# A Higher derivatives of the spacetime metric

1. Dec 30, 2016

### EagleH

I wonder if my following understanding is correct. Please, comment it.

Given a locally inertial frame (LIF), it is possible to define another LIF with new coordinates ya and metric gik where the partial derivatives of order n of the new metric in respect to the new coordinates depend on the totality of the partial derivatives from order https://www.physicsforums.com/file://localhost/Users/iliasgkanoutasleventis/Library/Caches/TemporaryItems/msoclip/0/clip_image001.png [Broken]to order n+1 plus 6 of the first order derivatives of the old coordinates xm in respect to the new coordinates ya. Making the calculations it results that for n larger or equal to 3, the number of derivatives of the metric are less than the number of the derivatives of the coordinates transformation. That is:

Σ(from k=3, to k=n+1)(2/3)(k+1)(k+2)(k+3)+6>(5/3)(n+1)(n+2)(n+3)

Hence, in a LIF all the partial derivatives of the metric of an arbitrarily chosen order larger or equal to 3, may be set to vanish with an adequate choice of coordinates. This is not possible only for the second order partial derivatives of the metric. The reason is:

There are 100 second order partial derivatives of the new metric in respect to the new coordinates, that depend on the values of 16 first order, 40 second order and 80 third order partial derivatives of the old coordinates in respect to the new coordinates. But, in a LIF only 6 of the first order and the 80 of the third order are available to this end, as 10 of the first order and the 40 of the second order of them are already well defined. This holds not only in a LIF at a point, but also in a Fermi LIF along the worldline of anything.
Sorry for not being able to use LaTex.
Thank you.

Last edited by a moderator: May 8, 2017