Energy Formulas in SR: Explained

[PWiz]

This is actually the reason I dislike the term "covariant vector" (and "contravariant vector"). The vectors themselves are either tangent vectors, which may be defined as directional derivatives or equivalence classes of curves, or covectors. Tangent vectors have a coordinate basis which transforms covariantly with components transforming contravariantly and covectors have contravariant coordinate bases and covariant components. The vectors themselves are not dependent on the choice of coordinate system and are either tangent vectors or covectors.

Well "covariant" and "contravariant" are old terms - they tend to mix up the expression of new vectors in terms of the old basis vectors and the expression of the same vector in terms of the new basis. I think we can agree that one-forms and vectors sound "nicer".

 

[Orodruin]

I actually prefer tangent vectors and co(tangent)vectors. The reason is that "one form" brings my mind to think of n-forms and not arbitrary tensors.

 

[ChrisVer]

The reason is that "one form" brings my mind to think of n-forms and not arbitrary tensors.

Is there a difference between n-forms and tensors?

 

[stevendaryl]

AFAIK, there is no way to actually "prove" why upper indices were chosen to be contravariant - it just seems like an arbitrary decision to me. All that really matters in the end is that the summation is correctly carried out when two identical upper and lower indices are seen in an expression, and that one-forms and vectors are clearly distinguishable and recognizable when seen together (to prevent tensor algebra from going topsy-turvy).

Well, in the Einstein convention, it's pretty hard to go wrong, because you always match a raised index with a lowered index. The only additional bits that need to be remembered is that whatever your convention for variables such as $x^\alpha$, derivatives count as the opposite: $\partial_\alpha$.

 

[stevendaryl]

Is there a difference between n-forms and tensors?

The way I understand it, an n-form is a special case of a tensor, namely a tensor whose components have all lowered indices.

 

[ChrisVer]

I should have written an (0 n)-tensor or (n 0)-tensor (I don't remember right now where the contra/co-variant rank goes in this notation)

 

[Orodruin]

An $n$-form is (equivalent to) a totally anti-symmetric $(0,n)$ tensor. A typical example of a $(0,2)$ tensor which is not a 2-form is the metric.

 

[PWiz]

Isn't an n-form a $\binom {0}{n}$ type tensor - a function of $n$ vectors into the real numbers which is linear in each of its $n$ arguments?

 

[ChrisVer]

So it takes the antisymmetry of the (0 n)-tensor...
The thing is that I have only encountered 0 or 1 forms,and this didn't let me see any distinction.

 

[Orodruin]

Of course, for 1-forms, there is nothing to be anti-symmetric with so they are equivalent to covectors.

 

[ChrisVer]

As for the $ict$...
One useful application of that in SR is when you want to look at the rotations and boosts that are there for the Lorentz Group SO(3,1) instead of pure rotations that would be for the SO(4). In fact it would change the trigonometric functions (rotations) to hyperbolic ones (boosts). It is useful because I think it doesn't intercept with the problems of GR set by Stevendaryl .
And generally then for Wick rotations (but that's another thing).

 

[PWiz]

Well, in the Einstein convention, it's pretty hard to go wrong, because you always match a raised index with a lowered index. The only additional bits that need to be remembered is that whatever your convention for variables such as $x^\alpha$, derivatives count as the opposite: $\partial_\alpha$.

It can sometimes be difficult to remember which indices go up and which go down. Take the tensor transformation law for example:
$$S^{μ'}\ _{ν\ 'ρ'}= \frac{∂x^{μ'}}{∂x^{μ}} \frac{∂x^{ν}}{∂x^{ν \ '}} \frac{∂x^{ρ}}{∂x^{ρ'}} S^{μ}\ _{νρ}$$
I'm trying to memorize this (and it's a little tricky) right now. If I were to mix up the order of even one index, everything would go for a toss.

 

[Orodruin]

It can sometimes be difficult to remember which indices go up and which go down. Take the tensor transformation law for example:
$$S^{μ'}\ _{ν\ 'ρ'}= \frac{∂x^{μ'}}{∂x^{μ}} \frac{∂x^{ν}}{∂x^{ν \ '}} \frac{∂x^{ρ}}{∂x^{ρ'}} S^{μ}\ _{νρ}$$
I'm trying to memorize this (and it's a little tricky) right now. If I were to mix up the order of even one index, everything would go for a toss.

There is really only one way of doing it and the rules to follow are very simple. Free upper indices need to be up on both sides and vice versa. Repeated (summation) indices need to appear one up and one down (partial derivatives count as down in terms of the coordinate they are derivatives with respect to).