Relativistic Notation in Waves: Confusion Solved

[kent davidge]

If we have a plane wave, usually in Relativity notation it is written as $A^\alpha = a^\alpha \exp(i x_\alpha k^\alpha)$. (I know we need to take the real part in the end). In cartesian coordinates, and two dimensions say, that $x_\alpha k^\alpha$ would be $x^\alpha k_\alpha = x k_x + y k_y$. How does it reads in polar coordinates? $x^\alpha' k_\alpha' = r k_r + \theta k_\theta = r [(\partial x/ \partial r)k_x + (\partial y/ \partial r)k_y] + \theta [(\partial x/\partial \theta )k_x + (\partial y/\partial \theta )k_y] = r (\cos\theta k_x + \sin\theta k_y) + \theta[(-r\sin\theta ) k_x + (r\cos\theta ) k_y]$. The first term is just equal to its version in Cartesian coordinates. But then there is that second term, which makes $x^\alpha' k_\alpha' \neq x^\alpha k_\alpha$. I already expected this result from the fact that $x^\alpha$ do not transform like, and thus are not, four vectors. So a product like $x^\alpha k_\alpha$ will not be a scalar.

But then what does this result mean? A field like $A^\alpha = a^\alpha \exp(i x_\alpha k^\alpha)$ will not transform as a vector if the exponent is not a scalar. But it should transform as a vector. So what we do?

 

[Orodruin]

If you want to do non-Minkowski coordinates, then you should replace $x^\alpha$ by the event vector (which in Minkowski coordinates have the coordinates as components - just like the position vector in Cartesian coordinates in Euclidean space has the coordinates as components).

 

[kent davidge]

event vector

Sorry, I never heard about the event vector. What is it?

 

[Orodruin]

Sorry, I never heard about the event vector. What is it?

It is the same thing in Minkowski space as the position vector is in Euclidean space.

 

[kent davidge]

It is the same thing in Minkowski space as the position vector is in Euclidean space.

Ok. That makes a lot of sense.

- we know that in Minkowski coordinates the term has the form $x^\alpha k_\alpha$;
- we also know that it should be a scalar;
- so we identify $x^\alpha$ as the components of a vector;

Do you know which symbol is used to denote this vector?