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Rigorous definitions in general relativity

  1. Feb 2, 2013 #1
    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
     
  2. jcsd
  3. Feb 2, 2013 #2

    WannabeNewton

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    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).
     
    Last edited: Feb 2, 2013
  4. Feb 2, 2013 #3

    CompuChip

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    Sorry to be nitpicking, but a collection of charts is an atlas, not an atals.
     
  5. Feb 2, 2013 #4

    WannabeNewton

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    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.
     
  6. Feb 2, 2013 #5

    CompuChip

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    No problem, but you were misspelling it so consistently that I started wondering :)
     
  7. Feb 2, 2013 #6
    Thank you for your answers

    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
     
    Last edited: Feb 2, 2013
  8. Feb 2, 2013 #7

    WannabeNewton

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    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!
     
  9. Feb 2, 2013 #8
    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 ?

    Thank you
     
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