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Events in relativity[SR or GR]are marked by coordinate values and not by physical values.We write a metric for motion along the x-axis: $$ds^2=g_{00}dt^2-g_{11}dx^2$$ ----------- (1)

For physical values we may write:

$$ds^2=dT^2-dL^2$$ ------------ (2)

Where $dT^2=g_{00}dt^2$ and $dL^2=g_{11}dx^2$

For the null geodesic $ds^2=0$

From (1) we have for the null geodesic,

$$\frac{dx}{dt}=\sqrt{\frac{g_{00}}{g_{11}}}\ne1$$ [Generally speaking]-----(3)

Relation (2) provides a SR picture in the local context. since, $$\frac{dL}{dT}=1$$ ------------ (4)

[c=1 in the natural units]

Since events are marked by coordinate values our light cone in GR should correspond to equation (3). It should be a flickering one in a time varying field[ and one with a distorted surface in a stationary field] since the metric coefficients go on changing in a time varying field.

As I advance long the time axis the distorted surface of my light cone goes on changing.Points which I expect to be at space-like separation in the future are now at a time-like separation[or vice versa]

Query:Does this relate to changes in the causal structure of the light cone?

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  • $\begingroup$ What metric are you using here? Also, I don't understand how you're getting the conclusion that the light cone flickers in a time-varying field. Which field is that, anyway? $\endgroup$
    – David Z
    Commented Apr 28, 2012 at 0:48
  • $\begingroup$ A time varying gravitational field has been considered.The metric has been provided by equation (1) $\endgroup$
    – Anamitra Palit
    Commented Apr 28, 2012 at 1:01
  • $\begingroup$ Actually while plotting the cone with coordinate vales we have to use the physical values.For example while plotting the coordinate label x1 we have to consider the physical distance of x1 from the origin.The flickering cone is an euclidean memory.That could solve the problem.But then again for planetary orbits we consider trajectories given by coordinate values. So physically the shape of these orbits may be different from what we perceive them to be. $\endgroup$
    – Anamitra Palit
    Commented Apr 28, 2012 at 1:07
  • $\begingroup$ @AnamitraPalit: The coordinate labels are not physical, you need a gauge condition to determine your coordinate choice. A good gauge choice for the solar system is by demanding that the metric is approximately diagonal. I didn't read your question as "how to choose coordiantes", I read it as "what happens to the causal structure in the presence of gravity". $\endgroup$
    – Ron Maimon
    Commented Apr 28, 2012 at 1:09
  • $\begingroup$ Sorry, I still don't understand what you're talking about with this flickering cone. What exactly do you mean by flickering, anyway? It seems like you're saying that the light cone exists at some times and not at other times, which doesn't make any sense. $\endgroup$
    – David Z
    Commented Apr 28, 2012 at 1:45

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The equation for determining the light-cone in the case where there are two varying metric coefficients is the one you wrote down---

$$ {dx\over dt} = \sqrt{g_{00}\over g_{11}} $$

And this does make a curved line in the coordinate-labelled spacetime. This is just stating that light will bend in response to gravity, so that the influence region is dynamical.

I am having a hard time understanding exactly what the question is--- you are saying correct things. The causal structure is dynamical in GR, and you need to understand how the future light-cones intersect each other to find the topological causal structure.

If all the light-cones are spreading outward in a curvy way (but not smooshing together too much, so gravity is not too strong at any point), the causal structure is qualitatively identical to ordinary SR. If the gravity is strong enough to bend the outgoing lightrays in some sphere so that they are really ingoing, you can tell this intrinsically by noticing that the area of the outgoing light-rays is shrinking instead of growing. When there is a sphere where the outgoing lightrays have shrinking area, you have a closed trapped surface, and the causal structure in the interior of the closed trapped surface can't reach infinity without passing through a singularity. The inevitable occurence of a singularity in the presence of a closed trapped surface is Penrose's celebrated theorem, and it is a consequence of the causal structure changing in the presence of enough matter to make a black hole.

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  • $\begingroup$ The bending of light as perceived by considering coordinate variables should be different from what one should get from physical variables--especially in the context of strongly curved spacetime.Here I am referring to the trajectories in terms of the coordinate values and the physical values $\endgroup$
    – Anamitra Palit
    Commented Apr 28, 2012 at 1:55
  • $\begingroup$ @AnamitraPalit: What are the "physical values"? The only way to define these is to have fields in the space-time that allow you to label positions. This is the way in which you make physical values--- you gauge fix the metric. $\endgroup$
    – Ron Maimon
    Commented Apr 28, 2012 at 2:27
  • $\begingroup$ The distance between a pair of labels is different in the Euclidean context and the curved spacetime context.While interpreting the trajectories[in curved space] we consider the Euclidean background in the curved spacetime context $\endgroup$
    – Anamitra Palit
    Commented Apr 28, 2012 at 2:59
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    $\begingroup$ @AnamitraPalit: You can pretend to do that, ok, but the Euclidean distances are not meaningful. If you want to figure out what's going on, just change coordinates in Euclidean Minkowski space to x'=x+cos(t). This will make the light-cones coordinate jiggle, but it will be an off-diagonal metric tensor. $\endgroup$
    – Ron Maimon
    Commented Apr 28, 2012 at 3:32

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