Why does the lock object have to be readonly?

Question

When implementing a lock, I used to create a private object inside of my class:

If I want to be sure that it is locked in the thread that created my class:

Answer 1

I guess what meant is "a variable referencing a lock object should be read only".

You lock on a lock object that variable references, not on a variable itself. I.e. having

private object Locker = new object();

you lock on that new object(), not on Locker field. Then, if you replace the value of the field with a reference to another object, say

Locker = new object();

and lock on it, you are locking on two different objects, and that defeats the purpose of the locking, because you do not get a synchronized access now.