Black Holes

Mathematical derivation

The observational consequences of this effect can be derived using the equations from general relativity that describe a homogeneous and isotropic universe.

To derive the redshift effect, use the geodesic equation for a light wave, which is

$ds^2=0=-c^2dt^2+\frac{a^2 dr^2}{1-kr^2}$

where

$ds^2$ is the spacetime interval
$dt^2$ is the time interval
$dr^2$ is the spatial interval
$c$ is the speed of light
$a$ is the time-dependent cosmic scale factor
$k$ is the curvature per unit area.

For an observer observing the crest of a light wave at a position $r=0$ and time $t=t_\mathrm{now}$ , the crest of the light wave was emitted at a time $t=t_\mathrm{then}$ in the past and a distant position $r=R$ . Integrating over the path in both space and time that the light wave travels yields:

$c \int_{t_\mathrm{then}}^{t_\mathrm{now}} \frac{dt}{a}\; = \int_{R}^{0} \frac{dr}{\sqrt{1-kr^2}}\,.$

In general, the wavelength of light is not the same for the two positions and times considered due to the changing properties of the metric. When the wave was emitted, it had a wavelength $\lambda_\mathrm{then}$ . The next crest of the light wave was emitted at a time

$t=t_\mathrm{then}+\lambda_\mathrm{then}/c\,.$

The observer sees the next crest of the observed light wave with a wavelength $\lambda_\mathrm{now}$ to arrive at a time

$t=t_\mathrm{now}+\lambda_\mathrm{now}/c\,.$

Since the subsequent crest is again emitted from $r=R$ and is observed at $r=0$ , the following equation can be written:

$c \int_{t_\mathrm{then}+\lambda_\mathrm{then}/c}^{t_\mathrm{now}+\lambda_\mathrm{now}/c} \frac{dt}{a}\; = \int_{R}^{0} \frac{dr}{\sqrt{1-kr^2}}\,.$

The right-hand side of the two integral equations above are identical which means

$c \int_{t_\mathrm{then}+\lambda_\mathrm{then}/c}^{t_\mathrm{now}+\lambda_\mathrm{now}/c} \frac{dt}{a}\; = c \int_{t_\mathrm{then}}^{t_\mathrm{now}} \frac{dt}{a}\,$

or, alternatively,

$\int_{t_\mathrm{now}}^{t_\mathrm{now}+\lambda_\mathrm{now}/c} \frac{dt}{a}\; = \int_{t_\mathrm{then}}^{t_\mathrm{then}+\lambda_\mathrm{then}/c} \frac{dt}{a}\,.$

For very small variations in time (over the period of one cycle of a light wave) the scale factor is essentially a constant ( $a=a_\mathrm{now}$ today and $a=a_\mathrm{then}$ previously). This yields

$\frac{t_\mathrm{now}+\lambda_\mathrm{now}/c}{a_\mathrm{now}}-\frac{t_\mathrm{now}}{a_\mathrm{now}}\; = \frac{t_\mathrm{then}+\lambda_\mathrm{then}/c}{a_\mathrm{then}}-\frac{t_\mathrm{then}}{a_\mathrm{then}}$

which can be rewritten as

$\frac{\lambda_\mathrm{now}}{\lambda_\mathrm{then}}=\frac{a_\mathrm{now}}{a_\mathrm{then}}\,.$

Using the definition of redshift provided above, the equation

$1+z = \frac{a_\mathrm{now}}{a_\mathrm{then}}$

is obtained. In an expanding universe such as the one we inhabit, the scale factor is monotonically increasing as time passes, thus, z is positive and distant galaxies appear redshifted.

Using a model of the expansion of the universe, redshift can be related to the age of an observed object, the so-called cosmic time–redshift relation. Denote a density ratio as Ω₀:

$\Omega_0 = \frac {\rho}{ \rho_{crit}} \ ,$

with ρ_crit the critical density demarcating a universe that eventually crunches from one that simply expands. This density is about three hydrogen atoms per thousand liters of space. At large redshifts one finds:

$t(z) = \frac {2}{3 H_0 {\Omega_0}^{1/2} (1+ z )^{3/2}} \ ,$

where H₀ is the present-day Hubble constant, and z is the redshift

Black Holes

Sunday, 9 June 2013

Mathematical derivation

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