Rigorous definitions in general relativity

[Nana Dutchou]

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

 

[WannabeNewton]

I don't know where you got your definition of (b) from. In the context of GR, a coordinate chart is a pair $(U,\varphi )$ such that $U$ is open, $\varphi :U \rightarrow U'$ is a homeomorphism where $U'\subseteq \mathbb{R}^{n}$ is open, and $\forall (V,\psi )$ another coordinate chart such that $V\cap U\neq \varnothing$ implies $\psi \circ \varphi ^{-1}:\varphi (U\cap V)\rightarrow \psi (U\cap V)$ 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 $C^{2}$ others may require $C^{\infty }$) $\gamma :I \rightarrow M$ 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).

 

[CompuChip]

Sorry to be nitpicking, but a collection of charts is an atlas, not an atals.

 

[WannabeNewton]

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 My spelling is quite horrid as you can see lol; I'll have it fixed.

 

[CompuChip]

No problem, but you were misspelling it so consistently that I started wondering :)

 

[Nana Dutchou]

Thank you for your answers

As for (c), a world - line is a regular curve (some may require just $C^{2}$ others may require $C^{\infty }$) $\gamma :I \rightarrow M$ 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

 

[WannabeNewton]

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!

 

[Nana Dutchou]

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 ?

(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