I am studying Special Theory of Relativity from the book "Special Relativity And Classical Field Theory" by Leonard Susskind. I am not being able to understand the following:
The three space components of a 4-vector may equal zero in your reference frame. You, in your frame, would say that this displacement is purely timelike. But this is not an invariant statement. In my frame, the space components would not all equal zero, and I would say that the object does move in space. However, if all four components of a displacement 4-vector are zero in your frame, they will also be zero in my frame and in every other frame. A statement that all four components of a 4-vector are zero is an invariant statement.
Firstly, if a displacement in my reference frame is purely timelike then why isn't an invariant statement? The author said in another chapter that
The property of being timelike is invariant: If an event is timelike in any frame, it is timelike in all frames.
Secondly, I do not understand the line:
In my frame, the space components would not all equal zero, and I would say that the object does move in space.
The specification of the space components denotes a particular point in space. How can I say whether an object is moving in space from just the information about the space coordinates? After all, from two different reference frames, an object can have two different sets of spatial coordinates and yet be at rest w.r.t. both of them.
Thirdly, I do not understand why
A statement that all four components of a 4-vector are zero is an invariant statement.
Please help me with these doubts.