Market›Null infinity
Company Profile

Null infinity

In theoretical physics, null infinity is a region at the boundary of asymptotically flat spacetimes. In general relativity, straight paths in spacetime, called geodesics, may be space-like, time-like, or light-like. The distinction between these paths stems from whether the spacetime interval of the path is positive, negative, or zero. Light-like paths physically correspond to physical phenomena which propagate through space at the speed of light, such as electromagnetic radiation and gravitational radiation. The boundary of a flat spacetime is known as conformal infinity, and can be thought of as the end points of all geodesics as they go off to infinity. The region of null infinity corresponds to the terminus of all null geodesics in a flat Minkowski space. The different regions of conformal infinity are most often visualized on a Penrose diagram, where they make up the boundary of the diagram. There are two distinct regions of null infinity, called past and future null infinity, which can be denoted using a script 'I' as 𝓘+ and 𝓘−. These two regions are often referred to as 'scri-plus' and 'scri-minus' respectively. Geometrically, each of these regions actually has the structure of a topologically cylindrical three dimensional region.

Conformal compactification in Minkowski spacetime
The metric for a flat Minkowski spacetime in spherical coordinates is ds^2=-dt^2+dr^2+r^2d\Omega^2. Conformal compactification induces a transformation which preserves angles, but changes the local structure of the metric and adds the boundary of the manifold, thus making it compact. For a given metric g_{ij}, a conformal compactification scales the entire metric by some conformal factor such that \overline{g_{ij}}=\Omega^2 g_{ij} such that all of the points at infinity are scaled down to a finite value. ds^2 = - dT^2 + dR^2 + (\sin^2 R) d\Omega^2. This is the metric on a Penrose diagram, illustrated. Unlike the original metric, this metric describes, a manifold with a boundary, given by the restrictions on R and T. There are two null surfaces on this boundary, corresponding to past and future null infinity. Specifically, future null infinity consists of all points where T= \pi -R and 0, and past null infinity consists of all points where T = R - \pi and 0. The construction given here is specific to the flat metric of Minkowski space. However, such a construction generalizes to other asymptotically flat spaces as well. In such scenarios, null infinity still exists as a three dimensional null surface at the boundary of the spacetime manifold, but the manifold's overall structure might be different. For instance, in Minkowski space, all null geodesics begin at past null infinity and end at future null infinity. However, in the Schwarzschild black hole spacetime, the black hole event horizon leads to two possibilities: geodesics may end at null infinity, but may also end at the black hole's future singularity. The presence of null infinity (along with the other regions of conformal infinity) guarantees geodesic completion on the spacetime manifold, where all geodesics terminate either at a true singularity or intersect the boundary of infinity. == Other physical applications ==
Other physical applications
The symmetries of null infinity are characteristically different from that of the typical regions of spacetime. While the symmetries of a flat Minkowski spacetime are given by the Poincaré group, the symmetries of null infinity are instead given by the Bondi–Metzner–Sachs (BMS) group. The work by Bondi, Metzner, and Sachs characterized gravitational radiation using analyses related to null infinity, whereas previous work such as the ADM framework dealt with characterizations of spacelike infinity. Using the BMS group, quanta on null infinity can be characterized as massless spin-2 particles, consistent with the quanta of general relativity being gravitons. == References ==
Source: Wikipedia ↗
tickerdossier.comtickerdossier.substack.com