Succeed in Learning Relativity: Transport, Frames, Tensors

In summary, understanding concepts such as different types of transport of vectors, frames, tensor densities, and integrations is essential in order to fully grasp the theory of general relativity and its applications. While some of these concepts may seem easy, they play an important role in understanding the theory and its implications. Omitting them may hinder one's success, especially for those interested in doing research in this field.
  • #1
kent davidge
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How much essential is it to learn concepts such as
- different types of transport of vectors, like Fermi Walker and Parallel transport;
- different frames (basis vectors);
- tensor densities and integrations
To put it another way, do they play a important role in the theory? If one for some reason does not know much about them (although they seem to be easy for me, so it would be easy to have a grasp), would this person still succeed?
 
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  • #2
I guess you are asking about general relativity. It depends on the course. I had a course that introduced the mathematics as needed. Sure, you needed to know what vector spaces are and so on, but nothing beyond these basics. Some courses expect that many things are known in advance, however.
 
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  • #3
It depends what you want to do with GR. Just understand some GR, and do some problems once in a while? Then just learn what you need to solve the problem!

Want to read papers, and eventually do research in GR? then you will be gated by your math if you don't learn some of these things. It may not be a computational aspect of it, but more so a "how is this connected to what i know". If this is the path you want to take, well, you have two options:
1) Struggle with the ideas now, forget them for the most part, but have some better intuition when discussing ideas with colleagues.
2) Have no idea when the topic is being discussed or brought up in a paper, having to go home and expecting to be able to discuss it in detail the next week.

If you have time, I'd go with option 1).
 
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  • #4
Parallel transport and basis vectors are fundamental.

The others are all good to know, but you don't have to know them right away.
 
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  • #5
pervect said:
Parallel transport and basis vectors are fundamental.
Is parallel transport more fundamental than other types of transport? Like Fermi-Walker transport.
 
  • #6
kent davidge said:
Is parallel transport more fundamental than other types of transport? Like Fermi-Walker transport.

In my opinion, yes. Parallel transport defines geodesic (geodesics are curves that parallel transport their tangent vectors), and the covariant derivative, both of which are vital.

Pre-requisite concepts are the notion of a vector (as far as I know, you can't talk about transporting vectors until you know what they are, though I can't rule out the possibility of some alternative that I've never studied), and the notion of a connection. Picking out a specific connection is also needed to make parallel transport unique. Any connection defines some notion of parallel transport and thus some notion of geodesics. Using the Levi-Civita connection makes geodesics curves of extremal length. It remains then to show that the Levi-Civita connection is the unique torsion free connection that is metric compatible (and to define metric compatible).

Fermi-walker transport isn't stressed as much, it's important for Fermi-normal coordinates. While Fermi-normal coordaintes are not tyhpically stressed, I find them very useful in providing physical interpretations of the math and as a way to define the notion of "an observer".
 
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  • #7
pervect said:
to define the notion of "an observer"
what is the more natural coordinate system used by an observer? if any..
 
  • #8
kent davidge said:
what is the more natural coordinate system used by an observer? if any..

SOme of this is a matter of opinion - I think Fermi normal coordinates are an important local coordinate system. They are useful in the region near a reference worldline. Harmonic coordinates also have some very nice properties, especially for weak fields.
 
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  • #9
kent davidge said:
How much essential is it to learn concepts such as
- different types of transport of vectors, like Fermi Walker and Parallel transport;
- different frames (basis vectors);
- tensor densities and integrations
To put it another way, do they play a important role in the theory? If one for some reason does not know much about them (although they seem to be easy for me, so it would be easy to have a grasp), would this person still succeed?
Parallel transport is needed to understand curvature, covariant derivative, and helps with geodesics.

Fermi-walker transport is essential to understand rotation versus non-rotation. Fermy-walker transport defines what happens to a gyroscope.

Basis vectors are needed to compute any physical observables.

Tensor densities are needed to state any integral relations. Arguably, if you stick to ultralocal physics, you don't need them.
 
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  • #10
kent davidge said:
How much essential is it to learn concepts such as
- different types of transport of vectors, like Fermi Walker and Parallel transport;
- different frames (basis vectors);
- tensor densities and integrations
To put it another way, do they play a important role in the theory? If one for some reason does not know much about them (although they seem to be easy for me, so it would be easy to have a grasp), would this person still succeed?
I guess you talk about the general theory of relativity. Then all these notions are very important. They are simply the language needed to formulate the theory! The only thing you can omit at a first glance is Fermi-Walker Transport, which becomes however important whenever it comes to things related with spin (e.g., to understand what's behind the Gravity-probe B experiment and frame dragging and all that).
 
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Related to Succeed in Learning Relativity: Transport, Frames, Tensors

1. What is relativity and why is it important to learn?

Relativity is a theory developed by Albert Einstein that explains the relationship between space and time. It is important to learn because it has revolutionized our understanding of the universe and is the basis for many modern technologies, such as GPS systems.

2. What are transport, frames, and tensors in the context of relativity?

Transport refers to the movement of objects through space and time. Frames are reference points used to measure this movement. Tensors are mathematical objects used to describe the properties of space and time in different frames of reference.

3. How does relativity impact our everyday lives?

Relativity has led to the development of technologies such as GPS and satellite communication, which rely on accurate measurements of time and space. It also plays a role in our understanding of the behavior of light and the structure of the universe.

4. What are some common misconceptions about relativity?

One common misconception is that relativity only applies to objects moving at very high speeds. In reality, it applies to all objects in motion, but its effects are only noticeable at high speeds or in extreme gravitational fields. Another misconception is that relativity is solely a theory of time and space, when in fact it also includes the concept of mass-energy equivalence.

5. How can I apply the principles of relativity in my own research or work?

Relativity has applications in many fields, including physics, astronomy, and engineering. It can be used to understand and predict the behavior of objects in motion, as well as to develop technologies that rely on precise measurements of time and space. If you are conducting research or work in any of these areas, a solid understanding of relativity is crucial.

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