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. 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. 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 $\bar t$. In the second part, we study the local isometric embedding of surfaces in R3 we discuss metrics with Gauss curvature which is everywhere positive, negative, nonnegative, nonpositive, as well as the case of mixed sign. 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. isometric embedding of Riemannian manifolds in Euclidean spaces these include the Janet-Cartan Theorem and Nash Embedding Theorem. 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?
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