Time Unit Vector: Calculate d'Alembert with 4 Vectors

[kareeemmuhammed]

Hello every one i have question what is the unit vector of time component like c^2 t^2 -x^2-y^2-z^2 = 0
OR
if we need to calculate the d'alembert cross some 4 vector have 3 space component x y z and unit vector i j k
and ct component what is the unit vector in Ct direction ?

 

[PeterDonis]

What do you think the unit vectors are in the x, y, and z directions?

 

[kareeemmuhammed]

The cross product yield vector represent the area between 2 vectors and volume in 3 vectors if we Cross the unit time vector and space vector what will Yield ??

 

[PeterDonis]

The cross product yield vector represent the area between 2 vectors and volume in 3 vectors if we Cross the unit time vector and space vector what will Yield ??

A cross product in general is not a vector; it's an antisymmetric second-rank tensor. It just so happens that, in ordinary 3-dimensional space, there is a one-to-one mapping between antisymmetric second-rank tensors and vectors, so you can interpret the cross product as a vector. But spacetime has four dimensions, not three, so you can't interpret the cross product as a vector. A cross product between a timelike and spacelike vector in spacetime is just an antisymmetric second-rank tensor representing an area in a plane with one timelike and one spacelike dimension.