On Monday, December 11, 2017 at 9:14:09 AM UTC, Russell Standish wrote: > > On Sun, Dec 10, 2017 at 07:06:00PM -0800, [email protected] > <javascript:> 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. >
*IIUC, you're saying that tensors transform covariantly, that is, independent of coordinate system, but you haven't addressed my question why Einstein would think accelerating frames are equivalent, or why a theory of gravity could be, or must be covariant. BTW, is it correct to say that "covariance" is a synonym for "Lorentz invariant"? TIA, AG* > > > -- > > ---------------------------------------------------------------------------- > > Dr Russell Standish Phone 0425 253119 (mobile) > Principal, High Performance Coders > Visiting Senior Research Fellow [email protected] > <javascript:> > Economics, Kingston University http://www.hpcoders.com.au > ---------------------------------------------------------------------------- > > -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
