On Dec 24, 8:43 pm, Eric Gisse wrote:
On Dec 24, 9:20 am, Roland PJ wrote:
On Dec 24, 8:06 pm, Eric Gisse wrote:
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?
Is the solar system spherically symmetric?
It's a model, not an exact model, but a model. Most of the solar
system's mass is in the centre, at the Sun, so it's not a completely
useless model.
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).
Uh, no.
Doppler shift is ONLY a function of velocity - not acceleration.
Knowing the doppler shift of the signals allows a determination of
velocity as a function of time, and from that, acceleration.
The shift in signal frequency is both a function of relative velocity
(Doppler), and a function of the 'strength' of the gravity field at
the transmitter and receiver (due to relative time dilation).
Unfortunately, these effects are impossible to distinguish w.r.t. only
the frequency shift.
So, an increasingly blue-shifted signal could be caused by the
transmitter accelerating towards the receiver, or it could be caused
by the transmitter climbing out of a gravity well, and, of course,
both together.
Your analysis is flawed because it has already been considered,
I couldn't find reference to it in the paper I read. They seemed to
calculate the Pioneer trajectory in ephemeris co-ords, but there was
no further mention of the fact that Pioneer would be moving out of the
Sun's well - i.e. Pioneer's time co-ords would change over time due to
the Sun's gravity field. Earth's co-ords, of course, don't change
much, but they seemed to take detailed account of the earth's slightly
asymmetric orbit, local earth gravity field changes, plate techtonics,
et al. In other words, a lot of detail was described about the
receiver end's time dilation correction (red-shift of the earth's
field mainly), but little on the Pioneer's end, which is less
constant.
because the Schwarzschild metric is NOT a valid model for the solar
system,
Wheeler et al seem to disagree with you in "Black Holes". In any case,
it is a model, the question is only how accurate the model is.
and because you didn't even do it right.
Almost certainly not
To get the total change, you have to integrate along the path. Which
you didn't bother doing.
I don't believe so. The Schwarzchild solution already solves the
relative time dilation between any two points in the field. It is a
solution to the field equations, _not_ the field equations themselves.
The time dilation effect on signal frequency is only dependent on the
relative time dilation between the end-points.
I can also tell you are wrong without even going through the analysis
by looking at your conclusions - you have concluded that _doppler
shift_ can make something appear to be accelerating. This is
completely and utterly wrong.
Nope. See above. It is impossible to determine from frequency shift
alone whether the frequency shift is caused by relative velocity or
relative time-dilation.
Oh, one more reason why you are wrong. The deceleration is, as far as
can be determined, constant. Would you like to make the claim that the
gravitational effects are the same regardless of how close the craft
is to the sun?
Actually, the value from the dPioneer ata is not quite constant, but
starts small (near earth, but dominated by solar wind pressure), hits
a peak, and then falls very slightly in the later readings.
But if you look at the equation for relative time dilation between
Pioneer and earth in my original post, you'll see that it changes very
slowly until Pioneer is at least 10^4 a.u. from the Sun. In fact in
the interesting region ( 10 a.u and 100 a.u.) the rate of change
with distance is nearly constant (due to the small mass of the Sun).
So, yes, I make the claim that the effect is almost constant in the
interesting region. And, unlike you I demonstrate this with a formula
based on the Schwarzchild solution, rather than just waving my hands.
Roland