Contravariant first index, covariant on second, Vice versa?

In summary, the conversation discusses the derivation of the Dirac matrix transformation properties and the use of tensors in the Lorentz transformation. The key points are that the Lorentz transformation coefficients are not the elements of a tensor, and that there is a difference between using the inverse matrix to transform covectors and using it to transform vectors. The reference provided, "Lessons in Particle Physics" by Luis Anchordoqui and Francis Halzen, does not focus on teaching relativity and may not be the most effective resource for learning this topic.
  • #36
pervect said:
My understanding is that @Orodruin suggests that there is a problem with what I quoted
No. What you quoted is fine. What you are not paying proper attention to is the difference between MTW’s ##L^{\alpha’}_{\phantom\alpha\beta}## and their ##L^{\alpha}_{\phantom\alpha\beta’}##, which are different sets of transformation coefficients. The primes are important in this notation, they tell you what transformation coefficients are intended. You cannot hope to properly recover the transformation rules unambiguously if you drop the primes on the indices unless you introduce different notation for the transformation coefficients.
 
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  • #37
pervect said:
For usage, we might transform a vector using the first transformation matrix, i.e ##x^{\mu} = \Lambda^{\mu}{}_{\nu} \, x^{\nu}## and covectors using the second, ##x_{\mu} = \Lambda^{\nu}{}_{\mu} x_{\nu}##. Which would be my guess as to what you wanted to do.

Just to be a bit clearer. You are using
$$
x_\mu = \Lambda^\nu_{\phantom\nu\mu} x_\nu,
$$
which is unclear for many reasons. First of all, it is not true taken at face value unless ##\Lambda^\nu_{\phantom\nu\mu} = \delta^\nu_\mu##. You need to prime one of the sets of coordinates. Second, if you do this, it is not clear how you intend to handle the indices. If you just leave the indices as they are without priming, you would obtain
$$
x'_\mu = \Lambda^\nu_{\phantom\nu\mu} x_\nu.
$$
There is now nothing that tells me that the ##\Lambda^\nu_{\phantom\nu\mu}## is in any way different from the ##\Lambda^\nu_{\phantom\nu\mu}## that appeared in the contravariant transformation rule. This is the reason for priming the indices belonging to the primed coordinate system. You would end up with
$$
x'_{\mu'} = \Lambda^\nu_{\phantom\nu\mu'} x_\nu
$$
as well as
$$
x'^{\mu'} = \Lambda^{\mu'}_{\phantom\mu\nu} x^\nu,
$$
where the transformation coefficients are clearly distinct.
 
<h2>1. What is the difference between contravariant and covariant indices?</h2><p>Contravariant indices are used to represent vectors, while covariant indices are used to represent covectors. In other words, contravariant indices indicate the direction of a vector, while covariant indices indicate the components of a covector.</p><h2>2. How does the placement of indices affect tensor calculations?</h2><p>The placement of indices determines how the tensor transforms under coordinate transformations. If the indices are contravariant, the tensor will transform as a vector, while if the indices are covariant, the tensor will transform as a covector.</p><h2>3. Can the placement of indices be interchanged?</h2><p>Yes, the placement of indices can be interchanged as long as the overall tensor remains unchanged. This is known as index raising and lowering, and is often used to simplify calculations.</p><h2>4. How is the metric tensor related to the placement of indices?</h2><p>The metric tensor is used to raise and lower indices in tensor calculations. For a given metric, the placement of indices can be changed by multiplying the tensor by the metric or its inverse.</p><h2>5. What is the physical significance of contravariant and covariant indices?</h2><p>Contravariant and covariant indices are used to describe the geometric properties of space and time. They allow us to represent physical quantities such as vectors and tensors in a coordinate-independent manner, making them essential in the study of physics and mathematics.</p>

1. What is the difference between contravariant and covariant indices?

Contravariant indices are used to represent vectors, while covariant indices are used to represent covectors. In other words, contravariant indices indicate the direction of a vector, while covariant indices indicate the components of a covector.

2. How does the placement of indices affect tensor calculations?

The placement of indices determines how the tensor transforms under coordinate transformations. If the indices are contravariant, the tensor will transform as a vector, while if the indices are covariant, the tensor will transform as a covector.

3. Can the placement of indices be interchanged?

Yes, the placement of indices can be interchanged as long as the overall tensor remains unchanged. This is known as index raising and lowering, and is often used to simplify calculations.

4. How is the metric tensor related to the placement of indices?

The metric tensor is used to raise and lower indices in tensor calculations. For a given metric, the placement of indices can be changed by multiplying the tensor by the metric or its inverse.

5. What is the physical significance of contravariant and covariant indices?

Contravariant and covariant indices are used to describe the geometric properties of space and time. They allow us to represent physical quantities such as vectors and tensors in a coordinate-independent manner, making them essential in the study of physics and mathematics.

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