Rolling ball
Encyclopedia
In topology
, quantum mechanics
and geometrodynamics
, rolling-ball arguments are used to describe how the perceived geometry
and connectivity of a surface can be scale-dependent.
If a researcher probes the shape of an intricately curved surface by rolling a ball across it, then features that are continually curved but whose curvature radius is smaller than the ball radius may appear in the ball's description of the geometry as abrupt points, barriers and singularities.
whose throat narrows to slightly less than the ball's diameter, the ball may be able to enter and explore each wormhole mouth, but will not be able to pass through the throat, and will produce a map in which the narrowing mouth walls each terminate in a sharp geometrical spike.
The smooth and multiply connected surface will be mapped by the physics of a "large" particle as being singly connected and including geometrical singularities
.
In this case, no real geometry-change occurs in the deduced shape of the underlying metric – the process identified and "caught" a wormhole candidate (getting the ball wedged in the throat), then modified the curvature of the metric over time, forcing the throat to inflate to dimensions that allowed it to be traversed.
's geometrodynamic description of quantum mechanics
, the small-scale structure of spacetime is described as a quantum foam
whose connectivities are not obvious part in large-scale physics, but whose behaviours become more apparent as we probe the surface at progressively smaller scales.
In wormhole theory, the idea of this "quantum foam" is sometimes invoked as a possible way of achieving large-scale wormholes without geometry change – instead of creating a wormhole from scratch, it may be theoretically possible to pluck an existing wormhole connection from the quantum foam and inflate it to a useful size.
Topology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
and geometrodynamics
Geometrodynamics
In theoretical physics, geometrodynamics generally denotes a program of reformulation and unification which was enthusiastically promoted by John Archibald Wheeler in the 1960s.-Einstein's geometrodynamics:...
, rolling-ball arguments are used to describe how the perceived geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
and connectivity of a surface can be scale-dependent.
If a researcher probes the shape of an intricately curved surface by rolling a ball across it, then features that are continually curved but whose curvature radius is smaller than the ball radius may appear in the ball's description of the geometry as abrupt points, barriers and singularities.
Scale-dependent topology
If the surface being probed contains connections whose scale is a smaller than the ball diameter, then these connections may not appear in the ball's map. If the surface contains a wormholeWormhole
In physics, a wormhole is a hypothetical topological feature of spacetime that would be, fundamentally, a "shortcut" through spacetime. For a simple visual explanation of a wormhole, consider spacetime visualized as a two-dimensional surface. If this surface is folded along a third dimension, it...
whose throat narrows to slightly less than the ball's diameter, the ball may be able to enter and explore each wormhole mouth, but will not be able to pass through the throat, and will produce a map in which the narrowing mouth walls each terminate in a sharp geometrical spike.
The smooth and multiply connected surface will be mapped by the physics of a "large" particle as being singly connected and including geometrical singularities
Gravitational singularity
A gravitational singularity or spacetime singularity is a location where the quantities that are used to measure the gravitational field become infinite in a way that does not depend on the coordinate system...
.
Topology change without topology change
If the surface being explored is flexible or elastic, the way the ball is used may affect the reported topology. If the ball is forced into a wormhole mouth that is slightly too small, and the ball and/or throat distorts to allow the ball through, then in the ball's description of the surface, a "new" wormhole connection has suddenly appeared and disappeared again, and the connectivity of the surface has fluctuated in an illegal way.In this case, no real geometry-change occurs in the deduced shape of the underlying metric – the process identified and "caught" a wormhole candidate (getting the ball wedged in the throat), then modified the curvature of the metric over time, forcing the throat to inflate to dimensions that allowed it to be traversed.
Quantum foam
In John WheelerJohn Archibald Wheeler
John Archibald Wheeler was an American theoretical physicist who was largely responsible for reviving interest in general relativity in the United States after World War II. Wheeler also worked with Niels Bohr in explaining the basic principles behind nuclear fission...
's geometrodynamic description of quantum mechanics
Quantum mechanics
Quantum mechanics, also known as quantum physics or quantum theory, is a branch of physics providing a mathematical description of much of the dual particle-like and wave-like behavior and interactions of energy and matter. It departs from classical mechanics primarily at the atomic and subatomic...
, the small-scale structure of spacetime is described as a quantum foam
Quantum foam
Quantum foam, also referred to as spacetime foam, is a concept in quantum mechanics, devised by John Wheeler in 1955. The foam is supposed to be the foundations of the fabric of the universe. Additionally, it can be used as a qualitative description of subatomic spacetime turbulence at extremely...
whose connectivities are not obvious part in large-scale physics, but whose behaviours become more apparent as we probe the surface at progressively smaller scales.
In wormhole theory, the idea of this "quantum foam" is sometimes invoked as a possible way of achieving large-scale wormholes without geometry change – instead of creating a wormhole from scratch, it may be theoretically possible to pluck an existing wormhole connection from the quantum foam and inflate it to a useful size.
See also
- Fractals
- WormholeWormholeIn physics, a wormhole is a hypothetical topological feature of spacetime that would be, fundamentally, a "shortcut" through spacetime. For a simple visual explanation of a wormhole, consider spacetime visualized as a two-dimensional surface. If this surface is folded along a third dimension, it...
s - John WheelerJohn Archibald WheelerJohn Archibald Wheeler was an American theoretical physicist who was largely responsible for reviving interest in general relativity in the United States after World War II. Wheeler also worked with Niels Bohr in explaining the basic principles behind nuclear fission...
- PregeometryPregeometry (physics)In physics, a pregeometry is a structure from which geometry develops. The term was championed by John Archibald Wheeler in the 1960s and 1970s as a possible route to a theory of quantum gravity...