On Dec 24, 8:06 pm, Eric Gisse wrote:
On Dec 24, 7:58 am, Roland PJ wrote:



On Dec 24, 6:51 pm, Eric Gisse wrote:

On Dec 24, 7:07 am, Roland PJ wrote:

The Pioneer Spacecraft have shown an anomalous acceleration towards the
Sun of ~8 x 10^-16 m/s^2.

Pioneer 10 and Pioneer 11 were traveling at about 1.22 x 10^4 m/s and
1.16 x 10^4 m/s away from the Sun at the end of their lifetimes.

Now, the Pioneers will have experienced time dilation in the correct
direction (blue-shift for us on earth) as they escape the gravitational
well of the Sun, as follows:

We will use Swarzschild time dilation, dt_shell = (1-2M/r)^1/2 dt, where
dt_shell is shell time (i.e. use on earth, or Pioneer's electronics),
and dt is ephemeris time (bookkeeper time) valid at a great distance.

...and your justification for this is what, exactly?

Remember what the Schwarzschild solution describes.

Modeling the Sun as a stationary point mass, and computing the time
dilation effects of the Sun within the Pioneers' paths ( 100 a.u.). A
point mass iss certainly accurate enough when we get as far away as
the earth, no?

Note that any mass will cause time dilation according to GR. With
black holes the time dilation is massive near the event horizon, but
common and garden bodies like the Sun (and even a tennis ball) will
cause similar (although much smaller) time dilation effects.

1) The Schwarzschild solution _still_ does not apply.

Why not? The Schwarzschild solution applies to all point (or
spherically symmetric) masses. Why doesn't it apply to the Sun?

2) Time dilation cannot be confused with acceleration, no matter how
badly you butcher the analysis.

Yes it can. The Doppler shift of signals from the Pioneer (which is
the basis of the inferred acceleration) is directly related to the
time dilation, and the apparent motion of a constant-velocity body
(viewed from earth) will appear to decelerate, as a direct result of
the time dilation due to distance from the Sun (the dominant mass in
the solar system).

You can make up arguments to suit your agenda (what is it?), but you
should show your workings (i.e. maths), or show where my analysis is
flawed. Otherwise you're just blowing smoke.

Roland