Visualizing curved spacetime

Abstract

I present a way to visualize the concept of curved spacetime. The result is a curved surface with local coordinate systems (Minkowski Systems) living on it, giving the local directions of space and time. Relative to these systems, special relativity holds. The method can be used to visualize gravitational time dilation, the horizon of black holes, and cosmological models. The idea underlying the illustrations is first to specify a field of timelike four-velocities. Then, at every point, one performs a coordinate transformation to a local Minkowski system comoving with the given four-velocity. In the local system, the sign of the spatial part of the metric is flipped to create a new metric of Euclidean signature. The new positive definite metric, called the absolute metric, can be covariantly related to the original Lorentzian metric. For the special case of a 2-dimensional original metric, the absolute metric may be embedded in 3-dimensional Euclidean space as a curved surface.Comment: 15 pages, 20 figure

Sector models—A toolkit for teaching general relativity: I. Curved spaces and spacetimes

“…Thus, the second and third of the fundamental questions stated in the introduction, namely the motion of particles and the connection between curvature and matter can also be addressed within this framework. Einstein, 1916;Gamow, 1961;Sartori, 1996;Tipler und Mosca, 2008;Stannard, 2008 Sartori, 1996;Hartle, 2003;Stannard, 2008;Natário, 2011 (Flamm, 1916;Epstein, 1994;Jonsson, 2001Jonsson, , 2005 sowie für raumzeitliche Unterräume (Marolf, 1998) Der zweite Schritt illustriert, wie eine gekrümmte Fläche durch kleine ebene Flächenstücke angenähert werden kann. Dazu dienen die beiden Bastelbögen in Abb.…”

“…Another pictorial but more sophisticated description of curved spaces and spacetimes uses embedding diagrams. Embeddings of two-dimensional subspaces in three-dimensional flat space have been described both for the spatial (Flamm, 1916;Epstein, 1994;Jonsson, 2001Jonsson, , 2005 and for the spatiotemporal case (Marolf, 1998). The significance of embeddings goes beyond that of analogies as mentioned in items 2 and 4 above.…”

Topology of the Misner space and its $$g$$ g -boundary

“…It is worth mentioning that quite often this space is not depicted completely right [1,9,11] as the circles Di are missing. Probably this lack of precision is not important for many purposes, but it is of capital importance if we want to understand in detail the Misner space.…”

Relativity, thermodynamics and entropic forces

“…As mentioned in the introduction, the information proposal asserts that gravitational and inertial forces are entropic forces caused by changes in information stored in space that is associated with the positions and movements of material bodies [1]. According to general relativity, however, gravitational and inertial forces are manifestations of space-time geometry [7][8][9][10][11][12][13][14][15]. As shown in [16], the force on a body in arbitrary coordinates, due either to the body's own acceleration or to being held fixed in position outside a large gravitational source, is given by…”

“…It will be noted that (10) indicates that for any given temperature, the entropy of the accelerating body decreases as positive work is performed on the body. At the same time, however, the entropy of the heat engine increases as it performs positive work to accelerate the body, which we may denote by ΔS he > 0.…”