Classification of

WormholesAyan Banerjee

Jadavpur UniversityDept. of Mathematics

At the beginning of the 20th century, Albert Einstein revolutionized the idea that space and time do not exist as separate , rather it is a combination of space and time to a single abstract universe i.e. space, consisting of three dimensions and time is said to have only one dimension.

Einstein said we can think of these space-time as fabric.

In physics, a wormhole is a hypothetical topological feature of space-time that is essentially a "shortcut" through space and time. A wormhole has at least two mouths which are connected to a single throat. If the wormhole is traversable, matter can 'travel' from one mouth to the other by passing through the throat.

Traveling through a wormhole takes less time than traveling between the same starting and ending regions in normal space .

The name "wormhole" comes from the following analogy used to explain the phenomenon: imagine that the universe is the skin of an apple, and a worm is travelling over its surface, then distance from one side of the apple to the other is equal to half of the apple's circumference. The worm has to travel is considerably less, if it burrows a wormhole directly through the apple, instead of staying on the apple's surface.

History of Wormhole

Einstein first proposed wormholes in 1935. He wrote a paper with Nathan Rosen in which they showed that general relativity allowed for what they called “bridges.” They theorized that there could be places where space/time is folded that allowed transfer of matter from one point to another in the universe, specifically in black holes.

Weight added to a piece of fabric will cause sag towards the center.

Super massive black holes cause a tear in space-time and this tear is the wormhole.

These two regions would meet and form a wormhole connection.

Types of Wormholes

Lorentzan wormholes (general relativity) Metric with signature (-1, +1, +1, +1) Mainly studied by experts in Einstein gravity

Euclidean wormholes (particle physics) Metric with signature (+1, +1, +1, +1) Mainly studied in quantum gravity theory

However, in 1962 John A. Wheeler and Robert W. Fuller showed that Einstein-Rosen bridge space-time structure was highly unstable in field-free space, and that it will pinch off too quickly before a single photon could be transmitted through it. This work lead to find different kinds of  wormholes. The possibility of traversable wormholes in general relativity gathered pace by the publication of a paper in 1987, “Wormholes in space-time and their use for Interstellar travel: A tool for teaching general relativity”, by Michael Morris and Kip Thorne which is known to us a Morris-Thorne wormhole.

“Morris-Thorne Wormholes”

Simplified their analysis by first assuming the existence of a suitably well-behaved geometry.

The solution must every where obey the Einstein field equations. It is assumed that GR is correct.

The matter and fields that generate the wormhole’s space-time curvature must have a physically reasonable stress-energy tensor or not.

Φ(r) called the “red shift” function since it determines the gravitational red shift.

The metric for a static spherically symmetric space-time is given by

b(r) called the “shape” function since it determines the spatial shape of the wormhole.

As is the throat radius, so =

Traversability criteria: For traversable wormhole, there should not be any horizons present i.e. must be finite everywhere.

Einstein Field Equations Einstein field equation, in an orthonormal frame, (with c = G = 1) we obtain the following stress-energy scenario which are

Where is the energy density, is the radial pressure, and is the lateral pressure measured in the orthogonal direction to the radial direction.

Stress-Energy at the Throat

at the throat, the tension exceeds the total mass-energy density.

Materials with the property is called, “exotic”.

fundamental condition in wormhole physics is the violation of the NEC. i.e. . Matter that violates the NEC is denoted as exotic matter.

Thank you