Metric tensor : raising/lowering indices

In summary, Tensor calculus can be a little confusing. It's helpful to have a good textbook when studying this subject.
  • #1
tb87
8
1
Hi everyone,

I'm currently studying Griffith's Intro to Elementary Particles and in chapter 7 about QED, there's one part of an operation on tensors I don't follow in applying Feynman's rules to electron-muon scattering :

## \gamma^\mu g_{\mu\nu} \gamma^\nu = \gamma^\mu \gamma_\mu##

My teacher spent very little time on tensors and I'm really not sure 1) what's the difference between ##\gamma^{\mu\nu}## and ##\gamma_{\mu\nu}##, 2) why is the second ##\gamma##'s indice switched from ##\nu## to ##\mu## (and was also lowered).

Besides, I'm still strugling to understand the general difference between lowered and raised indices (e.g. : why I'm never seeing ##\gamma_\mu^\nu##, but other tensors are written that way). In Griffith's electrodynamics, the author says that in ##\Lambda_\mu^\nu##, it is ##\mu## that represents lines and ##\nu## that represents columns. However, this isn't consistent with everything I'm seeing on tensors.

Thanks!
Alex
 
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  • #2
it is very bad idea to study tensor calculus by physics textbooks. I am sure participants of PF soon prompt you good textbook in differential geometry or in tensor calculus
 
  • #3
Yes, this notation is not good. As a general guidance, consider the following...

Suppose ##V## an arbitrary finite-dimensional vector space. Then there will always exist a dual space ##V^*:V \to \mathbb{R}## such that for any ##\varphi \in V^*## and any ##v \in V## that ##\varphi(v) = \alpha \in \mathbb{R}##.

And if there exists a bilinear form on ##V##, that is ##b:V \times V \to \mathbb{R},\,\,\,b(v,w) \in \mathbb{R}## ie. an inner product(or norm), then there will always be some particular ##\varphi_v \in V^*## such that ##\varphi_v(w)=b(v,w)## for any ##w \in V##

Now if I fix some ##v \in V## and write, say ##b(v,\,\cdot) \in \mathbb{R}## then I may have the equality ##b(v,\,\cdot)= \varphi_v## That is, ##b(v,\,\cdot)## acts as though it is a dual vector.

Just carry out the obvious notational substitutions, i.e ##b \mapsto g_{\mu\nu}##, and ##v \mapsto\gamma^{\mu},\,\varphi_v \mapsto \gamma_\nu## and you have your answer
 

1. What is the purpose of raising and lowering indices in a metric tensor?

The metric tensor is a mathematical object used in general relativity to describe the curvature of spacetime. Raising and lowering indices allows us to convert between covariant and contravariant vectors, which have different transformation properties under coordinate transformations. This is important because the metric tensor itself is a covariant object, but many physical quantities, such as velocities and momenta, are described by contravariant vectors.

2. How is the metric tensor used in general relativity?

The metric tensor is used to calculate the distance between two points in spacetime, which is necessary for understanding the curvature of spacetime caused by massive objects. It is also used to define geodesics, which are the paths that particles follow in curved spacetime.

3. What is the difference between a covariant and contravariant vector?

A covariant vector is a vector that transforms in the same way as the coordinates under a coordinate transformation. In contrast, a contravariant vector transforms in the inverse way of the coordinates. This distinction is important in general relativity because it allows us to properly define physical quantities in a curved spacetime.

4. How does raising and lowering indices affect the metric tensor?

Raising and lowering indices does not change the values of the metric tensor, but it changes the way that it transforms under coordinate transformations. When indices are raised or lowered, the metric tensor is multiplied by the inverse or the determinant of the metric tensor, respectively.

5. Can the metric tensor be used in other fields of science?

Yes, the metric tensor has applications in many other fields of science, such as physics, mathematics, and engineering. It is used to describe the geometry of spaces with varying curvature, and it is also used in the study of black holes, gravitational waves, and cosmology.

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