dt}{a(t)}
[/tex]
Thus one second measured in the future at time [itex]t[/itex] is equivalent to [itex]1/a(t)[/itex] seconds of our present time [itex]t_0[/itex].
Therefore in an expanding Universe clocks speed up from the perspective of our present time.
Is this effect real or apparent?
Are there any observations that could decide between the two?
The conventional view is that this effect is only apparent and is the cause of the cosmological redshift.
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