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Schwarzschild space-time
Schwarzschild spacetime (Schwarzschild spacetime) represents a solution in general relativity that describes the gravitational field of a spherically symmetric, non-rotating, non-charged celestial body and is also the first exact solution of Einstein’s equations for constructing a black hole. This solution was discovered by Carl Schwarzschild in 1916.
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Schwarzschild spacetime is characterised by spherical symmetry, where space is perfectly spherically symmetrical and the physical properties are the same in all directions, the gravitational source objects are assumed not to rotate and the objects are assumed to have no electric charge.
As a mathematical description, it is described by the following metric of space-time (a measure of distance).
\[
ds^2 = -\left(1 – \frac{2GM}{c^2 r}\right)c^2 dt^2 + \left(1 – \frac{2GM}{c^2 r}\right)^{-1}dr^2 + r^2 (d\theta^2 + \sin^2\theta \, d\phi^2)
\]
Variable Meaning:
– \( ds^2 \): space-time interval (distance).
– \( t \): time.
– \( r \): radial distance from the gravitational source.
– \( \theta, \phi \): angle in spherical coordinates.
– \( G \): gravitational constant.
– \( M \): mass of the central body.
– \(c \): Speed of light.
The Schwarzschild radius (Schwarzschild radius), defined by this Schwarzschild spacetime, defines the event horizon (Event Horizon) of the black hole as follows.
\[
r_s = \frac{2GM}{c^2}
\]
Meaning of this formula, at the Schwarzschild radius \( r_s \), the gravity becomes so strong that even light cannot escape from that region. Beyond this radius, the object behaves as a black hole.
The physical properties of Schwarzschild spacetime include the following
- External spacetime: in the region of the Schwarzschild radius ( r > r_s ), the spacetime is similar to normal space and is distorted by the masses of celestial bodies, e.g. gravitational lensing effects are observed.
- Event horizon: the boundary of the Schwarzschild radius ( r = r_s ) is the point of separation between the observable universe and the interior of a black hole. Matter and light beyond this boundary cannot travel outwards.
- Internal spacetime: in the region of ( r < r_s ), the extreme spacetime distortion predicted by general relativity occurs, and at the singularity ( r = 0 ) the spacetime distortion becomes infinite and difficult to describe with current physical theory.
The Schwarzschild spacetime is a simple but important solution that captures the essence of gravity and spacetime, and the Schwarzschild spacetime has been used as a model for non-rotating black holes, to accurately describe the motion of matter and light around celestial bodies (orbits, redshifts, etc.), to predict light bending due to gravitational fields They are treated as models.
In addition to this static Schwarzschild spacetime, there are four other models of black holes: one based on the Reissner-Nordstrom spacetime, which takes charge into account; one based on the Kerr spacetime, which takes rotation into account; and a combination of these.
ブラックホールは4種類？――宇宙はなぜブラックホールを造ったのか（5）より
Einstein-Rosen Bridge
In 1935, Einstein and Rosen studied this Schwarzschild spacetime as a solution to general relativity, and found that there was an infinitely dense point inside the Schwarzschild black hole, called a singularity, and Einstein and Rosen discovered another mathematical structure that could avoid this singularity They found that the singularity is not a black hole. As a result, they considered that a bridge-like structure could exist between the black hole and ‘another spacetime’ or ‘another universe’.
This structure came to be known as the ‘Einstein-Rosen bridge’ after its discoverer. The entrance to this bridge behaved as a black hole and the other as a white hole (a theoretical entity), defined as a black hole sucking in matter and energy, whereas a white hole emits them.
What is the meaning of ‘Wormhole’ which is commonly found in SF works?より
The bridge is depicted as a tunnel connecting two different regions of space-time, suggesting that it may be possible to reach another place and time by a shorter route than through normal space.
The Einstein-Rosen bridge is considered to be highly unstable, collapsing at the slightest disturbance, and any attempt by matter or light to pass through the tunnel would result in the bridge itself collapsing, making passage impractical.
It is also said that ‘exotic matter’ (hypothetical matter with negative energy density) is needed to stabilise the wormhole, rather than ordinary matter. Current physics does not confirm the existence of such matter.
Furthermore, white holes themselves have never been observed and their existence is entirely theoretical.
New interpretations and possibilities for these issues have been discussed using the latest quantum gravity theory (integration of general relativity and quantum mechanics), and recent research has focused on the relationship between the Einstein-Rosen Bridge (ER) and quantum mechanical ‘entanglement’ (EPR). This leads to theories suggesting that quantum entanglement and wormholes may be different expressions of the same physical phenomenon.
The Einstein-Rosen bridge has had an important impact on theoretical physics as an early model of wormholes, and although it is unclear whether this structure actually exists or is available in the universe, it has provided profound questions for understanding the fundamental structure of the universe and the nature of space-time.
Wormholes in film and television.
One means of visualising these various theories and hypotheses is science fiction prototyping, as described in ‘Science fiction prototyping and visualisation in DX’.
SF Prototyping is a method of using science fiction (SF) ideas and stories to explore the possibilities of new technologies and future societies, and has gained attention as a powerful tool for realistically exploring future technologies and their impact, rather than just creating stories. It has become a focus of attention as a
Here, the science fiction imagination is explored in the ‘Interstellar, Doctor Who and Foundation universes; from DNEG’s Space VFX showreel’ to the wormholes in Interstellar, supervised by theoretical physicist Kip Thorne, and the Tardis (a trans-dimensional space-time travel device in Doctor Who), supervised by theoretical physicist Kip Thorne. Transcendental space-time travel device), describing the Foundation wormhole travel scene described in ‘Foundation’.
These are basically in the form of creating a black hole with some high energy, starting to move in the form of being sucked into it, and then a white hole appears.
The common issue for these is whether it is possible to create a white hole for an arbitrary destination and how to select one when there are multiple bridges, even if the creation of a black hole can be made possible by some technological approach.
As regards the generation of white holes, they could be constructed by artificially creating a black hole and manipulating its structure, or by using the indeterminacy of quantum mechanics to link a part of space-time to a destination, or by using negative energy density (e.g. amplifying the Casimir effect) to set the exit to the destination, Several ideas have been considered.
Some science fiction ideas for arbitrary selection of wormholes include using artificial intelligence technology to simulate or predict exits, navigating through higher dimensional spaces to which wormholes are connected, and tagging wormhole networks and navigating using them as indicators. Ideas are being considered.
There are still many obstacles to overcome in order to realise these ideas.
reference book
The following reference books are listed to help deepen understanding of this topic.
Reference books for the general public:
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Specialist reference books (university level and above)
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Thematic Guide:
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