Basic question on co- and contra-variance

[sanctus]

The usual definition I saw for covariant and contravariant vectors, tensors etc. is the way they transform. But is there anything physical to it, I can't see beyond the math there but I'm sure there must be something very basic?

[Qfwfq]

The very basic thing is the non-euclidean geometry.

 

One approach, not much used, is to define the time component as imaginary so that its square is automatically of opposite sign. The usual method using g is a bit more machinous but keeps things more "real" ;).

 

Apart from the distinction of indices as being co or contra, Lorentz covariance is basically the same as Euclidean covariance. If the two sides of the equation are of the same type, it holds in all coordinates. If instead you propose an equation in which mass depends on some function of the x component of some vector, people are gonna ask "Hey, but in whose coordinates?". There's not more to it but it's so "obvious" that professors don't always spell it out very well!!! ;)

[Erasmus00]

The usual definition I saw for covariant and contravariant vectors, tensors etc. is the way they transform. But is there anything physical to it, I can't see beyond the math there but I'm sure there must be something very basic?

 

The contra varient tensors live in the actual vector space, while covarient tensors live in a dual space. (which is why you can combine co and contra varient tensors to get a scalar).

 

If your space is equipped with an inner product, then you can connect the two spaces via the metric.

-Will

[Qfwfq]

I agree that it can be stated in terms of duality but the terms covariant/contravariant are due to the coordinate transformations and are essentially geometric, more than algebric. Of course, the product isn't "inner" if you consider it as between V and V*, it only becomes inner if you consider V as being self-dual.

 

I think Sanctus was concerned less with the math than with finding something beyond it.

[sanctus]

But why one is co- and the other contra-?

Is it because the covariant varies without the need of the metric (ie you just do a lorentz transformation) and the contravariant need the metric (ie you first lower the indices and then you transform)? If yes, the introduction of contravariant vectors is only motivated by the fact that it simplifies the wrtitting of things like the scalar product etc?

 

And here comes he most stupid question (here I shouldn't say that I'm doing the master in physics :note: ) :

what is exactly meant by euclidian covariance (or lorentz covariance)? Just the way it transforms(ie in the euclidian case adding the speed of the new reference system and in the lorentz one just making a lorentz transformation with the betas and gammas).

 

I'm sure that you are right when you say that it is so trivial that nobody ever says it. That's why now that I started to ask myself now what's actually underneath it (and making my own hypothesis), I have nobody to ask to as this things seem now to be taken as granted from every prof...

[Qfwfq]

The co is the one that varies with the coordinates, i. e. geometrically. The contra is the one that varies the other way, or "against" so to speak. Of course, which is the change of coordinates and which is the other is arbitrary from a mathematical point of view, whereas physically you measure space and time coordinates of each event.

 

Covariance means that, if you write pµ = m uµ and you change the coordinate axes, the components of p will change but so will the those of m u because they are the same type of tensor object. It's exactly the same notion as with euclidean 3-vectors, pi = m vi, except that in this case the metric is all +1 and no -1 so there's no contra. Of course, the trick of using an imaginary time coordinate allows the use of euclidean metric for space-time (or space-itime you might call it). It just gains the same result in a different way, the minus sign is "split between" the two vectors in the scalar product.

 

In short, it's just a way of getting that darn '-' in dx^2 - dt^2.

 

here I shouldn't say that I'm doing the master in physics

Don't worry, we all found these things terribly confusing and mysterious and the professors all fail to spell them out more, because they become so obvious once you have reached enlightenment... :hyper: At that point you see the light but don't have the faintest idea how to get it across to the newbie.