General Relativity - geodesic - affine parameter

In summary, the conversation discusses the definition of abstract tensors and their relation to affine parameterisation in the context of geodesics. The conversation also mentions the use of affine connections and the role of affine parameters in parallel-transported tangent vectors. The conversation concludes by highlighting the importance of affinely parameterising geodesics in order for the velocity vector to remain parallel-transported.
  • #1
binbagsss
1,254
11

Homework Statement



Question attached:

geoafftens.png


Homework Equations


see below

The Attempt at a Solution


[/B]
my main question really is

1) what is meant by 'abstract tensors' as I have this for my definition:
to part a)
##V^u\nabla_uV^a=0##

but you do say that ##V^u=/dot{x^u}## ; x^u is a coordinate so, probably a stupid question, how is it really an 'abstract tensor'? I would say ##\nabla_u## is but not ##V^u##.

and

b) how this definition specifies/relies on being affinely parameterised. I have the definition as the tangent vector is parallel transported along itself. I would have thought that any tangent vector of a geodesic is parallel transported along itself- does it need to be affinely parameterised and what does this mean?

(I have seen the terminology 'affine connection' used elsewhere and that the Levi-Civita connection is one such connection, which is assumed in GR via the fundamental theorem of Riemann geometry - I posted a thread the other week on showing that definitions of geodesics agree due to this (the above and definition based on calculus of variations), and but in our lecture notes we have only defined a affine parameter as one such that dL/ds =0 , where L is the Lagrangian of a free-ly falling particle, we have not talked about affine connections and so I don't understand the definition of affine parameters in the context of parallel-transported tangent vector definition of geodesic as given above)- I guess the terms affine parameter and affine connection are related and this may cover it?

part b)

simply expand this out to get
##\ddot{x^u}+\Gamma_{ab}^{u}\dot{x^a}\dot{x^b}##

many thanks
 

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  • #2
binbagsss said:
I would have thought that any tangent vector of a geodesic is parallel transported along itself- does it need to be affinely parameterised and what does this mean?
The direction of the tangent (aka velocity) vector may be constant along the curve, but the magnitude will not, unless the geodesic is affinely parameterised. A curve whose velocity vector is always tangent but which varies in magnitude is not a geodesic. Changing the parameterisation changes the magnitude of the velocity vector. If the change is linear, a velocity vector that was parallel-transported will remain parallel-transported because it will be multiplied by the same constant everywhere - being the gradient of the linear transformation. But if the parameter transformation is nonlinear that will not be the case.
 

1. What is General Relativity?

General Relativity is a theory of gravity developed by Albert Einstein in the early 20th century. It describes the relationship between the curvature of space-time and the distribution of matter and energy.

2. What is a geodesic in General Relativity?

A geodesic is the path that a freely-moving object will follow in a curved space-time according to the laws of General Relativity. It is the shortest possible path between two points and is often described as the "straight line" in curved space-time.

3. What is an affine parameter in General Relativity?

An affine parameter is a mathematical quantity used to describe the motion along a geodesic in General Relativity. It is a measure of distance along the geodesic and is used to calculate the properties of the geodesic, such as its curvature and velocity.

4. How is the concept of geodesics related to the theory of General Relativity?

The concept of geodesics is central to the theory of General Relativity, as it is used to describe the motion of objects in a curved space-time. The theory predicts that objects will follow geodesic paths in the presence of gravity, which is caused by the curvature of space-time.

5. What are some real-world applications of geodesics and affine parameters in General Relativity?

Geodesics and affine parameters are used in various applications, such as in the navigation of spacecraft and satellites, in the development of GPS systems, and in the study of black holes and other massive objects in space. They also play a crucial role in the accurate prediction and understanding of gravitational effects on objects in the universe.

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