Gravitational singularity

In physics and astronomy, a singularity is a place where gravity is so strong that the curvature of space-time diverges, colloquially known as "infinity". This means that at these places the metric of spacetime also diverges and the singularity is not a component of spacetime. Physical quantities like the mass density, for whose calculation the metric is needed, are not defined there.

Geodesic lines hitting the singularity have finite length, so spacetime is causally geodesically incomplete.

According to general relativity there are singularities in spacetime under very general conditions, as Stephen Hawking and Roger Penrose showed in the 1960s (singularity theorem). The singularities can be formulated as mathematical singularities and depend, among other things, on special mass values , angular momentum or other parameters. Here the physical law in question is for the limit , where is a critical parameter value, undefined, invalid and unsuitable to describe the ratios. Singularities can be point-like, i.e. infinitesimally small, or non-point-like, in which case spacetime curves around the object so much that magnitudes cannot be meaningfully related to the metric of the surrounding space.

It is assumed that singularities show the limits of general relativity and that another model (for example quantum gravity) must be used to describe them.

Types of singularities

The singularities discussed in this article are also called real, intrinsic, or curvature singularities to emphasize that they are physical properties of spacetime. In them, a coordinate-independent quantity, the curvature of spacetime, diverges. They are to be distinguished from so-called coordinate singularities, which are merely a mathematical property of the chosen coordinates. The latter can be "transformed away" by a suitable coordinate transformation. For real, essential singularities this is not possible, here a new theory (a new physical law) is needed.

Singularities, for example inside a normal black hole, are surrounded by an event horizon, which in principle removes the object from observation. Whether singularities without event horizons (so-called naked singularities) also exist is unclear. That singularities are shielded by event horizons, i.e. that there are no naked singularities, is the subject of Roger Penrose's cosmic censor hypothesis. It is unproven and represents one of the great open problems of general relativity.

Astrophysics and Cosmology

In astrophysics and cosmology, the term singularity is often used synonymously with black hole or, in big bang theories, with initial singularity.

In both cases, Einstein's field equations are the physical laws used for explanation. However, the theory underlying these equations (Albert Einstein's general theory of relativity) is a "classical theory", not a quantum theory. Therefore, it loses its validity on very small length scales (Planck length) and that is where the realm of a theory of quantum gravity begins. However, very little is known about the internal state or structure of singularities within such a theory.

Initial singularity

In the big bang theories, space-time "starts" in a mathematical singularity. The first physically describable point in time is placed at the shortest possible time distance from this singularity, namely the Planck time of about 10-43seconds. The big bang theories therefore do not describe the big bang itself, but only the development of the universe since this world age. In the mathematical initial singularity, space and time do not yet exist. Information about expansion or duration are therefore defined out of physics.

In the initial singularity the laws of nature known to us could not have been valid. The initial singularity was not a black hole. It had no event horizon and no outer space surrounding it.

black holes

Black holes can be characterized by their effect on the surrounding spacetime. However, many properties of the singularity inside a black hole, such as its density, are similarly undefined as those of the initial singularity.

Karl Schwarzschild was the first who could give a solution (outer Schwarzschild solution) for the field equations. His solution describes uncharged non-rotating, i.e. static black holes, which in reality do not exist, and becomes singular in the central point (point singularity). In the Kruskal coordinates, the point singularity becomes a manifold described by a hyperboloid. Thus one sees explicitly that here at the event horizon itself no singularity occurs.

Only in 1963 the New Zealand mathematician Roy Kerr found another solution (Kerr solution) for rotating black holes, which becomes singular in a one-dimensional ring in the equatorial plane. The radius of the ring singularity corresponds to the Kerr parameter. An even more general solution with an additional electric point charge leads to the Kerr-Newman metric.

The outer Schwarzschild solution is a special case of the Kerr solution (Kerr parameter a = Jc/(GM²) = 0, i.e. no rotation). For maximally rotating black holes, i.e., when the event horizon rotates at the speed of light, on the other hand, a = 1. Objects with a spin of a > 1 must therefore have an expansion higher than the gravitational radius corresponding to their mass, since otherwise the event horizon would dissolve and a naked singularity would be visible from the outside at the poles and at the equator. That naked singularities are shielded from outside observers by event horizons is the subject of the Cosmic Censorship Hypothesis. It is generally unproven and may require an extension of known physical theories, but there is evidence for its validity from numerical simulations, mathematical analyses, and thought experiments.