I know that 4-velocity is defined as

$$u^\mu = \frac{dx^\mu }{d\tau}$$ $$u_\mu=g{_\mu}{_\nu}u^\nu$$

and

$$u_\mu u^\mu=c^2$$

1. Is it true that

$$\frac{d\tau}{dx^\mu} = \frac{u_\mu}{c^2} = \frac{1}{c^2}g{_\mu}{_\nu}u^\nu$$

1. If it is so, then does this chain rule make any sense?

$$\frac{dx^\mu}{dx^\nu}=\frac{dx^\mu}{d\tau}\frac{d\tau}{dx^\nu}$$

The derivatives are total, not partial. I'm pretty sure that it's not the kronecker delta because the trace is supposed to be 1. But then aren't the dx's supposed to be independent? My confusion is that with partial derivatives of the x's, it is the kronecker delta. Why would total derivatives give a different result?

• Where did you encounter these equations? Do you think they are true, or did you see them somewhere? – Hugo V Oct 11 at 12:09
• @HugoV I was just trying to study relativity and derive some equations, then I thought of the quantities above. I thought the dx's are supposed to be independent. – Bao Oct 11 at 12:31