Constant Jacobian transformation of an inertial frame

[hwl]

Suppose we do a constant Jacobian transformation (which is not Lorentz) of a SR (inertial)
frame, by using four linear change of variables equations. This defines an apparent field with a
constant metric (which is not the SR metric) in which there is relative acceleration of separation.
From the geodesic - metric equation we see that the acceleration vector depends on the first
partial derivatives of this constant metric and so at least some of these derivatives must be
non-zero. How can this be true?
Can anyone shed light on this puzzle?

 

[Bill_K]

the acceleration vector depends on the first partial derivatives of this constant metric and so at least some of these derivatives must be non-zero.

Why would the first partial derivatives of a constant metric be nonzero?

 

[hwl]

The acceleration vector in this field is NON-ZERO. But according to the geodesic-metric equation it should be
ZERO because the metric is constant with (presumably !) zero partial derivatives. The only way we can
reconcile these two conflicting values is if these derivatives were non-zero. How else can we explain this
contradiction ?