Basis Vector in GR: Cartesian Plane Comparisons

In summary, Hartle's book on GR explains that basis vectors in x,y,z,t coordinate axes are similar to the coordinate axis of a Cartesian plane in the sense that they point along the axes. However, coordinates in GR do not have many of the special properties that Cartesian coordinates have, such as orthonormality. This may be a difficult concept for beginners and a simpler text like "Exploring black holes" may be more suitable.
  • #1
Tony Stark
51
2
I Have been reading hartle's book on Gr which states that basis four vectors point in x,y,z,t coordinate axes. So are they similar to the coordinate axis of Cartesian plane.
 
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  • #2
Tony Stark said:
are they similar to the coordinate axis of Cartesian plane

Only in the very general sense that basis vectors point along coordinate axes. Coordinates in GR in general will not have many of the special properties that Cartesian coordinates on a plane have: a big one is that the basis vectors may not be orthonormal (which means the coordinate axes may not be perpendicular, and the lengths of the basis vectors may not all be the same).
 
  • #3
I'm getting the sense that Hartle may be too advanced a text for the OP. Perhaps something like "Exploring black holes" would be a better choice.
 

Related to Basis Vector in GR: Cartesian Plane Comparisons

What are basis vectors in general relativity?

Basis vectors in general relativity are a set of vectors used to define a coordinate system within a curved space. They are used to express the position and movement of objects within this space.

How do basis vectors in general relativity differ from those in a Cartesian plane?

In a Cartesian plane, the basis vectors are constant and do not change as you move around the space. In general relativity, the basis vectors can vary depending on the curvature of the space and the position of the observer.

Why are basis vectors important in general relativity?

Basis vectors are essential in general relativity because they allow us to define and describe the geometry of a curved space. They also enable us to express physical quantities, such as velocity and acceleration, in a meaningful way within this space.

How are basis vectors used in calculations within general relativity?

Basis vectors are used in calculations within general relativity to transform between different coordinate systems and to express physical quantities in a covariant form. They are also used to define the metric tensor, which is a fundamental concept in general relativity.

Can basis vectors be visualized in a curved space?

Yes, basis vectors can be visualized in a curved space. In a two-dimensional space, the basis vectors can be represented by two arrows that are tangent to the surface at a given point. In a three-dimensional space, three arrows are needed to represent the basis vectors, one for each dimension.

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