Co- and Contravariance

[deuteron]

TL;DR

What physical meaning does covariance and contravariance have?

Hi,
I am very confused about the mathematics related to special relativity.
I have understood, that a four-vector with an upper index has the form:
$$A^\mu = (A^0 , A^1, A^2, A^3)$$
where lowering the index would make the indices other than the $0$th negative:
$$A_\mu = (A_0, -A^1, -A^2, -A^3)$$
In order to lower the index, we would need to multiply the four-vector with the Minkowski metric.

I understand, that the four vector $x^\mu = (ct, x^1,x^2,x^3)$ gives the temporal and spatial positions of an event.
What I don't understand is what "multiplying by the Minkowski metric" physically does to the four vector, and what the physical difference between $x_0$ and $x^0$ is. What physical difference does it make when the four-vector has a lower index or an upper index?

I am in general very confused about this, so any recommendation of a source on the mathematics of special relativity is very welcome

 

[Orodruin]

That is a very good question. In fact, many (many!) students are confused by this. In fact, it is such a good question I might just take the time to write up a PF Insight on the topic …

In the meantime, know that you are not alone and that it is very confusing. It will be somewhat clearer if you go to curvilinear coordinates or curved spacetimes (ie, GR) as there the distinction between the tangent space and its dual is clearer.

 

[deuteron]

That is a very good question. In fact, many (many!) students are confused by this. In fact, it is such a good question I might just take the time to write up a PF Insight on the topic …

In the meantime, know that you are not alone and that it is very confusing. It will be somewhat clearer if you go to curvilinear coordinates or curved spacetimes (ie, GR) as there the distinction between the tangent space and its dual is clearer.

In case you write that Insight, can you please also explain what makes the d'Alembert operator "relativistic" and different from the classical $\frac 1 c \frac {\partial^2}{\partial t^2}-\Delta$ that we have in the "classical" wave equation?..

 

[pervect]

TL;DR Summary: What physical meaning does covariance and contravariance have?

Hi,
I am very confused about the mathematics related to special relativity.
I have understood, that a four-vector with an upper index has the form:
$$A^\mu = (A^0 , A^1, A^2, A^3)$$
where lowering the index would make the indices other than the $0$th negative:
$$A_\mu = (A_0, -A^1, -A^2, -A^3)$$
In order to lower the index, we would need to multiply the four-vector with the Minkowski metric.

I understand, that the four vector $x^\mu = (ct, x^1,x^2,x^3)$ gives the temporal and spatial positions of an event.
What I don't understand is what "multiplying by the Minkowski metric" physically does to the four vector, and what the physical difference between $x_0$ and $x^0$ is. What physical difference does it make when the four-vector has a lower index or an upper index?

I am in general very confused about this, so any recommendation of a source on the mathematics of special relativity is very welcome

It's rather like the difference between a row vector and a column vector in matrix notation, if that helps any. Graphically, vectors, represented with a superscript like $A^i$, are usually represented as little arrows. co-vectors, represented with a subscript, like $A_i$, are usually represented by a "stack of parallel plates". A co-vector maps a vector to a scalar, in the graphical representation this is the number of parallel plates the line with the arrow pierces.

As long as you have a metric tensor, you can convert vectors to co-vectors, the metric tensor maps vectors to co-vectors as you mentioned, and the inverse of the metric tensor, $g^{uv}$, maps co-vectors to vectors. In matrix notation, there is no metric tensor, the operation of taking the transpose makes a vector a co-vector and vica-vera.

I would say it's largely mathematical question rather than a physical one, though there are some standardized conventions as to which one is used in which circumstance.

If the stack-of-plates notion isn't familiar, https://math.stackexchange.com/questions/1078427/how-to-visualize-1-forms-and-p-forms has some illustrations.

Mathematically a co-vector, or "one-form", is a linear map from a vector to a scalar. The relation between vectors and co-vectors is known as duality - each is the "dual" of the other.

 

[Sagittarius A-Star]

I am in general very confused about this, so any recommendation of a source on the mathematics of special relativity is very welcome

The chapter about the metric tensor may help:
http://www.scholarpedia.org/article/Special_relativity:_electromagnetism#The_metric_tensor

 

[Ibix]

Ben Crowell (former mentor here) noted that the concept is similar to frequency and period. Both encode the same information but in different ways, and if you change coordinates (e.g. change from measuring time in seconds to minutes) they change in opposite directions. And the product of the two is invariant under coordinate change.