🚀 go-pugleaf

Koobee Wublee schrieb:
> On Jan 19, 5:46 am, Dieter Heidorn wrote:
> 
>>Koobee Wublee schrieb:
> 
>>>The equation [c = c_0 (1 + V/c^2)] is correct for gravitational redshift.
>>
>>Of course, because Einstein derived the c-equation in his 1911 work from
>>the correct equation for gravitational red-shift. 
>>
> Einstein did not derive anything.  In this case, he merely stated
> them.  <shrug>
> 

A look in his 1911 work will show you that this is not correct. This
work is available in english (see the repeated postings of Pentcho Valev):
" 'On the Influence of Gravitation on the Propagation of
Light,' Annalen der Physik, 35, 1911. You can find a copy beginning on
page 99 of the Dover book 'The Principle of Relativity.' "

>>>> [variable speed of light in GR:   c_A = c ( 1 + 2 V/c^2)]
>>>>
>>>This equation will lead to a wrong value of the gravitational
>>>redshift.  This is another example where GR is contradictory of itself
>>>because it does not agree with the metric derivation as you have shown
>>>above.
>>
>>You can take this as contradiction only, if gravitational red-shift
>>would depend on a variable speed of light. In fact this is not correct
>>in GR. 
>>The gravitational red-shift depends on the metric:
>>
>>    nu_B / nu_A = sqrt(g_00(r_A)/g_00(r_B))
>>
>>In weak fields the following approximation holds:
>>
>>    nu_B / nu_A = 1 + Delta V/c^2
> 
> 
> In physics in general, we have
> 
> C = nu lambda
> 
> Where
> 
> **  nu = frequency
> **  lambda = wavelength
> 
> There is no way you can get away with the red-shift not dependent on
> the speed if the speed does indeed vary.  Thus, I stand by my
> conclusion that the subject of gravitational redshift is one instance
> where this interpretation to the field equations is contradicting
> itself.  <shrug>
> 

In local inertial frames of reference the speed of light is constant.
Thus local measurements always result in

c ~ 3e8 m/s

So a shifted frequency leads to a shifted wavelength and you have

c = nu lambda = const

The variability of the speed of light in GR is not to be understood in
the sense of  Einstein's 1911 work. That was preliminary.
The meaning of the equation

c_A = c ( 1 + 2 V/c^2)

can be taken from Einstein's 1916 work:

>>Die Grundlage der allgemeinen Relativit�tstheorie.
>>AdP 49, 769 (1916)
> 
> Sorry, ich lese nicht Deutsch.
> 

But you can write it - fine :-)

If you're interested, here's an english textbook containing Einstein's
four lectures given at Princeton University (1921) that gave an overview
of his main ideas in his 1916 work:

"The Meaning of Relativity (ISBN: 9780691120270)
  University Presses of Ca."

I try to give a short overview of the part concerning the variable speed
of light in GR:

An observer at a point B outside the gravitational field of a mass M
(that means: in a sufficient great distance) uses a coordinate system CS
with coordinates (x^mu) = (ct, x^i). Using his coordinates he measures
the line element (or the metric)

ds^2 = g_mu,nu dx^mu dx^nu

In linear approximation for weak fields (V << c^2) the following
equation holds:

ds^2 = (1 + 2V/c^2) c^2dt^2 - (1 - 2V/c^2) dx^2

Another observer is located at a Point A near the mass M. He uses a 
local frame of reference (at rest relative to CS) with coordinates 
(X^alpha) = (cT, X^i). Using local coordinates he measures the line element

ds^2 = c^2 dT^2 - dX^2

The speed of light at A results as follows:

* measured by the observer at A:

   ds^2 = 0

   c^2 dT^2 = dX^2

   c_A(measured at A) = dX/dT = c

   That means as stated before:
   In local inertial frames of reference the speed of light is constant,
   and local measurements always result in

   c ~ 3e8 m/s

* measured by the observer at B:

   ds^2 = 0

   (1 + 2V/c^2) c^2dt^2 = (1 - 2V/c^2) dx^2

   c (1 + V/c^2) dt = (1 - V/c^2) dx

   c_A(measured at B) = dx/dt = c (1 + V/c^2) / (1 - V/c^2)

   c_A(measured at B) = c (1 + 2V/c^2)

Clearly the observer at A cannot use c_A = (1 + 2V/c^2), and of course 
he will not measure a local variability of c in his local frame of 
reference.

The difference between

c_A = c (1 + 2V/c^2)  (1916)

and

c_A = c (1 + V/c^2)   (1911)

is the factor 2 with the gravitational potential. Equation (1916) is
confirmed by experiment, for instance by:

* light deflection
   (using eq.(1916) Einstein calculates the correct deflection angle
    alpha = 4 GM / (c^2 R))
* radar time delay (Shapiro experiment)

So eq.(1911) is disproved by experiment. It's consistency with
gravitational redshift is based on the fact, that it was derived from
the correct equation for the frequency shift - however it's wrong 
because it was derived using flat space instead of curved space-time.

The connections can (very) simplified be illustrated by the following 
diagram:

            equivalence principle
              |                |
              | 1911           | 1916
              |                |
frequency shift <---------- metric ds^2
                               |
                            variable speed of light (non-local)
                               |
                            light deflection

Dieter Heidorn