Contravariant and covariant indices

• spookyfish
In summary: However, if you use the convention that ##\eta^{\mu\nu}## raises indices and ##\eta_{\mu\nu}## lowers them, then you get$$\Lambda^\mu_\nueq \Lambda^\mu{}_\nu.$$
spookyfish
When we write contravariant and covariant indices, for example for the Lorentz transformation, does it matter if we write $\Lambda^\mu\,_\nu$ or $\Lambda^\mu_\nu$?
i.e. if the $\nu$ index is to the right of the $\mu$ or they are at the same place with respect to left-right?

Lorentz transformations are linear operators on ##\mathbb R^4## (or ##\mathbb R^2## or ##\mathbb R^3##). So they can be represented by matrices. (See the https://www.physicsforums.com/showthread.php?t=694922 about matrix representations of linear transformations). I will not make any notational distinction between a linear operator and its matrix representation with respect to the standard basis.

Let ##\Lambda## be an arbitrary Lorentz transformation. By definition of Lorentz transformation, we have ##\Lambda^T\eta\Lambda=\eta##. This implies that ##\Lambda^{-1}=\eta^{-1}\Lambda^T\eta##. Let's use the notational convention that for all matrices X, we denote the entry on row ##\mu##, column ##\nu## by ##X^\mu{}_\nu##. If we use this convention, the definition of matrix multiplication, our formula for ##\Lambda^{-1}## and the convention that every index that appears twice is summed over, we get
$$(\Lambda^{-1})^\mu{}_\nu = (\eta^{-1})^\mu{}_\rho (\Lambda^T)^\rho{}_\sigma \eta^\sigma{}_\nu = (\eta^{-1})^\mu{}_\rho \Lambda^\sigma{}_\rho \eta^\sigma{}_\nu.$$ This is where things get funny. It's conventional to write ##\eta_{\mu\nu}## instead of ##\eta^\mu{}_\nu##, and ##\eta^{\mu\nu}## instead of ##(\eta^{-1})^\mu{}_\nu##. If we use this convention, we have
$$(\Lambda^{-1})^\mu{}_\nu = \eta^{\mu\rho} \Lambda^\sigma{}_\rho \eta_{\sigma\nu}.$$ Now if we also use the convention that ##\eta^{\mu\nu}## raises indices and ##\eta_{\mu\nu}## lowers them, we end up with
$$(\Lambda^{-1})^\mu{}_\nu = \Lambda_\nu{}^\mu.$$ So if ##\Lambda## isn't the identity transformation, we have
$$\Lambda_\nu{}^\mu = (\Lambda^{-1})^\mu{}_\nu \neq \Lambda^\mu{}_\nu.$$ As you can see, the inequality is a result of the definitions of ##\eta_{\mu\nu}## and ##\eta^{\mu\nu}##, so if you use a notational convention that denotes these things by something else, or doesn't use these things to raise and lower indices, it may be OK to write ##\Lambda^\mu_\nu##.

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

Contravariant and covariant indices are two types of index notation used in mathematics and physics. The main difference between them is the way they transform under coordinate transformations. Contravariant indices transform in the opposite direction of the coordinate system, while covariant indices transform in the same direction.

2. Why are contravariant and covariant indices important?

Contravariant and covariant indices are important because they allow us to express mathematical equations in a way that is independent of the coordinate system being used. This is essential in physics, where equations must hold true regardless of the chosen coordinate system.

3. How do contravariant and covariant indices relate to tensors?

In mathematics and physics, tensors are objects that can be described by both contravariant and covariant indices. The contravariant and covariant indices represent different aspects of the tensor, and they are related through a transformation rule. This allows us to perform operations on tensors, such as addition and multiplication, without worrying about the coordinate system.

4. Can you give an example of a contravariant and covariant index in a real-world application?

One example of contravariant and covariant indices in a real-world application is in general relativity. In this theory, the metric tensor is a fundamental object that relates the geometry of space-time to the distribution of matter and energy. The components of the metric tensor have both contravariant and covariant indices, which allows us to describe the curvature of space-time in a coordinate-independent way.

5. How do we know which type of index to use in a given situation?

The choice of using a contravariant or covariant index depends on the specific problem or equation being solved. In general, contravariant indices are used for vectors and covariant indices are used for dual vectors. However, there are situations where both types of indices may appear in the same equation. It is important to carefully consider the transformation properties of the indices and choose the appropriate type for the given problem.

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