A new visualization technique for visualizing three-dimensional symmetric positive definite tensor fields of rank two is described. It supports studying the spatial projection of a spacetime metric. The rendering technique is demonstrated upon the Schwarzschild metric of a static black hole, the Kerr metric of a rotating black hole in two different coordinate systems, and a numerically computed...

In spacetime physics, we frequently need to consider a set of all spaces (‘universes’) as a whole. In particular, the concept of ‘closeness’ between spaces is essential. However there has been no established mathematical theory so far which deals with a space of spaces in a suitable manner for spacetime physics. Based on the scheme of the spectral representation of geometry, we construct a spac...

Instead of conformal to flat spacetime, we take the metric conformal to a spacetime which can be thought of as “minimally” curved in the sense that free particles experience no gravitational force yet it has non-zero curvature. The base spacetime can be written in the Kerr-Schild form in spherical polar coordinates. The conformal metric then admits the unique three parameter family of perfect f...

The expectation value of the stress-energy tensor of a free conformally invariant scalar field is computed in a two-dimensional reduction of the Alcubierre “warp drive” spacetime. The stress-energy is found to diverge if the apparent velocity of the spaceship exceeds the speed of light. If such behavior occurs in four dimensions, then it appears implausible that “warp drive” behavior in a space...

The causal and metric structure of globally hyperbolic spacetimes is investigated from a C∗-algebra point of view related to part of Connes’ noncommutative geometry programme. No foliation of the spacetime by means of spacelike surfaces is employed, but the complete Lorentzian geometry is considered. Several results are produced. As a first result, Connes’ functional formula of the distance is ...

We use the duality between the local Cartezian coordinates and the solutions of the Klein-Gordon equation to parametrize locally the spacetime in terms of wave functions and prepotentials. The components of metric, metric connection, curvature as well as the Einstein equation are given in this parametrization. We also discuss the local duality between coordinates and quantum fields and the metr...

We show that the regularity of the gravitational metric tensor cannot be lifted from C to C by any C coordinate transformation in a neighborhood of a point of shock wave interaction in General Relativity, without forcing the determinant of the metric tensor to vanish at the point of interaction. This is in contrast to Israel’s Theorem [6] which states that such coordinate transformations always...

The geometry of curved ‘three-dimensional’ absolute intrinsic metric space (an Riemannian space) \(\varnothing\hat{\mathbb{M}}^3\), which is (as a hyper-surface) toward the time/absolute time ‘dimensions’ (along vertical), and projects flat three-dimensional proper \(\varnothing\hat{\mathbb{E}}^\prime\)ab3 its outward manifestation namely, 3-space \(\hat{\mathbb{E}}^\prime\)ab3, both as hyper-s...