Question

The shift vector is a part of the metric tensor in General Relativity (GR). It's g0i with i in [1,3]

.This post is related to this question. There, I ask whether a changing of coordinates is possible where the metric tensor gμν changes to g~μν
(see below) with g~0i =0 while making all g~ij independent (giving up the symmetry). The number of degrees of freedom stays the same (10). However, the new g~μν isn't a tensor in spacetime manifold, because if one changes the coordinates the shift vector possibly becomes something different than zero.
Now, I'm wondering why and how the shift vector can be different from zero at all.
For the coordinate system of an arbitrary observer on earth (me), is the shift vector zero or not zero? How about an observer moving in free fall? How about an observer on a planet which is much more compact than the earth?
What would be the consequences if we postulated that the shift vector in every coordinate system we chose has to be zero? As I understand it, this postulate would only restrict the coordinate system - not the degrees of freedom of the physical system. Right?

(Is it probably the case of free fall with shift vector = 0? No acceleration of the coordinate system? Or probably Droste coordinates, field-free observer? )

g~μν=(g00 0 0 0
0 g11 g21 g31
0 g12 g22 g32
0 g13 g23 g33)

Addendum: Is g~μν
necessarily symmetric if it is a valid tensor or not?

Answer 1

The shift vector in General Relativity can differ from zero depending on the coordinate system and observer's motion. On Earth, the shift vector can be non-zero if the observer is moving relative to the reference frame. If we postulated the shift vector to be zero in every coordinate system, it would restrict coordinate choices but not the degrees of freedom.

Step-by-step explanation:

The shift vector in General Relativity (GR) is a part of the metric tensor and is represented by g0i with i in [1,3]. The shift vector can differ from zero depending on the coordinate system and the observer's motion.

For an observer on Earth, the shift vector can be non-zero if the observer is moving relative to the reference frame being observed. On the other hand, an observer in free fall or on a planet with more compact dimensions may have a shift vector close to zero due to the absence of acceleration.

If we postulated that the shift vector in every coordinate system must be zero, it would restrict the choice of coordinate systems but not the degrees of freedom of the physical system.

The g~μν tensor does not necessarily need to be symmetric to be a valid tensor.