On Tuesday, October 13, 2020 at 2:26:14 PM UTC-6, Lawrence Crowell wrote:
>
> I will try to give a definitive answer. 
>


*So regardless of your subsequent corrections, will you now admit, as I was 
suggesting, that the exact solution can be determined solely by GR, and 
that Clark's introducing SR is confusing and mistaken. Thank you in advance 
for your honesty! AG*
 

> The Schwarzschild metric is
>
> ds^2 = c^2(1 – 2m/r)dt^2 – (1 – 2m/r)dr^2 – r^2(dθ^2 – sin^2θdφ^2)
>
> for m = GM/c^2. For the motion of a satellite in a circular orbit there is 
> no radial motion so dr = 0. We set this on a plane with θ = π/2 so dθ = 0 
> and this reduces this to
>
> ds^2 =c^2(1 – 2m/r)dt^2 – r^2dφ^2.
>
> For circular motion dφ/dt = ω and the velocity v = ωr means this is
>
> ds^2 = [c^2(1 – 2m/r) – r^2ω^2]dt^2
>
> and so ds^2 = [c^2(1 – 2m/r) – v^2]dt^2 the term Γ = 1/√[c^2(1 – 2m/r) – 
> v^2] is a general Lorentz gamma factor and in flat space with m = 0 reduces 
> the form we know. ds is an increment in the proper time on the orbiting 
> satellite and t is a coordinate time, say on the ground of the body.
>
> We can do more with this. The ds^2 = [c^2(1 – 2m/r) – r^2dφ^2]dt^2 can be 
> written as
>
> 1 = [c^2(1 – 2m/r) – r^2ω^2](dt/ds)^2
>
> Now take a variation on this, where obviously δ1 = 0 and
>
> 0 = [c^2δ(1 – 2m/r) – δ(r^2ω^2)](dt/ds)^2 + [c^2(1 – 2m/r) – 
> r^2ω^2]δ(dt/ds)^2.
>
> We think primarily of a variation in the radius and so
>
> 0 = -[ 2mc^2/r^2 – 2rω^2](dt/ds)^2δr + [c^2(1 – 2m/r) – r^2ω^2]δ(dt/ds)^2,
>
> where for the time I will ignore the last term.  The first term gives
>
> rω^2 = -GM/r,
>
> and this is just Newton’s second law with acceleration a = rω^2 with 
> gravity. Also this is Kepler's third law of planetary motion.
>
> Now I will hand wave a bit here. The term δ(dt/ds)^2 = 1 in the Newtonian 
> limit, but we can feed the general Lorentz gamma factor in that. This will 
> have a correction term to this dynamical equation. This correction is 
> general relativistic. The algebra gets a bit dense, but it is nothing 
> conceptually difficult. 
>
> LC
>
> On Tuesday, October 13, 2020 at 9:17:37 AM UTC-5 agrays...@gmail.com 
> wrote:
>
>>
>>
>> On Tuesday, October 13, 2020 at 8:06:30 AM UTC-6, Lawrence Crowell wrote:
>>>
>>> I am not sure why you have endless trouble with this. On the Avoid list 
>>> you repeatedly brought up this question, and in spite of dozens of 
>>> explanations you raise this question over and over. You need to read a text 
>>> on this. The old Taylor and Wheeler book on SR gives some reasoning on 
>>> this. Geroch's book on GR is not too hard to read.
>>>
>>> LC
>>>
>>
>> Actually, I think your memory is faulty, other than to express your 
>> annoyance with my question. In any event, if gravity and acceleration exist 
>> for a system under consideration, why is SR relevant? Why does Clark claim 
>> that the result of SR must be subtracted for the result of GR to determine 
>> an objective outcome, when the conditions of SR are non-existent?  AG
>>
>>>
>>>
>>> On Tuesday, October 13, 2020 at 12:20:44 AM UTC-5 agrays...@gmail.com 
>>> wrote:
>>>
>>>>
>>>>
>>>> On Monday, October 12, 2020 at 11:11:33 PM UTC-6, Brent wrote:
>>>>>
>>>>>
>>>>>
>>>>> On 10/12/2020 9:56 PM, Alan Grayson wrote: 
>>>>> > Why is it that in SR a stationary clock appears to advancing at a 
>>>>> more 
>>>>> > rapid rate than a moving clock, and vice versa -- so the effect is 
>>>>> > relative or symmetric, not absolute -- whereas in GR the effect 
>>>>> seems 
>>>>> > absolute; that is, a ground clock actually advances at a slower rate 
>>>>> > compared to an orbiting clock? AG 
>>>>>
>>>>> It's the same as the twin effect.  The clock on the ground is 
>>>>> following 
>>>>> a non-geodesic path thru spacetime and so measures less duration, 
>>>>> while 
>>>>> the orbiting clock is following a geodesic path.  In relativity the 
>>>>> minus sign in the metric means that the path that looks longer 
>>>>> projected 
>>>>> in space is shorter in spacetime. 
>>>>>
>>>>> Brent 
>>>>>
>>>>
>>>> How does gravity cause the difference between what the theories 
>>>> predict? AG 
>>>>
>>>

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