On Thursday, December 21, 2017 at 6:54:50 PM UTC-6, Bruce wrote:
>
> On 22/12/2017 11:22 am, [email protected] <javascript:> wrote:
>
> On Thursday, December 21, 2017 at 11:03:53 PM UTC, Brent wrote:
>>
>>
>> On 12/21/2017 2:04 PM, [email protected] wrote:
>>
>>
>>
>> *If Newton's Law of Gravitation is covariant -- that is, coordinate frame
>> independent -- I'd expect it to to be invariant between inertial frames,
>> but I don't believe it is. That is, I don't think a LT between inertial
>> frames will leave the form of the law unchanged. How do you resolve this
>> problem? TIA, AG *
>>
>>
>> Don't use a Lorentz transform between frames in a Galilean invariant
>> theory.
>>
>
> *OK, So why didn't Einstein do what he did for classical mechanics which
> is not Lorentz invariant, and directly modify Newton's Law of Gravitation?
> AG*
>
>
> Special relativity is kinematics, gravitation is a dynamical theory -- one
> doesn't go into the other. You need a new dynamical theory.
>
> Bruce
>
Special relativity as the kinematical theory is global. In isolation it
assumes spacetime is flat everywhere. However, this may not be the case.
Suppose we have two regions with bundle sections s and s'. This means there
are Lorentz vectors X and X' in two local regions that transform according
to special relativity separately. If these two regions intersect we have a
transition between them. Let g be the group such that gs → s' and gX → X'.
There is then a difference between these given by
X's' – Xs = d(Xs) = gXs' - Xs.
Which by g^{-1}g = 1 gives
d(Xs) = (gXg^{-1} - X)s.
The LHS gives us (X' + gdg^{-1})s and so we get
X' = gXg^{-1} - gdg^{-1}.
This is the transformation principle for vectors and what defines the
connection coefficients in general relativity.
LC
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