Is the four-velocity always normalized? In the book i have reading defines the four-velocity like $$U^\mu=dx^\mu/d\tau.$$ The metric used is $\eta_{\mu\nu}=diag(-1,1,1,1)$. It is straightforward to show that the norm of the four-velocity is normalized,
$$\eta_{\mu\nu}U^\mu U^\nu=\eta_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau}=-c^2$$ since, we have defined the proper time as
$$c^2d\tau^2=-ds^2=-\eta_{\mu\nu}dx^\mu dx^\nu.$$
My question is, what happens in the case of other more general curved metrics $g_{\mu\nu}$? Is still the norm of the four-velocity equal to $-c^2$? Or $-1$ if we put $c=1$?
 A: Yes, but it is more general than this. The norm of any 4-vector is invariant under arbitrary coordinate transformations in any spacetime. So showing that it is always equal to -1 in one coordinate system shows that it is equal to -1 in any coordinate system.
A: I have realized that indeed, the norm always be $-1$ under these considerations above. Because the definition of proper time $d\tau^2$.
A: In the case of a general curved metric $g_{\mu\nu}$, the four-velocity $U^\mu$ is defined as
$$U^\mu=\frac{dx^\mu}{d\tau},$$
where $\tau$ is the proper time along the particle's worldline. The proper time is defined by
$$d\tau^2=-ds^2=-g_{\mu\nu}dx^\mu dx^\nu.$$
The normalization condition of the four-velocity is
$$g_{\mu\nu}U^\mu U^\nu=-c^2.$$
Note that the normalization constant can be different from $-c^2$ depending on the choice of units.
To see why this is true, consider the interval between two nearby events along the particle's worldline. This interval is given by
$$ds^2=g_{\mu\nu}dx^\mu dx^\nu.$$
If we choose the coordinate system such that the particle is at rest at the origin, then the interval reduces to
$$ds^2=g_{00}dt^2,$$
where $t$ is the coordinate time. The proper time elapsed between the two events is given by
$$d\tau^2=-ds^2=-g_{00}dt^2,$$
so $d\tau/dt=\sqrt{-g_{00}}$. Therefore, the four-velocity is
$$U^\mu=\frac{dx^\mu}{d\tau}=\frac{dt}{d\tau}\frac{dx^\mu}{dt}=(\sqrt{-g_{00}},0,0,0),$$
which satisfies the normalization condition
$$g_{\mu\nu}U^\mu U^\nu=-g_{00}=-c^2.$$
In summary, the normalization condition for the four-velocity depends on the metric $g_{\mu\nu}$, and it is given by $g_{\mu\nu}U^\mu U^\nu=-c^2$ (or $-1$ if we use natural units $c=1$).
