On Sun, Dec 10, 2017 at 07:06:00PM -0800, [email protected] wrote:
> Excellent summary. TY. But why would Einstein think there could be a
> covariant theory for accelerating frames when an observer inside such a
> frame can do measurements to confirm acceleration and differences with
> other such frames, unlike the case for inertial frames which are clearly
> equivalent? AG
If I have a tensor equation like C=\sum_{ij} A_{ij}B_{ij} where C is a
scalar quantity, then the coefficients of A and B must "covary" with
each other as you select different coordinate systems, since C must
remain unchanged regardless of coordinate system. Given the
setting is spacetime, that includes rotations in the time dimension
too, which is equivalent to changes in inertial reference frame by
velocity boost.
This is really obvious if we use things like the 4 dimensional dot
product, but traditional tensor equations are written in component
form, so one must ensure the covariance property is preserved to have
a valid equation. Indeed, in the above equation, superscripts are used
to represent the covariant indices, ie
C = A_{ij}B^{ij}
where the summation sign is dropped, since it is obvious from the way
the equation is written.
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Dr Russell Standish Phone 0425 253119 (mobile)
Principal, High Performance Coders
Visiting Senior Research Fellow [email protected]
Economics, Kingston University http://www.hpcoders.com.au
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