What is nice is that it could also give an image of expanding universe and how particles with different world lines would envision this expansion.īut I am struggling with the math and I am not sure if this example actually makes sens. geodesics in these spaces can be regarded as intersections with planes going through the origin of the embedding space, and comment on the consequences. At a point with positive curvature the ball on the manifold will be smaller than the equivalent ball of radius in Euclidean space: a circle cut from a flat piece of paper will have to be folded or crumpled to cover the equivalent circle on the surface of a sphere. I thought that a paraboloid space-time could be a good example (the minimum referring to the big-bang event) and each observer would define their space from cross sections (which would be 1D objects without boundaries). We could then clearly see how the space $x$ of the first observer would change with $t$ and how the space of the second observer $\bar x$ would change with $\bar t$. Basically what I would like to visualise is :ġ)how the light cones are modified in presence of curvature.Ģ)how the coordinate transformation (similar to the Lorentz transformation in SR) applies in presence of curvature.ģ)how 2 observers evolving in 2 different inertial reference frames(not sure if that makes sens in curved spacetime) would define their 1D space $x$, $\bar x$ and time $t$, $\bar t$ in a curved space time. In fact I feel like the representation traditionally displayed in vulgarisation of GR is quite erroneous.īut the reason I am asking that about 2D spacetime is because it is the only case where we can see an isometric embedding (so here a 3D representation) of a curved space time (here 2D). I have read things like "in 1 1 and 2 1 dimensions the vacuum spacetimes are flat" so perhaps my question does not mean anything but if it does I think it would give a better and more faithful representation of what a curved space time actually is. I am wondering if there are basic examples of general relativity in 2D (1 1) space time to help visualising the concept of curved space time? One technique for visualizing the curvature of spacetime is to study the curvature of three-dimensional spacelike cross-sections, that is, surfaces of constant time.' In such cases, the intrinsic geometry of these surfaces represents physical space, whose curvature can be vi-sualized by an embedding into four-dimensional Euclidean space. Here's an example: The circle is a 1-dimensional manifold, and here it is embedded in the plane, which is the 2-dimensional Euclidean space.
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