Why Lowering Index on Components Affects Physical Variables

In summary, the conversation discusses the process of lowering the index of a vector in order to compute the scalar product using a metric. This is related to the invariance of certain equations in different coordinate systems.
  • #1
enomanus
7
0
Hi everyone! I am trying to self- study GR and I am (very slowly) working through Schutz at the moment ( not the clearest at times for a " first course" text). Anyway I have noticed that several times when working with components of physical variables e.g. momentum/ energy, the 'up' index in say p alpha is deliberately 'lowered' in the workings to get p alpha 'down'.
for example in - deriving the Consevation law on geodesics so that we can spot if the metric is independent of any x component or - in Shutz page 170 where we work out the scalar product - p.p to show that energy is conserved in a stationary gravity field.
i can follow the lines and the equations BUT I don't understand why we keep lowering the index?
Is this something to do with physical variables or with 1-forms??
The answer is probably something very simple but I'm asking anyway
Thanks!
 
Physics news on Phys.org
  • #2
We hypothesize that there is a metric - it eats two vectors and spits out a number - this number is the scalar product of the two vectors. When you compute the scalar product, you may choose to have the metric eat one vector before eating the second vector. This process is called lowering the index of the first vector.
 
  • #3
It is related to the fact that

[tex]\sum_{\alpha=0}^3dx_{\alpha}dx^{\alpha}=\sum_{\alpha=0}^3\sum_{\beta=0}^3g_{\alpha\beta}dx^{\alpha}dx^{\beta}[/tex]​

is invariant (the same in every coordinate system), whereas

[tex]\sum_{\alpha=0}^3dx^{\alpha}dx^{\alpha}[/tex]​

isn't.
 

1. Why is lowering the index on components important?

Lowering the index on components is important because it allows for the manipulation and control of physical variables. The index refers to the refractive index, which is a measure of how light bends when passing through a material. By lowering the index, we can change the way light interacts with a material, leading to changes in physical properties.

2. How does lowering the index on components affect physical variables?

Lowering the index on components can affect physical variables in several ways. For example, it can change the optical properties of a material, such as its transparency or reflectivity. It can also alter the electrical conductivity or thermal conductivity of a material, which can have a significant impact on its performance in various applications.

3. What are some real-world applications of lowering the index on components?

Lowering the index on components has numerous real-world applications, particularly in the field of optics and photonics. This includes the production of anti-reflective coatings for lenses, the creation of high-efficiency solar cells, and the development of optical fibers for telecommunication. It is also used in the production of electronic devices to improve their performance and efficiency.

4. Are there any potential drawbacks to lowering the index on components?

While lowering the index on components can lead to many benefits, there are also potential drawbacks to consider. For example, altering the refractive index of a material can also affect its mechanical strength, making it more susceptible to damage. It can also be challenging to control and maintain a specific index, which may require advanced manufacturing techniques.

5. How can scientists and engineers manipulate the index on components?

Scientists and engineers can manipulate the index on components in various ways, depending on the material and application. One common method is through the use of thin-film deposition techniques, where a layer of material with a specific index is deposited onto a substrate. Other methods include ion implantation, which involves bombarding a material with high-energy ions, and adding dopants to alter the index of a material.

Similar threads

  • Special and General Relativity
Replies
1
Views
619
  • Special and General Relativity
Replies
10
Views
1K
  • Special and General Relativity
2
Replies
35
Views
2K
Replies
3
Views
719
  • Special and General Relativity
Replies
1
Views
2K
  • Special and General Relativity
Replies
20
Views
972
  • Advanced Physics Homework Help
Replies
0
Views
520
  • Special and General Relativity
Replies
4
Views
2K
  • Special and General Relativity
Replies
1
Views
779
  • Programming and Computer Science
Replies
9
Views
2K
Back
Top