I would be interested in comments on the following corrected Doppler
formula (see gr-qc/0512019). First, recall the textbook derivation that
$d \tau _i = \sqrt{1-2M/r_i}dt$ (derived from the Schwarzschild metric
by setting $dr=d \phi = d \theta =0$ and taking the square root)
implies $\frac{d \tau _1}{d \tau _2} =
\frac{\sqrt{1-2M/r_1}}{\sqrt{1-2M/r_2}}$.

However, both of these equations are first order approximations. To
calculate the correct equation, note that a distant observer sees time
dilation at radius $r$ of
$\frac{1}{\sqrt{1-v^2}}=\frac{1}{\sqrt{1-2M/r}}$ which is equivalent to
velocity of $v=\sqrt{2M/r}$ under Fermi coordinates for an observer
free falling from infinity, that is, under the Painlev{\'e}-Gullstrand
metric (see for instance Taylor and Wheeler's textbook Black Holes)

$d \tau^2 = \left(1 - \frac{2M}{r} \right ) d t^2 - 2 \sqrt{2M/r} dt
dr- dr^2-r^2(d \theta^2+ \sin ^2\theta d \phi^2)$

Hamilton and Lisle interpret this metric as a ``river model" of black
holes: space itself appears to flow like a river through a flat
background, while objects move through the river according to the rules
of special relativity. The river flows inward at the Newtonian escape
velocity $\sqrt{2M/r}$ reaching the speed of light at the horizon.
Under this metric, let $u=\sqrt{2M/r_1}$ and $w=\sqrt{2M/r_2}$ and
calculate $v$ according to the following version of the velocity
composition law

$v=\frac{u-w}{1-uw}$

or equivalently

$\frac{1}{\sqrt{1-v^2}}=\frac{1}{\sqrt{1-\left(\frac{u-w}{1-uw}\right)^2}}=\frac{1-uw}{\sqrt{1-u^2}\sqrt{1-w^2}}$.

and setting $u=\sqrt{2M/r_1}$ and $w=\sqrt{2M/r_2}$ yields the exact
Doppler shift formula

$\frac{1}{\sqrt{1-v^2}} = \frac{1-2M/\sqrt{r_1
r_2}}{\sqrt{1-2M/r_2}\sqrt{1-2M/r_1}}=\frac{(1-2M/\sqrt{r_1
r_2})/\sqrt{1-2M/r_1}}{\sqrt{1-2M/r_2}}$.

Note that the corrected equation is approximately the same as the
textbook equation for $r_1 \approx r_2$. The second-order $uw$ term in
the numerator has the right sign to explain the Pioneer anomaly, if it
has been ignored.

"Hunter" wrote in message
ups.com...
I would be interested in comments on the following corrected Doppler
formula (see gr-qc/0512019). First, recall the textbook derivation that
$d \tau _i = \sqrt{1-2M/r_i}dt$ (derived from the Schwarzschild metric
by setting $dr=d \phi = d \theta =0$ and taking the square root)
implies $\frac{d \tau _1}{d \tau _2} =
\frac{\sqrt{1-2M/r_1}}{\sqrt{1-2M/r_2}}$.

However, both of these equations are first order approximations. To
calculate the correct equation, note that a distant observer sees time
dilation at radius $r$ of
$\frac{1}{\sqrt{1-v^2}}=\frac{1}{\sqrt{1-2M/r}}$ which is equivalent to
velocity of $v=\sqrt{2M/r}$ under Fermi coordinates for an observer
free falling from infinity, that is, under the Painlev{\'e}-Gullstrand
metric (see for instance Taylor and Wheeler's textbook Black Holes)

$d \tau^2 = \left(1 - \frac{2M}{r} \right ) d t^2 - 2 \sqrt{2M/r} dt
dr- dr^2-r^2(d \theta^2+ \sin ^2\theta d \phi^2)$

Hamilton and Lisle interpret this metric as a ``river model" of black
holes: space itself appears to flow like a river through a flat
background, while objects move through the river according to the rules
of special relativity. The river flows inward at the Newtonian escape
velocity $\sqrt{2M/r}$ reaching the speed of light at the horizon.
Under this metric, let $u=\sqrt{2M/r_1}$ and $w=\sqrt{2M/r_2}$ and
calculate $v$ according to the following version of the velocity
composition law

$v=\frac{u-w}{1-uw}$

or equivalently

$\frac{1}{\sqrt{1-v^2}}=\frac{1}{\sqrt{1-\left(\frac{u-w}{1-uw}\right)^2}}=\frac{1-uw}{\sqrt{1-u^2}\sqrt{1-w^2}}$.

and setting $u=\sqrt{2M/r_1}$ and $w=\sqrt{2M/r_2}$ yields the exact
Doppler shift formula

$\frac{1}{\sqrt{1-v^2}} = \frac{1-2M/\sqrt{r_1
r_2}}{\sqrt{1-2M/r_2}\sqrt{1-2M/r_1}}=\frac{(1-2M/\sqrt{r_1
r_2})/\sqrt{1-2M/r_1}}{\sqrt{1-2M/r_2}}$.

Note that the corrected equation is approximately the same as the
textbook equation for $r_1 \approx r_2$. The second-order $uw$ term in
the numerator has the right sign to explain the Pioneer anomaly, if it
has been ignored.

http://www.androcles01.pwp.blueyonde...er/Doppler.htm

Androcles.