On Friday, March 9, 2018 at 7:28:58 PM UTC-6, agrays...@gmail.com wrote:
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> On Friday, March 9, 2018 at 12:51:29 PM UTC-5, Lawrence Crowell wrote:
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>> I am not exactly sure why you are stuck on the idea that the advance of 
>> time causes motion. 
>>
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> *Because in the absence of a force causing non-geodesic motion, the 
> increase in time results in a change in spatial position since in the 
> equations of motion, spatial coordinates are not independent variables; 
> they depend on time IIUC. AG*
>

The metric defines the arc length and this is identified with the proper 
time. The interval s is then s = cτ, where τ is time measurement of this 
arc length. The metric defines the interval ds^2 = g_{ab}dx^adx^b as a 
fundamental invariant. In effect all this says is we are all moving 
forwards along this parametrized interval at the speed of light.

What we call motion or dynamics is the relative change in a vector 
separating two test masses. The geodesic equation 

dU^a/ds = Γ^a_{bc}U^bU^c

is a connection based equation of motion. It really only makes sense from 
the perspective of a distant observere far removed from a gravity field 
where spacetime is approximately flat. However, we can look at this 
according to a variation in this to define relative motion of two nearby 
masses. This means we have

δdU^a/ds = δΓ^a_{bc}U^bU^c.

where δU^a = V^a for V^a the spacetime vector separating the two test 
masses and δΓ^a_{bc} = R^a_{bcd}v^d, with V = dv/ds. We now have the 
geodesic deviation equation

dV^a/ds = R^a_{bcd}U^bU^cv^d,

which is the general equation for the relative dynamics between two masses.

We can compare this to Newton's laws. For a weak gravity field the 
coordinate time and the proper time are close enough so that ∂t/∂s = V^t ≈ 
1 and for V^r = dr/ds ≈ dr/dt, which means as well v^r = r. The Riemannian 
curvature component most relevant, or highest order in the speed of light 
is R^r_{trt} = GM/r^3 so that

d^2r/ds^2 = R^r_{trt} = GM/r^3(∂t/∂s)^2r 

and with our approximations

d^2r/ds^2 ≈ GM/r^2,

which recovers Newton's second law of motion with gravitation.

LC
 

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>> In one sense you can say that by being at rest in a frame one is moving 
>> at T = ct distance along a fourth dimension. But that is not really a cause 
>> for motion in spatial directions.
>>
>> LC
>>
>

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