Cauchy Problem in Convex Neighborhood

In summary, the conversation discusses the concept of the Green functional and its role in the calculation of solutions in curved spacetime. The focus is on Eq.(16.6) and Eq.(16.7) which deal with the evolution equation and a specific solution, respectively. It is mentioned that the separation of the Green functional is only valid in the convex neighborhood of a field point x, which is related to the Cauchy problem and the requirement for the points to be connected by a unique geodesic. This is to ensure agreement in calculations and explains the limitation to a convex neighborhood.
  • #1
PLuz
64
0
While reading the reference

Eric Poisson and Adam Pound and Ian Vega,The Motion of Point Particles in Curved Spacetime, available http://relativity.livingreviews.org/Articles/lrr-2011-7/fulltext.html,

there is something that I don't quite understand. Eq.(16.6) is an evolution equation for de Green functional. Then in Eq.(16.7) Poisson et. al. look for a specific solution and they state that the separation of the Green functional is valid only in the convex neighborhood of a field point x. I assume that is because the Cauchy problem is valid only in that neighborhood... My question is why? Why is the Cauchy problem related to the imposition that the two points must be connected by a unique geodesic?
 
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  • #2
From a glance, it looks like the calculation propagates the solution from one point to another by integrating along the geodesic that joins them. If they are connected by more than one geodesic, then the two calculations might not agree. Hence the restriction to a convex neighborhood.
 

1. What is the Cauchy Problem in Convex Neighborhood?

The Cauchy Problem in Convex Neighborhood is a mathematical concept that deals with finding solutions to partial differential equations within a convex neighborhood. It involves finding a solution that satisfies both the differential equation and certain boundary conditions within a specified region.

2. What makes a neighborhood convex?

A neighborhood is considered convex if any two points within it can be connected by a straight line that lies entirely within the neighborhood. In other words, all points between any two points in the neighborhood are also included in the neighborhood.

3. What is the importance of the Cauchy Problem in Convex Neighborhood?

The Cauchy Problem in Convex Neighborhood is important in mathematics because it allows us to find solutions to partial differential equations in a specific region, rather than over a larger domain. This makes it a useful tool in many mathematical and scientific fields, such as physics, engineering, and economics.

4. How is the Cauchy Problem in Convex Neighborhood solved?

The Cauchy Problem in Convex Neighborhood is typically solved using various mathematical techniques, such as Fourier series, power series, or Laplace transforms. These techniques involve finding a series of functions that satisfy the differential equation and boundary conditions within the neighborhood.

5. What are some applications of the Cauchy Problem in Convex Neighborhood?

The Cauchy Problem in Convex Neighborhood has many practical applications in fields such as fluid dynamics, heat transfer, and electromagnetism. It can also be used in the development of numerical methods for solving partial differential equations and in the study of chaos theory.

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