I How important are these concepts for studying Relativity?

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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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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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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 dont learn some of these things. It may not be a computational aspect of it, but moreso 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).
 

pervect

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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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Parallel transport and basis vectors are fundamental.
Is parallel transport more fundamental than other types of transport? Like Fermi-Walker transport.
 

pervect

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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".
 

pervect

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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.
 

PAllen

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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?
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.
 

vanhees71

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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?
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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