Rigorous definitions in general relativity

In summary: Thank you.In summary, the universe is a topological space with events and a coordinate system. A world line is a continuous function which is defined on an open subset of R and takes values in U. A generalized physical space is a family of world lines of material bodies.
  • #1
Nana Dutchou
14
0
Hello

(a) The universe U is a topological space whose elements are called events and as each event has a neighborhood homeomorphic to R^4.

(b) A local coordinate system is a homeomorphism between an open subset of U and a bounded subset of R.

(c) A world line segment is a continuous function which is defined on an open subset of R and takes values in U.

(d) A generalized physical space (a set of spatial positions) is a particular family of world lines of material bodies. For example, in general relativity, a generalized physical space of Rindler consists of world lines of a family of Rindler observers. http://en.wikipedia.org/wiki/Rindler_coordinates#The_Rindler_observers

(e) To define a time variable in a generalized physical space we just have to choose a particular parametrization along each of his world lines or (in a corpuscular model) we just have to choose a particular parametrization along the world line of the body whose movement is studied. For example, a Poincaré-Einstein dating carried out by an experimenter P is a temporal variable (t) and a Poincaré-Einstein dating carried out by an experimenter P' is another temporal variable (t'). A Poincaré-Einstein dating carried out by an experimenter P is a temporal variable obtained by this method : the date associated with an event A is the arithmetic mean of the dates of issuance and receipt by P of a light signal which is reflected in A.

These definitions are correct in general relativity ?

Thank you.
Rommel Nana Dutchou
 
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  • #2
I don't know where you got your definition of (b) from. In the context of GR, a coordinate chart is a pair [itex](U,\varphi )[/itex] such that [itex]U[/itex] is open, [itex]\varphi :U \rightarrow U'[/itex] is a homeomorphism where [itex]U'\subseteq \mathbb{R}^{n}[/itex] is open, and [itex]\forall (V,\psi )[/itex] another coordinate chart such that [itex]V\cap U\neq \varnothing [/itex] implies [itex]\psi \circ \varphi ^{-1}:\varphi (U\cap V)\rightarrow \psi (U\cap V)[/itex] is a diffeomorphism (such charts are said to be smoothly compatible). The maximal collection of all smoothly compatible coordinate charts on a manifold is called the smooth atlas (I say in the context of GR because outside of GR we can just talk about topological manifolds with no smooth atlas / smooth structure). Point (a) is incomplete if you are talking about space - times in GR. A space - time is a topological manifold (using Wald's convention this means it is a Hausdorff, second countable, locally euclidean topological space) that has a smooth atlas and has subsequently been endowed with a metric tensor that is a solution to the EFEs. As for (c), a world - line is a regular curve (some may require just [itex]C^{2}[/itex] others may require [itex]C^{\infty }[/itex]) [itex]\gamma :I \rightarrow M[/itex] where M is the space - time and I is a non empty interval in R (we take intervals so that the domain is connected, which we want for obvious reasons).
 
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  • #3
Sorry to be nitpicking, but a collection of charts is an atlas, not an atals.
 
  • #4
CompuChip said:
Sorry to be nitpicking, but a collection of charts is an atlas, not an atals.
Ah yes. I sincerely apologize for that. I don't know what gets into me sometimes :smile: My spelling is quite horrid as you can see lol; I'll have it fixed.
 
  • #5
No problem, but you were misspelling it so consistently that I started wondering :)
 
  • #6
Thank you for your answers

WannabeNewton said:
As for (c), a world - line is a regular curve (some may require just [itex]C^{2}[/itex] others may require [itex]C^{\infty }[/itex]) [itex]\gamma :I \rightarrow M[/itex] where M is the space - time and I is a non empty interval in R (we take intervals so that the domain is connected, which we want for obvious reasons).

These details do not bother me. I noted U what is noted M. The space-time is not just a topological manifold, we can say that this is a differentiable manifold.

Do you agree that (d) is an acceptable definition :

(d) A generalized physical space (a set of spatial positions) is a particular family of world lines of material bodies that never meet. For example, in general relativity, a generalized physical space of Rindler consists of world lines of a family of Rindler observers. http://en.wikipedia.org/wiki/Rindler_coordinates#The_Rindler_observers

Thank you
 
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  • #7
Nana Dutchou said:
Do you agree that (d) is an acceptable definition :

(d) A generalized physical space (a set of spatial positions) is a particular family of world lines of material bodies that never meet.
While time - like and null - like geodesic congruences are defined on proper open subsets of the space - time (Wald and Carroll define it this way for example) I don't know if you can extend the congruence globally to cover the whole space - time but what you said is definitely true locally. Hopefully someone else can answer that part. Cheers!
 
  • #8
WannabeNewton said:
While time - like and null - like geodesic congruences are defined on proper open subsets of the space - time (Wald and Carroll define it this way for example) I don't know if you can extend the congruence globally to cover the whole space - time but what you said is definitely true locally. Hopefully someone else can answer that part. Cheers!

I know that a world line of a material body is a time-like world line in GR. If we can locally define (on an open subset of space-time) a generalized physical space as a family of time-like world lines that never meet, maybe we can locally define a physical space as a family of time-like world lines which seems continuously immobile for a unique observer ?

Nana Dutchou said:
(e) To define a time variable in a generalized physical space we just have to choose a particular parametrization along each of his world lines or (in a corpuscular model) we just have to choose a particular parametrization along the world line of the body whose movement is studied.

For example, a Poincaré-Einstein dating carried out by an experimenter P is a temporal variable (t) and a Poincaré-Einstein dating carried out by an experimenter P' is another temporal variable (t'). A Poincaré-Einstein dating carried out by an experimenter P is a temporal variable obtained by this method : the date associated with an event A is the arithmetic mean of the dates of issuance and receipt by P of a light signal which is reflected in A.

Thank you
 

FAQ: Rigorous definitions in general relativity

1. What is a rigorous definition in general relativity?

A rigorous definition in general relativity refers to a precise and well-defined mathematical description of the theory. It is based on the principles of differential geometry and the equations of motion derived from Einstein's field equations.

2. Why is it important to have rigorous definitions in general relativity?

Rigorous definitions in general relativity are crucial to accurately describe and understand the behavior of spacetime and the effects of gravity. Without precise definitions, it would be difficult to make precise predictions and test the validity of the theory.

3. How are rigorous definitions in general relativity different from other scientific definitions?

Rigorous definitions in general relativity are based on the principles of differential geometry, which is a branch of mathematics that deals with curved spaces. This is different from other scientific definitions that may be based on simpler mathematical concepts or empirical observations.

4. Can rigorous definitions in general relativity be applied to all situations?

Rigorous definitions in general relativity are applicable to most situations, but they may break down in extreme cases such as singularities or regions where quantum effects become important. In these cases, new theories and definitions may be needed to accurately describe the behavior of spacetime.

5. How do scientists ensure the rigor of definitions in general relativity?

Scientists ensure the rigor of definitions in general relativity by using rigorous mathematical techniques, such as tensor calculus and differential geometry, to formulate and test the theory. They also compare the predictions of the theory with experimental observations to verify its accuracy.

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