Lower Indices Tensor in Special Relativity: What to Know?

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In summary, when given a tensor with upper indices in special relativity, you can find the corresponding tensor with lower indices by using contraction with the metric. This is because the components of the original tensor and its dual are related by a metric function. To find the lowered components, you can use 'inner multiplication' with the metric tensor and its dual.
  • #1
Palindrom
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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 [tex]$\varepsilon ^{iklm} $
[/tex], what is [tex]$\varepsilon _{iklm} $
[/tex]?
 
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  • #2
How do you always find the lowered components given the upper components? The answer is contraction with the metric.
 
  • #3
So now we're down to this question: What is contraction?

(thanks by the way)
 
  • #6
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.
 

1. What is a lower indices tensor in special relativity?

A lower indices tensor in special relativity is a mathematical object that describes the relationship between quantities measured in different reference frames in the theory of relativity. It is used to represent the physical laws of special relativity in a covariant manner, meaning that it remains unchanged under coordinate transformations.

2. How is a lower indices tensor different from an upper indices tensor?

A lower indices tensor is a dual object to an upper indices tensor, meaning that it represents the same physical quantity but in a different way. In special relativity, a lower indices tensor is used to represent contravariant vectors, while an upper indices tensor represents covariant vectors. The main difference is in how the components of the tensor transform under coordinate transformations.

3. What are some common examples of lower indices tensors in special relativity?

Some common examples of lower indices tensors in special relativity include the metric tensor, the stress-energy tensor, and the electromagnetic field tensor. The metric tensor describes the geometry of spacetime, the stress-energy tensor represents the energy and momentum of matter and radiation, and the electromagnetic field tensor describes the electromagnetic field in spacetime.

4. How is the contraction of a lower indices tensor calculated?

The contraction of a lower indices tensor is calculated by summing over the repeated indices in the tensor. This operation is also known as index contraction or Einstein summation convention. It is used to simplify tensor equations and represents a generalization of the dot product between vectors in Euclidean space.

5. What are some applications of lower indices tensors in special relativity?

Lower indices tensors in special relativity have various applications in physics, including in the theory of gravity, particle physics, and cosmology. They are used to describe the behavior of matter and energy in relativistic systems and are essential in formulating the equations of motion and conservation laws in these fields.

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