Dear Friends
Is there any coordinate tetrad in spacetime except Cartesian basis ?
since tetrad basis should be orthogonal (( In Lorentzian description ))and the only orthogonal basis is Cartesian ( the metric is (+1,-1,-1,-1 ) but in any other coordinate basis like Spherical metric is ( +1,-1,-r,-rsino ) and they aren't coordiante tetrad .

Chestermiller
Mentor
Dear Friends
Is there any coordinate tetrad in spacetime except Cartesian basis ?
since tetrad basis should be orthogonal (( In Lorentzian description ))and the only orthogonal basis is Cartesian ( the metric is (+1,-1,-1,-1 ) but in any other coordinate basis like Spherical metric is ( +1,-1,-r,-rsino ) and they aren't coordiante tetrad .

The tetrad refers to the coordinate basis vectors, rather than the coordinates. The 4 coordinate basis vectors for Spherical coordinates are all orthogonal to one another, as are the coordinate basis vectors for cylindrical coordinates. But, in cylindrical and spherical coordinates, some of the 3 spatial coordinate basis vectors vary with spatial location.

I know that but i mean are the only holonomic tetrad basis Cartesian basis ?
unit vector basis in spherical coordinate are tetrads but coordiante basis isn't .and the relations between tetrads and basis coordiante are :
e$_{r}$=∂$_{r}$
e$_{\theta}$=$\frac{1}{r}$∂$_{\theta}$
e$_{\varphi}$=$\frac{1}{rsin\varphi}$∂$_{\varphi}$
e$_{0}$=∂$_{r}$=$\frac{∂}{∂t}$
and we can see those unit vectors have non vanishing Lie's bracket (( so they are anholonomic or non coordiante basis ))
so i thought the cartecian basis are the only the coordinate tetrads (( or holonomic tetrads ))