Re-parametrization of Geodesics: Can You Confirm?

  • Thread starter bloby
  • Start date
  • Tags
    Geodesics
In summary, the conversation discusses the geodesic equation and how changing the parametrization does not affect the world line, but the solution may not satisfy the equation. It is noted that only parameters related by a linear function will satisfy the equation and that the invariant interval is one possible solution.
  • #1
bloby
112
8
Hello.

If I find a solution of the geodesic equation and I change the parametrization, the new function does not
satisfy this equation for a general re-parametrization. But the world line is the same.

Can you confirm it: does it come from the fact that we usually choose [itex]\nabla_VV=0[/itex] instead of
[itex]\nabla_VV=fV[/itex]for the geodesic equation?
 
Physics news on Phys.org
  • #2
bloby said:
Hello.

If I find a solution of the geodesic equation and I change the parametrization, the new function does not
satisfy this equation for a general re-parametrization. But the world line is the same.

Can you confirm it: does it come from the fact that we usually choose [itex]\nabla_VV=0[/itex] instead of
[itex]\nabla_VV=fV[/itex]for the geodesic equation?

Yes. Only parameters related by a linear function will satisfy the simple form of geodesic equation. Further, invariant interval (proper time or distance) is one of the possible solutions, so only e.g. aτ+b will work as another parametrization and still satisfy this equation.
 
  • #3
Ok, thank you.
 

Related to Re-parametrization of Geodesics: Can You Confirm?

1. What is the concept of re-parametrization of geodesics?

The concept of re-parametrization of geodesics refers to the process of changing the parameterization of a geodesic curve without altering its underlying geometry. In other words, it is a way to represent the same geodesic curve using a different set of variables.

2. Why is re-parametrization of geodesics important?

Re-parametrization of geodesics is important because it allows for easier calculation and analysis of geodesic curves. By changing the parameterization, we can simplify the equations and make them more manageable, without changing the fundamental properties of the geodesic curve.

3. How is re-parametrization of geodesics performed?

Re-parametrization of geodesics is typically performed by introducing a new parameter, such as arc length or time, and expressing the original equations in terms of this new parameter. This can be done using mathematical transformations such as re-scaling, substitution, or integration.

4. What are the benefits of re-parametrization of geodesics?

The main benefit of re-parametrization of geodesics is that it simplifies the equations and makes them easier to work with. This can lead to more efficient calculations and a better understanding of the underlying geometry. It can also help to uncover new insights and relationships between various geodesic curves.

5. Can you provide an example of re-parametrization of geodesics?

Sure, consider the geodesic curve given by the equation x(t) = t^2, y(t) = t^3. By re-parametrizing using arc length, we obtain the equations x(s) = s^2/2, y(s) = s^3/3. Both sets of equations represent the same curve, but the latter is easier to work with as it eliminates the parameter t and replaces it with the arc length s.

Similar threads

I A question about geodesics and notion of parallelism
    • Special and General Relativity
Replies
19
Views
1K
I How can a square-based lattice turn into a circular-based one?
    • Special and General Relativity
Replies
31
Views
869
I About global inertial frame in GR - revisited
    • Special and General Relativity
2
Replies
57
Views
2K
A Computing Null Geodesics in Schwarzschild Geometry
    • Special and General Relativity
Replies
4
Views
1K
A Re-writing the geodesic deviation eqn in matrix notation (3d only)
    • Special and General Relativity
Replies
1
Views
580
A Variation of CMFR mass density ##\rho## along a world line
    • Special and General Relativity
Replies
17
Views
426
I Geodesics using 2 Variables: Time & Radius from Mass
    • Special and General Relativity
Replies
1
Views
948
I Understanding Killing Vectors & Schwarzschild Geodesics
    • Special and General Relativity
Replies
1
Views
2K
A Two Questions about Novikov Coordinates
    • Special and General Relativity
Replies
8
Views
1K
I Lie derivative of hypersurface basis vectors along geodesic congruence
    • Special and General Relativity
Replies
15
Views
2K

Hot Threads

Recent Insights

Back
Top