Lower Indices Tensor in Special Relativity: What to Know?

[Palindrom]

Given a tensor with upper indices in special relativity, what do I know of the corresponding tensor with lower indices? Why?
For example, for the antisymmetric tensor $$\varepsilon ^{iklm}$$, what is $$\varepsilon _{iklm}$$?

 

[Physics Monkey]

How do you always find the lowered components given the upper components? The answer is contraction with the metric.

 

[Palindrom]

So now we're down to this question: What is contraction?

(thanks by the way)

 

[robphy]

http://mathworld.wolfram.com/IndexGymnastics.html

 

[Palindrom]

http://mathworld.wolfram.com/IndexGymnastics.html

Great, thanks.

 

[jcsd]

Well the relationship is that e^iklm are the commpoents of a (4,0) tensor and e_iklm are the components of it's dual. Clearly then there's a metric function (in this case a (4,4) tensor) that maps the first tensor to it's dual and unsuprisingly this metric is related to the metric on the space of (1,0) tensors.

All you need to do is use 'inner multiplication' the metric tensor and it's dual.